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National Technical University of Ukraine“Kyiv Polytechnic Institute”
Authors: Fedyay Artem, Volodymyr Moskaliuk, Olga Yaroshenko
Quantum transport simulation tool,supplied with GUI
Presented by: Fedyay Artem
13, April 2011 Kyiv, UkraineElNano XXXI
Department of physical and biomedical electronics
fedyay@phbme.ntu-kpi.kiev.ua 2
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
ψ( ) ( )χ( )nu
kr r r
i rχ( )=e krin case of homogeneous s/c
and flat bands (Bloch waves)
Envelope of what?
of the electron wave function:
What if not flat-band?
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Physical model. Type
ENVELOPE FUNCTION (EFFECTIVE MASS) METHOD
actual potential
hereafter will denote envelope of the wave function of electron in a crystal
*1m *
2m
its approximation within the method
ψ( ) χ( )r r
( ) ( )U Ur r
0 * m m
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Model’s restrictionh/s with band wraps of type I (II)
TYPE III
InAs-GaSb
Bandstructures sketches
TYPE I
GaAs – AlGaAs
GaSb – AlSb
GaAs – GaP
InGaAs – InAlAs InGaAs – InP
TYPE II
InP-Al0.48In0.52As
InP-InSb
BeTe–ZnSe
GaInP-GaAsP
Si-SiGe
za
Ec
Ev
Ez
DEc
DEv
za
Ec
Ev
Ez
DEc
DEv
za
Ez
A Б
Ec
Ev
DEc
DEv
A
Б
A
Б
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Physical model.Type
What do we combine?
leftreservoir
i-AlxGa1-xAs
bbс с
i-GaAs n+-GaAs
ND ND
device(active region)
rightreservoir
a
semiclassical
envelope-fucntion
we combine semiclassical and “quantum-mechanical”approaches for different regions
Sometimes referred to as “COMBINED”
(*) homogeneous ,(**) almost equilibrium high-doped
(*) 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=1022...1024 m-3ND=0
Material base n+ GaAs n+ GaAsi-GaAs, i-AlAs
Electron gas (e.g.)
3D 3Dquazi-2D
State Local equilibrium Local equilibriumNon-equilibrium
2 2 2 2
* *( ) ,
2 2zk k
E U zm m
Dispersion low2 2 2 2
* *( )
2 2zk k
E U zm m
2 2y zk k k
Wave nature of electron is taking into account by means of
Effective mass;Band wrappings
Effective mass;Band wraping;
Envelope wave function
Mean free pathMore then reference
dimensions
More then reference
dimensions
Less then reference
dimentions
Motion mechanism
Drift, diffusionBallistic, quazi-
ballistic
2 2 2 2
0(5) * *( ) ,
2 2if 0( 0)
z
z z
k kE U z
m mk k
Electron concentraion
Ф 1( )
,( )
1 expk
i
ii c
NU
E Un N dE
E E U
T
* 2 3/24 (2 / h )cN m
0 5
2
( ) Ф 1( )( )
Б( ) 0( 5)
( ( ), ) ( )ln 1 exp ,
i i
L R z z z NL R c z
U U z i i
E k z E E Un N dE
k TE U
* 3/2Б
2 3
2( ) k
(2 )c
m TN
Effective mass;Band wrappings
Drift, diffusion
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Physical model.Master equations.(1 band)
0 5
2* 3/2
( ) Ô 1( )Á( ) 2 3
Á( ) 0( 5)
1). Electronconcentrationinquantumregion:
( ( ), ( , ) ( )2( )( , ) ln 1 exp ,
(2 )
where ( , ( )) ( ) ( , ( ))
2
i i
L z z L R z NL R z
U U z i i
lattice
scf
scf
scf
E k z E E Um k Tz dE
k TE U
U z n z U z U z n
n
z
UUn
0
*( )
( )* 2
). Poissonequation :
1( ) ( , ( )) ( ) .
Wave functions are solutionsof Schrodinger equations
( )1 2 ( ( ))( ) 0
( )
scfscf
L RL R
ddz z z N z
dz dz
d zd m E U zz
dz m z
U
d
n U
z
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Physical model.Electrical current.Coherent component
Coherent component of current flow is well described
by Tsu-Esaki formulation:
5 0
*B
2 3max( , )
( )2
,where2
( )( )
i i
z z
U U
zz D Em ek T
T dE EJ e
B 1
Á
F
B
( )1 exp
ln( )
1 ex
( )
p
z
z N
z
E E Uk T
T
DE E U
E
k
is a transmission
coefficient through
quantumre io
( )
g n
zT E
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Physical model.Electrical current.(!) Coherent component
z
Ez
LFE
RFE
NU
0 z0 z50iU L
EF
EF
5iU
quantum region
electron states from
left reservoir
electron states from both reservoirs
no electron states
depends on
( )
( )zT
U z
E
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Physical model.
Which equation L and R are eigenfunction of?
We need |L|2 and |R |2 for calculation of CURRENT and CONCENTRAION
– Schrödinger equation with effective mass. H E , where
ˆ ˆ ( ) ( )c H opH T E z U z iW ,
where:
2 1ˆ2 *( )
Tz m z z
is kinetic energy operator;
( )cE z is a bottom of Г-valley;
HU is the Hartree potential;
opW is so-called “optical” potential, which is modeling escaping of electrons
from coherent channel due to interaction with optical phonons.
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2-band model. What for?
E
D
XL
AlA
sE
XE
D
0
0.2
0.4
0.6
0.8
E, эВ
L
X
LG
aAs
E
LEDX AlA
sE
a b
0 a a+b
b
AlAsGaAs GaAs GaAsAlAs
z
XG
aAs
E
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2-band model. What for?
0
0.2
0.4
0.6
0.8
E, эВ
X
Г-X-Г
Г-X-Г
Г-X mixing points
Г-X
Current re-distribution between valleys changing of a total current Electrons re-distribution changing potential
! [100]
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Physical model. Г, X
It was derived from k.p-method that instead of eff.m.Schr.eq. it must be a following system:
2
ÃÃ Ã Ã
2X X X
X
10
( ) 2 0( ) 1
02
H z i
i H z
U U E x z m zx U U E
z m z
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.
L
R
ikzLr eikze ik z
Lt e ikz
Rt eik ze ik z
Rr e z
2
( ) LzTk
tk
E
Transmission coefficient needs to be found for currentcalculation
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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
Seminar on Electronics Technology “ISSE 2010. 2. 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. - 2005. – Вып. 39. - 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.
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 methodsNumerical problems and solutions:
? Computation of concentration n(z) needs integration of stiff function
using adaptive Simpson algorithm;
? Schrodinger equation have to be represented as finite-difference scheme, assuring conservation, and needs prompt solution
using of conservative FD schemes and integro-interpolating Tikhonov-Samarskiy method;
? Algorithm of self-consistence with good convergence should be used to find VH
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
left.r.
i-Al0.33Ga0.67As
3 nm
i-GaAs n+-GaAs
ND=1024
device right.r.
4 nm3 nm
ND=1024
L=100 nm
10 nm 20 nm
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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)
mГ(x),
(!) Each layer supplied withthe following parameters:
x – molar rate inAlxGa1-xAs
DEc(x)=U00*x
mГ(x)=m00+km*x,
DEc(x) – band off-set
(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 g (Ez,z) (in new window with legend)
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Distribution function N (Ez,z) (tone gradation)
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I-V characteristic(non self-consistent case)
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Resonant tunneling diode(2 valley approach)
mX
DEХ-Г
(!) Each layer supplied withadditional parameters:
CB in Г and X-points
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LDOS in Г and X-valleys
X-valley:barriers wells
Г-valley
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Transmission coefficient2 valleys
Г – valley only
both Г and X valleys
(*) Fano resonances (**) additional channel of current
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Another example:supperlattice AlAs/GaAs 100 layers
LDOS
CB profile
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Try it for educational purposes!
Simulation tool corresponding to 1-band model w/o scatteringwill 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 resultsand will not be submitted heretofore
THANK YOU FOR YOUR ATTENTION