Influence of field dependent form of collision integral on kinetic coefficients
The kinetic equation is turned out in the form that contains collision integral obviously dependent on the value of external electric and magnetic fields. The correspondent calculation of kinetic coefficients shows that for the case considered here they depend evidently on the ratio of average deBro...
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| Zitieren: | Influence of field dependent form of collision integral on kinetic coefficients / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2016. — Т. 19, № 2. — С. 173-182. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1215572017-06-15T03:05:11Z Influence of field dependent form of collision integral on kinetic coefficients Boiko, I.I. The kinetic equation is turned out in the form that contains collision integral obviously dependent on the value of external electric and magnetic fields. The correspondent calculation of kinetic coefficients shows that for the case considered here they depend evidently on the ratio of average deBroighle wavelength and free-path length. Just here, the real possibility appears to reasonably separate physical kinetics by the classic and non-classic (quantum) ones. 2016 Article Influence of field dependent form of collision integral on kinetic coefficients / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2016. — Т. 19, № 2. — С. 173-182. — Бібліогр.: 12 назв. — англ. 1560-8034 DOI: 10.15407/spqeo19.02.173 PACS 72.10-d, 72.20−i http://dspace.nbuv.gov.ua/handle/123456789/121557 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The kinetic equation is turned out in the form that contains collision integral obviously dependent on the value of external electric and magnetic fields. The correspondent calculation of kinetic coefficients shows that for the case considered here they depend evidently on the ratio of average deBroighle wavelength and free-path length. Just here, the real possibility appears to reasonably separate physical kinetics by the classic and non-classic (quantum) ones. |
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Boiko, I.I. |
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Boiko, I.I. Influence of field dependent form of collision integral on kinetic coefficients Semiconductor Physics Quantum Electronics & Optoelectronics |
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Boiko, I.I. |
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Boiko, I.I. |
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Influence of field dependent form of collision integral on kinetic coefficients |
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Influence of field dependent form of collision integral on kinetic coefficients |
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Influence of field dependent form of collision integral on kinetic coefficients |
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Influence of field dependent form of collision integral on kinetic coefficients |
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Influence of field dependent form of collision integral on kinetic coefficients |
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influence of field dependent form of collision integral on kinetic coefficients |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2016 |
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Influence of field dependent form of collision integral on kinetic coefficients / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2016. — Т. 19, № 2. — С. 173-182. — Бібліогр.: 12 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT boikoii influenceoffielddependentformofcollisionintegralonkineticcoefficients |
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2025-07-08T20:07:08Z |
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1837110642658508800 |
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 173-182.
doi: 10.15407/spqeo19.02.173
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
173
PACS 72.10-d, 72.20−i
Influence of field dependent form of collision integral
on kinetic coefficients
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. The kinetic equation is turned out in the form that contains collision integral
obviously dependent on the value of external electric and magnetic fields. The
correspondent calculation of kinetic coefficients shows that for the case considered here
they depend evidently on the ratio of average deBroighle wavelength and free-path
length. Just here, the real possibility appears to reasonably separate physical kinetics by
the classic and non-classic (quantum) ones.
Keywords: kinetic equation, collision integral, mobility.
Manuscript received 07.12.15; revised version received 14.04.16; accepted for
publication 08.06.16; published online 06.07.16.
1. Introduction
Along the way of construction of kinetic equation from
some “first principles”, the rightful place belongs to the
influence of external macroscopic fields and microscopic
scattering fields on movement of band charged carriers.
The scattering fields give the main, principal contribution
to existence and form of collision integral. Evident
influence of macroscopic fields on the scattering system is
not usually taken into attention, because one supposes that
for the scheme of second order perturbation theory the
external field can be omitted (see [1-3]). Special
consideration shows that the latter approach is not
universal, and in some situations the direct influence of
macroscopic fields on the form of collision integral and of
corresponding non-equilibrium distribution function can
sufficiently change the value of kinetic coefficients.
2. One-particle density matrix of nonequiltbrium
many-particle system
Design by the symbols А, В etc. some set of quantum
numbers (for instant, components of the wave vector)
that characterizes a state of each separate band particle;
farther, for simplicity, we shall say about electrons. One
does not use the direct designations for spin variables
and spin quantum numbers, because processes of spin
overturn are not considered here. The act of averaging
we designate by angle brackets; formally that procedure
is performed using the non-equilibrium statistical
operator of total system of electrons and external system,
representing all scattering fields that interact with the
electron system (see [4-7]).
Define the one-particle density matrix ρAB(t) by
using the following mode:
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 173-182.
doi: 10.15407/spqeo19.02.173
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
174
( ) ( ) ( )tatat ABAB
+=ρ . (1)
Here, t is time, +
Aa and Aa are operators of
generation and annihilation of electron at the state A.
The averaged value of given matrix (we call it farther as
the one-particle density matrix) is
( ) ( ) ( ) ( )tatattf ABABAB
+=ρ= . (2)
The dynamic value ( )∑
=
=
N
n
nrCC
1
r
that belongs to
the additive type, in representation of secondary
quantization has the form ∑ +=
BA
ABBA aaCC
,
ˆ , where
( ) ( ) ( ) rdrrCrC ABBA
rrrr 3* ˆ ΨΨ= ∫ . In this formula, the
value ( )rA
r
Ψ is the wave function of separate band
particle, which belongs to the state A.
Deduce an equation, solution of which is the one-
particle density matrix ABf for the considered non-
equilibrium system of particles. As a start point, we use
the standard motion equation for operator ( )tABρ at
Heisenberg representation:
( )[ ] ( ) ( )tHHtHt
t
i AB
tottot
AB
tot
AB
AB ρ−ρ≡ρ=
∂
ρ∂ ,h . (3)
3. Total Hamiltonian of band electrons
and scattering system
One can represent the Hamiltonian of considered total
system totH as the sum of four parts: the Hamiltonian
eH for electrons non-interacting with microscopic
scattering fields, Hamiltonian eeH related to inter-
electron interaction, individual Hamiltonian SH of
external scattering system and Hamiltonian eSH which
relates to interaction of band electrons with the
scattering system:
eSeeSe
tot HHHHH +++= . (4)
In this paper, we assume only the point charged
impurities as external scattering system )( IS → .
At the presence of constant, uniform electrical E
r
and magnetic H
r
fields, the Hamiltonian of free charged
carriers is as follows
rEerA
c
epHe
rrrrr
−⎟
⎠
⎞
⎜
⎝
⎛ +ε= )(ˆ . (5)
In this formula,
r
ikp rh
r
h
r
∂
∂
−== ˆˆ is the momentum
operator; )( p
r
ε − dispersion law; )ˆ( p
r
ε − operator of
the kinetic energy; ])[2/1( rHA
rrr
×= − vector-potential
of magnetic field. Further, we suppose that the
dispersion law has the simple form:
( ) mkmpp 22 222 h
r
==ε , (6)
where m is the effective mass and kp
r
h
r
= is proper
value of the momentum operator.
The quantum limit of strong magnetic field in this
paper is not considered. Therefore, in the Hamiltonian
(5) we omit the terms of the order A2; the latter is
acceptable under the condition 〉ε〈<<mceH 2h . Let
))(2/1()( 122121 AAAAAA +=+ . Then, it follows from
Eq. (5):
( ) ( )( )+⋅+−ε=++= prA
mc
erEepHHHH HE
e
ˆˆ)()()0( rrrrrr . (7)
Assume the following orientation of fields:
)0,,( yx EEE =
r
; ),0,0( zHH =
r
;
)0,,( xHyHA zz−=
r
. (8)
For this case (see Eq. (6)), the separate terms of
Hamiltonian eH are
( ) ( ) mkmppH 2ˆ2ˆˆ 222)0(
r
h
rr
==ε= ;
( )yx
E yExEeH +−=)( ; ( )xy
zH kykx
mc
HeH ˆˆ
2
)( −=
h . (9)
In representation of secondary quantization, the
Hamiltonian of electrons that do not interact with
microscopic scattering fields is (see Eqs (7) and (9))
( ) ( )
( ) ( ) ( ){ } .)()(0∑
∑∑
ρ++=
=ρ== +
AB
BAAB
H
AB
E
AB
AB
BAABe
AB
BAABee
HHH
HaaHH
(10)
The Hamiltonian of interactions between band
electrons and charged impurities is
( ) ( )∑ ρ=
AB
BAABeIeI tHH . (11)
The Hamiltonian of Coulomb e-e-interaction has
the following form (see [8]):
( )
( ) ( ) ,,21
21
'
''
''
''
''
''''
∑∑
∑
ρ+ρρ−
==
+′′
++
ABA
BAAABA
BABA
ABABBABA
BABA
BBAABABAee
VV
aaaaVH
(12)
( ) ( ) ( ) ( )rr
rr
rrrdrdeV BBAA
L
BABA
rr
rr
rrr
ΨΨ
−
ΨΨ
ε
= ′
∗∗∫ ∫ '
'
1'' '
33
2
'' .
(13)
Here, Lε is the dielectric constant of the
considered crystal. Excluding the term that represents a
simple shift of the origin point for energy, one obtains
(see [9]):
( )
( ) ( ) .,21
21
''
''
''
''''
∑
∑
+′′
++
ρρ−=
==
BABA
ABABBABA
BABA
BBAABABAee
V
aaaaVH
(14)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 173-182.
doi: 10.15407/spqeo19.02.173
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
175
As a result, the total Hamiltonian presented in
Eq. (3) has the form
( ) ( ){ }
( ) ( ) .,21
''
''∑
∑
+′′ ρρ−
−ρ+=
BABA
ABABBABA
AB
BAABeIABe
tot
V
HHH
(15)
The plane waves are the natural basis for spatially
uniform system of electrons:
( ) ( )rkiLr AA
rrr
exp2/3−→Ψ ; (16)
then
( ) ( )
ABqABq
L
ABAB bb
q
qdeV
′′−′′ ∫επ
= rr
r
2
3
2
2
2
, (17)
( ) ( ) ( ) ( )
( ) ( ) .2
exp
,
3
3
qBkAkBA
BAABq
qkkL
rdrrqirb
rrr
r
rrr
rrrrr
+
∗
δ=−−δπ=
=ΨΨ= ∫ (18)
Substituting the expression (15) to (3) and
performing necessary commutations of Fermi-operators,
one obtains the following equation for the density
matrix:
( ) ( ) ( ){ }
( )( ) ( )( ) ( ) ( )( )( ){ }.,,
)()(
∑
∑
Γ
+ΓΓ+ΓΓ
Γ
ΓΓΓΓ
ρ−ρ+
+ρ−ρ=
∂
ρ∂
BABA
ABeBAe
AB
tHtttH
tHtH
t
tih
(19)
In this formula,
( )( ) ( )( ) ( )( )
( )( ) ( ) .∑
′′
′′′′ ρ+=
=+=
BA
BAABABABeI
ABeeABeIAB
tVtH
tHtHtH
(20)
4. Averaged values and fluctuations
Separate the density matrix ( )tABρ and Hamiltonian
( )tH by averaged values and fluctuations. One assumes
that the average scattering potential is zero. Therefore,
( ) ( ) ( ) ( ) ( )ttfttt ABABABABAB δρ+=δρ+ρ=ρ , (21)
( )( ) ( )( ) ( )( ) ( )( )ABABABAB tHtHtHtH δ=δ+= . (22)
Independence (or very weak dependence) of
electron density on spatial coordinates is provided by the
following condition:
( ) ( ) ( )tftftf AABAAABAB δ≡δ= . (23)
The fluctuations ( )tABδρ are considered as small
values. Then,
( )( ) ( )( ) ( )( )
( )( ) ( ) .tVtH
tHtHtH
BA
BA
ABABABeI
ABeeABeIAB
′′
′′
′′ δρ+δ=
=δ+δ=δ
∑ (24)
Accept also the following condition (See [6]):
tftt ABABAB ∂∂=∂〉ρ∂〈=∂ρ∂ . (25)
Introducing (21), (24) and (25) to (19), one obtains
the equation
{ }
( ){
( )}
( ) ( ){ }
( ) ( ){ } .)()(,)()(
)()(,)()(
)()()(
)()()(
)()(
∑
∑
∑
Γ
+ΓΓΓΓ
Γ
+ΓΓΓΓ
ΓΓΓ
Γ
ΓΓΓ
δρ+δ+−
−δρ+δ++
+δρ+−
−δρ+=
=δρ+
∂
∂
ttftHtH
ttftHtH
ttfH
ttfH
ttf
t
i
AABB
BBAA
AABe
BBAe
ABABh
(26)
Averaging the letter expression, we find:
( ) ( ) ( ) ( ){ } ( )∑
Γ
ΓΓΓΓ +−=
=
∂
∂
,St
)(
tfitfHtfH
t
tfi
ABABeBAe
AB
h
h
(27)
where
( )
( )( ) ( ) ( )( ) ( ){ }∑
Γ
+ΓΓ+ΓΓ δρδ−δρδ−=
=
.),),
St
ttHttHi
tf
ABBA
AB
h
.
(28)
It follows from here (See. (23)):
( ) ( ) ( ) ( )[ ] ( )∑ +−=
∂
∂
B
AABBAeBAABe
A tfitfHtfH
t
fi Sthh ,
(29)
( ){
( )( ) }.))(,)(
))(,)(St
+
+
δρδ−
−δρδ−= ∑
ttH
ttHif
ABBA
B
BAABA
h (30)
One calls Eq. (29) as the kinetic equation and
Eq. (30) as the collision integral.
Subtracting Eq. (27) from Eq. (29), we find (terms
of second order of trifle are omitted here):
( ){ } ( ) ( ) ( ) ( )[ ]
( ) ( )[ ] ( )( ) .
)
ABAB
ABeBAeAB
tHtftf
tHtHt
t
i
δ−+
+δρ−δρ=δρ
∂
∂ ∑
Γ
ΓΓΓΓh
(31)
Farther in Eq. (31), we consider the average
distribution function to be smooth in comparison with
fluctuating values. Using the Laplas transformation [10]
( ) ( ) ( )∫
∞
ωξ=ωξ
0
exp dttit ,
( ) ( ) ( )∫
+∞
+∞−
ωω−ωξ
π
=ξ
0
0
exp
2
1
i
i
dtit , (32)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 173-182.
doi: 10.15407/spqeo19.02.173
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
176
one obtains:
( ) ( )
( ) ( ) ( ) ( ){ } ( ) ( )( )∑
Γ
ΓΓΓΓ ωδ−+ωδρ−ωδρ
=ωωδρ+=δρ−
,)
0
ABABABeBAe
ABAB
HffHH
ti hh
(33)
[ ]{
( )( ) .)(,)(
)(expIm
2
1St 2
⎪⎭
⎪
⎬
⎫
ω′δωδρ×
×ω′+ω−ω′ω
π
−=
∑
∫ ∫
+
B
BAAB
A
H
iddf
h
(34)
5. Correlation of scattering potential for charged
impurities
Designate the non-screened electrical potential created
by a charge disposed in the point 0=r
r
by the symbol
( ) ( ) erHr eII
rr
=ϕ . The total potential created by all
centers is
( ) ( )∑
=
−ϕ=ϕ
N
j
jI
I rrr
1
)( rrr
, (35)
( ) ( ) ( ) ( )
( ) ( ) ( ) .exp2
exp0,,
1
1
)(
∑
∑
=
=
−ϕωπδ→
→−ϕω=ωϕ
N
j
jI
N
j
jI
I
rqiq
rqiqCq
rrr
rrrr
(36)
Here, jr
r is the radius-vector of j-th impurity,
( ) LI qeq επ=ϕ 24
r , N – total number of impurities in
crystal.
Calculating correlations over positions of
impurities, we have (See [9] and [11])
( ) ( )
( ) ( ) .4
,,
,
24
)()(
ω
δϕ′+δω′+ωδπ+
=〉′ω′δϕωδϕ〈
qI
II
qq
qq
r
rr
rr
(37)
Here,
( )
( )[ ] .322
;
422322
2
,
2
qneqn LIIIqI
qIqI
επ=ϕπ=δϕ
ωδ⋅δϕ=δϕ
ω
r
r
rr
(38)
Turn out to the equation for fluctuation of the
density matrix in the form (31) and use the following
approximation:
( ) AAAAAeH εδ=ωδ= ΓΓΓ h (39)
The standard approach (See [1-3]) is related with
neglecting the field terms in collision integral; in
particular one uses the form (7), where
)0(HHe → . (40)
Then
( ) ( ) AAAAAAe HH εδ=ωδ== ΓΓΓΓ h0 , (41)
and equation for the fluctuation of density matrix (33)
accepts the simplified form
( ) ( ) ( )
( ) ( )( ) .0
0
=ωδ−+
+ωδρε+ε−ω+=δρ−
ABBA
ABBAAB
Hff
ti hh
(42)
If field terms in the Hamiltonian eH are
maintained (this is non-standard or “field” approach),
using the form (39) gives such equation:
( ) ( )
( ) ( )( ) .0
)(0
=ωδ−+
+ωρε+ε−ω+=δρ−
ABBA
ABBAAB
Hff
ti hh
(43)
Then after formal transition )(2)0/( xixi πδ→+
and designation ABBAAB ω=ε−ε=ε h we obtain
( ) ( )
( ) ( )( ) ( )[ ] ./0
2
ωδ−−=δρ×
×ε−ωδπ=ωδρ
ABABAB
ABAB
Hffit h
hh
(44)
The standard approach can be considered as a
limited case of the non-standard approach. For this
approach, we apply the following approximation:
AA ε→ε . (45)
The distinction between Hamiltonians eH and
)0(H (See (7)) is the principal one, and as result the
essential difference can appear between coefficients
calculated by these two ways. Farther, it is convenient to
use the numerical factor χ :
0=χ for the case of standard variant (а), (46а)
1=χ for the case of non-standard (field)
variant (b). (46b)
Below, when calculating kinetic coefficients, we
will see at which condition both variants give practically
the same and at which the opposite condition when a
substantial difference appears.
As the initial form of density matrix ( )tABρ , we use
( ) ABAB aat +==ρ 0 . (47)
Here and farther, we don’t show the argument
0=t for Fermi-operators Aa and +
Aa .
Construct the correlator for ( ) ==δρ 0tAB
ABAB aaaa ++ −= and ( ) ABABBA aaaat ′
+
′′
+
′′′ −==δρ 0 :
( ) ( )
ABABABAB
BAAB
aaaaaaaa
tt
′
+
′
+
′
+
′
+
′′
−
==δρ=δρ 00
. (48)
Using Bogolubov’s principle of weakening of
correlations and performing Week-coupling for two-
particle correlator ABAB aaaa ′
+
′
+ , one obtains the
following expression:
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 173-182.
doi: 10.15407/spqeo19.02.173
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
177
( ) ( )( )
( ) ( )
( ) ( )[ ].1(1
2
,
2
)0()0(
ABBA
BABAAB
BAAB
ffff −+−×
×δδω−ωδω′+ωδπ
=ω′δρωδρ
′′
+′′
(49)
Completing the calculation of corresponding
correlators, represent collision integral in the form
AeeAeIA fff StStSt += , (50)
where
( )
( ) ( ) ( )
( ) ( ) ( ) ( )[ ] ,1111
,
4St
22
24
3
32
4
ABBABAAB
BAABBAqABq
BAB AB
Aee
ffffffff
bb
qq
qd
L
ef
′′′′
′′′′
′′
−−−−−×
×ω−ωδ×
×
ωε
= ∑ ∫
rr
r
r
h
(51)
( )
( ) ( )
( ) .
2
1St
2
23
23
qIAB
B
ABqABAeI
Hff
bqdf
r
r
r
h
δ−×
×ωδ
π
= ∑∫
(52)
Introducing the designations kA
r
→ and kB ′→
r
in Eq. (52), we obtain (here ( )kffk
r
r ≡ )
( )
( )
( )( ) .
2
1St
23
3
∫ δ−ε−εδ×
×
π
=
−− qIkqkqkk
eI
Hffqd
kf
r
rrrrrr
r
h
r
(53)
In practical calculations, we shall use in future the
following approximation for dielectric function (here
1/ 0q is screening length, ϑ is step-function):
( ) ( ) ( )01,1),(/1 qqqq L −ϑε→ωεωε
rr
. (54)
6. Calculation of the energies Aε
Accept the components of wave vectors k
r
as quantum
numbers:
( )AzAyAxA kkkkA ,,=→
r
. (55)
The set of matrix elements of Hamiltonian eH is
(See (7))
( ) ( ) ( ) ( ) ΓΓΓΓ ++= A
H
A
E
AAe HHHH )()()0( . (56)
Here,
( ) ( )
Γ
Γ ⎟
⎠
⎞
⎜
⎝
⎛=
A
A kmH 22)0( ˆ2
r
h ,
( ) ( ) ( )[ ]yAxAA
E EyExeH ΓΓΓ +−=)( ,
( ) ( )
ΓΓ −=
Axy
z
A
H kykx
mc
He
H ˆˆ
2
)( h . (57)
Note that the Hamiltonian eH containing field-
dependent terms is not arbitrary invariant in space. The
problem disappears when using the standard approach.
Usually, this approach is applied without sufficient basis
(see, for instants, [1] and [2]). The most convenient for
calculations are the following wave functions
( ) ( ) ( )wkiLwkw awawA exp2/1−=Ψ≡Ψ . (58)
Here and farther, zyxw ,,= and
2/2/ LwL <<− . The linear dimension L of the system
exceeds utmost an every characteristic length. These
functions are proper functions for the operator of
momentum k̂
r
h (and for the operator of kinetic energy):
( ) ( )wkkwki AwAwAww ;; Ψ=Ψ∇− ,
( ) ( )wkkwk AwAwAwx ;; 22 Ψ−=Ψ∇ .
When the parabolic law of dispersion takes place,
( ) ( ) ( ) ( ) ( )rkmkrkkrk AAAAA
rr
h
rr
h
rr
;2;;ˆ 22 Ψ=Ψε=Ψε .
Write this way:
ΓΓ δ= AA)1( , ( ) ΓΓ δ= AAwAw kk̂ ,
( ) ( ) ΓΓ δ=ε AAA mk 2ˆ 22h . (59)
Now consider the non-standard (field) variant. In
consequence of (46), the matrix elements of Hamiltonian
eH can be presented by the form
( ) ( ) ( ) ( )[ ]ΓΓΓΓ +χ+= A
H
A
E
AAe HHHH )()()0( . (60)
When one uses the field variant, the Hamiltonian
eH evidently depends on spatial coordinates. But at the
same time, all points of r
r
-space are equivalent. Note
that wave functions are invariant to the shift of argument
w on the length proportional to the deBroigle
wavelength. For a minimal shift
( ) ( )[ ]
( ) ( ) ,
exp2/1
wkw
wkiLw
awA
awawawA
Ψ=Ψ=
=λ+=λ+Ψ −
(61)
where
awaw kπ=λ 2 . (62)
Using (61) and shifting the area of itntegration to
awaw LwL λ+<<λ+− 2/2/ , calculate the matrix-
components of radius-vector:
( ) ( ) ( )
( ) ( ) .;;*
;;*
)/(
)(
)(
)/(
dwwkwkw
dwwkwkww
L
L
bwBwawAw
L
L
BwAwAB
∫
∫
+
−
+
−
λ+Ψλ+Ψ=
=ΨΨ=
(63)
The diagonal element ( AB → ) is
( ) ( ) ( ) ∫∫
+
−
−
+
−
=ΨΨ=
)(
)(
1
)(
)(
;;*
L
L
L
L
AwAwAA wdwLdwwkwkww . (64)
Here awLL λ+±=± 2/)( . As it follows from (64):
( ) awawAA kw π=λ= 2 . (65)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 173-182.
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178
Further, we assume the following relation to be
valid:
∞→π=λ 2awaw kLL (66)
and accept
( ) ΓΓ ≈ AA ww )Re( ,
( ) ( ) ( ) ΓΓΓ δπ=δ→ AawAAAA kww 2 . (67)
As a result of the limit ∞→L , one obtains the
formula (39), where
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−Ω+
π
−
π
−=ε
Ay
Ax
Ax
Ay
Ay
y
Ax
xA
A k
k
k
k
k
eE
k
eE
m
k
h
r
h 22
2
22
, (68)
mceH zπ=Ω . (69)
The transformation (63) is not possible only one.
We accept the shown form because just that gives
expected physical result (See below (124) and (125)).
Using the designations kkA
rr
→ and qkkB
rrr
−→ ,
one obtains the form
( )
( ) ( ) ( )
( ) ( ) .11
22
2
2
2
2
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
+
−
×
×−Ω+
−
π
+
−
π
+
+−=ε−ε→ε−ε=ε −
yyyxxx
xyyx
yyy
yy
xxx
xx
qkkBAAB
qkkqkk
qkqk
qkk
qEe
qkk
qeE
qqk
m
h
rrh
rrr
(70)
Simplify the calculations with the help of the
approximation ww kq << and such changes:
( )
( ) .3231
,3231
2222
2222
h
h
〉ε〈==→
〉ε〈==→
mkkk
mkkk
yy
xx
(71)
Here, )(2/)(3 2/12/3 ηη=〉ε〈 FFTkB is average
energy;
( ) ( )∫
∞
η−++Γ
=η
0
exp11
1)(
w
dww
r
F
r
r ; TkBFε=η . (72)
As a result,
( ) ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−Ω+
π
−
π
−=ε≡ε
y
x
x
y
y
y
x
x
k k
k
k
k
k
eE
k
eE
m
k
k h
r
hr
r
22
2
22
,
m
k
k 2
22
r
h
r =ε , (73)
( )
( ) ( )
.
33
2
2
32
2
2
〉ε〈
−Ω
++
〉ε〈
π
+−=ε−ε −
m
qkqk
qEqE
m
e
qqk
m
xyyx
yyxx
qkk
hh
rrh
rrr
(73a)
We don’t consider here quantazed magnetic field
(that is b << 1); therefore values of the order b2 will be
every case omitted. Under the designations
)(3 EkEe
rr
=〉ε〈π , b=〉ε〈Ωh3 (74)
the expression (73a) can be written as
( ){
( ) .
2
2
)(
)(
2
⎪⎭
⎪
⎬
⎫
−++++
+−+=ε−ε −
qqkqbkkk
qbkkk
m
zzyxy
E
y
xyx
E
xqkk
h
rrr
(75)
The latter expression prompts to introduce the new
vector ( )k
rr
κ :
( ) ( ) ( ) ( )( )kkkk zyx
rrrrr
κκκ=κ ,, . (76)
Here,
( ) y
E
xxx bkkkk −+=κ )(
r
; ( ) x
E
yyy bkkkk ++=κ )(
r
;
( ) zz kk =κ
r
. (77)
The reverse transformation (if using the inequality
b2 << 1) is
.
,
)()(
)()(
E
x
E
yxyy
E
y
E
xyxx
bkkbk
bkkbk
+−κ−κ=
−−κ+κ=
(78)
Then (for approximations shown before) one
obtains from (75)
( )
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
−κ=ω−ω=ε−ε −− 2
22 qq
mqkkqkk
rrh
h rrrrrr . (79)
Introducing the mechanical momentum kp
r
h
r
= ,
one can see that at (73a) two latter terms are quantum
amendment to classical part. Therefore, the retained field
terms in collision integral give the reason to name the
considered kinetic equation as the quantum kinetic one.
7. Balance of forces
For stationary spatially uniform system kinetic equation
(29) has the form
( )[ ] k
k f
k
f
Hkv
c
Ee
r
r
r
rrrr
h
St1
=
∂
∂
⎭
⎬
⎫
⎩
⎨
⎧
×+ . (80)
Construct the first moment of the equation (80),
applying there to both sides the operator
( ) ∫−π kdk
rr
3322 . (81)
Then, we obtain a vector equation, having the sense
of balance of dynamical and statistical fields forced in
all the system of band carriers:
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 173-182.
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179
( )( )[ ]
( )
( )( )[ ] .0/1
St
2
2/1 3
3
=+×+=
=
π
+×+ ∫ −
eI
kSe
FuHcEe
kdfk
n
uHcEe
rrrr
rrhrrr
r
(82)
Here, the value eIF
r
is resistant force acting from
the side of charged impurities (something as “friction
force”). The values
( ) ∫π
= kdfn k
r
r 3
32
2 ,
( ) ( )
( ) ( )∫∫
∫
π
== kdfkv
nkdkf
kdkfkv
u k
rrr
rr
rrrr
r
r 3
33
3
)2(
2 (83)
are density of electrons and drift velocity of whole band
electrons.
After non-complicated transformations of the
formula (53), we obtain the following expression:
( ) ( )∫∫ −−κδκ
επ
−= 4233
223
4
2 qqqqdqdkf
n
mne
F
L
I
eI
rrrrrr
h
r
. (84)
Performing here integration over components of
vector q
r
(see (78)) we find
( )( ) ( ) κκκκ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
επ
−= ∫ − rrrrr
h
r 33
0
223
4
ln
2
dkkf
q
q
n
nme
F M
L
I
eI . (85)
For standard variant 0=χ (See (46)), the
expression (85) transforms to
( ) kdkkkf
q
q
n
nme
F M
L
I
eI
rrr
h
r 33
0
223
4
ln
2
∫ −
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
επ
−= . (86)
8. The model of non-equilibrium distribution
function
As one can see, the friction forces (85) and (86) are
linear integral functionals of the non-equilibrium
distribution function ( )kf
r
. As the sufficiently simple
model of ( )kf
r
, we accept here Fermi-function with a
shifted argument:
( ) ( ) ( ) 1
22
0
2exp1
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ ε−−
+=−=
Tk
mumkkkfkf
B
F
u
h
rr
hrrr
.
(87)
Introduce three-dimensional vectors K
r
, )(uK
r
and
several dimensionless values:
( )0,, )()()( u
y
u
x
u KKK =
r
, hx
E
y
E
x
u
x mubkkK ++= )()()( ,
hy
E
x
E
y
u
y mubkkK +−= )()()( , (88)
( ) ( )zxyyxzyx bbKKKK κκ−κκ+κ== ,,,,
r
. (89)
tTmkK B
rr
h =2 ,
YTmkK B
u rr
h =2)( , η=ε TkBF . (90)
Тhen, the “friction” force (85) takes the form
( )
( )[ ]
( )
.
exp1
ln
2
3
2
1
0
233
42/12/32/13
∫
⎥⎦
⎤
⎢⎣
⎡
⎟
⎠
⎞⎜
⎝
⎛ η−−+
×χ+
×
×⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
επ
−=
−
td
Yt
tebtt
q
q
n
neTkm
F
z
p
M
L
IB
eI
r
rr
rrr
h
r
(91)
Introduce the dimensionless electric field and
current density:
TEEW /
rr
= , uTkmjjJ B
rrr
== 0 . (92)
Here,
heTmkE BT π〉ε〈= 32 , mTkenj B=0 . (93)
It follows from (90), (92) and (46) that
( )[ ]zeWbWJY
rrrrr
×+χ+= . (94)
Then the balance equation accepts the form
( ) ( )
( )[ ]
( )[ ]
,
exp1
2
31
2/3
)(
∫
⎭⎬
⎫
⎩⎨
⎧ η−−×χ−χ−+
×χ−
×
×
η
Θ
=×+
−
JeWbWt
tdetbtt
F
JebW
z
z
p
I
z
rrrrr
rrrr
rrr
(95)
where
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ε
π
=Θ
0
2/52/12
432/5
)( ln2
q
q
Tkm
ne M
BL
I
I
h . (96)
For 0qqM >>
( ) ( )
( ) ( )⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
ηηπ
ηε
≈
− 2/1
2
2/1
2
2
1
0
84
ln
2
1ln
FFme
FTk
qq BL
M
h
. (97)
If external magnetic field is absent, the equation
(95) is converted to the following equation:
( ) [ ]{ }∫
η−−χ−+η
Θ
=
−
2
31
2/3
)(
exp1 JWt
tdtt
F
W
p
I
rrr
rr
r
. (98)
Designate a mobility tensor by the symbol
)(ˆ χμ and write here:
Eenj
rr )(ˆ χμ= , ( )WJ
rr
min
)(ˆ μμ= χ , (99)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 173-182.
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180
( )
( )
( )
( ) .
2
2
3
2/3
2/1
2/3
2/1
0
min
η
η
=
η
ηπ
=
=
〉ε〈
π
==μ
F
F
M
F
F
Tmk
e
m
e
enE
j
B
T
h
h
(100)
Dimensionless magnetic field cHb z /minμ= . In
absence of magnetic field jiji ,
)()( δμ=μ χχ . In the system
CGSE and for 28102 −⋅=m , K100=T , we find:
CGSE101.8 5⋅=M .
Define the conditional free-path length L and
average length of the deBroighle wave λ by the
relations:
emL 〉ε〈μ= χχ 2ˆ )()( , 〉ε〈=λ m2h . (101)
Тhen the equality
λ>>χ)(L (102)
can be written as WJ >> or min
)(ˆ μ>>μ χ . Under the
condition λ≤χ)(L , or WJ
rr
≤ , the concept “mobility”,
how one will see below, loses its usual meaning, and
description of macroscopic movement of band carriers
requires other ways.
If electrical field and current density are weak, that is
1<<+ JW
rr
, (103)
one can linearise the model non-equilibrium distribution
( )kf
r
. In this case, the relation of dimensionless
electrical field W
r
with dimensionless density of current
J
r
becomes the linear equation
( ) ( )
( )[ ]
( )[ ]
( )[ ]{ } ( ) .exp
exp1
2
32
2232/3
)(
tdteWbWJt
tt
etbt
F
JebW
z
zI
z
rrrrrr
rrr
rrr
η−×χ+χ+⋅×
×
η−+
×χ+
η
Θ
=×+ ∫ (104)
For the case 0=χ , one obtains after performing
the integration over angles in the latter formula:
( ) ( )
( )
( )[ ] dt
t
ttJ
F
JebW
p
I
z ∫
η−+
η−
η
Θπ
=×+
−
22
25
2/3
)(
exp(1
exp
3
4 rrrr
. (105)
9. Current-voltage characteristics
in absence of magnetic fields
For b = 0 (that is at H = 0), the linear equation (105) has
the form (external macroscopic electrical field is
directed along the x-axis):
( ) ( ) ( )( )xxIIxxx WJQWJW χ+ηα⋅Θ=ηχ+= )()( , (106)
( ) ( )
( )
( )[ ]
( )
( ) [ ] ( )
.
)exp(13
4
3
4
exp1
exp
3
8
2/32/3
1
0
22
2
2/3
ηη−+
π
=
η
ηπ
=
=
η−+
η−
η
π
=ηα
−
∞
∫
FF
F
dt
t
tt
F
(107)
Accordingly to (99) and (106), the current-voltage
characteristic has the form
( )( )χ−η=μ
μ
= χ
)(
)(
min
11
Ixxx QWWJ . (108)
It follows from here:
)(min
)0( / IQμ=μ , ( ) )()(min
)1( 1 II QQ−μ=μ ; (109)
)(
)0()1( 1/ IQ−=μμ . (110)
One can see from the formula (123) that for field
variant ( )1=χ the concept “mobility” has a meaning
only under the condition:
1)( <IQ . (111)
In this case, )0()1(0 μ<μ< . It also follows from
(109) that distinction between results of calculations for
standard and field variants disappears under the
condition 1)( <<IQ or min
)0( μ>>μ .
10. Galvanomagnetic kinetic effects
10.1. Kinetic characteristics calculated for standard
linear equation of forces balance
Supposing 0=χ , we write the vector equation
(105) as the system
xIyx JQbJW )()( η=− ; (112а)
yIxy JQbJW )()( η=+ . (112b)
Here, we accept b2 << 1. One writes the solution of
the system (112) in the form (99). Components of
mobility tensor )0(μ̂ are
)(
)()( 22
)(
)(min)0()0(
HbQ
Q
HH
I
I
yyxx
+
μ
=μ=μ ,
)(
)()()( 22
)(
min)0()0(
HbQ
HbHH
I
yxxy
+
μ
−=μ−=μ . (113)
At presence of magnetic field zzeHH
rr
= and at
current )0,( xjj =
r
, the longitudinal component xj ,
transverse component of electrical field yE and Hall
constant )0(
HR are
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 173-182.
doi: 10.15407/spqeo19.02.173
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
181
( )
,)(
)(
)(
)()(
)0(
||
)0(
2)0(
)0(
x
x
xx
xy
xxx
EHen
E
H
H
HenHj
μ=
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
μ
μ
+μ=
(114)
)(
min)0(
|| )(
IQ
H μ
=μ ,
x
I
x
xx
yx
xy E
Q
HbE
H
H
EHHE
)(
)0(
)0(
)0( )(
)(
)(
)()( =
μ
μ
=ϑ= , (115)
encHjH
HE
HR
xz
y
H
1
)(
)(
)()0( == . (116)
As one can see, in the standard variant the
longitudinal conductivity )0(
||
)0(
|| μ=σ en and Hall
constant )0(
HR do not depend on the intensity of
magnetic field.
10.2. Kinetic characteristics calculated for non-standard
linear equation of forces balance
Write the components of linear vector equation for
1=χ :
( ) ( )yxxIyx bJWJQbJW −+η=− )( ; (117а)
( )( )xyyIxy bJWJQbJW ++η=+ )( . (117b)
Solving this system of equations, represent the
solution in the form (99). One applies the magnetic field
to be not quantized, that is 1/)0( <<μ= cHb ; but the
value b can be comparable with )(IQ and even exceeds
it. As a result,
( )
( )2)(
22
)(
)()(
min
)1()1(
1
1
II
II
yyxx
QbQ
QQ
−+
−
μ=μ=μ ,
( )
( )2)(
22
)(
2
)(
min
)1()1(
1
1
II
I
yxxy
QbQ
Q
b
−+
−
μ−=μ−=μ . (118)
Consider the case ( )0,xjj =
r
. Тhen,
( )
xx
xx
xy
xxx EenEenj )1(
||)1(
2)1(
)1( μ=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
μ
μ
+μ= ,
( )
)(
)(
min
)1(
||
1
)(
I
I
Q
Q
b
−
μ=μ , min
)0(
||
)1(
|| )()( μ−μ=μ bb , (119)
( ))(
)0(
)(
)(
)1(
)1(
)1( 1)(
1
)(
)()(
)( I
I
I
xx
yx
x
y Qb
Q
Q
b
b
b
E
bE
b −ϑ=
−
=
μ
μ
==ϑ ,
encbjH
bEbR
xz
x
H
1
)(
)()(
)1(
)1( =
ϑ
= . (120)
Comparing the results of standard and field
variants, we find:
( ))(
)0(
||
)1(
|| 1 IQ−μ=μ , ( ))(
)0()1( 1 IQ−ϑ=ϑ ,
HHH RbRbR == )()( )0()1( . (121)
It follows that formulae (119), (120) and the
meanings of mobility and Hall-angle have a sense at the
following condition only:
1)( <IQ . (122)
The free-path distance (see (101)) is
( ))(
)0()1( 1 IQLL −= . (123)
If for the inequality (122) the value )(IQ is
sufficiently close to unity, one can say about small
mobility or about definite “demobilization” of band
electrons due to extremely high intensity of scattering.
The limit of mobility 0)1(
|| =μ achieves at 1)( =IQ .
One obtains from (101) and (121):
( )χ
χ
−
π
=
λ )(
)(
)(
1
2
3
I
S
Q
Q
L ; )0(
)( 23 LQ I λπ= . (124)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 2. P. 173-182.
doi: 10.15407/spqeo19.02.173
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
182
It follows from here that retention of field terms in
collision integral is the reason of appearance of quantum
amendment to kinetic coefficients, for instance:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ λπ
−μ=μ )0(
)0(
||
)1(
|| 2
31
L
. (125)
As a result, we obtain the important conclusion: a
quantum kinetic equation distinguishes from classical
kinetic equation by retention of field terms in collision
integral.
Therefore, we can use the expression “classical
approach” instead of “standard variant” and “quantum
approach” instead of “field variant” (See the forms (46)).
Below in Fig. 1 there are presented several plots,
drawn using the formulae (115) and (119). One can see
that, under the field approach to collision integral
typical, the kinetic characteristics substantially differ by
the values corresponding to standard variant even at
1.0)( >IQ .
11. Discussion
One can make the conclusion that regard for the field
terms in collision integral results there in appearance of
deBroighle wavelength λ and comparability of that with
the free-path distance L. Taking into account the finite
ratio of λ and L, we can say about quantum kinetic
equation in total. If a consideration does not use directly
the field terms in collision integral, the kinetic equation
leaves to be the classic one.
References
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Current Carriers in Metals and Semiconductors.
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(in Russian).
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