Transcript T(k)

Computational Solid
State Physics
計算物性学特論 第１０回
10. Transport properties II:
Ballistic transport
Electron transport properties
le: electronic mean free path
lφ: phase coherence length
λF: Fermi wavelength
Tunneling transport
IL
Current in one-dimension


I L  2e
f [  ( k ),  L ]v ( k )T ( k )
UL
dk 
dk
d 
d
1
v

f [  ( k ),  L ]v ( k )T ( k )
UL

2e
h


2
d

I L  2e
dk
f [  ( k ),  L ]T ( k ) d 
UL
T(k): transmission coefficient
d
hv
Total current in one-dimension
IL 

2e
IR  
h

UL
2e
h
f [  ,  L ]T (  ) d 


f [ ,  R ]T (  ) d 
UR
I  IL  IR 
2e
h

 [ f ( , 
UL
L
)  f (  ,  R )]T (  ) d 
Low bias limit
I  IL  IR 

2e
h
 [ f ( , 
L
)  f (  ,  R )]T (  ) d 
UL
f (  ,  L )  f (  ,  R )  eV

2
I 
2e V
h

UL
f

G  I /V
G 
2e
h

f

G 
2
h

T ( ) d 
: conductance

f
    T ( ) d 
UL
  (   )
2e
 f ( ,  )
2
T ( )
at low temperatures
  eV
 f ( ,  )

Landauer’s formula
I 
2
2e
T (  )V
I: current, V: bias
h
2e
G 
2
T ( )
：Conductance
h
T ( )
G0 
R
0

：transmission coefficient
e
2
 38 . 7μS
：Quantum conductance
 25 . 8 k 
：Quantum resistance
h
h
e
2
Two- and four- terminal
measurements
Tow- and four- terminal
measurement
R mn , pq  V pq / I mn
R 21 ,12 
R 21 , 43 
h 1
2-terminal
measurement
2
2e T
h 1T
2e
2
T
4-terminal
measurement
Conductance of a quantum
point contact
Conductance of a quantum point
contact
Quantization of
transverse motion
Only one channel
(n=1) is open.
2
 (n, k z )   n ( z ) 
 kz (z)
2m
2
T ( )  1
for n=1
Nanowire of Au
Nanowire of Au
Nanowire of Au
Mechanically Controllable Break
Junction
Histogram of conductance of a relay
junction
Conductance through
a quantum dot
Tunneling current via
quantum dot
e
I  
T (E ) 
d
dE
h
T ( E )[ f ( E )  f ( E  eV ds )] dE

  (E  EN )
2
f (E )  
:Lorentzian broadening
of resonant tunneling
through quantized
energy EN of a dot
2
1
kT
2
exp(
[exp(
E  EF
kT
E  EF
kT
)
)  1]
2
:Thermal broadening
A bound state and
a resonant state
Transmission coefficient for
resonant tunneling
T pk
T (E ) 
1 (
 
v
2a
E  E pk
/2
(T L  T R )
If TL=TR
T ( E pk )  1
T pk 
)
2
4T L T R
(T L  T R )
2
Transmission coefficient of a
resonant-tunneling structure
Characteristics of resonant tunneling
diode
Resonant tunneling current
 k , k  exp( ik  r ) u k ( z )
z
2
 (k , k z )  U L 

IL  e
dk z
2
0
n2 D ( ) 
 k
2m
h
2

 kz
2
mk B T

2
d k
( 2 )
UL
2D
2
f (  ( k , k z ),  L )]
ln( 1  exp(  / k B T ))
 kz
2
2m

n
:energy
2m
v z ( k z )T ( k z )[ 2 
  L U L 
e
2
2
2
IL 
:wave function
z
(  L  E )T ( E ) dE
n2D
Total resonant tunneling current
2
E UL 
J 
h
e
h
2m

e
JL 
2
 kz
n
2D
(  L  E )T ( E ) dE
2D
(  L  E )  n 2 D (  R  E )] T ( E ) dE
UL

 [n
UL
Large bias and low temperature limit
J 
L
e m
h 
2
 (
UL
L
 E )T ( E ) dE
Transmission coefficient for
resonant tunneling
T pk
T (E ) 
1 (
 
v
2a
E  E pk
/2
(T L  T R )
If TL=TR
T ( E pk )  1
T pk 
)
2
4T L T R
(T L  T R )
2
Profile through a three-dimensional
resonant tunneling diode
L
Profile through a three-dimensional resonant-tunnelling diode. The bias increases
from (a) to (d), giving rise to the I(V) characteristic shown in (e). The shaded areas on
the left and right are the Fermi seas of electrons.
Problems 10
 Calculate the density of states for free
electrons in one, two and three dimensions.
 Calculate the ballistic current in two
dimensions.
 Calculate the transmission coefficient for a
square barrier potential.
 Calculate the transmission coefficient for a
double square barrier potential.