Department of physical and biomedical electronics
National Technical University of Ukraine
“Kyiv Polytechnic Institute”
Quantum transport simulation tool,
supplied with GUI
Authors: Fedyay Artem, Volodymyr Moskaliuk, Olga Yaroshenko
Presented by: Fedyay Artem
13, April 2011
ElNano XXXI
Kyiv, Ukraine
Overview





Objects of simulation
Physical model
Computational methods
Simulation tool
Examples of simulation
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Objects to be simulated
Layered structures with transverse electron transport:
resonant-tunneling diodes (RTD) with 1, 2, 3, … barriers;
Supperlattices
Reference
topology
(example):

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Physical model. Intro
ENVELOPE FUNCTION (EFFECTIVE MASS) METHOD

Envelope of what?
of the electron wave function:
ψ(r )  u (r )χ(r )
nk
χ(r )=e
ikr
in case of homogeneous s/c
and flat bands (Bloch waves)

What if not flat-band?
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Physical model. Type
ENVELOPE FUNCTION (EFFECTIVE MASS) METHOD
m1*
ψ(r)

χ(r)
U (r)

U (r)
m

m*
0
actual potential
its approximation
within the method
m2*
hereafter  will denote envelope of
the wave function of electron in a crystal
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Model’s restriction
h/s with band wraps of type I (II)
DEv
Б
Ev
a
A
Ev
a
z
Б
DEv
Б
A
Ec
Ec
DEc
DEc

Ez
Ec
DEv
Band
structures
sketches
Ez
z
Ev
A
a
TYPE I
TYPE II
TYPE III
GaAs – AlGaAs
GaSb – AlSb
GaAs – GaP
InGaAs – InAlAs
InGaAs – InP
InP-Al0.48In0.52As
InP-InSb
BeTe–ZnSe
GaInP-GaAsP
Si-SiGe
InAs-GaSb
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DEc
Ez
6
z
Physical model.
Type
Sometimes referred to as
“COMBINED”
 What do we combine?
“
we combine semiclassical and quantum-mechanical
”
approaches for different regions
device
(active region)
left
reservoir
right
reservoir
ND
ND
b
a
с
с
i-AlxGa1-xAs
b
i-GaAs
semiclassical
(*) homogeneous ,
(**) almost equilibrium high-doped
envelopefucntion
n+-GaAs
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(*) nanoscaled heterolayers ,
(**) non-equilibrium intrinsic
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Physical model.
Electron gas
Parameter
Value in l.r.
Value in a.r.
Value in r.r.
Donor’s
concentration
ND=1022...1024 m-3
ND=0
ND=1022...1024 m-3
Material base
n+ GaAs
i-GaAs, i-AlAs
n+ GaAs
Electron gas
(e.g.)
3D
quazi-2D
3D
State
Dispersion low
Local equilibrium
E  U (z) 
k   k y2  k z2
Wave nature of
electron is
taking into
account by
means of
Mean free path
Motion
mechanism
Electron
concentraion
2
2 2
k z2
k

,
*
2m
2m *
Non-equilibrium
2 2
k z2
k

,
*
2m
2m *
if k z  0(k z  0)
E  U ( z0(5) ) 
Effective mass;
Band wrappings
Less then
reference
dimentions
Ballistic, quaziballistic
Drift, diffusion
nL( R )  Nc


Ui 0 ( Ui 5 )
2
Effective mass;
Band wraping;
Envelope wave function
More then
reference
dimensions
 L( R ) (Ez (k z ), z )
Ez  Ui 0( i 5)
2
Local equilibrium
E  U (z) 
2
2 2
k z2
k

*
2m
2m *
Effective mass;
Band wrappings
More then
reference
dimensions
Drift, diffusion

 E  (EФ  U1( N ) )  
ln  1  exp   z
 dEz , Nc 
kБT



2(m * )3/2 k БT
(2)2 3
E  Ui
dE, Nc  4(2m * / h2 )3/2
E
 (EФ  U1( N ) ) 

Ui
1  exp 

kT



ni  Nc 
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Physical model.
Master equations.
(1 band)
1). Electronconcentrationinquantumregion:
nL( R ) ( z,Uscf ) 
* 3/2
2(m ) kÁT
(2)2 3
 L (Ez (k z ),Uscf ( z, nL( R ) )


Ui 0 (Ui 5 )
Ez  Ui 0( i 5)
2

 E  (EÔ  U1( N ) )  
ln  1  exp   z
 dEz ,

k
T

Á


where U ( z, n( z ))  Ulattice ( z )  Uscf ( z, n( z))
2). Poissonequation :
dU
d
1
( z ) scf    n( z,Uscf ( z ))  N ( z ).
dz
dz
0
Wave functions are solutionsof Schrodinger equations
d  1 d  L( R ) ( z )  2m * (E  U ( z ))
 L( R ) ( z )  0


2
dz  m * ( z )
dz

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Physical model.
Electrical current.
Coherent component
Coherent component of current flow is well described
by Tsu-Esaki formulation:
2m*ekBT
J  ez
(2 )2 3
T (Ez ) is a transmission
coefficient through


T (Ez )D(Ez )dEz ,where
max(Ui 5 ,Ui 0 )

 E z  (EB  U1 )  
 1  exp  

k
T

Á

D(E z )  ln 

 E z  (EF  UN )  
1

ex
p


 
k
T

B


quantumregion
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Physical model.
Electrical current.
(!) Coherent component
EFR
UN
T (E z ) depends on
EF
electron states
from both
reservoirs
Ez
U i5
U (z )
EF
electron
states from
left reservoir
EFL
Ui0 0
z0
z5
no
electron
states
L
quantum region
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z
Physical model. Y?
We need |YL|2 and |YR |2 for calculation of CURRENT and CONCENTRAION
 Which equation YL and YR are eigenfunction of?
– Schrödinger equation with effective mass.
Hˆ   E , where
Hˆ  Tˆ  Ec ( z )  UH ( z )  iWop ,
where:
2

1 
is kinetic energy operator;
2 z m *( z ) z
Ec ( z ) is a bottom of Г-valley;
UH is the Hartree potential;
Wop is so-called “optical” potential, which is modeling escaping of electrons
from coherent channel due to interaction with optical phonons.
Tˆ  
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2-band model. What for?
0.6
DE L
XL
EAlAs
GaAs

L
EGaAs
L
0.2
0
AlAs
X
X
EGaAs
0.4
GaAs
DE 
0.8
AlAs
X
EAlAs
GaAs
DE X
E,
эВ
b
a
0
b
a
a+b
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z
13
2-band model. What for?
E,
эВ
0.8
Г-X-Г
Г-X
0.6
! [100]
0.4
X
Г-X-Г
0.2
0

Г-X mixing points


Current re-distribution between valleys changing of a total current
Electrons re-distribution  changing potential
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Physical model. YГ, YX?
It was derived from k.p-method that instead of eff.m.Schr.eq. it must be a
following system:
UÃ  UH  E z
  ( x )

i
 2  1 

 ( xi )   à   2 z mà z

U X  UH  E z   X  
0


  
 Ã  0
2
 1    X 

2 z m X z 
0
which “turns on” Г-X mixing at heterointerfaces (points zi) by means of .
It of course reduces to 2 independent eff.m.Shcr.eqs. for X and Г-valley
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Physical model.
Boundary conditions for Schr. eq.
We have to formulate boundary conditions for Schrödinger equation. They
are quite natural (QuantumTransmissionBoundaryMethod). Wave functions in the
reservoirs are plane waves.
eikz  ikz
rL  e
t R  e  ikz
t L  eik z
e  ik z ik z
rR  e
z
L
R
k
T (E z ) 
tL
k
2
Transmission coefficient needs to be found for current
calculation
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Physical model. Features




Combined quazi-1D.
Self-consistent (Hartee approach).
Feasibility of 1 or 2-valley approach.
Scattering due to POP and Г-X mixing is taking
into acount.
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Scientific content circumstantial evidence:
direct use of works on modeling of nanostructures 1971-2010
1.
Moskaliuk V., Timofeev V., Fedyay A. Simulation of transverse electron transport in resonant tunneling diode // Abstracts Proceedings of 33nd International Spring
2.
Seminar on Electronics Technology “ISSE 2010.
Abramov I.I.; Goncharenko I.A.; Kolomejtseva N.V.; Shilov A.A. RTD Investigations using Two-Band Models of Wave Function Formalism // Microwave &
Telecommunication Technolog, CriMiCo 2007. 17th International Crimean Conference (10–14 Sept. 2007), 2007.–P.: 589–590.
3.
Абрамов И.И., Гончаренко И.А. Численная комбинированная модель резонансно-туннельного диода // Физика и техника полупроводников. - С. 1138-1145.
4.
Pinaud O. Transient simulation of resonant-tunneling diode // J. Appl. Phys. – 2002. – Vol. 92, No. 4. – P. 1987–1994.
5.
Sun J.P Mains R.K., Haddad G.I. “Resonant tunneling diodes: models and properties”, Proc. of IEEE, vol. 86, pp. 641-661, 1998.
6.
Sun J.P. Haddad G.I. Self-consistent scattering calculation of Resonant Tunneling Diode Characteristics // VLSI design. – 1998. – Vol. 6, P. 83–86.
7.
Васько Ф.Т. Электронные состояния и оптические переходы в полупроводниковых гетероструктурах. – К.: Наукова Думка, 1993. – 181 с.
8.
Zohta Y., Tanamoto T. Improved optical model for resonant tunneling diode // J.. Appl. Phys. – 1993. – Vol. 74, No. 11. – P. 6996–6998.
2005. – Вып. 39.
9. Mizuta H., Tanoue T. The physics and application of resonant tunnelling diode. – Cambridge University Press, 1993. – 245 c.
10. Sun J.P., Mains R.K., Yang K., Haddad G.I. A self-consistent model of Г-X mixing in GaAs/AlAs/GaAs quantum well using quantum transmitting boundary method // J.
Appl. Phys. – 1993. – Vol. 74, No. 8. – P. 5053–5060.
11. R. Lake and S. Datta. Nonequilibrium “Green’s function method applied to double barrier resonant tunneling diodes”, Phys. Review B, vol. 45, pp. 6670-6685, 1992.
12. Lent C. S. and Kirkner D. J. The quantum transmitting boundary method // Journal of Applied Physics. - 1990. - Vol. 67. - P. 6353–6359.
13. K.L. Jensen and F.A. Buot. “Effects of spacer layers on the Wigner function simulation of resonant tunneling diodes”, J. Appl. Phys., vol. 65, pp. 5248-8061, 1989.
14. Liu H.C. Resonant tunneling through single layer heterostructure // Appl. Phys. Letters – 1987. – Vol. 51, No. 13. – P. 1019–1021.
15. Пакет для моделювання поперечного транспорту в наноструктурах WinGreen http://www.fz-juelich.de/ibn/mbe/software.html
16. Хокни Р., Иствуд Дж. Численное моделирование методом частиц: Пер. с англ. – М.: Мир, 1987. – 640 с.
17. Нгуен Ван Хьюеу. Основы метода вторичного квантования. – М.: Энергоатомиздат, 1984. – 208 с.
18. R. Tsu and L. Esaki. “Tunneling in a finite superlattice”, Appl. Phys. Letters, vol. 22, pp. 562–564, 1973.
19. Самарский А.А. Введение в теорию разностных схем. – М.: «Наука», 1971. – 553 с.
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Computational methods
Numerical problems and solutions:
?
Computation of concentration n(z) needs integration of stiff function

?
Schrodinger equation have to be represented as finite-difference scheme, assuring conservation, and
needs prompt solution

?
using of conservative FD schemes and integrointerpolating Tikhonov-Samarskiy method;
Algorithm of self-consistence with good convergence should be used to find VH

?
using adaptive Simpson algorithm;
using linearizing Gummel’s method
Efficient method for FD scheme with 5-diagonal matrix solution
(appeared in 2-band model, corresponding to Schrödinger equation)

direct methods, using sparse matrix concept in Matlab (allowing significant memory
economy)
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Let’s try to simulate
Al0.33Ga0.64As/GaAs RTD
L=100 nm
device
left.r.
right.r.
ND=1024
ND=1024
3 nm
10 nm
i-Al0.33Ga0.67As
4 nm
3 nm
20 nm
i-GaAs
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n+-GaAs
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Application with GUI
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Emitter
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Quantum region
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Base
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Materials data-base
(1-valley case)
(!) Each layer supplied with
the following parameters:
mГ(x), x – molar rate in
AlxGa1-xAs
mГ(x)=m00+km*x,
DEc(x) – band off-set
DEc(x)=U00*x
(x) is dielectric permittivity
 (x)= e00+ke*x
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Settings
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Graphs
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Calculation: in progress
(few sec for nsc case)
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Calculation complete
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Concentration
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Potential (self-consistent)
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Concentration (self-consistent)
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Transmission probability
(self-consistent)
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Local density of states
g (Ez,z)
(self-consistent)
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Local density of states
(in new window with legend)
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g (Ez,z)
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Distribution function
(tone gradation)
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N (Ez,z)
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I-V characteristic
(non self-consistent case)
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Resonant tunneling diode
(2 valley approach)
(!) Each layer supplied with
additional parameters:
mX
DEХ-Г

CB in Г and X-points
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LDOS in Г and X-valleys
Г-valley
X-valley:
barriers  wells
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Transmission coefficient
2 valleys
Г – valley only

(*) Fano resonances
(**) additional channel of current
both Г and X valleys
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Another example:
supperlattice AlAs/GaAs 100 layers
CB profile
LDOS

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Try it for educational purposes!
Simulation tool corresponding to 1-band model w/o scattering
will be available soon at: www.phbme.ntu-kpi.kiev.ua/~fedyay
(!) Open source Matlab + theory + help
Today you can order it by e-mail: fedyay@phbme.ntu-kpi.kiev.ua
2-band model contains unpublished results
and will not be submitted heretofore
THANK YOU FOR YOUR ATTENTION
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