Stanford 11/10/11

Modeling the electronic structure of semiconductor devices

M. StopaHarvard University

Thanks to Blanka Magyari-Kope, Zhiyong Zhang and Roger Howe

Introduction

• Self-consistent electronic structure for nanoscale semiconductor devices requires calculation of charge density• Conceptually simple solutions (Solve the Schrödinger equation!) not practical in most cases (too many eigenstates).• Thomas-Fermi approaches can be developed in some cases, but even these are limited.

Nano by Numbers

Outline

• I will describe self-consistent electronic structure code SETE for density functional theory calculation of electronic structure for semiconductor devices. • Highlight the role of density calculation for increasingly complex systems.• Present various results for different systems.• Case of “exact diagonalization” and using Poisson’s equation to calculate Coulomb matrix elements.

SETE: Density functional calculation for heterostructures

Approximations(1) effective mass(2) effective single particle(3) exchange and correlation via a local spin density approximation

Allows full incorporation of (1) wafer profile(2) geometry and voltages of surfaces gates voltages(3) temperature and magnetic field

Self-consistent electronic structure of semiconductor heterostructures including quantum dots, quantum wires and nano-wires, quantum point contacts.

Outputs:1. electrostatic potential (r)2. charge density (r)3. for a confined region (i.e. a dot)

eigenvalues Ei, eigenfunctions I tunneling coefficients i

4. total free energy F(N,Vg,T,B)

define a meshdiscretize Poisson equationguess initial (r), Vxc(r)

rrrr bgionDEG 22

solve Poisson equation

Compute (r)1. Schrödinger equation2. Thomas-Fermi regions

in= out ?no

yes

adjust Vxc(r)

Vxc same ?

yes

DONE

no

Mesh must be inhomogeneous, encompassing wide simulation region so that boundary conditions are simple

Jacobiank

j

• Thomas-Fermi appx.• wave functions• N or fixed ?

Bank-Rose damping iii tV , F

convex

Optimize t by calculating several times for different t.

R. E. Bank and D. J. Rose, SIAM J. Numer. Anal., 17, 806 (1980)

2D Schrödinger equation• classically isolated region provided by gate potentials• fix either N or • cut off wave function in barrier regions (Dirichlet B.C.’s)• dot nearly circular expand in eigenfunctions (Bessel fns.); otherwise discretize on mesh (Arnoldi method)• use perturbation theory

details

zxyzzyxVzm

xyo

xy002

2

*

2

,,2

Adiabatic treatment of z AF 2,

iii t 1

iFAF

ii

Newton-Raphson

Density from potential

3D Thomas-Fermi zero temperature:

2

3

0

2

023

3

3

1

221

F

kk

k

k

k

kdkk

kd

V

N FFF

Quasi-2D Thomas-Fermi zero temperature:

2

1

221

2

0022

22 F

kk

k

k

k

kkdk

kdz

V

N FFF

Quasi-2D Thomas-Fermi T≠0:

FFk

k x

k

kk Vek

kdfz

V

N

022

22

exp1

1

22

rr

rr xVeB eTk

1ln

2

Sandia, NM 10/11/11

rr xF Vek

Only true under the assumption of parabolic bands

m

kk 2

22

Examples of SDFT results

Triple dot rectifier

M. Stopa, PRL 88, 146802 (2002)

Blue dots are donors, red

circles are ions

donor layer disorder/order

M. Stopa, Phys. Rev. B, 53, 9595 (1996)M. Stopa, Superlattices and Microstructures, 21, 493 (1997)

Statistics of quantum dot level spacings

Transition from Poisson statistics to Wigner statistics as disorder increases

Degenerate 2D electron gas (quantum Hall regime)

2

1 022

1

s

xss Vefn

rrr

Bsg Bs

2

1

eB

c 2

mc

eB

Density of states

3 2

1

Single photon detector

Evolution of magnetic field induced compressible and incompressible strips in a quantum dot

Magnetic terraces Quantum dot

Radial potential profile as B is increased

Komiyama et al. PRB 1998 Stopa et al. PRL 1996

rrrrA nnnxcB VzVec

ei

2

2

1

N

nnC rr

kdC fd

22

kr yxkk ,

Charge density in two parts:

(i) Thomas-Fermi density from adiabatic subband energies:

(ii) Schrödinger density, eigenvalue problem in restricted 2D region:

Eigenvalues in Quantum dots

Frequently divide 2DEG region into “dot” and “leads.Dot = small number of isolated electrons.

E=-0.5 Ry*

Coulomb interaction of scars

Schematic of wire simulation

Metallic leads

InP barriers

Wire length 100 nm (smaller than expt.)

Back gate 40 nm from wire

InAs wire simulation (SETEwire)

SPM tip

Complex band structure TF – in progress

Luttinger Hamiltonian for valence band (light holes and heavy holes)

replaces the Laplacian

No analytic relation between Fermi momentum and Fermi energy. Numerical relation has to be determined at each position in space! Tough problem.

Going beyond mean field theory – using Kohn-Sham states as a basis for Configuration Interaction calculation

Exact diagonalization in quantum dots

ji

ji

N

iexti VVtH rrri ,

1

Typical case: double dot potential with N=2

Coulomb interaction

RL ,

Simple single particle basis states:

LRLRRRLL ,,

LRLR

Two-particle basis states

Singlet, S=0

Triplet, S=1

exerraexerS VVV

tVVE

intint

2

int 42

exerT VVE int2

Singlet energy = single ptcls. + interdot Coulomb + exchange - delocalization

Triplet energy = single ptcls. + interdot Coulomb - exchange

LL

RR

rrrrrrA nnnnxcB hVzVec

ei

~2

doteeee 02

r

Kohn-Sham equations

rAr extVc

eih

2

exact diagonalization

),(2

1)(

2,1,2,1ji

jiii VhH rrr

Dirichlet boundary conditions on gates

DFT basis for exact diagonalization

Summary: exact diagonalization N=21. Solve DFT problem for spinless electrons with full device

fidelity.

2. Remove Coulomb interaction and exchange-correlation effects from Kohn-Sham levels.

3. Truncate basis to something manageable.4. Compute Coulomb matrix elements using Poisson’s equation.5. Diagonalize Hamiltonian.

Form all symmetric and anti-symmetric combinations of basis states for singlet and triplet two electron states, resp.

mnnmnmOS S 2

1

mnnmnmOA A 2

1

nm

mn mn Symmetric states

Anti-symmetric states

SETE solves Kohn-Sham problem, i.e. mean field

Modeling of electronic structure by configuration interaction (CI) with a basis of states from density functional theory (DFT)

1. Use DFT and realistic geometry (gate configuration, wafer profile, wide leads, magnetic field B) with N=2.

2. Resulting “Kohn-Sham” states used as basis for “exact diagonalization” (configuration interaction) of Coulomb interaction.

MAIN MESSAGE: capture both geometric effects and many-body correlation.

ADVANTAGES: 1. Fewer basis states needed because basis already includes potential profile and B.2. Coulomb matrix elements calculated with Poisson’s equation screening of gates

included automatically plus no 3D quadratures required.3. No artificial introduction of tunneling coefficient. Basis states are states of full

double dot.

NSECCECAM08

rsVnm

VddV srmnnmrs

||

),( 22212*

1*

21 rrrrrrrr

Dirichlet boundary conditions on gates

222121 , rrrrrr srrs Vd

21212 , rrrr V

1112 rrr srrs

11*

1*

1|| rrrr rsmndrsVnm

Calculating Coulomb matrix elements

POINT: calculated matrix element without ever knowing V(r1,r2) !POINT: inhomogeneous screening automatically included.

L R

Exact diagonalization calculation for realistic geometry double dot.

• We calculate the N=2 (many-body) spectrum, lowest two singlet and triplet states, near the transition from (1,1) to (0,2).• For ε<0 singlet and triplet ground states have one electron in each dot, singlet and triplet excited states have both electrons in right dot.• T1 must have occupancy of higher orbital in R

NSEC

M. Stopa and C. M. Marcus, NanoLetters 2008

GATE

nanoparticle/dots

11222111*

22*

21 , rrrrrrrr ehheF VddV

11 re

11 rh 22 rh

22 re

dot 1 dot 2

Exciton transfer via Förster process motivation

Similar to quantum dot, we can calculate electronic structure of confined excitons taking gate into account via boundary conditions on Poisson equation.

BABeBh

BBAeAh

AAF RRVRR

RRdRR

RRdDV ,22

*11

*2

BeBh

BBABAhe RR

RRRVdRR 22

*2,2 ,

21212 , rrrr V

AeAh

AAhe RR

RR 11

*1,1

2

AheAeAh

AAF RRR

RRdDV 2,211

*2

Conclusions

• In contrast to molecular systems, number of eigenstates in semiconductor systems is too great to calculate all states.• Thomas-Fermi is valuable, both 3D and effective 2D, in some cases• For complex band structure of inhomogeneous systems there is no systematic way to implement TF.• Finally, for isolated, small N systems, can go beyond even standard Kohn-Sham method to incorporate many-body correlation into self-consistent calculation in realistic environment.