Charge and Heat Transport Properties of Electrons
Figure 1 illustrates electron transport under a small electric field, temperature gradient,
and concentration gradient (and thus chemical potential gradient) along the Z direction.
Consider an infinitesimal point (red dot in Fig. 1) at height Z. At this point, the
distribution function of electrons is f, and the number of electrons with an energy
between E and E+dE is fD(E)dE. Since the electric field, temperature gradient and
concentration gradient are small, these electrons will have almost the same probability to
move toward any direction. Also because the solid angle of a sphere is 4 [Steradian or
sr], the probability for an electron to move in the (, ) direction within a solid angle
d sin dd ) will be d/4. A charge q ( = -e for electrons and +e for holes) moving
in the (, ) direction within a solid angle d causes a charge flux of qvcos and energy
flux Evcosin the Z direction, where is defined as the angel between the velocity
vector and the positive Z direction with a range between 0 to . Hence, the charge flux
and energy flux in the Z direction carried by all electrons moving toward the entire
sphere surrounding the point are respectively
JZ  
4
JE
Z
2 1


d 
d  sin  cos d  fD( E ) qvdE
 ( fD( E ))( qv cos  )dE  
4 E  0
  0 4
 0
E 0
 
4
(1a)
2 1


d 
d  sin  cos d  fD( E ) EvdE (1b)
 ( fD( E ))( Ev cos  )dE  
4 E  0
  0 4
 0
E 0
With the relaxation-time approximation, the Boltzmann Transport Equation for electrons
take the following form:
 f
f f
f 
 v  f  qE    0
t
p

(2)
where q=-e for electrons and +e for holes. For the steady state case with small
temperature/concentration gradient and electric field in the Z direction only, the variation
of the distribution function in time is much smaller than that in space, or
so that we can assume

f
 v  f ,
t
f
~0. The temperature gradient and electric field is small so that
t
the deviation from equilibrium distribution f 0 is small, i.e. f 0  f << f 0 ,
1
f dE  f
f f
f  f 0 , and   0  0   v 0 . With these assumptions, eqn. 2 becomes
p p E dp
E
 f
f f

v  [f 0  qE 0 ]  0
E

(3)
The equilibrium distribution of electrons is the Fermi-Dirac distribution

f 0 (k ) 
1
1
E


;
exp( )  1
k BT
E (k )  
exp(
) 1
k BT
(4)
where  is the chemical potential that depends strongly on carrier concentration and
weakly on temperature. Both E and  are measured from the band edge (e.g. EC for
conduction band). This reference system essentially sets EC = 0 at different locations
although the absolute value of EC measured from a global reference varies at different
location as shown in Fig. 6.9b in Chen (  =EF in Chen). In this reference system

corresponding to Fig. 6.9b in Chen, the same quantum state k = (kx, ky, kz) has the same


 2 ( k x2  k 2y  k z2 )
energy E (k )  E ( k )  EC 
at different locations. Hence this reference
2m

system yields the gradient E ( k )  0 , simplifying the following derivation. If we use a
global reference level as our zero energy reference point as in Fig. 6.9a in Chen, the same


 2 (k x2  k 2y  k z2 )
 EC because
quantum state k = (kx, ky, kz) has different energy E (k ) 
2m

EC changes with locations. In this case, E (k )  EC  0 , making the following
derivation somewhat inconvenient. However, both reference systems will yield the same
result.
From eq. 4,
f 0 df 0  df 0 1


; or
E d E d k B T
df 0
f
 k BT 0
d
E
(5)
From eq. 5,
f 0 
df 0
f
  k BT 0 
d
E
(6)

Also because E ( k )  0 for the reference system (Fig. 6.9b in Chen) that we are using
2
 

1
E
1
E
(E (k )   ) 

T





T
k BT
k BT
k BT 2
k BT 2
(7)
From eqs. 6-7,
f 0  
f 0
E
( 
T )
E
T
(8)
Combine eqs. 3 and 8, we obtain
 f

f f
E
v  [  
T  qE ] 0  0
T
E

(9)
Note that

E  e
(10)
where  e is the electrostatic potential (also called electrical potential, which is the

potential energy per unit of charge associated with a time-invariant electric field E );
From eqns. 9-10, we obtain

f
f f
E
v  [  
T  q e ] 0  0
T
E

(11)
From eqn. 11, we obtain
f

E
f  f 0  v  [ 
T ] 0
T
E
(12)
where     q e , is the electrochemical potential that combines the chemical potential
and electrostatic potential energy. This definition of the electrochemical potential is the
definition in Chen’s text multiplied by a factor of q. Both definitions are used in the
literature, with the definition here are used more widely. Electrochemical potential is the
driving force for current flow, which can be caused by the gradient in either chemical
potential (e.g. due to the gradient in carrier concentration) or the gradient in electrostatic
potential (i.e. electric field). When you measure voltage V across a solid using a
voltmeter, you actually measured the electrochemical potential difference  per unit
charge between the two ends of the solid, i.e. V   / q . If there is no temperature
gradient or concentration gradient in the solid, the measured voltage equals  e .

In the current case all the gradients and E are in the Z direction, so from eq. 12,
3
f  f 0  v cos  [ 
f
d E   dT
d E   dT f 0

 qE z ] 0  f 0  v cos  [ 

]
dZ
T dZ
E
dZ
T dZ E
(13)
Combine eqns. 1 and 13, we obtain the charge flux and energy flux respectively
2


1
d  sin  cos d  f 0 D ( E ) qvdE
  0 4
 0
E 0
JZ  
2

 f
1
d E   dT
0
 
d  sin  cos 2 d 
D ( E ) qv 2 (

 qE z )dE
dZ
T dZ
  0 4
 0
E  0 E
(14a)
and
2
JE
Z


1
d  sin  cos d  f 0 D ( E ) EvdE
  0 4
 0
E 0
 
2

 f
1
d E   dT
0
 
d  sin  cos 2 d 
D( E ) Ev 2 (

 qE z )dE
dZ
T dZ
  0 4
 0
E  0 E
(14b)
Note that the first term in the right hand of eqn. 14 side is zero and the second term
yields
JZ 
JE
Z

1  f 0
d E   dT
D( E )qv 2 (

 qE z )dE

3 E 0 E
dZ
T dZ
(15a)
1  f 0
d E   dT
D( E ) Ev 2 (

 qE z )dE

3 E 0 E
dZ
T dZ
(15b)
Note that
E
1 2
mv
2
(16)
Use eqn. 16 to eliminate v in eq. 15, we obtain
JZ 
2q  f 0
d E   dT
D( E ) E (

 qE z )dE

3m E  0 E
dZ
T dZ
2q  f 0
d E   dT

D( E ) E (

)dE

3m E  0 E
dZ
T dZ
JE
Z

2  f 0
d E   dT
D( E ) E 2 (

 qE z )dE

3m E 0 E
dZ
T dZ
(17a)
(17b)
The energy flux from Eq. 17b can be broken up into two terms as following
4
JE

Z

2  f 0
d E   dT
D ( E ) E 2 (

 qE z )dE

3m E  0 E
dZ
T dZ
2  f 0
d E   dT
D ( E ) E ( E   ) (

 qE z )dE

3m E  0 E
dZ
T dZ

2  f 0
d E   dT
D ( E ) E (

 qE z )dE

3m E  0 E
dZ
T dZ

2  f 0
d E   dT
J
D ( E ) E ( E   ) (

 qE z )dE  Z

3m E  0 E
dZ
T dZ
q

2  f 0
d E   dT
J
D ( E ) E ( E   ) (

)dE  Z

3m E  0 E
dZ
T dZ
q
(18)
where JZ is the current density or charge flux given by eq. 17a. At temperature T = 0 K,
the first term in the right hand side of eq. 18 is zero, so that the energy flux at T = 0 K is
J E (T  0 K ) 
J Z
Z
q
(19)
Because electrons don’t carry any thermal energy at T = 0 K, the thermal energy flux or
heat flux carried by the electrons at T ≠ 0 is
J q (T )  J E (T )  J E (T  0)
Z

Z
Z
2  f 0
d E   dT
D ( E ) E ( E   ) (

)dE

3m E 0 E
dZ
T dZ
(20)
Equations 17a and 20 can be rearranged as
J Z  L11 ( 
1 d
dT
)  L12 ( 
)
q dZ
dZ
(21a)
 L21 ( 
1 d
dT
)  L22 ( 
)
q dZ
dZ
(21b)
L11  
2q 2  f 0
D( E ) EdE

3m E 0 E
(22a)
L12  
2q  f 0
D( E ) E ( E   )dE

3mT E 0 E
(22b)
Jq
Z
where
L21  
2q  f 0
D( E ) E ( E   )dE  TL12

3m E  0 E
(22c)
5
L22  
2  f 0
D( E ) E ( E   ) 2dE

3mT E 0 E
(22d)
Electrical conductivity:
In the case of zero temperature gradient and zero carrier concentration gradient,
dT
d
 0 and
 0 , eqn. 21a becomes
dZ
dZ
J Z  L11 ( 
1 d
1 d
)  L11 ( 
 E z )  L11E z
q dZ
q dZ
(23)
The electrical conductivity is defined as
2q 2  f 0

 L11  
D ( E ) EdE

Ez
3m E 0 E
JZ
(24)
Equation 24 will be simplified further in the following section on Wiedemann-Franz law.
Seebeck Coefficient:
In the case of non-zero temperature gradient along the Z direction, a thermoelectric
voltage can be measured between the two ends of the solid with an open loop
electrometer, i.e. J Z  0 . Hence from eq. 21a we obtain
J Z  L11 ( 
1 d
dT
)  L12 ( 
) =0
q dZ
dZ
(25)
Therefore
 d 


 dZ    qL12
L11
 dT 


 dZ 
(26)
As discussed above, the voltage that the electrometer measure between the two ends of
the solid is V   / q . Similarly, dV  d / q . The Seebeck coefficient is defined as
the ratio between the voltage gradient and the temperature gradient for an open loop
configuration with zero net current flow
6
  f 0

 dV 
 d 
 
D( E ) E ( E   ) dE 




1 dZ  L12
1  E  0 E
dZ 

S  
 


 f

q  dT  L11 qT 
 dT 
0
D ( E ) EdE







 dZ 
 dZ 
E  0 E


 f


0

D ( E ) E 2 dE 

1 
E


  E 0


f 0
qT
D ( E ) EdE 


E  0 E


Combine eq. 27, 24, and 21 a, we can write J Z   ( 
(27)
1 d
dT
)  S ( 
)
q dZ
dZ
The scattering mean free time depends on the energy, and we can assume
   0E r
(28)
where 0 is a constant independent of E.
When E is measured from the band edge for either electrons or holes, the density of states
D( E ) 
(2m) 3 / 2
2 
2 3
E1 / 2
(29)
 f


0 2  r 1 / 2

E
dE 

   E  0 E

 f


0 1 r 1 / 2
E
dE 


E  0 E


(30)
Combine eqns. 27 & 29
 f


0

D( E ) E 2 dE 

1 
E
 1
S
  E 0


f 0
qT
qT
D( E ) EdE 


E  0 E


The integrals in Eq. 30 can be simplified using the product rule



f 0 s
E dE  f 0 E s |0  s  f 0 E s 1dE   s  f 0 E s 1dE
E  0 E
E 0
E 0

(31)
Using eq. 31 to reduce eq. 30 to



r  5 / 2  f 0 E r  3 / 2 dE 
1 

E 0
S



qT 
r  3 / 2  f 0 E r 1 / 2 dE 

E 0


(32)
The two integrals in eq. 32 can be simplified with the reduced energy   E / k BT
7


E 0
0
n 1
n 1
n
n
 f 0 ( E ,  ) E dE  k BT   f 0 ( , ) d  k BT  Fn ( );    / k BT
(33)
where the Fermi-Dirac integral is defined as

Fn ( )   f 0 ( , ) n d
(34)
0
Use Eq. 33 to reduce eq. 32 to


1 
S
  k BT
qT 


5


 r   Fr  3 / 2 ( ) 
2

   kB

3
q

 r   Fr 1 / 2 ( ) 
2


5




 r   Fr  3 / 2 ( ) 
2
  



3


 r   Fr 1 / 2 ( ) 

2



(35)
Seebeck coefficient for metals:
For metals with    / k BT  0 , the Fermi-Dirac integral can be expressed in the form
of a rapidly converging series

Fn ( )   f 0 n d  
0
1  f 0 n 1
 d

n  1 0 

1  f 0  n 1  d m  n 1


 

n  1 0  
d m
m 1




 
(   ) m 
d
m! 

(36)


1
(   )
0  n 1

 ( n  1) n (   )  ( n  1)n n 1
 ... d


n  1 0  
2

 n 1
n 1
 f
 n n 1
2
2
6
 ...
If we use only the first two terms of eq. 36 to express the two Fermi-Dirac integrals in eq.
35, we obtain the following (q = -e for electrons in metals)
8
5
3
5


 




   r   Fr 1 / 2 ( )   r   Fr  3 / 2 ( ) 
 r   Fr  3 / 2 ( ) 
k
2
2
2

  kB  

S   B   




3
3
q
e


 r   Fr 1 / 2 ( ) 
 r   Fr 1 / 2 ( )



2
2








   r 
 
kB 


e 





3   r  3 / 2 
 r 

2  r  3 

2



1  r 1 / 2  2  
5   r  5 / 2 
   r  
 r 

2
6  
2  r  5 


2


r 3/ 2
3

r  
2 r  3

2


3  r 1 / 2  2  


2
6 








 2 k B k BT 3
(
)(  r )
3e
 2
(37)

This value can be either positive or negative depending on r, or how the scattering rate
depends on electron energy. We can ignore the weak temperature dependence of  and
assume  = EF, the Fermi level that is the highest energy occupied by electrons at 0 K in
a metal.
Seebeck coefficient for nondegenerate semiconductors:
In non-degenerate semiconductors,  is located within the bandgap with a distance from
the conduction or valence band edges larger than 3kBT so that
E
     3 . This is
k BT
true for both electrons in the conduction band and holes in valence band. For holes in
valence band, the energy is higher at a position further down below the valence band
edge. When
E
     3 , the Fermi-Dirac integrals become
k BT



1
1
 n d  
 n d
exp(



)

1
exp(



)
0
0
Fn ( )   f 0 ( , ) n d  
0

(38)
 exp( )  exp(  ) d  exp( )( n  1)
n
0
where the gamma function has the property

( n  1)   exp(  ) n d  n( n )
(39)
0
9
We can use eq. 39 to reduce eq. 35 to obtain
5
5




 r   exp( )  r   
k
2
2

S   B   
3
3
q 


 r   exp( )  r   

2
2



k 
5 
1 
5 


  B    r     
   k BT  r   
q 
2 
qT 
2 


(40)
In this equation,  is measured from the conduction band edge EC for electrons and from
the valence band edge Ev for holes. Located within the bandgap,  is negative for
electrons, and is also negative for holes because the hole energy is higher when the
energy level is moved further down. Also q = -e for electrons and +e for holes, so that the
Seebeck coefficient is negative for electrons in the conduction band and positive for holes
in the valence band.
If  is measured from a global reference instead of the band edge as the zero energy
point, we can express eq. 40 for electrons and holes separately
Se  
Sh 
1
( Ec    ( r  5 / 2)k B T )  0,
eT
1
(   Ev  ( r  5 / 2)k BT )  0,
eT
for electrons
(41)
for holes
(42)
The effective Seebeck coefficient in a nondegenerate semiconductors have contribution
from both electrons and holes, i.e.
S
n e S e  p h S h
n e  p h
(43)
where n and p are electron and hole concentrations, respectively, and e and h the
mobility of electrons and holes, respectively. The mobility is defined in the following
section on Wiedemann-Franx law.
Thermal conductivity of electrons
From eq. 21a

L
1 d
dT
1
  12 ( 
)
J
q dZ
L11 dZ
L11 Z
(44)
10
Use eq. 44 to eliminate
Jq

Z
d
from eq. 21 b to obtain
dZ

L21
L L  dT
dT
J Z   L22  12 21 ( 
)  J Z  k e ( 
)
L11
L
dZ
dZ
11 

(45)
The Peltier coefficient  and thermal conductivity ke are defined in the following.
In the case of zero current JZ =0 and non-zero temperature gradient along the Z direction,
Jq
Z

L L  dT
  L22  12 21 ( 
)
L11  dZ

(45)
The thermal conductivity of electrons
Jq

L L 
  L22  12 21   L22  L21S 
dT
L11 

dZ
2
   f


0



D
(
E
)
E
(
E


)

dE


 f

2   E  0 E
 
2
0

D
(
E
)
E
(
E


)

dE



 f
3mT 
E  0 E
0

D( E ) EdE



E  0 E


ke  
Z
(46)
Eq. 46 can be reduced to the following by expanding the (E-) term in the two integrals,
2
   f


2
0



D
(
E
)
E

dE






f 0
2  E  0 E
 
3
ke 
D
(
E
)
E

dE



 f
3mT 
E  0 E
0

D ( E ) EdE
 E  0 E



(47)
For metals, S is usually very small so that from eq. 46
k e  L22  L21S   L22  
2  f 0
D( E ) E ( E   ) 2dE

3mT E 0 E
(48)
Note that
f 0 df 0 
df E  

 0
T d T
d k BT 2
(49)
Compare eq. 49 with eq. 5, we can obtain
f 0
T f 0

E
E   T
(50)
Combine Eqs. 49 & 50,
11
ke 
2  f 0
D( E ) E ( E   )dE

3m E 0 T
(51)
We can use E = mv2/2 to rewrite Eq. 51 as
ke 
1  f 0 ( E )
D( E )v 2 ( E   )dE

3 E 0 T
(52)
When E is far away from  , f 0 ( E ) remains to be either 0 or 1 as the temperature
changes, so that
f 0 ( E )
is non-zero only when E is close to  . Therefore, Eq. 52 can be
T
approximated by taking v = vF and  = F, i.e. the Fermi velocity and the scattering mean
free time of Fermi electrons,
ke 
 f ( E )
1 2
1
1
0
vF  F 
D( E )( E   )dE  v F 2 F C e  C e v F l F
3
3
3
E  0 T
(52)
This is essentially the Kinetic theory expression of the thermal conductivity.
Wiedemann-Franz law:
From eq. 24, the electrical conductivity is

JZ
Ez

2e 2  f 0
D( E ) EdE

3m E 0 E
(53)
f 0 ( E )
is non-zero only when E is close to  , and can be approximated to as a delta
E
function
f 0 ( E )
  ( E   )
E
(54)
Combine Eqs. 54 and 53,
2e 2 
2e 2
2e 2


DE    E   
DF  F
  ( E   ) D( E ) EdE 
Ez
3m E  0
3m
3m
JZ
(55)
We can use   EF  mvF 2 / 2 to reduce eq. 55 to

e2
DF v F 2 F
3
(56)
Note that the electron concentration can be calculated as
12


EF
n   f 0 ( E ) D( E )dE   f 0 ( E , T  0) D ( E )dE   D( E )dE
E 0
EF
1  2m 
 
2  2 
E  0 2   
E 0
2/3
E
1/ 2
E 0
1  2m 
dE  2  2 
3   
2/3
EF
3/ 2
(57)
2
2
 DF E F  DF 
3
3
Combine eqs. 57 and 55, we obtain

e2
n F
m
(58)
If we use the following definition of electron mobility
e 
e
F
m
(59)
we obtain from eq. 58
  nee
(60)
Note that e is electron mobility and is different from  that is chemical potential.
We can use eqs. 58 and 52 to calculate the ratio between the electron thermal
conductivity and electrical conductivity
1 2
v  C
k e 3 F F e mC e v F 2



e2
3ne 2
n F
m
(61)
Here we have assumed that the  F is the same in the thermal conductivity and electrical
conductivity expressions. As discussed in Chen, these two  F terms can be different.
Note that the electron specific heat of metals has been derived previously as
1 2
 nk BT  2 nk 2T
1 2
B
C e   nk BT / TF  2

2
2
mvF
mvF 2
2k B
(62)
Combine eqs. 61 and 62, we obtain
ke


mvF 2  2 nk B 2T
3ne 2
mvF 2

 2 k B 2T
3e 2
(63)
We define the Lorentz number
13
L
 2k B 2
3e
 2.45 x 10-8 ( W/K 2 )
2
(64)
So that we obtain the Wiedemann-Franz law
ke

 LT
(65)
Peltier coefficient:
If the current is not zero, but temperature gradient is zero. From eq. 45
Jq

Z
L21
JZ
L11
(66)
The ratio of the heat flux to the charge flux is defined as the Peltier coefficient, and can
be obtained from eqs. 66, 27, 22c to be

Jq
Z
Jz

L21 L12

T  ST
L11 L11
(70)
This relation between  and S is one of the Kevin relationships.
Thomson coefficient:
See eq. 6.105-6.106 on page 256 of Chen. This is a small effect due to the temperature
dependence of Seebeck coefficient, which is usually ignored.

E
T2, V2, n2
Z
T1, V1, n1
14