computational solid state physics 計算物性学特論 第１０回 10. transport properties ii:...
TRANSCRIPT
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
I L
Current in one-dimension
L
L
L
U
L
U
LL
U
LL
dkTkfh
e
hv
dkTkvkfeI
dv
dd
dkdk
dkkTkvkfeI
)(]),([2
)()(]),([2
1
2)()(]),([2
T(k): transmission coefficient
LU
LL dTfh
eI )(],[
2
RU
RR dTfh
eI )(],[
2
LU
RLRL dTffh
eIII )()],(),([
2
Total current in one-dimension
Low bias limit
),(),(),(),(
feV
feVff RL
dTf
h
eG
VIG
dTf
h
VeI
L
L
U
U
)(2
/
)(2
2
2
)(2
)(
2
Th
eG
f
LU
RLRL dTffh
eIII )()],(),([
2
: conductance
at low temperatures
Landauer’s formula
)(
)(2
)(2
2
2
T
Th
eG
VTh
eI
： transmission coefficient
kR
μS
0 8.25
7.38
2
2
0
e
hh
eG
：Conductance
： Quantum conductance
： Quantum resistance
I: current, V: bias
Two- and four- terminal measurements
Tow- and four- terminal measurement
T
T
e
hR
Te
hR
IVR mnpqpqmn
1
2
1
2
/
243,21
212,21
,
2-terminal measurement
4-terminal measurement
Conductance of a quantum point contact
m
zkzkn z
nz 2
)()(),(
22 1)( T
Only one channel (n=1) is open.
for n=1
Conductance of a quantum point contact
Quantization of transverse motion
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
dEeVEfEfETh
eI ds )]()()[(
2
22
2
]1)[exp(
)exp(1)(
)()(
kTEEkTEE
kTEf
dE
d
EEET
F
F
N
:Lorentzian broadening
of resonant tunneling through quantized energy EN of a dot:Thermal broadening
Tunneling current via quantum dot
A bound state and a resonant state
Transmission coefficient for resonant tunneling
)(2
)2/
(1)(
2
RL
pk
pk
TTa
v
EE
TET
2)(
4
RL
RLpk TT
TTT
1)( pkET
If TL=TR
Transmission coefficient of a resonant-tunneling structure
Characteristics of resonant tunneling diode
Resonant tunneling current
)]),,(()2(
2)[()(2
22),(
)()exp(
2
2
0
2222
,
Lzzzzz
L
zLz
kkk
kkfkd
kTkvdk
eI
m
k
m
kUkk
zurikzz
LU
LDL
zLL
BB
D
dEETEnh
eI
m
kU
TkTmk
n
)()(
2
))/exp(1ln()(
2
22
22
Dn2
:wave function
:energy
L
L
U
RDLD
U
LDL
zL
dEETEnEnh
eJ
dEETEnh
eJ
m
kUE
)()]()([
)()(
2
22
2
22
L
LU
L dEETEm
h
eJ
)()(2
Large bias and low temperature limit
Total resonant tunneling current
Transmission coefficient for resonant tunneling
)(2
)2/
(1)(
2
RL
pk
pk
TTa
v
EE
TET
2)(
4
RL
RLpk TT
TTT
1)( pkET
If TL=TR
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.
Profile through a three-dimensional resonant tunneling diode
L
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.