Download Construction of the exact solution of the stationary Boatman

Construction of the exact solution of the stationary Boatman equation for
the semiconductor with isotropic dispersion law
MALYK O.P.
Semiconductor Electronics Department
“Lviv Polytechnic” National University
Bandera Street. 12, 79013, Lviv,
UKRAINE
Abstract: - The way of construction of totality of the exact solutions of a stationary Boltzmann equation for the
homogeneous semiconductor with the isotropic dispersion law is proposed in case of Hall effect and electrical
conductivity. The criterion of selection of the physical solutions from totality of the mathematical solutions of
Boltzmann equation is determined.
Key-Words: - Transport Phenomena, Boltzmann Equation, Exact Solution
1 Introduction
The description of transport phenomena in
semiconductors is usually carried out on the base of
Boltzmann equations with the use of relaxation time
approximation or variational method [1,2]. However,
these methods are approximate and therefore do not
allow to answer the question: as far as the selected
quantum mechanical models of charge carriers
scattering describe correctly the physical reality? The
aim of the present work is the precise solution of the
stationary Boltzmann equation that will allow directly
to determine
the adequacy of charge carriers
scattering model to physical reality.
2 Solution of stationary Boltzmann
equation
Let's consider the stationary Boltzmann equation
for the homogeneous semiconductor in isothermal
conditions when the magnetic field is directed along
Z-axis B={0,0,B} and the electric field has
components E={E1, E2, 0} :
q E  q vB   f ( p )  W ( p, p ) f ( p ) 
p
 1  f ( p ) - W ( p , p  ) f ( p ) 1  f ( p  )dp ,
(1)

where
f ( p ) - distribution function; v 
p
charge carrier velocity; q – charge of the carrier.
Suppose that the charge transport is carried out by
two types of carriers (electrons and holes) with
isotropic dispersion law. Let’s find the solution of
equation (1) in the form:
 f ( p)
(2)
f a ( p )  f0 a ( p ) - 0 a
a( p ) ,


where a  1, 2 - respectively for electrons and
holes; f 0 a ( p ) - equilibrium Fermi-Dirac function;
 a ( p ) - unknown functions.
Substituting (2) in (1), using a principle of detail
equilibrium and relations:
 f0 1( p )
1
f 0 1 ( p ) 1  f 0 1 ( p ) ;

kB T
 f0 2 ( p )
1

f 0 2 ( p ) 1  f 0 2 ( p )
(3)

kB T
we shall receive a set of equations for functions
 a ( p ) ( a  1, 2 ) :
f ( p )
  a ( p ) f 0 a ( p )
qa E v a 0 a
 q a v a  B 


p

 qa E

f 0 a ( p )   a ( p )
 2 f0 a
 q a E v a a ( p )


p
2
2
1
b 1
B
  k T  W
ab (
p , p  ) f 0 a ( p ) 1  f 0 a ( p  ) 
  a ( p  ) -  a ( p )dp  
Wab ( p , p  ) 
kB T 2
 f 0 a ( p ) 1  f 0 a ( p  ) f 0 a ( p  )  f 0 a ( p )  a ( p  ) 
  a ( p ) dp  ,
(4)
where Wab ( p , p ) - the charge carrier transition
probability from initial state “p” in band “a” to final
state “p’ “ in band “b”; the upper sign before the
first integral in a right side of the equation relates to
electrons and the lower sign - to holes.
It should be noticed that equation (4) differs from
the known Bloch equation, obtained by a linearization
of Boltzmann equation, by the availability of two last
items in the left side of equation and by the availability
of a second item in the right side of the equation which
is absent in the case of elastic carriers scattering .
1
2

Let’s find the unknown functions  a ( p ) in
such a form:
a( p ) 
C na p (  )  n ,
(5)

n,

where C n a - unknown coefficients; n = 0,1,2,3..
, a = 1,2 respectively for electrons and holes;  =
1,2,3 ; p (  ) - quasi momentum component.
Let's substitute (5) in the Boltzmann equation
(4), multiply this equation by p (  )  m (m = 0,1,2,3
…,  = 1,2,3) and integrate its both parts over the
quasimomentum. At an integration of the left side of
the equation (4) we transfer to a spherical coordinate
system and then obtain:
 f0 a ( p )
2V 4
qEv
p  (  )  m dp 


2 3 3

f 0 a (  )
k ( ε ) 3 ε m dε  B ma ;
ε
  a ( p )  f0 a ( p )
 q v  B 
p  (  )  m dp 
p

2V 4


(  1  2    2   1 ) C nα a qB 
3
3
2 
n ,
   1 E1    2 E 2  q

a




f 0 a (  )
k ( ε ) 3 ε m  n dε   C nα a M βmαna ;
ε
n ,

f 0 a ( p )   a ( p )
p  (  )  m dp 

p
a

 qa E
 q a


 
n, ,
a
f 0 a ( p )


E C na     n  n-1


k

k k  
m n

dp  0 ;
 p (  ) 
k 

 q a E v a a ( p )
 q a
   E
n, ,
a
 f0 a ( p )
2
2
Wab ( p , p )  Wab ( p, p )  (       ) ,
(8)
where Wab ( p , p ) - some function;  - some
energy equal to null at the elastic scattering
mechanism.
For the calculation of the first integral in the
right side of the equation let’s transfer to a spherical
coordinate system concerning a vector p  having
guided an axis OZ along a vector p . Then the integral
over a state p’ in a band “b” becomes:
 W
ab (
p , p  ) f 0 a ( p ) 1  f 0 a ( p  ) a ( p  ) -
b
  a ( p ) dp  
V
( 2  ) 3

 C   W  ( p, p ) 
na
n,
ab
b

 f 0 a ( p ) 1  f 0 a ( p  ) p (   )   n  p (  )  n 
dp (   )
 p  2 (   ) sin   (       )
d d d  
d 

 C  F (  )  C 
na
1
na
F2 (  ) p (  ) ,
(9)
n,
where F1 (  ) and F2 (  ) - known functions of
an energy.
At a repeated integration of a relation (9) over
the state of the vector p we transfer to a spherical
coordinate system and then the first item at integration
over angles gives null. The integral for a second item
in the view of a manifestative aspect of function
F2 (  ) and relation (7b) becomes:
 W
ab (
p , p  ) f 0 a ( p ) 1  f 0 a ( p  ) a ( p  ) -
-  a ( p )p  (  )  m dp  dp  
4V 2
( 2  )6
 C 
na
n,
4

3
 (  ) f 0 a (  ) 1  f 0 a (    )  n  m p(  ) 4 
    Wab

p  (  )  dp 
m
 f0 a ( p )

k C k
p (  ) 
  na 
ma
2
2
  m  n dp  0 ,
where k( ) - module of a wave vector; V –
crystal volume;    - Kronecker deltas; ma - charge
carrier effective mass and at an integration the
relations are taken into account:

a( p )
  k k  
 C na     n  n-1
(7a)
;
 p
k
k 

n ,

4
 p (  ) p (  ) sin d d  3  | p |
2
a
 p(  )  p(    )
d ,
(10)


 (  ) is determined by relation :
where Wab
 p(    ) 2
 (  )   Wab
 ( p, p ) sin  d  d  .
Wab
(10a)
Passing from integration over a quasimomentum to
integration over a wave vector we discover final
expression obtained at an integration of the first item
in the right side of the Boltzmann equation:
1
Wab ( p , p ) f0 a ( p ) 1  f0 a ( p )

k BT

  a ( p ) -  a ( p ) p (  )  m dp dp  
   . (7b)
At an integration of the right side of the equation
(4) we take into account that the carrier transition
probability Wab ( p , p ) is proportional to a deltafunction:

 C
n,

na
4  2 
3 k BT
W  (  ) f
ab
a
0 a(
4V 2

( 2 )6
 ) 1  f0 a (    )
 n m k(  )4 k(    )2


C na K n m b a .
 f 0 a (  ) 1  f 0 a (    )  f 0 a (    ) - f 0 a (  ) 
 k(  )  k(    )
d 


(10b)
(    )  n m k (    )3

n,
We do similarly at an evaluation of the second
integral in the right side of the Boltzmann equation . In
this case the integral over a state p  in a band "b"
becomes:
Wab ( p , p  ) f 0 a ( p ) 1  f 0 a ( p  ) f 0 a ( p  ) -

b
 f 0 a ( p ) a ( p  )  a ( p ) dp  

V
( 2  ) 3
 C  C   W
a
na
n , , ,
ab (
p , p  ) f 0 a ( p ) 1 -
b
 f 0 a ( p  ) f 0 a ( p  )  f 0 a ( p ) p (   ) p (  )    
  n p  2 (   ) sin    (       )
C
3

 a Cn a
L m n b a .
 k(    )
 k(  )
k(  )4
d 


(14)
n,,
It should be noticed that for given dispersion laws
k(  ) and scattering probabilities Wab ( p , p ) the
values K, L, M, B can be always defined with any
given degree of accuracy.
Thus as a result of integration the Boltzmann
equations are reduced to the system of the nonlinear
algebraic equations concerning values C na for
electrons and holes:
C na K mn b a 
C 3a C na L mn b a 


n , , b
dp (   )
d  d  d    Bma 
d 
n ,  , , b
 C 
na
M mn a
(15)
n,
 ) f 0 a (  )1  f 0 a (    )  For the further analysis let's rewrite (15) in the other
form separating electronic and hole components and
n , ,
also  - components of the equations ):
  f 0 a (    ) - f 0 a (  ) p (  )(    )   n p(    ) 3 
mn
mn
3
mn
3
mn
C n1 1 ( K11
11  K 11 12  C  1 L11 11  C  2 L11 12 ) 
 p(    )


,
(11)
2
nm
m
 C n 1 M 12 1  B11

 2
mn
mn
3
mn
3
mn
 (  ) is determined by a relation:
Where Wab
C n 1 ( K 22 11  K 22 12  C  1 L22 11  C  2 L22 12 ) 

V
( 2  ) 3
 C  C
na
3
 (
 aW ab

 (  )  Wab ( p, p ) cos  sin d d  ,
Wab
(12)
and it is also taken into account that the scattering
probability depends only on an angle  .
Repeatedly integrating a relation (11) over the
states of a vector p we shall receive:
1
Wab ( p , p  ) f 0 a ( p ) 1  f 0 a ( p  ) 
2 2
kB T

  f 0 a ( p  ) - f 0 a ( p )  a ( p  )  a ( p ) p  (  ) m 
 dp dp 
4 V 2 4   
( 2 )6 3k B T
 C  C  W  (  ) 
na
n , ,
3
a
ab
a
 f 0 a (  ) 1  f 0 a (    ) f 0 a (    ) - f 0 a (  ) 
(    )   n  m p(    ) 3
 p(    )
p(  ) 4 

 p(  )
d .
(13)

Passing to integration over a quasimomentum
we receive for the second integral in the right side of
the Boltzmann equation:
1
Wab ( p , p ) f0 a ( p ) 1  f0 a ( p )
2 2
kB T


  f0 a ( p ) - f0 a ( p )  a ( p )  a ( p ) p (  ) m 
 dpdp 
3
4 V 2 4    
( 2 )6 3 k B 2T 2
C C W  (  ) 
na
n , ,
3
a
 C 1 M n m  B m
21
 n 1 21 1
3
m
n
mn
3
mn
C n 1 ( K 33 11  K 33
12  C  1 L33 11 

3
mn
(15a)
 C  2 L33 12 )  0
 3
mn
mn
3
mn
C n 2 ( K 33 22  K 33 21  C 1 L33 21 

3
mn
 C  2 L33 22 )  0
C 2 ( K m n  K m n  C 3 Lm n   C 3 Lm n  ) 
22 21
 1 22 21
 2 22 22
 n 2 22 22
nm
m
 C n1 2 M 21
2  B 22
 1
mn
mn
3
mn
3
mn
C n 2 ( K 11 22  K 11 21  C  1 L12 21  C  2 L12 22 ) 

nm
m
 C n2 2 M 12
1  B12
The system (15a) can be solved in two steps: 1) for the
fixed value m = 0,1,2 .. one can obtain С3n1 та С3n2 ( n
=0,1,2 ..) from 3 rd and 4 th system equations; 2) one
can obtain the values С 1n1 , C 2n1 and C 1n2 , C 2n2 (n
=0,1,2...) respectively from 1 st and 2 nd , 5 th and 6
th
system equations, which form two independent
linear systems equations. It should be noticed that
values С αn b (α=1,2; b=1,2; n=0,1,2..) are the linear
functions from the components of the electric field
intensity vector Е. The system of nonlinear algebraic
equations for values С 3n1 and С 3n2 ( n=0,1,2 ..) may
have such types of solutions: 1) С 3n1 = С 3n2 = 0 ; 2)
С 3n1 = 0, С 3n2  0; 3) С3n1  0 , С3n2 = 0; 4) С3n1 
0, С 3n2  0 . That’s why the necessity of choice of
ab
a

In relation (15a) the rule of summarizing over the indexes
which are repeated twice is used.
physical solutions between the totality of mathematical
solutions arises. To formulate the choice criterion let's
calculate beforehand the components of a current
density vector:
1
 f 0 a 
J   3 3

 q a v a   a dp 
4  a   

 f 0 a 
(16)
 k (  )3  n d
 
3 a , n
Let’s formulate the next criterion of choice:
Jz = Jz n + Jzp = 0 , where Jz n , Jz p are the electron
and hole contributions into the Z – component current
density vector. Only the solution of type 1) satisfies
this criterion. Other solution types under the value of
the magnetic field induction
B = 0 transform
determinants of the linear system equations for values
С 1n1 and С 1n2 ( 1 st and 6 th equations of system
(2a) ) into zero , thus they are physically senseless and
that’s why must be rejected.
For taking off the physical solutions among
solutions of type 1) it is necessary to determine at
fixed value n = 0,1,2 … the components of a
conductivity tensor
and
compare them with
experiment. For evaluation of conductivity tensor

1
2
 q C   
a
na
components let’s use that the values C n1 a (a =1,2)
are the linear functions of E1 and E2 then Jx will also
linearly depend upon them and the values attached to
them will be equal to respective conductivity tensor
components 11 and 12 for electrons and holes.
References:
[1] A.I. Anselm. Introduction to the Theory of
Semiconductors, Nauka,
Moscow, 1978 (in
Russian).
[2] I.M. Dykman , P.M. Tomchuk, Transport
Phenomena and Fluctuations in Semiconductors,
Naukova Dumka, Kiev, 1981 (in Russian).