Modelowanie Nanostruktur
Semester Zimowy 2011/2012 Wykład
Jacek A. Majewski Chair of Condensed Matter Physics Institute of Theoretical Physics
Faculty of Physics, Universityof Warsaw
E-mail: Jacek.Majewski@fuw.edu.pl
Coherentny Transport w Nanostrukturach
Przewodnosc poprzez transmisje Tunelowanie w nanostrukturach
Modelowanie Nanostruktur, 2011/2012
Jacek A. MajewskiWykład 5 – 8 XI 2011
2D-Electron Gas
Coherent quantum transport: Conductance from Transmission General expression for the current
Tunneling in semiconductors
Resonant Tunneling Diode Negative Differential Resistance
Recommended reading:
Supriyo Datta, “Quantum Transport, Atom to Transistor”
Cambridge University Press 2005
Supriyo Datta, “Electronic Transport in Mesoscopic Systems”
Cambridge University Press 1995
Coherent Transport
2D-Electron Gas
Electron mobility
Bulk GaAs at T=300 K – 9 000 cm2V-1 s-1 2DEG at T=300 K in HEMT –
10 000-12 000 cm2 V-1 s-1
Electron mobility in 2DEG at low temperatures ( < 1 K) can reach 20 000 000 cm2 V-1 s-1 !!!
EF
M AlGaAs GaAs
z
2DEG Ec
EF AlGaAs GaAs
z
2DEG Ec Gated AlGaAs/GaAs heterostructure AlGaAS/GaAs heterostructure
Classical in-plane transport
Ballistic, Coherent Quantum Transport
GaAs - substrate AlGaAs 2DEG
Mesoscopic devices – macroscopic with quantum effects present
Conductance from Transmission
( , ) ( , )
F L F F
f E
f E E
V
( , ) ( , )F R F F
f E f E E
Device
Lead Lead
Contact
Contact
m
L= EF + VL R m
R= EFmL- electrochemical potential = chemical potential + voltage V – external bias
EF – Fermi energy (chemical potential in the absence of bias)
The Landauer approach – very useful in describing mesoscopic transport
The current through a conductor (device) is expressed in terms of
probability that an electron can transmit through it
Leads are reservoirs of electrons in which energy- and momentum relaxation processes are so effective that the electron system remains in equilibrium even under a given applied voltage bias
( , ) 1
exp[ ( )] 1 fF E
E
The electron concentration in the leads is so high that the electrostatic potential in each lead is taken to be constant (as for the case of metal)
Fullerene-based molecular nanobridges
Ab-initio studies of the electrical transport in nanocontactsAl Al
C
60Conductance from Transmission
Time-independent transport; Inelastic processes are negligible Schrödinger Equation:
2 2
*
( , , ) ( , , ) ( , , ) 2 V x y z x y z E x y z
m
1 2
( , , ) ( )
( , )V x y z
V x
V y z ( , , ) x y z ( ) x
T( , ) y z
( , ) , ( , )
T y z n m y z
ETEn m,
( ), x E
||
The potential energy
Transverse modes Scattering states
Transport in x-direction
|| T
E E E
Asymptotic solutions for energy
I R
( )
T
l l
r
ik x ik x
l L
ik x
r R
e r e x x
t e x x
The wave function of electrons incoming from the left lead Incoming wave Reflected wave
Transmitted wave
E
|| V x1( ), x Conductance from Transmission
I R T
The wave function of electrons incoming from the right lead Incoming wave Reflected wave
Transmitted wave
( )
r r
l
ik x ik x
r R
ik x
l L
e r e x x
t e x x
( ) & ( )
Two linearly independent solutions for energyE
||0 x
divj j const j const Continuity equation for current (particle flow)
j x (
L) j x (
R)
: *( * *)2
i d d
j m dx dx
* *
(1 )
l l l r r r
k r r k t t
* *
(1 )
r r r l l l
k r r k t t
Continuity of current forscattering states
( )
( )* *
* *
( ) (2 2 ) (1 )
L 2 l l l l l l l
j x i ik ik r r k r r
m m
exp( ) R exp( )
R r r r r r
t ik x d ik t ik x
dx
*
* * *
exp( ) R exp( )
R r r r r r
t ik x d ik t ik x
dx
*
* *
* *
( ) ( )
2
R R
R R R r r r
d d
j x i k t t
m dx dx m
* *
(1 )
l l l r r r
k r r k t t
*
*
: *( )
2
i d d
j m dx dx
exp( ) exp( ) L exp( ) exp( )
L l l l l l l l l
ik x r ik x d ik ik x ik r ik x
dx
*
* * *
exp( ) exp( ) L exp( ) exp( )
L l l l l l l l l
ik x r ik x d ik ik x ik r ik x
dx
* * *
exp( 2 ) exp(2 )
L
L l l l l l l l l l l
d ik ik r ik x ik r ik x ik r r
dx
*
* *
exp( 2 ) exp(2 )
L
L l l l l l l l l l l
d ik ik r ik x ik r ik x ik r r
dx
( )Right lead Left lead
Conductance from Transmission
Proof of the relationk
l(1 r r
l l*) k t t
r r r* forConductance from Transmission
More relations for transmission and reflection coefficients
( ) ( )
&
Two linearly independent solutions for energy( )* & ( )*
Are also solutions of the Schrödinger Equation for energyE
||E
||Schrödinger equation – equation of the 2nd order
ONLY TWO LINEARLY INDEPENDENT SOLUTIONS
( )*
A
( )B
( )
( )* C
( ) D
( )( ) ( ) ( )
( ) ( ) ( )
* L A L B L
exp(ik xl )rl*exp(ik xl ) Aexp(ik xl )Arlexp(ik xl ))Btlexp(ik xl )
1
Ar
lBt
lA
r
l*( ) ( ) ( )
( ) ( ) ( )
* R A R B R
*
exp( ) exp( ) exp( ) exp( )
r r r r r r r
t
ik x
At ik x
B
ik x
Br ik x
0 At
r Br
rB t
r*1 r r
l l* t t
r*lr t
l*r t r
r*r 0
(I) (II)
From Eq. (II) one obtains
* *
,
l r
C
t D
r 1 r r
r r* t t
r l*rt
l l* t r
l r* 0
Conductance from Transmission
More relations for transmission and reflection coefficients
* *
1r rl l t tr*l r tl rt rr*r0 1r rr r*t tr l* r tl l*t rl r*0
* *
l r r r 0 r t t r
*
* r r l
r
r t r t
*
* l r l
l
r t r t
* *
* *
l r
r l l r
r l
t
t t t t t
t t
(*) & (**) (**)
* * 2 2
| | | |
l l r r l r
r r r r r r
Previously it was proved
* 2
1
l l r|
r|
l
r r k t
k
* 2
1
r r l|
l|
r
r r k t
k
2 2 2 2
| | | |
r r l l
k t k t
2 2
| |
l| |
r
r l
l r
k
k t t
k
k
(*)
Conductance from Transmission
Definition of transmission coefficients*
*
: *( )
2
i d d
j m dx dx
exp( )
in ik xl
outtrexp(ik xr ) reflrlexp(ik xl )
2 2
| | | |
*
r
out r r r
j k t v t
m
/ *
in l l
j k m v
| |2 l
R L l
r
T k t
k
2 2
( )
refl|
l| |
r|
in
R E j r r
j
( ) ( ) 1
T E
R E
2 2
| | | |
out r r
L R r r
in l l
j k v
T t t
j k v
Incoming and outgoing electron flows
ratio of outgoing to incoming flow
( )
Components of state
Transmission coefficient through the device for state ( )
( )
Transmission coefficient through the device for state
(
||)
R L L R
T
T
T E
|| 2||22 *
E k
m Kinetic energy Reflection coefficient=the ratio of reflected and incoming flows
Conductance from Transmission
Electric current through the device||
, , k n m
|| ||
( , , ) T( , )
EE k n m EE n m
||
|| || ||
, 0
2 ( ) ( ( , , ) )
L L
n m k c
I e v T E f E k n m
L
||
|| || ||
, 0
2 ( ) ( ( , , ) )
R R
n m k c
I e v T E f E k n m
L
||
|| || || ||
, 0
2 ( )[ ( ( , , ) ) ( ( , , ) )]
L R R L
n m k c
I I I e v T E f E k n m f E k n m
L
v
||The number of electrons in state per unit length 2 ( ( , ,f E k n m|| )L) /Lc The total contribution to the electric current from the electrons entering from the left
Similarly, for the electrons entering from the right
The total electric current through the device
It is convenient to introduce the function || ||
,
( ) : 2 ( ( , , ) )
n m
F E
f E k n m ||
|| ||
||
{ } { } { }
2 2
c c
k
dk dE
L L
v
||
|| || ||
( )[ ( ) ( )]
2
R LI e dE T E F E F E
The total electric current independent on
L EF V
R EF
n-GaAs substrate GaAs QW
GaAs GaAs AlGaAs
Barriers
EC Z
Example of Quantum Transport: Coherent Tunneling Resonant Tunneling Diode (RTD)
Esaki, Chang, Tsu (IBM, 1974)
A
