Influence of mutual drag of light and heavy holes on magnetoresistivity and Hall-effect of p-silicon and p-germanium
Hall-effect and magnetoresistivity of holes in silicon and germanium are considered with due regard for mutual drag of light and heavy band carriers. Search of contribution of this drag shows that this interaction has a sufficient influence on both effects.
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| Опубліковано в: : | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Дата: | 2011 |
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| Мова: | English |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2011
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| Цитувати: | Influence of mutual drag of light and heavy holes on magnetoresistivity and Hall-effect of p-silicon and p-germanium / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 4. — С. 437-440. — Бібліогр.: 10 назв. — англ. |
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Boiko, I.I. 2017-05-26T17:43:02Z 2017-05-26T17:43:02Z 2011 Influence of mutual drag of light and heavy holes on magnetoresistivity and Hall-effect of p-silicon and p-germanium / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 4. — С. 437-440. — Бібліогр.: 10 назв. — англ. 1560-8034 PACS 61.72, 72.20 https://nasplib.isofts.kiev.ua/handle/123456789/117793 Hall-effect and magnetoresistivity of holes in silicon and germanium are considered with due regard for mutual drag of light and heavy band carriers. Search of contribution of this drag shows that this interaction has a sufficient influence on both effects. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Influence of mutual drag of light and heavy holes on magnetoresistivity and Hall-effect of p-silicon and p-germanium Article published earlier |
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Influence of mutual drag of light and heavy holes on magnetoresistivity and Hall-effect of p-silicon and p-germanium |
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Influence of mutual drag of light and heavy holes on magnetoresistivity and Hall-effect of p-silicon and p-germanium Boiko, I.I. |
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Influence of mutual drag of light and heavy holes on magnetoresistivity and Hall-effect of p-silicon and p-germanium |
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Influence of mutual drag of light and heavy holes on magnetoresistivity and Hall-effect of p-silicon and p-germanium |
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Influence of mutual drag of light and heavy holes on magnetoresistivity and Hall-effect of p-silicon and p-germanium |
| title_full_unstemmed |
Influence of mutual drag of light and heavy holes on magnetoresistivity and Hall-effect of p-silicon and p-germanium |
| title_sort |
influence of mutual drag of light and heavy holes on magnetoresistivity and hall-effect of p-silicon and p-germanium |
| author |
Boiko, I.I. |
| author_facet |
Boiko, I.I. |
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2011 |
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English |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Article |
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Hall-effect and magnetoresistivity of holes in silicon and germanium are
considered with due regard for mutual drag of light and heavy band carriers. Search of
contribution of this drag shows that this interaction has a sufficient influence on both
effects.
|
| issn |
1560-8034 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/117793 |
| citation_txt |
Influence of mutual drag of light and heavy holes on magnetoresistivity and Hall-effect of p-silicon and p-germanium / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 4. — С. 437-440. — Бібліогр.: 10 назв. — англ. |
| work_keys_str_mv |
AT boikoii influenceofmutualdragoflightandheavyholesonmagnetoresistivityandhalleffectofpsiliconandpgermanium |
| first_indexed |
2025-11-24T16:29:27Z |
| last_indexed |
2025-11-24T16:29:27Z |
| _version_ |
1850482365437050880 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 437-440.
PACS 61.72, 72.20
Influence of mutual drag of light and heavy holes
on magnetoresistivity and Hall-effect of p-silicon and p-germanium
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. Hall-effect and magnetoresistivity of holes in silicon and germanium are
considered with due regard for mutual drag of light and heavy band carriers. Search of
contribution of this drag shows that this interaction has a sufficient influence on both
effects.
Keywords: quantum kinetic equation, Hall-effect, magnetoresistivity, interband drag.
Manuscript received 29.03.11; revised manuscript received 29.08.11; accepted for
publication 14.09.11; published online 30.11.11.
1. Introduction
In the previous work (see Ref. [1]), we investigated the
influence of mutual drag of heavy and light holes on
conductivity of p-germanium and p-silicon. It was
shown that this drag significantly diminishes the total
conductivity of holes. In practice, this effect attracted
attention of a great number of earlier investigators. One
of the reasons was vain attempts to describe mutual drag
of band carriers belonging to different groups (see, for
example, Refs. [2, 3]) by using the wide-spread tau-
approximation in the course of solution of kinetic
equation (see Refs. [4, 5]). In this work, we have used
the method of balance equation (see Refs. [6-8]), which
allows to introduce into consideration mutual drag of
carriers from two bands that close up in the center of
wavevector space. For simplicity of calculations, we
accept here spherical bands approximation. So, the
dispersion law for holes has the following simple form:
a
a
k
mk 2/22)( hr =ε (a = 1 or 2). (1)
In this formula, is the effective masses ( is
mass of light holes, and is mass of heavy holes).
am 1m
2m
2. Balance equations
Let us consider the set of two balance equations obtained
as a first momentum of quantum kinetic equations (see
Refs. [7, 9]):
0)]()/1([
2
1
),()()( =++×+ ∑
=b
baaa FFuHcEe
rrrrr
(а = 1, 2).
(2)
Here vectors E
r
and H
r
represent electrical and
magnetic fields, the values )(au
r
are drift velocities of
light and heavy holes, )(aF
r
is the resistant force related
to an external scattering system, ),( baF
r
is the force
related with Coulomb interaction of heavy and light
holes.
We restrict here our consideration by external
scattering system containing charged impurities and
acoustic phonons.
Accepting the model of non-equilibrium
distribution functions of holes from different groups as
Fermi functions with argument containing for a-group
the shift of velocity )/()( )(1)( kkv a
k
a r
h
rr
r ∂∂= − ε on
correspondent velocity )(au
r
))(( )()()(0)( aaaa
k ukvff
rrr
r −= (a = 1, 2) (3)
(here ))(( )()(0 kvf aa rr
is equilibrium distribution
function for a-carriers) we obtain the following
expressions for forces presented in Eq. (2):
)()()( aaa ueF
rr
β−= ; ( ))()(),(),( bababa uueF
rrr
−−= ξ .
(4)
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
437
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 437-440.
Fig. 1. Influence of interband drag on the Hall coefficients. (a) p-silicon: p = 1017 cm–3; (b) p-germanium: p = 1014 cm–3.
1 – T = 50 K, 2 – 100 K.
Fig. 2. Dependence of the relative Hall coefficient of p-silicon on the dimensionless magnetic field. p = 1014 cm–3.
(a) T = 50 K, (b) 100 K. 1 – ξ = 0, 2 – ξ ≠ 0.
In what follows, we shall consider only
nondegenerate band carriers. Then (see Ref. [10])
In what follows, we shall consider only
nondegenerate band carriers. Then (see Ref. [10])
)()()( aaa χλβ += ; (5) ; (5)
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
)()()( aaa χλβ +=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
+
= ∫
∞
Tkm
q
qq
dqq
Tk
nem
BaLB
Iaa
8
exp
)()(3
24 22
22
0
2
3
0
22/3
3
)( h
ε
π
λ ;
(6)
242/3
2/52/32
)(
3
)(28
se
mTk aBAa
ρπ
χ
h
Ξ
= ; (7)
;11
8
exp
)(
3
8
0 21
22
22
0
2
2
23
),(
∫
∞
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−
+
×
×=
mmTk
q
qq
dqq
mTk
pme
B
aB
abba
h
h
γ
ξ
(8)
;29.3
sinh 2
2
∫
∞
∞−
≈=
w
dwwγ (9)
Tk
ppe
q
BLε
π )(4 21
2
2
0
+
= . (10)
Here, is the density of a-holes, is the
density of charged impurities. We assume
ap In
21 pppnI +== .
For nondegenerate carriers
2/3
2
1
2
1
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
==
m
mw
p
p . (11)
Fig. 3. Dependence of the relative Hall coefficient of p-
germanium on the dimensionless magnetic field. p = 1014 cm–3.
T = 50 K: 1 – ξ = 0, 2 – ξ ≠ 0. T = 100 K: 3 – ξ = 0, 4 – ξ ≠ 0.
438
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 437-440.
Fig. 4. Dependence of the relative magnetoconductivity of p-silicon on the dimensionless magnetic field. (a) T = 50 K; (b) T =
100 K. 1 – ξ = 0, 2 – ξ ≠ 0; p = 1014 cm–3.
Fig. 5. Dependence of the relative magnetoconductivity of p-germanium on the dimensionless magnetic field. (a) p = 1014 cm–3;
T = 50 K: 1 – ξ = 0, 2 – ξ ≠ 0. T = 100 K: 3 – ξ = 0, 4 – ξ ≠ 0. (b) p = 1017 cm–3; T = 100 K: 1 – ξ = 0, 2 – ξ ≠ 0.
For p-germanium w = 0.042, for p-silicon
w = 0.153 (see Ref. [5]).
For p-germanium w = 0.042, for p-silicon
w = 0.153 (see Ref. [5]).
From the formulae (2) and (4), one obtains the
system of equations for drift velocities:
From the formulae (2) and (4), one obtains the
system of equations for drift velocities:
( ) 2). 1, = (.0
)()/1(
2
1
)()(),(
)()()(
аuu
uuHcE
b
baba
aaa
=−−
−−×+
∑
=
rr
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
( ) 2). 1, = (.0
)()/1(
2
1
)()(),(
)()()(
аuu
uuHcE
b
baba
aaa
=−−
−−×+
∑
=
rr
rr rr
ξ
β
(12)
The total density of current is
∑∑
==
==
2
1
)(
2
1
)(
a
a
a
a
a upejj
rrr
. (13)
3. Hall-coefficient and magnetoresistivity
Let us consider the case when the external magnetic
field H
r
is directed along z-axis and total current −
along x-axis:
),0,0( zHH =
r
. (14) )0,0,( xjj =
r
Then the electrical field E
r
has the following
components:
)0,,( yx EEE =
r
. (15)
Here the component is related to applied
electrical field.
xE
The relation between measured components of the
total current and electrical field can be represented in the
form
xx EHj )(*σ= . (16)
The scalar value )(* Hσ is called by us as
magnetoresistivity.
Let us introduce the Hall coefficient by using
the following relation:
HR
xz
y
H jH
E
R = . (17)
To calculate the values )(* Hσ and , the
system of equations (12) should be solved at the first
stage. For the case (14) and (15), we have the system of
five equations:
HR
( ))2()1()1()1()1( )()/1( uuuuHcE
rrrrrr
−+=×+ ξβ ;
( ;
)()/1(
)1()2(1)2()2(
)2(
uuwu
uHcE
rrr )
rrr
−+=
=×+
−ξβ
(18)
0)2(
2
)1(
1 =+ yy upup .
439
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 437-440.
Here, (see Eqs. (8) and (11)); )2,1(ξξ =
)0,,( )()()( a
y
a
x
a uuu =
r
(a = 1, 2). The unknown values are
, , , and . )1(
xu )1(
yu )2(
xu )2(
yu yE
Analytical solution of the system (18) is very
simple but rather complicated in the form. Therefore we
present here the results of our numerical calculations
only by figures (here ). To carry out the
calculations, we used the following numerical values:
)1(
0 βcH =
eV2.4,Pa1066.1,12 112 −=Ξ⋅== AL sρε for p-
silicon and
eV9.1,Pa1026.1,16 112 =Ξ⋅== AL sρε for p-
germanium.
4. Results of calculations
Fig. 1 gives the possibility to compare Hall coefficients
calculated for the case when interband drag is taken into
account ( ),( ξHRH ) and is not taken ( ). One
can see that
)0,(HRH
)0,(),( HRHR HH <ξ , and the difference is
especially significant at small magnetic field.
Figs. 2 and 3 show dependence of the Hall
coefficient on the intensity of magnetic field. It follows
that drag makes this dependence more smooth (in some
cases this dependence practically disappears; see Fig. 3,
the curve 2).
Figs. 4 and 5 represent the dependence of relative
magnetoconductivity of p-silicon and p-germanium on
dimensionless magnetic field. Calculations show that the
influence of intensity of the field on conductivity is
significantly moderated by interband drag (especially in
germanium where the difference between effective
masses of light and heavy holes is aparently high).
References
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(1978).
3. C.A. Kukkonnen, P.M. Maldague, Electron-hole
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potential theory for many-valley semiconductors
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© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
440
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