Impurity scattering of band carriers
Mobility of band carriers scattered on donors, partially ionized, partially neutral, at low temperatures, is considered in general and calculated for AIII-BV group crystals. It is shown that temperature dependence of mobility is determined by relationship between number of ionized and neutral don...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2010
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irk-123456789-1182382017-05-30T03:05:33Z Impurity scattering of band carriers Boiko, I.I. Mobility of band carriers scattered on donors, partially ionized, partially neutral, at low temperatures, is considered in general and calculated for AIII-BV group crystals. It is shown that temperature dependence of mobility is determined by relationship between number of ionized and neutral donors and by average energy of electrons. 2010 Article Impurity scattering of band carriers/ I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 2. — С. 214-220. — Бібліогр.: 15 назв. — англ. 1560-8034 PACS 71.20. 72.20 Dp http://dspace.nbuv.gov.ua/handle/123456789/118238 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Mobility of band carriers scattered on donors, partially ionized, partially
neutral, at low temperatures, is considered in general and calculated for AIII-BV group
crystals. It is shown that temperature dependence of mobility is determined by
relationship between number of ionized and neutral donors and by average energy of
electrons. |
| format |
Article |
| author |
Boiko, I.I. |
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Boiko, I.I. Impurity scattering of band carriers Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Boiko, I.I. |
| author_sort |
Boiko, I.I. |
| title |
Impurity scattering of band carriers |
| title_short |
Impurity scattering of band carriers |
| title_full |
Impurity scattering of band carriers |
| title_fullStr |
Impurity scattering of band carriers |
| title_full_unstemmed |
Impurity scattering of band carriers |
| title_sort |
impurity scattering of band carriers |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/118238 |
| citation_txt |
Impurity scattering of band carriers/ I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 2. — С. 214-220. — Бібліогр.: 15 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT boikoii impurityscatteringofbandcarriers |
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2025-07-08T13:36:35Z |
| last_indexed |
2025-07-08T13:36:35Z |
| _version_ |
1837086065432723456 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 2. P. 214-220.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
214
PACS 71.20. 72.20 Dp
Band carriers scattering on impurities
I.I. Boiko
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
41, prospect Naukyt, 03028 Kyiv, Ukraine
E-mail: igorboiko@yandex.ru
Phone: 38 (044) 236-5422
Abstract. Mobility of band carriers scattered on donors, partially ionized, partially
neutral, at low temperatures, is considered in general and calculated for AIII-BV group
crystals. It is shown that temperature dependence of mobility is determined by
relationship between number of ionized and neutral donors and by average energy of
electrons.
Keywords: impurities, scattering potential, quantum kinetic equation, mobility.
Manuscript received 23.11.09; accepted for publication 25.03.10; published online 30.04.10.
1. Introduction
Investigations of scattering of band carriers by neutral
impurities have no noticeable advance for a long time
[1–5]. Several approaches to the problem of neutral
impurity scattering were used, but only one received
wide recognition. There was consideration of interaction
of electrons with shallow neutral impurity, which
imitates spherically symmetrical hydrogen atom. In this
case relaxation time for momentum of carriers was
constructed on the base of cross-section for scattering
process [6]. This time has well-known Erginsoy’s form
[1]
022
3201
n
em
L
. (1)
Here L is dielectric constant of lattice, m is
effective mass, 0n is concentration of scattering centers.
However, there are serious claims to the method of
scalar relaxation time as a whole [7, 8]. The attempts to
improve agreement of theory and experimental data by
the way of introducing some adjusting factor in Eq. (1)
(see, for instance, [4]) should be considered as very
naive only.
Other direction of investigations uses model for
scattering potential of neutral impurity as rectangular
spherically isotropic hole [9]. Limit case of this model is
delta-shaped function in space [10, 11]). In this case
there is no possibility to evaluate amplitude of
interaction. There is also no way to derive rectangular or
delta-shaped potential as well-reasoned limit case of
physically grounded interaction.
Bellow we shall consider mobility of band carriers,
scattered by charged and neutral impurities; calculations
will be based on quantum kinetic equation [10, 11]. For
simplicity we use here only a model of simple isotropic
parabolic dispersion law for band carriers.
We consider shallow donors, which are partially
ionized, as a scattering system; degree of donors
ionization depends on temperature. So, generally we
have both neutral and charged scattering centers; relation
between their concentrations depends on temperature.
We consider here only low temperatures and do not take
into account phonon scattering.
2. Scattering potential
2.1. Delta-shaped potential
The formulation “scattering of band electron on neutral
point defect” is completely conditional, because
Coulomb interaction of charged particle with really
neutral point object does not take place. Therefore
neutral scattering center has to be some compact
complex structure containing several different charges
and has to be neutral as a whole only. In this case range
of forces is practically limited by geometrical size of the
complex center.
Let us consider delta-shaped potential as the
simplest model of a neutral scattering center:
)()( rrI
. (2)
Fourier component of this potential is:
rdrrqiq II
3)()exp()( . (3)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 2. P. 214-220.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
215
Let us note that the value )(qI
does not depend
on wave vector q
.
2.2. Charged impurity
Fourier component of Coulomb potential generated by
charged impurity has the form (see [10, 11]):
)(
4
),(
2
L
I
IC
q
e
q
. (4)
Taking into account screening of potential of
scattering by band carriers, one can obtain:
)(
)]0,([
4
),(
2
qq
e
q
L
I
IC
. (5)
Here L is dielectric constant of lattice, ),( q
is dielectric function of band carriers.
Correlator of screened potentials is:
)(
)]0,([
32
42
23
,
2
qq
Ne
L
IC
qIC , (6),
where (see [10]):
Fig. 1
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 2. P. 214-220.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
216
kqk
qkk
ff
kdP
q
e
q
)()(
),0(
00
3
22
2
, (7)
and )(0
kf is equilibrium distribution function;
mkk 2/22 is dispersion line for this section (we
assume that it is simple isotropic parabolic relation).
Carrying out integration similar to that in Eq. (3),
we obtain:
)(,,
4
),(
1
1
2
1
c
q
q
q
e
q
L
I
IC
. (8)
Here /81 Tkmq B , TkBF / ,
Tkmec BL 8/2 , and
.
)exp(1/
/
ln
),,/(
0 1
1
1
2
1
1
z
dz
qqz
qqz
q
q
c
q
q
cqq
(9)
The case 0c corresponds to absence of
screening of Coulomb potential by band carriers. Figure
1 represents dependence of Fourier component of
potential generated by screened charged impurity on
dimensionless wave vector 1/ qq for different values of
dimensionless Fermi energy and screening constant c.
Comparison of these figures and dependence presented
by Eq. (3) shows that screened Coulomb potential
cannot imitate delta-shaped potential. The reason for that
is evident: the screening cuts Coulomb interaction at
long distances and is not important for short distance
interaction.
Using Eqs. (8) and (9), let us rewrite correlator (6)
in the following form:
.)(
)],,/([
32
)(
2
1
2
1
2
23
,
2
,
2
cqqq
Ne
L
IC
qICqIC
(10)
2.3. Hydrogen-shape neutral impurity
Let us consider donor impurity having the structure
similar to the spherically symmetrical hydrogen atom.
Space density of negative charge can be presented by
the following relation:
2
)()( e . (11)
Here )( )/exp( Br is wave function of
electron of shallow donor;
22 / mer LB (12)
rB is Bohr radius of exterior donor electron; m is
effective mass. The charge density )( is normalized
by the relation
0
2)(4 ed . (13)
Electrostatic potential of the positive kernel of
impurity atom in crystal is
.)(
r
e
r
L
(14)
Electrostatic potential generated by distributed
negative charge of exterior donor electron is
.
)/2exp(
)/2exp(
.)(
0
2
0
2
dr
dr
r
e
r
B
r
B
L
(15)
Total scattering potential of neutral center is:
.)/2exp(221
)/2exp(
)/2exp(
1
)()()(
2
2
0
2
0
2
B
BBL
B
r
B
L
IN
rr
r
r
r
r
r
e
r
r
r
e
rrr
(16)
Several examples for the space distribution of
potential )(rN are presented in Fig. 2. Here
2
0 /)(),( errrK INL ; the curves (a), (b), (c), (d)
refer to 4,2,1,0/0 Brr respectively. The value
of radius Br determines range of action for scattering
center.
Fourier component of potential (16) has the
following form:
Fig. 2.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 2. P. 214-220.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
217
Fig. 3.
.)()/(
4
)(
)(
22/3
1
4
),(
2
222
4
22
2
22
B
BL
B
B
B
B
BL
IN
qq
q
e
qq
q
qq
q
qq
e
q
(17)
Here
])1(2)1)(2/3(1[)1()( 221212 pppp
(see Fig. 3), and 22 /2/2 LBB merq .
Fig. 4.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 2. P. 214-220.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
218
One can see that wave vector Bq is natural
measure for distribution of scattering potential,
generated by neutral impurity, in q-space. For n-GaAs
we have: 0067.0 mm , and 5.12L ; therefore we
obtain: 2122 10108.4 cmqB . Hence it follows that
noticeable screening of the short-range potential (16) by
band electrons takes place in this crystal at
concentrations 31710 cmn . Assuming 0Bq (that is
Br ) in expression (17), we obtain form (4).
As follows from Eq. (17), the correlator of
scattering potentials for neutral centers is:
.)(
)(
22/3
1
)(
32
)(
2
222
4
22
2
2222
23
2
,
2
B
B
B
B
BL
IN
qINqIN
qq
q
qq
q
qq
Ne
(18)
Here INN is concentration of neutral impurities.
Due to short range of considered scattering center, there
is no need to involve screening of scattering potential by
band electrons into consideration.
3. Mobility of band carriers
Let us consider impurity system as partially ionized,
partially neutral donors. The degree of ionization depends
on temperature T. Let us write the relation between
concentrations of ionized and neutral impurities as
ICINIDNDD NNNNN . (19)
Below we shall assume that band electrons
concentration n is equal to concentration of ionized
donors
1
exp1
Tk
NNNn
B
DF
DICID . (20)
Here F is Fermi energy, 0D is energy level
for donors.
We calculate mobility of band carriers using the
formula (see Ref. [11])
INIC
1
. (21)
Here
qIC
B
IC
Tkmq
qdq
n
em
2
0
22
3
34
2
8/exp124
. (22)
qIN
B
IN
Tkmq
qdq
n
em
2
0
22
3
34
2
8/exp124
.
(23)
The value IC represents contribution of charged
impurities in reverse mobility of band carriers, the value
IN refers to neutral impurities.
We carried out the following numerical
calculations for set of AIII-BV-group crystals (see Ref.
[12] and Table 1):
Table 1
AIII-BV m/m0 L D (eV)
1 GaAs 0.067 12.5 0.008
2 GaSb 0.05 15 0.003
3 InP 0.07 14 0.008
4 InSb 0.013 17 0.0007
5 InAs 0.02 14 0.002
Results of calculations of mobility based on
formulae (21)(23) are presented in Fig. 4 (a l). Here
M
me
L
23
32
2
3
. (24)
Numbers on curves correspond to numbers in
Table 1.
Fig. 5
Fig. 6
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 2. P. 214-220.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
219
Temperature dependence of mobility for
considered crystal appears as a result of competition of
two processes: change of ionized centres number and
change of average energy of electrons. The first process
dominates at lower temperatures, the second one — at
higher temperatures. Therefore calculated dependence of
dimensionless mobility M on temperature T is non-
monotonous.
Figure 5 is presented here for comparison, which
reproduces Fig. 4.5(b) from Ref. [9]. Here curve 1 is
constructed on the base of Erginsoy’s theory [1], curves
2 and 3 — on the base of theoretical calculations of N.
Sclar [12] and T. McGill with R. Baron [13]
respectively. Our curves shown in Fig. 4 have the same
shape as curves 2 and 3 in Fig. 5.
Fig. 7.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 2. P. 214-220.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
220
Fig. 8.
4. Discussion
To compare contributions of neutral and charged
impurities in mobility, let us introduce border
temperature *T by the relationship
*)(*)( TT INIC . Connection between temperature
*T and donor concentration DN is presented in Fig. 6
by five lines (corresponding to five different crystals).
These lines divide the plane {ND, T} in two areas. In top
area (T > T*) the scattering on charged donor prevails; in
lower area (T < T*) scattering on neutral impurities
dominates. The numbers of curves correspond to
Table 1.
Let us now compare results obtained in this article
with results which can be obtained on the base of
calculations carried out on the base of tau-approximation
(see Ref. [5]). Result of comparison is shown in Fig. 7.
Here B-lines refer to the calculations of this article, A-
line are constructed with the help of corresponding
formulae represented in monograph of Anselm (see Ref.
[5]). One can see that their divergence is quite
noticeable.
In Fig. 8 our theoretical curve (solid line) and
experimental curve (dashed line) obtained for InSb by
H.J. Hrostowski et al. are presented (see Refs. [14, 15]).
It is seen that these lines are in gratifying agreement.
Acknowledgements
The essential help of Dr. E. B. Kaganovich is
gratefully acknowledged.
References
1. C. Erginsoy, Neutral impurity scattering // Phys.
Rev. 79, 1013, 1950.
2. K. Zeeger, Semiconductor Physics, Springer
Verlag, Wien, 1973.
3. V.L. Bonch-Bruevich, S. G. Kalashnikov: Physics
of semiconductors, Nauka, M., 1977 (in Russian).
4. P. Norton, T. Braggins, H. Levinstein. Impurity and
lattice scattering parameters as determined from
Hall and mobility analysis in n-type silicon. Phys.
Rev. B8, p. 5632 (1973).
5. A.I. Anselm, Introduction to the Theory of
Semiconductors. Nauka, Moscow, 1978 (in
Russian).
6. L.I. Shiff, Quantum Mechanics. McGraw-Hill,
New York, 1968.
7. I.I. Boiko, Specific thermoemf in crystals with
monopolar conductivity // Semiconductor Physics,
Quantum Electronics & Optoelectronics 12(1), p.
47-52 (2009).
8. I.I. Boiko, Electron-electron drag in crystals with
many-valley band // Semiconductor Physics,
Quantum Electronics & Optoelectronics 12(3), p.
212-217 (2009).
9. B.K. Ridley. Quantum processes in
semiconductors. Clarendon Press, Oxford, 1982.
10. I. I. Boiko: Kinetics of Electron Gas Interacting
with Fluctuating Potential. Naukova dumka, Kyiv,
1993 (in Russian).
11. I.I. Boiko: Transport of Carriers in
Semiconductors. To be published, Kyiv, 2009 (in
Russian).
12. N. Sclar, // Phys. Rev. 104, p. 1548, 1559 (1956).
13. T. McGill, R. Baron, // Phys. Rev. B11, p. 5208
(1975).
14. H.J. Horstowski et al, // Phys. Rev. 100, p. 1672,
(1955).
15. C. Hilsum, A.C. Rose-Innes: Semiconducting III-V-
compounds. Pergamon Press, 1961.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 2. P. 214-220.
PACS 71.20. 72.20 Dp
Band carriers scattering on impurities
I.I. Boiko
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
41, prospect Naukyt, 03028 Kyiv, Ukraine
E-mail: igorboiko@yandex.ru
Phone: 38 (044) 236-5422
Abstract. Mobility of band carriers scattered on donors, partially ionized, partially neutral, at low temperatures, is considered in general and calculated for AIII-BV group crystals. It is shown that temperature dependence of mobility is determined by relationship between number of ionized and neutral donors and by average energy of electrons.
Keywords: impurities, scattering potential, quantum kinetic equation, mobility.
Manuscript received 23.11.09; accepted for publication 25.03.10; published online 30.04.10.
1. Introduction
Investigations of scattering of band carriers by neutral impurities have no noticeable advance for a long time [1–5]. Several approaches to the problem of neutral impurity scattering were used, but only one received wide recognition. There was consideration of interaction of electrons with shallow neutral impurity, which imitates spherically symmetrical hydrogen atom. In this case relaxation time ( for momentum of carriers was constructed on the base of cross-section for scattering process [6]. This time has well-known Erginsoy’s form [1]
0
2
2
3
20
1
n
e
m
L
h
e
=
t
.
(1)
Here
L
e
is dielectric constant of lattice,
m
is effective mass,
0
n
is concentration of scattering centers. However, there are serious claims to the method of scalar relaxation time as a whole [7, 8]. The attempts to improve agreement of theory and experimental data by the way of introducing some adjusting factor in Eq. (1) (see, for instance, [4]) should be considered as very naive only.
Other direction of investigations uses model for scattering potential of neutral impurity as rectangular spherically isotropic hole [9]. Limit case of this model is delta-shaped function in space [10, 11]). In this case there is no possibility to evaluate amplitude of interaction. There is also no way to derive rectangular or delta-shaped potential as well-reasoned limit case of physically grounded interaction.
Bellow we shall consider mobility of band carriers, scattered by charged and neutral impurities; calculations will be based on quantum kinetic equation [10, 11]. For simplicity we use here only a model of simple isotropic parabolic dispersion law for band carriers.
We consider shallow donors, which are partially ionized, as a scattering system; degree of donors ionization depends on temperature. So, generally we have both neutral and charged scattering centers; relation between their concentrations depends on temperature. We consider here only low temperatures and do not take into account phonon scattering.
2. Scattering potential
2.1. Delta-shaped potential
The formulation “scattering of band electron on neutral point defect” is completely conditional, because Coulomb interaction of charged particle with really neutral point object does not take place. Therefore neutral scattering center has to be some compact complex structure containing several different charges and has to be neutral as a whole only. In this case range of forces is practically limited by geometrical size of the complex center.
Let us consider delta-shaped potential as the simplest model of a neutral scattering center:
)
(
)
(
r
r
I
v
v
d
U
=
j
.
(2)
Fourier component of this potential is:
U
=
j
-
=
j
ò
¥
¥
-
r
d
r
r
q
i
q
I
I
r
r
r
r
v
3
)
(
)
exp(
)
(
.
(3)
Let us note that the value
)
(
q
I
r
j
does not depend on wave vector
q
r
.
2.2. Charged impurity
Fourier component of Coulomb potential generated by charged impurity has the form (see [10, 11]):
)
(
4
)
,
(
2
w
d
e
p
=
w
j
L
I
I
C
q
e
q
r
.
(4)
Taking into account screening of potential of scattering by band carriers, one can obtain:
)
(
)]
0
,
(
[
4
)
,
(
2
w
d
e
D
+
e
p
=
w
j
q
q
e
q
L
I
I
C
r
r
.
(5)
Here
L
e
is dielectric constant of lattice,
)
,
(
w
e
D
q
r
is dielectric function of band carriers.
Correlator of screened potentials is:
)
(
)]
0
,
(
[
32
4
2
2
3
,
2
w
d
e
D
+
e
p
=
ñ
j
á
w
q
q
N
e
L
I
C
q
I
C
r
r
,
(6),
where (see [10]):
ò
e
-
e
e
-
e
p
=
e
D
-
-
k
q
k
q
k
k
f
f
k
d
P
q
e
q
r
r
r
r
r
r
r
r
)
(
)
(
)
,
0
(
0
0
3
2
2
2
, (7)
and
)
(
0
k
f
e
is equilibrium distribution function;
m
k
k
2
/
2
2
h
=
e
is dispersion line for this section (we assume that it is simple isotropic parabolic relation).
Carrying out integration similar to that in Eq. (3), we obtain:
)
(
,
,
4
)
,
(
1
1
2
1
w
d
ú
ú
û
ù
ê
ê
ë
é
÷
÷
ø
ö
ç
ç
è
æ
h
F
e
p
=
w
j
-
c
q
q
q
e
q
L
I
I
C
r
.
(8)
(8)
Here
h
/
8
1
T
k
m
q
B
=
,
T
k
B
F
/
e
=
h
,
T
k
m
e
c
B
L
h
e
p
=
8
/
2
, and
.
)
exp(
1
/
/
ln
)
,
,
/
(
0
1
1
1
2
1
1
h
-
+
÷
÷
ø
ö
ç
ç
è
æ
-
+
´
´
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
=
h
F
ò
¥
z
dz
q
q
z
q
q
z
q
q
c
q
q
c
q
q
(9)
The case
0
=
c
corresponds to absence of screening of Coulomb potential by band carriers. Figure 1 represents dependence of Fourier component of potential generated by screened charged impurity on dimensionless wave vector
1
/
q
q
for different values of dimensionless Fermi energy
h
and screening constant c. Comparison of these figures and dependence presented by Eq. (3) shows that screened Coulomb potential cannot imitate delta-shaped potential. The reason for that is evident: the screening cuts Coulomb interaction at long distances and is not important for short distance interaction.
Using Eqs. (8) and (9), let us rewrite correlator (6) in the following form:
.
)
(
)]
,
,
/
(
[
32
)
(
2
1
2
1
2
2
3
,
2
,
2
w
d
h
F
e
p
=
=
w
d
ñ
j
á
=
ñ
j
á
w
w
c
q
q
q
N
e
L
I
C
q
I
C
q
I
C
r
(10)
2.3. Hydrogen-shape neutral impurity
Let us consider donor impurity having the structure similar to the spherically symmetrical hydrogen atom. Space density of negative charge
c
can be presented by the following relation:
2
)
(
)
(
r
y
-
=
r
c
e
.
(11)
Here
)
(
r
y
(
)
/
exp(
B
r
r
-
is wave function of electron of shallow donor;
2
2
/
me
r
L
B
e
=
h
(12)
rB is Bohr radius of exterior donor electron;
m
is effective mass. The charge density
)
(
r
c
is normalized by the relation
ò
¥
-
=
r
r
r
c
p
0
2
)
(
4
e
d
.
(13)
Electrostatic potential of the positive kernel of impurity atom in crystal is
.
)
(
r
e
r
L
e
=
j
+
(14)
Electrostatic potential generated by distributed negative charge of exterior donor electron is
.
)
/
2
exp(
)
/
2
exp(
.
)
(
0
2
0
2
ò
ò
¥
-
r
r
-
r
r
r
-
r
e
-
=
j
d
r
d
r
r
e
r
B
r
B
L
(15)
Total scattering potential of neutral center is:
.
)
/
2
exp(
2
2
1
)
/
2
exp(
)
/
2
exp(
1
)
(
)
(
)
(
2
2
0
2
0
2
B
B
B
L
B
r
B
L
I
N
r
r
r
r
r
r
r
e
r
r
r
e
r
r
r
-
ú
ú
û
ù
ê
ê
ë
é
+
+
e
=
=
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ë
é
r
-
r
r
-
r
-
e
=
=
j
+
j
=
j
ò
ò
¥
-
+
(16)
Several examples for the space distribution of potential
)
(
r
N
j
are presented in Fig. 2. Here
2
0
/
)
(
)
,
(
e
r
r
r
K
I
N
L
j
e
=
g
; the curves (a), (b), (c), (d) refer to
4
,
2
,
1
,
0
/
0
=
=
g
B
r
r
respectively. The value of radius
B
r
determines range of action for scattering center.
Fourier component of potential (16) has the following form:
Fig. 2.
Fig. 3.
.
)
(
)
/
(
4
)
(
)
(
2
2
/
3
1
4
)
,
(
2
2
2
2
4
2
2
2
2
2
w
d
W
e
p
º
º
w
d
ú
ú
û
ù
ê
ê
ë
é
+
+
+
+
´
´
+
e
p
=
w
j
B
B
L
B
B
B
B
B
L
I
N
q
q
q
e
q
q
q
q
q
q
q
q
e
q
r
(17)
Here
]
)
1
(
2
)
1
)(
2
/
3
(
1
[
)
1
(
)
(
2
2
1
2
1
2
-
-
-
+
+
+
+
+
=
W
p
p
p
p
(see Fig. 3), and
2
2
/
2
/
2
h
L
B
B
me
r
q
e
=
=
.
One can see that wave vector
B
q
is natural measure for distribution of scattering potential, generated by neutral impurity, in q-space. For n-GaAs we have:
0
067
.
0
m
m
=
, and
5
.
12
=
e
L
; therefore we obtain:
2
12
2
10
108
.
4
-
×
=
cm
q
B
. Hence it follows that noticeable screening of the short-range potential (16) by band electrons takes place in this crystal at concentrations
3
17
10
-
>
cm
n
. Assuming
0
=
B
q
(that is
¥
®
B
r
) in expression (17), we obtain form (4).
As follows from Eq. (17), the correlator of scattering potentials for neutral centers is:
.
)
(
)
(
2
2
/
3
1
)
(
32
)
(
2
2
2
2
4
2
2
2
2
2
2
2
2
3
2
,
2
w
d
ú
ú
û
ù
ê
ê
ë
é
+
+
+
+
´
´
+
e
p
=
w
d
ñ
j
á
=
ñ
j
á
w
B
B
B
B
B
L
I
N
q
I
N
q
I
N
q
q
q
q
q
q
q
q
N
e
r
(18)
Here
I
N
N
is concentration of neutral impurities.
Due to short range of considered scattering center, there is no need to involve screening of scattering potential by band electrons into consideration.
3. Mobility of band carriers
Let us consider impurity system as partially ionized, partially neutral donors. The degree of ionization depends on temperature T. Let us write the relation between concentrations of ionized and neutral impurities as
I
C
I
N
I
D
N
D
D
N
N
N
N
N
+
=
+
=
. (19)
Below we shall assume that band electrons concentration
n
is equal to concentration of ionized donors
1
exp
1
-
ú
ú
û
ù
ê
ê
ë
é
÷
÷
ø
ö
ç
ç
è
æ
e
-
e
+
=
=
=
T
k
N
N
N
n
B
D
F
D
I
C
I
D
.
(20)
Here
F
e
is Fermi energy,
0
<
e
D
is energy level for donors.
We calculate mobility
m
of band carriers using the formula (see Ref. [11])
I
N
I
C
b
+
b
=
m
=
b
1
.
(21)
Here
(
)
q
I
C
B
I
C
T
k
m
q
q
d
q
n
em
ñ
j
á
h
-
+
p
=
b
ò
¥
2
0
2
2
3
3
4
2
8
/
exp
1
24
h
h
.
(22)
(
)
q
I
N
B
I
N
T
k
m
q
q
d
q
n
em
ñ
j
á
h
-
+
p
=
b
ò
¥
2
0
2
2
3
3
4
2
8
/
exp
1
24
h
h
.
(23)
The value
I
C
b
represents contribution of charged impurities in reverse mobility of band carriers, the value
I
N
b
refers to neutral impurities.
We carried out the following numerical calculations for set of AIII-BV-group crystals (see Ref. [12] and Table 1):
Table 1
AIII-BV
m/m0
(L
((D (eV)
1
GaAs
0.067
12.5
0.008
2
GaSb
0.05
15
0.003
3
InP
0.07
14
0.008
4
InSb
0.013
17
0.0007
5
InAs
0.02
14
0.002
Results of calculations of mobility based on formulae (21)((23) are presented in Fig. 4 (a ( l). Here
M
m
e
L
2
3
3
2
2
3
h
e
p
=
m
.
(24)
Numbers on curves correspond to numbers in Table 1.
Fig. 5
Fig. 6
Temperature dependence of mobility
m
for considered crystal appears as a result of competition of two processes: change of ionized centres number and change of average energy of electrons. The first process dominates at lower temperatures, the second one — at higher temperatures. Therefore calculated dependence of dimensionless mobility
M
on temperature T is non-monotonous.
Figure 5 is presented here for comparison, which reproduces Fig. 4.5(b) from Ref. [9]. Here curve 1 is constructed on the base of Erginsoy’s theory [1], curves 2 and 3 — on the base of theoretical calculations of N. Sclar [12] and T. McGill with R. Baron [13] respectively. Our curves shown in Fig. 4 have the same shape as curves 2 and 3 in Fig. 5.
Fig. 8.
4. Discussion
To compare contributions of neutral and charged impurities in mobility, let us introduce border temperature
*
T
by the relationship
*)
(
*)
(
T
T
I
N
I
C
b
=
b
. Connection between temperature
*
T
and donor concentration
D
N
is presented in Fig. 6 by five lines (corresponding to five different crystals). These lines divide the plane {ND, T} in two areas. In top area (T > T*) the scattering on charged donor prevails; in lower area (T < T*) scattering on neutral impurities dominates. The numbers of curves correspond to Table 1.
Let us now compare results obtained in this article with results which can be obtained on the base of calculations carried out on the base of tau-approximation (see Ref. [5]). Result of comparison is shown in Fig. 7. Here B-lines refer to the calculations of this article, A-line are constructed with the help of corresponding formulae represented in monograph of Anselm (see Ref. [5]). One can see that their divergence is quite noticeable.
In Fig. 8 our theoretical curve (solid line) and experimental curve (dashed line) obtained for InSb by H.J. Hrostowski et al. are presented (see Refs. [14, 15]). It is seen that these lines are in gratifying agreement.
Acknowledgements
The essential help of Dr. E. B. Kaganovich is gratefully acknowledged.
References
1.
C. Erginsoy, Neutral impurity scattering // Phys. Rev. 79, 1013, 1950.
2.
K. Zeeger, Semiconductor Physics, Springer Verlag, Wien, 1973.
3.
V.L. Bonch-Bruevich, S. G. Kalashnikov: Physics of semiconductors, Nauka, M., 1977 (in Russian).
4.
P. Norton, T. Braggins, H. Levinstein. Impurity and lattice scattering parameters as determined from Hall and mobility analysis in n-type silicon. Phys. Rev. B8, p. 5632 (1973).
5.
A.I. Anselm, Introduction to the Theory of Semiconductors. Nauka, Moscow, 1978 (in Russian).
6.
L.I. Shiff, Quantum Mechanics. McGraw-Hill, New York, 1968.
7.
I.I. Boiko, Specific thermoemf in crystals with monopolar conductivity // Semiconductor Physics, Quantum Electronics & Optoelectronics 12(1), p. 47-52 (2009).
8.
I.I. Boiko, Electron-electron drag in crystals with many-valley band // Semiconductor Physics, Quantum Electronics & Optoelectronics 12(3), p. 212-217 (2009).
9.
B.K. Ridley. Quantum processes in semiconductors. Clarendon Press, Oxford, 1982.
10.
I. I. Boiko: Kinetics of Electron Gas Interacting with Fluctuating Potential. Naukova dumka, Kyiv, 1993 (in Russian).
11.
I.I. Boiko: Transport of Carriers in Semiconductors. To be published, Kyiv, 2009 (in Russian).
12.
N. Sclar, // Phys. Rev. 104, p. 1548, 1559 (1956).
13.
T. McGill, R. Baron, // Phys. Rev. B11, p. 5208 (1975).
14.
H.J. Horstowski et al, // Phys. Rev. 100, p. 1672, (1955).
15.
C. Hilsum, A.C. Rose-Innes: Semiconducting III-V-compounds. Pergamon Press, 1961.
�
�
�
Fig. 7.
�
�
�
Fig. 4.
�
��
Fig. 1
a)�b) �
Fig. 9. Spectral dependences of the photocurrent on wavelength for Au/GaAs structures with thick (a) and thin (b) Au contacts. Curve 1 in the figure b is for structure with Au NPs and curve 2 without them, dotted curve is transmittance of the light into the GaAs substrate through the continuous film of Au with 21 nm thickness.
a)� b)�
Fig. 10. a) Calculated transmittance spectra of the light into the GaAs substrate with Au NPs on the top. The angle of light incidence is 0º. The numbers in the figure are the distance between NP in nm. Parameters for calculations: outer NP diameter is 55 nm, Au core diameter is 15 nm, shell is SiO2 with the refractive index n = 1.47. NPs are placed in triangular cell on GaAs substrate. b) Corresponding Au NPs absorption spectra.
�� � �
a
Surface distance 14.082 nm
Horiz distance 12.695 nm
Vert distance 4.418 nm
Angle 19.189 degree
Surface distance 10.923 nm
Horiz distance 10.742 nm
Vert distance 0.034 nm
Angle 0.182 degree
Surface distance 30.256 nm
Horiz distance 29.297 nm
Vert distance 3.780 nm
Angle 7.353 degree
b c
Fig. 1. Vertical film’s surfaces profile (a) with the indication of sizes between bench marks (b); the nuance of grey color corresponds to the nuance of bench marks; 3-D view of SnO2 film surface(c).
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
214
a
Surface distance 14.082 nm
Horiz distance 12.695 nm
Vert distance 4.418 nm
Angle 19.189 degree
Surface distance 10.923 nm
Horiz distance 10.742 nm
Vert distance 0.034 nm
Angle 0.182 degree
Surface distance 30.256 nm
Horiz distance 29.297 nm
Vert distance 3.780 nm
Angle 7.353 degree
b
c
Fig. 1. Vertical film’s surfaces profile (a) with the indication of sizes between bench marks (b); the nuance of grey color
corresponds to the nuance of bench marks; 3 -D view of SnO
2
film surface(c).
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