Dependence of the collision integral on electric field
Conductivity of charged band carriers is considered for the case when collision integral evidently depends on electric field. This dependence for the case of scattering by charged impurities results in decrease of carrier conductivity.
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2015
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| Цитувати: | Dependence of the collision integral on electric field / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 1. — С. 138-143. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1207252017-06-13T03:03:09Z Dependence of the collision integral on electric field Boiko, I.I. Conductivity of charged band carriers is considered for the case when collision integral evidently depends on electric field. This dependence for the case of scattering by charged impurities results in decrease of carrier conductivity. 2015 Article Dependence of the collision integral on electric field / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 1. — С. 138-143. — Бібліогр.: 7 назв. — англ. 1560-8034 DOI: 10.15407/spqeo18.02.138 PACS 72.10-d, 72.20-i http://dspace.nbuv.gov.ua/handle/123456789/120725 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Conductivity of charged band carriers is considered for the case when collision integral evidently depends on electric field. This dependence for the case of scattering by charged impurities results in decrease of carrier conductivity. |
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Boiko, I.I. |
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Boiko, I.I. Dependence of the collision integral on electric field Semiconductor Physics Quantum Electronics & Optoelectronics |
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Boiko, I.I. |
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Boiko, I.I. |
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Dependence of the collision integral on electric field |
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Dependence of the collision integral on electric field |
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Dependence of the collision integral on electric field |
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Dependence of the collision integral on electric field |
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Dependence of the collision integral on electric field |
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dependence of the collision integral on electric field |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/120725 |
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Dependence of the collision integral on electric field / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 1. — С. 138-143. — Бібліогр.: 7 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT boikoii dependenceofthecollisionintegralonelectricfield |
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2025-07-08T18:28:22Z |
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2025-07-08T18:28:22Z |
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1837104421760139264 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 138-143.
doi: 10.15407/spqeo18.02.138
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
138
PACS 72.10-d, 72.20-i
Dependence of the collision integral on electric field
I.I. Boiko
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
45, prospect Nauky, 03028 Kyiv, Ukraine,
E-mail: igor.boiko.35@mail.ru
Phone: (380-44) 236-5422
Abstract. Conductivity of charged band carriers is considered for the case when collision
integral evidently depends on electric field. This dependence for the case of scattering by
charged impurities results in decrease of carrier conductivity.
Keywords: kinetic equation, collision integral, conductivity.
Manuscript received 13.11.14; revised version received 23.03.15; accepted for
publication 27.05.15; published online 08.06.15.
1. Introduction
The collision integral (CI) is the principal term of kinetic
equation. Its structure determines the whole form of the
non-equilibrium function f of many-body system.
Usually, this integral is constructed for the second order
of the perturbation theory, considering the interaction of
charged carriers with stochastic microfields. On the way
of consecutive deduction of CI from the “first
principles” the specific term E appears as an
appendix to the particle energy; here, E is the amplitude
of electric field and – fluctuation of the density
matrix that is proportional to the scattering micro-
potential . Most often, this term is omitted as the
small one, together with terms of conformable higher
orders for fluctuations. But any severe evidence of this
neglect has not been represented and this trouble has not
been overcome completely. The noted here problem is
connected naturally with the specific question arising in
quantum mechanics for the state of electron interacting
with the constant uniform force F (there was not
represented the continuous analytical transition from the
state at 0F to the state at 0F ). As a result, the
influence of the maintained E term on the value of
kinetic coefficients has not been well investigated up to
this time.
2. Fluctuative scattering potential
and fluctuation of density matrix
Introduce the one-particle density matrix of many-
particle system:
)()()( tatat ABAB
, (1)
where
Aa and Aa are operators of generation and
annihilation of a particle at the A state. Heisenberg
equation of motion for the density matrix is
)()(],)([ tHHtHt
t
i AB
tottot
AB
tot
AB
AB
. (2)
For the simplicity, restrict here our consideration
by the case of electrons moving in uniform space at the
presence of constant electric field E
and set of charged
impurities homogeneously distributed over the crystal.
Here, total Hamiltonian is
A
AAA
totH +
AB
BA
E
ABe )(
+
AB
BA
S
ABe )(
–
''
'''' ],[)2/1(
BABA
BAABBABAV . (3)
Here, A is the kinetic energy of a particle in the
A-state,
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 138-143.
doi: 10.15407/spqeo18.02.138
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
139
;)()(
)()()(
3
3)()(
rdrrrEe
rdrrr
BA
B
E
A
E
AB
(4)
)()'(
'
)'()(' ''
2
'
33
'' rr
rr
e
rrrdrdV AB
L
AABABA
; (5)
)(rA
is the wave-function of a free particle, L –
dielectric constant of the considered crystal,
)ˆˆˆˆ)(2/1(]ˆ,ˆ[ abbaba ; )(S
AB – matrix element of the
scattering micropotential. For example, for impurities
with the charge Ie
j
B
jL
I
A
I
AB r
rr
e
rrd )()(3)(
. (6)
Introducing Hamiltonian (3) into the equation (2)
and using standard commutation rules for fermions
operators of generation and annihilation, one obtains this
equation for the fluctuative part of the density matrix
(see [1-3]):
)()( )()()(
)( E
BAB
E
AABAB
AB ttet
t
t
i
)](,)([)](,)([ tttte BABA ; (7)
BAAB . (8)
The matrix element of the screened scattering
potential is
BA
BAAABB
S
ABAB tVett )(/1)()( ''''
)( ;
qdbb
q
e
V
ABqABq
L
AABB
3
''22
2
''
1
2
; (9)
rdrrqirb BAABq
3exp
. (10)
Separate the density matrix into two parts: an
averaged one and a fluctuation (here, angular brackets
show the average with the help of non-equilibrium
statistical operator (see, for instance, [4])
.)()()()(
)()()()(
ttfttf
ttatat
ABABAABAB
ABABAB
(11)
For spacially uniform system,
ABAAB tftf )()( . (12)
The average scattering potential is assumed to be
zero:
)()()()( SSSS . (13)
Using Eq. (12) and the supposition
tttt ABAB /)(/)( ,
perform the average of fundamental Eq. (7). One
obtains:
A
E
AAA
E
A
A fitftfe
t
tf
i St)()(
)( )()(
; (14)
B
BAABA ttef )](),([Im/2St . (14a)
Composing the difference of Eqs. (7) and (14), we
find the equation for the fluctuation of the density
matrix:
,
)()(
ABBABA
E
BAB
E
A
ABBA
AB
Gffe
e
t
i
(15)
where
,],[],[
],[],[
BABA
BABAAB eG
(15a)
rErE
)()( ; AB
E
AB rE
)( . (15b)
For ),0,0( zEE
we have ABz
E
AB zE )( .
Usually, one neglects in Eq. (15) the bilinear over
fluctuations term GAB (that is so-called collisionless
approximation for fluctuations). The field term in
Eq. (15), which has the form
)()( E
BAB
E
Ae , (16)
is also usually omitted (see, for example, [2]). S, one
declares that the electric field E is small and neglect the
conformable field term as negligible. But there appears a
need to show some corresponding criterion which
permits to omit this term. Usually, this criterion is not
represented. Here, we don’t neglect the discussed term.
If one uses the obvious representation of plane
waves, he obtains:
AzAAzAyAxA kkkkkkA ,,,
etc.;
.)(exp
exp)()(
1
2/3
zrkiL
rkiLrr
kzAA
AAkA
(17)
Here, zikLz AzkzA exp)( 2/1 . It follows from
here:
A
E
A
)( , (18)
and we come to the formal record:
.)()(
)(
)(
)(
)()(
tGte
tffe
t
t
t
i
ABAB
E
BB
E
AA
ABAB
ABBA
AB
(19)
We can consider the term GAB (t) as a source of
some part of a total “friction” force.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 138-143.
doi: 10.15407/spqeo18.02.138
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
140
Introduce the designation
BkBkAkAkz
E
BB
E
AA zzE
,,
)()( . (20)
Represent Eq. (19) using the form
,)()(
)],([
)(
tffet
BAReE
t
t
i
ABABAB
zBA
AB
(21)
or
)()(
)(
tffet
t
t
i ABABABBA
AB
.(21a)
Let zzA kk and zzzB qkk . Then the general
structure of the form R(A,B) (see Eq. (20)) and
dimensionship of the considered term prompt the form
close to
2
,,
/ zzzBkBkAkAkz kqEzzE . (22)
The omitted here factor of proportionality requires
a separate special calculation. That is found as
nontrivial. If external electric field E
(let that is
homogeneous and constant in space) is applied to the
system of noninteracting charged particles, the use of
presentation of plane waves is not totally convenient and
can be a source of some disturbance that appears in the
limit case 0E
(see [7]).
For the purpose to construct the appropriate form of
the term R(A,B), we proceed our consideration using
some possible artificial way. At the fixed quantum
number kzA, choose the area AA Lzl and accept the
model wave function rA
as
zrki
L
rr kzAAAkA
exp
1
. (23)
Here,
zkkCzLlzz zAzAAAkzA cos)( ; (24)
)(z is the step function and
1
2
2sin2sin
1
2
)(
AAzA
AzAAzA
AA
zA
lLk
lkLk
lL
kC .(24a)
The function (24) is normalized to unity:
1)(2 dzz
LA
Al
kzA .
Note that, inside the shown area, every function
having the form (23), (24) is the self-function of the
kinetic energy operator m2/22 ; the corresponding
self-value for which is mkk zAAA 2222 . But for
the whole set, all the proposed functions zk are not
mutually orthogonal. We use these functions only to
calculate the value R(A,B).
Using the form (24), calculate the diagonal matrix
element. Choose the lengths LA and lA, as well аs LB and
lB, satisfying the conditions
AzALk2sin AzAlk2sin
12sin2sin BzBBzB lkLk .
In addition, we accept the following relation:
LA + lA = LB + lB. Then after several transformations and
simplifications (LA >> lA), we come to the following
expression (see (22)):
.
11
222
),(
22
zBzA
zBA
zABAB
kk
eE
m
k
m
k
BAReE
(25)
The latter expression does not depend on the
lengths L and l. Naturally, the form (25) is not strictly
correct, because we used the model wave-function. But
we can consider it as acceptable at a sufficiently long
length LA lA.
Here, we have to notice that the form (24) of
accepted wave-function is totally intuitive. Usage of this
form is justified by final results for the calculated
mobility that are totally transparent.
With the use of Eq. (7) and formulae (21)-(25), we
execute a succession of calculations that repeat the
standard order of steps (see, for instance, [3, 5]) and
come to the collision integral in the form:
.exp
)2(
St
2
2
3
3
2
BA
q
S
B
ABBA
ASe
ffrqiqd
e
f
(26)
Here,
q
S
2 is the correlator for Fourier-
components of scattering potentials.
Let kA
and qkB
; then in the
representation of plane waves
AB
rqi
exp
qkk
,
. In
the course of the following calculations, we shall
consider the case when scattering takes place with small
transfer of momentum ( kq ). Then it follows
from Eq. (25):
.
222
)]()([
2
222
z
zz
zBABA
k
qeE
m
qk
m
k
BRAReE
(27)
3. Calculation of mobility
For k
-representation, apply to both sides of kinetic
equation (14) the operator
kdk
33)2(2 . Let zu is z-
component of the macroscopic drift velocity u
.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 138-143.
doi: 10.15407/spqeo18.02.138
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
141
Introducing mobility μ by the determination zz Eu ,
we obtain the equation of the forces balance:
kdfk
n
eu
Ee
kz
z
z
3
3
St
)2(
2
. (28)
Using the forms (26), (28) and the designation
12
)(
2 r
S
q
S q , one finds after uncomplicated
transformations:
1233
6
)(
)2(
21 r
kqkkz
z
S
qfqdqkd
un
e
. (29)
Here, n is the density of band particles, forms for
the functions )(S and numerical values of the
parameter r are determined by the choice of the
mechanism of momentum relaxation.
Further, we restrict our consideration by the case of
scattering of band carriers by uniformly distributed
charged impurities with the density nCI. For this case
(see refs. [5, 6])
22332;2/3 LCICIS ner . (30)
Moreover, we will use the simplicative procedure
for construction of the screened scattering potential. It
introduces in (29) under the sign of integral the step-
function )( 0qq , where 1/q0 is the length of
screening.
Implement the new designations:
222/; zzzz kmeEkk
. (31)
Then, the expression (27) takes the form
mqmq
qkk
2222
. (32)
At further calculation, one uses such simplification:
.432
4
22
22
zzzzz
zzzzz
eEkkmeEk
kmeEk
(31a)
Here, the average energy is
2/1
2/330
3 2
3
4
1
F
FTk
kdf
n
B
kk
, (33)
TkBF is the dimensionless Fermi-energy,
0
exp11
1
w
dww
r
F
r
r .
Carrying out in Eq. (29) the integration over
components of vector q
, we obtain:
kdf
kq
q
un
mne
k
zM
zL
CI
3
3
0
223
3
ln
1
, (34)
Where mqM 22 . Considering the case
n = nCI, we come to the following record:
2/3
22
1
13
2
/
KEEK
EEK
KfKdEE
zz
zz
z
. (35)
Further, the following designations are accepted:
TmkkK B2
;
323
0
33
2
ln2
L
MB qqTkme
E
;
2/1
2/3
01
F
F
EE ;
e
Tkm
E
B
3
0
22
. (36)
Introduce into the formula (35) the model form of
the non-equilibrium function of distribution as the
Fermi-function with the “shifted” argument:
)()( 00 umkfuvff
k
. (37)
Then,
TmkkTkumTmkk
f
BBzBz
k
222exp1
1
22
2
,
22
exp1
1
)(
KJK
Kf
z
.
Here, 002 jjjenuTkumJ zzBz .
Below, one uses the designations:
0EEW z ;
n
n
Tk
qqme
E
E
T CI
BL
M
222
0
4
0
2
2
ln
)(
. (38)
Then, the relation between the dimensionless
electric field W and dimensionless current density J
takes the form:
,
exp1
1
)()(
)()(
)(
22
2/322
2/32/1
2/32/13
KJK
KFWFK
FWFK
KdTW
z
z
z
(39)
or
.
exp1
)(
)(
)(
0
2
2/32
2/3
2/1
gp
pgp
dgdp
T
F
F
J
(40)
Here,
JW
F
F
JW
)(
)(
,
2/3
2/1 . (41)
Assuming the electric field to be small, perform
linearization of the expression (40). One finds:
S
F
F
JWJ
)(
)(
1,
2/3
2/1 , (42)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 138-143.
doi: 10.15407/spqeo18.02.138
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
142
where
.
)(exp1
)(exp
)(2)(
22
2
0
2/32
2
Tgp
Tgp
pg
p
dgdpTTS
(43)
As a result, we come to the Ohm law in the
following form ( 1WW ):
)()(1 2/32/1
1
FSF
S
JW .
Here, one obtains dimensionless specific
conductivity (taking into account the evident dependence
of the collisional integral on electric field) in the form:
S
FFS
j
E
j
E
E
j
W
J
z
z )()(1 2/32/1
0
0
1
0
0
1
1
. (44)
For the case of standard calculation, where the
evident dependence of the collisional integral on electric
field is not considered (so in the formula (35), one uses
the change zzz KEEK 1 ), the other record appears:
.
exp1
1
)(
2
0
2/32
2
gJp
pg
p
dgdpTW
(45)
After linearization of the expression (45), we obtain
other form of Ohm law:
,
exp1
exp
)(2
22
2
0
2/32
2
2
gp
gp
pg
p
dgdpJTW
(46)
and come to the following expression for dimensionless
specific conductivity:
Sj
E
j
E
E
j
W
J
z
z 1
0
0
2
0
0
2
2 . (47)
One obtains from the formulae (44) and (47) the
ratio of conductivities calculated by both mentioned
ways:
)()( 21 TT
= )()()(1)()( 2/32/121 FFTSTT . (48)
4. Conclusion
Figure shows the results of calculations of the ratio of
two specific conductivities. One of them (the value 1 )
relates to the consideration that takes into account the
evident dependence of the collision integral on electric
field. The other one (the value 2 ) relates to traditional
approach that neglects the mentioned field dependence.
At construction of the shown figure, the following
numerical values were used: g10 28m , 1502 L ,
5.2ln 0 qqM . The curve 1 relates to the case
316 cm10 CIn , the curve 2 to the case
317 cm10 CIn . It is seen for scattering with a small
transfer of momentum that the introduced evident
dependence of the collision integral results in decrease
of the calculated conductivity, but doesn’t disturb the
Ohm law. The distinction of conductivities 1 and 2 is
expressed clearer, if the temperature of carriers is lower
and the density of scattering impurities is higher
(practically, the less is the free path length). At a high
temperature and low current density, the influence of the
evident electric field, entered into collision integral, is
not essential.
One can see the vast area of temperatures in
presented figure, where neglecting the electrical term
)()()( te AB
E
BB
E
AA in Eq. (21) and renouncement of
the following complication of the collision integral is
totally satisfactory. It follows from Eqs. (48) and (33)
that possibility to neglect the evident dependence of the
collision integral on the electric field E is not bound with
the amplitude of the field (as that is usually supposed,
see [1, 2]), but with mobility μ of carriers.
Introduce the designations:
en11 ; en22 ; Tmke B20 .
Then, it follows from Eqs. (44) and (47):
2
3
1
)2(
0
)2()1(
TkB ; (49)
)2()1( at
m
eTkB
4
3
2
3
0
. (50)
Represent the length of free motion l(Sc) and
deBroighle length :
m
e
vl Sc 2 ;
m
l dB
2
;
m
e
l
l
Sc
dB
2
. (51)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 138-143.
doi: 10.15407/spqeo18.02.138
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
143
Then, the condition (50) can be presented in the
form:
dBSc
ll . (52)
Therefore, the inequality (50) can be considered as
the criterion of neglect by the evident dependence of the
collision integral on the electric field. One can see that
the usual concept of mobility is related to the small ratio
ScdB
ll . Really, the concept of mobility, as it follows
from (49) and (51), has a sense only at
1)()()( 2/32/1 FFTS . (53)
From the physical viewpoint, the obtained results
are evident. So, we can consider the model (24), as
intuitive (like to every model) and as acceptable one.
References
1. Yu.L. Klimontovich, Statistical Physics. Nauka,
Moscow, 1978 (in Russian).
2. E.M. Lifshits and L.P. Pitaevskiy, Physical
Kinetics. Nauka, Moscow, 1979 (in Russian).
3. I.I. Boiko, Kinetics of Electron Gas Interacting
with Fluctuating Potential. Naukova dumka, Kyiv,
1993 (in Russian).
4. D.N. Zubarev, Non-equilibrium Statistical Thermo-
dynamics. Nauka, Moscow, 1971 (in Russian).
5. I.I. Boiko, Transport of Carriers in Semiconductors,
V. Lashkaryov Institute of Semiconductor Physics,
NAS of Ukraine, Kyiv, 2009 (in Russian).
6. A.I. Anselm, Introduction to the Theory of Semi-
conductors. Nauka, Moscow, 1978 (in Russian).
7. L.D. Landau and E.M. Lifshits, Quantum
Mechanics. Nauka, Moscow, 1963 (in Russian).
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