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Physics of Dielectrics and DRAMThomas Schroeder

IHP Im Technologiepark 25 15236 Frankfurt (Oder) GermanyIHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany www.ihp-microelectronics.com

Physics of Dielectrics and DRAM

Dielectrics

IHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany

www.ihp-microelectronics.com

Description of a dielectric material Dielectric behavior in a nut shell: A dielectric material is a non-conducting substance whose bound charges are polarized under the influence of an externally applied electric field. The figure of merrit to describe the degree of polarization in a given material is the dielectric constant It is clear that the degree of polarization is related to the structure of the material. In consequence, dielectric behavior in electrostatic and alternating electric fields depends on static and dynamical properties of the structure. Dielectric behavior must be specified with respect to the time or frequency domainIHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany www.ihp-microelectronics.com

Electrostatics: Basic experiment to get the central idea

Plate capacitor with oppositely charged plates and no material inserted. According to the surface charge density, a certain electric field E is created inside.

If dielectric material is inserted, polarized charges neutralizes some of the charges on the plates. In this way, one talks about free (unneutralized) and bound charges (neutralized) on the plates. As only free charges create electric field, a current must raise the free charge density and keep E constant.

If dielectric material is inserted and current source disconnectedm the polarized charges neutralize some of the free charges on the plates. In consequence, a constant D results in a decrease of the electric field between the plates.

IHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany

www.ihp-microelectronics.com

Electrostatics: Macroscopic description of dielectrics Poisson Equationelectric field E

divD = free

Polarisation

Dielectric Displacement DElectric Susceptibility

D = 0E + P

Polarisation

P = 0eE

Putting things together:

D = 0 (1 + e ) E = 0 r E

Result: dielectric displacement D is linearly related to the electric field and the dielectric constant is the linear coefficient of the relationshipIHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany www.ihp-microelectronics.com

Dielectric Constant

Electrostatics: Microscopic Approach and the local field Polarisation: Local electric field:density of induced dipoles applied electric field polarisability

P = N a p = N a ElocEloc = Ea + Edipole

The local electric field can be calculated according to the crystal structure by the method of Clausius and Mosotti. For example, for cubic structures the Clausius Mosotti equation reads:

Na r 1 = 3 0 r + 2

For example, to design a material with high dielectric constant: - ions with high polarisability - material with high density of induced dipolesIHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany www.ihp-microelectronics.com

Microscopic Mechanism of dielectric polarisation and frequency dependenceRelaxation Space charge polarisation materials with space charge inhomogeneities (ceramics with conducting grains and insulating boundaries)

Different mechanism show different dynamic behavior in time domain. In consequence, adsorption occurs at different windows in frequency domain

Relaxation

Orientation polarization alignment of permanent dipoles in a material

Resonance Resonance

Ionic polarization mutual displacement of negative and positive sublattice in ionic crystals

Electronic polarization displacement of electron shell against positive nucleusIHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany www.ihp-microelectronics.com

Dielectric Behaviour in alternating electric fieldsIn alternating electric fields, a frequency dependent phase shift occurs between applied electric field and displacement of charges in the material To express this mathematically, a complex extension of dielectric function and susceptibilty is introduced:

r = r + i r'

ur

''

uu r

e = e + i e'

''

r' + ir'' r = r' + i r''urFigure of merrit for a dielectric material: the quality factor Q

r = r + i r'

ur

r tan := ' r''

''

r'' tan := ' r

Q :=

1 tan

A low quality factor states that a material heavily dissipates energy from the alternating electric field by adsorbing mechanisms

tan = (tan ) dipole + (tan )condIHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany www.ihp-microelectronics.com

Description of ionic and electronic resonance phenomenafriction Equation of motion acting force acting force

2 d u du * * 2 mi 2 + mi i u = qi Eloc + mi* 0, i dt dt

displacement from equilibrium

resonance frequency

restoring force

Case 1: Acting force is a dc field and switched off at a given moment

Restoring force pulls charges back in equilibrium position a) If friction force is negligible, we arrive at undamped oscillations b) If friction force is not negligible, we arrive at damped oscillation. (behavior similar to relaxation process with certain time constant)

IHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany

www.ihp-microelectronics.com

Description of ionic and electronic resonance phenomenafriction Equation of motion acting force acting force

2 d u du * * 2 mi 2 + mi i u = qi Eloc + mi* 0, i dt dt

displacement from equilibrium

resonance frequency

restoring force

Case 2: Acting force is an ac field Electric ac field:

Ansatz:

Result: Frequency dependent amplitude u of oscillations and dipole field:

uuu r i ( kr t ) Eloc = Eloc ,0 e r ur i ( kr t ) u = u0 e

ur (qi / mi* ) Eloc ,0 u0 = 2 0,i 2 + i iIHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany

uu r uu r pi = qiu0 ei ( kr t ) = i Eoc ,0 ei ( kr t )www.ihp-microelectronics.com

Description of ionic and electronic resonance phenomenafriction Equation of motion acting force acting force

2 d u du * * 2 mi 2 + mi i u = qi Eloc + mi* 0, i dt dt

displacement from equilibrium

resonance frequency

restoring force

qi i i ( ) = 2 2 mi ( 0,i 2 ) 2 + i2 22 ''

qi2 / mi* ' '' i ( ) = 2 = + ( ) i ( ) i i 2 0,i + i i 2 2 2 q i' ( ) = i 2 2 0,i 2 2 Reduces in the static case to: mi ( 0,i ) + i2 2 q i ,s := i (0) =' 2 2 mi2 0, i i

uu r

Case 2: Acting force is an ac field Frequency dependent complex polarisability for ionic and electronic mechanisms (f > 1011 Hz):

and i'' (0) = 0

IHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany

www.ihp-microelectronics.com

restoring force resonance displacement from equilibrium frequency Case 2: Acting force is an ac field

2 d u du * * 2 mi 2 + mi i u = qi Eloc + mi* 0, i dt dt

With the frequency dependent complex polarisability and the Clausius-Mosotti equation, we get the frequency dependent complex dielectric function in the frequency range > 1011Hz :' ' ( ) ( 0+ ) ' r r 0 r ( ) = r ( 0+ ) + 1 ( / 0 ) 2 + i / 02

ur

In the case of an ideal insulator with negligible magnetisation, the optical refractive index is given by Maxwell`s law

r ur n( ) = r ( )

Optical properties of matters: ectric properties under the influence of alternating electric fields with f > 1011 HzIHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany www.ihp-microelectronics.com

Description of relaxation phenomena: orientation and space charge contributionsDebye Relaxation Debye relaxation denotes a system with a single relaxation time t. Example: Medium with one type of oriented dipole which can be oriented by external field Such a system can be directly described by the general expression when the term of the restoring force is omitted:' ' ( ) ( 0+ ) 0 r r r ( ) = r' ( 0+ ) + 1 ( / 0 ) 2 + i / 02

ur

We get:' ' ' ( ) ( ) ' 0+ r r r ( ) = r' ( 0+ ) + r 0 = ( ) + 0+ r 1 + i / 02 1 + i

ur

The relaxation step describes the part of the permittivity due to a relaxation process:

r := r ( 0 ) r ( 0+ ) and := / 0' ' 'IHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany www.ihp-microelectronics.com

2

Description of relaxation phenomena: orientation and space charge contributionsDebye Relaxation The relaxation step describes the part of the permittivity due to a relaxation process:

r := r ( 0 ) r ( 0+ ) and := / 0' ' '

2

For microelectronics this means: There is no high-k material without energy dissipating process. Find a material which is well behavin in the frequency range of interest ! The real and imaginary part of the dielectric function for Debye relaxation than read:' r r' ( ) = r' ( 0+ ) + 2 2 1+ ' r '' r ( ) = 1 + 2 2IHP Im Technologiepark 25 15236 Frankfurt (Oder) Germany www.ihp-microelectronics.com

Description of relaxation phenomena: orientation and space charge contributionsHow to identify Debye relaxation ? 1) Thermal behaviour

= 0 eW0 / kBTCertainly, this results in an exponential dependance of the relaxation frequency with temperature 2) Cole Cole diagram In case of true Debye behavior (only one relaxation time) plotting imaginary against real part forms a semi-circle This examp