Abstract: The model of inelastic electron scattering on polar optical phonons is proposed in which the scattering probability does not depend on macroscopic parameter – crystal permittivity. The reviewed model gives the good agreement between the theory and experiment in temperature range 77 - 300
К.
Key-Words: Electronic Transport , Inelastic Scattering , Mercury Telluride
The electron-polar optical phonon scattering was considered in relaxation time approximation in [1]. In
[3, 4] this scattering mechanism was considered in view of inelastic character of scattering within the framework of a precise solution of the stationary
Boltzmann equation. There was exhibited that the crystal oscillations; b q ,

and b
 q ,

- operators of phonons annihilation and birth respectively of

-th branch with wave vector q ;
  i ( n
1
, n
2 n
2

, n
3 n
3
) a
0
2

1 , 2 ...
 j (
n
1 a
0
 n
3
) a
0
2
 k ( n
2
 n
1
) a
0
2
, i , j , k
usage of standard model of electron - polar optical phonon scattering reduces to a disagreement between
the theory and experiment in temperature range Т>
100 К. According to our opinion this model has following shortages: 1) the use of macroscopic parameter - permittivity- is not reasonable in
Under optical oscillations in a unit cell a polarization vector arises:
P
 e ( Q
1
V
0

Q
2
)
,
(2) microscopic processes; 2) the interaction potential of an electron with optical oscillations of a crystal lattice is long-range that contradicts the special relativity. The purpose of the present work is the build-up of such a model of scattering which at first would well match with experiment and secondly would not have the where V
0
 a
4
0
- the volume of the unit cell; e - elementary charge.
Using (1) and taking into account only the long wave ( q

0 ) oscillations one can obtain: above mentioned shortages.

4 e
P

 b q a
,

3
0 e
 q ,
 i qρ



2 M



 b q

,
 e
( q
 i qρ
)




1
2
( ξ
1
( q ,

)
 ξ
2
( q ,

) )

.
(3)
Let's consider a displacement of j-th ( j =1, 2) atom in a unit cell of a crystal with zinc blende structure under the influence of optical oscillations [5]:
It must be noticed that the polarization vector is a function of discrete variables P

P ( n
1
, n
2
, n
3
) . To calculate the bound charge
 
div P let's make following replacement of a partial derivative of a
Q

  * j
(
1
G q ,

 q ,

) b q




,

2 M e

 i q
 q ) 


1
2

 j
( q ,

) b q ,
 e i q
 


(

, (1) where G – a number of unit cells in a crystal volume; M = M
Hg
+ M
Te
- the mass of the unit cell; q and


( q ) - wave vector and angular frequency of

-th branch of a crystal optical oscillations respectively (
 
4 , 5 , 6 ) ;
 j
- polarization vector of polarization vector on coordinates:


P x x

P x
( n
1

1 , n
2
, n
3

) x

P x
( n
1

P x
( n
1
, n
2

1 , n
3
)

P x
( n
1
, n
2
, n
3
, n
2
)

P x
( n
1
, n
2
, n
3

 x
1 )

 x
P x
( n
1
, n
2
, n
3
)

, n
3
)

, (4) where structure.
 x
 a
0
2
for a unit cell of the zinc blende

The similar relations can be written for partial derivatives

P y
 y
and


P z z with
 y
  z
 a
2
0
.
Then Poisson equation for a scalar potential bound up with crystal oscillations becomes:
 2   


0

8 i e a
0
3 
0
 q




2 G


( q ) 

 1 2




 

M
 b q
M
Hg e
Hg i qρ

M
M
Te
Te
 b q


 
 e
1 2
(
 i qρ q x
 q y
 q z
)
2
 q

, (5) where the relation q i a
0
2

1 ( i
 x , y , z ) is used and only optical longitudinal vibrations are taken into account,

0
- dielectric constant.
To solve the equation (5) let’s substitute the unit cell by an orb of effective radius R
  a
0
the magnitude which lays within the limits from half of smaller diagonal up to half of greater diagonal of a unit cell ( 0.5
< γ < 3 2 ). Magnitude γ =0.628 is picked so to adjust the theory with experiment.
Spherically symmetric solution of a Poisson equation looks like:
 

2

0
( R
2  r
3
) , ( 0
 r

R ) . (6)
Then the interaction energy of an electron with polar optical oscillations of a lattice is determined from expression:
U
  e
 
4 i e a
0
3 
2
0
( R
2  r
3
)
 q




2 G


( q ) 

 1 2




 

 b
M q
M e
Hg
Hg
 i qρ
M
M
Te
Te
 b q


 
 e
1 2
(
 i qρ q x
 q y
 q z
)
2
 q

.
(7)
Let's mark that the potential (7) is short-range as it takes into account interaction of an electron only with one unit cell. To calculate the transition probability connected with electron – phonon interaction let’s write the wave function of the system “electron + phonons” in a form:
 
1 exp( i kr )

( x
1
, x
1
...
x n
) , (8)
V where V - crystal volume;

( x
1
, x
1
...
x n
) - wave function of the system of independent harmonic oscillators.
Then the transition matrix element from interaction energy looks like:

N
 q
, k

| U | N q
, k
  a
0
4 i e
3 
0
2
V
 exp( i s r ) ( R
2  r
3
) d r


 q




2 G


  
(
( q )


 1 x
1
, x
1
...
x n
2
)

 

 b q
 dx
1 dx
2
...
dx n
, s
M
M

Hg e k
Hg i qρ


M
M
Te
 b
Te
 q

 
 e
1 2
 i qρ
( q x


(
 x
1 q
, y q x
1
 q z
...
x n k

.
(9)
)
2
)


The integration over the electron coordinates is carried out in the limits of unit cell and gives:
I ( s )
  exp( i s r ) ( R
2  r
) d r

3


( 8 sin Rs

8 Rs cos Rs

8 3 R
3 s
3 cos Rs )
(10) s
5
The calculation shows that the electron wave vector ( and s together with it) varies within the limits from 0 up to 10 9 м -1 at energy changing from 0 up to 10 к
В
Т ( к
В
– Boltzmann constant) at the temperature range 4.2 – 300 К .
I ( s ) / I ( 0 )
1,0
0,5
0,0
10
7
10
8 s
10
9
10
10
10
11
Fig. 1. The dependence of the function I ( s ) from s

| k
 k

| .
I ( 0 )
As introduced at fig. 1 the dependence of the value
I ( s ) I ( 0 ) from s it is seen that at the indicated limits of varying of wave vector the following relation is well fulfilled:
I ( s )

I ( 0 )

16 15

R
5 
16 15
 a
0
5  5
The integration over the harmonic oscillators coordinates gives the factors N q
and N q

1
( N q
– the number of phonons with a frequency

( q )
 
0 at q

0 ) for phonon annihilation and birth operators respectively. To

calculate the sum over the vector q let’s do the following simplifications –1) taking into consideration quasi continuous character of varying of wave vector let’s pass from summation to integration over q ; 2) let's pass from an integration on a cube with a crossbar
2
 a
0
to an integration on an orb with effective radius
 a
0
:

 q
...

(
V
2


2
 
 
0 0 0 a
0
...
q
2
)
3


 a
0


 a
0

   a


0 a
0 a
0
...
dq x a
0 dq y dq z
 sin
 dq d
 d

.
(11)
Then we obtain for the sum the following expression:
 q
...

F (

)

 
8 cos

Q

8

Q sin

Q
 4

4
 2
Q
2 cos

Q

8




N q
N q
 absorption ;

1
 radiation.
Q
  a
0
.
(12)
As introduced at fig. 2 the dependence of the value
F (

) / F ( 0 ) from

it is seen that the function
F (

) can be approximated by the expression:
F ( 0 )
 
Q
4 


N q
N q

1
(12a)
F (

) / F ( 0 )
1,0
0,5
0,0
10
-11
1x10
-10

1x10
-9
1x10
-8
Fig. 2. The dependence of the function
F (

) / F ( 0 ) from

.
After the calculations we can obtain the expression for electron transition probability connected with phonon absorption and radiation:
W ( k , k

)

64
225


0
2
7 a

0
10
4 e
G
4

0
M
M
Hg
Hg

M
M
Te
Te



N q

(
     

0
)

( N q
 where

- electron energy.
1 )

(
     

0
)

, (13)
10
7
10
6
10
5
2
1
10
4
T , K
100
Fig. 3. The temperature dependence of electron mobility in HgTe : solid line –mixed scattering mechanism; 1 –intraband scattering; 2 –interband scattering. Experiment – [2].
On the base of this for intra- and interband electron transitions the values K n m
  a b
figuring in a method of a precise solution of the stationary
Boltzmann equation can be obtained [3,4] :
K n m
 
1 1
 
2
2 V
 
3
64
675


0
6
2 e a
4
0


2
0
 k
10
B
T
M
M
Hg
Hg

M
M
Te
Te




N
 k (
 q f
0

 
(



0
)

1
)


( f
0
N
( q



1


)

0
(

)
 k

2
(



0

)

 f
0
(
0

)
)




1
 f
0
(
  

0
)
 k
2
(
  

0
)
 k (
 
 


0
) k
4


(

)



 k (
 

)
 n
 m d

;
K n m
 
1 2
 
2
2 V
 
3
32

675

0
6 e
4 
2 a
0
2  10

0 k
B
T
M
M
Hg
Hg

M
M
Te
Te 


(



2 m h h
 
 2
  g



3
2 where



( N q

1 ) f
0
(

)

1
 f
0
(
 


0
)


 

0
)
1
2 k
4
(

)
 k (
 

)



- Kronecker delta ; n
 m f
0
( d


,
) - Fermi –
Dirac function;

( x ) - step function.

The calculation of the temperature dependence of electron mobility was made for acceptor concentration
N
A
= 3 x 10 15 сm -3 thus it is possible to neglect the contribution of heavy holes ( about 1%). At calculations the same scattering mechanisms as in
[3,4] were taken into consideration. As it seen from
Fig. 3 the theoretical curve well coincides with experimental data at the temperature range T >100 K.
It testifies that the model, offered by us, adequately describes the electron-polar optical phonon scattering process as against model introduced in [1]. From figure it is also seen that the basic scattering mechanism in the interval T > 100 K is intraband scattering on polar optical phonons. The contribution of the interband scattering is negligible and can be neglected.
The model of inelastic electron-polar optical phonon scattering in HgTe is designed which in the framework of a precise solution of the stationary
Boltzmann equation well coincides with experiment at the temperature range T > 100 K.
References:
[1] W. Szymanska, T. Dietl, Electron scattering and transport phenomena in small-gap zinc-blende semiconductors, J. Phys. Chem. Solids, Vol. 39,
1978, pp. 1025-1040.
[2] J. Dubowski, T. Dietl, W. Szymanska, Electron scattering in Cd x
Hg
1-x
Te, J. Phys. Chem. Solids,
Vol. 42, 1981, pp. 351-362.
[3] O.P.Malyk, Nonelastic electron scattering in mercury telluride. Ukr. Phys. Zhurn., Vol.47,
2002, pp. 842-845.
[4] O.P. Malyk, Nonelastic charge carrier scattering in mercury telluride, Accepted to publication in
Journal of Alloys and Compounds.
[5] V.L. Bonch-Bruevich, S.G. Kalashnikov, Physics of Semiconductors, Nauka, Moscow, 1977 (in
Russian).