Download Hall effect

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
key quantities
le : mean free path of electrons
lφ: phase coherence length
λF: Fermi wavelength
single electron charging
Point contact: ballistic
quantum
conductance
2e 2
Go 
h
Aharanov Bohm effect:
phase coherent
quantum
magnetic flux
0  h / e
Quantum dot:
single electron charging
Shubnikov-de Haas oscillations
and quantum Hall effect
Diffusive transport
Equation of motion for electrons
dk

F
k: wave vecot of Bloch electron
dt
d
1 
r  Vg 
 (k )
dt
 k
Scattering Rate
2
2
Pkk  
 k (r ) Vscat (r )  k  (r )   (k )   (k ) 

1
  Pkk'
 k k'
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
 2k 2
 (k )   c 
2m
m: effective mass
1 dk d 2 (k ) 1

 F
2
dt
 dt dk
m
dvg
equation of motion
m
dv g
 F m
vg
dt

F   e( E  v g  B )
j  env g
E: electric field
B: magnetic field
current density
n: electron concentration
Drude model: B=0
m
dvg
dt
 eE  m
vg

E  E0 exp( it )
 e
1
: drift velocity
vg 
E
m 1  i
ne 2
1
j  envg 
E
j


E
m 1  i
ne 2
1
conductivity  ( ) 
m 1  i
Drude model: steady state
solution in magnetic field
m
dv g
 F m
vg
dt

F   e( E  v g  B )
eB
c 
m
: cyclotron
frequency
: B is assumed parallel to z.
F   e( E x  v y B , E y  v x B , E z )
drift velocity
 v gx 
 1
 

e
1

 v gy   
2 2  c
m 1  c  
v 
 0
 gz 
 c
1
0
 E x 
 
 E y 
1  c2 2  E z 
0
0
Conductivity tensor in
magnetic field
j  env g  E
1


ne 2
1


2 2  c
m 1  c  
 0
eB
c 
m
 c
0


1
0

2 2
0
1  c  
Hall effect
B // z
I // x
 1

ne 
1
j

2 2  c
m 1  c  
 0
2
eB
c 
m
 c
0
 Ex 
 
1
0
 E y 
2
0
1  c  2  Ez 
jx
B
ne
jy  0
E y  cE x 
ne 2
jx 
Ex
m
no transverse magneto-resistance
Hall effect
Monte Carlo simulation for
electron motion
dk
F
dt
d
1 
r  Vg 
 (k )
dt
 k
2
Pkk  
 k (r ) Vscat (r )  k  (r )

  (k )   (k ) 

2
dk

F
dt
Scattering
d
1 
r  Vg 
 (k )
dt
  k Drift
2
Pkk  
 k (r ) Vscat (r )  k (r )

  (k )   (k ) 
Drift
Scattering
2
Drift velocity as a function of time
Vx
1 t
  Vx (t )dt 
t 0
jx  en  vx 
: current
Boltzmann equation
Motion of electrons in r-k space during
infinitesimal time Interval Δt
k
r  v g t
r
F
k  t

Equation of motion for
distribution function
equation of motion for electron distribution
function fk(r,t).
f k (r , t  dt )  f k (r , t )
f k (r , t  dt )  f k (r , t ) 
f k | force 
f k |diff 
dt
dt
f Fdt (r , t )  f k (r , t )
f k (r  v g dt , t )  f k (r , t )
k



dt
dt
f
F f k
 v g k

r
 k

Boltzmann equation




f k  f k |diff  f k | force  f k |scatt
Steady state



f k |diff  f k | field   f k |scatt

F
   k f k  v g   r f k   f k |scatt

Boltzmann equation
Electron scattering

f k |scatt 

1
(2 )
3
1
[ f
(1  f k ) Pkk '  f k (1  f k ' ) Pk 'k ]dk '
(2 )
3
[ f
 f k ]Pkk ' dk '
k'
k'
detailed balance condition for transition probability
Pkk '  Pk 'k
Scattering term
assume: elastic scattering, spherical symmetry
1
[ f
 f k ' ]Pkk ' dk '
(2 )
1
0
0

[(
f

f
)

(
f

f
k
k
k'
k ' )] Pkk ' dk '
3 
(2 )
1
0
 ( fk  fk )
3
k
k
1
k

1
(2 )
3
P
kk '
(1  cos  kk ' )dk '
Pkk '  P(k , kk ' )
Transport scattering time
1
k

1
(2 )
3
P
kk '
Pkk '  P(k , kk ' )
(1  cos  kk ' )dk '
k’
Θkk’
k
Contribution of forward scattering is not efficient.
Contribution of backward scattering is efficient.
Linearized Boltzmann equation
f k0
f k0
1
0
 vk 
T  vk 
F  ( fk  fk )
T

k
0
0

f

f
f k  f k0  vk  ( k T  k F ) k
T

1
1

P (1  cos  kk ' )dk '
3  kk '
 k (2 )
Pkk '  P(k , kk ' )
Fermi sphere is shifted by electric field.
Current density and
conductivity
j
2
(2 )3
3

evf
d
 k k
f k0
3
j
ev
(
v

eE
)

d
k
k
k
k
3 
(2 )

2
j x   xx E x
 xx
0

f
e2
2
  3  vx  k k d 3k
4

Electron mobility in GaAs
Energy flux and thermal
conductivity
f k0
3
U 

(
k
)
v
(
v


T
)

d
k
k
k
k
3 
(2 )
T
2
f k0
3


(
k
)
v
v

d
k  T
k k
k
3 
(2 )
T
2
thermal conductivity
U  KT
f k0
3
K 

(
k
)
v
v

d
k
k k
k
3 
(2 )
T
2
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.