Quantum Transport

Quantum Transport

Outline:

What is Computational Electronics?

Semi-Classical Transport Theory Drift-Diffusion Simulations Hydrodynamic Simulations Particle-Based Device Simulations

Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical Simulators Tunneling Effect: WKB Approximation and Transfer Matrix Approach Quantum-Mechanical Size Quantization Effect

Drift-Diffusion and Hydrodynamics: Quantum Correction and Quantum Moment Methods

Particle-Based Device Simulations: Effective Potential Approach

Quantum Transport Direct Solution of the Schrodinger Equation (Usuki Method) and Theoretical

Basis of the Green’s Functions Approach (NEGF) NEGF: Recursive Green’s Function Technique and CBR Approach Atomistic Simulations – The Future

Prologue

Transport Properties of system/device using Green’s functions formalism

Low field transport Linear response theory (ASU)

High field transport Bulk systems – Airy approach (Rita Bertoncini,

ASU, PhD Thesis) Devices:

Recursive Green’s Functions Approach (ASU, Purdue)

CBR Approach (ASU, WSI, Purdue)

Linear Response Theory

Only the retarded Green’s function is needed as it includes the collisional broadening of the states

In the ASU’s simulator for low-field mobility calculation in silicon inversion layers, strained-Si layers and InGaAs/InAlAs heterostructures the following features have been implemented:

Realistic treatmet of scattering within the self-consistent Born approximation

Modification of the density of states function is accounted for due to the collisional broadening of the states and the intersubband scattering

Random phase approximation in its full implementation is included to properly treat static screening of Coulomb and Interface-Roughness scattering

Bethe-Salpether integral equation is solved in the calculation of the conductivity

Excellent agreement is obtained with measured low-field mobility data in silicon inversion layers and predictions were made for the mobility behavior in Strained-Si layers and InGaAs/InAlAs heterostructures that were later confirmed with experimental measurements

Relevant Literature

D. Vasileska, P. Bordone, T. Eldridge and D.K. Ferry, “Calculation of the average interface field in inversion layers using zero-temperature Green’s functions formalism”, J. Vac. Sci. Technol. B 13, 1841-7 (1995).

P. Bordone, D. Vasileska and D.K. Ferry, “Collision duration time for optical phonon emission in semiconductors”, Physical Review B 53, 3846-55 (1996).

D. Vasileska, T. Eldridge and D.K. Ferry, “Quantum transport: Silicon inversion layers and InAlAs-InGaAs heterostructures”, J. Vac. Sci. Technol. B 14, 2780-5 (1996).

D. Vasileska, P. Bordone, T. Eldridge and D. K. Ferry, “Quantum transport calculations for silicon inversion layers in MOS structures”, Physica B 227, 333-5 (1996).

D. Vasileska and D. K. Ferry, “Scaled silicon MOSFET’s: Part I - Universal mobility behavior”, IEEE Trans. Electron Devices 44, 577-83 (1997).

G. Formicone, D. Vasileska and D.K. Ferry, “Transport in the surface channel of strained Si on a relaxed Si1-xGex substrate”, Solid State Electronics 41, 879-886 (1997).

Proposed Strained-Si and Strained-SiGe Devices

Strained-Sin+ n+

Source DrainSiO2

n+poly-Si

Gate

Relaxed Si1-xGex

SiGe Graded Buffer5% x% of Ge

(a)

Strained-Si

n+ n+

Source DrainSiO2

n+poly-Si

Gate

Relaxed Si1-xGex

SiGe Graded Buffer5% x% of Ge

Si1-xGex

(b) Gate

Strained-Si

Strained-Si

n+ n+

Source DrainSiO2

n+poly-Si

Relaxed Si1-xGex

SiGe Graded Buffer5% x% of Ge

Si1-xGex

(c)

Strained-Si1-xGex

p+ p+

Source DrainSiO2

Metal

Gate

n- Si Substrate

(d)

SiStrained-Si1-xGexp+ p+

Source DrainSiO2

Metal

Gate

n- Si Substrate

(e)

Si

p+ modulation doping

Is Strain Beneficial in Nanoscale MOSFETs With High Channel Doping Densities?

2-band

Regular SiliconBiaxial tension

Strained Silicon

00’+

4-band

00’

’

’

’

’

’

’

1

1.2

1.4

1.6

1.8

2

1016 1017 1018

x=0.1x=0.2x=0.4

stra

ined

-Si/

Si

Substrate doping NA [cm-3]

0

500

1000

1500

2000

2500

1016 1017 1018

Exp. dataSilicon

x=0.1x=0.2

x=0.4

Mob

ility

[cm

2/V

-s]

Substrate doping NA [cm-3]

1

1.2

1.4

1.6

1.8

2

2.2

1012 1013

NA=1x1017 cm-3

NA=2x1017 cm-3

NA=5x1017 cm-3

NA=7x1017 cm-3

NA=1x1018 cm-3

Mob

ility

enh

an

cem

ent

ratio

Inversion charge density Ns [cm-2]

High Field Transport in Devices:Recursive Green’s Functions Approach The most complete 1D transport in resonant tunneling diodes

(RTDs) that operate on purely quantum mechanical principles was accomplished with the NEMO1D Code

The NEMO 1D Code was developed by Roger Lake, Gerhard Klimeck, Chris Bowen and Dejan Jovanovich while working at Texas Instruments/Raytion

It solves the retarded Green’s function (spectral function) in conjuction with less-than Green’s function (occupation function) self-consistently

References for NEMO1D:

Roger. K. Lake, Gerhard Klimeck, R. Chris Bowen, Dejan Jovanovic, Paul Sotirelis and William R. Frensley,"A Generalized Tunneling Formula for Quantum Device Modeling",VLSI Design, Vol. 6, pg 9 (1998).

Roger Lake, Gerhard Klimeck, R. Chris Bowen and Dejan Jovanovic,"Single and multiband modeling of quantum electron transport through layered semiconductor devices", J. of Appl. Phys. 81, 7845 (1997).

The Philosophy Behind the Recursive Green’s Function Approach

K. B. Kahen, Recursive-Green’s-function analysis of wave propagation in two-dimensional nonhomogeneous media, .Phys. Rev. E 47, 2927 - 2933 (1993).

S. Datta, From Atom to Transistor, 2008.

Representative Simulation Results

High Field Transport in Devices:Contact Block Reduction MethodThe retarded Green’s function of an open system:

( ) [ ] 1 1[ ]R E E E- -= - = - -0G I H I H S

To determine Green’s function of an open system we need to invert a huge matrix

The Dyson equation,

( )E0G describes closed system (decoupled device)

( ) ( ) ( ) ( )1

,R E E E E-é ù= -ë û

0 0G I G GS

( )10

0

E E iE ia a h

a ah

e h +

-

=

é ùº - + =ë û - +å0G I Hwhere , Eaa a=0H

where closed system Hamiltonian , self-energy matrix0H S

D. Mamaluy, D. Vasileska, M. Sabathil, T. Zibold, and P. Vogl, “Contact block reduction method for ballistic transport and carrier densities of open nanostructures”, Phys. Rev. B 71, 245321 (2005).

Retarded Green’s Function of an open system in CBR formalism:

RG1 0 1 0

1 0 0 1 0 0

= ,

RCD C C C CDR

RDC C C DC DC C

RC

RCD DDDC

- -

- -

é ù é ùê ú ê ú= ê ú ê ú- + - +ê ú ê úë ûë û

G A G A GG

A A G G A GG G

G

AG

0 0C,C C C C DC DC= - =-A 1 G A GS S

where, index D denotes the interior device region index C denotes the contact ( boundary ) region

The left upper block fully determine the transmission functionRCG

The left lower block determines density of states, charge density etc.RDCG

All elements of GR can be determined from inversion of small matrix AC

D

C

is the contact portion of the 0CG 0G

Transmission Function and Local Density of States Calculation

Transmission Function

CBR Formalism

Local Density of States Function

CBR Formalism

' †'( ) ( )R RT E Tr l l

l l = G GG G

where'

'0 1 0 ††( ) , [ ]( ) [ ]R

CR R

C C C CC C C C C CT E iTr l lll l

-= - = -= 1G G S G S S G GG GG

( ) †, | | 2R REr p=r r G G rG

( )

0 1 1

', 0

2,

| | '| |

1,

= '| |Rm DC C

Rm mm

m

Cm

mm m m

E i

E

a a h

a a

rp

e h +

- -

=

=2

=- +å

å

r

r

rr G B B

r G

G

G

Properties of Widely Acceptable 2D Simulators

Exactness Exactness -- Accomplished with comparison Accomplished with comparison with experiments with experiments

Speed (Optimization and Process Variation)Speed (Optimization and Process Variation)

Buried Oxide

Gate

AA′

B′

3D view

X

ZY

Source Drain

B

Experimental FinFET*Gate length Lg = 10 nmFin width tSi = 12 nm, Gate oxide thickness tox= 1.7 nm(110) channel orientation

*Bin Yu et. al., “FinFET Scaling to 10 nm Gate Length”, IEDM Tech. Digest, 2002

H. R. Khan, D. Mamaluy and D. Vasileska, “Approaching Optimal Characteristics of 10 nm High Performance Devices” a Quantum Transport Simulation Study of Si FinFET, IEEE Trans. Electron Devices, Vol. 55(1), pp. 743-753 (2008).H. R. Khan, D. Mamaluy and D. Vasileska, “Simulation of the Impact of Process Variation on the Optimized 10-nm FinFET”, IEEE Transactions Electron Dev. Vol. 55(8), pp. 2134 – 2141, August 2008.

Top Gate

Side Gate

Y

Side Gateh

tox

tSi

Z

Top Gate

Side Gate

Z Y

Side Gateh

tox

tSi

Z

Top Gate

Side Gate

Y

Side Gateh

tox

tSi

Z

Top Gate

Side Gate

Z Y

Side Gateh

tox

tSi

Z

Top Gate

Side Gate

Y

Side Gateh

tox

tSi

Z

Top Gate

Side Gate

Z

Y

Side Gateh

tox

tSi

Z

Top Gate

Side Gate

Y

Side Gateh

tox

tSi

Z

Top Gate

Side Gate

Z

Y

Side Gateh

tox

tSi

Z

Buried Oxide

Gate

B'

Z X

Y

Source Drain

Lg

h

tSiB

Buried Oxide

Gate

B'

Z X

Y

Source Drain

Lg

h

tSiB

Gate length =10 nm Fin width = 4 nm Gate oxide thickness = 1.2 nm Gate dielectric – SiO2 Fin height = 4 nm ~ 8 nm

Source/drain doping = n type-2×1019 cm-3

Body doping = 2×1015 cm-3

Doping gradient = 1.25 nm/dec Gate doping = uniform, n type-2×1019 cm-3

Y

Y

View along B-B'

Simulated geometry

oxide

Need for 3D Device SimulationsNeed for 3D Device Simulations

2-D simulator does not give us the opportunity to analyze:the effect of fin height on carrier transportdevice characteristics of tri-gate structurethe effect of an unintentional dopant on

device characteristics

0.0 0.1 0.2 0.3 0.4

0

1

2

3

4

Dra

in c

urr

ent

[A

]

Drain voltage [V]

DG FinFET TG FinFET

VGS

= 0.1V

Need for 3D Device Simulations

DG vs. TG FinFET

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.16

-0.12

-0.08

-0.04

0.00

0.04

0.08

Net

gate

leakag

e [

nA

]

Gate voltage [V]

DG FinFET TG FinFET

VDS

= 0.4V

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

0

2

4

6

8

10

12

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Dra

in c

urr

ent,

ID

S [

µA

]

Dra

in c

urr

ent,

ID

S [

µA

]

Gate voltage [V]

DG FinFET TG FinFET

VDS

= 0.4V

0.1220.027|IG| @ VDS = 0.4V, VGS = -0.4V [nA]

7073Subthreshold swing [mV/dec]

0.00480.0319ISD,LEAK = IDS @ VDS = 0.4V, VGS = -0.4V [nA]

10.187.57ION = IDS @ VDS = 0.4V, VGS = 0.3V [μA]

TGDGParameter

0.1220.027|IG| @ VDS = 0.4V, VGS = -0.4V [nA]

7073Subthreshold swing [mV/dec]

0.00480.0319ISD,LEAK = IDS @ VDS = 0.4V, VGS = -0.4V [nA]

10.187.57ION = IDS @ VDS = 0.4V, VGS = 0.3V [μA]

TGDGParameter

0

2

4

Y [

nm

]

2.0E18

6.0E18

1.2E19

1.8E19

2.4E19

3.0E19

3.6E19

4.2E19

<2.0E18

>

tox

0 2 4 6 8 10 120

2

4

6

8

Z [nm]Y

[n

m] t

ox

024 Y [nm]

2.0E

18

6.0E

18

1.2E

19

1.8E

19

2.4E

19

3.0E

19

3.6E

19

4.2E

19

<2.0

E18

>

t ox

02

46

810

1202468

Z [nm

]

Y [nm]

t ox

ON-State Electron density along the dotted line

VGS=0.2, VDS=0.4V

Electron density (TG) > electron density (DG)

Atomistic Simulations – The Future of Nano-Devices

Examples of devices for which atomistic simulations will be necessary include: Devices in which local Strain exists Alloy Disorder has to be properly described

• Group of Gerhard Klimeck, Purdue University, West Lafayette, IN, USA• Group of Aldo di Carlo, Tor Vergata, Rome, Italy.

Why Tight-Binding ?

Allows us to describe the band structure over the entire Brillouin zone

Relaxes all the approximations of Envelope Function approaches

Allows us to describe thin layer perturbation (few Å)

Describes correctly band mixing

Gives atomic details

The computational cost is low

It is a real space approach

Molecular dynamics

Scalability (from empirical to ab-initio)

Scalability of TB approaches

DFT local basis approaches provide transferable and accurate interaction potentials. The numerical efficiency of the method allows for molecular dynamics simulations in large super cells, containing several hundreds of atoms.

Density Functional based Tight-Binding (DFTB, FIREBALL, SIESTA)

Empirical Tight-Binding

Semi-Empirical Hartree-Fock

Hamiltonian matrix elements are obtained by comparison of calculated quantities with experiments or ab-initio results. Very efficient, Poor transferability.

Methods used in the chemistry context (INDO, PM3 etc.). Medium transferability.

The sp3s* Hamiltonian [Vogl et al. J. Phys. Chem Sol. 44, 365 (1983)]

In order to reproduce both valence and conduction band of covalently bounded semiconductors a s* orbital is introduced to account for high energy orbitals (d, f etc.)

The sp3d5s* Hamiltonian[Jancu et al. PRB 57 (1998)]

Many parameters, but works quite well !

Tight-Binding sp3d5s* model for nitrides

Ab-Inito Plane Wave DFT-LDA Band Structure for GaN Wurtzite

Ab-Inito Plane Wave DFT-LDA Band Structure for GaN Wurtzite

TB Wurtzite GaN Band Structure

Nearest-neighbours sp3d5s* tight-binding parametrization for wurtzite GaN, AlN and InN compare well with Ab-Initio results.

Boundary conditions

Finite chain

Periodic

Open boundary conditions

After P planes the structure repeats itself. Suitable for superlattices

H=

After P planes the structure end.Suitable for quantum wells

H=

After P planes there is a semiinfinite crystalSuitable for current calculations BULK BULKP

P

P

∞ ∞

Where do we put the atoms ?

To describe the electronic and optical properties of a nanostructure we need to know where the atoms are.

1) We know “a priori” the atom positions (for example X-ray information)

2) We need to calculate the atomic positions

Simple analytic espressions

Full calculation

Classical calculations

Quantum calculation

Continuum theoryAtomistic (Valence Force Field)

Example: Strain and Pseudomorphic growthAn epitaxial layer is grown, on a substrate with different lattice constant.The epilayer deforms (strain)

011

120 2 aa

C

Caa s

as

as

a0

a0

as

as

as

a

asas

as

RR )1(' Strain tensor

Strain in a AlGaN/GaN Nanocolumnz,

[00

01]

, [1010]x, [1210]y

GaN

Al0.28Ga0.72 N

20nm

Calleja’s pillars

AlGaN/GaN Nanocolumns

( 4 ( )) 4pz py P P

piezo-electricpolarization

pyro-electric polarization

pzi ijk jkP d

piezo-electric moduli tensor

The Poisson equation

Potential

How do we describe alloys ?

Usually, tight-binding parameterizations are made for single elments and binary compounds (Si, Ge, GaAs, InAs etc.). However, nanostructure are usually build by using also ternary (AlGaAs etc.) and quatrnary (InGaAsP etc.) alloys.

1) Supercell calculations

A0.5B0.5C

Average over an ensamble of configurations

2) Virtual crystal approximation

P(AxB1-xC)=x P(AC) + (1-x) P(BC)

A new crystal is defined with averaged properties (P)

3) Other methods (Modified VCA, CPA, T-matrix etc.)

Self-Consistent Tight-Binding

With the aim of Self-Consistent treatment of external electrostatic potential, Tight-Binding can be applied to semiconductor device simulations.

Full self consistent approach only suitable for small systems like molecules

Self-consistent approach for only the free charge

Schrodinger Poisson

Charge transfer is important in semiconductor nanostructures.

Self-consistent solution of Schredinger and Poisson equations are common in envelope function approaches

Tight-binding allows for a full (with all the electrons) self-consistent solutionof the nanostructure problem

Self-Consistent Tight-Binding

x

y z

The electron and hole densities in each 2D layer are given by:

The influence of free carrier charge redistribution and macroscopic polarization fields are included by solving the Poisson equation:

boundary conditions

[A. Di Carlo et. al., Solid State Comm. 98, 803 (1996); APL 74, 2002 (1999)]

ADH NNnpePV

dz

d

dz

dzD

dz

d )(

HC VHH

+

Summary

Linear response and solution of the Beth-Salpether equation in conjunction with the Dyson equation for the retarded Green’s function is useful when modeling low-field mobility of inversion layers

When modeling high field transport both Dyson equation for the retarded Green’s function and the kinetic equation for the less-than Green’s function have to be solved self-consistently

CBR approach and recursive Green’s function method have both their advantages and their disadvantages

When local strains and stresses have to be accounted for in ultra-nano-scale devices then atomistic approaches become crucial

Prologue

What are the lessons that we have learned? Semi-classical simulation is still a very important part of Today’s

semiconductor device modeling as power devices and solar cells (traditional ones) operate on semi-classical principles

Quantum corrections can quite accurately account for the quantum-mechanical size quantization effect which gives about 10% correction to the gate capacitance

For modeling ultra-nano scale devices one can successfully utilize both Poisson-Monte Carlo-Schrodinger solvers and fully quantum-mechanical approaches based on NEGF (tunelling + size quantization)

Full NEGF is a MUST when quantum interference effects need to be captured and play crucial role in the overall device behavior

For a subset of ultra-nano scale devices that are in the focus of the scientific community now, in which band-structure, local strain and stresses, play significant role, atomistic simulations are necessary.

Simulation Strategy for Ultra-Nano-Scale Devices

Calibrate semi-empirical approaches with ab-initio band

structure simulations

Perform BANDSTRUCTURE/TRANSPORT calculations on systems containing

millions of atoms

1000 atoms

Millions of atoms

Atomistic Simulations Selected Literature

Mathieu Luisier and Gerhard Klimeck,"A multi-level parallel simulation approach to electron transport in nano-scale transistors", Supercomputing 2008, Austin TX, Nov. 15-21 2008. Regular paper - 59 accepted papers, 277.

Mathieu Luisier, Neophytos Neophytou, Neerav Kharche, and Gerhard Klimeck,"Full-Band and Atomistic Simulation of Realistic 40 nm InAs HEMT", IEEE IEDM, San Francisco, USA, Dec. 15-17, 2008, DOI : 10.1109/IEDM.2008.4796842,

Mathieu Luisier, and Gerhard Klimeck,"Performance analysis of statistical samples of graphene nanoribbon tunneling transistors with line edge roughness", Applied Physics Letters, Vol. 94, 223505 (2009), DOI:10.1063/1.3140505,

Mathieu Luisier, and Gerhard Klimeck, "Atomistic, Full-Band Design Study of InAs Band-to-Band Tunneling Field-Effect Transistors ", IEEE Electron Device Letters, Vol. 30, pp. 602-604 (2009), DOI:10.1109/LED.2009.2020442.

Questions or Comments?