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

Quantum Transport

Direct Solution of the Schrodinger Equation: Usuki Method (equivalent to Recursive Green’s Functions Approach in the ballistic limit)

NEGF (Scattering): Recursive Green’s Function Technique, and CBR approach

Atomistic Simulations – The Future of Nano Devices

i=0 i=N

j=0

j=M+1

y

x

incidentwaves

transmittedwaves

reflectedwaves

Wavefunction and potential defined ondiscrete grid points i,j

i th slice in x direction - discrete problem involves translating from one slice to the next.

Grid spacing: a<< F

Description of the Usuki Method

Usuki Method slidesprovided by RichardAkis.

Obtaining transfer matrices from the discrete SEapply Dirichlet boundary conditions on upper and lower boundary:

01,0, Mjiji

Wave function on ith slice can be expressed as a vector

1,

1,

,

.

.

.

i

Mi

Mi

i

j=0

j=M+1

j=1

j=M

i

Discrete SE now becomes a matrix equation relating the wavefunction on adjacent slices:

iiii Ett 110iH

where:

)4(0

)4(

)4(

0)4(

1,

2,

,

,

tVt

ttVt

ttVt

ttV

i

i

Mi

Mi

0iH

(1b)

(1b) can be rewritten as:

Combining this with the trivial equation one obtains:

11

iii t

E 0iH

ii

Modification for a perpendicular magnetic field (0,0,B) :

0

2

,2

,

2

//

,

)(0

eh

Ba

eP

t

E

jiji

ji

0iiHP

P

IT

B enters into phase factorsimportant quantity:

flux per unit cell

i

i

i

i

1

1iT(2)

where

t

E0iiH

I

IT

0 Is the transfer matrix relating adjacent slices

)()(

)(

mm

m

u

u

Mode eigenvectors have the generic form:

redundant

There will be M modes that propagates to the right (+) with eigenvalues:

Mqme

qmeam

m

amikm

,,1,)(

,,1,)(

propagating

evanescent

There will be M modes that propagates to the left (+) with eigenvalues:

Mqme

qmeam

m

amikm

,,1,)(

,,1,)(

propagating

evanescent

)()(1 muu

U )()(1 mdiag

UU

UUUtot

anddefining

Complete matrix of eigenvectors:

Solving the eigenvalue problem:

0

1

0

11

T yields the modes on the left side of the system

Transfer matrix equation for translation across entire system

r

IUTTTU

0

ttotNNtot 121

1

Transmission matrix

Zero matrixno waves incident from right

Unit matrixwaves incident from left have unitamplitude

reflection matrix

Converts from mode basisto site basis

Converts back to mode basis

2

,,

22 nm

mnm

n tv

v

h

eGRecall:

In general, the velocities must be determined numerically

0CI,C (0,0)2

(0,0)1 Boundary condition- waves of

unit amplitude incident from right

Variation on the cascading scattering matrix technique methodUsuki et al. Phys. Rev. B 52, 8244 (1995)

1i22

(i,0)2i21i2

(i,0)1i21i2i1

i2i1i

i

(i,0)2

(i,0)1

i

1,0)(i2

1,0)(i1

]TC[TP

,CTPP

,PP

0IP

PI0

CCT

I0

CC

plays an analogous role to Dyson’s equation inRecursive Greens Function approach

Iteration schemefor interior slices

Final transmission matrix for entire structure is given by

A similar iteration gives the reflection matrix

111λUUCλUt

11N

After the transmission problem has been solved, the wave function can be reconstructed

MNkNNNN ,,1,2

ψP

wave function on column N resulting from the kth mode

121 iiii ψPPψ

q

kijkjinyxn

1

2),(),(

The electron density at each point is then given by:

One can then iterate backwards through the structure:

It can be shown that:

First propagating mode for an irregular potential

confiningpotential

u1(+) for B=0.7 T

j0 40 80

u1(+) for B=0 T

u1j

2)22sin(2 mjj

mm ujkth

ev

Mode functions no longersimple sine functions

general formula for velocity of mode m obtained by taking the expectation value of the velocity operator with respect to the basis vector.

)( yu nn

Vg= -1.0 V Vg= -0.9 V Vg= -0.7 V

Potential felt by 2DEG- maximum of electron distribution ~7nm below interface

Potential evolves smoothly- calculate a few as a function of Vg, and create the rest by interpolation

-0.2

0.0

0.2

0.4

0.6

0.8

0.00 0.02 0.04 0.06 0.08 0.10

Co

nd

uct

ion

ba

nd

[e

V]

z-axis [m]

Fermi level EF

Conduction band profile Ec

Energy of theground subband

Simulation gives comparable2D electron density to that measured experimentally

2110

3*

2

104~)(2

cmEEm

N DF

Example – Quantum Dot Conductance as a Function of Gate voltage

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.9 -0.8 -0.7 -0.6

co

nd

uct

an

ce

flu

ctu

atio

n (

e2/h

)

gate voltage (volts)

EXPERIMENT (+0.6)

THEORY

0.01 K

0.4 m

-0.951 V-0.923 V

-0.897 V

Subtracting out a background that removes the underlying steps you get periodic

fluctuations as a function of gate voltage. Theory and experiment agree very well

Same simulations also reveal that certain scars may

RECUR as gate voltage is varied. The resulting

periodicity agrees WELL with that of the conductance

oscillations

* Persistence of the scarring at zero magnetic field

indicates its INTRINSIC nature

The scarring is NOT induced by the application of

the magnetic field

Magnetoconductance

Conductance as a function of magnetic field also shows fluctuations that are virtually periodic- why?

B field is perpendicular to plane of dot

classically, the electron trajectories are bent by the Lorentz force

Green’s Function Approach: Fundamentals The Non-Equilibrium Green’s function approach for device

modeling is due to Keldysh, Kadanoff and Baym It is a formalism that uses second quantization and a

concept of Field Operators It is best described in the so-called interaction

representation In the calculation of the self-energies (where the scattering

comes into the picture) it uses the concept of the partial summation method according to which dominant self-energy terms are accounted for up to infinite order

For the generation of the perturbation series of the time evolution operator it utilizes Wick’s theorem and the concepts of time ordered operators, normal ordered operators and contractions

Relevant Literature

A Guide to Feynman Diagrams in the Many-Body Problem, 2nd Ed. R. D. Mattuck, Dover (1992).

Quantum Theory of Many-Particle Systems, A. L. Fetter and J. D. Walecka, Dover (2003).

Many-Body Theory of Solids: An Introduction, J. C. Inkson, Plenum Press (1984).

Green’s Functions and Condensed Matter, G. Rickaysen, Academic Press (1991).

Many-Body TheoryG. D. Mahan (2007, third edition).

L. V. Keldysh, Sov. Phys. JETP (1962).

Schrödinger, Heisenberg and Interaction Representation

Schrödinger picture

Interaction picture

Heisenberg picture

HHHSHH

oIIo

So

II1I

SSSS1oS

H,O(t)Ot

tHeOtHe(t)O 0 (t)t

H,O(t)Ot

tH

eOtH

e(t)O (t)tH (t)t

0Ot

OO (t)HH (t)t

iiii

iii

i

ii

Ut

UH

U (0)(t,0)U(t) HS

i

operator evolution time

Time Evolution Operator

Time evolution operator representationas a time-ordered product

Contractions and Normal Ordered Products

21

212k1l1

2k1l1l2k

1212k

kl

1l2kkllk

122k1l1

1l2k1l2k

t t 0

t t)(ta)(ta1--)(ta)(ta)(ta)(ta

t ttt

eδ

te

teaa aa

t t)(ta)(ta1--)(ta)(ta)(ta)(ta

AB - ABBA

i

ii

NT

Wick’s Theorem

Contraction (contracted product) of operators

For more operators (F 83) all possible pairwise contractions of operators

Uncontracted, all singly contracted, all doubly contracted, …

Take matrix element over Fermi vacuum

All terms zero except fully contracted products

211l2k

1212k

kl1l2k

t t 0)(tb)(tb

t ttt

eδ)(tb)(tb

i

]Z.XY..W.UV[...Z]XYW...UV[

Z]XY[UVW...W...XYZ]UV[][UVW...XYZ][UVW...XYZ

NN

NNNT

0]Z.XY..W.UV[0...0][UVW...XYZ00][UVW...XYZ0 NNT

Propagator

Partial Summation Method

Example: Ground State Calculation

GW Results for the Band Gap

Correlation functions Direct access to observable expectation values

Retarded, Advanced Simple analitycal structure

and spectral analysis

Time ordered Allows perturbation theory (Wick’s theorem)

* 1 = x1,t1

Definitions of Green’s Functions

Just one indipendent GFJust one indipendent GF

General identities

Spectral function

Fluctuation-dissipation th.

Gr, Ga, G<, G> are enough to evaluate all the GF’s and are connected by physical relations

See eg: H. Haug, A.-P. Jauho A.L. Fetter, J.D. Walecka

Equilibrium Properties of the System

Contour-ordered perturbation theory:Contour-ordered perturbation theory:

No fluctuation dissipation

theorem

Gr, Ga, G<, G> are all involved in the PT

• Time dep. phenomena• Electric fields • Coupling to contacts at

different chemical potentials

2 of them are indipendentContour ordering

See eg: D. Ferry, S.M. Goodnick H.Haug, A.-P. Jauho J. Hammer, H. Smith, RMP (1986) G. Stefanucci, C.-O. Almbladh, PRB (2004)

Non-Equilibrium Green’s Functions

Dyson Equation

Two Equations of MotionTwo Equations of Motion

Keldysh Equation

Computing the (coupled) Gr, G< functions

allows for the evaluation of transport properties

In the time-indipendent limit

Gr, G< coupled via the self-energies

Constitutive Equations

Summary

This section first outlined the Usuki method as a direct way of solving the Schrodinger equation in real space

In subsequent slides the Green’s function approach was outlined with emphasis on the partial summation method and the self-energy calculation and what are the appropriate Green’s functions to be solved for in equilibrium, near equilibrium (linear response) and high-field transport conditions