Network for Computational Nanotechnology (NCN)Purdue, Norfolk State, Northwestern, MIT, Molecular Foundry, UC Berkeley, Univ. of Illinois, UTEP

NEGF Simulation of Electron Transport in Resonant Tunneling and

Resonant Interband Tunneling Diodes

A. Arun GoudNetwork for Computational Nanotechnology (NCN)

Electrical and Computer Engineering

11/28/2011

A.Arun Goud

Beyond CMOS

Scaling challenges Leakage effects – High k dielectrics Gate control – Non-planar structures Variability – Process improvement Mobility – Strain, III-V

For the last 3 decades CMOS scaling driven by Moore’s law has been the norm

ITRS 2009 - Emerging Research Devices

Another line of thought…Quantum mechanical effects Tunneling Interference Quantization, etc.

Emerging devices will have to utilize these effects while delivering high performance (high speed, low power consumption)

A.Arun Goud

Outline

Example of a quantum device – Resonant tunneling diode (RTD) Characteristics Applications…Why show interest in RTDs? Shortcomings...Why RTDs are not common? Simulation tool using NEMO5…To understand Physics behind RTDs NEGF formalism…A quantum formalism to calculate charge and current

Resonant interband tunneling diode (RITD) Alternative to RTDs Overcomes some drawbacks with RTDs Modeling of RITDs

Two other simulation tools –1dheteroBrillouin zone viewer

A.Arun Goud

Quantum device – RTD (GaAs/AlGaAs)

First demonstrated by Chang, Esaki and Tsu (1974)Grown using MBEVertical devices current flows along growth direction

n GaAsn GaAs

AlxGa1-xAsAlxGa1-xAs

GaAsGaAsAlxGa1-xAsAlxGa1-xAs

n+ GaAsn+ GaAs

n GaAsn GaAs

n+ GaAsn+ GaAs

L < Phase coherence length

z

V

I

(a) (b) (c)

(a)

(b)

(c)

VvVp

Ip

Iv

Peak to Valley Current Ratio = Ip/Iv (figure of merit)Requirements Large Ip, low Iv.

IV characteristics showing NDR

A.Arun Goud

Motivation - RTDs for digital applications

RTDs have been used for microwave circuits such as oscillators due to NDR.

Oscillations as high as 2.5 THz! (TCLG Sollner, Applied Physics Letters 43: 588)

6T SRAM memory cellRTD latch

(a) Ultra-high switching speeds (b) Not transit time limited(c) Low voltage

Digital circuit applications?

Multi-Functional devices -

YES!

Peak current should be larger than leakage currents of read/write FETsElse there is unwanted state transition

Simulation models needed.Should be Physics driven instead of compact model

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So why are RTDs not widespread

Compatibility with mainstream Si technology?

2 terminal No isolation

Low drive capabilities. Peak current, PVR must be increased

More importantly,

AlGaAs/GaAs, InGaAs/InAlAs, etc are popular choices but not compatible with Si technology and are expensive Si/SiGe RTDs have been demonstrated. Tend to have poor PVRs at 300K…

Advances in MBE, integration techniques Viable way to integrate RTDs with mainstream processes is likely (InP based RTD/HEMTs already exist)

Device variations from die to die

Perfect Lab for studying quantum phenomena - Physics involved and Simulation techniques devised will be useful for analyzing other devices too

So is the emphasis laid on RTDs totally unfounded?

A.Arun Goud

Contribution - RTD NEGF tool

Features - Coherent simulation of GaAs/AlGaAs RTDs - Charge density 1. Semiclassically (Thomas-Fermi)

2. Quantum self-consistent (Hartree)

- Effective mass Hamiltonian - NEGF formalism for transport Scattering/Relaxation in emitter reservoir NEMO5 driven

Output -• Energy band diagram, Resonance levels• Transmission coefficient• Well, Emitter quasi-bound |Ψ|2 • Current density• IV• Charge & sheet density profiles• Resonances vs voltage• Energy resolved charge profiles

a) Charge - 1. Thomas-Fermi method 2. Hartree method (NEGF)b) Transport - NEGF

A.Arun Goud

RTD modeling – Thomas-Fermi

Free charge density non-zero only in reservoirs

Thomas-Fermi expression

Solved iteratively with Poisson’s equation. BCs are φ(z=L)=V and φ(z=0)=0

The converged potential is used by NEGF solver to calculate current

A.Arun Goud

RTD modeling - Hartree

Charge treated semiclassically in terminalsQuantum charge calculated in Quantum regionCurrent calculated only in Non-equilibrium region

A.Arun Goud

NEGF - Quantum Charge and Current

gN,N = GN,N

Only 1st and Nth column of G are needed

1. RGF method2. Dyson’s equation3. iη relaxation model

EQ

NEQ

(Right contact will be ignored in thisexplanation )

Mimics broadening just as imaginary part of

A.Arun Goud

Simulation flow – Thomas-Fermi

Described in previous slide

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Simulation flow - Hartree

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Thomas-Fermi vs Hartree

Hartree

Thomas-Fermi

Quantization Low charge density => Low potential energy

Well charge CB raises to block further flow of charges into well

Hartree Vp > TF Vp

IVCB profile

Well charge vs Bias

Resonance drops below Ec slower w.r.t bias in Hartree method than in Thomas-Fermi method

PVR = 2

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Approximations made

Parabolic transverse dispersion• Higher order subband minima are overestimated => 2nd and further turn-on voltages are overestimated

Transverse energy and momentum are separable • T(E,k||) T(Ez) => Current calculation involves integration over only Ez

Full transverse dispersion and integration over k|| for exact analysis of coherent RTDsScattering self-energies also for incoherent simulation

J. Appl. Phys. 81 (7), 1997

A.Arun Goud

Recap

Resonant tunneling diode (RTD) Characteristics…NDR Applications…Memory Shortcomings...Low PVR at 300K Simulation tool…To understand Physics behind RTDs NEGF formalism…To calculate current

Is there a way to increase PVR?...We can draw inspiration from the Esaki diode

A.Arun Goud

From Esaki diodes to RTDs to RITDs

Esaki diode operation -

1) High peak to valley current ratio due to drastic reduction in valley current

2) Major drawback - Heavily doped junctions difficult to produce - High capacitance which degrades speed of operation

V

I

In the case of RTD’s,

Barriers are not effective in reducing valley current – low PVR Barriers and well are undoped – low capacitance

We need a mix

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Esaki diode + RTD = RITDs

InAs/AlSb/GaSb RITD

Multiband model is needed for proper description.

- Type II broken gap- Interband like Esaki diode

Exhibit larger PVR at 300K than RTDs by reducing valley current.

InAs non-parabolicityMixing of CB, VB states

A.Arun Goud

Tight binding Hamiltonian

Form Bloch sum of localized orbitals in the transverse plane

z

||α Cation or anion orbitals (10 for sp3s*)σ Layer index

Δ=a0/2

σ1 σ2

v

… …

Wavefunction is expressed in terms of planar orbitals in each layer

Real space Schroedinger equation

can be transformed to this basis using

CationAnion

Open boundary conditions using NEGF

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RITD multiband simulation IV

Valley region is broad because effectively electrons see bandgap of AlSb+GaSb+AlSb layers

1. Thomas-Fermi charge model

2. sp3s* TB model with spin orbit coupling

3. Numerical k|| integration to compute current

PVR = 50

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0 1 2 3 4

J(kx) at Vp and Vv

a = 0.6058 nm2π/a = 10.37 /nm

kx,ky grid (0.15,0.15) * 2π/a

kx

ky

Majority of the current is due to tunneling through Г state

A.Arun Goud

Energy resolved electron density

At peak voltage

At valley voltage

A.Arun Goud

1dhetero

Features Schroedinger-Poisson solver 3 options for Hamiltonian- Single band- TB sp3s* with spin-orbit coupling- TB sp3d5s* with spin-orbit coupling Semiclassical density-Poisson option Choice of substrates

ApplicationDesign and study of electrostatics within HEMTs

Sheet charge density Analytical method – Parabolic transverse dispersion Numerical – Transverse dispersion from TB calculation used

Outputs1. Energy band diagram2. Potential3. Resonances4. Wavefunction magnitude squared5. Sheet density, doping density6. Resonance vs voltage

Gate Bulk

Schroedinger domain

Poisson domain

Users 281

SimulationSessions

1421 (WCT– 104 days)

http://nanohub.org/1dhetero/usage

A.Arun Goud

Brillouin Zone viewer

ApplicationVisualization of 1st Brillouin zones for lattice system Cubic (SC,BCC,FCC) Hexagonal (Wurtzite) Honeycomb (Graphene) Rhombohedral (Bi2Te3)

InputTranslational vectorsLattice constant

Output1st Brillouin zoneReal space unit cell Users 61

SimulationSessions

157(WCT – 5 days)

http://nanohub.org/brillouin/usage

A.Arun Goud

Summary

RTD NEGF Coherent simulation of GaAs/AlGaAs RTDs using effective mass model and NEGF for transport Relaxation in equilibrium reservoir modeled using imaginary optical potential term iη Future work – Implementation of self energy expressions for various scattering mechanisms, (111) wafer orientation

RITD multiband simulationA coherent InAs/AlSb/GaSb RITD was simulated using NEMO5 with sp3s* SO model

1dhetero toolSimulation tool for the study and design of 1D heterostructures using a choice of substrates

Brillouin zone viewerSimulation tool for visualizing the 1st Brillouin zones of cubic, hexagonal, honeycomb and rhombohedral lattice systems.

A.Arun Goud

Acknowledgements

Advisory committeeProf. Gerhard Klimeck Profs. Mark Lundstrom, Vladimir Shalaev

NEMO5 developersSebastian Steiger – 1dhetero, Brillouin and for answering other questions Hong-Hyun & Zhengping Jiang – RTD NEGF, NEGF simulation technqiuesTillmann Kubis & Michael Povolotskyi – NEMO5 simulation issues

All other members of the Nanoelectronic modeling group…Presentation skills

Xufeng Wang, JM Sellier – For code that went into 1dhetero

Steven Clark – Tool installationDerrick KearneyGeorge Howlett

Cheryl HainesVicky Johnson

Funding agencies – NSF, SRC, NRI

Rappture support

Scheduling appointments, handling paperwork

A.Arun Goud

Thank You!

A.Arun Goud

Coherent tunneling

Coherent tunneling –

Translational periodicity in the transverse direction

Two rules should be satisfied –

1) Total energy is conserved 2) Transverse momentum is conserved

In Emitter In Well(Bulk like) (2D subband)

Shaded disk in Fermi sphere indicates kx, ky states in emitter that take part in tunneling for a particular subband min. Eo in the well

A.Arun Goud

IV at 0K

Under equilibrium

CB Profile, resonance position

kx, ky that take part in tunneling

Contribution to current

Relative position of Well subband & E-kx dispersion in emitter

No overlap between well suband level & emitter bulk level => No tunneling channel

A.Arun Goud

IV at 0K

V < Peak voltage Vp

CB Profile, resonance position

Contribution to current

kx, ky that take part in tunneling

Relative position of Well subband & E-kx dispersion in emitter

Some well suband levels & emitter bulk levels overlap => Tunneling channel

A.Arun Goud

IV at 0K

V = Peak voltage Vp

CB Profile, resonance position

kx, ky that take part in tunneling

Contribution to current

Relative position of Well subband & E-kx dispersion in emitter

Maximum overlap of well suband levels & emitter bulk levels => Current is at its max.