Electron-electron drag in crystals with multivalley band
Mobility of electrons in multivalley bands of Si and Ge is considered with due regard for intervalley drag. It is shown that drag sufficiently diminishes electron mobility at low temperatures. This effect is clearer pronounced in germanium than in silicon.
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2009
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| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Цитувати: | Electron-electron drag in crystals with multivalley band / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 3. — С. 212-217. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1188642017-06-01T03:04:35Z Electron-electron drag in crystals with multivalley band Boiko, I.I. Mobility of electrons in multivalley bands of Si and Ge is considered with due regard for intervalley drag. It is shown that drag sufficiently diminishes electron mobility at low temperatures. This effect is clearer pronounced in germanium than in silicon. 2009 Article Electron-electron drag in crystals with multivalley band / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 3. — С. 212-217. — Бібліогр.: 8 назв. — англ. 1560-8034 PACS 61.72, 72.20 http://dspace.nbuv.gov.ua/handle/123456789/118864 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Mobility of electrons in multivalley bands of Si and Ge is considered with due
regard for intervalley drag. It is shown that drag sufficiently diminishes electron mobility
at low temperatures. This effect is clearer pronounced in germanium than in silicon. |
| format |
Article |
| author |
Boiko, I.I. |
| spellingShingle |
Boiko, I.I. Electron-electron drag in crystals with multivalley band Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Boiko, I.I. |
| author_sort |
Boiko, I.I. |
| title |
Electron-electron drag in crystals with multivalley band |
| title_short |
Electron-electron drag in crystals with multivalley band |
| title_full |
Electron-electron drag in crystals with multivalley band |
| title_fullStr |
Electron-electron drag in crystals with multivalley band |
| title_full_unstemmed |
Electron-electron drag in crystals with multivalley band |
| title_sort |
electron-electron drag in crystals with multivalley band |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118864 |
| citation_txt |
Electron-electron drag in crystals with multivalley band / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 3. — С. 212-217. — Бібліогр.: 8 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT boikoii electronelectrondragincrystalswithmultivalleyband |
| first_indexed |
2025-07-08T14:48:05Z |
| last_indexed |
2025-07-08T14:48:05Z |
| _version_ |
1837090565221515264 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 212-217.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
212
PACS 61.72, 72.20
Electron-electron drag in crystals with multivalley band
I.I. Boiko
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine, 45, prospect Nauky, 03028 Kyiv, Ukraine
E-mail: igorboiko@yandex.ru; phone: (38-044)236-5422
Abstract. Mobility of electrons in multivalley bands of Si and Ge is considered with due
regard for intervalley drag. It is shown that drag sufficiently diminishes electron mobility
at low temperatures. This effect is clearer pronounced in germanium than in silicon.
Keywords: quantum kinetic equation, mobility, intervalley drag.
Manuscript received 03.04.09; accepted for publication 14.05.09; published online 15.05.09.
1. Introduction
In absence of external electric field, electron-electron
interaction in uniform medium is usually considered as a
principal mechanism of electron gas thermalization and
formation of a dielectric function for the studied system.
In crystals with simple band electron-electron scattering,
the latter does not change the total momentum of carriers
and, therefore, has no direct, independent contribution to
conductivity (see [1-3]). For non-equilibrium electron
gas that is represented by only one group, the mentioned
interaction manifests itself mainly as a mechanism of
formation of symmetrical part for the distribution
function. But in the case when mobile carriers belong to
different groups with rare transitions between these
groups, quite specific contribution of e-e-interaction to
conductivity appears. This effect is related with a
difference of drift velocities for different groups and
manifests as intergroup drag. The most interesting
objects for consideration of the drag are electrons in
multivalley semiconductor with equivalent anisotropic
valleys (all groups have the same population and that
favours to effect). In details, we will examine the
intervalley drag in n-Si and n-Ge. This phenomenon
better displays in the region of sufficiently low
temperatures where Coulomb scattering is not damped
by collisions with phonons.
2. Quantum kinetic equation
Let us consider here a uniform crystal in constant
uniform electric field E
. In this case, the stationary
quantum kinetic equation for a distribution function
)(a
k
f for carriers from a-group can be presented in the
following form (see, for instance, Refs [4-7]):
).,2,1,,(
1
)(
)()()(
)(
MbafSt
fStfStfSt
k
f
E
e
M
b
a
kba
a
kphe
a
kIe
a
k
a
ka
(1)
Here, M = 6 for n-Si, and M = 4 for n-Ge. In
Eq. (1) we suppose the same electric charge for mobile
carriers from different groups. The collision integral in
the right part of this equation involves scattering of a-
carriers by charged impurities (e-I process), phonons (e-
ph process) and band carriers belonging to all the
considered groups (a-b process). Below, we restrict our
consideration with quasi-elastic scattering by
longitudinal acoustic phonons. Intervalley transitions are
not taken in consideration. We also assume that inter-
groups transitions can be neglected in comparison with
direct Coulomb scattering by band carriers.
The screening dielectric function ),( q for quasi-
elastic collisions has the form
),0(),( qq eL
, (2)
where L is the dielectric constant of crystal lattice and
),( qe
is contribution of band electrons to total
dielectric function. Farther we accept the following
designation:
22
0 /)(),0( qqqq L
. (3)
Using the simplified model (2) and form (3) for the
screening dielectric function, we write (see Ref. [4])
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 212-217.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
213
;)]1()1(
)1()1([
)]([
)(
'
)()()(
'
)(
'
)(
'
)(
'
)()(
22
0
22
)(
'
)(
'
)()(
33
3
4
)(
a
qk
a
k
b
k
b
qk
b
qk
b
k
a
k
a
qk
L
b
k
b
qk
a
qk
a
ka
kba
ffff
ffff
qqq
qdkd
e
fSt
(4)
.
)
2
tanh
))1)()1(
)(
)2(
3
,
2
)(,
2
)(
)()(
)()()()(
3
2
)()(
qd
ff
Tk
ffff
d
e
fStfSt
qphqI
a
qk
a
k
B
a
k
a
qk
a
qk
a
k
qkk
a
kphe
a
kIe
(5)
Here
)(
2
)(
2
)(
22
)(
2 a
zz
z
a
yy
y
a
xx
xa
k m
q
m
q
m
q
is the dispersion law for a-carriers with ellipsoidal
surfaces of constant energy (for corresponding suitable
system of axes);
)(
)]([
32
22
0
22
23
,
2
)(
qqq
ne
L
I
qI ;
)(
2
)(
)(
22
2
2
0
2
2
,
2
)(
se
Tk
qqq
q
q B
Aqph
. (6)
Below, we will perform numerical calculations
using this approximation:
udA q )2/1()( . (7)
Here (see, for example, Refs [5] and [6]), d and
u are dilatation and shear deformation potential
constants.
Designing the distribution function for equilibrium
gas of a-carriers as )( )(0 a
k
f , we write (see Refs [3, 4])
.
0
)()(
),(
)()(
)(0)(0
3
22
2
0
)(
i
ff
kd
q
e
q
a
k
a
qk
a
qk
a
k
a
(8)
Then
,
)(
)(
ln
)exp(1)(
1
2
)(
1 0
)(
)(
)(
3
2/1
||
2
2
0
M
a
a
a
a
a
L
B
qQ
qQd
qQ
Tkmme
qq
(9)
where
2/1
)(
2
)(
2
)(
2
)(
8
)(
a
zz
z
a
yy
y
a
xx
x
B
a
m
q
m
q
m
q
Tk
qQ
.
For farther numerical calculations, we will simplify
the form (9) to
.
)(
)(
ln
)exp(1
)(
2
)()(
0
3
2
||
2
2
0
2
0
qw
qwd
qw
TkmmMe
qqqq
L
B
(10)
Here
|||| 3
1212
3
1
*
1
m
L
mmm
,
Tkmqqw B*8/)( 22 , TkBF / (11)
( F is the Fermi energy). The dimensionless energy
is related with the concentration of mobile electrons n by
the relation
0
2/1
32/7
||
2/3
)exp(12
)2(
w
dwwmmTkM
n
B
.
3. Balance equation and kinetic coefficients
Applying to both sides of Eq. (1) the operator
kdk
334/1 ,
we obtain balance equations for forces:
),,2,1,(
0
1
),()()(
Ma
FFFEe
M
b
baa
phe
a
Ie
(12)
where
.][
)}2/tanh()]1()1([
{)(
)2(
,
2
)(,
2
)(
)()()()(
)()()()(3
3
6
2
)()(
qphqI
B
a
k
a
qk
a
qk
a
k
a
qk
a
k
a
qk
a
k
a
a
phe
a
Ie
Tkffff
ffdqdq
kd
n
e
FF
(13)
;),',(
),0(
)(1
'
4
2
)()()()(
4
3
33
)(6
4
),(
qkk
qq
qd
kdkdk
n
e
F
ab
L
b
qk
b
k
a
qk
a
k
a
ba
.)1()1(
)1()1(),',(
)(
'
)(
'
)()(
)(
'
)(
'
)()(
b
k
b
qk
a
qk
a
k
b
qk
b
k
a
k
a
qkab
ffff
ffffqkkY
(14)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 212-217.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
214
As an appropriate model of the non-equilibrium
distribution function for carriers of a-group, we accept
the Fermi function with the argument containing shift of
velocity )/()( )(1)( kkv a
k
a
on the correspondent
drift velocity )(au
of whole a-group:
))(( )()()(0)( aaaa
k
ukvff
. (15)
Here ))(( )()(0 kvf aa
is the equilibrium distribution
function for a-carriers.
Carrying out linearization of forces in Eq. (12), we
obtain
)()()()( ~ aaa
phe
a
Ie ueFF
,
)()(),(),( ~ bababa uueF
, (16)
where kinetic coefficients can be presented in the
following forms:
,][
),(Im
sinh
)2(2
,
2
)(,
2
)(
0
)(
32
)(5
)(
qphqI
a
B
B
a
a
qqq
Tk
qdq
d
Tkne
, (17)
,),(Im),(Im
)]([
)2/(sinh)2(2
)1(
0
)(
0
)(22
0
2
34
2)(24
2
),(
qq
qqq
qdqqq
Tk
d
Tkne
ba
BB
a
L
abba
(18)
)(an is the density of a-carriers; zyx ,,, .
From Eqs (12) and (16), one obtains the system of
equations for drift velocities:
).,2,1,(
0
~~
1
)()(),()()(
Ma
uuuE
M
b
babaaa
(19)
Here, terms containing matrices ),(~ ba respond for
the intervalley drag. The value 1)( )
~
( a is the mobility
tensor for a-carriers, if one neglects the mentioned drag
( 0
~ ),( ba ).
4. Mobility of band carriers in n-Si and n-Ge
The structure of valleys in n-Si and n-Ge is shown in
Fig. 1. At the beginning, let us consider the electron
band in silicon.
Fig. 1.
For its six valleys
;
00
00
00
~~
;
00
00
00
~~
;
00
00
00
~~
||
)6()3(
||
)5()2(
||
)4()1(
.
00
00
00
~
;
00
00
00
~
;
00
00
00
~
2
1
1
)3,2(
1
1
2
)3,1(
1
2
1
)2,1(
Here
;]~~[
),(Im
sinh
)2(
3
,
2
)(,
2
)(
0
)1(
32
,
2
5,||
qphqI
B
xz
B
q
Tk
qdqq
d
Tkne
(20)
.
)]([
),(Im),(Im
)2/(sinh)2(
3
32
,22
0
2
0
)2(
0
)1(
4
224
2
2,1
qdq
qqq
qqq
Tk
d
Tken
yx
BBL
(21)
A solution of the system (19) for n-Si can be
represented in the form Eu aa
)()( ~ , where
;
00
00
00
~
;
00
00
00
~;
00
00
00
~
||
)6,3(
||
)5,2(
||
)4,1(
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 212-217.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
215
1||||
1||
)2(2
6
;
1||||
1
||
)2(2
6
. (22)
The tensor of conductivity for totally mobile
electrons:
nene ijij
a
a
ij
a
ij
6
1
)()( ,
.
/)2(21
)21(/181
3
21
)6/()(4
)6/()(23
3
1
||1
||1
||
1||1||
1||
)Si(
L
LLL
(23)
Note that the value 2 does not enter to the
formula (23); this value appears when galvano-magnetic
effects are considered.
Analogous consideration for n-Ge gives a similar
result:
)3/4()2(1
)21/()/12(1
3
21
||1
||1
||
)Ge(
L
LLL
. (24)
One can see that the reason of intervalley drag (this
effect is represented in formulae (23) and (24) by the
value 1 ) is anisotropy of valleys ).1or,( || L
We carry out farther calculations of the values
,|| and 1 on the base of a simplifying procedure of
preliminary partial average over angle expressions
represented by three-dimensional integrals (see also Eqs
(20), (21) and (10)). As a result, we obtain (here In is
the concentration of charged impurities):
L|| , )(
||
)(
||||
phI ; (25)
2
0
223
2
||
3
)(
||
)]([
1
)exp(1)1(
16
wwww
dww
Ln
mne
scrL
II
;
(26)
;
)]([
1
)exp(1
)(])2/1([16
0
2
3
2273
4
||
32
)(
||
wwww
dww
nLse
mTk
scr
Budph
(27)
;
)]([
1
)exp(1
12
3)(2
0
22
2/7
||
263
2/33
1
wwww
dww
L
m
n
Tke
scr
L
B
(28)
Fig. 2.
;ln
)exp(1
*8
2
)(
0
3
||
2
w
wd
TkwmL
mMe
ww
BL
scr
(29)
29.3
sinh 2
2d
. (30)
Results of numerical calculations of the mobility at
Inn are presented for n-Si in Fig. 2, and for n-Ge in
Fig. 3. Here,
1
00 )/( me ,
where s10,g101066.9 12
0
28
0
m . In this case,
1 corresponds to sV/cm10758.1 23 . Carrying
out the calculations, we have accepted 6M , 8.4L ,
12L , 92.0)/( 0|| mm , Pa1066.1 112 s ,
eV2.4)2/1( ud for n-Si, and 4M ,
4.19L , 16L , 59.1)/( 0|| mm , Pa1026.1 112 s ,
eV9.1)2/1( ud for n-Ge.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 212-217.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
216
Fig. 3
Curves 1 in Figs 2 to 5 show the mobility of
electrons calculated on the base of formulae (23) or (24)
at 01 (that is the case when the intervalley drag was
not accepted in consideration). Curves 2 relate to the
same formulae and represent the mobility with due
regard for the intervalley drag ( 01 ). One can see
from these figures that the intervalley drag in n-Si and
n-Ge introduces an appreciable contribution to the
mobility at sufficiently low temperatures. In germanium
crystals, the considered effect manifests itself more
clearly, because anisotropy of values in germanium
considerably exceeds that in silicon.
Fig. 4.
Fig. 5.
One can see from Fig. 4 that the intervalley drag
practically disappears for high concentrations of carriers
(in this case, the charge of every band carrier is
substantially screened). This effect significantly weakens
at a small concentration, because electron-electron
scattering is rare and therefore unimportant.
Fig. 5 shows as calculated here curves as
experimental data represented in Ref. [7]. The latter are
shown by small circles. The author thinks that the
curve 2 (intervalley drag is taken into consideration)
better coordinates with experimental points than the
curve 1 (intervalley drag is not taken into account). It
was pointed out in Ref. [8] that the mobility in n-Ge
calculated on the base of -approximation substantially
exceeds the experimentally measured one.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 212-217.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
217
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