stanford 11/10/11 modeling the electronic structure of semiconductor devices m. stopa harvard...
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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
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.
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.