computational solid state physics 計算物性学特論 第9回
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Computational Solid State Physics 計算物性学特論 第9回. 9. Transport properties I: Diffusive transport. Electron transport properties. l e : mean free path of electrons l φ : phase coherence length λ F : Fermi wavelength. Examples of quantum transport. key quantities. - PowerPoint PPT PresentationTRANSCRIPT
Computational Solid State Physics
計算物性学特論 第9回
9. Transport properties I:
Diffusive transport
Electron transport properties
le: mean free path of electrons
lφ: phase coherence length
λF: Fermi wavelength
Examples of quantum transport
le : mean free path of electrons lφ: phase coherence lengthλF: Fermi wavelength single electron charging
key quantities
Point contact: ballistic
h
eGo
22
quantum conductance
Aharanov Bohm effect: phase coherent
eh /0
quantum magnetic flux
Quantum dot: single electron charging
Shubnikov-de Haas oscillations and quantum Hall effect
Diffusive transport
'
2
scat
g
1
)()()()(V)(2
P
)(1
dt
ddt
d
kkkk'
kkkk
P
kkrrr
kk
Vr
Fk
Equation of motion for electrons
Scattering Rate
k: wave vecot of Bloch electron
How to solve equations of motion for electrons with scattering?
Relaxation time approximation for scattering Direct numerical solution:
Monte Carlo simulation Boltzmann equation for distribution function
of electrons
Relaxation time approximation
m
kk c 2
)(22
Fmdk
kd
dt
dk
dt
dvg 1)(12
2
)( BvEeF
vmF
dt
dvm
g
gg
equation of motion
genvj current density
m: effective mass
n: electron concentration
E: electric field
B: magnetic field
Drude model: B=0
gg v
meEdt
dvm )exp(0 tiEE
Eim
neenvj
Eim
ev
g
g
1
1
1
1
2
Ej
im
ne
1
1)(
2
conductivity
: drift velocity
Drude model: steady state solution in magnetic field
)( BvEeF
vmF
dt
dvm
g
gg
),,( zxyyx EBvEBvEeF
z
y
x
c
c
c
cgz
gy
gx
E
E
E
m
e
v
v
v
2222
100
01
01
1
1
: B is assumed parallel to z.
m
eBc : cyclotron
frequency
drift velocity
Conductivity tensor in magnetic field
2222
2
100
01
01
1
1
c
c
c
cm
ne
m
eBc Eenvj g
zB //
xI //
z
y
x
c
c
c
c E
E
E
m
nej
2222
2
100
01
01
1
1
xx
y
Em
nej
j
20
no transverse magneto-resistance
Bne
jEE xxcy Hall effect
Hall effect
m
eBc
Monte Carlo simulation for electron motion
)()(
)()(V)(2
P
)(1
dt
ddt
d
2
scat
g
kk
rrr
kk
Vr
Fk
kkkk
)()(
)()(V)(2
P
)(1
dt
ddt
d
2
scat
g
kk
rrr
kk
Vr
Fk
kkkk
Drift
Scattering
Drift
Scattering
tdtVt
Vt
xx )(1
0 xx venj : current
Drift velocity as a function of time
Boltzmann equation
r
k
tF
k
tvr g
Motion of electrons in r-k space during infinitesimal time Interval Δt
Equation of motion for distribution function
r
fv
dt
trftdtvrfdt
trfdttrff
kg
kgk
kkdiffk
),(),(
),(),(|
equation of motion for electron distribution function fk(r,t).
k
fFdt
trftrfdt
trfdttrff
k
kFdtk
kkforcek
),(),(
),(),(|
Boltzmann equation
scattkforcekdiffkk ffff |||
scattkfieldkdiffk fff |||
Steady state
scattkkrgkk ffvfF
|
Boltzmann equation
'][)2(
1
'])1()1([)2(
1|
''3
''''3
dkPff
dkPffPfff
kkkk
kkkkkkkkscattk
Electron scattering
detailed balance condition for transition probability
kkkk PP ''
Scattering term
)(1
')]()[()2(
1
'][)2(
1
0
'0''
0
3
''3
kkk
kkkkkk
kkkk
ff
dkPffff
dkPff
assume: elastic scattering, spherical symmetry
')cos1()2(
11''3dkP kkkk
k
),( '' kkkk kPP
Transport scattering time
')cos1()2(
11''3dkP kkkk
k
),( '' kkkk kPP k
k’
Θkk’
Contribution of forward scattering is not efficient.Contribution of backward scattering is efficient.
Linearized Boltzmann equation
)(1 0
00
kkk
kk
kk ffF
fvT
T
fv
')cos1()2(
11
)(
''3
000
dkP
Ff
TT
fvff
kkkkk
kkk
kkk
),( '' kkkk kPP
Fermi sphere is shifted by electric field.
Current density and conductivity
kdeEf
vevj kk
kk3
0
3)(
)2(
2
kdevfj k3
3)2(
2
kdf
ve
Ej
kkxxx
xxxx
30
2
3
2
4
Electron mobility in GaAs
Energy flux and thermal conductivity
TkdT
fvvk
kdTT
fvvkU
kk
kk
kk
kk
3
0
3
30
3
)()2(
2
)()()2(
2
kdT
fvvkK k
kkk
30
3)(
)2(
2
TKU
thermal conductivity
Problems 9
Calculate both the conductivity and the resistivity tensors in the static magnetic fields, by solving the equation of motion in the relaxation time approximation.
Study the temperature dependence of electron mobility in n-type Si.
Calculate the electron mobility in n-type silicon for both impurity scattering and acoustic phonon scattering mechanisms, by using the linearized Boltzmann equation.