n ational technical university of ukraine “kyiv polytechnic institute”
DESCRIPTION
NTUU "KPI" 1898. Department of physical and biomedical electronics. N ational Technical University of Ukraine “Kyiv Polytechnic Institute”. Q uantum transport simulation tool, supplied with GUI. Authors: Fedyay Artem, Volodymyr Moskaliuk, Olga Yaroshenko. Presented by: Fedyay Artem. - PowerPoint PPT PresentationTRANSCRIPT
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
Overview
Objects of simulation Physical model Computational methods Simulation tool Examples of simulation
Objects to be simulated
Layered structures with transverse electron transport:- resonant-tunneling diodes (RTD) with 1, 2, 3, … barriers;- Supperlattices
Reference topology (example):
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?
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
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
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
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
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
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
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
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.
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
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]
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
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
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.
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.
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)
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
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
Resonant tunneling diode(2 valley approach)
mX
DEХ-Г
(!) Each layer supplied withadditional parameters:
CB in Г and X-points
Transmission coefficient2 valleys
Г – valley only
both Г and X valleys
(*) Fano resonances (**) additional channel of current
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: [email protected]
2-band model contains unpublished resultsand will not be submitted heretofore
THANK YOU FOR YOUR ATTENTION