NEGF SIMULATION OF ELECTRON TRANSPORT ... - Purdue Engineering · NEGF SIMULATION OF ELECTRON TRANSPORT IN RESONANT TUNNELING AND RESONANT INTERBAND TUNNELING DIODES A Thesis Submitted

NEGF SIMULATION OF ELECTRON TRANSPORT IN RESONANT

TUNNELING AND RESONANT INTERBAND TUNNELING DIODES

A Thesis

Submitted to the Faculty

of

Purdue University

by

Arun Goud Akkala

In Partial Fulfillment of the

Requirements for the Degree

of

Master of Science in Electrical and Computer Engineering

December 2011

Purdue University

West Lafayette, Indiana

ii

To my family...

iii

ACKNOWLEDGMENTS

I’m grateful to my advisor prof. Gerhard Klimeck for giving me the opportunity

to be his student and conduct research in his group. His encouragement as well as

criticism have been a strong motivation for me to try harder and come up with my

best.

I also wish to thank prof. Mark Lundstrom and prof. Vladimir Shalaev for their

role as my advisory committee members. Prof. Lundstrom’s excellent tutorship in

ECE 606 Solid State Devices was the key in inspiring me to pursue this track. There

were times when ploughing through the vast literature and the complex numerical

techniques employed in device modeling left me feeling lost. In times like these,

prof. Datta’s brilliant lectures hosted on Nanohub often came to my rescue. I also

acknowledge his great patience and willingness to answer questions I had during

ECE 495 and 659 class hours, two courses that undoubtedly raised my awareness of

Quantum effects and transport in Nanoelectronic Devices.

Most of the projects that I was involved with used the atomistic simulation tool

NEMO5. The developers Sebastian Steiger, Hong-Hyun Park, Tillmann Kubis and

Michael Povolotskyi offered tremendous support to me in helping understand the

various solvers built into NEMO5. The time spent on the 1dhetero, Brillouin zone

viewer tool with Sebastian and RTD NEGF with Hong-Hyun and Zhengping were

particularly the most fruitful to me in terms of learning about numerical techniques

used in device modeling.

The weekly group meetings and presentations by my fellow group members taught

me some of the challenges of device modelling and effective presentation of research

work. There are too many students to be named here, so instead I’ll thank them

collectively and wish them the best for their respective degrees.

iv

I’ve enjoyed being a contributor to NanoHUB, using its simulation tools for home-

work and going through its treasure trove of educational resources. NanoHUB’s ad-

ministrator Steven Clark and Rappture support team consisting of Derrick Kearney

and George Howlett were quick to answer questions relating to tool installation on

Nanohub and other computational issues I came across. I express my gratitude to

them as well as to NCN, RCAC for the computational resources and to Vicki Johnson

and Cheryl Haines for help with scheduling appointments. Special thanks is also due

to Xufeng Wang who shared his prior experience working with Rappture when I was

stuck on the Brillouin zone viewer tool development and for sharing his custom TCL

shell code that went into the 1dhetero tool’s code.

Lastly, it would be unfair if I didn’t thank my parents and my sister for their

support when my morale was down. They always stood by my side and encouraged

me to pursue what I wanted to. I’m indebted to them for their belief in my abilities...

v

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Beyond Si CMOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 RTDs for digital logic and memory circuits . . . . . . . . . . . . . . 2

1.3 Motivation for studying RTDs . . . . . . . . . . . . . . . . . . . . . 4

1.4 Computational Modeling of RTDs . . . . . . . . . . . . . . . . . . . 5

1.5 Need for Atomistic Device Simulation . . . . . . . . . . . . . . . . . 5

1.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 PHYSICS OF RTDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Structure of a GaAs/AlGaAs RTD . . . . . . . . . . . . . . . . . . 8

2.3 Quasi-bound states and resonant transmission . . . . . . . . . . . . 9

2.4 Global coherent tunneling . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Tunneling current density . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Transfer matrix method . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7 Esaki-Tsu Current formula . . . . . . . . . . . . . . . . . . . . . . . 17

2.8 Valley current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.8.1 Minimizing thermionic emission current . . . . . . . . . . . . 18

2.8.2 Minimizing current due to higher resonances . . . . . . . . . 18

2.8.3 Effect of scattering . . . . . . . . . . . . . . . . . . . . . . . 18

vi

Page

3 MODELING OF RTDS IN RTD NEGF . . . . . . . . . . . . . . . . . . 21

3.1 Intoduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Thomas-Fermi Model (Semiclassical treatment) . . . . . . . 21

3.2.2 Hartree Model (Quantum Mechanical treatment) . . . . . . 24

3.3 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.1 Transmission and Current . . . . . . . . . . . . . . . . . . . 32

3.4 Self-consistent electrostatic potential calculation . . . . . . . . . . . 33

3.5 Thomas-Fermi vs Hartree method . . . . . . . . . . . . . . . . . . . 35

3.5.1 Conduction band profile . . . . . . . . . . . . . . . . . . . . 35

3.5.2 Free charge density . . . . . . . . . . . . . . . . . . . . . . . 36

3.5.3 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Cumulative current density . . . . . . . . . . . . . . . . . . . . . . . 43

3.7 2D-2D Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 RESONANT INTERBAND TUNNELING DIODES . . . . . . . . . . . . 47

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Esaki diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 InAs/AlSb/GaSb RITD . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 Multiband modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Current density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.6.1 IV characteristics . . . . . . . . . . . . . . . . . . . . . . . . 53

4.6.2 At peak and valley voltage . . . . . . . . . . . . . . . . . . . 55

4.6.3 At high bias (1.1V) . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 1DHETERO AND BRILLOUIN ZONE VIEWER . . . . . . . . . . . . . 64

vii

Page

5.1 1dhetero tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Brillouin Zone Viewer tool . . . . . . . . . . . . . . . . . . . . . . . 70

6 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A RTD NEGF - USER OPTIONS . . . . . . . . . . . . . . . . . . . . . . . 76

A.1 Basic options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A.2 Multiscale Domains options . . . . . . . . . . . . . . . . . . . . . . 78

A.3 Advanced options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

A.4 Resonance Finder options . . . . . . . . . . . . . . . . . . . . . . . 80

viii

LIST OF TABLES

Table Page

3.1 GaAs/AlGaAs DBRTD structure used in the simulations. . . . . . . . 35

3.2 Simulation parameters for GaAs/AlGaAs DBRTD structure used in thesimulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 InAs/AlSb/GaSb RITD structure used in the simulations. . . . . . . . 54

5.1 Substrates and materials supported by 1dheterostructure tool . . . . . 64

ix

LIST OF FIGURES

Figure Page

1.1 Simulated current voltage characteristics at 300K of a GaAs/AlGaAs RTDstructure (used in Chapter 3) showing negative differential resistance. 2

1.2 A 6 transistor SRAM memory cell (top), an RTD latch based SRAMmemory cell (bottom, left) and its load line diagram (bottom, right) [4],[5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Conduction band diagram (at Γ point) for a GaAs/AlGaAs DBRTD underequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Quasi Bound states of a quantum well . . . . . . . . . . . . . . . . . . 10

2.3 The transmission coefficient plot shows peaks at energies equal to the quasibound state energies of the GaAs quantum well. . . . . . . . . . . . . 10

2.4 Current voltage characteristics when Eo >> EF in emitter and when nobias is applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Current voltage characteristics when Ec < Eo < EF in emitter, i.e., ap-plied bias is smaller than peak voltage. . . . . . . . . . . . . . . . . . 13

2.6 Current voltage characteristics when Eo = Ec in emitter,i.e., when appliedbias is equal to the peak voltage. . . . . . . . . . . . . . . . . . . . . . 14

2.7 Expected IV characteristics at 0K . . . . . . . . . . . . . . . . . . . . 14

2.8 Small potential steps are used for Transfer matrix calculation. . . . . . 15

3.1 Partitioning of regions for semiclassical simulation. The charge density inthe well is set to 0. The potential variation is therefore linear in the well. 23

3.2 Simulation flow for semiclassical Thomas-Fermi simulation for one biasvoltage value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Quantum modeling of regions. . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Computation of left connected Green’s function and fully connected Green’sfunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 Simulation flow for Hartree self-consistent simulation for one bias voltagevalue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

x

Figure Page

3.6 Conduction band edge profile obtained from semiclassical Thomas Fermimodel (blue) and Hartree quantum self-consistent method (red). The up-ward shift for the Hartree result in the well region is due to the presenceof charge in the well which pulls the conduction band edge and the reso-nance level upwards in energy as against the applied bias which tries topull them down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7 Comparison of electron density obtained from a Thomas-Fermi and Hartreequantum self-consistent caclculations for different bias voltages applied tothe DBRTD. The 2nd plot is at peak voltage Vc = 0.2V . . . . . . . . 39

3.8 IV characteristics computed from semiclassical Thomas Fermi model (blue)and Hartree quantum self-consistent method (red) . . . . . . . . . . . . 40

3.9 Variation of first 2 resonances (in red) for Thomas-Fermi (left) and Hartreequantum self-consistent (right) methods. The emitter Fermi level (inblue), the conduction band edge at the left spacer/barrier interface (bot-tom black line) and the peak conduction band edge to the left of the barrier(top black line) are also shown. . . . . . . . . . . . . . . . . . . . . . . 41

3.10 Conduction band edge profile (top), quantum charge density (center, blue)at 0.475 V and variation of quantum charge (bottom, orange), semiclassicalcharge (bottom, red) with bias in the non-equilibrium device region. Asthe 2nd barrier becomes shorter with bias, the well charge diminishes. 42

3.11 Conduction band profile (left),transmission coefficient (center,blue), cur-rent density (center, dark green), normalized cumulative current density(center, light green) and resonance energy vs voltage at peak voltage of0.2V. The normalized cumulative current density shows that the 1st reso-nance level is the major contributor to the total current. . . . . . . . . 43

3.12 Conduction band profile (left),transmission coefficient (center,blue), cur-rent density (center, dark green), normalized cumulative current density(center, light green) and IV curve at a voltage of 0.375V. The normalizedcumulative current density shows that the 1st resonance level is no longercontributing significantly to the total current. . . . . . . . . . . . . . . 44

3.13 Emitter quasi-bound state (left, orange) at an applied bias of 0.68V. Tun-neling from this quasi-bound state into well resonance contributes somecurrent although its magnitude is much smaller than the current arisingfrom the higher order well resonance. . . . . . . . . . . . . . . . . . . 45

4.1 Esaki diode operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Equilibrium band diagram for an InAs/AlSb/GaSb RITD. . . . . . . . 49

xi

Figure Page

4.3 Equilibrium band diagram and overlap between InAs emitter bands andGaSb valence subband dispersions along kx for an InAs/AlSb/GaSb RITD. 51

4.4 Band diagram for voltages greater than peak voltage and overlap betweenInAs emitter bands and GaSb valence subband dispersions along kx for anInAs/AlSb/GaSb RITD. . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.5 IV curve for the InAs/AlSb/GaSb RITD at 300K showing a PVR of 50. 54

4.6 Energy integrated current density J(kx, ky) along kx direction at peakvoltage (top) and at valley voltage (bottom). . . . . . . . . . . . . . . 56

4.7 Energy integrated current density J(kx, ky) along ky direction at peakvoltage (top) and at valley voltage (bottom). . . . . . . . . . . . . . . 57

4.8 Energy integrated current density J(kx, ky) along kx = ky direction atpeak voltage (top) and at valley voltage (bottom). . . . . . . . . . . . 58

4.9 Cumulative current density along ky = 0 at peak voltage (top) and atvalley voltage (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.10 LDOS profile along with current density plot for ky = 0 direction showingthe energy at which the current is flowing for peak voltage (top) and atvalley voltage (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.11 LDOS and current density at 1.1V. . . . . . . . . . . . . . . . . . . . . 61

4.12 LDOS and current density at 1.1V. . . . . . . . . . . . . . . . . . . . . 62

5.1 Design section of the 1dhetero tool where the dimensions and doping den-sity of the materials and substrate constituing the heterostructure can bespecified. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Equilibrium conduction band profile and bound state resonances for thedefault AlGaAs/GaAs heterostructure calculated from a single band sim-ulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Equilibrium potential profile for the default AlGaAs/GaAs heterostructurecalculated from a single band simulation. The potential peak is due todepletion of carriers in the doped AlGaAs barrier. . . . . . . . . . . . . 68

5.4 Equilibrium sheet density profile for the default AlGaAs/GaAs heterostruc-ture calculated from a single band simulation. A 2DEG is formed near theAlGaAs-GaAs interface resulting in a very high sheet concentration of8.8× 1012/cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

xii

Figure Page

5.5 Equilibrium sheet density profile for the default AlGaAs/GaAs heterostruc-ture calculated from a semiclassical Thomas-Fermi simulation. The semi-classical method predicts maximum sheet density at the AlGaAs-GaAsinterface whereas the single band calculation result of Fig.5.4 shows max-imum sheet concentration at the center of the inversion layer, away fromthe AlGaAs-GaAs interface. . . . . . . . . . . . . . . . . . . . . . . . . 69

5.6 Brillouin zone viewer tool showing the 1st Brillouin zone for the cubicFCC lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.1 RTD NEGF simulation tool showing Basic tab options. . . . . . . . . 77

A.2 Multiscale Domains tab options. . . . . . . . . . . . . . . . . . . . . . 78

A.3 Advanced tab options. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

A.4 Resonance Finder tab options. . . . . . . . . . . . . . . . . . . . . . . 80

xiii

ABBREVIATIONS

RTD Resonant tunneling diode

RITD Resonant interband tunneling diode

DBRTD Double barrier resonant tunneling diode

NEGF Non-equilibrium Green’s function

MBE Molecular beam epitaxy

MOCVD Metal organic chemical vapour deposition

LDOS Local density of states

EMA Effective mass approximation

PVR Peak to valley current ratio

IV Current voltage characteristics

NDR Negative differential resistance

NEMO5 NanoElectronic MOdeling program

xiv

NOMENCLATURE

GaAs Gallium Arsenide

AlGaAs Aluminium Gallium Arsenide

AlAs Aluminium Arsenide

InAs Indium Arsenide

GaSb Gallium Antimonide

AlSb Aluminium Antimonide

InGaAs Indium Gallium Arsenide

InAlAs Indium Aluminium Arsenide

InP Indium Phosphide

GaP Gallium Phosphide

AlP Aluminium Phosphide

Si Silicon

Ge Germanium

SiO2 Silicon Dioxide

xv

ABSTRACT

Akkala, Arun Goud M.S.E.C.E., Purdue University, December 2011. NEGF Simula-tion of Electron Transport in Resonant Tunneling and Resonant Interband TunnelingDiodes. Major Professor: Gerhard Klimeck.

The challenges due to continuous scaling of CMOS has prompted research into

alternative structures for future logic devices that are capable of high speed opera-

tion with reduced power consumption. One such contender in the emerging devices

category, the Resonant tunneling diode (RTD), has attracted considerable interest

due to its low voltage operation, THz capabilities and negative differential resistance.

RTDs operate on the principle of quantum mechanical tunneling of electrons through

a potential barrier into quantized well states resulting in resonances in the transmis-

sion characteristics. Due to the quantum mechanical nature of the tunneling process,

quantum transport simulation models are needed to describe RTD characteristics. In

this work, a simulation tool based on the Non-Equilibrium Green’s Function formal-

ism using effective mass approach has been developed to study GaAs/AlGaAs RTD

characteristics. Scattering in the emitter reservoir has been treated in an approximate

manner to ease computational burden.

However the room temperature peak to valley current ratio (PVR), which is the

figure of merit for RTDs, has to be improved in order to make them a suitable

candidate for digital applications. Resonant Interband Tunneling diodes (RITD) are

capable of achieving higher PVRs by reducing valley current. A multiband model is

required for RITDs since both conduction and valence band play a role. An sp3s*

spin-orbit based tight binding model along with an NEGF based transport solver has

been used to study a coherent InAs/AlSb/GaSb RITD and the simulated IV shows

a high PVR of 50.

1

1. INTRODUCTION

1.1 Beyond Si CMOS

Silicon based CMOS has been the dominant technology of the semiconductor

industry for the last 50 years. The chip density and operating speed of Si based

ICs have shown a regular trend of growth while the operating voltage has steadily

decreased over the past decades. This has been achieved through continuous scaling

of transistor dimensions. However as dimensions approach close to the wavelength of

an electron, quantum effects such as tunneling, interference, quantization that arise

due to wave nature of electron start playing an important role in determining device

performance.

Novel devices with a different operational paradigm over conventional field effect

based devices can be built by utilizing these quantum mechanical effects. With the

help of high mobility III-V compound semiconductors and advances in heteroepitax-

ial growth techniques such as Molecular Beam Epitaxy (MBE) and Metal-Organic

Chemical Vapour Deposition, it has become possible to realize devices with very fast

switching speeds and that are also capable of operating at low voltages, two very

critical requirements for future digital logic technologies. Resonant tunneling diodes

(RTD) hold a lot of promise in this regard. The negative differential resistance (NDR)

characteristics exhibited by RTDs due to the resonant tunneling phenomenon have

resulted in their usage in some niche applications. Since 1974, when the first GaAs-

AlGaAs based RTD was demonstrated by Chang, Esaki and Tsu, [1] RTDs have

been explored as alternatives to transistors for high frequency applications [2] such

as microwave circuits and even logic circuits [3].

2

Fig. 1.1. Simulated current voltage characteristics at 300K of aGaAs/AlGaAs RTD structure (used in Chapter 3) showing negativedifferential resistance.

1.2 RTDs for digital logic and memory circuits

The ultra high speed capabilities and low voltage operation of RTDs make them

a suitable future choice for digital logic devices. RTDs are considered as multi-

functional devices in that operations and functionalities that could be realized using

conventional techniques such as using transistors can be implemented differently using

RTDs and usually using fewer number of components. For instance Fig.1.2 depicts

a conventional 6 transistor SRAM memory cell which consists of 2 cross-coupled

inverters that acts as a latch to store data indefinitely. The same latching operation

can be implemented using 2 series connected RTDs as is clear from the load line

characteristics [4], [5].

3

Fig. 1.2. A 6 transistor SRAM memory cell (top), an RTD latch basedSRAM memory cell (bottom, left) and its load line diagram (bottom,right) [4], [5].

4

There are 3 operating points of which only A and B are stable. A can be treated

as the logic LOW state and B the logic HIGH state. By thus implementing the SRAM

function using RTDs, a significant reduction in the total size of the chip is possible.

More importantly, scaling of transistors to nanometer regime results in undesirable

effects such as leakage currents but for resonant tunneling to occur, RTDs require

dimensions that are already in the nanometer regime and are therefore conducive for

development of memory and even logic circuits with reduced areas.

1.3 Motivation for studying RTDs

Despite the high speed and low voltage operation advantages offered by RTDs

there are some drawbacks too due to which RTDs have not become widespread.

These are

1. RTDs are 2 terminal devices and cannot offer input to output isolation. They

can therefore only augment transistors but cannot replace them.

2. The load driving capabilities of RTDs are poor and hence overall throughput

would be affected. To increase the drive capabilities of RTDs, the peak current

and the PVR need to be improved.

3. The most critical shortcoming is that RTDs with excellent PVRs can be grown

using GaAs/AlGaAs, InGaAs/InAlAs and other III-V compound semiconduc-

tors all of which are incompatible with the mainstream Si technology and are

expensive to manufacture. Si/SiGe RTDs that are compatible with Si technol-

ogy have been demonstrated but they tend to have low PVRs due to the short

barrier height of SiGe.

So far, RTDs have been commonly used alongside III-V based HEMTs [6]. But

as advances in fabrication techniques such as MBE continue, in the near future, we

may have a viable and inexpensive way of manufacturing RTDs and integrating them

with Si based circuits as well as III-V based HEMTs.

5

The immediate goal then is to understand and develop simulation models that

can capture the underlying Physics of operation of RTDs accurately. Beginning with

simple approximations, the models are then refined to account for effects happening

in realistic devices such as scattering thereby giving us a better understanding and a

deeper insight into RTDs.

1.4 Computational Modeling of RTDs

The computational modeling of RTDs has remained a challenging problem and

has been addressed several times. Because of the quantum mechanical nature of the

tunneling process, these devices require quantum transport models to describe their

current voltage characteristics. Models that take into account atomistic effects and

band bending generally produce better results over those that treat the device as a

continuum and assume a linear drop in applied bias.

The need for an accurate modeling of RTDs has resulted in different formalisms

being applied to the modeling of RTDs. The first calculations performed by Esaki

and Tsu utilized the Transfer Matrix method to obtain the transmission and the

Esaki-Tsu current formula to calculate the current density.

However, scattering in contacts and the device region plays an important role

in the problem of electron transport in RTDs. Non-equilibrium Green’s function

(NEGF) is a convenient method to capture the overall effect of coupling of the device

to the contacts and also provides an extensible framework to build upon such as

by introduction of the various elastic and inelastic scattering mechanisms through

self-energies.

1.5 Need for Atomistic Device Simulation

As semiconductor processes scale down, the costs and challenges involved in fabri-

cation tend to increase and hence it can be prohibitively costly to fabricate individual

samples in order to assess device performance. Computational modeling of devices

6

provides a quick and inexpensive alternative of studying device characteristics over

fabricating and testing real physical samples. Devices that are beyond the reach of

current processing technologies can be analysed. However as agressive scaling of de-

vices continues, this also presents considerable challenges. Unlike the semiclassical

drift-diffusion models that were employed for devices upto 100nm, sophisticated quan-

tum transport models are needed. Effects such as bandstructure, confinement, device

dimensions, orientation, interface quality, strain, etc will play a significant part in

determining the device performance. Full atomistic simulation techniques that take

into account the fine details of positioning of atoms within the device will be required.

Atomistic Simulation tools that can capture these effects can be very helpful for

engineers and researchers working in the field of nanoelectronic device design and

even for students who want to learn about these devices. TCAD tools are also a great

enabler for experimentalists who wish to compare their results with those predicted

by Physics based simulation models. With these interests in mind, NanoHUB.org an

outreach effort of the Network for Computational Nanotechnology was started. More

than 200 simulation tools are available for public use on NanoHUB.org.

1.6 Outline

In this work, a simulation tool named RTD NEGF has been developed using the

NEMO5 simulator to study GaAs/AlGaAs RTD characteristics. NEMO5 is an atom-

istic simulator for studying nanoscale devices and was developed by (in alphabetical

order) - Dr. Kubis, Dr. Park, Dr. Povolotskyi and Dr. Steiger under the guidance

of prof. Klimeck. The NEMO5 based RTD NEGF tool replaces the previous Mat-

lab based version that existed on NanoHUB.org. Transport has been treated using

NEGF solver that was built into NEMO5 by Dr. Park with assistance from Zheng-

ping Jiang. An approximate way of including scattering through optical potential

term in the reservoir regions has been used. The accomplishment of the author in-

cludes modifying the front end to make the RTD NEGF tool work with NEMO5 and

7

developing an algorithm to relate the resonances obtained for different bias values

with the purpose of displaying a resonance vs bias curve.

The layout of this thesis is as follows. Chapters 2 and 3 will present an overview

of the Physics of RTDs and how they are modeled in RTD NEGF. An alternative

to RTDs, the Resonant Interband Tunneling diode (RITD) is described in Chapter

4 and results of an InAs/AlSb/GaSb structure studied using NEMO5’s atomistic

techniques such as tight binding are presented. Apart from the RTD NEGF tool,

two other NEMO5 powered tools, the modified 1dheterostructure tool for studying 1

dimensional heterostructures and Brillouin zone viewer, a new tool to visualize the

first Brillouin zones of common crystal lattice systems were developed during the

course of this work and are also hosted on NanoHUB. Features of these two tools will

be presented in Chapter 5. Future work will be discussed in the concluding chapter. It

is hoped that the Nanoelectronics research community as well as students interested

in topics such as Semiconductor Device Physics and Quantum Transport will be able

to benefit from these simulation tools.

8

2. PHYSICS OF RTDS

2.1 Introduction

In this chapter, the principle of operation of RTDs will be described. The Global

Coherent Tunneling model will be explained and used to estimate the general shape

of the current voltage characteristics. The cause for the appearance of the negative

differential resistance feature in the IV curve follows from this. For the following

discussion, a GaAs/AlGaAs RTD which is the most popular type of RTD will be

used as the reference.

2.2 Structure of a GaAs/AlGaAs RTD

A typical GaAs/AlGaAs resonant tunneling structure or diode is a 2 terminal het-

erostructure device formed by sandwiching the narrow bandgap GaAs layer between

two wide bandgap AlGaAs layers. The wide band gap layers act as potential barriers

for electrons in the conduction band. Molecular Beam Epitaxy is a commonly used

technique to grow RTDs. Fig.2.1 shows the conduction band edge profile for a typical

Double barrier resonant tunneling diode (DBRTD).

Fig. 2.1. Conduction band diagram (at Γ point) for a GaAs/AlGaAsDBRTD under equilibrium

9

The band gap of GaAs is 1.42eV while that of AlxGa1−xAs varies from 1.42 to

2.16eV as the molefraction of Al, x is varied from 0 to 1. Both GaAs and AlxGa1−xAs

(for x < 0.3) are direct bandgap semiconductors which means their conduction band

minimum and valence band maximum occur at the Brillouin zone center (Γ point).

When GaAs and AlGaAs are brought together, they form a type I heterostructure

with the conduction and valence band edge of GaAs lying between those of AlGaAs.

The conduction band offset is around 0.27 eV.

To increase the current density through the device, heavily doped contacts are

used which can supply large number of electrons. High doping gives rise to high

levels of impurity scattering which can destroy the wave coherence of electrons in the

well that is necessary for resonant transmission. Therefore, low doped spacer layers

are used in between the undoped barrier/well/barrier region and the doped contacts

to prevent diffusion of impurity atoms into the barriers and well.

2.3 Quasi-bound states and resonant transmission

The DBRT structure is assumed to be translationally invariant along the trans-

verse direction, i.e., in the plane normal to the growth direction. Due to partial

confinement along the longitudinal direction, i.e., the width of the GaAs well, quasi

bound states arise in the well. The bulk bands are quantized in to 2D subbands

in the well region. The resonance energy levels correspond to the minima of these

subbands. They can be roughly estimated using the expression for the eigen values

corresponding to the particle in a 1D box problem with the width of the box replaced

by the effective width which takes into account the portion of the wavefunction inside

the barriers.

When the energy of an incident electron wave coincides with the quasi bound state

in the well, it can excite the occupancy of the well state. For these energies, the am-

plitude of the wavefunction in the well can build up leading to enhanced transmission

through the structure via quantum mechanical tunneling through barriers. This is

10

Fig. 2.2. Quasi Bound states of a quantum well

shown as peaks in the transmission coefficient curve, which is the ratio of transmit-

ted electron flux on the right(collector) to that of the incident electron flux on the

left(emitter). A peak transmission of unity arises for symmetrical barriers, implying

that the potential barriers appear completely transparent at some energies of the in-

cident electron wave. However applying a bias results in asymmetry in the structure

and so the transmission is never always unity under non-equilibrium condition. [7]

Fig. 2.3. The transmission coefficient plot shows peaks at energiesequal to the quasi bound state energies of the GaAs quantum well.

11

2.4 Global coherent tunneling

The Global coherent tunneling model neglects all phase coherence breaking pro-

cesses within the device. The electrons in the emitter Fermi sea tunnel through the

AlGaAs barrier into empty subband states in the GaAs well at the same energy. This

tunneling process should satisfy the following two rules

1. Total electron energy E is conserved.

2. Transverse momentum k|| is conserved

In the presence of phase-breaking elastic and inelastic scaterring, condition 2 will

be violated. In the coherent tunneling model, the behaviour of current with applied

bias can be deduced from these 2 conditions as explained in the next section.

2.5 Tunneling current density

The variation of the tunneling current density with bias voltage can be estimated

from the overlap of the well subband states with the bulk emitter states for different

values of applied bias. For the sake of simplicity an operating temperature of 0K will

be assumed in this section.

Due to energy conservation and transverse momentum conservation,

Ec +h2k2

x

2m∗+h2k2

y

2m∗+h2k2

z

2m∗= Eo +

h2k2x

2m∗+h2k2

y

2m∗(2.1)

kz =

√2m∗(Eo − Ec)

h(2.2)

In the above equation, parabolic effective mass dispersion has been assumed and

the electron effective mass in the bulk emitter and the narrow well are assumed to

be equal. The subband minimum Eo in the well is measured relative to the emitter

conduction band edge Ec.

Eq.(2.2) specifies the longitudinal wave vector states in the bulk emitter that take

part in tunneling for a given value of the resonance Eo. The corresponding kx and ky

12

electron states can be found from the Fermi sphere of electron states (assuming the

temperature is 0K) in the emitter.

When Eo is high above the Fermi level EF in the emitter (Fig.2.4), the kz value

given by Eq.(2.2) is such that there is no corresponding occupied kx, ky electron

state in the emitter that can take part in the tunneling process. This can be seen in

Fig. where the kz plane is outside the Fermi sphere. So no current flows under this

condition.

Fig. 2.4. Current voltage characteristics when Eo >> EF in emitterand when no bias is applied.

When Eo drops below EF , the kz value given by Eq.(2.2) lies within the Fermi

sphere (Fig.2.5). The intersection of the kz plane and the Fermi sphere gives the

occupied kx, ky electron states in the emitter that take part in tunneling. As Eo

13

drops with applied bias, the number of transverse k states that take part in tunneling

steadily increases as can be seen from the area of the shaded disc.

Fig. 2.5. Current voltage characteristics when Ec < Eo < EF inemitter, i.e., applied bias is smaller than peak voltage.

When Eo is aligned with the emitter conduction band edge (Fig.2.6), kz = 0 and

the transverse states that are of interest lie on the disc passing through the centre of

the Fermi sphere. This is the largest number of states that can take part in tunneling

and the corresponding current is the peak current on the IV characteristics.

For further bias voltages, the kz value predicted by Eq.(2.2) becomes imaginary

and so there are no more transverse states that can participate in tunneling. The

current then abruptly drops to zero.

The tunneling current density is thus proportional to the density of states indi-

cated by the intersecting disc. If the transmission through the resonant state remains

constant for all values of applied bias then [8]

J ∝ π(k2F − k′2z ) ∝ (EL

F − Eo) (2.3)

14

Fig. 2.6. Current voltage characteristics when Eo = Ec in emitter,i.e.,when applied bias is equal to the peak voltage.

where

k′z =

√2m∗(Eo − Ec)

h(2.4)

The ideal IV as per the above argument is shown in Fig.2.7.

Fig. 2.7. Expected IV characteristics at 0K

15

The transmission function that was assumed to be unity in this discussion is

commonly found numerically using the Transfer Matrix approach.

2.6 Transfer matrix method

The transfer matrix method provides a means of computing the transmission

probability function T (Ez) numerically. The maxima of this function correspond to

the quasi-bound states in the quantum well. This technique involves solving the 1D

Schroedinger equation with scattering boundary conditions for the wavefunctions and

constructing the transfer matrices for the heterostructure. The 1D, time-independent

effective mass Schroedinger equation is

− h2

2

d

dz

(1

m∗(z)

d

dz

)Ψ(z) + V (z)Ψ(z) = EzΨ(z) (2.5)

Fig. 2.8. Small potential steps are used for Transfer matrix calculation.

The potential profile V (z) is approximated using small steps. In each section the

wavefunction is expressed as

Ψi(z) = Aiejkiz +Bie

−jkiz (2.6)

16

where

ki =

√2m∗i (Ez − Vi)

h(2.7)

At the boundaries of adjacent sections, the following continuity relations are used

Ψi(zi) = Ψi+1(zi+1) (2.8)

1

m∗i

dΨi(z)

dz

∣∣∣∣z=zi+1

=1

m∗i+1

dΨi+1(z)

dz

∣∣∣∣z=zi+1

(2.9)

The coefficients in adjacent sections can then be related as Ai+1

Bi+1

= Ti

Ai

Bi

(2.10)

where Ti is defined as

Ti =1

2

(1 +

m∗i+1

m∗i

kiki+1

)ej(ki−ki+1)zi+1

(1− m∗i+1

m∗i

kiki+1

)e−j(ki+ki+1)zi+1(

1− m∗i+1

m∗i

kiki+1

)ej(ki+ki+1)zi+1

(1 +

m∗i+1

m∗i

kiki+1

)e−j(ki−ki+1)zi+1

(2.11)

By subsequent multiplication of transfer matrices, the coefficients on emitter and

collector side can be related.

AN

BN

= T

A1

B1

(2.12)

where

T = TN−1TN−2...T2T1 =

T11 T12

T21 T22

(2.13)

By using the scattering boundary condition AN

0

=

T11 T12

T21 T22

1

B1

(2.14)

the transmission probability T (Ez) follows

T (Ez) =JNJ1

=vNv1

|AN |2 (2.15)

=m∗1m∗N

kNk1

|AN |2 (2.16)

17

2.7 Esaki-Tsu Current formula

Once T (Ez) has been calculated, current density J can be obtained using [9], [10],

[11]

J =m∗ekBT

2π2h3

∫ ∞0

dEzT (Ez)ln

1 + eELF−EzkBT

1 + eERF−Ez

kBT

(2.17)

Eq.(2.17) is the Esaki-Tsu current density formula and is arrived at by assuming a

parabolic dispersion in the transverse plane and the transmission probability to be a

function of only Ez.

Historically, the transfer matrix method and the Esaki-Tsu current density formula

were used to study RTDs. However the NEGF approach of computing transmission

and current density for open systems is now a popular technique for studying RTDs.

This technique considers coupling to contacts and provides a framework for including

scattering.

2.8 Valley current

The actual IV characteristics of DBRTD at room temperatures do not show zero

current beyond the peak current. This discrepancy from the IV shown in Fig.2.7 is

attributed to the valley current flow off-resonance. The major contributors to valley

current are -

1. Thermionic emission current flowing over the barriers

2. Tunneling through higher order subband levels at increased temperatures

3. Inelastic scattering processes that provide alternative tunneling channels

By using the right combination of material systems and dimensions for the layers,

some fraction of the valley current can be minimized.

18

2.8.1 Minimizing thermionic emission current

The thermionic emission current component can be minimized by using tall barri-

ers, such as by using AlAs instead of AlGaAs. However, increasing the barrier heights

will result in sharp resonances with a reduced transmission probability off-resonance

and scattering can then broaden the transmission causing a drop in the current. The

barriers will then have to be made thin in order to improve the current density.

2.8.2 Minimizing current due to higher resonances

The component of current due to tunneling from higher order resonance levels

can be reduced by shifting these resonances upward in energy and away from the first

resonance. This can be done by using a low effective mass material for the well or by

using narrow wells as can be seen from the following expression

En =n2π2h2

2m∗L2(2.18)

2.8.3 Effect of scattering

The various scattering mechanisms that can contribute to valley current are opti-

cal and acoustic phonon scattering,intervalley scattering, scattering due to impurity

atoms, interface roughness and alloy disorder in the case of AlxGa1−xAs barriers.

Polar optical phonon scattering is the dominant phonon scattering mechanism in

polar semiconductors such as GaAs. When the resonance level Eo drops below the

emitter conduction band edge, tunneling can still occur if the electron loses energy

corresponding to the difference Ec − Eo by emitting a phonon.

In the presence of phase-breaking scattering, resonant tunneling can be described

as two continuous tunneling processs - tunneling from emitter into the quantum well

followed by tunneling to the collector. Between these two processes, electrons suf-

fer phase-breaking scattering in the quantum well and are relaxed into local quasi-

19

equilibrium states [8]. This is the sequential tunneling model and is an alternative to

the global coherent tunneling model when scattering is present [12], [13], [14].

Due to the open nature of the system, the resonances in the well exhibit an intrinsic

broadening Γ. The transmission probability for energies close to resonances can be

approximated using the following Lorentzian form

T (Ez) =Γ2

(Ez − Eo)2 + Γ2(2.19)

The dwell time of the electrons in the quantized well states is related to the intrinsic

broadening as

td =h

Γ(2.20)

The effect of scattering is to broaden the resonance levels in the well further and

thus the transmission.

T (Ez) =Γ

Γtot

Γ2tot

(Ez − Eo)2 + Γ2tot

(2.21)

where Γtot = Γ + Γs, is the sum of intrinsic and extrinsic scattering broadening.

As in Eq.(2.20) a phase coherence breaking time ts corresponding to Γs can be

defined

ts =h

Γs(2.22)

The ratio Γs

Γacts as boundary between global coherent and sequential tunneling.

When Γs

Γ> 1, the transmission peak decreases and becomes broadeer. If steps are not

taken to enhance the peak current through the DBRTD, then scattering can degrade

the peak to valley ratio (PVR), which is the figure of merit for these devices, by

increasing the valley current.

Two important measures that are taken to minimize phase-breaking scattering

include usage of high quality interfaces to minimize roughness and alloy disorder

scattering and undoped layers for barriers and well to minimize impurity scattering.

However to enhance the current density peak, heavily doped contacts are regularly

20

used. This introduces unwanted diffusion of impurities into the barrier and well layers

and hence impurity as well as electron-electron scattering is always present in RTDs

that are operated at nominal temperatures. An interesting technique to minimize

valley current is then through using RITDs. This will be presented in Chapter 4.

The next chapter addresses the critical question of how to model RTDs in order

to understand their characteristics and to devise techniques to improve their PVR.

21

3. MODELING OF RTDS IN RTD NEGF

3.1 Intoduction

In this chapter, the device modeling techniques used in the RTD NEGF simulation

tool, which is powered by NEMO5 simulator, will be described. The RTD NEGF tool

can be used to study the characteristics of GaAs/AlGaAs RTDs.

The computational modeling of RTDs can be divided into two sections, one is

determining the electrostatics within the device by taking into account the space

charge effect and the other is 1D transport of charge carriers.

3.2 Electrostatics

The problem of electrostatics involves finding the electron density and the charge

self-consistent electrostatic potential profile in the device.

Two different techniques have been employed

1. Thomas-Fermi model - Treats electron density in a semiclassical fashion using

the Thomas-Fermi approximation. Fast but not accurate.

2. Hartree model - Quantum charge is solved self-conisistently with the electro-

static potential. Takes longer to complete but results are closer to experiment.

3.2.1 Thomas-Fermi Model (Semiclassical treatment)

In this model, the electron charge density within the barriers and the well is

set to zero. The charge density n(z) in the emitter and collector and the electro-

static potential φ(z) throughout the structure can then be determined by solving

22

the Thomas-Fermi semiclassical charge density equation and the Poisson equations

self-consistently.

n(z) =

Nc2√πF1/2

(EF−(Ec(z)−qΦ)

KBT

), z ∈ (0, b1), (b2, L)

0 , z ∈ (b1, b2)

(3.1)

d

dz

(εrdΦ

dz

)=e

ε

(n(z)−N+

D

), z ∈ (0, L) (3.2)

where F1/2 is the Fermi-Dirac integral of order 1/2 and is defined as

F1/2(η) =

∫ ∞0

E1/2dE

1 + eE−η(3.3)

The other terms are

EF → Quasi-Fermi level in emitter, collector

EC(z) → Equilibrium conduction band edge profile

εr → Dielectric constant of the layers which is position dependent

NC → Effective density of conduction band states, 2[m∗nkT

2πh2

]3/2

N+D → Donor doping concentration

b1, b2, L → Boundary regions as defined in Fig. 3.1

A finite element mesh is used to solve the resulting Non-linear Poisson’s equation

with the boundary conditions being

Φ(z = L) = VC , (Dirichlet B.C.) (3.4)

Φ(z = 0) = 0 , (Dirichlet B.C.) (3.5)

where VC is the bias voltage applied to the collector.

23

Fig. 3.1. Partitioning of regions for semiclassical simulation. Thecharge density in the well is set to 0. The potential variation is there-fore linear in the well.

Fig. 3.2. Simulation flow for semiclassical Thomas-Fermi simulationfor one bias voltage value.

24

The advantage of this method is that it is computationally inexpensive in terms

of computation time and memory needed and the potential profile so obtained is

not very different from the Hartree quantum charge self-consistent method which is

presented next.

3.2.2 Hartree Model (Quantum Mechanical treatment)

The device is partitioned into a central region consisting of the two barriers and

the well and reservoirs that inject electrons and draw them out from the central

region. The portion of emitter and collector where flat band conditions exist will be

referred to as terminals while the region between the terminals and the central region,

that includes the spacer, will be referred to as reservoirs. The emitter terminals and

reservoirs are assumed to be in strong equilibrium with an occupation factor f1(E −

EF1) and the collector terminal and reservoir is also assumed to be in equilibrium

but with an occupation factor f2(E−EF2). The central region is in non-equilibrium.

Fig.3.3 demonstrates how the RTD is sectioned

Green’s functions will be computed for the quantum region consisting of the reser-

voirs and the non-equilibrium region. This region also serves as the range within

which charge will be treated quantum mechanically while in the flat band terminal

regions, the charge will be obtained from the semiclassical expression. This reduces

computational burden since the terminals can be very large and including them will

only increase the complexity of the problem.

At high biases, the band bending resulting from the non-uniform doping profile

within the structure gives rise to a triangular potential energy well near the emit-

ter/barrier interface. Quasi bound states are formed in this emitter well and injection

from these quasi bound states must also be considered. The electron sheet density in

this region can be very high and therefore considerable electron-electron and electron-

phonon scattering can be expected here. As a first approximation, the relaxation due

25

Fig. 3.3. Quantum modeling of regions.

to scattering in this region can be treated by an optical potential term which will be

described in the following sections.

Effective Mass Hamiltonian

The 1D nearest neighbor effective mass Hamiltonian for the whole device is

H = − h2

2

d

dz

(1

m∗d

dz

)+ Vk(z) +

h2k2

2m∗L(3.6)

where m∗L is the effective mass in the left lead and

Vk(z) = V (z) +h2k2

2m∗L

(m∗Lm∗(z)

− 1

)(3.7)

26

The H matrix is obtained by discretizing the above operator and is of the form

• •

• • •

−t1,0 ε1 −t1,2−t2,1 ε2 −t2,3

−t3,2 ε3 −t3,3• • •

• •

(3.8)

where

εi =h2

∆2

(1

m∗i−1 +m∗i+

1

m∗i +m∗i+1

)+ Vi(k) (3.9)

ti,j =h2

∆2

1

m∗i +m∗j(3.10)

As the discretization is done in real space, the matrix elements are scalar quan-

tities. However, if orbital basis functions are used then each of the matrix elements

would be a sub-matrix of size equal to the number of orbital basis functions used.

Open boundary conditions have to be applied to Eq.(3.8) to account for charge

flow into and out of the non-equilibrium region.

Non-Equlibrium Green’s Function method

NEGF is a convenient approach for treating open systems and their interaction

with contacts. An exact treatment of semi-infinite contacts is possible.

In the sections that follow, the notations used in [15] will be adopted. The deriva-

tion follows the approach in [16].

The Green’s function for the whole device in the absence of scattering is

G = [EI −H]−1 (3.11)

Let

A′ = [EI −H] (3.12)

27

Since the total Hamiltonian can be partitioned into sub-Hamiltonians of left emit-

ter terminal (L), right collector terminal (R) and the non-equilibrium central device

region (D), A’ and G can be related as

A′G = I →

A′LL A′LD 0

A′DL A′DD A′DR

0 A′RD A′RR

GLL GLD GLR

GDL GDD GDR

GRL GRD GRR

(3.13)

From Eq.(3.13) it can be shown[A′DD − A′DLA′−1

LLA′LD − A′DRA′−1

RRA′RD

]GDD = I (3.14)

The 2nd and 3rd matrices on the left hand side of Eq.(3.14) will have only one

non-zero entry on the main diagonal as can be observed from the definition of the A′

sub-matrices.

A′LL =

• • •

• • •

−T †l4,l3 A′l3 −Tl3,l2−T †l3,l2 A′l2 −Tl2,l1

−T †l2,l1 A′l1

(3.15)

A′RR =

A′r1 −Tr1,r2−T †r1,r2 A′r2 −Tr2,r3

−T †r2,r3 A′r3 −Tr3,r4• • •

• • •

(3.16)

A′LD = A′†DL =

0 0 • • 0 0

0 0 • • 0 0

0 0 • • 0 0

0 0 • • 0 0

−TLD 0 0 • • 0

(3.17)

28

A′RD = A′†DR =

0 0 • • 0 −T ′RD0 0 • • 0 0

0 0 • • 0 0

0 0 • • 0 0

0 0 0 • • 0

(3.18)

Next the Green’s function for the isolated semi-infinite terminals is defined as

gL = A′−1LL and gR = A′−1

RR (3.19)

The surface Green’s function for the left and right terminals are Green’s function

elements corresponding to the edge layers l1 and r1 respectively.

gLl1,l1 = (A′−1LL )1,1 and gRr1,r1 = (A′−1

RR)1,1 (3.20)

Eq.(3.14) can be written using Eq.(3.20) as

[A′DD − ΣL − ΣR]GDD = I (3.21)

[EI −HDD − ΣL − ΣR]GDD = I (3.22)

where

ΣL = TDLgLl1,l1TLD (3.23)

ΣR = TDRgRr1,r1TRD (3.24)

are the self-energies for left and right terminals respectively.

Green’s function and Correlation function

The definition of Green’s function for the central non-equilibrium plus the equi-

librium reservoir region (D) follows from Eq.(3.22).

G′ = [EI −HDD − ΣL − ΣR]−1 (3.25)

29

Fig. 3.4. Computation of left connected Green’s function and fullyconnected Green’s function.

where the effect of the terminals has been folded into region D using ΣL and ΣR.

The submatrix of G′ corresponding to the non-equilibrium region alone would be

G = [EI −Hd − Σ1 − Σ2]−1 (3.26)

where Hd is the submatrix of Hamiltonian H corresponding to the non-equilibrium

region and Σ1 and Σ2 are the self-energies that take into account the coupling to the

left and right equilibrium reservoirs. Σ1 and Σ2 are related to the surface Green’s

function at layers l1 and r1 respectively.

Σ1 = Tl1,1gLl1,l1T1,l1 (3.27)

Σ2 = TN,r1gRr1,r1Tr1,N (3.28)

30

The in-scattering self-energies due to equilibrium reservoirs Σ1 and Σ2 are defined

as

Σin1 (E) = −2Im[Σ1(E)]f1(E) = Γ1(E)f1(E) (3.29)

Σin2 (E) = −2Im[Σ2(E)]f1(E) = Γ2(E)f2(E) (3.30)

where Γ1 and Γ2 are the broadening functions. The electron correlation function Gn

for the non-equilibrium region is given by

Gn = G(Σin1 + Σin

2 )G† (3.31)

In the next section a technique to quickly generate the necessary elements of G′

(of which G is a submatrix) and Gn will be presented.

Recursive Green’s function method

In order to determine the elements of G′, Eq.(3.24) and Eq.(3.25) show that gLl1,l1

and qRr1,r1 need to be computed first.

In the following derivation, only the left terminal will be treated. The right

terminal can be treated in a similar fashion. To determine gLl1,l1, the following

equation needs to be solved recursively.[A′l − T

†l gLl1,l1Tl

]gLl1,l1 = I (3.32)

Once we have gLl1,l1 the elements down the main diagonal of the left-connected

Green’s functions can be determined successively from Dyson’s equation

gLq+1q+1,q+1 =

(Aq+1,q+1 − Aq+1,qg

Lqq,qAq,q+1

)−1(3.33)

where the site index q+1 varies from ln to N. For indices ln to l1, an energy dependent

optical potential term iη is added to the diagonal elements of H appearing in A. The

energy dependence can be chosen to be either exponential or Lorentzian or a constant

value. This imaginary term determines the scattering induced broadening in the

31

equilibrium reservoir region just as the imaginary part of the self-energy (Eq.(3.30))

determines the intrinsic broadening due to coupling to the reservoirs. Because of η,

the density of states around the emitter quasi-bound state in the equilibrium reservoir

is broadened and takes the form

D(E) =η(E)/2π

(E − ε)2 + η(E)2(3.34)

When the index q + 1 reaches l1, the self-energy terms Σ1 and Σ2 defined in

Eq.(3.28) can be constructed.

The last element obtained from above step, gLNN,N is same as the last diagonal

element of the fully connected Green’s function, GN,N (or G′N,N) for the quantum

region.

Starting from the last element GN,N the other elements of G can be determined

successively using

Gq+1,q = −Gq+1,q+1Aq+1,qgLqq,q (3.35)

Gq,q+1 = −gLqq,qAq,q+1Gq+1,q+1 (3.36)

Gq,q = gLqq,q − gLqq,qAq,q+1Gq+1,q (3.37)

For the equilibrium reservoir region, Eq.(3.37) can be used to get the diagonal

elements of G′ which are the only ones needed for computing quantum charge in the

equilibrium region.

The correlation function for the non-equilibrium region can be obtained from G

through

Gn = G (Γ1f1 + Γ2f2)G† (3.38)

The above equation shows that only the 1st and Nth column of G are required to

compute Gn.

Similarly, for the equilibrium reservoir region, the diagonal elements of the corre-

lation function are

Gnq,q = if1

(G′q,q −G′†q,q

)(3.39)

32

Charge density

The electron density in the quantum region is obtained through

nq(E) = 2Gnq,q(E)

2π(3.40)

nq = 2

∫dE

2πGnq,q(E) (3.41)

3.3 Transport

For both Thomas-Fermi and Hartree models, transmission and current are calcu-

lated by using the NEGF formalism. The difference lies in the kind of self-consistent

potential that is passed on to the NEGF solver.

• In Thomas-Fermi method, the electrostatic potential is solved self-consistently

with semiclassical charge and then finally passed to the NEGF solver for deter-

mining the tranmission and current.

• In Hartree method, the electrostatic potential that is passed to the NEGF solver

is calculated self-consistently with the quantum charge obtained from a previous

NEGF calculation. Once the quantum charge and potential have converged,

the transmission and current from the last NEGF calculation are treated as the

result.

3.3.1 Transmission and Current

Both transmission and current are computed only within the non-equilibrium

region. The transmission at energy E is

T (E) = Tr[Γ1(E)G(E)Γ2(E)G†(E)

](3.42)

33

The above equation shows that the off-diagonal (1, N) element ofG is the only element

that is needed. An efficient technique to compute T(E) that uses only the (1, 1)

element is

A = GΓ1G† +GΓ2G

† (3.43)

T (E) = Tr{

Γ1

[A−GΓ1G

†]} (3.44)

In the phase coherent limit, the current can be calculated using

I =2e

h

∫dE

2πT (E) [f1(E)− f2(E)] (3.45)

The energy grid used for numerically evaluating the integrals in Eq.(3.41) and

Eq.(3.45) are determined based on the resonance levels found by the resonance finder.

Since the transmission probability is peaked at energies around the resonance level,

the contribution to the integrals will be more at these energies. So a finer grid would

be required to resolve the contribution at these energies.

Once the quantum charge density for the non-equilibrium and the equilibrium

reservoir region is obtained, it is concatenated with the semiclassical density obtained

for the flat band region in the terminals. From this charge profile, a quasi Fermi level

is extracted and a semiclassical density-Poisson self-consistent calculation is carried

out.

3.4 Self-consistent electrostatic potential calculation

The Poisson equation

d

dz

(ε(z)

dΦ

dz

)= q

(n(z)−N+

D

)(3.46)

is discretized to get

Fi =1

a2

(Φi−1ε

− − Φi(ε− + ε+) + Φi+1ε

+)

+ q(ND

+i − ni

)= 0 (3.47)

where

ε+ =εi + εi+1

2(3.48)

ε− =εi + εi−1

2(3.49)

34

Fig. 3.5. Simulation flow for Hartree self-consistent simulation for onebias voltage value.

Eq.(3.47) can be solved for Φ using Newton-Raphson iteration technique∑j

∂Fmi

∂Φmj

δΦm+1j = −Fm

i (3.50)

where

Φm+1 = Φm + δΦm+1 (3.51)

Eq.(3.50) requires ni and ∂ni/∂Φj

ni = NcF1/2

(EF i − Eci + qΦi

kBT

)(3.52)

∂ni∂Φj

= δi,jq

kBTNcF−1/2

(EF i − Eci + qΦi

kBT

)(3.53)

35

For Hartree method, using the quantum mechanical charge, Eq.(3.52) can be

inverted to obtain a quasi-Fermi level which is then used in the Jacobian calculation

step (Eq.(3.53).

3.5 Thomas-Fermi vs Hartree method

The potential profiles and current-voltage characteristics obtained from Thomas-

Fermi calculation and from Hartree self-consistent calculation differ mainly due to the

well charge that is taken into account in the case of Hartree self-consistent method.

The structure described in Table 3.1 will be used for the following discussion.

Layer Material Length Doping

(nm) (/cm3)

Lead1 GaAs 30 1× 1018

Spacer1 GaAs 10 1× 1015

Barrier1 AlGaAs 5 1× 1015

Well1 GaAs 5 1× 1015

Barrier2 AlGaAs 5 1× 1015

Spacer2 GaAs 10 1× 1015

Lead2 GaAs 30 1× 1018

Table 3.1GaAs/AlGaAs DBRTD structure used in the simulations.

3.5.1 Conduction band profile

The conduction band profile under non-equilibrium conditions is obtained by

adding the selfconsistent potential energy to the equilibrium conduction band profile.

Fig.3.6 shows the conductions band profile at 0.175 V obtained from a Thomas-Fermi

semiclassical and Hartree quantum self-consistent calculation. There is not much de-

36

Parameter Value

Temperature 300 K

Mesh size 0.2833nm

Cross-sectional area 100nm× 100nm

m∗GaAs 0.067mo

m∗AlGaAs 0.0919mo

εrGaAs 13.18

εrAlGaAs 12.0105

ECGaAs 1.424 eV

ECAlGaAs 1.7019 eV

Table 3.2Simulation parameters for GaAs/AlGaAs DBRTD structure used inthe simulations.

viation except that in the well region the Hartree method shows that the conduction

band and resonances have been raised in energy over the Thomas-Fermi result. This

is due to the influence of the electrons tunneling into the well.

3.5.2 Free charge density

A comparison of free charge density obtained from the 2 techniques is shown

in Fig.3.7. The blue curve labelled Thomas-Fermi + Hartree(1 pass) is obtained by

using the slef-consistent electrostatic potential obtained at the end of the Thomas-

Fermi semiclassical calculation in the NEGF quantum charge equation.

The semiclassical free charge density (in black) is zero in the barriers and quantum

well and it shows a trend of accumulation near the emitter/barrier interface and

depletion near the collector/barrier interface.

37

Fig. 3.6. Conduction band edge profile obtained from semiclassi-cal Thomas Fermi model (blue) and Hartree quantum self-consistentmethod (red). The upward shift for the Hartree result in the wellregion is due to the presence of charge in the well which pulls theconduction band edge and the resonance level upwards in energy asagainst the applied bias which tries to pull them down

38

The quantum charge (in red) in the center of the quantum well increases with

bias till the peak voltage is reached and then for higher biases it drops as the second

barrier’s effective height reduces.

39

Fig. 3.7. Comparison of electron density obtained from a Thomas-Fermi and Hartree quantum self-consistent caclculations for differentbias voltages applied to the DBRTD. The 2nd plot is at peak voltageVc = 0.2V

40

Fig. 3.8. IV characteristics computed from semiclassical ThomasFermi model (blue) and Hartree quantum self-consistent method (red)

3.5.3 Current

A comparison of the IV characteristics for the Thomas-Fermi and Hartree quantum

self-consistent techniques is shown in Fig. 3.8.

The Hartree model predicts a larger peak current and peak voltage. The reason

for this can be deduced from the conduction band edge profile for an intermediate

voltage point as shown in Fig. 3.6

A larger voltage is needed in the Hartree case to pull down the conduction band

and resonance level within the well, therefore, resulting in a larger peak voltage over

the semiclassically predicted one.

41

Fig. 3.9. Variation of first 2 resonances (in red) for Thomas-Fermi(left) and Hartree quantum self-consistent (right) methods. Theemitter Fermi level (in blue), the conduction band edge at the leftspacer/barrier interface (bottom black line) and the peak conductionband edge to the left of the barrier (top black line) are also shown.

This same observation can also be made from the resonance vs voltage plots

obtained from the two methods.

The bump in the resonance vs. voltage plot for the Hartree self-consistent method

is due to the charge in the well that is trying to pull the resonance up in energy.

Beyond the peak voltage, the resonance starts dropping rapidly with applied bias

similar to the case of Thomas-Fermi calculation. This is due to reduction in the

effective height of the second barrier as the bias is increased (Fig.3.10). The 2nd

barrier becomes more and more transparent and hence the well charge moves into the

collector side.

42

Fig. 3.10. Conduction band edge profile (top), quantum charge den-sity (center, blue) at 0.475 V and variation of quantum charge (bot-tom, orange), semiclassical charge (bottom, red) with bias in the non-equilibrium device region. As the 2nd barrier becomes shorter withbias, the well charge diminishes.

43

Fig. 3.11. Conduction band profile (left),transmission coefficient (cen-ter,blue), current density (center, dark green), normalized cumulativecurrent density (center, light green) and resonance energy vs voltageat peak voltage of 0.2V. The normalized cumulative current densityshows that the 1st resonance level is the major contributor to thetotal current.

3.6 Cumulative current density

The cumulative current density plot gives information about each resonance level’s

contribution to the total current that flows for a particular applied voltage. At peak

voltage, it can be seen from Fig.3.11 that nearly 92% of the total current is carried by

the 1st resonance. As the bias increases, the 1st resonance drops below the emitter

conduction band and ceases to contribute to current flow (Fig.3.12). The current that

flows then is entirely due to higher order resonances.

44

Fig. 3.12. Conduction band profile (left),transmission coefficient (cen-ter,blue), current density (center, dark green), normalized cumulativecurrent density (center, light green) and IV curve at a voltage of0.375V. The normalized cumulative current density shows that the1st resonance level is no longer contributing significantly to the totalcurrent.

45

3.7 2D-2D Tunneling

For high biases, as explained before, the band bending in emitter region close to

the barrier results in a triangular potential well. The narrow subbands that result

in this emitter notch can act as sources of electrons for the tunneling process. The

tunneling between these emitter quasi-bound states and the well quasi-bound states

is referred to as 2D-2D tunneling in contrast with the 3D-2D tunneling between the

bulk emitter states and the well subbands.

Fig. 3.13. Emitter quasi-bound state (left, orange) at an applied biasof 0.68V. Tunneling from this quasi-bound state into well resonancecontributes some current although its magnitude is much smaller thanthe current arising from the higher order well resonance.

46

Electrons from the emitter terminal are injected into the quasi-bound states in the

emitter by scattering processes resulting in some broadening which has been mimicked

by adding an imaginary potential term (iη) to the on-site Hamiltonian elements for

the emitter notch region. By including the emitter spacer region close to the barrier

in the NEGF equations, the quasi-bound states resulting from confinement within

the emitter notch can be accounted for automatically.

3.8 Summary

In this chapter, the modeling techniques used in the NEMO5 based RTD NEGF

simulation tool were discussed. The energy band diagrams resulting from a self-

consistent semiclassical Thomas-Fermi and Hartree quantum charge methods were

presented and it was seen that they were nearly identical but for the contribution of

the well charge in the Hartree method.

47

4. RESONANT INTERBAND TUNNELING DIODES

4.1 Introduction

The low PVR of RTDs has necessiated research into techniques to reduce the valley

current. Resonant interband tunneling diodes which combine the resonant transmis-

sion phenomenon through undoped layers of RTDs and the band-to-band tunneling

behaviour of Esaki diodes have been shown to offer good PVRs. An overview of Esaki

diode is presented first followed by the operation of an InAs/AlSb/GaSb RITD and

the NEGF simulation results with an sp3s* tight binding model.

4.2 Esaki diode

The Esaki diode consists of a degenerately doped n type semiconductor separated

from a degenerately doped p side by a very thin depletion region [17]. Electrons from

the conduction band on the n side can tunnel through the thin barrier into empty

valence band states on the p side at the same energy.

As can be seen from Fig.4.1, starting from 0 bias the electrons in the conduction

band on the n side see more and more empty valence band states at the same energy

until the bias reaches the peak voltage. Consequently the tunneling current increases

with voltage. Beyond the peak voltage the electrons in the conduction band on n

side see the bandgap of the p side material and the tunneling current drops. Beyond

the valley voltage the current again increases due to the usual diode like behaviour,

namely, injection of electrons into the conduction band on the p side.

The degenerate doping needed by the Esaki diode is a clear disadvantage since it

results in a high capacitance and can degrade the speed of operation. In RTDs this is

overcome by making the barrier and well regions undoped and using doped contacts

48

Fig. 4.1. Esaki diode operation.

49

to supply the carriers. The PVR of Esaki diodes on the other hand is usually much

better than RTDs.

RITDs combine the desirable features of Esaki diodes and RTDs in that their

PVR is good like Esaki diodes and like RTDs the barrier and well are undoped

resulting in a low capacitance and hence high switching speed as well as being easier

to fabricate [18], [19], [20].

4.3 InAs/AlSb/GaSb RITD

The conduction band profile of a typical RITD with AlSb barriers, GaSb quantum

well and InAs contact layers is shown in Fig.4.2.

Fig. 4.2. Equilibrium band diagram for an InAs/AlSb/GaSb RITD.

The InAs/AlSb/GaSb system form a Type II heterostructure with the conduction

offsets such that the valence band maximum of GaSb is above the conduction band

minimum of InAs. The electrons from the InAs emitter can tunnel through the AlSb

50

barrier into quantized valence band states in GaSb and then into the collector through

the 2nd AlSb barrier. Since the tunneling process involves conduction band states

in emitter and valence subbands in the well, it is referred to as interband tunneling.

The width of the quantum well is chosen such that a quantized valence band state is

present in the well at an energy above the conduction band edge of the emitter. The

overlap between the well valence subband and the emitter bands for energies below the

emitter Fermi level (assuming 0K) gives the number of electron k states particpating

in the tunneling process. At equilibrium there is an overlap between emitter bands

and the valence subband in the well (Fig.4.3). But because of the Fermi levels in the

emitter and collector being at the same energy, no current flows. When the positive

bias applied to the RITD increases, the valence subband minimum in the well drops.

For a particular value of the applied bias, the well subband minimum falls below the

conduction band edge in the emitter. This is when the current starts to decay. For an

intermediate value of the bias, the current rreaches a peak value. For more positive

biases, the tunneling electrons see the bandgap of GaSb and so the transmission

probability is greatly reduced. This is illustrated in Fig.4.4.

If the bias voltage is increased further, then the electrons from the conduction

band of emitter can tunnel into conduction subbands in the GaSb quantum well for

electrons. This is the regular resonant tunneling phenomenon that was described in

the preceding chapters.

The advantage of RITDs is that in the valley current regime, the electrons have

to tunnel through not only the AlSb barriers but also through the bandgap of GaSb

layer which is relatively thick. Therefore the transmission probability is reduced for

the voltages in the valley regime and this translates into a higher PVR.

51

Fig. 4.3. Equilibrium band diagram and overlap between InAs emit-ter bands and GaSb valence subband dispersions along kx for anInAs/AlSb/GaSb RITD.

Fig. 4.4. Band diagram for voltages greater than peak voltage andoverlap between InAs emitter bands and GaSb valence subband dis-persions along kx for an InAs/AlSb/GaSb RITD.

52

4.4 Multiband modeling

As the operation of RITDs involves both the conduction band and the valence

band, the single band effective mass model described in the previous chapter is

inadequate. Atleast a 2 band model is needed to compute the IV characteristics

[21], [22], [23]. The NEMO5 simulator in conjunction with an sp3s* tight binding

model that takes into account electron spin-orbit coupling has been used to study an

InAs/AlSb/GaSb RITD.

The RITD is treated as a sequence of monolayers that are parallel to the interfaces.

Let M represent the number of orbitals per unit cell in the tight binding basis set

(M = 20 for sp3s* model with spin orbit coupling included). The basis orbitals can

be written as∣∣R||σα⟩, where σ = 1, 2, ...,M is the integer monolayer label, R|| is the

in-plane component of unit cell coordinate and α = 1, 2, ...,M represents the orbitals

within a unit cell.

The wave function can be written as

|Ψ〉 =∑σ,α

Cσα∣∣σα,k||⟩ (4.1)

where∣∣σα,k||⟩ is a planar orbital formed by taking Bloch sums of tight-binding

orbitals over the N|| unit cells in the σth monolayer∣∣σα,k||⟩ =1√N||

∑R||

ejk||.R||∣∣R||σα⟩ (4.2)

The Schrodinger equation (H −E) |Ψ〉 = 0 can be expressed in the planar orbital

basis as

Hσ,σ−1Cσ−1 + Hσ,σCσ + Hσ,σ+1Cσ+1 = 0 (4.3)

where Cσ is a vector of length M ,

Cσ =

Cσ1

Cσ2

...

CσM

(4.4)

53

and Hσ,σ′ and Hσ,σ are M ×M matrices whose elements are given by

(Hσ,σ′)α,α′ =⟨σα,k|| |H|σ′α′,k||

⟩(4.5)

and

(Hσ,σ

)α,α′

=⟨σα,k|| |(H − E)|σα′,k||

⟩(4.6)

where the resultantH is the tight binding Hamiltonian whose matrix elements (orbital

on-site energies and overlap elements) are chosen in such a way that they reproduce

the bulk band gaps along the important crystal symmetry directions [24], [25].

Once the Hamiltonian has been expressed in the planar orbital basis, the Green’s

function for the device and self-energies can be derived and the charge density com-

puted in a manner analogous to that described in Chapter 3 for effective mass mod-

eling. The matrix elements in this case would be of size M ×M .

4.5 Current density

The expression for current Eq.(3.45) assumes a parabolic dispersion in the trans-

verse direction. In the case of multiband calculation, the bandstructure along the

transverse direction will appear in the current calculation through the dependence of

transmission probability and occupation factor on k|| [10].

J =2q

(2π2)h

∫ ∫T (E,k||)[fE(E)− fC(E)]dEdk|| (4.7)

4.6 Simulation Results

4.6.1 IV characteristics

The RITD structure simulated using NEMO5 simulator is described in Table 4.1

A Thomas-Fermi semiclassical calculation was performed for this structure and the

IV curve obtained is shown in Fig.4.5 The simulated IV curve shows that the PVR is

50 and the valley current has been reduced considerably compared to the peak current.

54

Layer Material Monolayers Doping

(/cm3)

Lead1 InAs 100 1× 1018

Spacer1 InAs 16 1× 1015

Barrier1 AlSb 4 1× 1015

Well1 GaSb 16 1× 1015

Barrier2 AlSb 4 1× 1015

Spacer2 InAs 16 1× 1015

Lead2 InAs 100 1× 1018

Table 4.1InAs/AlSb/GaSb RITD structure used in the simulations.

Fig. 4.5. IV curve for the InAs/AlSb/GaSb RITD at 300K showing a PVR of 50.

55

The valley current region is also relatively broad extending from around 0.3V to 0.7V.

The rise in the current beyond the valley region is attributed to resonant tunneling of

InAs emitter valence band electrons through valence subband in GaSb well and into

empty conduction band states in the InAs collector. In this simulation, only ballistic

transport of electrons is treated. When scattering mechanisms are included, the PVR

is expected to be less than the value obtained from this simulation.

4.6.2 At peak and valley voltage

Current density

The current density J(kx, ky) at peak and valley voltage for 3 different radial k

directions of ky = 0,kx = 0 and kx = ky are shown in Figs.4.6,4.7 and 4.8. The

plots corresponding to peak voltage for all 3 directions look similar. This is due to

isotropic nature of the transverse dispersion. Likewise the plots corresponding to

valley voltage are also identical. For peak voltage, majority of the current flow is due

to the overlap of states around the Γ point as pictured in Fig.4.3. At valley voltage,

the overlap between the energy broadened Γ states (Fig.4.4) is the major contributor

to the current.

Cumulative current density

The cumulative current density at peak voltage and valley voltage along kx is

shown in Fig.4.9. For a particular kx, additional overlap of states between the InAs

conduction and GaSb valence subbands occurs at higher energies. This leads to a

step-like increase in the cumulative current density.

Local density of states

Fig.4.10 shows the energies where the current is flowing. This appears as dark

regions in the local density of states plot.

56

Fig. 4.6. Energy integrated current density J(kx, ky) along kx direc-tion at peak voltage (top) and at valley voltage (bottom).

57

Fig. 4.7. Energy integrated current density J(kx, ky) along ky direc-tion at peak voltage (top) and at valley voltage (bottom).

58

Fig. 4.8. Energy integrated current density J(kx, ky) along kx = kydirection at peak voltage (top) and at valley voltage (bottom).

59

Fig. 4.9. Cumulative current density along ky = 0 at peak voltage(top) and at valley voltage (bottom).

60

Fig. 4.10. LDOS profile along with current density plot for ky = 0direction showing the energy at which the current is flowing for peakvoltage (top) and at valley voltage (bottom).

61

4.6.3 At high bias (1.1V)

Current density

The current density along with the LDOS plot shows that the current is from

tunneling of valence band electrons in InAs emitter through GaSb valence subband

to InAs collector conduction band.

Fig. 4.11. LDOS and current density at 1.1V.

Cumulative current density

The cumulative current density plot at 1.1V shows the k point values where the

current is concentrated for different k directions. Again as in the case of peak current,

most of the current is due to overlap between the subbands away from the Γ point.

62

Fig. 4.12. LDOS and current density at 1.1V along kx (top), ky (mid-dle) and kx = ky (bottom).

63

4.7 Applications

Due to their superior PVRs compared to RTDs, InAs/AlSb/GaSb RITDs have

been used to realize logic circuits along with Schottky diodes [26] and also by inte-

grating with HEMTs [27].

4.8 Summary

In this chapter, the operation of InAs/AlSb/GaSb RITDs was described. Due

to the involvement of conduction and valence band a multiband modeling method

is required for RITDs. An sp3s* tight binding model with spin orbit coupling and

NEGF was used to study the characteristics of a coherent RITD. The IV curve for

the considered RITD structure does show a high PVR of the order of 50.

64

5. 1DHETERO AND BRILLOUIN ZONE VIEWER

5.1 1dhetero tool

The 1dhetero tool is a NEMO5 based atomistic simulation tool for studying the

electrostatic properties of 1 dimensional heterostructures [28]. It can be used for

studying the equilibrium properties of channels in HEMTs built using different semi-

conductor materials. The substrate materials that are supported are listed in Table

5.1.

Substrate Materials supported

GaAs GaAs

AlGaAs

AlAs

GaP GaP

AlP

GaSb GaSb

AlSb

InP InP

In53GaAs

In52AlAs

Si Si

Ge

SiO2

Table 5.1Substrates and materials supported by 1dheterostructure tool

65

The tool solves the coupled 1D Schrodinger and Poisson equations self-consistently

along the growth direction for each value of gate voltage. The source and drain

are assumed to be in equilibrium. Hardwall boundary conditions are used for the

Schrodinger equation. For Poisson equation a Dirichlet boundary condition is used

for the gate side and for the bulk side.

Φ(z = 0) = VG (5.1)

Φz = L = 0 (5.2)

The domain for Poisson equation includes the substrate while for computing wave-

functions using Schrodinger equation it is excluded meaning it is assumed that the

wavefunctions decay to 0 along the position axis before reaching the substrate region.

Three types of Hamiltonian models are available -

1. Single band model

2. Tight binding sp3s* model with spin orbit coupling

3. Tight binding sp3d5s* model with sping orbit coupling

Apart from this there is also a semiclassical option that performs a selfconsistent

Thomas-Fermi semiclassical density-Poisson calculation. Since no wave function for-

malism is employed for this option, no Schrodinger equation is solved and hence no

resonances and wave functions will be displayed for this option. The advantage of this

semiclassical approach is that it is very fast even though the electrostatic potential

profile and the free charge density obtained may not be physically correct (Fig.5.5).

Free charge density

The free charge density can be computed using two different techniques

1. The analytical expression for the free charge density is

n(z) =m∗kBT

πh2

∑i

|Ψi(z)|2ln[1 + e

EF−EikBT

](5.3)

66

where Ei is the subband minimum energy. The above equation assumes the

subband dispersion is parabolic in the transverse k plane as is the norm for

effective mass simulation.

2. The numerical k-space summation that considers the exact dispersion in

the transverse k plane, E(kx, ky) is a more rigorous way to account for non-

parabolicity.

n(z) = 2∑i

∑kx,ky

|Ψi(z)|2fo(Ei + E(kx, ky)− EF ) (5.4)

where Ei + E(kx, ky) is the exact dispersion for the subband. The kx, ky grid

for the summation can be set by the user. This technique is used for the tight

binding options.

Outputs

Visualization of the following output results is possible -

1. Conduction/Valence band profile, Fermi level and inversion layer resonances

2. Electrostatic potential

3. Sheet density

4. Resonance energy vs gate voltage

5. Wavefunctions

6. Charge density

7. Doping density

67

Fig. 5.1. Design section of the 1dhetero tool where the dimensionsand doping density of the materials and substrate constituing theheterostructure can be specified.

68

Fig. 5.2. Equilibrium conduction band profile and bound state reso-nances for the default AlGaAs/GaAs heterostructure calculated froma single band simulation.

Fig. 5.3. Equilibrium potential profile for the default AlGaAs/GaAsheterostructure calculated from a single band simulation. The poten-tial peak is due to depletion of carriers in the doped AlGaAs barrier.

69

Fig. 5.4. Equilibrium sheet density profile for the default Al-GaAs/GaAs heterostructure calculated from a single band simulation.A 2DEG is formed near the AlGaAs-GaAs interface resulting in a veryhigh sheet concentration of 8.8× 1012/cm2.

Fig. 5.5. Equilibrium sheet density profile for the default Al-GaAs/GaAs heterostructure calculated from a semiclassical Thomas-Fermi simulation. The semiclassical method predicts maximum sheetdensity at the AlGaAs-GaAs interface whereas the single band cal-culation result of Fig.5.4 shows maximum sheet concentration at thecenter of the inversion layer, away from the AlGaAs-GaAs interface.

70

5.2 Brillouin Zone Viewer tool

This is a visualization tool that computes and displays the first Brillouin zone for

various crystal lattice systems [29]. The Brillouin zone is the primitive or Wigner-

Seitz cell of the reciprocal lattice. It includes all the k-points that are closer to a

reciprocal lattice point than its neighbours. The Bloch wavefunctions and the energy

band for periodic crystals can be completely characterized by their behaviour within

a Brillouin zone.

The lattice systems available are-

1. Cubic systems - Simple cubic, FCC, BCC

2. Wurtzite - Hexagonal

3. Honeycomb - Graphene

4. Rhombohedral

The inputs to the tool are the lattice translational vectors which don’t necessarily

have to be that of the conventional unit cell. For cubic crystal systems, the lattice

vectors can be specified either in terms of the standard orthogonal co-ordinate sys-

tem basis or in the basis of the primitive translational vectors. For example, the

translational vectors for FCC lattice are

−→a1 =a0

2(−→ax +−→ay) (5.5)

−→a2 =a0

2(−→ay +−→az) (5.6)

−→a3 =a0

2(−→ax +−→az) (5.7)

where a0 is the lattice constant. In the standard xyz orthogonal co-ordinate sys-

tem basis, the above translational vectors can be specified using the components of

−→ax,−→ay ,−→az , i.e., as (110),(011) and (101) respectively.

In the basis of the primitive vectors, the same vectors can be specified as com-

ponents of −→a1 ,−→a2 ,−→a3 , i.e., as (100),(010) and (001) respectively. This is the default

specification option for non-cubic systems used in the tool.

71

Fig. 5.6. Brillouin zone viewer tool showing the 1st Brillouin zone forthe cubic FCC lattice.

72

6. SUMMARY

In this thesis, the NEMO5 based RTD NEGF simulation tool was presented for

GaAs/AlGaAs RTDs. The procedure used and results obtained from a coherent

simulation of 1D transport in GaAs/AlGaAs RTDs using effective mass model and

NEGF were demonstrated. The effect of scattering in the emitter reservoir region has

been treated by including an optical potential term in the self-energy calculation. This

technique eases the computational burden that would otherwise have to be tolerated if

scattering were treated in a rigorous fashion. For improved results and a closer match

to experimental IV, a realistic treatment of electron-phonon interaction throughout

the device via self-energies would be necessary.

Future work for this tool would be the addition of tight binding option to account

for atomistic effects. Material system apart from GaAs/AlGaAs such as GaAs/AlAs,

InGaAs/InAlAs, etc. could be a useful addition to the tool. The wafer orientation

for the RTDs, which has been set to (100) here, could be included as an option.

(111) oriented GaAs/AlAs RTDs have been reported with PVRs larger than (100)

counterparts [30]. A simulation tool capabling of handling different wafer orientations

of RTDs would be a valuable aid.

Apart from effective mass simulation of RTDs, an sp3s* tight binding model was

used to simulate an InAs/AlSb/GaSb coherent RITD. The PVR for RITDs tend to

be higher due to the reduction in the valley current by introduction of the GaSb layer.

A simulation tool similar to RTD NEGF can be developed exclusively for studying

the characteristics of InAs/AlSb/GaSb RITDs.

Two other simulation tools - 1dheterostructure and Brillouin zone viewer powered

using NEMO5 were also developed during the course of this work and the modeling

techniques used were described in the concluding chapter. The 1dheterostructure tool

can be used for simulation of heterostructures such as HEMT structures with only a

73

gate voltage applied and no source to drain voltage. A natural extension to this could

be the addition of source to drain voltage and the option of determining the current

voltage characteristics. An accurate treatment of contacts will have to be included

for this. As for the Brillouin zone viewer, crystal lattice systems apart from cubic,

honeycomb, hexagonal and rhombohedral could be added.

LIST OF REFERENCES

74

LIST OF REFERENCES

[1] L.L. Chang, L. Esaki, and R. Tsu, “Resonant tunneling in semiconductor doublebarriers,” Applied Physics Letters, vol. 24, p. 593, 1974.

[2] T.C.L.G. Sollner, W.D. Goodhue, P.E. Tannenwald, C.D. Parker, and D.D. Peck,“Resonant tunneling through quantum wells at frequencies up to 2.5 THz,” Ap-plied Physics Letters, vol. 43, p. 588, 1983.

[3] Pinaki Mazumder, Shriram Kulkarni, Mayukh Bhattacharya, Jian Ping Sun, andGeorge I. Haddad, “Digital Circuit Applications of Resonant Tunneling Devices,”Proceedings of the IEEE, vol. 86, April 1998.

[4] J.P.A. van der Wagt, A.C. Seabaugh, and E.A. Beam, III, “RTD/HFET LowStandby Power SRAM Gain Cell,” IEEE Electron Device Letters, vol. 19, Jan-uary 1998.

[5] B. Prince, Emerging memories: technologies and trends. Kluwer Academic Pub-lishers, 2002.

[6] A. Seabaugh, B. Brar, T. Broekaert, F. Morris, P. van der Wagt, and G. Frazier,“Resonant-tunneling mixed-signal circuit technology,” Solid-State Electronics,vol. 43, pp. 1355–1365, 1999.

[7] B.Ricco and M.Ya.Azbel, “Physics of resonant tunneling. The one-dimensionaldouble-barrier case,” Physical Review B, vol. 29, p. 1970, February 1984.

[8] H. Mizuta and T. Tanoue, The Physics and Applications of Resonant TunellingDiodes. Cambridge University Press, 1995.

[9] Jian Ping Sun, George I. Haddad, Pinaki Mazumder, and Joel N. Schulman,“Resonant Tunneling Diodes: Models and Properties,” Proceedings of the IEEE,vol. 86, p. 641, April 1998.

[10] R.C. Bowen, G. Klimeck, and R.K. Lake, “Quantitative simulation of a resonanttunneling diode,” Journal of Applied Physics, vol. 81, no. 7, 1996.

[11] R. Tsu and L. Esaki, “Tunneling in a finite superlattice,” Applied Physics Letters,vol. 22, pp. 562–564, 1973.

[12] S. Luryi, “Frequency limit of double-barrier resonant-tunneling oscillators,” Ap-plied Physics Letters, vol. 47, p. 490, 1985.

[13] S. Luryi, “Mechanism of operation of double-barrier resonant-tunneling ocsilla-tors,” International Electron Devices Meeting, 1985, pp. 666–669, 1985.

[14] S. Luryi, “Coherent versus incoherent resonant tunneling and implications forfast devices,” Superlattices and Microstructures, vol. 5, p. 375, 1989.

75

[15] S. Datta, Quantum Transport: Atom to Transistor. Cambridge University Press,2005.

[16] M.P. Anantram, Mark S. Lundstrom, and Dmitri E. Nikonov, “Modeling ofNanoscale Devices,” Proceedings of the IEEE, vol. 96, September 2008.

[17] L. Esaki, “New Phenomenon in Narrow Germanium p-n Junctions,” PhysicalReview, vol. 109, p. 603, 1958.

[18] M. Sweeny and J. Xu, “Resonant interband tunnel diode,” Applied Physics Let-ters, vol. 54, p. 546, 1989.

[19] J.R. Soderstrom, D.H. Chow, and T.C. McGill, “A new negative differential re-sistance device based on resonant interband tunneling,” Applied Physics Letters,vol. 55, p. 1094, 1989.

[20] L.F. Luo, R. Beresford, and W.I. Wang, “Interband tunneling in polytypeGaSb/AlSb/InAs heterostrutures,” Applied Physics Letters, vol. 55, p. 2023,1989.

[21] M. Ogawa, T. Sugano, and T. Miyoshi, “Multi-band simulation of quantumtransport in resonant interband tunneling devices,” Physica E: Low-dimensionalSystems and Nanostructures, vol. 7, p. 840, 2000.

[22] M. Ogawa, T. Sugano, and T. Miyoshi, “Tight binding simulation of quantumelectron transport in type II resonant tunneling devices,” Extended Abstracts of1998 Sixth International Workshop on Computational Electronics, p. 152, 1998.

[23] Roger Lake, Gerhard Klimeck, R. Chris Bowen, and Dejan Jovanovic, “Singleand multiband modeling of quantum electron transport through layered semi-conductor devices,” Journal of Applied Physics, vol. 15, June 1997.

[24] G. Klimeck, R.C. Bowen, T. Boykin, and T.A. Cwik, “sp3s* Tight-Binding Pa-rameters for Transport Simulations in Compound Semiconductors,” Superlatticesand Microstructures, vol. 27, no. 5-6, pp. 519–524, 2000.

[25] P. Vogl, H. P. Hjalmarson, J. D. Dow, and T.A. Cwik, “sp3s* Tight-BindingParameters for Transport Simulations in Compound Semiconductors,” J. Phys.Chem. Solids, vol. 44, no. 5, pp. 365–378, 1983.

[26] D.H. Chow, H. L. Dunlap, W. Williamson, S. Enquist, B. K. Gilbert, S. Subra-maniam, P.M. Lei, and G. H. Bernstein, “InAs/AISb/GaSb Resonant InterbandTunneling Diodes and Au-on-InAs/A1Sb-Superlattice Schottky Diodes for LogicCircuits,” Electron Device Letters, vol. 17, Febuary 1996.

[27] B.R. Bennett, A.S. Bracker, R. Magno, J.B. Boos, R. Bass, and D. Park, “Mono-lithic integration of resonant interband tunneling diodes and high electron mo-bility transistors in the InAs/GaSb/AlSb material system,” JVST B - Micro-electronics and Nanometer Structures, vol. 18, no. 3, 2000.

[28] 1D Heterostructure Tool, 2011. https://nanohub.org/tools/1dhetero.

[29] Brillouin Zone Viewer, 2011. https://nanohub.org/tools/brillouin.

[30] L.F. Luo, R. Beresford, W.I. Wang, and E.E. Mendez, “Inelastic tunnelingin (111) oriented AlAs/GaAs/AlAs double-barrier heterostructures,” AppliedPhysics Letters, vol. 54, no. 21, 1989.

APPENDIX

76

A. RTD NEGF - USER OPTIONS

The GUI of RTD NEGF simulation tool was developed using the Rappture program-

ming framework. The user options available on the tool will be described in the

following sections.

A.1 Basic options

1. Ambient Temperature - Temperature value that will be used for the Fermi

functions. Default options include 300K (room temperature) and 77K (liquid

Nitrogen boiling point).

2. Starting Bias - Bias voltage at which the emitter terminal will be held fixed.

3. Ending Bias - The collector terminal voltage will be ramped up from the

starting bias voltage value to this value.

4. No. of Points - Determines the bias voltage steps.

5. Potential Model - Thomas-Fermi or Hartree.

6. Quantum Charge - On/off option that is enabled only for Thomas- Fermi

simulation. If this option is set to be OFF then only semiclassical Thomas-

Fermi charge will be displayed. If set to be ON then quantum charge resulting

from NEGF transport calculation will also be displayed.

7. Lattice Constant - Determines real space grid size. Normally half the lattice

constant of the unit cell. In other words thickness of one monolayer of the

substrate.

77

Fig. A.1. RTD NEGF simulation tool showing Basic tab options.

78

Fig. A.2. Multiscale Domains tab options.

A.2 Multiscale Domains options

1. Semiclassical Charge Region - Region where charge is treated semiclassically.

This is used for calculating charge for Hartree model and also for transport

calculation for both Thomas-Fermi and Hartree models. Usually the region

defined as terminal with flat band conditions and in equilibrium.

2. Equilibrium Region - The dimension of the equilibrium reservoir region.

A.3 Advanced options

1. Not-normalized Current Plot - Switching this option to OFF will show the

normalized current density plot.

2. Resonances Scatter Plot - The resonance vs voltage plot will appear as a

scatter plot.

3. Carrier Distribution Surface Plots - Turning this option ON will display

the energy resolved electron density profile.

4. Reservoir Relaxation Model - Option to choose the energy dependence of

η used to account for scattering in the equilibrium reservoir. η can be made to

79

Fig. A.3. Advanced tab options.

reduce exponentially with energy below the emitter quasi-bound state or in a

Lorentzian fashion or in a fixed manner.

5. Reservoir Relaxation Energy - For exponential and Lorentzian roll-off, this

determines the maximum energy value of η.

6. Decay Length - For exponential and Lorentzian roll-off, this determines how

fast with energy η reduces.

7. Use Adaptive Energy Grid - Can be used to control whether an adaptive

energy grid or a fixed energy grid is used for calculation of transmission and

current.

8. Poisson Criterion - Convergence criterion for Poisson-NEGF charge self-consistent

computation (Hartree) or Poisson-semiclassical charge self-consistent computa-

tion (Thomas-Fermi and inner loop of Hartree).

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Fig. A.4. Resonance Finder tab options.

A.4 Resonance Finder options

1. Resonance Finder - Option to turn ON/OFF display of resonances and wave-

functions.

2. No. of homogeneous grid points - Number of energy grid points used by

resonance finder

3. No. of points per resonance

4. No. of points for Ec EF

5. Lanczos iteration step size - Option for algorithm used to roughly estimate

the location of resonances along energy axis.

6. Lanczos iteration limit - Option for algorithm used to roughly estimate the

location of resonances along energy axis.

7. Newton iteration step size - Option for algorithm used to narrow down the

energy location of the estimated resonance.

8. Newton solver convergence condition - Option for algorithm used to narrow

down the energy location of the estimated resonance.

9. Left exclusion region - Region to the left of the 1st barrier that will be

excluded from the spatial domain used for determining resonance wavefunctions.

81

10. Right exclusion region - Region to the right of the last barrier that will be

excluded from the spatial domain used for determining resonance wavefunctions.

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NEGF SIMULATION OF ELECTRON TRANSPORT ... - Purdue Engineering · NEGF SIMULATION OF ELECTRON TRANSPORT IN RESONANT TUNNELING AND RESONANT INTERBAND TUNNELING DIODES A Thesis Submitted