nextnano
A large software project
for the simulation of
3D nanometer semiconductor structures
Walter Schottky Institut, TU München
Current team: T. Zibold, T. Andlauer, R.
Morschl, K. Smith, T. Kubis, P. Vogl
Training and support: S. Birner
Thanks to many contributors worIdwide: Vienna, Rome, ASU,...
www.nextnano.de
nextnano overview
Goal: Provide quick global insight into basic physical properties of
mesoscopic semiconductor structures
Simulation
software for 3D
semiconductor
nanostructures
Si/Ge and III-V materials, Nitrides, alloys, zb and wz
Flexible structures and geometries
Quantum mechanical electronic structure
Equilibrium properties and carrier transport
Typically 5-10 downloads/day worldwide
Calculation of electronic structure
8-band k.p-Schrödinger (+LDA) and Poisson equation
Global strain minimization
Piezoelectric, pyroelectric charges, deformation potentials
Exciton energies and optical matrix elements
Magnetic field and spin effects
ISFET: Surface reactions @ semicond./electrolyte interfaces
Calculation of charge current
Quantum-drift-diffusion method : DD eq's + quantum densities
Ballistic current through open systems: Contact Block Reduction
Full quantum transport with scattering: NEGF for quasi-1D
Examples of nanostructures
1000 nm
In |1
Out |1
In |0
2DEG GaAs
Out |0
300 nm
Mach-Zehnderinterferometer
z=4 nm
Self-assembled
quantum dot array
y=500 nm
x=500 nm
InGaAs collector
InAs wetting layers
InP barrier
InGaAs emitter
Individual QD with e+h wave functions
Quantum dotresonant
tunneling
diode
Silicon 25 nm
Triple-gate FET
nextnano Program flow
Input:
structure, options
Database:
material parameters
INITIALIZATION
Bulk band structures, strain, def. pot's and piezo/pyro
POISSON EQUATION
Determination of potential
CURRENT EQUATION
QDD: Determination of quasi-Fermi levels
Many-band kp SCHRÖDINGER (LDA) EQUATION
Determination of wave functions and bound states
OUTPUT
MATRIX ELEMENTS
WAVE FUNCTIONS
OPTICAL SPECTRA,...
Numerical principles and techniques
Use state-of-the-art sparse linear systems solvers, CG methods
for N > 106
Example: eigenvalue problems
Hu=lu
Problem: H huge (N > 106) + sparse, need only few l
Solution: Subspace projection method
Power method: u0  H u0  H2 u0  ... <u|H|u  >= lmax
ARPACK: very robust, degenerate evals, but fairly slow.
Electronic structure principles and techniques
Multiband k.p envelope function approach
 Based on "patching up" bulk Hamiltonians to build Hamiltonian
for mesoscopic structures, is efficient and sufficiently accurate
 Method has built-in ambiguities that can lead to ghost states, spikes
in density,...
 Spatial discretization can lead to instabilities and wrong oscillatory
solutions
Spikes
Y 12 (S=0)
Y21
Ghost states
Have eliminated artifacts in k.p+envelope function theory by
 careful treatment of far-band contributions
 using operator orderings that are manifestly self-adjoint
 employing upwinding scheme for discretizing derivatives
Multivalued operator ordering
k·p
HVol(k) =
Ec
iPk
-i P k
Ev+Lk2
P(x) /x or
HVol() : /x P(x) or
(P(x) /x + /x P(x))
 Ordering unclear because of position dependent parameter P
 Different orderings yield Hermitian Ham., but cause
contradictory boundary conditions (Non-self-adjointness)
Solution of Problem:
Self-adjoint H
HVol(x) =
Ref: B. A. Foreman, Phys. Rev. B 56, R12748 (1997)
Ec
P /x
- /x P
Ev- /x L /x
Electronic structure principles and techniques
Example: Eliminating oscillatory solutions
Hbulk(k)  H(): Discretization of 1. derivates is not unique.
F(n) = F(n+1) - F(n-1) is compatible with
n-1
n+1
n
Solution:
F(n) = [F(n) - F(n1)] excludes unphysical oscillatory solutions
ForwardDifferencing
H=
BackwardDifferencing
Ref: Andlauer et al, to be publ.
equivalent to
upwinding scheme
Electronic structure principles and techniques
Problem: How to solve Schrodinger equation for nanodevice in B-field?
Vector potential A(x) diverges with x  Discretized version of
H violates gauge invariance  arbitrary results
(-i + A(x))2
H=
+V(x)
2m
Solution:*
Invariance under
gauge transformation is violated
if f = lim f(x+e) - f(x)
e
e0
x
y
Define U(x,y) = exp( -i  A(z) dz )
x
Define D = 1e [f(x+e) - U(x+e,x) f(x)]
Use Hamiltonian H =
guarantees local
gauge invariance
D2
+V(x)
2m
 This Hamiltonian is gauge invariant and suitable for discretizing
the Schrodinger equation in magnetic fields
 Works for any multi-band, relativistic k.p Hamilonian for nanostructures
*) Morschl et al, to be publ.
Carrier transport in nextnano
Quantum drift-diffusion (QDD) equations:
  j ( x )      n ( x ) E F ( x )   0


n(x) 

i
2
 E F ( x ) -E i 


k BT


 i ( x ) f 
 WKB-type approach, suitable for diffusive transport near equilibrium
 Good for barrier-limited transport
 Misses quantum resonances and interference effects
Mamaluy, Sabathil, V., PRB 05
Contact block reduction-method (CBR):
 Efficient method to calculate strictly ballistic transport through
open device with arbitrary number of leads
 Scales with N2 rather than N3
 Suitable for very short quantum devices close to resonance
Non-equilibrium Green’s function method (NEGF):
Kubis, V., subm.
 Full quantum transport with all relevant sccattering mechanisms
 Only for vertical transport (quasi-1D)
Assessment of QDD
Tunneling through thin barrier
Fully SC-NEGF
-0.05
Ballistic
20
30
40
Position [nm]
*) Kubis et al, Poster #96
50
Fully SC-NEGF
nextnano
-0.10
0V
-0.18
10
20
30
40
50
Position [nm]
0.00
10
-0.02
0
0.05
-0.10
n-GaAs (1018 cm-3)
0.06
n-Si
V=0
nextnano
InGaAs
n-GaAs
60
Conduction band [eV]
Conduction band [eV]
0.10
n-Si (1018 cm-3) SiO2
Carrier capture by quantum well
Conduction band [eV]
Comparison of QDD with
fully self-consistent NEGF*)
shows good agreement...
 close to equilibrium
 in situations where interference
effects are weak
0.06
nextnano
Fully SC-NEGF
-0.02
-0.10
0.08 V
-0.18
0
10
20
30
Position [nm]
40
50
1D Results: Optical absorption in
SiGe QCL-structures
Si/SiGe p-type Quantum Cascade Laser Structure
Collector
-0,70
HH1
Energy [eV]
-0,80
Injector
-0,90
-1,00
-1,10
SiGe
Si
-1,20
HH2
50 kV/cm
-10
0
10
20
30
40
Distance [nm]
QC structure:
G. Dehlinger et al., Science 290, 2277 (2000)
 HH1  HH2
Exp: 125 meV
Nextnano: 124.5 meV
 Lateral non-parabolicity
plays important role for
optical spectra: accurate
6-band kp-model necessary
2D Results: Equilibrium + QDD for Si DG-FET
Effect of el-el interaction:
exchange-correlation potential
Local density functional theory adds VXC=VX+VC to VHartree
-1 0
-2 0
y
VX >> VC
VX = a n1/3 = 50 meV for n=1020 cm-3
E xc h an g e
C o rr ela tio n
G
D
S
G
x
-3 0
VS D = 0 .0 5 V
VS G = 0 V
2
-4 0
-5 0
10
S o u rc e
D rain
H a rtre e + X C
H a rtre e
-6 0
-1
-7 0
0
5
10
15
20
25
30
35
40
45
50
X a x is [n m ]
H artree
H a rtre e + X C
-0 .0 2
-0 .0 4
Potential [eV]
S o u rc e -d ra in c u rren t
1
10
Current [Acm ]
Potential energy [meV]
0
G a te
G a te
-0 .0 6
-0 .0 8
0
10
VSD=0.05 V
-1
10
-2
10
-3
10
-4
G a te c u rren t
10
-0 .1 0
-5
-0 .1 2
y
-0 .1 4
G
D
S
G
x
-0 .1 6
0 .0
2 .5
5 .0
7 .5 1 0.0 1 2.5 1 5.0 1 7.5
Y ax is [n m ]
10
-0 .6
-0 .5
-0 .4 -0 .3 -0 .2 -0 .1
G a te Vo lta g e VS G [V ]
0 .0
0 .1
VXC has very large effect for small VSD
B
InP
InAs
Nanowire
InP
53 nm
32 nm
Effective g-factor
3D Results: Prediction of magnetic g-tensors in nanowires
10
8
nextnano
g^
6
nextnano
4
g//
Experiment (Björk et al, 05)
2
8
Ground state density (front view)
B=20 T
B=0 T
g//
10
12
14
16
18
20
Length [nm]
Excellent agreement between
calculated g-factor and experiment
3D results: Self-assembled buried QD
ex x
G aAs
ex x
ex z
G aAs
ex z
In A s
In A s
Material strain
20 n m
20 n m
-
+
+
-
Hole
Piezoelectric
polarization charge
Hole
Electron
Efficient light emission
Electron
Modified exciton
states
No light emission
3D results: Quantum Dot Molecule
Vertically stacked
Data: QD
InGaAs/GaAs
vertically stacked QD
Strain field (nextnano)
exx
WL
6 nm
WL
Electron wave functions:
Electron & hole wave functions
bonding
antibonding
Calculate exciton energies
3D Results: Neutral excitons in QD-Molecule
Quantum coupling + strain + Coulomb interaction
 Large separation: direct and indirect excitons
 Small separation: el-dominated bonding and antibonding excitons
Exciton Energy [eV]
1.29
antibonding
1.28
1.27
1.26
indirect Ex
bonding
Coulomb
interaction [~20 meV]
1.25
direct Ex
1.24
2
4
6
8
Dot separation [nm]
10
3D Results: Anticrossing of direct + indirect states
Energy [meV]
indirect
direct
1285
*
Nextnano
1280
Experiment
1275
14
16 18 20 22 24
Applied Field [kV/cm]
*) P.W. Fry et al, PRL 84, 733 (2000), G. Ortner et al., PRL 94, 157401 (2005)
H. J. Krenner et al., PRL 94, 057402 (2005), G. Bester et al., cond-mat/0502184
3D results: ballistic transport:
resonant tunneling through QD-molecule
InGaAs/InP dots
InGaAs collector
InAs wetting layers
J
InP barrier
40 nm
InGaAs emitter
250
x 10
200
150
Current
[pA] 100
Exp.
Theory
50
0
0.2
0.3 0.4 0.5
Voltage [V]
Bryllert et al, APL 82 , 2655(2003)
 Ballistic current (CBR)
 Good agreement in resonance position
 Absolute value of J off by factor of 10
 Current depends strongly on inter-dot distance
and lateral misalignment
Resonant tunneling through QD-molecule
30
25
20
15
10
5
0
7
Line width [meV]
Line width [meV]
The resonance line width ...
11 12 13 14 15
Distance between WLs [nm]
...decreases exponentially
with increasing inter-dot distance
6
5
4
25
50
75
100
Overlap of base areas [%]
...and is proportional to
overlap of base areas
How to get nextnano?
 Software including source is free (nextnano)
 Online documentation is free
 Online registration is free
 Support, customized input files + on-site training
available on request (by S. Birner)
 Some complex tutorial files (QCLs, MOSFETs) are not
free
www.nextnano.de
Summary
Nextnano provides base for physics of 1D, 2D, and 3D
semiconductor nanostructures
Handles equilibrium electronic structure, optics, magnetic fields
Nonequilibrium: QDD approach, ballistic current, and NEGF
Successful application to 2D+3D nano-MOS, QD molecules,
excitons, magnetic field effects, QCL‘s,...