Characteristics of Thin and Ultrathin Ferroelectric Capacitor Structures

Characteristics of Thin and Ultrathin

Film Ferroelectric Capacitor Structures

Thesis submitted for the degree of

Doctor of Philosophy

in the

Faculty of Science and Agriculture

by

Jonathan McAneney B.Sc.

Department of Physics and Astronomy

Queen’s University

Belfast

September 2005

It does not do harm to the mystery to

know a little more about it.

Richard P. Feynman

Acknowledgements

There are many people to whom I am indebted for their help and support during

these past four years. First and foremost, my supervisor Marty Gregg, without

whose guidance, and most importantly, motivation this work would still be a long

way from completion. Marty has trained me to be a better scientist, often keeping

my feet grounded when my head was bordering on science fiction. The most impor-

tant lesson he taught me was to have faith in the ‘principle of least astonishment’,

a lesson which is applicable to almost all avenues of experience.

The technical experience of Robert Bowman have been invaluable to my work,

in particular his help in keeping the laser and PLD system running and operational,

which has been invaluable to this research.

I owe much to two previous students of the group, Lesley Grattan (nee Sinna-

mon) and Niall Donnelly. When Lesley left, I inherited her vast legacy of films and

data. It is from this inheritance that Chapter 3 of this work was possible. Niall

was always on hand to fix that piece of equipment that would mysteriously cease

to function just as I was using it. He would also be there to impart a quick word

of advice to obtain more accurate data, or to explain how a particular piece of kit

worked.

I extend my gratitude to Gustau Catalan, and Beatriz Noheda, for inviting

me to enjoy a week’s worth of sleep deprivation with them in Hamburg. The trip

away was the catalyst that made Chapter 5 possible, as was Gustau’s enthusiastic

help with obtaining some of the data. Also, thanks go to Jim Scott for his ency-

clopedic knowledge. I must also thank Akeela Lookman of the group for her help

in obtaining the LSCO data presented in Chapter 3, and Matt Dawber for useful

discussions on the electric field penetration model used in Chapter 4. Thank you

also to Stephen McFarland and Jackie Patrick in the EMU for their assistance and

expertise on the microscopes, and for taking the time to coat my films in gold as

I needed them.

A lot of fun has been had in the group along the way, at least for me anyway.

i

Thank you to Akeela, Alison McMullan, Mohamed Saad, and Stephen Campbell,

who have made these past four years seem more like four months. Together we

have enjoyed the fun of liquid nitrogen, helium, and the general misuse of cellotape

and lab equipment. Oh, and let’s not forget all the cake and mountains of food

that have been consumed (mostly by Steve!).

I would like to thank all the new circle of physics friends Tony, Phil, Adam and

Jean for making lunch times a little more interesting and lively. Special thanks go

to Claire ‘Tubby’ Harper, for making me smile and laugh, and for always asking

‘why?!?’ when neither of us knew the answer. She has helped with this thesis

more than she may realise.

I would like to express my eternal thanks to my wife Helen, for always being

there for me when I needed her. Her selfless support has helped me through many

difficult times, particularly during the writing of this thesis, from the little things

like a cuddle or a cup of coffee, to bigger things like the typing of the references.

Finally I’d especially like to thank my dad and mum, Billy and Flo for giving me

all their love, encouragement, and support over the years.

ii

Abstract

Thin and ultrathin ferroelectric capacitors were fabricated using Pulsed Laser De-

position, and characterised both functionally and structurally to investigate the

thickness dependence of the permittivity, and the effects of mechanical boundary

conditions on the ferroelectric layer’s structural phase.

The series capacitor model was implemented to investigate the frequency and

temperature characteristics of thin film ferroelectric SrRuO3/Ba0.5Sr0.5TiO3/Au

and (La,Sr)CoO3/Ba0.5Sr0.5TiO3/Au capacitors. The extracted bulk component

was observed to be similar to bulk ceramics, displaying little frequency dependence

and a large peak permittivity at the expected temperature of 250 K. The extracted

interfacial component in the LSCO/BST was observed to have a little frequency

and temperature dependence, but the SRO/BST system displayed large frequency

and temperature dependence above T = 300 K. This was attributed to the thermal

de-trapping of charge carriers from defects in a thin layer parallel to the electrodes.

Ultrathin BST films (d = 5 − 16 nm) were successfully grown on LSCO elec-

trodes, and exhibited excellent functional properties. The thickness dependence of

the measured permittivity of these films was found to adhere to the series capacitor

model down to 5 nm, thereby reducing the upper limit of the total ‘dead-layer’

thickness from 7.5 nm as determined by Sinnamon et al (Appl. Phys. Lett. 78,

1724 (2001)) to 5 nm. High-resolution transmission electron microscopy of the

ultrathin films showed no evidence for a distinct interfacial ‘dead-layer’. A model

based on the space charge induced within the electrodes when the applied electric

field penetrates into its surface was used to calculate an interfacial capacitance of

di/εi = 0.47 nm for the LSCO/BST/Au system, which is close to the experimental

value of di/εi = 0.50± 0.06 nm.

The evolution of the structural phases of 2-dimensional mechanically clamped

LSCO/BaTiO3/Au thin films, experiencing zero misfit strain, has been investi-

gated using high resolution x-ray diffraction (XRD), and functional characterisa-

tion. Dielectric anomalies observed in the functional response of ‘non-virgin’ films

iii

corresponded well to the temperatures of expected bulk phase transitions, but

the anomaly associated with the tetragonal-cubic phase transition was suppressed

in ‘virgin’ films. The change in this functional behaviour was attributed to out-

of-plane ferreoelectric domains induced by an internal bias field associated with

asymmetric electrodes. Using XRD, the structural phase was determined to be

orthorhombic, with the longest axis in-plane, for T < 290 K, and tetragonal with

a = b < c for T > 290 K. Overall, the structural behaviour was observed to behave

similar to a bulk ceramic, and not as predicted by Pertsev et al (Phys. Rev. Lett.

80, 1988 (1998)) and Dieguez et al (Phys. Rev. B 69, 212101 (2004)).

iv

Publications

Some of the work outlined in this thesis has resulted in several publications of

articles in peer reviewed journals. Listed below is a summary of all publications

of work that the author has directly been involved.

Publications

L. J. Sinnamon, J. McAneney, R. M. Bowman and J. M. Gregg,

“Dependence of the interfacial capacitance on measurement regime used for inves-

tigation of thin film ferroelectric capacitors”, J. Appl. Phys., 93(1), 736 (2003).

J. McAneney, L. J. Sinammon, R. M. Bowman and J. M. Gregg,

“Temperature and frequency characteristics of the interfacial capacitance in thin-

film barium-strontium-titanate capacitors”, J. Appl. Phys., 94(7), 4566 (2003).

J. McAneney, L. J. Sinnamon, A. Lookman, R. M. Bowman, and J. M. Gregg,

“Characteristics of the interfacial capacitance in thin film Ba0.5Sr0.5TiO3 capaci-

tors with SrRuO3 and (La,Sr)CoO3 bottom electrodes”, Integr. Ferroelectr, 60, 79

(2004).

A. Lookman J. McAneney, R. M. Bowman, J. M. Gregg, J. Kut, S. Rios, A.

Rudiger, M. Dawber, and J. F. Scott,

“Effects of poling, and implications for metastable phase behavior in barium stron-

tium titanate thin film capacitors”, Appl. Phys. Lett, 85, 5010 (2004).

v

G. Catalan, B. Noheda, J. McAneney, L. J. Sinnamon, and J. M. Gregg,

“Strain gradients in epitaxial ferroelectrics”, Phys. Rev. B, 72, 020102(R) (2005).

S. Rios, J. F. Scott, A. Lookman, J. McAneney, R. M. Bowman, and J. M.

Gregg,

“Phase transitions in epitaxial Ba0.5Sr0.5TiO3 thin films”, J. Appl. Phys., 99,

024107 (2006).

M. M. Saad, P. Baxter, J. McAneney, A. Lookman, L. J. Sinnamon, P. Evans,

A. Schilling, T. Adams, X. Zhu, R. J. Pollard, R. M. Bowman, J. M. Gregg, P.

Zubko, D. J. Jung, F. D. Morrison and J. F. Scott

“Investigating the effects of reduced size on the properties of ferroelectrics”, IEEE

Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, in press (2006).

vi

Contents

Acknowledgements i

Abstract iii

Publications v

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Dielectrics and the Relationships between Permittivity and

Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.4 Ferroelectricity . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.5 Perovskites . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.6 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.7 Barium Titanate . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2 Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.1 Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.2 Collapse of Dielectric Constant . . . . . . . . . . . . . . . . 18

1.2.3 Interfacial Capacitance . . . . . . . . . . . . . . . . . . . . . 22

1.2.4 Characteristics of the Interfacial Capacitance . . . . . . . . . 23

1.3 Models of the Interfacial Capacitance . . . . . . . . . . . . . . . . . 25

1.3.1 The ‘Dead-layer’ . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3.2 Electrode Screening . . . . . . . . . . . . . . . . . . . . . . . 30

1.3.3 Interfacial Strain . . . . . . . . . . . . . . . . . . . . . . . . 35

1.4 Effect of Mechanical Boundary Condition on Phase Diagrams . . . 37

2 Experimental Methods 44

2.1 Capacitor Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . 44

vii

CONTENTS

2.1.1 Pulsed Laser Deposition . . . . . . . . . . . . . . . . . . . . 44

2.1.2 Target Preparation . . . . . . . . . . . . . . . . . . . . . . . 47

2.1.3 Deposition Procedure . . . . . . . . . . . . . . . . . . . . . . 49

2.2 Functional Measurements . . . . . . . . . . . . . . . . . . . . . . . 50

2.2.1 Functional Measurements of Thin Films . . . . . . . . . . . 51

2.2.2 Functional Measurements of Ultrathin films . . . . . . . . . 53

2.2.3 Polarisation Hysteresis Loops . . . . . . . . . . . . . . . . . 54

2.2.4 Measurement of Depolarisation Current . . . . . . . . . . . . 55

2.3 Transmission Electron Microscope . . . . . . . . . . . . . . . . . . . 55

2.3.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . 56

2.3.2 TEM Image Acquisition . . . . . . . . . . . . . . . . . . . . 58

2.3.3 Energy Dispersive X-ray Spectroscopy . . . . . . . . . . . . 60

2.4 X-ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.4.1 Bragg Law of Crystal Diffraction . . . . . . . . . . . . . . . 61

2.4.2 X-Ray Diffractometer . . . . . . . . . . . . . . . . . . . . . . 63

2.4.3 Sample Alignment . . . . . . . . . . . . . . . . . . . . . . . 64

2.4.4 Determination of Lattice Parameters . . . . . . . . . . . . . 66

2.4.5 Synchrotron Diffractometer . . . . . . . . . . . . . . . . . . 67

2.4.6 Grazing Incidence X-ray Analysis . . . . . . . . . . . . . . . 69

3 Characterisation of Bulk and Interfacial Properties 71

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2 Thickness Dependence . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2.1 SRO/BST system . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2.2 LSCO/BST System . . . . . . . . . . . . . . . . . . . . . . . 75

3.3 Series Capacitor model . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4 Behaviour of Bulk Component . . . . . . . . . . . . . . . . . . . . . 79

3.5 Behaviour of Interfacial Component . . . . . . . . . . . . . . . . . . 82

3.5.1 SRO/BST System . . . . . . . . . . . . . . . . . . . . . . . 82

3.5.2 LSCO/BST System . . . . . . . . . . . . . . . . . . . . . . . 85

3.6 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4 Extension of Series Capacitor Model to the Ultrathin Regime 91

4.1 Characterisation of Ultrathin capacitors . . . . . . . . . . . . . . . 92

4.1.1 Structural Characterisation and Thickness Determination . . 92

4.1.2 Functional Characterisation . . . . . . . . . . . . . . . . . . 94

viii

CONTENTS

4.1.3 Capacitance-Voltage Measurements . . . . . . . . . . . . . . 96

4.1.4 Thickness Dependence of Ultrathin Permittivity . . . . . . . 98

4.2 Extension of Series Capacitor Model . . . . . . . . . . . . . . . . . 99

4.3 Electrode Field Penetration . . . . . . . . . . . . . . . . . . . . . . 102

4.3.1 Derivation of Series Capacitance . . . . . . . . . . . . . . . . 103

4.3.2 Calculation of Electrode Properties . . . . . . . . . . . . . . 107

4.3.3 Application of Model . . . . . . . . . . . . . . . . . . . . . . 110

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5 Phase Transitions in Thin Film Barium Titanate 114

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2 Functional Measurements . . . . . . . . . . . . . . . . . . . . . . . 116

5.2.1 Capacitance Results . . . . . . . . . . . . . . . . . . . . . . 117

5.2.2 Relaxation Analysis . . . . . . . . . . . . . . . . . . . . . . . 120

5.2.3 Depolarisation Current . . . . . . . . . . . . . . . . . . . . . 121

5.2.4 Polarisation Hysteresis . . . . . . . . . . . . . . . . . . . . . 123

5.3 XRD Structural Phase Determination . . . . . . . . . . . . . . . . . 124

5.3.1 Synchrotron XRD . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3.2 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . 129

5.4 XRD Temperature Investigation . . . . . . . . . . . . . . . . . . . . 130

5.4.1 Measurement of Temperature Dependence of Structural Be-

haviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.4.2 Temperature Dependence of Structural Properties . . . . . . 131

5.4.3 Influence of Apparatus on Structural Measurements . . . . . 133

5.4.4 Room Temperature Phase Determination . . . . . . . . . . . 133

5.4.5 Effect of Internal Bias on Structural Properties. . . . . . . . 136

5.4.6 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . 137

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6 Conclusions and Further Work 140

6.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 140

6.2 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

ix

Chapter 1

Introduction

The aim of this chapter is to give a brief introduction to the properties of dielectric

materials, and the special group of dielectrics known as ferroelectrics. The first

few sections illustrate the physical properties of these materials, in particular the

concepts of capacitance and dielectric constant, as well as briefly examining their

ferroelectric aspects. Later sections concentrate on the problems associated with

these materials when their dimensions are reduced to the thin (d < 1µm), and

ultrathin (d < 10) nm regime. The bulk of this chapter, however, is a review of

the literature devoted to thin and ultrathin film ferroelectric research and spans a

large time period from the 1950’s to the most recent work of this year.

1.1 Background

1.1.1 Dielectrics and the Relationships between Permittiv-

ity and Conductance

Broadly speaking, all materials in nature can be classified as either a conductor,

or a dielectric, according to their response to an applied electric field. When

an electric potential difference is applied across a conductor (or semiconductor)

there is a net flow of charge within the material, due to an excess of free charge

carriers. In a dielectric material, however, although charge carriers are not free,

they can become displaced from their equilibrium positions by an applied electric

field, resulting in a net dipole across the material. For example, if we separate

two charges −Q and +Q by a distance d, then there will exist a dipole moment

between the two charges of magnitude p = Qd. This is a very simplistic view,

and in reality the boundary between these two classes of materials can be quite

1

1.1 Background

blurred. Indeed, it is impossible to find a perfect dielectric in nature, since most

dielectric materials can be considered as wide band gap semiconductors, and so

applying an electric field will produce a limited degree of conduction.

The polarisation P of a dielectric is defined as the net dipole moment per unit

volume and is related to the electric field E by

P = ε0χE, (1.1)

where ε0 is the permittivity of free space and

P = (ε− 1)ε0E, (1.2)

since the susceptibility χ = ε − 1. Here ε denotes the relative dielectric constant

of the material, with a value of unity corresponding to that of free space. The

dielectric displacement, D, can now be defined as

D = ε0E + P ≡ ε0εE. (1.3)

So far this description is adequate for describing a steady state electric field acting

upon a perfectly insulating dielectric. If an alternating field is applied, with angular

frequency ω, then there will exist a displacement current density, ∂D/∂t, associated

with the reorientation of the bound charges, as well as an electric current flux j,

due to any free charge carriers. These can be linked via Maxwell’s equation,

∇×H = j +∂D

∂t. (1.4)

The time dependent fields are related, via Fourier transforms, to the corresponding

frequency dependent quantities which determine the spectral behaviour of the

dielectric response:

E(t) =

∫ ∞

−∞E(ω)e−iωtdω, (1.5a)

D(t) =

∫ ∞

−∞D(ω)e−iωtdω. (1.5b)

Since these fields are real quantities, E(ω) = E∗(−ω) and D(ω) = D∗(−ω) [1],

where the star denotes the complex conjugate. Now by considering that j is related

2

1.1 Background

to the field E by the conductivity σ, equation (1.4) reduces to

∇×H(ω) = σ∗(ω)E(ω)− iωε0ε∗(ω)E(ω), (1.6)

where the star denotes a complex quantity. By treating the response entirely

in conductivity terms, a general expression for the conductivity (which includes

dielectric contributions) can be defined as

σ = σ∗(ω)− iωε0ε∗(ω). (1.7)

Also, by considering the contributions to ∇ × H being entirely due to the dis-

placement current density ∂D/∂t, then the generalised dielectric constant can be

defined as

ε = ε∗(ω) +iσ∗(ω)

ε0ω. (1.8)

Thus, it is apparent that only for static applied fields can a true distinction between

bound and free charge carriers be made, since for alternating fields, alternating

motion of bound charges contributes to the a.c. conductivity, and the oscillating

motion of free charges contributes to the frequency-dependent dielectric constant.

In other words, for a real dielectric within an a.c. field, the measured dielectric

function will be a combination of the true dielectric response, and other mobile

charge carrying phenomena. For an oscillating electric field, E ∝ E0 exp(−iωt),

the complex dielectric constant in terms of real (ε′) and imaginary components

(ε′′) is expressed as,

ε∗ = ε′(ω) + iε′′(ω), (1.9)

and thus, the displacement current density D is given by

D =∂D

∂t= (−iωε0ε

′ + ωε0ε′′)E, (1.10)

where the first term in the brackets (−iωε0ε′) is a reactive component and the

second term (ωε0ε′′) is a resistive component of the total current density. Equation

(1.10) can be illustrated in an Argand diagram of resistance versus reactance as

shown in figure 1.1 and demonstrates that, for a real dielectric, there will exist

an angle δ, known as the loss angle that relates both the real and imaginary

3

1.1 Background

Figure 1.1: Argand diagram for displacement current density D

components of ε and σ by the equation,

tan δ =ε′′

ε′=

σ′

σ′′ =σ∗

ωε0ε′. (1.11)

Hence, from equation (1.11) the following relations can be derived ,

σ∗(ω) = σ′(ω) + iσ′′(ω), (1.12)

σ∗ = ε0ε′′ω, (1.13)

σ′′ = ε0ε∗ω. (1.14)

1.1.2 Relaxation

As discussed in the previous section, the magnitude of the dielectric constant, is de-

pendent upon the frequency of the applied electric field. The reason for this lies in

the various mechanisms of displacement current contributing to the total dielectric

function. For example, the small inertial mass of electrons means that electronic

mechanisms, i.e. atomic or bond polarisation, contribute to the dielectric constant

at high frequencies, whereas the polarisation mechanisms involving the motions of

more inertially massive ions will only contribute at much lower frequencies. These

observations can effectively be modelled for all mechanisms, except orientational

dipoles, by using a driven damped oscillator model mr + mγr + kr = qE, where

m is the reduced mass of an oscillating object of charge q, r is the displacement

associated with polarisation, γ is a damping constant arising from coupling to

other excitations, and k is the restoring force spring constant to the driving field

E. As illustrated in figure 1.2(a), as the frequency of the oscillating field increases

from static values, the real part of the dielectric constant, (ε′) decreases in a series

4

1.1 Background

Figure 1.2: a) The frequency dependence of the real part of the dielectricconstant ε′, showing the relative contributions of different mechanisms. b)The frequency dependence of the real and imaginary parts of the dielectricconstant for atomic polarisability using the damped driven oscilator model.

of discontinuous steps as the more inertially massive mechanisms become ‘frozen’

due to their inability to keep pace with the applied field. At each of these steps ε′′

assumes a Lorentzian peak, near the resonant frequency ω0 corresponding to the

resonant absorption of energy, as illustrated in figure 1.2(b).

The frequency response of the orientational dipoles was first considered by

Debye [2] in 1929. The Debye dipolar relaxation model (figure 1.3) assumes

that the sudden application of an electric field at t = 0 results in a polarisation

P∞ = χ(∞)ε0E that is instantaneous on the timescale of dipolar rotation. Dipolar

relaxation then causes the polarisation to increase further with a time-dependent

component P′(t) until the static value PS is attained, where PS = P∞+P′(t = ∞).

Figure 1.3: The time dependent response of the polarisation P(t), after theapplication of an electric field at t = 0.

5

1.1 Background

The Debye relaxation equation is then given by

dP′

dt=

P′(∞)−P′(t)

τD

, (1.15)

where τD is the Debye relaxation time and P(t) = P∞ + P′(t). The time depen-

dence of the polarisation is then found by integrating equation (1.15):

P′(t) = (PS −P∞)

(1− exp

(− t

τD

)). (1.16)

For a sinusoidal applied field E ∝ E0 exp(−iωt), and remembering the instanta-

neous response at t = 0, equation (1.15) can be transformed into the frequency

dependent complex dielectric constant

ε†(ω) = ε(∞) +(ε(0)− ε(∞))

1− iωτD

, (1.17)

where the real and imaginary parts are given by

ε′(ω) = ε(∞) +(ε(0)− ε(∞))

1 + ω2τ 2D

, (1.18a)

ε′′(ω) =(ε(0)− ε(∞))ωτD

1 + ω2τ 2D

. (1.18b)

Figure 1.4: (a) Frequency response of the dielectric constant associatedwith dipolar orientation, using the Debye model, (b) a Cole-Cole plot forthe same relaxation function.

6

1.1 Background

The frequency domain representation of the Debye equations is illustrated in

figure 1.4(a). It can clearly be seen that the ε′ is equal to the static dielectric

constant, εS, at low frequencies, but will decrease to ε∞ in the high frequency

regime. The imaginary component (or dielectric loss) ε′′ is zero outside this dis-

persion region but peaks at a characteristic frequency ω = 1/τD. A useful way of

plotting dielectric data is in the form of a Cole-Cole plot. As illustrated in figure

1.4(b), a complex plane, Cole-Cole plot of ε′′ versus ε′ as an implicit function of fre-

quency, results in a simple semicircle, for a perfect Debye relaxor. Unfortunately,

perfect Debye behaviour is absent in nature, due primarily to the exponential rela-

tion in equation (1.16) being inaccurate in real materials. Nonetheless, the Debye

formulation is a useful starting point for describing more complex systems.

1.1.3 Capacitance

Figure 1.5: The parallel plate capacitor.

Consider two parallel conducting plates of area A, separated by a dielectric

material of dielectric constant εε0 and of thickness d. If d is very much less than

the dimensions of the plates, then fringe effects can be ignored and the charge

induced on the plates will be proportional to the applied potential V across the

dielectric, i.e.

Q = CV, (1.19)

where C is the constant of proportionality, or capacitance. If Gauss’ Law is applied

to the parallel plate geometry, then the surface charge density Q/A induced by

the applied field E = V/d isQ

A= εε0E. (1.20)

The capacitance can easily be calculated by combining equations (1.19) and (1.20),

7

1.1 Background

Figure 1.6: Simple circuit diagrams for (a) two capacitors in series and (b)two capacitors in parallel.

and is given by

C =εε0A

d. (1.21)

If two capacitors C1 and C2 are connected in series (as in figure 1.6(a)), then

the potential dropped across the combination is simply the sum of the potential

difference across each capacitor, (i.e. VT = V1 + V2); however the total charge of

the combination is equal to the charge on each capacitor (i.e. QT = Q1 = Q2).

Thus it can be easily shown that the total capacitance CT is determined from the

relation1

CT

=1

C1

+1

C2

, (1.22)

and hence, the total capacitance will always be less than the capacitance of each

capacitor. For an array of n capacitors in series, equation (1.22) is generalised as

1

CT

=n∑

k=1

1

Ck

. (1.23)

For two capacitors connected in parallel (as demonstrated in figure 1.6(b)), the

total charge across the combination is split between the two capacitors (QT =

Q1 + Q2), whereas the potential across each capacitor is the same. This means

that the total capacitance is simply the linear combination of each capacitor, and

is generalised as

CT =n∑

k=1

Ck. (1.24)

8

1.1 Background

1.1.4 Ferroelectricity

Ferroelectrics are a subclass of pyroelectric materials, which possess at least one

polar axis, and demonstrate a thermally dependent spontaneous polarisation PS

without an applied field. However, the definition of a ferroelectric requires that

the direction of PS can be reversed upon application of an electric field. Ferroelec-

tricity was first discovered in 1920 by Valasek [3] in the compound Rochelle Salt

(NaKC4H4O6 · 4H2O), but the observation was difficult to reproduce, since any

small deviation in the chemical composition destroyed the ferroelectricity. Thus,

it was widely believed that this phenomena was a curious anomaly of this sub-

stance, and was experimentally ignored until ferroelectricity was discovered in the

relatively simple perovskite, Barium Titanate (BaTiO3), in 1945.

Figure 1.7: Hysteresis loop for a typical ferroelectric. PS and Pr are thespontaneous and remnant polarisations and EC is the coercive field.

The phenomenon of ferroelectricity is usually associated with a deformation within

the crystal lattice, which often results in a displacement of the overall charge den-

sity of the material. It is this shift in charge density that induces PS, which

generally forms in domains that nucleate parallel to the crystallographic axes, re-

ducing the free energy of the system. However, in the absence of an electric field

and /or external stress, the overall orientations of these domains is random, and so

the net polarisation across the crystal is zero; if an external stimulus is applied in

the form of an electric, or biaxial stress field, it induces the domains to renucleate

parallel to the applied field, resulting in measurable charge on the surface of the

crystal. Figure 1.7 shows the response of a ferroelectric crystal to a cyclic electric

field, that is of sufficient strength to switch the direction of PS. In this figure, EC

9

1.1 Background

is the coercive field required to switch the direction of the polarisation P , whilst

Pr is the remnant polarisation when the electric field is removed.

As mentioned previously, PS is a thermally dependent variable (pyroelectricity),

and in most ferroelectric materials will tend to decrease as the temperature of

the crystal increases, until at a temperature T = TC , PS = 0 and the material

becomes paraelectric. TC is known as the Curie temperature, in analogy with

ferromagnatism, since in the paraelectric regime the dielectric constant ε exhibits

a Curie-Weiss behaviour,

ε =C

T − T0

, (1.25)

where C is the Curie constant of the material. In common with other types of

phase transition (and ignoring tri-critical cases), ferroelectric transitions can be

classified as either first-order, or second-order transitions. First-order transitions

display a discontinuity in the first derivative of the free energy with respect to T

or P , and are characterised by a discontinuity as PS → 0 at T = TC . Second-order

transitions are continuous in the first derivative, but discontinuous in the second

derivative of the free energy, resulting in a continuous function as PS → 0, but a

singularity in ε at T = TC .

Figure 1.8: First and second order phase transitions in the vicinity of theCurie temperature TC and their implications of PS(left) and ε′(right) acrossthe transition (after [4]).

10

1.1 Background

1.1.5 Perovskites

The perovskite structure, named after the mineral CaTiO3, is a simple structure

adopting the chemical formula ABO3 where A is a monovalent, divalent, or triva-

lent metal, and B is a trivalent, tetravalent or pentavalent one. The ideal structure

is shown in figure 1.9, and has A cations at the corners of a cubic unit cell, with

the B cations occupying the body centre. The oxygen atoms are located at each of

the face centres of the cube, forming an octahedral cage around the B atom. The

BO6 octahedra form corner sharing bonds with the BO6 octahedra in the neigh-

bouring cells, resulting in a crystal with an infinite corner sharing array. In this

ideal cubic form, the ratio of the A-O to the B-O bond lengths must equal√

2 [6].

However, when this condition cannot be met, the structure distorts by changing

the shape of the octahedral cages, either by moving the cations off centre, or by

twisting the linkages of the octahedra themselves, or even by both mechanisms.

These distortions invariably reduce the symmetry of the unit cell, resulting in a

structural phase transition from a cubic phase to a non-cubic phase. At high

temperatures, and with no applied field, these structures remain cubic since there

is enough thermal energy to create large vibrational amplitudes of the A and B

cations. Once the temperature is reduced below a critical temperature TC , the

vibrational amplitudes are no longer large enough to sustain the high symmetry

cubic form, resulting in the distortion of the prototypical unit cell. The nature

of this distortion depends upon the size of the A and B cations in relation to the

’hard’ O6 octahedra [6]. Inevitably, these distortions deform the crystal lattice

Figure 1.9: left) The perovskite structure unit cell and, right) as viewed asa network of oxygen octahedra, after [5].

11

1.1 Background

causing the migration of the overall charge density, and inducing a spontaneous

polarisation; if this can then be reversed by the application of an electric field the

crystal becomes ferroelectric.

Since the discovery of ferroelectricity in BaTiO3, the number of known ferro-

electric materials, having the perovskite structure, has increased dramatically, due

to the ability to complicate the ABO3 formula by substitution or doping of the

A and B lattice sites with other types of cations. A popular example of this is

the solid solution BaxSr1−xTiO3 which is achieved by the isovalent substitution of

the Ba cation in BaTiO3 for the smaller Sr cation. This causes the destabilisation

of the tetragonal (ferroelectric) phase of the perovskite, effectively reducing the

ferroelectric/paraelectric transition temperature as the concentration of Sr cations

increases.

1.1.6 Phenomenology

The phenomenological theory of dielectrics and ferroelectrics treats the materials

in question as a single macroscopic entity, whose physical properties can effectively

be described using only the principles of thermodynamics and classical mechanics.

Although it is a relatively simple approach, and is capable of describing many

experimental observations, it is limited by two factors: firstly, it largely ignores

the physics of the atomic structure of the material, and does not describe the

atomic displacements which accompany the process of polarisation and switching

in ferroelectrics; secondly, it is only valid for a material in equilibrium, and thus

cannot describe non-equilibrium conditions, such as those that occur during the

switching of a ferroelectric. The phenomenological description of a ferroelectric was

first considered by Devonshire [7, 8, 9], who expanded on the work of Landau and

Ginzburg, to explain ferroelectricity in BaTiO3. This analysis is now commonly

referred to as Landau-Ginzburg-Devonshire (LGD) theory [10].

Using the first law of thermodynamics, the change in internal energy dU , can

be expressed in terms of three conjugate state variables, temperature (T ) and

entropy (S); stress (Xi) and strain (xi); and displacement (Di) and electric field

(Ei):

dU = SdT + Xidxi + EidDi. (1.26)

It is possible to define other thermodynamic potentials using the conjugate pairs,

since there are three independent variables that can be combined in eight different

12

1.1 Background

ways. One such potential is the elastic Gibbs free energy

G = U − TS −Xixi, (1.27)

where U is the internal energy as defined by equation (1.26). It is common for the

free energy to be expressed as G1 but here it is simply called G. Using equation

(1.26), the differential form of equation (1.27) is given by

dG = −SdT − xidXi + EidDi. (1.28)

For a free crystal experiencing zero stress (dXi = 0), and held at a constant

temperature (dT = 0), equation 1.28 simplifies to

dG = EidDi. (1.29)

Recalling that D = ε0E+P , D can be replaced with P in equation (1.29), since for

a ferroelectric the polarisation P is much larger than the free space component ε0E.

In his theories on phase transitions, Landau introduced the concept of an order

parameter η, which is related to the change of some macroscopic property through

the phase transition, and is thus a measure of the deviation of the low temperature

phase from that of the high temperature phase. Near the phase transition, the free

energy can be expanded in terms of a polynomial of η,

G = g1η +1

2g2η

2 +1

3g3η

3 +1

4g4η

4 + . . . . (1.30)

For a ferroelectric it is customary to chose the macroscopic polarisation as the

order parameter, leading to the LGD theory. If the ferroelectric crystal has a high

temperature centro-symmetric phase, as in perovskite materials, then odd powers

in the expansion are neglected since the change of P → −P leaves G unchanged.

Considering a free perovskite crystal with polarisation along one of the crystal axis,

the elastic Gibbs free energy, as defined with respect to the cubic phase, becomes

G =1

2g2P

2 +1

4g4P

4 +1

6g6P

6 + . . . , (1.31)

where the coefficients of P are temperature dependent, in particular g2 which is

13

1.1 Background

given by g2 = C(T−T0)

with C being the materials Curie constant. Stable states are

characterised by the minima of G with the necessary and sufficient conditions

∂G

∂P= E = g2P + g4P

3 + g6P5 = 0, (1.32a)

∂2G

∂P 2= χ−1 = g2 + 3g4P

2 + 5g6P4 > 0. (1.32b)

It is apparent that when the crystal is in the non polar phase, the dielectric stiffness

χ−1 as determined by equation (1.32b) is identical to the Curie-Weiss law,

χ−1 ≈ 1

εε0

=1

Cε0

(T − T0). (1.33)

Inspection of equation (1.31) shows that there are two distinct cases to consider.

Firstly, if the coefficient g4 > 0 then no new physics is introduced and the sixth

order term can be neglected [11] resulting in a phase transition that is second order

in nature with T0 = TC . However, if g4 < 0 then g6 > 0 and the phase transition

becomes first order in nature and T0 < TC . The free energy curves for each case

is illustrated in figure 1.10. Clearly, a second order phase transition would appear

less complex than a first order phase transition, since the former is characterised

by a single temperature TC , which indicates the temperature at which PS becomes

unstable, and the ferroelectric becomes paraelectric. A first order phase transition

has four characteristic temperatures T0, TC , T1, and T2. Below T = T0, the crystal

is ferroelectric and thus exhibits a spontaneous polarisation, however at T = T0 an

extra minimum corresponding to the non-polar paraelectric phase appears. This

parelectric phase is metastable and coexists with the ferroelectric phase up to

T = TC at which point the non-polar phase becomes stable whilst the polar phase

coexists in a metastable state. This behaviour continues until T = T1 at which

point, the minima corresponding to the metastable polar states all but disappears,

leaving behind two points of inflection on the free energy curve, until finally at

T = T2, these vanish resulting in a purely parabolic curve corresponding to a truly

paraelectric crystal.

In this discussion, the effects of stress or strain on the free energy of the mate-

rial has been neglected, but obviously, when a structural phase transition occurs,

the unit cell will distort, inducing a degree of stress or strain which accompanies

the induced polarisation. Conversely, an external stress or strain can be impressed

upon the crystal, which will then cause an appreciable change in the free energy, re-

sulting in either a stabilisation, or destabilisation of the spontaneous polarisation,

14

1.1 Background

depending upon the direction and magnitude of the stress or strain fields. This

latter is very important in the fabrication of epitaxial thin films (see Sections 1.3.3

and 1.4), where strain can be introduced into the film by many mechanisms, which

can include ionic vacancies and changes in film stoichiometry, as well as the pres-

ence of grain boundaries. By far the most important strain-inducing mechanism

for thin films, is that induced by the mechanical boundary conditions associated

with the underlying substrate. The strain in this case arises from the mechanical

clamping of the thin film to the thick substrate, and this can be increased further

by a large mismatch in the lattice parameters and thermal expansion coefficients

of the two materials.

Figure 1.10: a) First and b) second order phase transitions in the vicinityof the Curie temperature TC (after [12]).

15

1.1 Background

1.1.7 Barium Titanate

BaTiO3, and its family of solid solutions (BaxSr1−xTiO3 etc.), is one of the most

extensively studied ferroelectric materials. Above 130 C, a pure BaTiO3 ceramic

is characterised by the prototypical cubic perovskite structure, but will undergo

three first order structural phase transitions as the temperature decreases, each of

Figure 1.11: Various properties of BaTiO3 as a function of temperature,illustrating the discontinuous change in a) lattice constant, b) spontaneouspolarisation, and c) relative permittivity. Anisotropic properties are shownwith respect to the lattice direction [13].

16

1.1 Background

which is accompanied by a discontinuous change in the crystal lattice constant,

spontaneous polarisation and permittivity (Figure 1.11 (a), (b) and (c) respec-

tively). Below T = 130 C, BaTiO3 becomes ferroelectric as its structure changes

from cubic to tetragonal, with PS directed along the [001] direction. As the tem-

perature decreases further, the structure becomes orthorhombic at T = 5 C,

and finally transforms to rhombohedral at T = −90 C , with PS directed along

[011] pseudocubic and [111] pseudocubic directions respectively. There does ex-

ist a non-ferroelectric hexagonal phase above T = 1460 C and also one at room

temperature, but this latter phase is metastable.

The temperatures at which these transitions occur is of course influenced by

conditions imposed upon the crystal. The effects of both electric and strain fields

on the ferroelectric transition temperature has previously been discussed. The

purity of the material can also influence the phase transition temperatures. The

transition temperatures of the end-members of a solid solution can be increased or

decreased by a change in the composition of the crystal. When an impurity atom

is substituted into the crystal, it induces a local strain field around that atom.

When sufficient atoms have been substituted, the strain fields coalesce to produce

an effective strain term in the free energy of the material, which raises or lowers

the transition temperature.

BaxSr1−xTiO3 (BST) is a widely studied thin film system, largely due to

the fact that its material properties such as transition temperatures and lat-

tice constants can be ‘tuned’ by the control of the Ba/Sr ratio. For example,

when the Ba/Sr ratio is 0.5, the ferroelectric transition temperature is reduced

to T = −25 C, and the structure is cubic at room temperature, with a lattice

constant of a = 3.957 A. Indeed, it has been observed that transition temperature

and lattice constant are approximately linearly dependent upon the the Sr concen-

tration. This ‘tuning’ makes BST solid solutions extremely attractive for memory

device applications [14].

17

1.2 Size Effects

1.2 Size Effects

1.2.1 Thin Films

The physical properties of ferroelectric materials, such as their high dielectric con-

stant, and spontaneous polarisations have made them highly desirable for appli-

cations in integrated devices. For example, their high dielectric constant makes

ferroelectrics attractive for high density charge storage devices (e.g. Dynamic

Random Access Memories (DRAM) and high density capacitors), whereas the

presence of a switchable remnant polarisation can be successfully implemented in

Non-Volatile memory applications (e.g. NVRAM).

Indeed, industry is looking to integrate ferroelectric materials into many silicon

semiconductor devices [15]. Integrated ferroelectric devices must necessarily be of

reduced dimensions (< 1µm) if they are to work in conjunction with conventional

semiconductor technology. It is for this reason that a large volume of research

has been performed on thin film ferroelectrics, both experimentally and theoreti-

cally. It has been widely observed that a thin film ferroelectric behaves in a vastly

different manner to that of its bulk counterpart. Some of these observed effects

include

• Increased polarisation fatigue under continuously cycling field [16].

• Dramatically reduced ε and alterations in TC [17].

• Change in the order of phase transition, and the appearance of structural

phases forbidden in bulk [18].

For the purpose of this work, a thin film is defined as a material whose thickness

d < 1 µm, although literature would seem to indicate that this limit should be of

the order of a few hundred nanometers. Following on from this, an ultrathin film

could be defined as a film whose thickness d < 10 nm.

1.2.2 Collapse of Dielectric Constant

The dielectric constant of bulk ceramic ferroelectric materials is very large, and it

is not uncommon for its value to exceed 1000 at room temperature, but can be

observed to be as much as 20,000 in the vicinity of the Curie temperature [19].

However, when the dimensions of the material are reduced to that of thin films, ε

is observed to be drastically decreased by orders of magnitude (figure 1.12). This

18

1.2 Size Effects

decrease in permittivity is observed to be highly thickness dependent (figure 1.13).

This effect has been observed in many perovskite titanates, in particular in PbTiO3

[20], Pb(Zr,TiO)3 [21], (Ba,Sr)TiO3 [22, 23], and its end-member compositions,

BaTiO3 and SrTiO3 [24].

Naturally, the permittivity of a thin film can be affected by the processing

conditions during the film fabrication. Thin perovskite films have been deposited

using various techniques, such as Pulsed Laser Deposition [26], Ion Sputtering [27],

Chemical Vapour Deposition [28] to name but a few, but invariably, there is no one

method that eliminates this size effect. Obviously, the microstructural quality of a

thin film will play a role in the suppression of ε. Komatsu and Abe [29] have shown

that thin films with mixed [001]/[011] crystal orientations in polycrystaline SrTiO3,

will cause a further reduction in ε compared with single [001] orientated films.

Fujisawa et al [30] note that thin polycrystalline Pb(Zr,TiO)3 films demonstrate

lower permittivities than their epitaxial counterparts, but that the epitaxial films

still show a strong thickness dependence of ε. Another processing factor to consider

is the effect of the film stoichiometry on the value of ε. Yamamichi et al [27] have

demonstrated that by varying the ratio of (Ba+Sr):Ti in (Ba,Sr)TiO3 the value

of ε can effectively be decreased. They noted that a 5% excess in Ti resulted in

a maximum of ε, but that a thickness dependence of the dielectric constant still

existed.

The microstructural quality of thin films has undoubtedly increased over the

last decade due to refinements in thin film fabrication processes. Indeed, films

Figure 1.12: Measured dielectric con-stant for BST ceramic and a 100 nm thinfilm (after [19]).

Figure 1.13: Thickness depen-dence of the measured dielectricconstant (after [25]).

19

1.2 Size Effects

with thicknesses less than 50 nm have been reported that demonstrate high de-

grees of crystallinity, controlled stoichiometries, and low dielectric losses, but still

demonstrate thickness dependent permittivites that are drastically smaller than

bulk values. This has led to an increase in the volume of work dedicated to re-

solving the nature of the origin of the dielectric collapse, both experimentally and

theoretically.

The most popular explanation for the collapse of ε is the presence of low per-

mittivity ‘dead-layers’ located at the electrode/dielectric interfaces [20, 21, 22, 23,

27, 31, 32, 33]. These parasitic layers will act in series with the bulk dielectric

function, resulting in an overall decrease in the measured dielectric constant. This

shall be discussed in greater detail in the following section.

There have been other suggestions as to the cause of reduced permittivity,

which do not rely on a functional ‘dead layer’ at the film surface. Sirenko et al [34]

attribute the reduction of ε to the hardening of the soft modes as the thickness

of the film reduces. The Lyddane-Sachs-Teller (LST) relation for phonon modes

in ferroelectrics relates the ratio of the static to high frequency ε to the ratio of

the eigenfrequencies of the longitudinal (LO) and transverse (TO) optical phonon

modes i.e.ε(0)

ε(∞)=

N∏j

ω2LOj

ω2TOj

. (1.34)

Thus, it is apparent that, should the frequency of the TO mode increase in magni-

tude, then the value of ε(0) would consequently decrease (assuming of course that

there is no sizeable change in ε(∞)). By investigating the lattice dynamics of the

lowest frequency TO phonon mode in SrTiO3 films, they observed that the LST

relation held for films as thin as 500 nm, and that the frequency of the TO mode

increased (i.e. hardened) as the thickness of the film decreased, which was closely

correlated with the increase of the films measured dielectric stiffness (figure 1.14).

However, data from Fedorov et al [35] illustrated that the LST relation no longer

held for a 280 nm film, leading Sirenko et al to conclude that an additional mech-

anism, possibly due to an intrinsic dead layer at the surface of the films, played a

dominant role in the suppression of ε in very thin films.

Another aspect of thin films that has been investigated is the influence of grain

size on dielectric properties. It has been observed that as the thickness of a film

decreases, there is a corresponding decrease in the size of the film’s grains. In-

deed, this correspondence results in a linear dependence of the dielectric constant

20

1.2 Size Effects

Figure 1.14: Comparison of the soft-mode frequencies and dielectric con-stants of bulk and thin-film STO. a) The square of the soft-mode TO1phonon frequency versus temperature for a 2 µm STO film (filled squares)and a STO single crystal. The single-crystal data is from Sirenko et al [34](open squares) and the hyper-Raman results from Vogt [36] (stars). b) Theinverse dielectric constant versus temperature for a 2 µm STO fillm (filledsquares) and a STO single crystal (open squares). The hardening of the softmode in the thin film clearly correlates to a lower static dielectric constantas predicted by the LST relation (after Sirenko et al [34]).

on the grain size [37], accentuated all the more by the work of Zhu et al [38]

which demonstrates the absence of a thickness dependence of ε in films of con-

stant grain size. The cause of the reduction of ε has been postulated to be due to

a functionally disrupted region of low permittivity adjacent to or within the grain

boundaries. The grains (and hence grain boundaries) in thin film ferroelectrics

tend to grow perpendicular to the surface, thus the effect on ε due to grain bound-

aries would only have a weak thickness dependence due, in principal, to a dilution

effect. However, Sinnamon et al [39] proposed a model that showed that the pres-

ence of low permittivity columnar grain boundaries could successfully explain the

observed collapse of ε for (Ba, Sr)TiO3, but this model only worked for purely

21

1.2 Size Effects

columnar grains. Recently, Visinoiu et al [40] have applied this model to dielectric

data from thin BaTiO3 films, resulting in a calculated grain boundary ‘dead-layer’

thickness of di = 12 nm. High resolution TEM analysis of a 150 nm thick BaTiO3

film revealed the presence of structurally modified grain-boundary layers, approx-

imately 8 nm thick, which could conceivably possess a lower permittivity than the

interior of the grains. However for films < 75 nm thick, a thickness dependence

of the dielectric constant is still observed even though columnar grains were only

observed to form above 75 nm thick.

1.2.3 Interfacial Capacitance

It has been widely postulated that the reduction of the dielectric constant in thin

film capacitors is due to the presence of a low permittivity ’dead-layer’ within

the electrode/dielectric interface. The effect of this layer is to introduce a small

interfacial capacitance which will then act in series with the capacitance of the

bulk material. Inspection of equation (1.22) shows that, if a low permittivity layer

exists, then the total measured capacitance, and thus effective dielectric constant,

will be much less than that of the normally behaved perovskite ferroelectric. If a

thin dielectric film of total thickness d and bulk permittivity of εb, has an interfacial

layer of thickness di and low permittivity εi, then by combining equations (1.21)

and (1.22), we can express the total capacitance in terms of material parameters

thus:d

ε=

d− di

εb

+di

εi

. (1.35)

If di is independent of the total thickness d and also di << d, and similarly εi << εb

then equation (1.35) is simplified to

d

ε=

d

εb

+di

εi

. (1.36)

This equation defines the essence of what is invariably known as the series capacitor

model. The final term in the equation encompasses all contributions to the series

capacitance, including the influence of both top and bottom electrode/dielectric

interfaces. By fitting a straight line to the plot of d/ε versus film thickness d, it is

possible to extract the value of the bulk dielectric constant, εb, from the reciprocal

of the slope, and the value of the interfacial capacitance from the intercept with the

y-axis. Unfortunately since di/εi is coupled, it is impossible to determine either

22

1.2 Size Effects

the thickness of the interfacial layer, or its permittivity without knowing one or

the other.

The physical origin of this interfacial layer is unknown, but there have been

numerous possibilities discussed in the literature. Surface layers that differed sig-

nificantly from bulk were first observed in 1954 by Anliker et al [41] in fine BaTiO3

powders in which the surface layers remained tetragonal above TC even though the

bulk structure was cubic. This was attributed to the presence of large electric fields

at the surface of the crystal. However, these surface anomalies were not observed

on etched single crystals [42], and the observation of Anliker et al was attributed

to a high temperature surface decomposition. In 1961, Schlosser and Drougard [43]

demonstrated that the dielectric constant and dielectric loss of thin BaTiO3 single

crystal wafers was very much dependent upon the thickness of the wafer. These

results were interpreted by assuming the existence of a surface layer with lower

permittivity than that of bulk, the former acting in series with the latter, reducing

the measured capacitance. Fatuzzo and Merz [12] suggested that the cause of this

low permittivity layer was due to high electric fields and mechanical strains within

the layer. Also, Bhide et al [44] inferred from electroluminescence measurements,

that a space charge layer with ε ∼ 200 and thickness ∼ 1 µm existed on BaTiO3

single crystals.

However, it has only been within the last fifteen years that serious interest into

the nature of the interfacial capacitance has taken off. This explosion in interest

was largely due to the desire to integrate thin film ferroelectric materials into

electronic devices; this brought the degree of the severity of the depression of the

permittivity into sharp focus. There have been a variety of mechanisms invoked

to explain the origin of the interfacial capacitance.

1.2.4 Characteristics of the Interfacial Capacitance

Most reports of an observed interfacial capacitance, tend to be from data taken at a

single temperature and measurement frequency [31, 46]. However, there have been

few investigations on the functional characteristics of the interfacial capacitance

with varying temperature and frequency. Basceri et al [25] observed that the mea-

sured interfacial capacitance of Pt/Ba0.7Sr0.3TiO3/Pt capacitors remains relatively

constant over a temperature range of 300-480 K (figure 1.15(a)). Later Park and

Hwang [45] demonstrated that the interfacial capacitance of a similar capacitor

system, Pt/Ba0.48Sr0.52TiO3/Pt, also remained constant within the temperature

23

1.2 Size Effects

Figure 1.15: Series capacitor plots demonstrating the temperature inde-pendence of the interfacial capacitance from a) Basceri et al [25], and b)Park and Hwang [45]. The inset of b) presents the interfacial capacitance asa function of thickness.

range of 480-580 K (figure 1.15(b)). However, Li et al [47], investigating the func-

tional behaviour of STO/YBCO capacitors, present data which would seem to

indicate a significant difference between the values of the interfacial capacitance

measured at 77 K and 280 K (figure 1.16(a)). Zafar et al [48] have performed an

extensive investigation of the frequency dependence of the interfacial capacitance

of Pt/Ba0.5Sr0.5TiO3/Pt at five temperatures (-40 C, -20 C, 25 C, 75 C and 125C), and have found that it remains constant over a frequency range of 102 − 106

Hz. They do note that below 25 C, the interfacial capacitance is constant, but

does increase by < 10% for temperatures > 25 C (figure 1.16(b)).

Figure 1.16: a) Series capacitor plot from Li et al [47], demonstrating anapparent temperature dependence of the interfacial capacitance (N.B. in thisplot the interfacial capacitance is determined from the slope of the best fitline). b) Frequency response of the interfacial capacitance, measured at fivetemperatures from Zafar et al [48]. The interfacial capacitance is frequencyindependent, but exhibits a temperature dependence for T > 25 C.

24

1.3 Models of the Interfacial Capacitance


The interfacial capacitance has invariably been observed in many thin film systems,

but its origin has yet to be conclusively identified. This section is dedicated to a

brief explanation of the more popular models proposed to explain the origin of the

interfacial capacitance.

1.3.1 The ‘Dead-layer’

The most widely attributed origin of the interfacial capacitance is the so called

‘dead-layer’. This layer is postulated to lie within the dielectric and adjacent to

the electrodes, and consists of a functionally disrupted material which exhibits a

much lower permittivity with respect to that within the interior of the film. The

origin of the layer is of course debated and a variety of mechanisms have been

discussed by Shaw et al [49],

Stolichnov et al [50] have shown that the localised interdiffusion of the electrode

material into the dielectric plays a significant role in the electrical properties of

SrRuO3/PLZT/Pt capacitors. Choi et al [51] have discussed similar diffusion

effects in a BaRuO3/BST. What is apparent is that the choice of electrodes of

the capacitor can introduce extrinsic effects into the system. For example Lee

and Desu [46] investigated the thickness dependence of the dielectric constant in

PZT/Pt capacitors, with different top electrodes of Al, Ag, and Pt. They observed

that the thickness dependence of ε for the capacitors incorporating Ag, and Pt top

electrodes was very similar, but that those capacitors utilising Al top electrodes

demonstrated a stronger thickness dependence. Al is a poor choice of electrode for

oxide materials, since it can readily react with the surface oxygen to form Al2O3,

which for these ferroelectric capacitors, would create a thin dielectric layer of much

lower permittivity between the metal and film. It is also concievable that oxygen

vacancies could be created in the surface region of the PZT (by oxygen diffusing

into the Al) during the Al oxidisation, which would lower the permittivity of this

region by local inhomogeneous strain fields at the vacancy sites or by changing the

material’s inherent structure.

Oxygen vacancies can be created quite easily in perovskite materials. Mehara

et al [52] have measured the concentration of such vacancies to be of the order 1018

cm−3 in the interior of the film, but can have a concentration of 5×1020 cm−3 at the

surface. A space charge distribution corresponding to a concentration of 5× 1020

cm−3 at a distance of 10 nm from the interface has also been measured by Dey

25


[15]. The role that oxygen vacancies play in perovskite ferroelectrics is extremely

important, and it has been suggested by Dawber and Scott [53, 54], that the

self ordering of vacancies at the interface is the dominating factor of polarisation

fatigue in thin films. Under the influence of an electric field, oxygen vacancies

will migrate toward the electrodes and aggregate within the interfacial region.

However, the perovskite structure cannot sustain a large density of point defects

[55], and so collapses with a shear vector of 〈111〉/2, resulting in the formation

of a Ruddlesden-Popper (RP) planar fault layer [56] next to the electrode. Since

the RP fault accommodates two consecutive A-O layers (figure (1.17b)) the local

lattice parameter in the region will be larger than the parent perovskite structure.

A possible RP planar fault has been observed by Jin et al [57] at BST/Pt

interface, using High Resolution Transmission Electron Microscopy (HRTEM).

The interfacial region was observed to be highly crystalline right up to the Pt

electrode, with a lattice constant of 0.39 nm, consistent with BST. However, they

found that over horizontally extended areas of the BST, the lattice was severely

distorted in the atomic layers next to the electrode, with the lattice constant

increased to 0.48 nm. In a previous paper [58], the same authors speculated

that this RP planar fault would have a lower dielectric constant than the rest

of the film and hence would decrease the measured ε of the film. It would also

cause the phenomena of fatigue (consistent with the model of Dawber and Scott).

To strengthen their argument, they compared many different capacitor systems

reported in the literature incorporating either metal or conducting oxide electrodes

(e.g. SrRuO3, IrO2 etc.), and comment that those films exhibiting a thickness

dependence of ε also demonstrate polarisation fatigue, but that crucially, these

phenomena are suppressed in those films with oxide electrodes. This could be due

to the interfacial oxygen vacancies being fed by oxygen from the electrode, thus

preventing the formation of the extended planar defect. Lee and Hwang [31] have

noted that annealing Pt/BST/Pt films in oxygen reduces the ‘dead-layer’ effect,

presumably due to the reduction of the concentration of oxygen vacancies within

the interfacial regions.

These problems can give rise to a low permittivity ‘dead-layer’, but with careful

processing of the films, they can in theory be minimised, if not completely elimi-

nated. Indeed, often HRTEM of thin films have failed to observe any microstruc-

turally distinct functionally disrupted regions within films exhibiting strong ‘dead-

layer’ effects [59, 32, 33, 60].

The presence of a surface or interface can have a dramatic impact upon the

26


Figure 1.17: Diagram showing the formation of a Ruddlesden-Popper (RP)planar fault from the perovskite structure; a) ABO3 structure; b) structureof the RP planar fault; c) projection of the perovskite structured ferroelectricalong [100] direction with RP planar fault formed at the ferroelectric/metalinterface [58].

Figure 1.18: High resolution TEM image of a BST/Pt interface. As indi-cated at the white lines, the separation of the ‘bb’ planes is larger next tothe Pt electrode than the separation of the planes far away, similar to anRP planar fault [57].

27


dipole-dipole interactions involved in ferroelectric ordering. Zhou and Newns [61]

have developed a model which predicts the presence of an intrinsic ‘dead-layer’ at

the surface of a ferroelectric thin film. By applying the Thomas theory for bulk

ferroelectrics1 [11, 62] to a free surface, they found that there would be a region near

the surface where the polarisability was much less than that of a bulk ferroelectric.

This conformed to a region near the surface where the soft modes corresponding

to large ionic contributions to the permittivity harden, thus resulting in a local

reduction in ionic contribution to ε. The thickness of this ‘dead-layer’ is estimated

to be 1-3 nm for a SrTiO3 film.

A similar effect has been predicted by Natori et al [63] using a slightly differ-

ent approach. By modelling the dielectric medium as a lattice of atomic dipole

moments, they considered the fact that dipoles near the surface of a paraelec-

tric material experience a different environment to those in the bulk interior, and

hence would experience a different local field. The local field for the dipoles near

the surface was calculated by the summation of all the dipoles within the material

and the field from the charges on the film’s electrodes. The effective dielectric

constant could then be calculated using Lorentz’s local field approach [64]. This

model predicted that there would be a thickness dependence of ε, due to a low

permittivity layer 2-3 unit cells thick at the electrode interfaces.

Depolarisation fields, and charge screening at the interface has also been sug-

gested as a potential cause for the thickness dependence of ε. Wurfel and Batra

[65] demonstrate that depolarisation fields cannot be significantly reduced by do-

main formation, and thus can only manifest themselves by reducing the intrinsic

polarisation of the thin films with respect to the bulk value. Wang et al [66] found

a similar effect due to the presence of large depolarisation fields at the surface of

the film.

An unavoidable effect in ferroelectric capacitors, is the alteration of their elec-

tronic band structure when in contact with their metal electrodes [15]. Ferro-

electrics can be considered as wide band gap semiconductors (Eg ∼ 3 eV), and

when in intimate contact with a metal, they will experience distortions of their

conduction and valence bands, due to the mismatch of the metal-ferroelectric en-

ergy bands. Figure 1.19 demonstrates this effect for a n-type semiconductor before

and after contact with the metal. When the semiconductor contacts the metal,

the Fermi energy, EF of the two materials equilise to preserve thermal equilibrium,

1This is a dynamical theory, based on a vibrational degree of freedom in each unit cell of theferroelectric, which carries the ferroelectric polarization.

28


Figure 1.19: The ideal energy-band diagram for a metal and semiconduc-tor a) before contact and b) after contact. φm and φs, are the metal andsemiconductor work functions, φB the Schottky barrier height, χ is the elec-tron affinity, W is the depletion width, and EF , EFi, EC , EV are the Fermienergy, intrinsic Fermi energy, and conduction and valance energy levelsrespectively (after [67]).

forming a Schottky barrier [68]. This is achieved by the movement of electrons

from the semiconductor to the lower energy states in the metal, resulting in a dis-

tortion of the energy bands near the metal-semiconductor interface. The electric

field within this region would be considerably large, and is associated with a region

of positive space charge, resulting in a finite region that is depleted of the negative

conduction electrons. It is possible that these regions would have a considerably

lower permittivity than that of the interior of the film, and could give rise to the

interfacial capacitance [59, 69, 70]. The size of these depletion widths is a matter

of debate. Some authors argue these regions can be as thick as 180 nm [71] imply-

ing that for many thin film systems the film would be fully depleted. On the the

29


other hand, Scott [72] has demonstrated that the width of the depletion region in

SrTiO3 should be of the order of 3-4 nm.

Chen et al [73] believe that the interfacial capacitance observed in Pt/BST/YBCO

structures originates from the low permittivity depletion region associated with the

Schottky barrier. They measured the di/εi value of the films using a series capac-

itor plot for capacitance measurements of a series of films. They then measured

the leakage current density, J , of these films and fitted the data to the Schottky

equation,

J = A∗∗T 2 exp

(−qφB

kT

)exp

(q

kT

√qV

4πε0εt

), (1.37)

where A∗∗, φB, V , ε, and t correspond to the Richardson constant, barrier height,

applied voltage, depletion width permittivity, and depletion width, respectively

with q, k and T , and ε0, denoting the usual roles of electronic charge, Boltzmann

constant, temperature, and permittivity of free space. Using equation 1.37, Chen et

al extract the value of εt and combine this with di/εi, to calculate the permittivity

and thickness of the interfacial region to be, εi = 42.6 and di = 2.8 nm. In reality,

this method of determining εi is questionable since the data obtained from these

two measurements may not be compatible. As pointed out by Scott [72] and

Zafar et al [74], the value of the parameter ε used in equation (1.37) corresponds

to the high frequency optical dielectric constant (∼ 5.5) and not the near static

permittivity value obtained from Chen et al’s 100 kHz capacitance measurements.

1.3.2 Electrode Screening

There is a growing volume of work that suggests that the origin of the interfacial

capacitance is not located within the dielectric, but instead is due to the finite

screening abilities of imperfect electrodes.

Vendik et al [75] has utilised a phenomenological model to study the effects of

the spacial correlation of the ferroelectric polarisation, and the boundary condi-

tions imposed upon this polarisation by the presence of the capacitor electrodes.

They state that for zero boundary conditions (as in the case of Pt), the dielectric

functionality of the dielectric disappears at the interface (figure 1.20(a)), but in the

case of free boundary conditions (for conducting oxides), the dielectric function-

ality is non-zero at the interface (figure 1.20(b)), and the polarisation penetrates

into the electrode, and is gradually screened over a finite distance. They applied

30


Figure 1.20: Distribution of the displacement D(x), the ferroelectric po-larisation P (x) and the electric field E(x) in a thin capacitor in the casesof a) zero boundary conditions, and b) free boundary conditions for theferroelectric polarisation (after [75]).

their model to data from two 200 nm thick Pt/STO/SRO and SRO/BST/SRO ca-

pacitors, obtained by Izuha et al [76], and observed a close correlation between the

model and data (Figure 1.21). There is a clear decrease of the zero field dielectric

constant from ∼ 700 in the SRO/BST/SRO system, to ∼ 300 in the SRO/BST/Pt

Figure 1.21: Dielectric constant as a function of applied voltage, of two200 nm BST capacitors incorporating different electrode configurations. Thesolid line is calculated by Vendik et al [75], whilst the dots show the exper-imental data of Izuha et al [76] (after Vendik et al [75]).

31


system. The implication is that it is the boundary condition that generates series

capacitor behaviour. The exact nature of the boundary conditions are often used

in Vendik’s work for pseudo-fitting purposes such that boundary conditions are

altered to fit experimental observation. Indeed, Vendik et al [75] also assume that

the substitution of a small amount of Ba (x = 0.12) for Sr in the BST formula,

would not have significant impact upon the bulk permittivity, in comparison with

STO. Also this change of stoichiometry, as well as the change of electrode ma-

terial, would most certainly change the interfacial environment with respect to

extrinsic contributions to the ‘dead layer’ effect. In general, this seems a some-

what dangerous approach, and consequently literature has not uniformly accepted

electrode-ferroelectric boundary conditions as entirely responsible for interfacial

capacitance.

Another proposed model dependent upon the electrode screening ability, pre-

dicts that the origin of the interfacial capacitance lies within a thin space charge

region below the electrode surface, associated with the finite Thomas-Fermi screen-

ing length of the metal [77, 78, 79].

Consider the simple capacitor structure illustrated in figure 1.22(a). When

Figure 1.22: a) Schematic of a metal-dielectric-metal thin film capacitor.b) The charge distribution for (ideal) perfect electrodes. The charge formsin an infinitely thin plane at the electrode/dielectric interface. c) The chargedistribution for imperfect (realistic) electrodes. The charge is screened grad-ually over a distance L within the electrode. N.B. These diagrams are for aperfectly insulating dielectric (after [79]).

32


an electric field is applied across the capacitor, it can penetrate into the elec-

trode surface. Normally, one assumes that the metals used are ideal, and thus

the penetrating electric field is screened immediately at the surface, resulting in

an infinitely thin sheet of charge at the metal-dielectric interface (figure 1.22(b)).

Unfortunately, when the electric field penetrates into the surface of an imperfect

metal, it is screened over a small, but finite distance, resulting in a charge distri-

bution of a finite spacial extent within the metal (figure 1.22(c)). This thin space

charge layer will have an associated capacitance, which will then act in series with

the capacitance of the dielectric film, reducing the measured dielectric constant of

the capacitor.

This model was first adopted by Ku and Ullman [77] in 1964 to explain the pres-

ence of an interfacial capacitance observed by Mead [80] in ultra thin Ta/Ta2O5/Bi

tunnel junctions (figure 1.23), which is the first application of the series capacitor

plot to a thin film system. Using degenerate Fermi statistics, they were able to

demonstrate numerically that the applied electric field would penetrate a short

distance into the electrode surface, due to the metal’s inability to instantaneously

screen the induced surface charge. Simmons [78] refined this model and expressed

Ku and Ullman’s equations in analytical form. A detailed account of this model is

given in Section 4.3.1 of this thesis. Later, Dawber et al [81], utilised this model

to obtain a value for the interfacial capacitance that agreed very well with that

Figure 1.23: Reciprocal capacitance as a function of tunneling voltage.Tunneling voltage was used as an accurate method to measure of the filmthickness (after Mead [80]).

33


obtained experimentally by Sinnamon et al [82] in Au/BST/SRO capacitors.

The value of the interfacial capacitance of this model, depends upon the dis-

tance the field penetrates into the metal, and on the permittivity of the spacial

extent in the metal for which this occurs, which depends upon the nature of the

screening method of the electrode. There is great debate as to whether the screen-

ing of the electrode is due entirely to the electron free ionic lattice of the metal,

or due solely to the screening of a free electron gas.

Black and Welser [79] believe that the magnitude of this permittivity should

depend upon the polarisability of the underlying metallic lattice, which only in-

cludes the response of the those electrons bound to the ionic cores. They quote

the work of Ehrenreich and Phillip [83] who previously measured the static dielec-

tric constants of Cu and Ag to be εm ∼ 5 and εm ∼ 2.5 respectively. Therefore

they conclude that εm for other metallic elements would have similar values (of

the order of 1 − 10). They also suggest that since the underlying crystal struc-

tures of conducting oxide electrodes are similar to those of insulating perovskites,

then the permittivity of the electrodes should be of the same magnitude as those

found in their non-conducting counterparts (i.e. εm ∼ 100 − 1000) due solely to

ionic displacements. This would fit very well with the observation that capaci-

tors incorporating conducting oxide electrodes often show little or no thickness

dependence (Hieda et al [84]) of ε, since the larger εm increases the interfacial

capacitance thus reducing the dead layer effect. Hwang [85] applied this idea to

the data of Hieda et al [84] who measured the ε on SRO/BST/SRO capacitors

down to 20 nm without seeing any thickness dependence. Hwang found that the

extracted interfacial capacitance for these films was enormous, resulting in a value

of εm ∼ 30, 000, which at first inspection would imply that the electrodes would be

better dielectrics than the actual film, were it not for all the conducting electrons.

Hwang did question the validity of this value, but assumed the large permittivity

was due an enhancement caused by the misfit strain imposed upon the electrode,

by the overlying film, even though his assumption could not be be confirmed by

diffraction studies of the electrode.

On the other hand, Dawber and Scott [86] believe that the approach of Black

and Welser, and Hwang is completely incorrect. By using the Drude Free Electron

Theory, they derive the equation that calculates the DC limit of the Thomas-Fermi

screening length. From this derivation they state that the appropriate dielectric

constant of the metal should be that not of an electron free metal (as suggested

above), but instead that of a free electron gas (i.e. εm = 1). The implication of

34


this is that the thickness dependence of ε is stronger for capacitors incorporating

conducting oxide electrodes, as opposed to capacitors utilising elemental metals,

due to the former’s lower concentration of free charge carriers.

1.3.3 Interfacial Strain

The observed interfacial capacitance is attributed to strain fields generated at the

interface due to a mismatch in lattice constants of the ferroelectric and substrate.

This strain does not create a ‘dead-layer’ per se but instead would appear to reduce

the permittivity of the entire film. Using an LGD formalism the free energy which

includes the elastic strain and electrostriction of a film is given by [87]

∆G =1

2αP 2 +

1

4βP 4 − 1

2sijXiXj −QijXiP

2, (1.38)

where sij are the elastic compliance coefficients, Xi,j the stress tensor, Qij the

electrostrictive coefficients, and α and β are constants for a given temperature.

To simplify the problem one makes the assumptions that the film is epitaxially

clamped to a rigid substrate generating in-plane stresses X1 and X2 , there is no

out-of-plane stress component and both the polarization and electric field lie along

the out-of-plane direction only. In this scenario the above expression becomes,

∆G =1

2αP 2 +

1

4βP 4− 1

2s11(X1 +X2)−

1

2s12(X1X2)−Q12(X1P

2 +X2P2), (1.39)

The inverse dielectric susceptibility measured along the out-of-plane direction,

χ−1 is the second derivative of the free energy with respect to polarisation

χ−1 =∂G2

∂2P= α + 3βP 3 − 2Q12(X1 + X2), (1.40)

and it follows that the stress dependence of the susceptibility is

∂χ−1

∂X1

=∂χ−1

∂X2

= −2Q12 (1.41)

Given that Q12 is a negative constant, this expression implies that any in-

crease in tensile stress relative to an initial state results in an increase in inverse

susceptibility (i.e. a reduction in permittivity).

35


An apparent thickness dependence of ε arises through the homogeneous relax-

ation of this strain as the thickness of thin film increases. Experimentally, Kim et

al [88], and Sinnamon et al [89], have observed the homogeneous strain to relax

exponentially as the thickness of the film increases, by monitoring the out-of-plane

lattice constants of the film. As the strain in the film relaxes, the dielectric con-

stant will increase back to its zero strain value, thus creating the impression of an

interfacial capacitance.

Figure 1.24: Dielectric response of a 200nm BST epitaxial film as a function oftemperature (after Dittman et al [90]).

Figure 1.25: Thickness depen-dence of out-of-plane latticeconstant and inverse capacitancedensity as a function of filmthickness (after Dittman et al [90]).

Dittman et al [90], demonstrate this point with a series of thin SRO/BST/SRO

capacitors with thickness ranging from 10-200 nm. The measured permittivity

of these films are probably the largest reported for films of these dimensions,

exhibiting ε ∼ 5000 for a 200 nm film at 300 K. However, a decrease in ε was

observed, which coincided with an increase in the out-of-plane lattice constant.

When plotted using the series capacitor model, an interfacial capacitance of Ci = 1

F/m2 was observed which the authors showed could be linked with the induced

strain in the film. To strengthen this argument HRTEM was employed to show

their structures are beautifully crystalline within the interfacial regions, and indeed

across the whole film, ruling out any influence from extrinsic size effects.

Recently, Catalan et al [91] considered the effects of inhomogeneous relaxing

misfit strain on the permittivity of thin film ferroelectric capacitors. If a film’s

strain state relaxes inhomogeneously with distance from the strain-inducing inter-

face, then strain gradients would present themselves across the film. These strain

gradients can then couple to the polarisation of the film via the flexoelectric effect,

36

1.4 Effect of Mechanical Boundary Condition on Phase Diagrams

Figure 1.26: a) Measured inhomogeneous strain and average strain(inset)of BST films as a function of thickness. b) Calculated (top) and measured(bottom) permittivity as a function of temperature for BST films of thickness950, 660, 340, 280, 220, 145 nm. (inset) Calculated (line) and measured(dots) of the temperature of maximum permittivity TM (after Catalan et al[92]).

and can thus cause a reduction in the measured permittivity of the films.

Catalan et al [92] furthered this study by using XRD to measure inhomoge-

neous strain from a series of SRO/BST/Au capacitors provided by Sinnamon et al

[82]. Using a thermodynamic approach, they demonstrated that strain gradients

associated with inhomogeneous strain could account for the observed thickness de-

pendence of the measured dielectric constant, without needing to invoke the series

capacitor model.

1.4 Effect of Mechanical Boundary Condition on

Phase Diagrams

Figure 1.26(b)(inset) demonstrates how the compressive strain state imposed upon

the film by the mismatch of lattice constants at the interface, could increase TM

with decreasing thickness (and hence increasing average strain), which is closely

associated with the ferroelectric-paraelectric phase transition. Pertsev et al [18]

have demonstrated that the mechanical clamping of a thin film by a much thicker

substrate can have a dramatic effect on the phase transition temperatures, as well

as the symmetry of the phase state at a given temperature. They derived a new

37


form of the thermodynamic potential which incorporates the influence of the 2D

clamping of a thin ferroelectric film on a sufficiently thick substrate. Considering a

ferroelectric film that is grown epitaxially in a paraelectric cubic state, on a cubic

(001) substrate, and that the interface between the two materials is commensurate,

then the new thermodynamic Gibbs function can be written as;

G = a∗1(P 2

1 + P 22

)+ a∗3P

23 + a∗11

(P 4

1 + P 42

)+ a∗33P

43 + a∗13

(P 2

1 P 23 + P 2

2 P 23

)+a∗12P

21 P 2

2 + a111

(P 6

1 + P 62 + P 6

3

)+a112

[p4

1

(P 2

2 + P 23

)+ P 4

3

(P 2

1 + P 22

)+ P 4

2

(P 2

1 + P 23

)]+a123P

21 P 2

2 P 23 +

u2m

s11 + s12

(1.42)

where,

a∗1 = a1 − umQ11 + Q12

s11 + s12

a∗3 = a1 − um2Q12

s11 + s12

a∗11 = a11 +1

2

1

s211 + s2

12

[(Q2

11 + Q212

)s11 − 2Q11Q12s12

]a∗33 = a11 +

Q212

s11 + s12

a∗12 = a12 −1

s211 − s2

12

[(Q2

11 + Q212

)s12 − 2Q11Q12s11

]+

Q244

2s44

a∗13 = a12 +Q12 (Q11 + Q12)

s11 + s12

.

In this notation sij and Qij are the elastic compliance and electrostrictive ten-

sors as before, a1, a11 and a123 are variables which are linearly dependent on tem-

perature, and um is the misfit strain at the interface, and is determined by the

lattice parameters of the substrate a0 and film b, by the relation um = (a0 − b)/b.

The 2D clamping of the film by the substrate, lowers the symmetry of the

cubic phase to tetragonal, resulting in a total of five possible low-temperature

phases, instead of three in the bulk, free material. For these materials the following

notation is introduced:

(i) the c phase, where P3 6= 0 and P1 = P2 = 0;

(ii) the a phase, where P1 6= 0 and P3 = P2 = 0;

(iii) the ac phase, where P1 6= 0, P3 6= 0 and P1 = 0;

38


(iv) the aa phase, where P1 = P2 6= 0 and P3 = 0;

(v) the r phase, where P1 = P2 6= 0 and P3 6= 0;

The equilibrium value for the states of BaTiO3 were determined by calculating

all of the minima of equation (1.42) with respect to the components of the polari-

sation, using parameters of the Gibbs function taken from Refs. [93, 94] and then

selecting the phase which corresponded to the minima minimorum. The resultant

temperature-misfit phase diagram is presented in figure 1.27.

Figure 1.27: Calculated misfit phase diagram for BaTiO3 thin films (afterPertsev et al [18]).

This diagram clearly shows that the phase transition temperatures of a ferro-

electric thin film can be changed depending upon the strain state imposed upon the

film by the substrate. Also, for compressive in-plane strains at most temperatures,

the tetragonal c phase is energetically more favourable than the orthorhombic ac

phase, with the polar axis out-of-plane, which is mirrored for tensile strains, except

the polar axis of the tetragonal aa phase is confined to the in-plane direction.

What is interesting to note is that even at zero misfit strain, the mechanical

boundary conditions would seem to impose a change in the order of the phase

transitions on cooling/heating. Recalling from figure 1.11, the order of the bulk

phase transitions is rhombohedral-orthorhombic-tetragonal-cubic, with increasing

temperature. In the Pertsev phase diagram, the zero strain condition predicts

the order of the phase transitions to be orthorhombic-rhombohedral-cubic, with

increasing temperature.

39


Figure 1.28: Calculated misfit phase diagram for Ba0.7Sr0.3TiO3 (solid line)and Ba0.6Sr0.4TiO3 (dashed line) (after Ban and Alpay [95])

Ban and Alpay [95] have used a similar technique to theoretically analyse the

misfit phase diagrams of Ba0.7Sr0.3TiO3 and Ba0.6Sr0.4TiO3, obtaining similar re-

sults (figure 1.28). However, they show no evidence for the presence of a low tem-

perature ac phase, but whether this is due to it being less energetically favourable

or if it simply occurs at a lower temperature than shown is not made clear.

A similar phase diagram for BaTiO3 has been constructed by Dieguez et al [96],

using ab initio calculations (figure 1.29). The basic elements of this phase diagram

are similar to that of Pertsev et al except they find the r phase is energetically

favourable over the ac phase for all temperatures and strains. They do point

out however, if one were to use the thermodynamical parameters for the Gibbs

function in Ref. [97], then the ac phase as predicted using the Pertsev technique,

all but vanishes, and can only be observed within a small temperature window at

large compressive strains. The phase transition temperatures would also appear

to be dramatically reduced, in particular the r − p transition at zero strain is

underestimated by approximately 100 C. This however is not taken as being

intrinsic to the 2D clamping, but is due to the artifact of using a first-principles

approach in the calculation of the misfit phase diagram. Indeed, it was found that

the effective Hamiltonian used to calculate the phase diagram underestimated the

cubic-tetragonal phase transition of a free bulk sample by approximately the same

amount.

Another first principles study by Lai et al [98] looks at the misfit strain phase

40


Figure 1.29: Calculated misfit phase diagram for BaTiO3 thin films (afterDieguez et al [96]).

diagram for ultrathin BaTiO3 films, under ideal short circuit boundary condi-

tions, and also under electrical boundary condition of screening of 96.3% of the

polarisation-induced surface/interface charge. Importantly they found the four-

phase crossing point was displaced from the zero misfit strain value, resulting in

a phase diagram that is asymmetric with respect to this zero misfit strain. They

also introduce a splitting of the r phase into two sub phases rc and raa, which

correspond to P3 > P1 = P2 6= 0 and P1 = P2 6= 0 > P3 respectively. There are a

few other important points worth noting with this study. The first is that the c

phase can be induced at small values of tensile strain, although this becomes less

so when the thickness of the film increases, since the degree of asymmetry of the

phase diagram with respect to zero strain was observed to decrease as the thickness

of the film increased. This observation is in contradiction with that of Pertsev et

al and Dieguez et al. Secondly, the temperature at which the p − c and aa − raa

transitions occur decreases for films incorporating the screening of the polarisation

by induced surface charge. This has the effect of shifting the four-phase crossing

point to lower temperatures and more negative strains.

The experimental exploration of these misfit phase diagrams has been excep-

tionally limited, due in part to the difficulty in finding suitable substrates to induce

the desired strains. As a result, most experimental work has been performed on

films which experience compressive in-plane strains. In most cases, these inves-

tigations simply map an increasing out-of-plane lattice constant with decreasing

41


Figure 1.30: First principle calculation of the misfit phase diagram forBaTiO3 of Lai et al [98] for the conitions (top) 5 unit cells and short circuitconditions, (middle) 7 unit cells and short circuit conditions, and (bottom)5 unit cells and the electrical boundary condition of screening of 96.3% ofthe polarisation-induced surface/interface charge [98].

42


thickness (corresponding to an increasing homogeneously relaxing strain) [88], but

a few have commented on the high temperature stabilisation of the ferroelectric

state [89]. Recently, Choi et al [51], have used compressive in-plane strain to

increase the ferroelectric phase transition temperature of BaTiO3 to ∼ 600 C,

whilst in an earlier paper Haeni et al [99] succeeded in obtaining room tempera-

ture ferroelectricity in SrTiO3.

There has however, been little or no experimental work devoted to investigating

the effect of tensile strain, and more importantly, the zero misfit strain regions of

the misfit phase diagrams, which would ascertain which models, if any, are correct

in their prediction of the occurrence of the exotic r phase.

This thesis is primarily concerned with the investigation of the nature of the

ferroelectric/electrode interface, and its influence on the functional and struc-

tural properties of thin and ultrathin ferroelectric capacitors. In Chapter 3,

the temperature and frequency characteristics of the interfacial capacitance of

SRO/BST/Au and LSCO/BST/Au thin film capacitors are investigated using the

series capacitor model. Chapter 4 reports on the successful deposition of ultrathin

LSCO/BST/Au capacitors and extends the series capacitor model to these ultra-

thin dimensions. Finally Chapter 5 investigates the phase state of ‘zero’-misfiit

strained LSCO/BaTiO3/Au thin film capacitors as a function of temperature.

43

Chapter 2

Experimental Methods

2.1 Capacitor Fabrication

2.1.1 Pulsed Laser Deposition

A burst of pulsed laser radiation of sufficient fluence, repetition rate and pulse

width, incident on the surface of a suitable material, will cause atoms to dissociate

and be ejected from the surface. The resulting plume of plasmatised material

may contain particulates, atoms or highly ionised species of the target material.

Pulsed Laser Deposition (PLD) uses this principle to grow thin film materials and

was first used in 1965 by Smith and Turner [100] to deposit a thin film using a

ruby laser. However, PLD was not widely implemented until the 1980’s when new

developments in high powered short pulsed lasers provided effective means with

which to deposit high quality complex high TC superconducting films [101].

PLD has many advantages as a thin film growth process. The deposition rates

are high and extremely controllable, allowing for better precision in fabrication of

films of a particular thickness, and thus PLD is a desirable method for deposition

of very thin films, including films of single atomic monolayers. The ability to easily

fabricate multilayered hetrostructures and superlattices is also a highly desirable

and advantageous attribute of PLD. Finally, at sufficiently large fluence, typically

> 2 J/cm2, the rapid heating of the target surface leads to the uniform evaporation

of all constituent elements regardless of specific evaporation points. This permits

stoichiometric transfer of the target material to the deposited film, although some

refinements must be considered to prevent loss of volatile species (e.g. lead and

bismuth) from the deposited film.

PLD does suffer the disadvantage that it has a small coverage area compared

44


to other deposition techniques, such as chemical vapour deposition, but for the

current work, the dimensions of the substrates (12 x 5, and 10 x 10 mm) means

that small film coverage areas do not pose a great problem, provided the substrate

is positioned correctly with respect to the plasma plume.

The laser used in the current work is a KrF (λ = 248 nm) Lambda Physik

COMPex 205i model excimer laser. It operates with a maximum energy of 700

mJ, a maximum power of 35 W, and a pulse width of 34 ns with a maximum

repetition rate of 50 Hz.

Figure 2.1: Schematic of the PLD external optics (after [5]).

A schematic of the external optics used to direct and focus the laser pulses into

a vacuum chamber is illustrated in figure 2.1. The laser beam pulse exits the laser

and is directed through a series of quartz absorber plates onto a mirror, which

then directs it through a focusing lens and into the entrance port of the vacuum

chamber which houses the target and substrate. The absorber plates are used to

reduce the transmitted power of the pulses, thereby allowing a degree of tuning to

the beam intensity. The lens is used to focus the beam onto the target to increase

the fluence to a level sufficient to plasmatise the target surface, typically 1 − 2

J/cm2 in the case of ceramics. In reality, the beam is focused to a point a few

centimetres in front of the target to encourage the dissociation of oxygen molecules

within the oxygen ambient, and thus minimise loss of oxygen from the deposited

45


film. The energy of the laser is measured before each deposition by inserting an

Ophir energy meter into the beam path, and measuring the voltage induced by

the laser pulses using an oscilliscope.

Figure 2.2 presents a schematic of the vacuum chamber used for PLD. Within

this chamber is a commercial Neocera Inc. multi-target carousel which permits

the deposition of up to six materials within a single vacuum cycle, without the

need to open the chamber. Directly facing the carousel is a commercial Neocera

Inc. heater block, capable of a maximum temperature of 900 oC, onto which the

substrate is attached. Substrates used to be held to the heating element using

screwed down clips, but this has since been replaced by the use of silver paste as

an adhesive to adhere the substrate directly to the heating element. The heating

element is controlled by a programmable Eurotherm 818p Temperature Controller.

Figure 2.2: Schematic of the PLD deposition chamber (after [102]).

A shield is positioned between the heater block and the target holder to protect

the substrate during the pre-cleaning of the target surface with the laser, prior to

the thin film deposition, and is rotated out of the way once deposition is ready to

commence.

Before deposition, the chamber is evacuated using a two pump system (illus-

trated in figure 2.3) comprising of a roughing rotary vane pump, and a turbo-

molecular pump backed by the rotary pump. The rotary pump is used to evacuate

the chamber to a pressure of < 0.5 mbar, after which, the turbomolecular pump is

used to continue the evacuation to a base pressure of 1× 10−5 mbar. The pressure

in the chamber is monitored using two pressure gauges: a MKS Type 107B Bara-

tron Absolute Pressure Transducer which operates effectively from ∼ 0.01− 1000

46


mbar, and a series 423 I-Mag Cold Cathode Ionization Gauge which operates from

10−2 − 10−9 mbar.

Figure 2.3: Schematic of the vacuum pump system for the PLD depositionchamber (after [102]).

The atmosphere inside the deposition chamber is controlled using a MKS 146A

Vacuum Gauge Measurement and Control system panel. Since the current work is

concerned with oxide materials, the atmosphere of choice for deposition is research

grade oxygen (99.999% purity), which is fed into the chamber by one of two gas

lines which is regulated by a Mass Flow Controller, operated on a feedback loop

system from the control panel. A third gas line is used for the venting of the

chamber with nitrogen.

2.1.2 Target Preparation

In the investigations of the current work, four different materials were used to

fabricate thin film capacitors: the ferroelectrics BaTiO3, and Ba0.5Sr0.5TiO3, and

the conducting oxides (La,Sr)CoO3 and SrRuO3. For each of these materials a

target was manufactured in-house, except for the SRO, which was commercially

obtained (Superconductive Components Inc., 99.9%), due to the difficulty of pro-

cessing ruthenium oxides which have low melting points. However, the commercial

targets were relatively low in density, which gave problems with surface roughness

of the deposited films. This was rectified by increasing the density through in-

house sintering at 1700 oC for 30 hours.

Of the remaining targets, BaTiO3 was the simplest to make, requiring only

BaTiO3 powder (Aldrich 99.9%). This powder was ground thoroughly with a

pestle and mortar before being transfered to a high purity alumina crucible, and

47


sintered at 1100 oC for 6 hours at a heating and cooling rate of 5 o/min. When

cooled, the powder was then removed and reground until it adopted a cornflower

like texture, after which it was pressed into a 1 inch disc, using a steel die set,

with an applied pressure equivalent to 7 tonnes for 60 seconds. The pressure was

then gradually released over a few minutes to avoid cracking. The disc was then

heated at 5 o/min to 1400 oC where it remained for 3 hours before cooling at the

same rate.

A similar method was employed for the BST target, except that equimolar

quantities of BaTiO3 (Aldrich 99.9%) and SrTiO3 (Aldrich 99%) powders were

ground together with the pestle and mortar and then sintered at a higher temper-

ature of 1210 oC to encourage the solid state reaction.

Finally the LSCO material required a little more work and attention than the

previous materials listed, requiring more steps for the fabrication of a robust, and

dense target. This was perfected in-house by Niall Donnelly [5], and his preparation

methodology is detailed below.

Cationic stoichiometric amounts of the oxides La2O3 (Reacton 99.9%), SrO

(Aldrich 99.9%), and Co3O4 (Alfa Aesar 99.7%) were mixed thoroughly and fired

in a crucible at 800 C for 3 hours using a temperature ramp rate of 30 C per

minute for both heating and cooling. The powders were then reground using a

pestle and mortar with a few drops of methanol and fired again at 1000 C for 6

hours. Once cooled the powders were again reground with methanol, and pressed

into a 1 inch disc as described previously. The disc was then sintered three times

to obtain a suitably dense target for use in PLD. The details of each of these three

sinters are sumarised in table 2.1. It is important to note that after the first and

second stages the target was reground and repressed.

Sinter Temperature / C Duration / mins1 1150 1802 1150 1203 1225 240

Table 2.1: Summary of sintering conditions for the fabrication of LSCOtargets. The temperature ramp rate was 5 C/min when heating and 10C/min cooling.

After the fabrication of each target, a small fragment from the designated

bottom was removed and its stoichiometry measured using EDX, to verify the

desired material stoichiometry had been obtained. Also, some of material from

48


the top surface could be removed with a razor blade to form a powder, which was

adhered with vaseline to a glass slide. X-ray powder diffraction could then be

performed as another phase verification technique.

2.1.3 Deposition Procedure

Target Cleaning

Prior to each deposition, the surface of the target was ground, using silicon carbide

paper. This was done to remove the burns from the surface of the targets from

previous depositions, and to give a smooth surface for the laser to interact with.

Before deposition commenced each of the targets pre-cleaned with the laser for

approximately 1000 shots, to remove any surface contamination.

SRO/BST

The SRO/BST capacitors used in this work were fabricated by Lesley Sinnamon

and as such the conditions for PLD fabrication of these structures have been de-

tailed elsewhere [82], but are summarised as follows:

The substrates were attached to the heater element using copper clips, and

heated to 800 C for 10 minutes to anneal the substrate and burn of volatile

contaminants. The SRO and BST were deposited at 800 C within an oxygen

ambient of 0.15 mbar, and then annealed at 600 C in 1000 mbar of oxygen. The

BST deposition temperature was adjusted to 750 C for films < 100 nm, whilst

films thicker than 500 nm were annealed for longer.

Thick LSCO/BST Films

The growth conditions for this system are very similar to that of the SRO/BST

described previously, except that both deposition temperature and annealing tem-

perature for optimal growth were determined to be 150 C less than before. This

may be due to the new method of adhering the substrate to the heating element.

For the SRO/BST system the substrate was clipped directly to the heating element,

with a thin copper sheet between it and the substrate, to aid thermal conductiv-

ity. In the LSCO/BST system, the substrate is glued to the heating element using

silver paste, thus reducing the degree of heat loss within the system due to the

clips. When using the silver paste, it was necessary to heat the substrate to 300C in air for 10 minutes, to evaporate organic compounds within the silver paste,

49

2.2 Functional Measurements

which also had the added benefit of helping the paste to dry quicker. After this

stage the substrate was placed within the chamber, and the chamber evacuated to

base pressure. The substrate was then heated to 700 C for 10 minutes to remove

any surface contamination, after which an oxygen atmosphere of 0.15 mbar was

bled into the chamber. The substrate was kept at 700 C for a further 15 minutes,

before being cooled to the deposition temperature of 650 C.

Each layer was deposited with a pulse repetition rate of 10 Hz, with the energy

of each pulse being ∼ 200 mJ, with a spot size of ∼ 8 mm. This means that the

fluence at the target surface was approximately 2 J/cm2. After deposition, the

film was cooled to 500 C, and allowed to anneal at this temperature for 15 - 30

minutes (depending on thickness), in ∼ 950 mbar of oxygen.

Ultrathin LSCO/BST films

The preparation method for fabrication of ultrathin films differed slightly from

that of their thicker counterparts. The overall temperature for deposition was

reduced to 600 C, and, in an attempt to reduce the surface roughness of the

LSCO layer, the deposition rate for the LSCO electrodes was increased from 10

Hz, to 20 Hz, whilst the thickness of the electrode was reduced by approximately

half. This technique proved highly successful in obtaining good quality ultrathin

films.

LSCO/BTO

The deposition conditions for the fabrication of capacitors with the BaTiO3 fer-

roelectric layer was identical to that of the thick LSCO/BST capacitors. How-

ever, the LSCO electrodes were deposited in the same manner as the ultrathin

capacitors, since the original purpose was to study the ferroelectricity at ultrathin

dimensions.


The majority of this thesis concerns the change of the functional properties of

thin and ultrathin film ferroelectrics with varying thickness, temperature, and

frequency. To this end capacitors were fabricated using PLD as detailed above.

However, to complete the capacitors, and to make electrical contact, top electrodes

of Au were thermally evaporated through a hard mask onto the surface of the fer-

50


roelectric layer. The shape and size of the electrodes varied depending on whether

or not the ferroelectric layer was thin, or ultrathin.

For thin films, the top electrodes were round and typically between 1 - 1.5 mm in

diameter, giving areas within the range of 7.85×10−3−55.5×10−3 cm2. The smaller

area electrodes were introduced later when it was observed that smaller electrodes

gave lower loss, and less frequency dispersion at high frequencies. The reason

for this is that with smaller electrodes one reduces the probability of sampling

major structural defects such as large particulates thrown off during the deposition

process, or electrical shorts through conduction paths along gaps in the grain

structure. It also reduces the measured capacitance, thus reducing the RC response

of the circuit.

The electrode arrangement on the surface of the film is illustrated in figures

2.4 and 2.5. Originally the substrates were supplied in imperial units of 12

′′x 1

4

′′

but were later resized to 10 x 10 mm by the supplier when they changed all the

measurements to metric units. It therefore became possible to fit twice as many

electrodes onto the surface of the film.

2.2.1 Functional Measurements of Thin Films

Figure 2.4: left) Arrangement of surface electrodes for thin LSCO/BSTcapacitors. right) Illustration of the method of functional characterisationusing the gold probes of the cryostat bridge.

The measurement of the functional properties with variation of temperature

and frequency is often fruitful in terms of the quantity of information gained. To

obtain such data the samples were mounted upon a small heater block, under a

custom built bridge, and then placed within an Oxford Instruments cryostat. The

cryostat consists of an inner cavity in which the sample is placed, and evacuated

before being filled with helium gas, and an outer jacket which is filled with liquid

nitrogen. The helium gas within the inner cavity acts as a heat exchange, transfer-

51


ring heat from the sample to the surrounding liquid nitrogen reservoir. Originally

nitrogen gas was used in place of the helium, but there were concerns that the

nitrogen gas would condense on the surface of the film with the addition of the

external reservoir.

The bridge contains four gold, spring loaded probes which are spaced the same

distance apart as the electrode pattern. These allow for the electrical contact

with the film, and permit functional characterisation. One problem is that due

to this circuit arrangement, any measurement performed will be a function of two

elements in series. For example the measured capacitance of the film would be

given by the the sum of the reciprocals of the capacitance of each element as given

by

1

Cmeas

=1

C1

+1

C2

. (2.1)

Since there is a minimum of four electrodes, and thus six different permutations

of measurement at any one time, it becomes a simple but tedious task to obtain

the capacitance of each element through simultaneous equations. A crude, but

highly effective strategy is to assume that the thickness and permittivity of the

film is homogeneous across the whole film, thus C1 = C2 = C, and therefore the

total measured capacitance is simply half the capacitance of a single element, i.e.

1

Cmeas

=1

C1

+1

C2

=2

C. (2.2)

The temperature of the film is regulated using a Lakeshore 330 autotuning

temperature controller. A Hewlett Packard HP4284B Precision LCR meter was

used to measure the film capacitance and dielectric loss over a frequency range

of 102 − 105 Hz in logarithmic increments. For more detailed frequency analysis

a Hewlett Packard HP4284A Precission LCR meter was employed allowing for

an increased frequency range 20 Hz to 1 MHz, as well as a finer step size in the

frequency increments.

The measurement process was automated using an in-house program imple-

menting the HPVee package, so that as the sample increased in temperature, the

LCR meter measured the capacitance and loss over a specified frequency range,

whilst the Lakeshore maintained a constant heating rate and recorded the temper-

ature at which each measurement was taken.

The measurement process has numerous parameters that can be varied, such

52


as the ac sensing voltage and dc bias level, temperature ramp rate, and the choice

of measuring while heating or cooling. Only the measurement on heating option

is used in this thesis, with a typical ramp rate of 3 K/min for capacitance mea-

surements. The ac sensing voltage was typically 50 mV.

2.2.2 Functional Measurements of Ultrathin films

Figure 2.5: left) Arrangement of the surface electrodes for ultrathinLSCO/BST capacitors each electrode is 200 x 200 µm2. right) Illustrationof the method of functional characterisation using the tungsten microma-nipulated probe.

The measurement of the functional properties of ultrathin films was different

from that described above. The main difference was the use of smaller electrodes

to make electrical contact with the film. The electrodes for this were typically

∼200 x 200 µm, giving an area of approximately 4 × 10−4 cm2. A mesh of small

electrodes was obtained by evaporating Au through TEM copper grids. For the 10

x 10 mm sized substrates, five of these grids could fit upon one film, increasing the

probability of gaining reliable functional data. However, due to the relative sizes of

the electrodes and spring loaded pins used with the cryostat, good electrical con-

tact was difficult to obtain, and so a micromanipulated tungsten probe was used in

conjunction with a microscope to make contact with a single top electrode. Con-

tact with the bottom electrode was achieved by scrapping the side of the capacitor

with a razor blade, and gluing a thin wire to the exposed bottom electrode us-

ing conductive silver paste. This unfortunately meant that temperature variation

could no longer be performed upon the ultrathin films.

However, this sacrifice is not without its merits, since any performed mea-

surement is now through a single element only, as opposed to two elements in

series, as in the case of the thicker films. This means that Capacitance-Voltage

measurements can be performed by using an external power source to apply a dc

53


bias level to the capacitor, and measuring the capacitance with the LCR meter.

The power source used for these measurements was an Agilent E3641A DC Power

Supply, which was again controlled by the HPVee program. The measurements

were limited to ±2.5 V, since this is the input limit to the LCR meter, but since

the thickness of the ultrathin films was ∼ 10 nm, this maximum voltage would

correspond to an applied field of 2.5 MV/cm.

2.2.3 Polarisation Hysteresis Loops

A ferroelectric material will display a polarisation hysteresis loop when cycled with

an electric field greater than the coercive field of the material. Traditionally, this

measurement was performed using a Sawer-Tower circuit [103], but has since been

replaced by commercially available test stations specifically designed to measure

polarisation hysteresis.

Polarisation hysteresis in this thesis was measured using a Radiant Technologies

Inc. Precision Materials Analyzer RS6000T, which is a commercial unit designed

for the express purpose of measuring various parameters related to ferroelectricity.

Instead of having a sense capacitor in series with the device under test, as for the

Sawyer-Tower method, this system uses a virtual earth circuit to keep the input

signal at 0 V to eliminate noise and cable capacitances. A standard bipolar trian-

gular voltage waveform defined by its maximum voltage and duration was applied

to the samples in a series of small voltage steps. The current induced in the sample

at each step was integrated and converted to a meaningful polarisation value, with

the first value assumed to be zero so that all other values were relative to this.

The values for remnant polarization calculated by the software were always based

on centred loops, although the loops could be displayed as centred or uncentred

plots.

The important measurement parameters to consider are the maximum applied

voltage, and the period of the measurement. It is considered good practice to

apply peak fields at least three times that of the coercive field, if any measurement

is to be considered reliable [5]. Also higher accuracy is achieved by using longer

measurement periods (in the region of 20 ms), but in the case of many thin films,

this is not always possible, due to movement of space charge at the frequency cor-

responding to this measurement period (50 Hz). At high periods (low frequencies)

space charge can easily keep pace with the driving field, and therefore, provide

an extra conduction contribution to the measured polarisation current, causing

54

2.3 Transmission Electron Microscope

the observed polarisation loops to become ‘bubbly’. To counteract this effect, all

hysteresis measurements in this work were taken using a 1 ms (1 kHz) period. An

option to use a preset loop was used, which performs one field cycle before the

actual measurement, which permits the measurement of the film in a predefined

state.

2.2.4 Measurement of Depolarisation Current

Ferroelectric materials belong to the pyroelectric family of materials which exhibit

a changing spontaneous polarisation with changing temperature. The change of

polarisation causes the redistribution of surface charge upon the capacitor elec-

trodes, which can be measured as a current within an external circuit. This is a

useful technique for mapping phase transition temperatures, since the polarisation

in ferroelectric materials can change rapidly in the vicinity of a phase transition.

As the temperature of the film is increased, the depolarisation current is mea-

sured by a Kiethley 6514 System Electrometer, capable of picoAmp resolution. A

large temperature ramp rate will increase the current signal, and an 8 K/min rate

was chosen to give balance between a strong signal and thermal lag. The recording

of temperature and current was again performed using a HPVee program.


The majority of this thesis is concerned with the investigation of the thickness de-

pendence of the dielectric constant, and therefore it is pertinent that the thickness

of the dielectric layers is known with high accuracy, particularly since calculation

of the permittivity requires knowledge of both the measured capacitance and the

dielectric thickness. To this end, a FEI Tecnai F20 field emission Transmission

Electron Microscope (TEM) was instrumental in determining the thickness of the

ferroelectric layer in each capacitor.

A TEM works by passing a beam of high energy electrons through a thin

sample. The wave-particle duality of quantum mechanics states that electrons

can exist simultaneously as both a particle and a wave, in which the de Brolgie

wavelength, λ, is determined by the momentum p of the particle as λ = h/p, where

h is Planck’s constant. However, the velocity of an electron accelerated through

a potential difference ≥ 105 V is comparable to the velocity of light c, and thus a

55


relativistic correction has to be made to the wavelength, i.e.

λ = h

[2meeV

(1 +

eV

2mec2

)]−1/2

, (2.3)

where e, and me are the charge and rest mass of the electron respectively. Thus,

an electron accelerated through a potential of 105 V, will have a wavelength of the

order of 0.05 A, permitting resolutions orders of magnitude greater than that of

visible wavelengths. Also, since a crystalline solid is made up of a periodic array

of atoms, in which the periodicity is typically of the order of Angstroms, then

electrons passing through a thin sample will be diffracted by the crystallographic

structure. Therefore, a TEM is capable of providing both high resolution images

of the morphology of a film, and obtaining crystallographic information.

2.3.1 Sample Preparation

To measure the thickness of the dielectric layer, cross sectional TEM samples were

prepared using a FEI FIB200TEM Focused Ion Beam microscope (FIB). The FIB

works in a similar way to a scanning electron microscope, but rather than using

electrons, it uses a beam of gallium ions. Operating at low beam currents (∼ 30

pA), the FIB operates as a scanning electron microscope, by detecting the emission

of secondary electrons from the interaction of the gallium with the surface. At

higher beam currents the gallium ions can be used to remove material from the

sample, effectively allowing the user to mill sub-micron sized patterns.

The FEI FIB200TEM has a set of automated programs which are used for

preparation of cross sectional TEM images. To begin the process, a selected ca-

pacitor structure is adhered to an aluminium stub using sticky carbon pads, and a

small dab of conductive carbon placed on one corner, connecting the surface with

the stub, to drain away excess charge induced by the gallium ion bombardment.

However, for the ferroelectric films studied, it was found that the films would

still become highly charged, causing problems with image capture, and inaccuracy

when milling. This was rectified by the sputtering of a very thin layer of Au over

the entire surface, which unfortunately rendered the capacitors useless for further

functional studies.

TEM samples would always be prepared from the electrodes from which func-

tional data were obtained. This was particularly important for the case of the

ultrathin films, since there was ∼ 300 electrodes to choose from. Before milling

56


commences, a platinum bar 10 µm long, 1.5 µm wide, and 1.5 µm thick, is deposited

over the selected site to protect the underlying material from gallium damage. This

step is important since gallium damage has been seen to form an amorphous layer

near the surface of the ferroelectric sample, which could easily be mistaken for the

‘dead-layer’ thought to cause the collapse of the dielectric constant with reducing

film thickness.

Once the sample is protected, a trench up to 10 µm deep is excavated either

side of the platinumised area. This exposes a vertical lamella which will become

the cross-sectional TEM specimen. Further milling parallel to the vertical sides

is performed to thin the lamella further, after which, the whole sample is tilted

45 and the lamella is cut away from its bulk material via a ‘u-cut’ (figure 2.6) ,

all except for two thin joints which are left to connect the lamella with its parent

body to supply stability during the final polishing stage. The sample is then tilted

back to ±1, to account for the non-parallel milling profile, and the sides of the

lamella polished further with the ion beam. This has the result of both polishing,

and thinning the lamella further.

Figure 2.6: A typical cross-sectional TEM specimen fabricated using FIB,just before the ‘u-cut’. The X pattern on either side is used by the automatedprogram for reference purposes.

A TEM sample must be relatively transparent to electrons, which for most

materials requires the specimen to be of the order of 100-150 nm. However, the

final polishing and thinning of the lamella over its whole length, introduces a

residual stress which creates a strain in the lamella, causing it to increasingly bow,

and bend, before eventually snapping. This stress is relieved by the partial removal

of the lamella material. The lamella is gradually made thinner as one approaches

the middle of the sample, whilst the depth of milling is also decreased, resulting

57


in the lamela profile illustrated in figure 2.7.

Figure 2.7: Illustration of the FIB thinning and polishing technique. Asthe thickness of the lamella decreases, the depth of the polishing is alsodecreased. This method prevents stress damage to the TEM sample.

Once all milling is complete, the lamella is freed by milling the small joins

connecting it to its parent material. The TEM specimen is now removed from the

trench using an ultra fine tipped glass needle, which is manoeuvered with the aid

of micromanipulators and a high powered microscope. The lamella is lifted from

the trench through electrostatic interaction, and is deposited upon a TEM grid.

2.3.2 TEM Image Acquisition

A basic schematic of a TEM is illustrated in figure 2.8. In the FEI Tecnai F20

TEM a small tungsten needle, coated in zirconia, is used as a field emission elec-

tron source with a maximum acceleration voltage of a 200 kV. The beam consists

of virtually monochromatic electrons nearly parallel to the optic axis since the

emission angle range is small. It is controlled down the evacuated TEM column

by a series of deflection coils and electromagnetic lenses, the focal lengths of which

can be changed by altering the current applied to the coils. Condenser lenses

control the spot size and intensity of the electron beam onto the sample, which

then diffracts or scatters the incident electrons from the atomic species within the

sample. The transmitted electron beams are then focused by the objective and

projector lenses to produce a highly magnified image, or diffraction pattern onto a

phosphor screen at the bottom of the column. An electron striking the phosphor

screen causes the emission of light, so bright areas represent electron transmission

through the sample. In addition, the FEI TEM has the option of image capture

using a CCD (charge coupled device).

58


Figure 2.8: A schematic of the standard layout of the transmission electronmicroscope (after [104]).

There are three main problems associated with the lens system: astigmatism,

chromatic aberration and spherical aberration [105]. Astigmatism is caused by

asymmetry in the lens fields, which reduces their rotational symmetry. It can be

corrected by the use of stigmators, which are lenses on which the astigmatism can

be continuously adjusted. Chromatic aberration arises from a spread of electron

energies, however the field emission design of the FEI TEM creates a virtually

monochromatic beam, thus minimising this effect. The most important defect is

spherical aberration which causes rays passing through the outer part of the lens

to be bent further, and hence focused sooner, than rays traveling near the principal

axis of the lens. This can easily be reduced by decreasing the objective aperture,

but at the cost of decreased resolution of the acquired image.

The most common imaging technique used on the TEM, and the method used

in this thesis, is diffraction contrast. In this mode the crystal is oriented so that

only one beam is used to form the image. There are two types of diffraction contrast

imaging: bright field and dark field. Bright field uses only the undiffracted beam

which is achieved by centring an aperture on the 000 reflection in the diffraction

pattern to allow only the central beam through. Dark areas on these images

indicate regions where strong electron scattering has occurred. In contrast, dark

59


field imaging the aperture is centred on one of the diffracted (elastically scattered)

beams and bright areas of the image correspond to regions of strong diffraction of

the chosen beam. In this thesis, only bright field imaging was performed.

2.3.3 Energy Dispersive X-ray Spectroscopy

The FEI TEM is equipped to perform Energy Dispersive X-ray Spectroscopy

(EDX) analysis, which is a technique used to obtain quantitative measurements of

the chemical composition of deposited films and of targets. Typically, it is carried

out within a dedicated Scanning Electron Microscope (SEM), but the FEI TEM

can perform the same measurements.

Figure 2.9: Illustration of the principle of energy dispersive x-ray analysis(EDX). (1) An incident electron interacts with the inner electron shell of anatom, removing one electron. (2) An outer shell electron then undergoes atransition to fill the electron vacancy, emitting a photon of x-ray wavelengthin the process. The wavelength of the emitted electron will be characteristicto the specific element.

When high energy electrons interact with matter, they can remove electrons

from the inner orbitals of an atom, thereby forcing the atom into a higher excited

energy state. An electron from an outer shell will then make a transition to the

now vacant inner shell, and in doing so will emit a photon, typically within the

x-ray region, thereby returning the atom to its ground state. The emitted photon

60

2.4 X-ray Diffraction

will have an energy that is specific to the element from which it was emitted, and

so analysis of the spectrum of emitted photons will impart information on the

chemical composition of the material being studied.

A detector within the FEI TEM records the emitted spectrum of x-rays and

compares it to standard spectra taken from single element specimens, and deter-

mines the constituent species, and their relative abundance within the material.

Crucially, EDX can be used to map the spacial variance of the abundance of par-

ticular elements, and therefore is an invaluable tool for determining the thickness

of ultrathin films.

Unfortunately the EDX system used in this thesis cannot detect the soft x-rays

emitted from elements lighter than sodium, and therefore the oxygen content could

not be determined using this technique, which could prove useful for determining

the oxygen vacancy concentrations within interfacial regions and of the interior of

the films studied.


2.4.1 Bragg Law of Crystal Diffraction

X-ray diffraction is a very powerful tool for the structural characterisation of thin

film systems. When a material crystallises, the atoms often order into a three

dimensional periodic lattice, in which the spacing between each atomic plane is

comparable to that of x-ray radiation. This means that the crystal structure acts

exactly like a diffraction grating to incident x-rays, and will produce a diffraction

pattern of bright and dark fringes, from which the interatomic spacings of the

material can be determined.

Consider a beam of x-rays which are in phase with each other incident on a

crystalline material, and intersecting a series of atomic planes at an angle θ to the

surface. Upon entering the medium, they will interact with the electron clouds in

many ways, one of which is to cause the electrons to oscillate with a sympathetic

frequency, resulting in the emission of secondary x-rays of the same wavelength and

phase as the incident wave. The primary and secondary waves then interfere with

each other producing a characteristic diffraction pattern. Since the primary waves

are in phase with the secondary waves, and since the secondary waves produced

from each atomic plane are in phase with each other, then one can conceptualise the

problem as if the x-rays are reflected from successive atomic planes [6], as in figure

61


Figure 2.10: The diffraction of x-rays from a crystalline material can beconceptualised as a series of Bragg reflections from each atomic plane withinthe crystal. For the condition of constructive interference the total pathdifference 2d sin θ must be a whole number of wavelengths. (after [106]).

2.10. This reflection geometry allows for the easy determination of the condition

for constructive interference. Clearly the lower ray in figure 2.10 has had to travel

further than the upper ray, with the path difference being equal to 2d sin θ, where

d is the distance between atomic planes. For constructive interference to occur,

the path difference of each ‘reflection’ from each successive plane must necessarily

equal an integer number of wavelengths n, i.e.

nλ = 2d sin θ. (2.4)

The parameter n, is often known as the order of diffraction, and can be incor-

porated into equation 2.4 to give the Bragg equation

λ = 2dhkl sin θ, (2.5)

where dhkl denotes the periodicity of the planes perpendicular to the direction

[hkl ].

62


2.4.2 X-Ray Diffractometer

The x-ray diffraction data presented in this work have been performed using a

Bruker-AXS D8 diffractometer, and on the synchrotron diffractometer in HASY-

LAB, at DESY, Hamburg. Regardless of the type, the principle components of

the diffractometer are the same, illustrated in figure 2.11. This figure shows a

top-down view of the D8 apparatus, such that the film’s surface lies in the vertical

plane.

Figure 2.11: A schematic showing the geometry of x-ray diffraction asviewed from above (after [5]).

X-rays are generated by 40 kV accelerated electrons which strike a copper target

within the housing of the x-ray source. Electrons in the K-shell of the copper atoms

are ejected from the atom, and are replaced by the transition of electrons from

higher energy orbitals, similar to figure 2.9. The resultant transitions produces

three main radiation wavelengths, Kα1 = 1.5406 A, Kα2 = 1.5444 A, and Kβ =

1.3922 A. Ordinarily, the Kβ radiation is absorbed from the beam by a Nickel

filter placed at the exit of the x-ray source, but in the case of Bruker-AXS D8,

Gobel mirrors placed just before the exiting slit absorb the Kβ wavelengths, and

thus a Nickel filter is not required. Only a small fraction of the Kβ radiation

remains, which is only observable in the diffraction of highly crystalline single

crystal structures, and even then the intensity ratio is of the order Kα1/Kβ =

1000/1.

The remaining Kα wavelengths are used in the diffraction measurement. Since

the ratio the intensities of the two α radiations is Kα1/Kα2 = 2/1, then a weighted

average x-ray wavelength of λ = 1.54184 A can be used for the calculation of the

63


lattice constants from the diffraction peaks. However, in the case of highly crys-

talline structures, α1-α2 splitting of high order diffraction peaks can be observed,

and in this case the individual wavelengths can be used for calculation of lattice

parameters.

2.4.3 Sample Alignment

It is essential that the film be positioned at the centre of rotation to obtain as

high as possible intensity of the diffraction maximum. To this end, before any

meaningful measurement could be performed, the film had to be aligned as close

as possible to the centre of rotation.

Figure 2.12: A simple diagram explaining the relation of the variables x,y, z, χ and φ, to the XRD stage and the verticle plane.

There are seven main drives which are used to centre the sample and align the

detector; θ, 2θ, x, y, z, χ and φ which are defined as follows. θ and 2θ describe

the angle that the stage and detector makes with respect to the plane of emitted

x-rays, respectively. x, y and z describe the position of the stage with respect to

a Cartesian axis, where z is the height of the stage, and is perpendicular to the

stage surface, x and y are directions perpendicular to each other, and parallel to

the stage surface. χ describes the angle that the stage makes with the vertical

plane, with respect to the x-ray source, and φ is the angle of rotation of the stage

about its normal axis z.

The sample is mounted on the stage, and the detector is aligned with the

straight through x-ray beam. In the straight through configuration, the film is

64


moved in the direction of z, into the beam, until the measured intensity falls to

half the straight through intensity. A crystal may exhibit a slope, such that its

surface may not lie parallel to the stage, and so the film must be ‘flattened’. This

is achieved by performing a rocking curve on a diffraction peak of the substrate.

For MgO, this was performed using the 002 reflection. A rocking curve involves

positioning 2θ at the theoretical position of the corresponding diffraction peak,

and ‘rocking’ the stage around a value of θ approximately half of the theoretical

2θ value. For MgO, using the average Kα wavelength, the value 2θ = 43 was used.

The deviation of θ from the expected value of 21.5 due to any slope or substrate

miscut, is corrected for by the XRD software effectively flattening the surface of

the film so that the incident x-rays intersect the actual lattice planes at an angle

angle θ.

However it was noticed in the course of this work that the use of this value for

2θ was in fact in error. When a rocking curve is performed with 2θ = 43, two

peaks are observed corresponding to α1-α2 splitting. When the computer is then

asked to flatten the surface, it will use the more intense α1 peak, and therefore a

slope will still remain in the film. This error is simply corrected by using the α1

theoretical value of 2θ = 42.9093 and rocking around θ = 21.45465. When this is

done, only one peak is observed. This method was found to improve the intensity,

the width, and accuracy of the studied diffraction peaks.

Further refinements to the alignment are performed by moving the stage so that

the film is fully illuminated by the beam. This is achieved by fixing θ = 21.45465

and 2θ = 42.9093, and moving the sample in the x and y directions until the

intensity of the detected beam is maximum.

The final alignment is relatively minor, and is only really necessary for accurate

calculation of in-plane lattice parameters. The previous rocking curve effectively

flattens the crystal in the horizontal plane, but not the vertical plane (both with

respect to the x-ray source). To do this, the angle χ is moved to 45, and θ

and 2θ fixed at 31.14925 and 62.2985 respectively, which corresponds to the 022

reflection of the cubic MgO substrate as calculated using Kα1 radiation. A φ-scan

is then performed around φ = 0 to locate the peak, which will be displaced from

zero by a magnitude depending on how parallel the substrate’s edges are with the

vertical plane. Once the peak has been found, φ is set to this position, and a χ

scan is performed from 42 to 48. Since the substrate is cubic, the maximum

intensity should occur at 45 and any deviation from this angle is due to the slope

of the film. The computer is then able to adjust for this discrepancy, effectively

65


flattening the sample in the vertical plane. The sample is now aligned for high

resolution diffraction.

2.4.4 Determination of Lattice Parameters

Once the alignment of the crystal is complete, then a θ − 2θ scan will give infor-

mation of the periodicities of the the atomic planes. With χ = 0, the detector

is rotated through an angle 2θ while the stage follows at half the angular speed,

moving through an angle θ. The peaks in this scan will correspond to periodic

planes which are present in the deposited film, parallel to the planes of the MgO

substrate. It is a simple matter then to calculate the periodicity of these atomic

planes by determining the angle θ at which the peak occurs, and applying the

Bragg law (equation 2.5). In this arrangement only those planes parallel to the

surface are detected, and consequently one can only infer information about the

out-of-plane periodicity.

The in-plane periodicities can be measured in two ways. One method is to mea-

sure the periodicities directly using the technique of Grazing Incidence Diffraction

(GID), which is described in detail in a later section. Unfortunately, the geometry

of the Bruker AXS D8 does not permit this type of technique to be used, and so

one must use the second method of measuring periodicities which share an in-plane

component. This is achieved by setting χ = 45, and measuring the periodicity of

the (0kl) planes, for example measurement of the 011 or 022 reflections.

Using this measurement, one can determine the in-plane lattice parameter,

provided an out-of-plane periodicity is known, by implementation of simultane-

ous equations. Of course this will give the lattice parameter of only one of the

crystallographic axes, and in order to measure the other the film must be rotated

through φ = 90, which will then access the h0l type reflections. Indeed one could

actually perform a 360 φ scan for each of the materials in the capacitor which

gives information on the epitaxy of the layers.

The one disadvantage of this method is that it is not as accurate as measuring

the periodicities directly, as in the case of GID. The width of the peaks is often

large, and coupled with a non-monochromatic x-ray beam, the degree of error is

increased. Whilst the out-of-plane measurements can give periodicities accurate

to at least 3 decimal places, the periodicities determined by the above method

are found to be accurate to only two decimal places. Also, if there are two or

more different periodicities of similar sizes, then due to the dual wavelengths, it is

66


impossible to separate them. A monochromator was available to remove the Kα2

wavelength, but it is used at the expense of dramatically reducing the intensity of

the reflections from the deposited films.

2.4.5 Synchrotron Diffractometer

Some of the work presented in this thesis was performed at the Hamburger Syn-

chrotronstrahlungslabor (HASYLAB at DESY), which implements a large scale

diffractometer located at the end of a synchrotron radiation source. Synchrotron

radiation occurs when electrons/positrons, travelling at close to the speed of light

within a storage ring, are accelerated by bending magnets within the ring (figure

2.13(a)). Normally, an accelerating charge will emit radiation in a random direc-

tion, but due to the relativistic speeds of the charged particles, the emitted photons

are confined to a cone of small solid angle in the direction tangential to the accel-

erating charge. What is more, the charged particles can be forced to radiate even

more by introducing ‘wigglers’ and ‘undulators’ into the storage ring. Measuring

several metres in length, these special magnets consist of a series of alternating

north and south poles. Due to the many magnetic poles in succession, relativistic

charged particles entering the ‘wiggler’ are forced into a rapid zig-zag course which

dramatically increases the number of photons emitted (figure 2.13(b)).

The brilliance of a synchrotron beam can be 2-3 orders of magnitude greater

than a standard x-ray tube, and when combined with a wiggler, as in the case

at HASYLAB, the brilliance can be increased by a further 6 orders of magnitude

Figure 2.13: a) the production of synchrotron radiation from the bendingof relativistic electrons by magnets. The radiation is emitted tangentiallyto the curved path. b) A wiggler composed of many magnets will cause theelectron beam to zig-zag, increasing the emission of photons (after [107]).

67


Figure 2.14: The average brilliance of various x-ray sources. Comparingthe characteristic radiation from sealed tubes with radiation from bendingmagnets gives a gain of 2-3 orders of magnitude. Wigglers and undulatorsincrease the brilliance by a further 5-6 orders of magnitude [108].

(figure 2.14). This means that diffraction experiments can be performed faster,

and with greater angular resolution than with a standard x-ray tube.

The synchrotron source at DESY uses relativistic positrons in combination

with a wiggler to generate synchrotron radiation for x-ray diffraction. The x-ray

radiation generated is highly collimated, and has an energy of 9.8 keV, which

converts to a wavelength of 1.26515 A. Although the diffractometer in HASYLAB

is extremely large and cumbersome, it is exceptionally versatile and permits for

the direct measurement of the out-of-plane, as well as in-plane lattice constants.

A basic schematic of it is illustrated in figure 2.15.

68


Figure 2.15: General set-up of a high resolution diffractometer similar tothat in HASYLAB in DESY (after [108]).

2.4.6 Grazing Incidence X-ray Analysis

By using the Grazing Incidence Diffraction (GID) technique with monochromatic

synchrotron x-rays, the in-plane lattice parameter of a material can be directly

measured. GID is much the same as normal diffraction, except, as its name sug-

Figure 2.16: The effective penetration depth below a GaAs surface for aGID experiment, calculated for different incidence angles αi, and exit angleαf . The penetration depth depends on the density of the material (after[108]).

69


gests, the x-rays enter the film almost parallel to the surface at an angle αi. Owing

to refraction of the incoming beam at the air-film interface, the penetration depth

of the probing x-ray can be controlled by varying αi to be either smaller or larger

than a critical angle αc (figure 2.16). In the first case, a portion of the incoming

beam becomes evanescent and propagates parallel to and close below the film sur-

face. The minimum penetration depth is of the order of 4-10 nm, depending upon

the density of the material. On increasing αi, the penetration depth within the

film increases up to about 400-600 nm.

When this condition is met, and the sample is aligned in the centre of rota-

tion, GID measurements can be performed. These are similar to the out of plane

measurements except that the sample is rotated an angle θ (traditionally called ω)

about the axis perpendicular to the film surface, and the detector moves at twice

the angular speed corresponding to 2θ in the same plane. The peaks in this scan

will correspond to the periodicities of the atomic planes parallel to the surface of

the film.

70

Chapter 3

Characterisation of Bulk and

Interfacial Properties

This chapter investigates the temperature and frequency characteristics of the

observed interfacial capacitance in two separate thin film ferroelectric capacitor

systems. Implementation of the series capacitor model facilitates the extraction of

the bulk-like and interfacial components from the dielectric measurements of a set

of capacitors within a thickness regime of ∼ 100 nm - ∼ 1 µm. The extracted bulk-

like component for both systems is found to behave like a bulk ceramic, with little

frequency dependence of the dielectric constant, whilst exhibiting a strong Curie

Weiss behaviour. For the SRO/BST system, the extracted interfacial component

is observed to be relatively temperature and frequency independent at low tem-

peratures, whereas at higher temperatures a thermally activated contribution to

this dielectric component dominates, exhibiting a large degree of temperature, and

frequency dependence. This latter behaviour is not observed in the LSCO/BST

system. The origin of this behaviour is discussed, and is attributed to a localised

region of defects, located within, or at least parallel to the electrode/dielectric

interface.

3.1 Introduction

The decrease in the measured dielectric constant with decreasing film thickness

has been widely attributed to the presence of an interfacial capacitance located

at the film/electrode interface. There has been a large volume of research into

the possible origin of this interfacial capacitance, but as yet, there has been lit-

71

3.2 Thickness Dependence

tle functional characterisation of this component. One study by Zafar et al [48]

has demonstrated, that the interfacial capacitance of a Pt/BST/Pt structure was

frequency independent, but showed a small temperature dependence, whereas the

bulk-like component was highly temperature dependent, whilst also showing a

power law dependence on frequency. Basceri et al [25], studying a similar system

at five temperatures, observed virtually no temperature dependence in the interfa-

cial capacitance, but a large dependence on temperature on the bulk component.

Unfortunately, Basceri et al, performed their measurements at one frequency, and

so could not comment of any frequency dependence of the extracted components.

It is apparent that too few studies have been performed on the functional

properties of the interfacial capacitance. Certainly, Zafar et al conducted detailed

frequency analysis, but of the few studies performed, most have been confined to

only a few temperatures, or within a narrow temperature range. In this thesis,

the functional characteristics of the interfacial capacitance were studied at 10 K

intervals from 100-400 K, and at four frequencies from 102 − 105 Hz.


To investigate the nature of the thickness dependence of ε, two sets of capacitor

structures were fabricated using pulsed laser deposition as described in Chap-

ter 2. The two capacitor structures consisted of SrRuO3/Ba0.5Sr0.5TiO3/Au and

(La, Sr)CoO3/Ba0.5Sr0.5TiO3/Au grown on commercial single-crystal 001 MgO

substrates. Since it is widely believed that the thickness dependent ε is due to an

interfacial capacitance, then changing the material of the bottom electrode would

conceivably alter the interfacial environment, and hence affect the observed thick-

ness dependence. Thus, functional behaviour of a series of films with thicknesses

ranging from ∼ 100 nm - ∼ 1 µm in both systems were investigated, over a wide

temperature range (100 K - 400 K) and a frequency range of between 102 and 105

Hz.

In order to effectively investigate the influence of the interfacial capacitance,

it is important to ascertain the quality of the films. As discussed in Chapter 1,

the permittivity of a thin film can be suppressed by microstructural aspects such

as poorly orientated crystallography [29, 30], and deviation from stoichiometry

[27, 110]. Using X-ray diffraction, the crystallography of the two capacitor systems

were verified to be be highly orientated with MgO001‖LSCO001‖BST001, and only

a little presence of BST011 orientation (figure 3.1(a) and (b)). By measuring the

72


Figure 3.1: X-ray diffraction of a) SRO/BST system (after [109]) and b)LSCO/BST system. The thickness of the BST layer in each case is∼ 270 nm.Clearly the films are highly oriented, with only a small 011 element present.The peak indexed with an asterix in a) is believed to be an impurity in theMgO substrate [109].

peak position, and applying the Bragg equation, the out-of-plane lattice constant

of each layer was calculated to be 3.935±0.006 A and 3.966±0.006 A for the SRO

and BST in the SRO/BST system, and 3.803 ± 0.006 A and 3.957 ± 0.006 A for

the LSCO and BST in the LSCO/BST system. These values agree very well with

accepted values, except that the lattice parameter of the BST in the SRO/BST

system is highly elongated, which has been shown by Sinnamon et al [89] to be

due to strain coupling at the SRO/BST interface. Also, the stoichiometry of the

BST was held constant at Ba : Sr = 1.14± 0.05, and (Ba + Sr) : Ti = 1.00± 0.03.

3.2.1 SRO/BST system

The SRO/BST system of capacitors was grown by Lesley Sinnamon, and their

functional properties have been extensively characterised by her, with the results

having been presented previously (see Ref. [82, 89, 109]). However, for the sake of

clarity, the relevant functional data is summarised below:

The low-field dielectric properties measured at 10 kHz, and as a function of

temperature for a number of capacitors, in which thickness of the dielectric layer

ranged from 145 to 950 nm, is shown in figure 3.2.

It is clear from this figure that the measured dielectric constant is dramatically

reduced, as the thickness of the BST layer decreases. Also, it is noted that as the

dielectric thickness is reduced, the peak in the permittivity is progressively sup-

pressed, whilst the temperature at which this peak occurs (TM) moves to higher

temperatures. This peak is assumed to be associated with the ferroelectric to

73


Figure 3.2: a) Selected low field dielectric data for the SRO/BST seriesof capacitors. b) Measured dielectric constant and tanδ at 400 K. All datashown is measured at 10 kHz (after [82]).

paraelectric phase transition, and its movement to higher temperatures has been

attributed to the increase of the average strain within the film, created at the

SRO/BST interface, due to the mismatch in lattice parameters of the two ma-

terials. Since the BST has a larger lattice constant than SRO, there will exist

a compressive in-plane strain within the BST, which causes an out-of-plane dis-

tortion, the magnitude of which depends on the Poisson’s ratio for the material,

resulting in stablisation of the ferroelectric phase. The degree of out-of-plane elon-

Figure 3.3: Normalised frequency response of the low field dielectric con-stant of the SRO/BST system measured at 400 K (after [82]).

74


gation depends on the thickness of the film, since the strain is found to relax

exponentially with film thickness [88, 89]. Thus the deformation of the unit cell

will be more pronounced for thinner films, resulting in a progressive increase in

TM .

The lower graph of figure 3.2(a) illustrates the corresponding dielectric loss of

each of these films. In general, the loss was maintained below tan δ ∼ 0.05 for all

values of thickness, and frequency, except for 100 kHz and the higher-temperature

low frequency response in the thinner films. Also, the frequency dispersion of

the dielectric constant of these films was limited, with ε′100kHz/ε′100Hz > 0.8, as

illustrated in figure 3.3.

3.2.2 LSCO/BST System

As for the previous system, the low field dielectric properties of the LSCO/BST

capacitors were measured as a function of the film thickness. Figure 3.4 shows the

10 kHz frequency response of the dielectric constant as a function of temperature,

for a selection of the LSCO/BST capacitor structures studied. Again there is a

distinct reduction of the value of ε as the dimensions of the dielectric layer decrease.

Similarly, as before, the tan δ of the films remained typically less than 0.05 for all

temperatures, frequencies and thicknesses. The dielectric loss measured at lower

frequencies demonstrated a larger degree of noise, the cause of which was later

found to originate from stray capacitance due to shorting of the bottom electrode

with the silver paste used to adhere the substrate to the heater block during the

film deposition. Subsequent removal of the paste from the substrate’s bottom

and edges with a razor blade resulted in cleaner data. Figure 3.5 illustrates the

frequency response of the LSCO/BST system at 400 K, again normalised to the

100 Hz measurement, demonstrating limited dispersion with ε′100kHz/ε′100Hz > 0.8,

similar to the SRO/BST system.

It is noted that, as the thickness of the films decrease, the degree of suppression

of the peak permittivity increases as seen in the SRO/BST system, but that the

position of TM remains relatively unchanged, occurring at T ∼ 250 K. The position

of this peak is particularly noticeable in the dielectric loss. Since there is no

migration of TM , it is possible to conclude that there is relatively little, or indeed

no strain coupling present within these films. Indeed, an induced misfit strain

would be difficult to maintain by elastic deformation alone within the film since

the lattice constant of the BST is so much larger than that of the LSCO, resulting

75


Figure 3.4: a) Selected low field dielectric data for the LSCO/BST seriesof capacitors. b) Measured dielectric constant and tanδ at 400 K. All datashown is measured at 10 kHz.

in an incoherent or semi-coherent interface. This is verified by Lookman et al [111],

who have measured the out-of-plane lattice constant of these films as a function

of thickness and found the lattice parameter to be constant down to 100 nm.

Comparison of the measured dielectric constants of the LSCO/BST and SRO/BST

systems shows that the permittivity of the former is much smaller than that ob-

Figure 3.5: Normalised frequency response of the low field dielectric con-stant of the LSCO/BST system measured at 400 K.

76

3.3 Series Capacitor model

served in the latter for a given thickness. For example, at its maximum, a 975

nm film exhibits a ε ∼ 2000 in the SRO/BST capacitors, whereas in the LSCO

system the permittivity is measured to be ε ∼ 700 for a similar thickness of 1035

nm. Clearly, changing the material of the bottom electrode would seem to have

had a large impact on the magnitude of the measured dielectric constant.


As discussed in Section 1.2.3, the thickness dependence of the dielectric constant

can be modelled using the series capacitor model. Recalling equation (1.23), the

reciprocal of the measured capacitance of n capacitors in series is the sum of the

reciprocal capacitance of each capacitor, i.e.

1

CT

=n∑

k=1

1

Ck

. (3.1)

Also, recall that this expression can be given in terms of material parameters

thusd

ε=

d

εb

+

(di

εi

)1

+

(di

εi

)2

, (3.2)

where d and di are the thickness of the bulk and interfacial layers, and ε, εb

and εi are the dielectric constants of the measured, bulk and interface components

respectively, with the numbers indicating each interface. Unfortunately, functional

measurements cannot distinguish the individual contributions from each interface,

and so the last two terms in equation (3.2) can be combined together, resulting in

the expressiond

ε=

d

εb

+1

K. (3.3)

It is convenient to represent the total capacitance of the interfacial regions of the

thin film capacitor by the parameter K, which in this notation would have units

of reciprocal length. The use of this single parameter K, is justifiable since from

a purely functional perspective it is impossible to uncouple the di/εi term in the

series capacitor equation.

Using the kinds of data sets illustrated in figures 3.2 and 3.4 above, the real

and imaginary components (ε′ and ε′′) of the dielectric constant at each measure-

ment frequency were sampled at 10 K intervals between 100 and 400 K. At each

77


Figure 3.6: a) d/ε′ and b) d/ε′′

as a function of thickness d at 10kHz, for data taken at 200 K and400 K for the SRO/BST system[102].

Figure 3.7: a) d/ε′ and b) d/ε′′

as a function of thickness d at 10kHz, for data taken at 200 K and400 K for the LSCO/BST system.

temperature, a plot of d/ε′eff and d/ε′′eff against d was created and the parameters

for ε′b and ε′′b , and K ′ and K ′′ in equation (3.3) were extracted from the gradient

and intercept of the best fit straight line. Figures (3.6) and (3.7) show examples

of such plots at 200 K and 400 K measured at 10 kHz, for both the capacitor

systems. The large error bars in these figures correspond to a 5% error in determi-

nation of the film thickness, due to an uncertainty in the multiplication factor of

the aquired TEM image. However a reasonable linear fit to the data was observed

for all cases, demonstrating a clear temperature dependence of both εb and K for

the two capacitor systems. The quality of the straight line fits, lends strength to

the general applicability of the series capacitor model (equation 3.3), such that it

can be confidently applied to the real and imaginary components of the dielectric

function.

78

3.4 Behaviour of Bulk Component


The extracted bulk dielectric behaviour as a function of temperature for both

types of capacitor are illustrated in figure 3.8. As can clearly be seen, the recon-

structed data assumes the expected form of dielectric behaviour for ceramic/single

crystal (Ba,Sr)TiO3 [112], with a clear permittivity peak occurring at ∼ 250 K.

Comparison of the magnitude of the permittivity peaks, reveals that whereas the

peak dielectric constant of the SRO/BST system is of the same magnitude as that

expected in bulk BST of the same composition, the magnitude of the LSCO/BST

dielectric constant is severely depressed, by an order of magnitude. The cause of

this depression can possibly be attributed to the presence of an internal voltage

(∼ 1 V) within the dielectric, due to the difference in the work functions of the

capacitor electrodes. The known work functions of Au, SRO and LSCO are φm =

5.1, 5.2 and 4.1-4.6 eV respectively [113].

Figure 3.8: Extracted bulk components of the dielectric constant and tan δfor a) SRO/BST and b) LSCO/BST systems.

To preserve equilibrium, when two materials are intimately in contact, charge

carriers will flow from one material to the other until the Fermi energies of the

two materials are equal. However, when a dielectric or semiconductor is sand-

wiched between two metals of differing work functions, its attempt to equalise its

Fermi energy with that of the two metals, results in a tilting or bending of the

conduction and valence bands. Since these bands define the potential energy of

the charge carriers of the material, a tilting of the bands will result in a potential

difference across the dielectric, which is observed as a voltage equal in magnitude

to the difference of the work functions of the two electrodes. Thus, the SRO/BST

system experiences little or no internal voltage, whereas the LSCO/BST system

exhibits an appreciable voltage, and since the dielectric measurements were taken

79


Figure 3.9: Simple band structure diagram illustrating how an internalvoltage is created within a dielectric due to a mismatch in the electrodework functions, φm1 and φm2 . A potential difference V12, is formed when theconduction band EC tilts, due to the equalisation of the the three materialFermi energies, EF (which for a metal is equal to the work function φ asmeasured from the vacuum level). χ is the electron affinity of the dielectric.

at zero externally applied bias field, the internal bias field could cause a significant

reduction in the extracted ε′b.

Curie-Wiess behaviour is also observed in the extracted bulk component for

temperatures significantly above the permittivity peak (figure 3.10), suggesting an

inherent Curie temperature of TC ∼ 300 K for the SRO system and TC ∼ 210 K for

the LSCO system. Why these values should differ significantly from the accepted

TC ∼ 248 K is not clear. For the LSCO system, the presence of the internal

bias field, which decreases the bulk permittivity, would increase the bulk dielectric

Figure 3.10: Extracted bulk permittivity, expressed in a Curie-Wiess plot,for a) SRO system, and b) LSCO system. The straight line demonstratesa strong linear dependence above the peak permittivity, implying TC = 300K and 210 K for the SRO and LSCO respectively.

80


Figure 3.11: Hypothetical illustration of how a bias field can shift the ap-parent TC . left) A sufficient bias field will reduce the permittivity of thematerial. right) Since the dielectric stiffness is determined by the recipro-cal of the permittivity, a decrease in εb, corresponds to an increase in 1/εb.This results in a decrease of the temperature axis intercept and hence anapparant reduced TC .

stiffness, which could result in an apparent shift of TC to lower temperatures, as

illustrated in figure 3.11.

However, there would be little or no field suppression of εb for the SRO sys-

tem, which would of course only reduce the apparent TC . Instead, the shift in

TC could be an artifact from the misfit strain induced ferroelectric stablisation.

Normally, one would not expect an interfacial phenomenon to be observed in the

extracted bulk component, however, εb was extracted from a series of films where

TM increased with decreasing thickness. Therefore, the extracted bulk TC could

represent an average of the thickness dependent TC of the thin films, and as such

be an artifact of the fitting of the series capacitance model to the experimental

data.

Finally there is very little observed frequency dispersion in the extracted bulk

permittivity of the two capacitor sets (figure 3.12), which one would normally as-

sociate with high quality bulk ceramics. This is in stark contrast to the results

of Zafar et al [48] who found a significant frequency dispersion in their extracted

bulk permittivities. On the whole, the behaviour of the bulk dielectric properties

extracted from equation (3.3) is strongly reminiscent of the kind of dielectric re-

sponse that would be expected of bulk material at this composition of BST. The

implication is first an affirmation of the general applicability of the series capac-

itor model, and second that the associated information on the behaviour of the

interfacial capacitance should be meaningful.

81

3.5 Behaviour of Interfacial Component

Figure 3.12: Dispersion of the extracted bulk dielectric constant for a)SRO system, and b) LSCO system.


3.5.1 SRO/BST System

The extracted real and imaginary interfacial components (K ′ and K ′′) for the

SRO/BST system are plotted as a function of temperature and frequency in figure

3.13. As can be seen in figure (3.13(a)), the real component of the interfacial ca-

pacitance increases slowly with increasing temperature in an approximately linear

fashion from 130 K to ∼ 300 K, then rapidly increases. There is also a moderate

frequency dispersion below 300 K, becoming more pronounced for T > 300 K.

Figure (3.13(b)) shows the behaviour of the imaginary part (K ′′) of the interfacial

Figure 3.13: a) Real (K ′), and b) imaginary (K ′′) components of the ex-tracted interfacial capacitance, illustrated as a function of temperature andfrequency.

82


component. The behaviour of K ′′ as a function of temperature and frequency is

similar to that of K ′, but, for the most part, demonstrates a smaller linear temper-

ature dependence and smaller frequency dispersion for T < 300 K. The behaviour

of K ′′ is particularly instructive as it can be related to the real part of conductivity:

1

Ri

= ωε0AK ′′, (3.4)

andσi

di

= ωε0K′′, (3.5)

where Ri, σi, and di denote the resistance, conductivity, and effective thickness,

respectively, of the interfacial layer; ω is the angular frequency of the applied ac

field, A is the electrode area, and ε0 the permittivity of free space.

For a semiconductor, conductivity is a thermally activated process:

σ = σ0 exp

(−EA

kBT

), (3.6)

where σ0 is a constant, and EA, kB, and T are the activation energy for conduction,

Boltzman constant, and temperature, respectively. By combining equations (3.5)

and (3.6), K ′′ can be defined as a thermally activated variable:

K ′′ =σ0

diωε0

exp

(−EA

kBT

). (3.7)

Arrhenius plots of ln(K ′′) against 1/T are demonstrated in figure (3.14). It is

clear that there are two regimes of behaviour, one at low temperatures (I), and

one dominating at high temperatures (II). The slope of the low-temperature re-

gion is extremely shallow (implied activation energy of 0.006 ± 0.001 eV) and, in

this regime, the imaginary interfacial capacitance could be considered as approx-

imately independent of temperature, and conduction is probably not thermally

activated. Direct measurement of the activation energy in the high temperature

region from figure 3.14, yields values of EA between 0.14 and 0.32 eV depending

upon measurement frequency. A thermally activated conduction process whose

activation energy is dependent upon the frequency of the applied measurement

field would seem unphysical, and could imply that use of equation (3.7) to model

the data is erroneous. However, inspection of equation (3.7) indicates that when

83


Figure 3.14: Arrhenius plots investigating the thermal response of theimaginary component of the interfacial capacitance, K ′′. Clearly, one candistinguish a difference between low temperature (region (I)) and high tem-perature (region (II)) behaviour.

T → 0 then K ′′ → 0, which is clearly not observed in figure 3.14. Therefore, one

could conclude that there are two mechanisms which contribute to the interfacial

component, with region (I) behaviour due to the inherent interfacial capacitance

of the system, and region (II) behaviour due to a thermally activated space charge

component superposed upon this background.

Figure 3.15: Arrhenius plots of the high temperature response, after cor-recting for the low temperature background. An activation energy ofEA ∼ 0.6 eV, independent of measurement frequency is observed.

84


If the temperature-independent background component of region (I) is sub-

tracted before generating Arrhenius plots, then the high-temperature activation

energy for conduction is determined to be EA ∼ 0.60±0.05 eV for all measurement

frequencies, as illustrated in figure 3.15. Since this component acts in series with

the bulk component, the analysis implies that the thermally activated component

of the imaginary permittivity must lie in a plane parallel to the electrodes, in the

parallel-plate configuration. The nature of this component is discussed in a later

section.

3.5.2 LSCO/BST System

Figure 3.16(a) demonstrates the reconstructed temperature and frequency char-

acteristics of the extracted K ′ for the LSCO/BST system. On the whole there

would seem to be very little frequency and temperature dependence, both in

the high and low temperature regions. There is a small temperature depen-

dence for T < 250 K in which K ′ decreases almost linearly with increasing

temperature, and also for T > 250 K where K ′ begins to increase with in-

creasing temperature, again in an almost linear fashion. Comparison of the fre-

quency dispersion of K ′ in the SRO and LSCO systems at 400 K, indicates

that the LSCO has K ′(100 kHz)/K ′(100 Hz) = 0.80 compared to that SRO of

K ′(100 kHz)/K ′(100 Hz) = 0.60.

Figure 3.16: a) Extracted real component (K ′) of the interfacial capaci-tance for the LSCO system, as a function of temperature and measurementfrequency. b) Comparison of extracted K ′ for the SRO and LSCO systems.Clearly the LSCO system varies little with temperature and frequency, com-pared to the SRO system, particularly at high temperature.

85


Figure 3.17: Extracted real component (K ′′) of the interfacial capacitancefor the LSCO system, as a function of temperature and measurement fre-quency. Clearly, there is little evidence of a thermally activated process, asobserved in figure 3.13(b).

A direct comparison of frequency and temperature behaviour of K ′ (figure

3.16(b)) illustrates that on the whole, the LSCO system does not share the same

high temperature space charge phenomena as its SRO system counterpart. This is

confirmed in the extracted K ′′ data as shown in figure 3.17. Clearly there would

appear to be little temperature dependence of K ′′, particularly at lower frequencies

where it was observed in the SRO system, and thus there is little evidence for the

presence of any thermally activated behaviour. Curiously, K ′′ becomes negative

in a discontinuous manner for temperatures above ∼ 300 K, the reason for which

is not completely certain. This change of sign implies that the conductivity, and

hence resistance of the interfacial layer becomes negative and may therefore purely

be an artifact from the fitting of the imaginary dielectric data to equation (3.3).

86

3.6 Discussion of Results


Whereas the interfacial capacitance of the LSCO/BST system was found to have

little dependence on temperature, the real component of the interfacial capaci-

tance (K ′) of the SRO/BST system was found to increase significantly at high

temperature. This increase was found to be associated with an increase of the

conductive imaginary component (K ′′) of the interfacial capacitance, suggesting

an increase in the concentration of charge carriers due to a thermally activated

process exhibiting an activation energy of EA ∼ 0.6 eV.

The movement of the charged species within the crystal lattice can contribute

significantly to both the low-frequency dielectric constant, and conductivity of per-

ovskite oxides. Frequently, such charged species are related to oxygen deficiency.

However, the activation energy for oxygen vacancy migration is typically ∼ 1.1 eV

[114, 115], which is too large to be considered as the predominant mechanism for

the conduction seen here. Alternatively, electrons can be trapped by Ti4+ ions or

oxygen vacancies, creating colour centres (in Kroger-Vink notation):

VO ⇔ V ·O + e′, (3.8)

V ·O ⇔ V ··

O + e′, (3.9)

Ti4+ + e′ ⇔ Ti3+. (3.10)

Such colour-centre electrons can be easily thermally activated to take part in

conduction as indicated in previous work: Ang et al [116], investigating high-

temperature dielectric relaxation in Bi-doped SrTiO3, found a relaxation peak

between 350 and 600 K which they attributed to a thermally activated second

ionization of conduction electrons from oxygen vacancies V ··O . They measured the

activation energy for this process to be EA = 0.59− 0.78 eV depending upon the

concentration of Bi within the film. Since the band gap of BST is Eg = 3.2 eV

[15], the energy level for V ··O (Ed = 1.3 − 1.4 eV [117]), occurs near the middle

of the band gap. Ang et al [116] measured Ed ∼ 1.3 eV using optical absorption

spectra, implying that the activation energy for conduction is EA = Ed/2 = 0.65

eV. It is worth noting that the activation energy of conduction electrons from Ti3+

(equation 3.10) is also EA ∼ 0.7 eV [53], and since Ti3+ is often associated with

oxygen vacancies, it is difficult to determine which donor species the conduction

electrons are associated with [118]. Nevertheless, it is clear that the observed

activation energy of ∼ 0.6 eV could easily be associated with oxygen-vacancy-

87


Figure 3.18: Simple band diagram illustrating the the thermal activation ofelectrons from a shallow trap level associated with oxygen vacancies defects.The introduction of a donor level Ed, 1.3 eV below the conduction bandEC , causes the formation of a new Fermi level EF = Ed/2 = 0.65 eV belowEC from which electrons can be thermally excited. Ei is the intrinsic Fermilevel of the material without defects.

related conduction.

The presence of defects also explains the increase in the real part of the in-

terfacial capacitance (K ′) above 300 K, since mobile carriers create an additional

space-charge component to the dielectric polarisability, as described in Chapter 1.

It appears then, that the behaviour of the interfacial capacitance at relatively

high temperatures in the SRO system is dominated by the presence of carriers in

shallow defect-related traps. However, several points should be made:

1. The dielectric behaviour that is well rationalized by the thermal activation of

carriers from defect traps is only associated with the interfacial component.

Defect trapped carriers must, therefore, be acting electrically in series with

the bulk-like component in the thin-film capacitors. This implies that they

lie in a discrete band parallel to the electrodes, and are probably situated

adjacent to the dielectric-electrode boundary. The detection of this defect

band has been implied in previous work [57] but it may have serious impli-

cations for ferroelectric fatigue [53, 54]. HRTEM work has been carried out

on the interfaces of the SRO/BST capacitors used in this study [89], but

unfortunately the resolution was not high enough to detect any Ruddlesden-

Popper planar faults next to the electrodes which would form due to ordering

of large concentrations of oxygen vacancies, as suggested by Jin et al [58].

Nevertheless, the detection of a defect band through series capacitor analysis

88

3.7 Summary

may be useful in the future rationalization of the mechanics of fatigue.

2. Examination of equation (3.3) demonstrates that if the thermally activated

species alone contributed to the imaginary interfacial capacitance, it would

be expected to tend to zero at 0 K. This is not apparent from the extracted

data (figure 3.17). Therefore, although the thermally activated release of

carriers dominates the interfacial capacitance at high temperatures (for the

capacitors examined in this study), at all temperatures, there is also a back-

ground interfacial capacitance that shows only a weak temperature depen-

dence. This background is apparent in most interfacial capacitance studies

and since it has been found here to be separable from defect-related effects, it

seems that interface defects cannot be generally responsible for the so called

‘dead layer’.

3.7 Summary

The bulk and interfacial functional properties of SRO/BST and LSCO/BST thin

film capacitors was studied as a function of temperature and measurement fre-

quency, by implementation of the series capacitor model. As far as the author is

aware, this is the first significant, and detailed study of the functional character-

istics of the interfacial capacitance.

The extracted bulk properties for both systems demonstrated ceramic-like char-

acteristics, displaying a permittivity peak at the temperature of 250 K, with very

little frequency dispersion and good Curie-Weiss behaviour. The magnitude of

the bulk permitivitty of the LSCO/BST system was observed to be highly de-

pressed, which could be attributed to an internal bias within the film, due to the

asymmetric electrodes.

The interfacial component showed little temperature dependence and a mod-

erate frequency dispersion for both systems, but the SRO/BST capacitors in par-

ticular displayed a large temperature and frequency dependence above T ∼ 300

K. This was attributed to a thin layer of oxygen vacancy related defects, lying

parallel to, and possibly next to the electrode interface. At elevated temperatures,

trapped electrons, associated with these defects, can be freed, so that they can

contribute to the dielectric response of the thin layer.

However, this mechanism only dominates at high temperatures, and was not

evident in the LSCO/BST capacitors, implying that this system was relatively free

89

3.7 Summary

of defects, but still demonstrated a significant interfacial capacitance. Crucially

this shows that in the absence of large concentrations of interfacial defects, there

still exists an unexplained interfacial component.

Thus, defects can play an important role in the collapse of the dielectric con-

stant with decreasing thickness, particularly at high temperatures, but they are

not the fundamental origin of the interfacial capacitance.

90

Chapter 4

Extension of Series Capacitor

Model to the Ultrathin Regime

In the work of Sinnamon et al [82], the investigation of the thickness dependence

of the dielectric constant was extended down to ultrathin films, the thinnest film

being 7.5 nm. They observed that the reduction of the measured permittivity

still adhered to the series capacitor model, down to 7.5 nm, thus leading to the

conclusion that if a dielectric ‘dead layer’ of finite thickness exists within the BST

film, then its total thickness must necessarily be less than 7.5 nm. This would

mean that if the dead layer at each electrode interface is identical in size, that

each would be < 3.75 nm thick.

It is an easy task to make ultrathin BST films, but the difficulty arises in

the fabrication of ultrathin films which display good dielectric behaviour. Indeed

Sinnamon et al had great difficulty in fabricating films thinner than 7.5 nm which

exhibited low dielectric loss and good frequency dispersion [102].

This chapter reports on the successful extension of the series capacitor model

to thicknesses < 7.5 nm, by the fabrication of good quality BST ultrathin films

within the thickness range of 5 − 16 nm, thus redefining the upper limit of the

‘dead-layer’ thickness. It will also be demonstrated that the thickness dependence

of the dielectric constant can be explained by an interfacial capacitance originating

not from a distinct ‘dead layer’ within the dielectric, but instead by a thin space

charge region within the electrodes.

91

4.1 Characterisation of Ultrathin capacitors


4.1.1 Structural Characterisation and Thickness Determi-

nation

Ultrathin BST films were deposited upon LSCO bottom electrodes using Pulsed

Laser Deposition, as described in Chapter 2. It was found that in order to grow

ultrathin films that exhibited good quality dielectric behaviour (such as low di-

electric loss, and minimum frequency dispersion), the deposition and annealing

temperature, had to be reduced by approximately 50 C, which could potentially

affect the crystalline quality of the film. Figure 4.1 shows the out-of-plane diffrac-

tion peaks of a 12 nm BST film. It is clear that the film is still highly orientated

with BST001‖LSCO001‖MgO001.

Figure 4.1: Out-of plane diffraction pattern for a 12 nm BST film. Thepeak indexed with an asterix is believed to be a contaminant of the MgOsubstrate.

The thickness of the ultrathin films was determined using High Resolution

TEM, and verified using Energy Dispersive X-Ray analysis (EDX). Figures 4.2(a)

and 4.2(b) show the TEM images of a 14 nm and a 5 nm film. One can discern

in parts a high degree of crystallinity through the whole film, with little evidence

of any amorphous regions anywhere within the film, and in particular next to the

electrodes, where one would expect to find the ’dead-layer’, if one existed.

Figure 4.3 shows a typical EDX depth profile across the capacitor, which was

used as another method to determine the thickness of the BST film. This figure

shows the EDX results for the 5 nm film that is presented in figure 4.2(b).

92


Figure 4.2: Transmission Electron Microscope images of a) 14 nm film,and b) a 5 nm film. The interfaces have been marked with a dashed line.The BST layer demonstrates good crystallinity up to each interface.

Figure 4.3: Energy Dispersive X-ray analysis of a 5 nm film. The abun-dance of Au, Ba and Co, was monitored across the capacitor to create adepth profile. The apparent appearance of Co in the Au electrode is anartifact due to the overlapping of the spectral peaks.

93


4.1.2 Functional Characterisation

The method for measurement of the dielectric constant of the ultrathin films varied

from that used for the thicker capacitors in one important aspect. As detailed in

Section 2.2.2, the area of the top Au electrodes were significantly reduced to 200 x

200 µm2. The use of such small electrodes has its advantages and disadvantages.

The prime disadvantage to using such small electrodes, is that a micromanipulated

tungsten probe, in conjunction with an optical microscope had to be used to

make contact with the top electrodes, which not only introduces a large contact

resistance to the effective circuit, but also means that the permittivity of the

ultrathin films could not be investigated as a function of temperature, as was seen

to be extremely useful in Section 3.5. However, the use of such small electrode

areas is advantageous since far more capacitor structures can be tested on a single

sample. Also, since the measured capacitance is proportional to the electrode

area, the effective time constant of the circuit τ = RC will be reduced, thus

increasing the frequency at which ‘Debye’ features from RC relaxation occur, and

also minimising the dielectric loss from circuit artifacts.

The measured real and imaginary dielectric constant for a selection of ultrathin

films is shown in figure 4.4. The dispersion in ε′ is minimal for frequencies < 104

Hz, but begins to dramatically increase as the frequency increases further above

this value. This is mirrored in the imaginary dielectric constant where ε′′ begins

to rise sharply at ∼ 104 Hz. Inspection of the frequency response of ε′ and ε′′

in the 14 nm film clearly shows a peak occurring in the imaginary response at

f ∼ 2 × 105 Hz, corresponding to approximately half the low frequency value of

ε′. This is indicative of Debye relaxation, and indeed a Cole-Cole plot of this

data results in the expected semi-circle figure (4.5). The characteristic time for

Debye relaxation (τD) in BST films is many orders of magnitude smaller than the

value of τ ∼ 5× 10−6 s. Also perfect Debye relaxation is never observed in nature

[119], rather such behaviour can be attributed to an increased time constant of the

circuit.

If the resistance of the equivalent circuit increases, then the associated time

constant for charging and discharging the capacitor will also increase since τ = RC.

If the time constant of a capacitor increases, then intuitively the time taken for a

capacitor to fully charge or discharge will also necessarily increase since

Q = Q0 exp

(−t

τ

). (4.1)

94


Figure 4.4: The frequency dependence of the real (ε′) and imaginary (ε′′)dielectric constants of a few of the ultrathin capacitors.

Therefore, when an ac field of an angular period similar to the RC of the circuit

is applied, the induced charge density on the capacitor plates can no longer keep

pace with the driving field, and hence the ratio of Q/V decreases, resulting in a

decrease of the measured capacitance, and thus permittivity.

It is possible to determine the series resistance RS that gives rise to this be-

haviour, by measuring the capacitance Cτ−1 at the frequency corresponding to τ−1,

and using relation τ = RSCτ−1 . For the ultrathin films illustrated in figure 4.4 the

series resitance was found to be quite large (RS ∼ 10 kΩ) and may originate from

a number of factors. One such factor to consider is the contact resistance of the

tungsten micromanipulator probe with the Au top electrode. Unfortunately this

contact resistance was not measured, but it could be appreciably large, depending

upon the geometry and roughness of the contact junction, as well as the resistance

of any impurities adsorbed onto the metal contacts. A large contact resistance may

also exist between the LSCO bottom electrode and the thin wire used to complete

the ciruit, which was crudely glued to a substrate edge in which the electrode had

95


Figure 4.5: Cole-Cole plot for the 14 nm thick film. The solid line is a semi-circular fit to the high-frequency data, which demonstrates almost perfectDebye relaxation behaviour.

been exposed with a razor blade. One must also consider the LSCO electrode

itself as a source of the increased series resistance. (La1−xSrx)CoO3−δ allows for a

degree of non-stoichiometry, which can change the resistivity of the material. The

resistivity of LSCO has been studied by several groups, and has been found to be

highly dependent upon the value of δ. Liu and Ong [120], and Sun et al [121],

have shown that oxygen deficiency within the LSCO can lead to an increase in

its resistivity by several orders of magnitude, and can even change the conduction

nature of the material from metallic to semiconducting.

Whatever the cause of the series resistance, it can justifiably be ignored since

it only really dominates above > 104 Hz. One can be sure, that the measured

functional response of the ultrathin capacitors for frequencies less than 104 Hz

should be intrinsic to the capacitor.

4.1.3 Capacitance-Voltage Measurements

When using the cryostat for the temperature measurements of the thicker films, it

is necessary to use two probes at the surface of the film to complete the electrical

circuit. This means the measured capacitance is actually the total capacitance

of two capacitors in series. However, when using the micromanipulator probe,

as in the case with the ultrathin films, the electrical circuit only comprises of a

single capacitor. It thus allows for the effective investigation of the films dielectric

96


Figure 4.6: Dependence of the dielectric constant on applied field for a 12nm ultrathin film. Positive voltage is in reference to the top electrode beingpositively biased. Arrows show the direction of the voltage sweep. Thereis a clear shift in the dielectric maximum from 0 V, due to an internal biasfield.

constant as a function of bias field.

Figure 4.6 presents the dielectric response of a 12 nm BST film at 10 kHz as a

function of applied DC field, and results from other ultrathin films with different

thickness share similar behaviour.

There is a clear displacement of the maximum permittivity from the zero field

point, which indicates that there is an internal bias within the film of approximately

1 V. As explained in Section 3.4, this internal bias arises from the mismatch in work

functions of the two asymmetric electrodes. There is also an observed hysteresis

in the measured permittivity over one ‘up’ and ‘down’ cycle of the applied field. A

qualitative explanation for such a hysteresis loop has been provided by Park and

Cho [113], as follows:

If one assumes that the BST films are slightly oxygen deficient, then the films

can be regarded as n-type semiconductors, and thus the interfaces of the BST

film between the Au and LSCO electrodes can form Schottky barriers by ionized

donors in BST. The density of the ionized donors of both interfaces can be dif-

ferent, due to the different growth nature of the two interfaces. Thus, during the

permittivity sweep, electrons that recombined to donor sites during the forward

bias of one interface cannot be activated to the conduction band at zero bias, due

to a small thermal activation energy, and only the reversely biased interface can

97


form a Schottky barrier. Therefore, the exaggerated hysteresis of the permittivity

at low temperatures results from the different values of the interface capacitance

formed by the Schottky barrier, where the voltage sweep from negative to zero

bias forms the Schottky barrier at the top interface, and the voltage sweep from

positive to zero bias forms the Schottky barrier at the bottom interface.

4.1.4 Thickness Dependence of Ultrathin Permittivity

The measured 10 kHz dielectric constant and corresponding dielectric loss, mea-

sured at zero applied field, of the ultrathin BST films, is presented as a function

of thickness in figure 4.7. The dielectric constant is still observed to decrease with

decreasing thickness within the ultrathin regime, indicating that the origin of the

collapse of the permittivity is still dominant down to 5 nm thickness. The dielectric

loss generally remains tan δ < 0.08, with the loss of the 5 nm film is tan δ = 0.065,

comparable with that observed in much thicker films (see Sections 3.2.1 and 3.2.2).

Figure 4.7: a) The thickness dependence of the dielectric constant of theultrathin BST capacitors. b) The corresponding loss tangent for the ultra-thin films.

98

4.2 Extension of Series Capacitor Model


Sinnamon et al [82], have demonstrated that the functional data, for SRO/BST/Au

capacitors, did not deviate from the series capacitor model, down to a film thickness

of 7.5 nm. Also, using HRTEM, they demonstrated that the SRO/BST interface

is completely coherent [89], with no observed microstructural regions which may

constitute an obvious functional ‘dead-layer’. They concluded that, if an interfacial

low permittivity ‘dead-layer’ exists within the capacitors then the total thickness

of this layer cannot exceed 7.5 nm.

In this section, the functional data obtained from the ultrathin LSCO/BST/Au

capacitors are analysed within a series capacitor context, to attempt to determine

if an interfacial ‘dead-layer’ exists within the dielectric film, and if so, ascertain

the maximum thickness of this layer. As such, a similar analysis as that used by

Sinnamon et al [82] shall be employed.

Recall from Section 1.2.3, that for capacitors in series, the reciprocal of the

measured capacitance can be expressed in terms of the material parameters

d

εeff

=d

εb

+di

εi

, (4.2)

where εeff is the measured permittivity, and εb and εi are the dielectric constants

of the bulk and interfacial components of thickness d − di and di respectively.

Therefore, by plotting the ratio of d/εeff versus d, one can extract the magnitudes

of the bulk permittivity and interfacial capacitance from the slope and intercept

of the best fit line to the data.

Figure 4.8 illustrates the adherence of functional data for LSCO/BST/Au ca-

pacitors to the series capacitor model at a temperature of 400 K, and a measure-

ment frequency of 10 kHz. The best fit line to this data implies that the bulk

dielectric constant of εb = 380 ± 20 and a reciprocal interfacial capacitance of

di/εi = 0.52 ± 0.11 nm. Comparison of these values with those of SRO/BST/Au

system (Chapter 3), shows that εb is much less for the LSCO system than the SRO

system (εb ∼ 1000). The reason for such a marked difference has been discussed

previously in Chapter 3. Also, di/εi is observed to be ∼ 25% larger in the LSCO

system than in the SRO system (di/εi = 0.40 nm). To the first approximation, this

would imply that if the permittivity of the ‘dead-layer’ of the two systems were

roughly equal, then the thickness of the ‘dead-layer’ in the LSCO system would

be ∼ 25% larger than the SRO.

99


Figure 4.8: Application of the series capacitor model for LSCO/BST filmsfor thickness d > 100 nm.

If the thickness of the ‘dead-layer’ is independent of the thickness of the capac-

itor structure, and assuming that εi does not vary with thickness within that layer,

then when d = di there will be a deviation from the series capacitor model, since

d/εb = 0 i.e. the series capacitor equation reduces to a single capacitor equation.

This is illustrated in figure 4.9(a). Using the parameters extracted from figure 4.8

for εb and di/εi to model this step function, it is apparent that there is a change in

behaviour in the functional response when d < di (figure 4.9(b)). This step model

may seem to be simplistic, but it illustrates rather well the most extreme scenario

for the functional behaviour.

Figure 4.9: a) Simple diagram illustrating a step model for the dielectricconstant in thin film capacitors, and b) the modelled functional response ofd/εeff for ‘dead-layer’ thickness of di= 0, 5, 10, 20 nm.

100


Using the functional data from the ultrathin capacitors, the applicability of

this model can be tested. It is important to note that the functional data for the

ultrathin capacitors were obtained at room temperature, whereas the functional

data presented in figure 4.8 was collected at 400 K. However, one may recall from

Sections 3.2.1 and 3.2.2 that as the thickness of the film decreases, the peak in

the dielectric constant is suppressed to such an extent that for the very thinnest

films, there is very little temperature dependence of εeff . Therefore, it is justifiable

to approximate the room temperature data with that of 400 K for the ultrathin

films. Figure 4.10 clearly shows that the functional data adheres well to the series

capacitor model down to a thickness of d = 5 nm. This then would be the max-

imum total thickness of any ‘dead-layer’ within the dielectric, the permittivity of

which is calculated to be εi = 10. However, if one assumes that a dead layer exists

at each interface, and that the thickness of each layer is equal, then the thickness

of each layer would be 2.5 nm. A layer of this thickness would be highly visible

with TEM analysis, but investigation of the ultrathin films using TEM (figure 4.2)

shows no discernible microstructurally distinct layers, less than 2.5 nm thick, at

either interface This could imply that no visible, low permittivity dead-layer exists

within the BST, within the resolution limit of these images.

Figure 4.10: Comparison of the step model (solid line) with dielectric dataincluding ultrathin films. The thickness of the dead layer was assumed tobe di = 5 nm.

Having established that the ultrathin functional data adheres to the series

capacitance model, it becomes justifiable to include this data with the those of the

thicker films. Replotting this data using the series capacitor equation results in a

more accurate value of εb = 375± 15 and di/εi = 0.50± 0.06 nm. Although these

101

4.3 Electrode Field Penetration

Figure 4.11: Application of the series capacitor model to the completeseries of films.

values are more accurate, they unfortunately do not reveal any new information

pertaining to the nature of the interfacial capacitance.

This analysis has lowered the maximum total thickness of the the proposed

low permittivity dead layer from 7.5 nm, to 5 nm, implying that the thickness of

each interfacial dead-layer is of the order of less than 2.5 nm exhibiting a dielectric

constant of εi = 10. This thickness value is close to the thickness of the low

permittivity depletion region as suggested by Scott [72], which, may not make its

presence known in the microstructure, since it is a purely electronic phenomena.

There is another idea that the interfacial capacitance is not due to a region with

the dielectric, but instead is due to a thin layer of space charge within the electrode

[86, 79]. This is investigated in the remainder of this chapter.


The previous section has demonstrated that the origin of the interfacial capacitance

may not be due to a low permittivity ‘dead-layer’ within the dielectric, or if it is,

its total thickness must be ≤ 5 nm. In this section, the premise that the interfacial

capacitance is due to a thin layer of space charge within the electrode is investigated.

This space charge layer arises due to the imperfect screening of the applied electric

field within the metal, which contributes an additional capacitive component to

the thin film capacitors. This idea was first considered by Ku and Ullman [77],

to explain the results of Mead [80] and later was refined by Simmons [78], who

102


expressed the model of Ku and Ullman in analytical form. Dawber et al [86] have

implemented this model to successfully calculate the interfacial capacitance of the

SRO/BST capacitors presented in Chapter 3.

Black and Welser [79], have applied a similar electric field penetration model to

BST capacitors, but consider the screening of the field to be due to the underlying

ionic lattice, as opposed to a purely electronic screening mechanism. Similarly,

Hwang [85] has applied the same model to BST capacitors incorporating conduc-

tive oxide electrodes, and argues that since the underlying ionic structure of many

oxide electrodes are similar to that of high permittivity dielectrics, the electrode

capacitance will be significantly large. This clearly implies that the use of conduct-

ing oxide electrodes in thin film capacitors will suppress the thickness dependence

of the measured permittivity. Clearly, this contradicts Dawber et al [86], who state

that since conducting oxides are poor metals in comparison to elemental metals,

their ability to screen the penetrating field is much less, thereby implying that

the electrode capacitance will be smaller, thus increasing the suppression of the

measured permittivity of the thin film. The question then is which mechanism of

screening is correct.

This section introduces the model of Ku and Ullman, as modified by Sim-

mons, and demonstrates that the interfacial capacitance could be due to incom-

plete screening of the electric field within the electrode. The problem of which

screening mechanism to use (ionic or electronic) is also addressed, before finally,

the model is applied to the LSCO/BST capacitor system. As such, the theoretical

treatment adopted over the next sections has been extensively discussed by Daw-

ber et al [81, 86], Simmons [78] and Ku and Ullman [77]. This theory will be used

to model the interfacial capacitance of a LSCO/BST/Au system, and compared

with the experimental data.

4.3.1 Derivation of Series Capacitance

The derivation by Simmons [78] of the series capacitance, due to the incomplete

screening of the electric field within the electrode, is detailed as follows:

The charge Q induced on the surface of a capacitor by an applied voltage Va

is given by Q = CVa, where C in this case denotes the capacitance per unit area.

From Gauss’ law the charge can be related to the dielectric displacement D by

Q = D =εb(V2 − V1)

d, (4.3)

103


where, εb is the permittivity of the dielectric material of thickness d, and V1 and

V2 is the potential applied to the negative and positive sides of the ferroelectric

material, respectively. Thus the reciprocal capacitance can be given by

1

C=

d

εb

(V2 − V1)

Va

, (4.4)

or1

C=

d

εb

[1 +

Va − (V2 − V1)

(V2 − V1)

]. (4.5)

If there is no electric field penetration into the electrode surface, then (V2− V1) =

Va, and the capacitance is given by the geometrical capacitance C = εb/d. How-

ever, if the electric field does penetrate into the electrodes, then there will exist

a small finite region of space charge near the metal surface, which arises due to

the movement of charge carriers as the metal tries to screen the field. Therefore,

a fraction of the applied electric field will be dropped over this interfacial region,

resulting in V2 − V1 < Va, and hence a decrease in the measured capacitance. Ef-

fectively the second term in equation (4.5) can be regarded as the capacitance of

the electrodes, and thus the problem reduces to expressing V1 and V2 as a function

of the applied field Va.

Using degenerate Fermi-Dirac statistics, Ku and Ullman [77], derived the fol-

Figure 4.12: Energy diagram illustrating the distribution of potentials fora parallel plate capacitor exhibiting electric field penetration into the elec-trodes. All energies are measured with respect to the negatively biasedelectrode Fermi level (after Simmons [78]).

104


lowing simultaneous equations connecting V1 and V2 to Va:

v2 − v1 = K

[2

5(1 + v1)

5/2 − v1 −2

5

]1/2

, (4.6)

v2 − v1 = K

[2

5[1− (va − v2)]

5/2 + (va − v2)−2

5

]1/2

, (4.7)

where all the potentials are connected by a dimensionless parameter v = eV/EF

relating the potential to the Fermi energy EF , and K = εmd/εbL; εm and L being

the permittivity and characteristic screening length of the metal, respectively.

In their current forms, equations (4.6) and (4.7) are intractable to analytical

solution, unless one assumes the condition that the difference in electrostatic po-

tential between the surface and interior of the the electrode is much less than the

Fermi energy EF , i.e.

v1, (va − v2) << 1. (4.8)

Such an assumption permits the expansion of equations (4.6) and (4.7), which

upon neglecting higher than second order terms gives the simple expressions,

V2 − V1 =

√3

2KV1, (4.9)

V2 − V1 =

√3

2K(Va − V2). (4.10)

From equations (4.9) and (4.10) one can obtain the expressions for relative poten-

tial of each interface to be

V1 =2Va√

3K + 4, (4.11)

Va − V2 =2Va√

3K + 4, (4.12)

and henceVa − (V2 − V1)

V2 − V1

=4√3K

. (4.13)

Combining equations (4.5) and (4.6) and recalling that K = εmd/εbL, the mea-

sured reciprocal capacitance can be expressed as a function of the material param-

105


eters,

d

εeff

=d

εb

+4√3

L

εm

. (4.14)

Clearly the magnitude of the interfacial capacitance is directly influenced by the

depth of penetration of the electric field into the metal. An analytical expression

can be derived for the change of the electrode surface potential with the depth

of penetration of the field into the electrode. Implementing the dimensionless

parameter v, the spacial variance of the penetrating electric field can be expressed

in the form of Poisson equations

d2v

dr2=

v(r)

L2, (4.15)

andd2v

dr2=

v(r)− va

L2, (4.16)

where r denotes the distance from the surface of the electrode. The solutions to

equations (4.15) and (4.16) are obtained by considering the boundary conditions

v = v1 at r = 0 and dv/dr = 0 at r = ∞, and v = v2 at r = 0 and dv/dr = 0 at

r = ∞. They are found to be

V (r) = V1 exp

(−√

3

2

r

L

)

=2Va√

3K + 4exp

(−√

3

2

r

L

), (4.17)

and,

V (r) = Va − (Va − V2) exp

(−√

3

2

r

L

)

= Va

[1− 2Va√

3K + 4exp

(−√

3

2

r

L

)]. (4.18)

Inspection of equations (4.17) and (4.18) would suggest that a more appropriate

choice of characteristic penetration length would be ro = 2L/√

3. Substituting this

106


into equation (4.14) gives

d

εeff

=d

εb

+8

3

L

εm

. (4.19)

4.3.2 Calculation of Electrode Properties

In the previous discussion, the electrode capacitance was derived using electrostatic

potentials, and was shown to depend upon the characteristic penetration length

of the metal, as well as the dielectric constant of the electrode. Therefore, if one

can determine the values of these parameters, then the magnitude of interfacial

capacitance could be calculated. The derivation of the Thomas-Fermi screening

length as well as the mechanism for screening (i.e. electronic or ionic) was discussed

by Dawber et al [86], but is summarised here as follows:

In the Drude Free Electron Theory of metals, the ac dielectric constant and

conductivity of a metal are given by;

εm =iσ

ε0ωσ =

σ0

1 + iωτ, (4.20)

where, ω is the frequency of the applied field, and τ is the time between electron

collisions. The charge distribution within a metal ρ(z) can be described by three

key equations:

Poisson’s equation, which relates the charge distribution to the gradient of the

applied electric field,

ρ(z) = εmε0∂E(z)

∂z. (4.21)

The continuity equation which relates the charge distribution to the gradient of

the current density of the metal,

−iωρ(z) =∂j(z)

∂z. (4.22)

The Einstein transport equation, which relates the current density, and field gra-

dients to the charge distribution via a diffusion coefficient, D,

j = σE −D∂ρ

∂z. (4.23)

107


These equations can be effectively combined to give:

∂2ρ(z)

∂z2=

σ

ε0D

(1

εm

+iωε0

σ

)ρ(z), (4.24)

from which the characteristic screening length L, is defined as;

L2 =

(σ

ε0D

(1

εm

+iωε0

σ

))−1

. (4.25)

In the DC limit (ω → 0), which applies to the frequencies of this work, this length

will be the Thomas-Fermi Screening length;

L2 =

(σ

εε0D

)−1

. (4.26)

Equation (4.21) is particularly instructive, as it essentially states that if there

exists an electric field within the metal, then it must necessarily decay exponen-

tially over a distance defined by the screening length L, which is determined by

the conductivity of the metal. Therefore, one would expect the screening lengths

in metals such as Au or Pt to be quite small (L ∼ 0.5 A), but that those associated

with conducting oxides, which are poor metals, to be considerably larger (L > 1

A). Since the screening length of the metal defines the size of the interfacial ca-

pacitance (equation 4.19), conducting oxide electrodes would conceivably have a

lower capacitance, and hence have a more detrimental effect upon the measured

permittivity of the capacitor. This is in disagreement with work of Vendik et al

[75], Dittmann et al [90] and Hwang et al [85], who categorically state that the use

of conducting oxides actually reduces the degree of suppression of the dielectric

constant in thin films.

Black and Welser [79], and later Hwang [85], state that the reason for the

improved functional properties of thin film capacitors incorporating conducting

oxide electrodes, lies in the increased permittivity of the metal. Both groups

argue that the underlying ionic lattice contributes to the effective screening of the

applied electric field. This means, that the value used for εm in equation (4.26)

would have a magnitude of ε ∼ 1 − 10 for an elemental metal, but would be

considerably larger for conducting oxides. Hwang [85] argues that, since SrRuO3

has a perovskite structure similar to BaTiO3, the εm would conceivably be orders of

108


magnitude larger than pure metals. However, the analysis described above would

indicate that using a dielectric constant of the ionic lattice stripped of electrons, is

indeed erroneous. It is true that the penetration of the electric field into the metal

produces a finite region within the electrode from which electrons are pushed out,

but the correct dielectric constant is not that of an electron free metal, but rather

that of a free electron gas [86]. Thus the correct permittivity to use is the free

electron dielectric constant, i.e. εm = 1.

As detailed above the screening length is dependent upon the conductivity of

the metal, as described in equation (4.26). The magnitude of L can be calculated

by considering the following. The conductivity of a metal is given by,

σ =N0e

2τ

me

, (4.27)

where N0 is the free carrier density, and e and me are the charge and mass of an

electron respectively. The diffusion coefficient D of equation (4.26), is given by

D = l2/τ , where l is the mean free path length between electron collisions, and is

related to the Fermi velocity vF , by l = vF τ . The diffusion coefficient can therefore

be expressed as,

D = v2F τ. (4.28)

Combining equations (4.26), (4.27), and(4.28) gives,

L2 =

(N0e

2

mev2F εm

)−1

. (4.29)

Free electrons moving at the Fermi velocity, exhibiting three degrees of freedom

will have an Fermi energy of EF = 32mev

2F and hence,

L2 =

(3N0e

2

2EF εm

)−1

. (4.30)

Equation (4.30) now provides sufficient information to calculate the screening

length of each electrode. By determining the value of EF , one can determine

109


N0 (or vice versa) from the relation,

EF =3

2

~me

(3π2N0

)2/3. (4.31)

4.3.3 Application of Model

Assuming that the interfacial capacitance observed in the LSCO/BST/Au capac-

itors is due only to the penetration of the applied electric field into the electrodes,

the magnitude of the total di/εi can be calculated if either the free carrier density,

or the Fermi energy of the two electrodes are known. The derivation in the pre-

vious section assumed that the two electrodes consist of the same metal, but it is

trivial to modify equation 4.19 to account for the dissimilar electrodes,

d

εeff

=d

εb

+8

3

(LAu

εm

+LLSCO

εm

). (4.32)

Although the Fermi energy for Au is well known (EF = 5.53 eV [122]), there

is very little work pertaining to the carrier density, or Fermi energy of LSCO. Yin

et al [123] measured the carrier concentration to be 1 × 1021 cm−3, whereas Liu

and Ong [120] found a room temperature value of ∼ 1019 cm−3. These two carrier

concentrations, when substituted into equation (4.31) result in Fermi energies of

EF = 0.37 eV and EF = 0.017 eV, respectively. In comparison Prokhorov et al

[124] estimated the Fermi temperature to be TF = 3700 K, which corresponds to

a Fermi energy EF = 0.31 eV, and a carrier concentration of N0 = 8.2 × 1020

cm−3. It is important to note that Prokhorov is cautious about stating this value,

and suggests that this is an estimate of the order of magnitude of the TF , and

could therefore be interpreted as having a value between TF = 1000 − 10000

which corresponds to the respective values of EF = 0.086 − 0.86 eV and N0 =

1.15× 1020 − 3.63× 1021 cm−3. With such a broad range of values, it is a difficult

choice to know which value is most viable for the model, however it would not be

unreasonable to assume for this model at least, that LSCO is a good metal, and

attribute a values of EF ∼ 0.37 eV and N0 ∼ 1× 1021 cm−3. Substitution of these

values into equations (4.30) and (4.32), and using the value of εm = 1, results in a

value of di/εi = 0.47 nm.

The experimental value for the LSCO/BST/Au capacitors, incorporating the

ultrathin functional data, was determined to be di/εi = 0.50 ± 0.06 nm. Clearly,

the calculated value agrees extremely well with the experimentally determined

110


value. Of course, one must consider that when calculating this value, the LSCO

was assumed to be a good metal, and attributed values accordingly. One could

use the extremes of the values estimated by Prokhorov et al, which would result

in calculated di/εi = 0.41− 0.60 nm, which would still be in reasonable agreement

with the experimental value.

As a further example, the value of di/εi can be calculated for the SRO/BST/Au

capacitors presented in Chapter 3, similar to Dawber et al [81]. By using a Fermi

energy EF = 2.0 eV for SrRuO3, di/εi = 0.36 nm is obtained, which again, would

agree very well with the measured 400 K, 10 kHz experimental value, di/εi = 0.40±0.05 nm. However, there is one problem with this result. In Chapter 3, the 400 K,

10 kHz value for the extracted real interfacial component was indeed found to be

1/K ′ = 0.40± 0.05, as reported by Sinnamon et al, but the interfacial component

was observed to be temperature and frequency dependent, particularly for T >

300 K. This behaviour was attributed to a thin layer of oxygen vacancy related

defects, superposed upon a temperature and frequency independent background

component. So to test the validity of the field penetration model, the thermally

activated behaviour must be subtracted from the extracted interfacial component.

The extracted di/εi for the background component of the SRO/BST/Au system

was found to be 0.45 < di/εi < 0.53 nm (depending on the slight temperature and

frequency dependence) which is much greater than the calculated value of 0.36

nm. However, as observed by Sinnamon et al [89, 109], these films exhibit strain

coupling from the interface, which could reduce the permittivity of the entire film

(see Section 1.3.3). Since the misfit strain is observed to relax with increasing film

thickness, it could appear as an interfacial capacitance [90], and thus modify the

apparent interfacial capacitance associated with field penetration.

As a final test of this model, the fraction of the applied field that is dropped

across the film, and each interfacial region is calculated and compared with the

fractional decrease of the films’ permittivity εeff , normalised by the extracted bulk

permittivity εb. This is simply determined by the equations (as given by Simmons)

V1

Va

=2K2

2K1 +√

3K1K2 + 2K2

, (4.33a)

V2

Va

=2K1

2K1 +√

3K1K2 + 2K2

, (4.33b)

where Va is the applied voltage, V1 and V2 is the potential dropped across each

111

4.4 Summary

Figure 4.13: Calculated fraction of the applied potential dropped acrossthe BST film and the interfacial regions as function of BST thickness (solidlines). A comparison of the fractional decrease of the measured permittivityεeff/εbulk (diamonds) shows a close correlation to the fraction of potentialdropped within the BST film.

electrode interface, and K1,2 = (2dεm)(3L1,2εb). The fraction of applied voltage

dropped across the film is Vd = Va−(V1+V2). Figure 4.13 illustrates the calculated

fraction of applied voltage dropped across each of the capacitors as a function of

film thickness, and compares the proportion of potential dropped across the film

with the ratio of εeff /εb. There is a clear correspondence between the fractional

decrease of the permittivity and the fractional decrease in potential dropped within

the film.

On the whole, the field penetration model as the origin of the series capaci-

tance would appear to be quite compelling. However, one must be cautious when

testifying that it is the only cause since, as Chapter 3 illustrates, the interfacial

capacitance can have many contributing factors.

4.4 Summary

Successful fabrication of ultrathin capacitors, exhibiting reliable functional charac-

teristics, has permitted the extension of the series capacitor model to the ultrathin

regime. It was found that there was no observed deviation of the dielectric data,

from the series capacitor model, down to a thickness of 5 nm. This therefore rede-

fines the maximum total thickness of the proposed low permittivity dead-layer to

be di < 5 nm. TEM analysis of the ultra thin capacitors show no discernible mi-

112

4.4 Summary

crostructurally distinct interfacial layers, implying that the interfacial capacitance

may not be located within the dielectric.

Instead the interfacial capacitance could be due to a thin space charge region

within the electrodes, formed by the incomplete screening of the electric field. A

model, based on the electronic screening of the applied field successfully predicts

a value of di/εi = 0.47 nm which is in close agreement to the experimental value

of the interfacial capacitance of di/εi = 0.50± 0.06 nm.

113

Chapter 5

Phase Transitions in Thin Film

Barium Titanate

As detailed in Chapter 1, the strain field induced within a thin BaTiO3 film due

to the mismatch of material lattice constants, will induce a change in the expected

phase transition order. In particular the theoretical models of Pertsev et al and

Dieguez et al [18, 96] predict a new ‘exotic’ rhombohedral or monoclinic phase

(the so-called ‘r -phase’) within the thin film under zero misfit strain, i.e. a perfect

lattice match. So far, there has been a plentiful volume of experimental work on

systems incorporating compressive in-plane strains [51, 89, 99], but there has been

little or no investigation of the zero strain system, or the tensile strain system,

due in part to the difficulty in using substrates with similar, but larger lattice

constants than that of the overlying film. In this chapter, the structural phases

of the ‘zero-strain’ system (La,Sr)CoO3/BaTiO3, is investigated using functional

characterisation, and high resolution X-ray diffraction.

5.1 Introduction

A free bulk BaTiO3 crystal exhibits a cubic perovskite structure above T ∼ 400

K, which progressively goes through a series of structural phase transitions, as

the temperature of the crystal is reduced (figure 5.1), with each phase transition

accompanied by a reorientation of the spontaneous polarisation with respect to

the crystallographic axes.

However, Pertsev et al [18] have calculated using LGD theory, that a thin

BaTiO3 film that is mechanically clamped to a thick substrate, will exhibit a dif-

114

5.1 Introduction

Figure 5.1: Top Temperature evolution of cell parameters for a free bulkBaTiO3 crystal [112] illustrating the structural phase transitions. BottomThe direction of the spontaneous polarisation for each structural phase, withrespect to the crystallographic axes.

ferent order of structural phase transitions to that of bulk, due to the misfit strain

imposed upon the film through differences in the in-plane lattice constants of the

substrate and overlying thin film (figure 5.2(a)). Probably the most surprising

result of this work, is that even at zero misfit strain, the mechanical boundary

conditions cause the thin film to exhibit a different order of phase transition than

that observed in bulk. As the temperature of the crystal increases from 0 K,

Figure 5.2: Calculated misfit phase diagrams of a) Pertsev et al [18] andb) Dieguez et al [96].

115


the crystal structure transforms from an orthorhombic (ac) phase to a rhombohe-

dral/monoclinic (r) phase to finally a cubic (p) phase, with an apparent eradication

of the tetragonal phase.

Dieguez et al [96], have found a similar misfit phase diagram for BaTiO3 using

ab initio calculations (figure 5.2(b)), but theoretically determine that the r -phase

is energetically more favourable than the ac-phase at zero misfit strain, for all

temperatures below TC .

Experimentally there has been no investigation of phase transition order in

a zero strained system. This is due primarily in the difficulty of finding cubic

substrates that share the same in-plane lattice constant of BaTiO3. However, one

could assume that if the difference in the two materials’ lattice constants are large

enough, then the misfit strain cannot be supported within the overlying film, and

will rapidly relax close to the interface. This would mean that a BaTiO3 thin

film experiencing such an interface would be predominately free from the influence

of misfit strain. This has been observed in the LSCO/BST system investigated

in Chapters 3 and 4 [111], where the BST room temperature out-of plane lattice

parameter was found to be identical to the bulk value for all films ∼ 100 nm or

thicker. Therefore, since BaTiO3 has an even greater lattice constant than that of

BST, one could assume that the it too would would experience a zero misfit strain

when deposited upon LSCO electrodes. Essentially, the nature of the interface is

incoherent when the lattice mismatch is large enough.


The measurement of the functional properties of a material can be extremely useful

in identifying phase transitions. Often when a ferroelectric material transforms

from one structural phase to another, the magnitude of the permittivity of the

material is observed to change also, particularly at the ferroelectric to paraelectric

transition temperature. This is very easily observed in bulk ceramic materials, but

can be quite obscured within thin films. However, Lookman et al [111, 125] have

demonstrated that if a field larger than the coercive field of the material is applied

at a low temperature, then upon heating, anomalies associated with structural

phase transitions appear or become more pronounced in the functional response.

116


Figure 5.3: left) Typical frequency and temperature dependence of thecapacitance and, right) dielectric loss of the ‘virgin’ BaTiO3 capacitors.Dashed lines indicate the temperature of bulk structural phase transitions.

5.2.1 Capacitance Results

Figure 5.3, shows the functional response with varying temperature and measure-

ment frequency typical of all the LSCO/BaTiO3/Au capacitors immediately after

deposition of the Au electrodes. The thickness of each BaTiO3 was approximately

175± 25 nm, as calculated from the PLD deposition rate. In this figure, the solid

lines indicate the approximate temperatures at which one would expect the struc-

tural phase transitions in a free bulk crystal. This so-called ‘virgin’ measurement

is characterised by a broad plateau-like anomaly in the capacitance, which could

be indicative of a diffuse phase transition. However, it may seem more likely that

this is not a single anomaly associated with a single phase transition, but instead

could be a combination of two peaks originating from two separate phase tran-

sitions occurring at ∼ 300 K and ∼ 350 K. There would certainly seem to be a

suppression of the peak that is associated with the tetragonal-cubic transition at

∼ 400 K, or it could be that this peak has been shifted to the lower temperature

of ∼ 350 K.

The dielectric loss of this film is shown on the right of figure 5.3. The loss

is typically below tan δ ∼ 0.05 for frequencies < 100 kHz, except for a marked

increase in the 100 Hz value for higher temperatures, due to space charge within

the film. The increased values of tan δ at 100 kHz is due to a large series resistance,

leading to an increase in time constant of the circuit, τ = RC, which is comparable

to the measurement frequency, as discussed in Chapter 4.

This data clearly shows a migration of a low temperature loss peak to higher

117


Figure 5.4: left) Typical frequency and temperature dependence of thecapacitance and, right) dielectric loss of the ‘non-virgin’ BaTiO3 capacitors.Dashed lines indicate the temperature of bulk structural phase transitions.

temperatures as the measurement frequency is increased. This relaxation be-

haviour is discussed in greater detail in section 5.2.2., and it is likely that it is

related to the increased dispersion observed in the capacitance below 300 K. A

small shoulder at T ∼ 350 K can be observed, particularly at the highest frequen-

cies, which is likely to be associated with the anomaly observed in the capacitance

at approximately the same temperature. As highlighted by Lookman et al [125]

and Rios et al [126], non-dispersive loss peaks can correspond to structural phase

transitions, and therefore this observed anomaly could be associated with a change

of structure at ∼ 350 K.

After these measurements, the film was immediately cooled from 500 K to 80

K, and the functional response was measured again, exactly as before. Figure 5.4,

illustrates the results of this so called ‘non-virgin’ measurement. The behaviour of

the capacitance as a function of temperature has changed dramatically. Although

an anomaly still appears at ∼ 300 K, the one that appeared at ∼ 350 K has

disappeared, and there is now a broad peak observed at ∼ 410 K. This latter

peak could be associated with the tetragonal-cubic phase transition, whilst the

former 300 K peak could be due to the orthorhombic-tetragonal phase transition

as observed in bulk.

The dielectric loss however, has changed only slightly. There still exists the

same migration of the loss peak with increasing frequency, but the the shoulder

that was previously observed at ∼ 350 K has disappeared, and a new peak at

∼ 400 K has appeared (again most noticeable in the higher frequencies), which

118


Figure 5.5: left) Typical frequency and temperature dependence of the ca-pacitance and, right) dielectric loss of the ‘poled’ BaTiO3 capacitors. Dashedlines indicate the temperature of bulk structural phase transitions.

occurs at ∼ 410 K, close to the expected temperature for the bulk tetragonal-cubic

phase transition.

Finally, the sample was again cooled from 500 K to 80 K. This time, before

beginning the temperature run, the sample was ‘poled’, using a 1 ms cycling 50

V peak voltage from the Radiant Precision work station. Upon poling, the ca-

pacitance and dielectric loss were measured as before, the results of which are

presented in figure 5.5. The behaviour of the capacitance is very similar to the

previous non-virgin run, displaying clear anomalies at 300 K and 410 K, only now

there appears a small shoulder at T ∼ 130 K. Inspection of the dielectric loss shows

the same similar behaviour, with again the appearance of the 130 K anomaly.

As previously observed, non-dispersive loss peaks can be associated with struc-

tural phase transitions, one could associate this peak with the rhombohedral-

orthorhombic phase transition that is observed in bulk. However, the measured

capacitance of another similarly poled BaTiO3 film shows quite clearly (figure 5.6)

the appearance of this 130 K peak, as well as another low temperature peak at

T ∼ 190 K, which is more likely to be related to the rhombohedral-orthorhombic

transition.

It is worth reiterating that after a virgin film has been functionally charac-

terised, any subsequent functional measurements do not reproduce the same be-

haviour, but will be identical to any other subsequent measurement. The reason for

the observed difference in behaviour of the virgin and non-virgin runs is discussed

in section 5.3.2.

119


Figure 5.6: Frequency and temperature dependence of the capacitance of apoled BaTiO3 film. Dashed lines indicate the temperature of bulk structuralphase transitions.

5.2.2 Relaxation Analysis

In the previous section, the dielectric loss of the film was found to display a distinct

relaxation behaviour in the low temperature loss peaks, which is observed to be

identical in the virgin, non-virgin, and poled measurement regimes. This behaviour

can be attributed to the dielectric relaxation of defects within the film. Often this

behaviour is observed to be a thermally activated process,

ω = ω0 exp

(−EA

kT

), (5.1)

where, ω is the frequency of the applied field, EA is the activation energy of the

relaxing species, T is the temperature and k is the Boltzmann constant. Thus, by

measuring the temperature at which the loss peak is maximum for each frequency,

it is possible to determine the thermal activation energy EA, of the relaxing species,

via an Arrhenius plot of ln(ω) versus 1/T . Figure 5.7, shows the Arrhenius plot

for twenty five frequencies within a frequency range of 100 Hz to 50 kHz. The

reciprocal of the gradient of the best fit line through this data gives an activation

energy of EA = 0.391± 0.003 eV.

It is often convenient to blame any defect related mechanism on oxygen vacan-

cies, which are common defects within perovskite materials. However, it is clear

that in this case, the movement of oxygen vacancies is not the culprit, since the

activation energy for oxygen vacancy migration is ∼1.1 eV [114].

The value of EA ∼ 0.4 eV has been obtained by many research groups, and

120


Figure 5.7: Arrhenius plot of the dielectric loss peaks. The extracted acti-vation energy is EA ∼ 0.4 eV

has been attributed to many mechanisms. Ang et al [127], obtained a value of

EA = 0.32 − 0.49 eV for Bi doped SrTiO3 ceramics, which they attribute to a

dipolar interaction between the off-centre Bi and Ti ions and thermally activated

conduction of electrons from oxygen vacancies. However, Morii et al [128] point

out that Ti4+ always exhibits an activation energy of 0.20− 0.50 eV within a wide

range of materials. Recently Jung et al [129] obtained EA = 0.38±0.02 eV in PZT,

and have quoted Suchaneck et al [130] who found a value of 0.36 eV for enthalpy

of migration of Ti vacancies, based indirectly on oxygen vacancy migration during

fatigue. Finally, the work of Bharadwaja and Krupanidhi [131] found a value of

0.36 eV from ac conductivity studies of PZT, whereas the dc conductivity studies

of by Sudhama et al [132] gave EA = 0.35 eV for a bulk-limited hopping model.

However, as noted by Robertson and Warren [133], there is a Pb3+ defect in PZT

which is believed to have a value of 0.3 eV.

It is apparent that the chemical nature of the species involved in the relaxation

in the present work, is by no means certain. Indeed, as noted by Scott [15], only

a few traps have been unambiguously identified in these materials.

5.2.3 Depolarisation Current

A useful technique for investigating phase transitions relies on the exploitation of

the pyroelectric effect. Ferroelectrics belong to the pyroelectric class of materials

in which the spontaneous polarisation can be changed by changing the tempera-

121


ture of the crystal. The spontaneous polarisation induces screening charge on the

surfaces of the crystal, which changes with the alteration of polarisataion, therefore

producing a flow of current within a closed circuit since,

i =dQ

dt= A

dD

dt= A

dD

dT

dT

dt= Apx dT

dt. (5.2)

Equation (5.2) relates the current i that flows through the film when the po-

larisation (which is related to the displacement D) changes with temperature T .

The quantities of A, px and dT/dt are the area, pyroelectric coefficient, and rate

of change of temperature respectively.

The depolarisation current for the BaTiO3 films in this work was measured

using a Kiethly electrometer. The film was first cooled to 80 K, and then poled

at this temperature using a dc voltage of ∼ 35 V, before being heated to 450 K

using a rate of temperature change dT/dt = 8 Kmin−1. This poling was found

to be necessary to increase the magnitude of the depolarisation current, and it is

important to note that the applied voltage was removed before heating.

Figure 5.8, shows the typical depolarisation current of the BaTiO3 films. There

are clear anomalies (as indicated by the dashed lines) at temperatures of 130 K,

180 K, 280 K, and 410 K. The latter three temperatures correspond very well to the

phase transition temperatures expected in bulk ceramics (figure 5.1), however there

Figure 5.8: Magnitude of the depolarisation current as a function of tem-perature. Three anomalies correspond closely with bulk-like structural phasetransition temperatures rhombohedral-orthorhombic (R-O) orthorhombic-tetragonal (O-T) and tetragonal-cubic (T-C).

122


is no known phase transition in BaTiO3 associated with the the 130 K anomaly.

The origin of this anomaly is unknown, but its appearance in only the poled non-

virgin capacitance data (figures 5.4 and 5.5), as well as the depolarisation current

measurements would suggest that it could be associated with the application of a

large electric field.

It should be noted that for the sake of clarity, figure 5.8 shows only the magni-

tude of the depolaristaion current and not the direction. In reality, the current is

observed to become negative (with respect to the low temperature bahaviour) at

T ∼ 350K, and continues to remain negative up to 450 K. This would imply that

the the direction of the spontaneous polarisation for T > 350 K is in the opposite

direction to that of T < 350 K. Above 350 K the current increases rapidly with in-

creasing temperature up to 410 K, at which point, it collapses to almost zero. This

is highly indicative of a first order phase transition, which occurs at TC ∼ 400 K in

bulk BaTiO3. The temperature at which this phase transition occurs in the thin

film, may be slightly larger due to the ∼ 1 V internal bias of the film, caused by

the mismatch in work functions of the top and bottom electrodes. Alternatively,

the difference in the observed transition temperature could be due a thermal lag

in the sample, due to the large heating rate of 8 Kmin−1.

The depolarisation current measurements indicate that there is a sequence of

three phase transitions in these BaTiO3 films which occur at the same temperature

as one would expect in a bulk ceramic. This observation is consistent with the

results of non-virgin capacitance measurements, with the 190 K anomaly in the

latter only appearing when the sample is poled at 80 K. This is in stark contrast

to the phase diagram of Pertsev et al [18] and Dieguez et al [96], in which the

tetragonal phase is suppressed at zero strain. Furthermore, inspection of these

phase diagrams indicates that, at no one strain value can there be three phase

transitions, the maximum possible number being two.

5.2.4 Polarisation Hysteresis

BaTiO3 is a ferroelectric material, and thus should display a hysteresis loop when

cycled with an electric field whose magnitude is greater than its coercive field.

Figures 5.9(a) and 5.9(b) show the results of hysteresis measurements performed

on a non-virgin BaTiO3 film at 80 K and 300 K respectively, using the Radiant

Precision workstation. Clearly the film exhibits strong ferroelectricity at 80 K

with a remnant polarisation of ∼ 4 µC/cm2, and a coercive voltage of ∼ 10 V.

123

5.3 XRD Structural Phase Determination

However, the room temperature measurement shows a dramatic reduction in the

coercive voltage, and looks similar to a non-linear dielectric.

The reason for such a change is due to the temperature at which the hysteresis

measurement is recorded. Recall that figure 5.8 showed a phase transition at

∼ 280 K. At a structural phase transition, the lattice becomes less rigid, and hence

susceptible to external stimulus. Therefore the film would require a smaller electric

field in order to switch the direction of the polarisation, creating a hysteresis loop

similar to that observed in 5.9(b).

Figure 5.9: Polarisation hysteresis loops measured at a) 80 K and b) 300K


Another way to determine the structural phase of the BaTiO3 thin films is by

direct measurement of the film lattice parameters using XRD.

Using a razor blade, the surface of the BaTiO3 target was scrapped away, and

the resultant powder mounted onto a glass slide with the aid of a thin film of

vaseline. XRD was then used to investigate the structural form of this powder,

and hence the target. Figure 5.10 demonstrates that the target used in this in-

vestigation displayed a room temperature tetragonal structure, with the lattice

parameters determined from the 002 type reflections as 5.10(inset) 4.03 A and

4.00 A ±0.01 A, consistent with the accepted bulk values of 4.036 A and 3.993 A.

Figure 5.11 shows the out-of-plane peaks, typical of the LSCO/BaTiO3/Au

capacitors. The capacitors are highly oriented with MgO001‖LSCO001‖BTO001,

and exhibit little or no mixed orientations. A φ scan (figure 5.12) of each layer of

124


Figure 5.10: Powder diffraction of the BaTiO3 target used to deposit thefilms studied. inset. 002 type reflections demonstrating room temperaturetetragonality.

the capacitors shows that the growth mechanism is effectively ‘cube on cube’. The

out of plane lattice parameter for the particular BaTiO3 film shown in figure 5.11

was calculated to be 4.011 A, but other films gave values of 4.007 A and 4.004 A,

which are all much less than the c-axis lattice parameter of bulk.

Figure 5.11: Typical out-of-plane diffraction reflections (00l) of thin filmcapacitors.

125


Figure 5.12: φ scans of the 022 type reflections for each layer, demonstrat-ing ‘cube on cube’ capacitor growth.

126


5.3.1 Synchrotron XRD

The synchrotron diffractometer in HASYLAB of DESY, Hamburg, was used to

directly measure the lattice constants of a BaTiO3 film. The advantage of using

this equipment lies in the brightness of the synchrotron source, and in the adapt-

ability of the XRD apparatus. This allows for extremely high resolution and direct

measurement of both out-of-plane and in-plane lattice parameters, the latter using

grazing angle incidence.

Figure 5.13: In-plane 3D ω − 2θ contour map around the 200 reflection.

Figure 5.13 shows an in-plane ω − 2θ scan around the 200 LSCO and BaTiO3

peaks, with the BaTiO3 being centred at ω = 0, with the horizontal dashed

lines representing the 2θ values for the LSCO and BaTiO3 peaks. This three

dimensional contour map is characterised by a sharp peak at ω = 0 corresponding

to the BaTiO3, but strangely there is absolutely no evidence of a LSCO peak at

this same ω value. There are however two broad ‘ω-ridges’ separated by a ‘2θ-

valley’ centered at ω = −3. The 2θ values of these ridges correspond to those of

the LSCO and the BaTiO3 layers.

This strange behaviour would seem to suggest the sharp BaTiO3 peak has

been twisted by approximately three degrees with respect to the LSCO peak, in

the plane perpendicular to the film surface. This could imply that during the

deposition of the BaTiO3, the film tries to grow epitaxially on the LSCO, but that

a short distance from the interface, possibly to minimise the interfacial strain, the

BaTiO3 experiences a twist, after which the epitaxial growth of the BaTiO3 layer

continues.

127


Figures 5.14(a) and (b) show the diffraction peaks obtained from the 003 and

200 reflections respectively. There is clearly only one peak observed for the 003

reflection, whereas there is a distinct shoulder present in the 200 reflection, which

can be interpreted as a secondary peak. The solid lines in figures (a) and (b),

represent a Lorentzian curve fit to the experimental data, from which the lattice

parameters of the film were calculated to be c = 4.011 A, b = 4.020 A and a = 4.002

A, with the error in each measurement of ±0.003A. Landolt-Bornstein [112] gives

the structure of BaTiO3 at T = 263 K as orthorhombic with pseudocubic lattice

constants of c = 4.018 A, b = 4.009 A, and a = 3.99 A. This would imply the

structure of this BaTiO3 thin film is orthorhombic at room temperature, with the

c-axis in plane, and b-axis out of plane, as defined with respect to the tetragonal

polar axis.

Figure 5.14: Synchrotron high resolution XRD for the a) 003 reflection,and b) 200 reflection. The solid lines indicate Lorentzian fits to the data.

These measurements have been independently verified on a similar BaTiO3

film, by PANanalytical using a X’pert XRD capable of grazing angle incidence

in-plane measurements. PANanalytical found c = 4.007 A, b = 4.012 A, and

a = 4.004 A, where the c-axis in this case is defined as the direction perpendicular

to the film surface.

The observation that these ‘zero-strain’ BaTiO3 films possess orthorhombic

structures at room temperature permits the following conclusions. Either:

a) The models of Dieguez et al and Lai et al are in error, as they both state that

the orthorhombic phase is energetically unfavourable at all temperatures,

and hence the phase diagram of Pertsev et al is correct.

OR

128


b) The misfit strain phase diagrams, which are calculated from the equilibrium

state, do not accurately describe real ferroelectric films which can be sub-

jected to kinetic processes.

OR

c) An observed orthorhombic phase at room temperature is not that surprising

since the bulk orthorhombic-tetragonal phase transition occurs at T ∼ 280

K. Therefore the film may behave as bulk (as indicated previously by the

functional data), thus implying that the zero strain state imposed upon the

film from a totally coherent interface, differs significantly from the zero strain

state of an incoherent interface.

5.3.2 Discussion of Results

In the previous sections, functional characterisation reveals that non-virgin BaTiO3

thin films exhibit the same structural phase transitions as its bulk ceramic counter-

part, which is in direct contradiction to the models of Pertsev et al [18] and Dieguez

et al [96]. Also there is a distinct change in the functional behaviour between the

virgin capacitance measurements and subsequent non-virgin measurements. This

change of behaviour could actually be caused by the presence of the internal bias

field originating from the asymmetric electrodes.

Consider a thin film that has just cooled to room temperature after a high

temperature deposition. If it is assumed that this film displays bulk like phase

transitions, then at TC a spontaneous polarisation will appear. It is conceivable

that this polarisation will present itself in randomly orientated domains, but will

display a net polarisation in a direction parallel to the surface of the film, as this

would be energetically more favourable in the thin film. This is confirmed by the

observation that the longer c-axis, which is the polar axis, is oriented parallel to

the surface.

When the film cools to room temperature, Au top electrodes are deposited

whilst the film is in the orthorhombic phase. This will now impose a ∼ 1 V

potential across the film perpendicular to the surface of the film. The magnitude

of this potential, even across a thin film, may not be large enough to cause a

reorientation of the net polarisation to a direction perpendicular to surface, and

as such, the dominant polar axis will remain in-plane. The film is now cooled to

80 K, after which functional data is recorded whilst the film is heated to 500 K.

129

5.4 XRD Temperature Investigation

Since the polar axis is parallel to the film surface in the tetragonal phase, and

since the measurement of the permittivity is perpendicular to the film surface,

the dielectric anomaly associated with the tetragonal-cubic phase transition may

appear suppressed in the virgin measurements.

However, upon cooling the film from 500 K, back to 80 K, the volume directly

beneath the electrodes is effectively field-cooled due to the presence of the internal

bias field, which could encourage the formation of domains perpendicular to the

film surface to form. Thus, when the permittivity of the film is measured upon

heating, the presence of the out-of-plane orientated domains may result in the

appearance of the anomaly associated with the tetragonal-cubic phase transition.

Also, since the internal bias is always present, then any subsequent functional

characterisation, will produce results similar to a non-virgin film. Only by negating

the internal bias, could it be possible to return the film to its virgin state.


The previous sections have demonstrated that the functional behaviour of non-

virgin BaTiO3 thin films exhibit anomalies at temperatures close to the phase

transition temperatures expected in bulk BaTiO3 ceramics. The behaviour of the

vigin films however, is completely different, exhibiting an anomaly at T ∼ 350 K,

and an apparent suppression of the tetragonal-cubic permittivity peak at T ∼ 400

K. The cause of this change in behaviour has been attributed to the internal

bias field across the film, due to the asymmetric electrodes, and as such should

only cause this behaviour change in the volume directly under the electrodes.

Therefore, XRD was used to investigate the the structure of the film as a function

of temperature.

5.4.1 Measurement of Temperature Dependence of Struc-

tural Behaviour

The structure of the BaTiO3 films was investigated using XRD to measuring the

out-of-plane lattice parameter as a function of temperature. To heat the film, the

sample was attached to a Peltier heater stack using silver paste, which in turn was

mounted on a large copper disc to provide a large thermal mass. The temperature

of the sample was monitored using a K-type thermocouple thermally bonded to

the surface of the Peltier stack, close to the film being investigated. Using this

130


method, the lattice parameter could be investigate within an effective temperature

range of ∼ 270−420 K, the limitations being dictated by the formation of ice on the

film surface at low temperature, and the melting of the Peltier electrical contacts

at high temperature.

Using the Bragg equation, the out-of-plane lattice parameter of the BaTiO3

was calculated from the peak position of a Gaussian fit to the data. The fits to the

data were typically very good, especially around the peak at lower temperatures,

but were observed to become slightly worse at high temperatures, resulting in the

fit that gave a higher 2θ value than the peak in the data. The intensity of this

peak position was also recorded.

5.4.2 Temperature Dependence of Structural Properties

Figure 5.15(a) illustrates the temperature dependence of the parameters extracted

from monitoring the 002 BTO reflection. There is a clear, sudden increase in the

lattice parameter at ∼ 290 K after which there is a gradual fall off, until at T ∼ 350

K it begins to rise again in an approximately linear manner, with the smallest hint

of a change at T ∼ 390 K. A similar behaviour is observed in the peak intensity

(figure 5.15(b)), with a sudden decrease in intensity at ∼ 290 K, and also at ∼ 390

K, (there is little change in behaviour at ∼ 350 K). The monitoring of the peak

intensity can be highly informative in investigating the structure of materials, as

it can give a qualitative indication of a change in structural phase associated with

a structure factor change.

Figure 5.15: Temperature dependence of a) the out-of-plane lattice param-eter, and b) peak intensity. The dashed lines indicate anomalies which maybe due to phase transitions

131


An important point to note is that the slit used at the x-ray source is 6 mm

wide, and therefore, the majority, if not the whole of the BaTiO3 film is illumi-

nated, and thus the diffracted data is from the whole of the sample. Since the

electrodes cover a small fraction of this area (∼ 1%), then one can assume that

the area investigated will be identical to a virgin film (assuming that the changes

in the observed functional behaviour in sequential thermal cycles was really due

to internal bias from the electrodes). Indeed, comparison of the measured out of

plane lattice constant with that of the measured virgin capacitance (figure 5.16

shows that the changes in the lattice parameters coincide with the anomalies ob-

served in the capacitance at ∼ 290 K and ∼ 350 K. Also, the measurement of the

peak intensity would seem to suggest that the 290 K anomaly is associated with a

structural phase transition, possibly the orthorhombic-tetragonal transition, but

that the 350 K anomaly may be due to a change of lattice behaviour, but may not

be associated with a distinct structural phase transition.

One must note that there appears to be a structural phase transition at T ∼ 390

K, which occurs at the approximate temperature of the expected bulk tetragonal-

cubic phase transition. It cannot be concluded at this point whether this transition

occurs in just the non virgin film of the volume under the electrodes, or if it is

from the whole of the film.

Figure 5.16: Comparison of the behaviours of the out-of-plane lattice pa-rameter with the capacitance of the virgin film. The anomalies in the ca-pacitance would seem to coincide with structural anomalies.

132


5.4.3 Influence of Apparatus on Structural Measurements

It is possible that the observed behaviour of the XRD data could be an artifact from

the heating of the apparatus. To check this, the lattice constant and peak intensity

of each layer of a similar capacitor structure was investigated simultaneously, with

the results presented in figure 5.17. It is apparent that the properties for the

BaTiO3 layer are very similar to that observed in the previous sample, sharing

the same linear increase in the lattice parameter above 350 K, and a change in

the peak intensity at 390 K. The LSCO layer on the other hand demonstrates

a completely linear expansion in the lattice parameter, and a relatively constant

value of peak intensity with increasing temperature, whilst the MgO substrate also

shows a similar linear expansion behaviour in its lattice constant, but a constant

decrease in the peak intensity with increasing intensity. Overall, these results

would suggest that the thermal properties of the apparatus have little influence

on the observed behaviour of the BaTiO3 film, since the other layers demonstrate

linear expansion with no apparent structural phase changes.

5.4.4 Room Temperature Phase Determination

The room temperature lattice parameters for this BaTiO3 film were determined

from the 002 and 202 type reflections, and were found to be c = 4.004 A and

a = b = 4.011 A. This is identical to the results obtained by Tenne et al [134], for

thin film BaTiO3/SrRuO3 structures on both LaAlO3 and SrTiO3 substrates. They

attribute this ‘orthorhombic’ phase, in which the in-plane lattice parameters are

larger than the out-of-plane lattice parameter, to a tensile in-plane strain imposed

upon the film from the SrRuO3 buffer layer. This is a bold assumption, since the

conventional belief is that the smaller lattice constant of the SrRuO3 (a = 3.93 A)

should impose a compressive in-plane strain on the film, thus elongating the out-of-

plane axis. Tenne et al acknowledge this, but state that the tensile strain originates

from the difference in the thermal expansion coefficients of the two layers, with

the coefficient of BaTiO3 being larger than that of SrRuO3. They use phase-field

modelling to obtain a domain stability map (figure 5.18) to show that for tensile

strain of 0.58%, the room temperature structural phase will be orthorhombic with

the polarisation direction in-plane. Also, they estimate that as the temperature

of the film decreases, the magnitude of the thermal strain increases (dashed line

in figure 5.18) and hence there is no observed phase transitions, a theoretical

result which they verify using Raman spectroscopy within a temperature window

133


Figure 5.17: Comaparison of the out-of-plane lattice parameters and peakintensities of each layer within the capacitor structure. The behaviour of theLSCO and MgO layers suggest that the anomalies observed in the BaTiO3

films are not induced by the apparatus.

of 5− 315 K.

There is a degree of similarity between the room temperature structures of the

films studied by Tenne et al, and the ones presented in the current work. Tenne et

al fail to observe an elongation in the out-of-plane lattice parameter, which would

134


Figure 5.18: Domain stability map for BaTiO3 under biaxial strain. Thestars indicate the strain at room temperature, measured using XRD, andestimated at 5 K. After Tenne et al [134].

be expected due to compressive in-plane strain at the BaTiO3/SrRuO3 interface,

which could suggest that their interface is incoherent. Yanase et al [135], have

demonstrated that BaTiO3 can be successfully grown on SrRuO3, with the former

exhibiting a highly elongated out-of-plane lattice parameter, suggestive of an in-

plane compressive strain from the lower electrode. Therefore the films of Tenne et

al could have an assumed zero misfit strain condition, similar to the LSCO/BaTiO3

system of this work.

If the LSCO electrode has a similar thermal expansion coefficient to that of

SrRuO3, then one would expect to see an absence of the low temperature phase

transitions in the LSCO/BaTiO3 system within the 5−315 K temperature window

that Tenne et al probed. Clearly this behaviour is not observed in the system

with LSCO, since a distinct phase transition at 290 K is detected in the functional

measurements of both virgin and non-virgin films, and in the data obtained from

x-ray diffraction.

135


5.4.5 Effect of Internal Bias on Structural Properties.

It has been suggested that when the BaTiO3 films are deposited, the polarisation

axis will form in-plane, but that when Au electrodes are deposited, and the film

is effectively field cooled from above TC to low temperatures, a polarisation per-

pendicular to the film surface may form, provided by the incentive of the small

internal bias from the asymmetric electrodes. However, this effect should only

occur in the volume directly under the electrodes, and as such may be masked by

the larger volume of the virgin film when investigated using XRD. Therefore, to

investigate any potential effect that this field may have on the observed structure,

a BaTiO3 film, identical to before, was depostited upon a LSCO electrode, and

the structural behaviour measured without the application of Au electrodes. After

this, the sample was removed from the Peltier stack and completely coated with

Au. The edges of the sample were then carefully scrapped away with a razor blade

to prevent any shorting of the Au, with the LSCO bottom electrode. If any gold

from the top electrode came into contact with the LSCO bottom electrode, then

there would no longer exist a potential difference between the two metals and thus

no internal bias field could exist. The sample was mounted in an evacuated cryo-

stat, and heated to 500 K, before cooling to room temperature. Once cool, the

sample was removed from the cryostat and mounted back upon the Peltier stack,

and the the structural behaviour investigated again.

Figure 5.19 compares the out of plane lattice constants and the peak intensities

for the BaTiO3 film for before (figures 5.19(a) and (b)), and after (figures 5.19(c)

and (d)) Au treatment. There is very little difference in the overall behaviour and

magnitude of the lattice parameter, and the phase transition at 290 K is clearly

seen, consistent with previous results. There does however appear a very slight

‘dislocation’ in lattice parameter of the Au coated film at T ∼ 403 K, similar to

figure 5.15(a), which is not observed in the non-Au coated film. Investigation of

the peak intensity on the other hand shows a dramatic difference between the Au

and non-Au coated films. In the non-Au coated film, the peak intensity rises in

an almost linear manner with a slight turning over at high temperature. The Au

coated film exhibits a similar linear trend, but has a distinct change in behaviour

at T ∼ 390 K. This value is remarkably close to the bulk tetragonal-cubic phase

transition at 393 K, and the sudden change of behaviour of the measured peak

intensity would seem to suggest that 390 K defines a structural phase transition

temperature.

136


Figure 5.19: Temperature dependence of the out-of-plane lattice parameterand peak intensity for a) and b) the non-Au coated, and c) and d) Au coatedBaTiO3 film.

Although the peak intensity measurements would seem to indicate a structural

phase transition at 390 K, the measured out-of-plane lattice constant does not

change significantly at this point. Indeed, calculation of the in-plane lattice pa-

rameter from the 202 reflection at different selected temperatures also shows little

deviation from linear behaviour at high temperatures (figure 5.20).

5.4.6 Discussion of Results

There is no doubt that a BaTiO3 thin film having undergone a field cooling with

Au electrodes, behaves differently than that without Au electrodes. The evidence

from the XRD data would suggest that there exists a distinct structural phase tran-

sition at 290 K, and a second possible transition at 390 K. However, in this latter

case, although there is a distinct change in peak intensity, which may suggest a

symmetry change, there is relatively little variance in the measured lattice param-

137


Figure 5.20: Temperature dependence of the in-plane lattice parameter forthe Au coated BaTiO3 film.

eters. These contradictory observations are quite puzzling. However, the previous

films presented in sections 5.4.2 and 5.4.4 did show very similar behaviour of the

peak intensity and lattice parameter, and so it may be the case that the majority

of this film remains in a virgin state with a only small volume of the film exhibiting

non-virgin behaviour.

In figure 5.16, the anomalous peak at 350 K in the virgin capacitance results was

found to coincide with the beginning of the linear expansion of the lattice constant,

as well as coinciding with the point in which the depolarisation current becomes

zero in figure 5.8. Previously it was suggested that the 350 K peak observed

in virgin capacitance measurements disappeared in the non-virgin measurements.

However, since the structural behaviour of a non-Au coated (virgin), and a Au

coated (non-virgin) film is virtually identical around < 390 K, then it could be

that the 350 K capacitance peak is only masked by the ferroelectric-paraelectric

capacitance peak in non-virgin films.

The observed decrease of the out-of-plane lattice constant from 290− ∼ 350 K,

may be due to the slow rotation of the polarisation axis from a [011] direction to a

[001] direction lying in-plane, within the virgin like film. However, in a non-virgin

film, an out-of-plane polarisation axis is encouraged to form under the electrodes,

by the internal bias field, as the film cools through TC . Thus in a non-virgin

film, the polarisation still rotates to an in-plane direction as before, but the polar

regions under the electrodes will still remain predominantly out-of-plane, above

138

5.5 Conclusions

350 K, until all polarisation disappears at T = TC .

It is inferred from the functional data that the ferroelectric-paraelectric phase

transition should occur around 400 K, which for bulk BaTiO3 is accompanied by

a symmetry change from tetragonal to cubic. Unfortunately, the in-plane lattice

parameter data does not go high enough in temperature to allow the determination

of the symmetry state, but inspection of figure 5.19(c) and (d) shows a slight

anomaly at ∼ 403 K, which could suggest a change in symmetry of the Au coated

film.

5.5 Conclusions

The work in this chapter would indicate that the misfit phase diagrams of Pertsev

et al and Dieguez et al do not describe the observed phase transition order of a

zero misfit strained, ∼ 175 nm thick BaTiO3 film. Although the functional charac-

terisation of non-virgin films show anomalies at temperatures which coincide with

temperatures of known bulk structural phase transitions, the virgin film behaviour

exhibits a suppression of the anomaly associated with the ferroelectric-paraelectric

phase transition. The difference in the two behaviours was attributed to the for-

mation of out-of-plane domains, by the internal bias produced by the asymmetric

electrodes. This was verified by x-ray diffraction, which samples the virgin film,

and showed that at room temperature the films exhibited an orthorhombic struc-

ture with the longest axis (corresponding to the polar axis in the tetragonal state)

in plane. Also, a comparison of the structural properties of the same film, before

and after application of a Au top layer, showed marked differences, with the data

from the Au coated film exhibiting evidence of a structural phase transition at a

temperature consistent with the bulk ferroelectric-paraelectric phase transition.

139

Chapter 6

Conclusions and Further Work

6.1 Summary and Conclusions

In chapter 3, the series capacitor model was implemented to examine the tem-

perature and frequency functional characteristics of the extracted bulk-like and

interfacial components of two different capacitor systems: SRO/BST/Au and

LSCO/BST/Au. The extracted bulk behaviour was similar to that expected in

ceramic materials, displaying large peak dielectric constants at TM = 250 K, and

low frequency dispersions, in disagreement with the observations of Zafar et al

[48]. However, the extracted bulk permittivity for the system incorporating LSCO

electrodes was observed to be highly depressed, which could be due to an internal

bias field present in the films, originating from the different work functions of the

asymmetric electrodes.

The extracted interfacial component demonstrated little frequency and tem-

perature dependence. However, the SRO/BST system did exhibit a temperature

dependence and large frequency dispersion of the real functional component, above

T = 300 K. Analysis of the imaginary functional component revealed a thermally

activated conduction mechanism, with an activation energy of EA ∼ 0.6 eV, asso-

ciated with the de-trapping of charge carriers from a thin defect layer parallel to

the electrodes. The freed charge carriers could then contribute to the functional

response of the interfacial capacitance resulting in the observed temperature and

frequency characteristics of the real interfacial component at elevated tempera-

tures. Since these defects contribute to the interfacial capacitance at elevated

temperatures only, they cannot be the sole cause for the interfacial capacitance.

Chapter 4 extended the investigation of the interfacial capacitance to ultra-

140


thin dimensions (d < 10 nm) . BST films exhibiting a thickness range of 5-16 nm

displaying little frequency dispersion, and low dielectric loss, were successfully fab-

ricated using Pulsed Laser Deposition. Perfect Debye-like behaviour was observed

in these films, but was attributed to an artifact of the increased time constant of

the measurement circuit, due to an increased resistance, but this was observed only

for frequencies greater than 104 Hz. Capacitance-Voltage measurements explicitly

showed the presence of the ∼ 1 V internal bias field attributed to the mismatch of

the electrode work functions.

Previous results of Sinnamon et al [82] had placed the upper limit of the total

thickness of any low permittivity dead layer within the dielectric at 7.5 nm, imply-

ing a thickness ≤ 3.75 nm dead-layer at each interface. The functional data from

these ultrathin films showed no deviation from the series capacitor model down

to 5 nm, implying that if a dead-layer exists with in the BST film, then its total

thickness cannot exceed 5 nm, with the thickness of the layer at each interface be-

ing ≤ 2.5 nm. TEM analysis of the ultrathin films detect no discernible dead-layer

at either interface, even though di/εi = 0.50± 0.06 nm for these films could imply

a highly visible layer.

A model pioneered by Ku and Ullman [77], and Simmons [78], and later used

by Dawber et al [81], states that the source of the interfacial capacitance is not

due to a dead-layer within the dielectric, but instead originates from a thin layer of

space charge within the electrodes, due to the incomplete screening of the applied

electric field. This screening model was used to calculate the magnitude of the

interfacial capacitance for a LSCO/BST/Au capacitor, and gave a theoretical value

of di/εi = 0.47 nm, which compared remarkably well, to the experimental value of

di/εi = 0.50± 0.06 nm, for the same system. Therefore, the effects of incomplete

screening of the electric field by imperfect electrodes could be considered the culprit

for the interfacial capacitance.

Chapter 5 investigated the misfit phase diagrams of BaTiO3 as calculated by

Pertsev et al [18] and Dieguez et al [96]. In these diagrams, the structural phase

transitions for zero misfit strain films, radically differ from that in bulk, with the

appearance of an exotic r-phase, forbidden in bulk. To this end, a series of ∼ 175

nm BaTiO3 films were deposited upon LSCO electrodes, with Au top electrodes,

and their structural phases as a function of temperature were investigated by

monitoring their functional behaviour, and by direct measurement of their lattice

parameters using X-ray diffraction.

Since the lattice mismatch between the LSCO and BaTiO3 is so large, one can

141


assume that the interface is incoherent, and thus the BaTiO3 will exhibit a zero

misfit strain like state. XRD also revealed a distinct phase transition at T ∼ 290 K.

Below this temperature, high resolution XRD using synchrotron radiation, revealed

the structure to be orthorhombic, with the longest axis in plane. Above this

temperature the films exhibited a tetragonal structure with the lattice parameters

a = b > c. The out-of-plane lattice constant was also observed to decrease from 290

K until at 350 K, it begins to increase, and continues to do so in an approximately

linear manner.

Non-virgin functional measurements showed anomalies at the approximate tem-

peratures one would expect the bulk structural phase transitions, as did depolar-

isation current measurements. However, virgin functional measurements demon-

strated an apparent suppression of the high temperature anomaly, but the appear-

ance of an anomaly at T ∼ 350 K. This anomaly seemed to coincide with the

sudden increase in lattice parameter of the film.

It was hypothesised that when the films are deposited the polar axis orients

predominately in-plane, but after the virgin measurements, where the film is heated

to a temperature above TC , it effectively field-cools back to room temperature, due

to the presence of the internal bias field, upon which an out of plane polar axis

is formed, directly below the electrodes. To test this hypothesis, XRD was used

to measure the out-of-plane lattice parameter, and the peak intensity of the 002

reflection, as a function of temperature for a film before being totally covered in

Au, and after being covered in Au, and allowing to cool through TC . Whilst the

lattice parameter of the film remained the same for both the non-coated and coated

film, the peak intensity differed significantly. The peak intensity was observed to

share the same behaviour up to T ∼ 390 K, at which point the behavior of the

Au-coated film changed significantly, implying a possible change of structure.

Overall, the zero misfit strain BaTiO3 thin films would seem to behave similar

to that of a bulk sample, and not at all like that predicted by Pertsev et al or

Dieguez et al. Certainly, there did not seem to be any evidence of the exotic r

phase. However, there may be a distinct difference between zero misfit strain from a

perfectly lattice matched interface, and a zero strain from a totally homogeneously

relaxed film.

142

6.2 Further Work

6.2 Further Work

It is clear that there is still much to learn about the properties of thin and ultrathin

ferroelectric films. Some of the possible avenues of interesting research, associated

with the work of this thesis are briefly explored below:

• The majority of capacitance measurements of thin film ferroelectrics tend to

be performed at low frequencies typically ≤ 106 Hz, which of course sam-

ples within the dipolar frequency regime (and for lower frequencies the space

charge and electrode polarisation regimes [136, 137]). An investigation of the

thickness dependence of the dielectric constant performed at higher frequen-

cies, e.g. optical frequencies, may prove fruitful in identifying the origin of

the apparent interfacial capacitance. This could be combined with the exten-

sion of the series capacitor model to films thinner than 5 nm. The ultimate

goal would be to obtain a dielectric film one unit cell thick. Should there

remain no deviation from the series capacitor model down to this limit, then

one can safely conclude that the origin of the interfacial capacitance may not

lie within the dielectric. If a deviation should occur, then one would then

know the thickness of the ‘dead-layer’ and thus determine its permittivity,

aiding the identification of the origin of the interfacial capacitance.

• For the above suggested research, it would be necessary to reduce the di-

mensions of the electrodes, which could be achieved through photolithog-

raphy. The small electrodes could then be wire bonded to larger contact

pads which would then permit temperature measurements within a cryo-

stat. Photolithography would also permit the use of symmetric electrodes,

which would remove the internal bias field observed in this work, and also

improve the interface, since the entire capacitor could be fabricated within

one vacuum cycle. Finally, photolithography could be used to create elec-

trodes perpendicular to the surface, thereby permitting the investigation of

in-plane functional properties.

• The work of Saad et al [138, 139] on single crystal SrTiO3 and BaTiO3 ca-

pacitors fabricated using the focused ion beam microscope, demonstrates no

thickness dependence in films down to 70 nm, implying that the interfacial

capacitance may actually be extrinsic, and is only introduced into capacitor

systems through film processing. Using Au electrodes, if the electrode field

143

6.2 Further Work

penetration model is correct, then one would still expect an interfacial ca-

pacitance of di/εi ∼ 0.2 nm, which is clearly not observed by Saad et al. This

work is fundamentally important to the dead layer debate, and its applica-

tion to other aspects of thin film ferroelectric research (such as polarisation

fatigue, and strain effects) is paramount.

• Ghosez and Junqera [140] have theoretically predicted that the critical thick-

ness for ferroelectricity in BaTiO3 is approximately 2.5 nm. With the ability

to successfully grow dielectrically functional ultrathin films of 5 nm, it may

be possible to investigate this prediction by decreasing this thickness further

in a BaTiO3 system.

• The calculated misfit strain phase diagrams of Pertsev et al [18] and Dieguez

et al [96] still remain largely uninvestigated within the tensile or zero strain

regime. This is due to the lack of substrates possessing lattice parameters,

equal to or greater than the lattice parameters of the overlying film. One

possible material that could be used as a buffer layer for BaTiO3 or BST is

SrxNbO3, which displays a cubic perovskite structure at room temperature

for x = 0.7 − 0.95 [141]. By varying the Sr content the lattice parameter

of the buffer layer could be controlled [142], thereby permitting a degree

of tunning of the misfit strain imposed upon the film. A second system

of choice is PbTiO3 deposited upon a BST buffer layer. Again, the lattice

parameter of the BST (and hence misfit strain imparted to the PbTiO3)

can be controlled by varying the Sr content of the material. Unfortunately,

these investigations could only be performed using XRD, and TEM, since

the buffer layers are non-conductive, however, it may be possible to engineer

a conductive SrxNbO3 by altering its oxygen content [143] or other dopants,

thereby permitting functional measurements.

144

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Characteristics of Thin and Ultrathin Ferroelectric Capacitor Structures