Transcript Modeling the electronic structure of semiconductor devices M. Stopa Harvard University Thanks to Blanka Magyari-Kope, Zhiyong Zhang and Roger Howe Stanford 11/10/11

Modeling the electronic structure of
semiconductor devices
M. Stopa
Harvard University
Thanks to Blanka Magyari-Kope,
Zhiyong Zhang and Roger Howe
Stanford 11/10/11
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
Self-consistent electronic structure of semiconductor heterostructures including
quantum dots, quantum wires and nano-wires, quantum point contacts.
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
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)
details
define a mesh
discretize Poisson equation
guess initial (r), Vxc(r)
Newton-Raphson
F ,    2    A  
F i 
 
   A   i  F i 

 

Mesh must be inhomogeneous,
encompassing wide simulation region so
that boundary conditions are simple
Adiabatic treatment of z
solve Poisson equation
 i 1   i  t i
 2 r  2 DEG r  ion r   bg r 
 j
 k
Compute (r)
1.
Schrödinger equation
2.
Thomas-Fermi regions
Jacobian
• Thomas-Fermi appx.
• wave functions
• N or  fixed ?
in= out
?
yes
Bank-Rose damping


V  F   t ,  
convex
i
i
i
Optimize t by calculating 
several times for different t.
adjust Vxc(r)
no
Vxc same ?
yes
R. E. Bank and D. J. Rose, SIAM J.
Numer. Anal., 17, 806 (1980)
DONE
no
  2 2
 xy
xy
 * 2  V x, y, z  0 z    o xy 0 z 
 2m z

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
Density from potential
3D Thomas-Fermi zero temperature:
k
k F    e r   Vx r 
k
F
N kF
d 3k
1 F 2
k F3
 1  2 
 2  k dk  2
3
V
 0
3
k
k 0 2 
Only true under the
assumption of parabolic
2 2
bands  k   k
Quasi-2D Thomas-Fermi zero temperature:
2m
k
k
kF
F
N
d 2k
1 F
k F2
2
   z  1  2 
 2  kdk 
2
V
 0
2
k
k  0 2 
Quasi-2D Thomas-Fermi T≠0:
k
kF
F
N
d 2k
1
2
  z   f  k   2 
2
2
V
1

exp




k
 e r   Vx r 


2

k
k 0
 

k BT

ln 1  e  e r Vx r 
2


Sandia, NM 10/11/11
Examples of SDFT results
Triple dot rectifier
M. Stopa, PRL 88,
146802 (2002)
donor layer disorder/order
Statistics of quantum dot level spacings
Blue dots are
donors, red
circles are ions
M. Stopa, Phys. Rev. B, 53, 9595 (1996)
Transition from Poisson
statistics to Wigner
statistics as disorder
increases
M. Stopa, Superlattices and Microstructures, 21, 493 (1997)
Degenerate 2D electron gas (quantum Hall regime)
1
nr  
2 2

f  


s 


1 0
2
s
 e r   Vxs r    
1
2
 s       g B Bs
c
 
eB
2
Density of states
eB

mc
Magnetic terraces
Single photon detector
Quantum dot
Radial potential profile
as B is increased
 1
 3
 2
Evolution of magnetic field
induced compressible and
incompressible strips in a
quantum dot
Komiyama et al. PRB 1998
Stopa et al. PRL 1996
Eigenvalues in Quantum
dots
Frequently divide 2DEG region into
“dot” and “leads.
Dot = small number of isolated
electrons.
Charge density in two parts:
(i) Thomas-Fermi density from adiabatic subband energies:
 k   k  x, y 
 C r   2
dk
f     k 
d
2 
(ii) Schrödinger density, eigenvalue problem in restricted 2D region:
2


e 







i


A

e

r

V
z

V
r


 n r    n n r 
B
xc
c 


N
 C r     n r 
n 1
2
Coulomb interaction of scars
E=-0.5 Ry*
InAs wire simulation (SETEwire)
Schematic of wire simulation
Wire length 100 nm
(smaller than expt.)
SPM tip
Metallic leads
Back gate 40 nm from wire
InP barriers
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
H   ti  Vext ri   V ri , r j 
N
i 1
i j
Coulomb interaction
Typical case: double dot
potential with N=2
Simple single particle basis states:
L, R
Two-particle basis states
LL , RR ,  LR  LR

 LR  LR 
Singlet, S=0
Triplet, S=1
Singlet energy = single ptcls. + interdot Coulomb + exchange - delocalization
ES  2  Vint er  Vex  4
Vint ra
t2
 Vint er  Vex
LL
RR
Triplet energy = single ptcls. + interdot Coulomb - exchange
ET  2  Vint er  Vex
DFT basis for exact diagonalization
Kohn-Sham equations
2


~
e 
  i  A   e r   VB  z   Vxc r  n r   h r  n r    n n r 
c 


  2 r   e   e   e 0  e dot
exact diagonalization
1
H   h(ri )   V (ri , r j )
2 i , j 1, 2
i 1, 2
2
e 

hr     i  A   Vext r 
c 

Dirichlet boundary
conditions on gates
Summary: exact diagonalization N=2
1.
Solve DFT problem for spinless electrons with full device
fidelity.
SETE solves Kohn-Sham
problem, i.e. mean field
Form all symmetric and anti-symmetric combinations of basis states for singlet
and triplet two electron states, resp.
 S  O
S
Symmetric states
Anti-symmetric states
2.
3.
4.
5.
nm 
1
 nm  m n 
2
 nm
 A  O
A
nm 
nm
nm
1
 nm  m n 
2
Remove Coulomb interaction and exchange-correlation effects
from Kohn-Sham levels.
Truncate basis to something manageable.
Compute Coulomb matrix elements using Poisson’s equation.
Diagonalize Hamiltonian.
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.
Calculating Coulomb matrix elements
Vnmrs   dr1  dr2 n* r1  * m r2 V (r1 , r2 ) r r2  s r2 
 nm | V | rs 
 rs r1    dr2V r1 , r2  r r2  s r2 
2V r1 , r2    r1  r2 
 2rs r1    r r1  s r1 
Dirichlet boundary
conditions on gates
 nm | V | rs   dr1 n* r1  m* r1  rs r1 
POINT: calculated matrix element without ever knowing V(r1,r2) !
POINT: inhomogeneous screening automatically included.
CECAM08
NSEC
Exact diagonalization calculation for realistic
geometry double dot.
NSEC
M. Stopa and C. M. Marcus, NanoLetters 2008
• We calculate the N=2 (many-body) spectrum,
lowest two singlet and triplet states, near the
transition from (1,1) to (0,2).
L R
• 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
Exciton transfer via Förster process
motivation
GATE
dot 1
dot 2
 e1 r1 
 e 2 r2 
 h1 r1 
 h 2 r2 
nanoparticle/dots
VF   dr1  dr2 *e 2 r2  *h1 r1 V r1 , r2  h 2 r2  e1 r1 
Similar to quantum dot, we can calculate electronic structure of
confined excitons taking gate into account via boundary conditions
on Poisson equation.
VF  D 2  d RA


 *h1 RA  e1 RA   d RB
 *h 2 RB e 2 RB  V RA , RB 
RA
RB


e 2,h 2 RA    dRBV RA , RB 



 *h 2 RB  e 2 RB 
RB


2V r1 , r2    r1  r2 
  2e1,h1 RA  
VF  D 2  d RA

 *h1 RA e1 RA 
RA



 *h1 RA e1 RA   e 2,h 2 RA 
RA


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.