Seebeck’s effect in p-SiGe whisker samples
p-SiGe whisker samples with a diameter of ~40 μm, grown by chemical precipitation from the vapor phase, have been investigated. Temperature dependences of the thermal e.m.f. and conductivity within the temperature interval 20…400 K have been measured. It has been shown that the mobility of holes...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2011
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1177832017-05-27T03:05:15Z Seebeck’s effect in p-SiGe whisker samples Dolgolenko, A.P. Druzhinin, A.A. Karpenko, A.Ya. Nichkalo, S.I. Ostrovsky, I.P. Litovchenko, P.G. Litovchenko, A.P. p-SiGe whisker samples with a diameter of ~40 μm, grown by chemical precipitation from the vapor phase, have been investigated. Temperature dependences of the thermal e.m.f. and conductivity within the temperature interval 20…400 K have been measured. It has been shown that the mobility of holes in p - SiGe whiskers upon the average is 1.5 times higher than that in bulk p - Si samples. p - SiGe whiskers possess smaller phonon scattering and larger phonon dragging in comparison with the bulk p - Si samples. 2011 Article Seebeck’s effect in p-SiGe whisker samples / A.P. Dolgolenko, A.A. Druzhinin, A.Ya. Karpenko, S.I. Nichkalo, I.P. Ostrovsky, P.G. Litovchenko, A.P. Litovchenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 4. — С. 456-460 — Бібліогр.: 6 назв. — англ. 1560-8034 PACS 72.15.Jf, 72.20.Pa http://dspace.nbuv.gov.ua/handle/123456789/117783 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
p-SiGe whisker samples with a diameter of ~40 μm, grown by chemical
precipitation from the vapor phase, have been investigated. Temperature dependences of
the thermal e.m.f. and conductivity within the temperature interval 20…400 K have been
measured. It has been shown that the mobility of holes in p - SiGe whiskers upon the
average is 1.5 times higher than that in bulk p - Si samples. p - SiGe whiskers possess
smaller phonon scattering and larger phonon dragging in comparison with the bulk
p - Si samples. |
| format |
Article |
| author |
Dolgolenko, A.P. Druzhinin, A.A. Karpenko, A.Ya. Nichkalo, S.I. Ostrovsky, I.P. Litovchenko, P.G. Litovchenko, A.P. |
| spellingShingle |
Dolgolenko, A.P. Druzhinin, A.A. Karpenko, A.Ya. Nichkalo, S.I. Ostrovsky, I.P. Litovchenko, P.G. Litovchenko, A.P. Seebeck’s effect in p-SiGe whisker samples Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Dolgolenko, A.P. Druzhinin, A.A. Karpenko, A.Ya. Nichkalo, S.I. Ostrovsky, I.P. Litovchenko, P.G. Litovchenko, A.P. |
| author_sort |
Dolgolenko, A.P. |
| title |
Seebeck’s effect in p-SiGe whisker samples |
| title_short |
Seebeck’s effect in p-SiGe whisker samples |
| title_full |
Seebeck’s effect in p-SiGe whisker samples |
| title_fullStr |
Seebeck’s effect in p-SiGe whisker samples |
| title_full_unstemmed |
Seebeck’s effect in p-SiGe whisker samples |
| title_sort |
seebeck’s effect in p-sige whisker samples |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117783 |
| citation_txt |
Seebeck’s effect in p-SiGe whisker samples / A.P. Dolgolenko, A.A. Druzhinin, A.Ya. Karpenko, S.I. Nichkalo, I.P. Ostrovsky, P.G. Litovchenko, A.P. Litovchenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 4. — С. 456-460 — Бібліогр.: 6 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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2025-07-08T12:47:35Z |
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2025-07-08T12:47:35Z |
| _version_ |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 456-460.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
456
PACS 72.15.Jf, 72.20.Pa
Seebeck’s effect in p-SiGe whisker samples
A.P. Dolgolenko1, A.A. Druzhinin2, A.Ya. Karpenko1, S.I. Nichkalo2, I.P. Ostrovsky2,
P.G. Litovchenko1, A.P. Litovchenko1
1Institute for Nuclear Researches, NAS of Ukraine, Kyiv
E-mail: Odolgolenko@kinr.kiev.ua; ayak@kinr.kiev.ua
2Scientific Center ‘L’vivs’ka Politechknika”, L’viv
E-mail: iostrov@polynet.lviv.ua
Abstract. p-SiGe whisker samples with a diameter of ~40 μm, grown by chemical
precipitation from the vapor phase, have been investigated. Temperature dependences of
the thermal e.m.f. and conductivity within the temperature interval 20…400 K have been
measured. It has been shown that the mobility of holes in SiGep whiskers upon the
average is 1.5 times higher than that in bulk Sip samples. SiGep whiskers possess
smaller phonon scattering and larger phonon dragging in comparison with the bulk
Sip samples.
Keywords: whisker semiconductor crystals, Seebeck’s effect, thermal e.m.f.,
thermoelectric effect, phonon drag, carrier scattering.
Manuscript received 11.07.11; revised manuscript received 15.08.11; accepted for
publication 14.09.11; published online 30.11.11.
1. Introduction
In recent years, nanotechnology achieved considerable
progress in the growth of semiconductor whisker
crystals [1] and creation of miniature sensors [2] that
have to work not only in the aggressive media but also in
high magnetic and radiation fields. Therefore,
determination of radiation and magnetic-field hardness
requires a reliable control of the electrophysical
parameters of whisker materials and instruments based
on them. It should be noted that in spite of significant
experience in studying the thermal e.m.f. as well as other
effects with high prospects of using xx1 GeSi as
thermoelectric material, the Seebeck effect of whisker
semiconductors is investigated insufficiently. It is related
with the fact that it is very difficult to calculate
contribution of mutual dragging the quasi-particles in
thermoelectric effects [3]. Difficulties exist also in
measurements of the Hall effect in whisker samples.
This forces to approach more correctly to measurements
of thermal e.m.f. and, all the more, to calculation of the
integral of collisions in the specific case. All this will
make it possible to reliably determine the temperature
dependence of carrier concentrations in the whisker
materials and to determine the performance
characteristics of sensors. In the case of bulk Sip
samples, measurements of the Hall effect [4] and
determination of the thermal e.m.f. in these materials [5]
were carried out reliably. The influence of the effects of
carrier dragging by phonons, their scattering by phonons
and at the boundaries of surface, and their influence on
the measured temperature dependence of thermal e.m.f.
were investigated.
2. Experimental details and results
Investigated in this work were SiGep whisker
samples with the specific resistance ρ = 0.018 –
0.03 Ohm·cm grown by the method of gas transport
reactions in the closed bromide system from the source
materials Si and Ge. The composition of solid solutions
was controlled using the method of microprobe analysis
and differed within the limits no more than 3%. The
crystals for investigations with the length of 10…20 mm
and with the transverse sizes of approximately 20 to
30 μm were selected. Our measurements of Seebeck’s
coefficient was carried out using the method of the
temperature gradient in whisker crystals created with the
aid of current transmission through the part of the
sample with the length ~0.3 to 0.5 mm. The temperature
of cold end was measured by AuPt thermocouple,
while the temperature of the heated part of the sample
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 456-460.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
457
was determined from the dependence R(T) [6]. The
accuracy of absolute thermal e.m.f. determination when
measuring the e.m.f. on Pt contacts proved to be ~3%.
Measurements of the conductivity with the accuracy 1%
and thermal e.m.f. 3% were performed within the
temperature interval 5 to 400 K with the gradient
1…3 K. The results of the carried out measurements and
calculations have been depicted in Figs 1 to 5. Shown in
Figs 1 and 2 are the experimental temperature
dependences for the thermal e.m.f. in bulk Sip
samples and whisker samples of SiGep . Also shown
are the computed values corresponding to them and
obtained with account of interaction between carriers
and phonons, determined by the integral of scattering.
Represented in Figs 3 and 4 are the temperature
dependences of the hole concentration in the bulk and
whisker samples, obtained by determining the Fermi
integral of ½ degree from the temperature dependences
of the thermal e.m.f., multiplied by the integral of
scattering. Shown in Fig. 5 are the dependences of the
hole mobility on temperature for two whisker SiGep
samples.
3. Theoretical analysis
The differential thermal e.m.f. () for semiconductor
with one type of charge carriers is defined as:
kT
Q
e
k
,
kT
F
,
where the sign coincides with the sign of charge carrier,
η is the reduced Fermi level, k – Boltzmann constant, T –
absolute temperature, Q* – transfer energy:
r
r
F
F
rkTQ 12 .
The value r is determined by a carrier scattering
mechanism. At temperatures higher than the Debye one
r = 1. Then:
r
r
F
F
r
e
k 12 ,
where Fr+1(η) and Fr(η) are the Fermi integrals of r
degree. In the case 1 , the Fermi integrals are
equal:
2
83
2 21
2
23
21F ;
32
1 2
2
1F ;
2
1
3
1 23
2 F .
But within the interval 1 for the integral
F1/2, it is correct to use the approximation:
exp
421F .
If scattering is determined by the only one
mechanism of scattering, then
n
N
p
e
k Vln
2
5
,
where p is the exponent of the energy dependence for the
carrier relaxation time, n – carrier concentration in the
sample, NV – density of states in the valence band. Then,
it is possible to construct a matrix and describe the
dependence F1/2 on the absolute thermal e.m.f. (Table),
as the thermal e.m.f. measurements were carried out
using Pt contacts.
Knowing F1/2, we can obtain the concentration of
carriers in a sample:
21
2
FNn V
.
According to the condition of the electroneutrality
of the extrinsic semiconductor, the concentration of
holes in dependence on temperature is equal:
,exp
2
1
exp
4
1
exp
2/1
2
d
a
V
a
aV
d
a
V
N
kT
E
TN
kT
E
NTN
N
kT
E
TNTn
,
01.0
exp51
1
300
105.2
23
2319
kT
T
mTN pV
(1)
where β is the degree of degeneracy of the boron
acceptor level (1/2); Na, Ea – concentration value and
energy position of boron levels in Sip .
Within the region of mixed scattering, the high
level of doping leads to the fact that the best adjustment
to the temperature dependence of the carrier
concentration is obtained with r = 1. With carrier
scattering by acoustic phonons 2/1r , and by
ionized impurities r = 3/2. Interaction of carriers with
phonons is determined by the integral of collisions:
p
ep
e
epep SSS .
The integral e
epS describes collisions with the
equilibrium phonons. The term p
epS considers the non-
equilibrium of phonons, and it is critical for drag of
carriers by phonons. Scattering the holes by phonons
gives not only mobility but also thermal e.m.f. to the
dependence 3.2T . Within the range of low
temperatures, thermal e.m.f. is proportional to the
specific heat capacity per unit of volume. Since the
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 456-460.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
458
Fig. 1. Temperature dependences of the thermal e.m.f. for p-type
silicon doped with boron in the samples with the sizes
0.12×0.16×1.1 cm [5]: 1 – experiment; 2 – with account of
phonon drag and carrier scattering by phonons; 3 – ~ T–2.32.
Fig. 2. Temperature dependences of the thermal e.m.f. for
p–SiGe doped with boron in the samples with the sizes: d =
40 μm, l = 1 cm: 1 – experiment; 2 – with account of phonon
drag and carrier scattering by phonons; 3 – 7.510–8 T–2.32.
specific heat capacity at temperatures considerably
smaller than the Debye temperature is proportional to Т3,
then (Т) will also be proportional to Т3. Therefore, we
will define the integral of collisions as:
32.23 ThTg
d
l
Sep . (2)
The contribution from the phonon drag to the
thermal e.m.f. of crystal should contain the parameter of
l /d, where d is the transverse size of a sample, l – path
length of carriers. At very low temperatures, phonon
scattering on the crystal boundaries is the factor that
limits the drag effect. On the other hand, in the near-
surface regions of crystal, there are gradients of non-
equilibrium carrier concentrations that collapse to the
bulk ones at the distances of the order of diffusion
length, which from the viewpoint of phonon drag should
be treated as the concentration de-compensation.
Phonon-phonon scattering also decreases the
effectiveness of the drag effect. For this very reason, the
carrier drag by phonons is negligibly small and in the
experiments with thermal e.m.f. measurements is weakly
pronounced near room temperatures, and at low
temperatures the thermal e.m.f. is reduced with
decreasing the crystal thickness.
Table. Technique of account of 2/1F dependence on .
8.8442 100.6720
7.4100 110.9355
6.5115 118.9494
5.4580 130.3624
4.4876 143.7049
3.6070 159.3120
2.8237 177.5778
2.1449 198.8875
1.5756 223.5701
1.1173 251.7649
0.7652 283.3738
0.5075 318.0719
0.3278 355.3125
0.2074 394.4944
0.1293 435.0507
0.0798 476.5373
0.0489 518.6230
0.0299 561.0935
0.0182 603.8098
0.00674 775.5611
0.00034 1034.0815
F1/2 :=
8·10–7
:=
1551.1223
4. Discussion
For p-type silicon samples, the temperature dependence of
the hole concentration [4] and thermal e.m.f. [5] were
experimentally obtained. After describing the carrier
concentration in Sip depending on temperature, Fig. 3,
and using the equation of electroneutrality (1), we
obtained the values of the ionization energy
(EV + 0.034 eV) and boron atom concentration
( 318
B cm105 aNN ) in the forbidden band of
silicon. Reduction in the ionization energy of boron in
silicon in comparison with the value (EV+0.046 eV)
accepted attests to the fact that the impurity band is
formed with this level of doping in the forbidden band of
silicon. In this case, the ionization energy of the doping
impurity is reduced. Formation of the impurity band is
confirmed by a low value of the carrier mobility and by
the fact that the thermal e.m.f. at T < 20 K reverses the
sign to the opposite one [4, 5]. The integral of collisions is
calculated by selecting the coefficients in the
expression (2) in such manner that the determined values
of the Fermi integral of degree ½ from its functional
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 456-460.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
459
dependence on the thermal e.m.f. (F1/2()) make it
possible to reliably determine the concentration of holes
(Fig. 3), which coincides with the experimental
temperature dependence [4]. The values of the
coefficients: l /d = 0.17; g = 1.010–8; h = 3.7·103 have
such dimensionality that the integral of collisions is
dimensionless. The effect of the integral of collisions on
the experimentally determined thermal e.m.f. is shown in
Fig. 1.
Fig. 3. Temperature dependences of the hole concentration in
the silicon sample doped with boron: – taken from the
work [4]; solid line – calculation of p(T) according to the
equation (1); □ – р(Т) NV(T)·F1/2.
Fig. 4. Temperature dependences of the hole concentration in
the p–SiGe whisker samples doped with boron: solid line –
calculation of р(Т) according to Eq.(1); , □ –
р(Т) NV(T)·F1/2.
Fig. 5. Temperature dependences of the hole drift mobility in
the boron doped p–SiGe whisker samples with the specific
resistances: 0.03 Ohm·cm (1), 0.018 Ohm·cm (2).
Unfortunately, measurements of the Hall-effect in
SiGep whisker samples are not currently possible.
Therefore, after being certified in the reliability of the
function F1/2(), and after determining the coefficients in
the integral of collisions: l /d = 0.21; 8105.3 g ;
h = 6.5·103, the temperature dependence of the hole
concentration in SiGep (Fig. 4) was calculated. The
influence of the integral of collisions on the
experimentally measured thermal e.m.f. is shown in
Fig. 2. The temperature dependence of the calculated
concentration of holes was described using the
equation (1). With the boron atom concentration
318 cm105 aN and concentration of the shallow
compensating centers 315 cm105 dN , the ionization
energy of boron atoms is equal (EV + 0.038 eV). An
increase of the ionization energy of boron atoms in
SiGep whisker samples indicates the smaller energy
extent of impurity band in the whisker samples in
comparison with that in bulk Sip samples. In the
whisker samples, in comparison with the bulk ones, the
coefficients in the integral of collisions have increased.
This only can mean that in the whisker samples, as
compared with the bulk Sip ones, the lower value of
the maximum thermal e.m.f. due to the lower cross-
section of samples is observed. An increase of g in the
whisker samples does testify that phonons transfer high
energy to charge carriers, and thus they strengthen the
phonon drag. The increase of h in the whisker samples
testifies about the decrease of carrier scattering by
phonons. Actually, in the whisker SiGep samples, in
comparison with bulk Sip , the mobility of holes is
increased, which indicates the decrease of charge carrier
scattering by phonons in the whisker samples (Fig. 5).
5. Conclusion
The concentration of boron atoms and mobility of charge
carriers in the whisker p – SiGe samples have been
determined. It is shown that on the average the mobility
of holes is ~1.5 times higher in the whisker p – SiGe
samples in comparison with the bulk p – Si ones. Using
selection of coefficients in the integral of collisions,
values of the thermal e.m.f. corresponding to the charge
carrier concentration have been calculated. The great
magnitude of the integral of collisions in the whisker
samples, in comparison with that in the bulk Sip
samples, is indicative of a decrease in carrier scattering
by phonons and increase in the phonon drag in SiGep
whisker samples.
References
1. V.G. Dubrovsky, G.E. Tsyrlin, V.M. Ustinov,
Semiconductor whisker nanocrystals: synthesis,
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 456-460.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
460
properties, application // Fizika tekhnika i
poluprovodnikov, 43(12), p. 1585-1628 (2009), in
Russian.
2. Ya.S. Budzhak, S.S. Varshava, I.P. Ostrovskii,
Theoretical and experimental base of Seebeck
elements design by use of semiconductor whiskers
// 8th Intern. Symp. on Temperature and Thermal
Measurements in Industry and Science:
Proceedings. Berlin, Germany, p. 121-126 (2001).
3. Yu.V. Ivanov, V.K. Zaitsev, M.I. Fedorov,
Contribution of non-equilibrium optical phonons to
the Peltier and Seebeck effects in polar
semiconductors // Fizika tverdogo tela, 40(7),
p. 1209-1215 (1998), in Russian.
4. F.J. Morin and J.P. Maita, Electrical properties of
silicon containing arsenic and boron // Phys. Rev.
96(1), p. 28-35 (1954).
5. T.H. Geballe and G.W. Hull, Seebeck effect in
silicon // Phys. Rev. 98(4), p. 940-947 (1955).
6. A.O. Druzhinin, I.P. Ostrovskii, N.S. Liakh,
S.M. Matviyenko, The way to determine the
coefficient of thermal e.m.f. for whiskers crystals //
Declaration Patent for Useful Model №3531 from
15.11.2004. Certificate №2004042607 from
06.04.2004. Bulletin №11 from 15.11.2004 (in
Ukrainian).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 456-460.
PACS 72.15.Jf, 72.20.Pa
Seebeck’s effect in p-SiGe whisker samples
A.P. Dolgolenko1, A.A. Druzhinin2, A.Ya. Karpenko1, S.I. Nichkalo2, I.P. Ostrovsky2,
P.G. Litovchenko1, A.P. Litovchenko1
1Institute for Nuclear Researches, NAS of Ukraine, Kyiv
E-mail: Odolgolenko@kinr.kiev.ua; ayak@kinr.kiev.ua
2Scientific Center ‘L’vivs’ka Politechknika”, L’viv
E-mail: iostrov@polynet.lviv.ua
Abstract. p-SiGe whisker samples with a diameter of ~40 μm, grown by chemical precipitation from the vapor phase, have been investigated. Temperature dependences of the thermal e.m.f. and conductivity within the temperature interval 20…400 K have been measured. It has been shown that the mobility of holes in
SiGe
-
p
whiskers upon the average is 1.5 times higher than that in bulk
Si
-
p
samples.
SiGe
-
p
whiskers possess smaller phonon scattering and larger phonon dragging in comparison with the bulk
Si
-
p
samples.
Keywords: whisker semiconductor crystals, Seebeck’s effect, thermal e.m.f., thermoelectric effect, phonon drag, carrier scattering.
Manuscript received 11.07.11; revised manuscript received 15.08.11; accepted for publication 14.09.11; published online 30.11.11.
1. Introduction
In recent years, nanotechnology achieved considerable progress in the growth of semiconductor whisker crystals [1] and creation of miniature sensors [2] that have to work not only in the aggressive media but also in high magnetic and radiation fields. Therefore, determination of radiation and magnetic-field hardness requires a reliable control of the electrophysical parameters of whisker materials and instruments based on them. It should be noted that in spite of significant experience in studying the thermal e.m.f. as well as other effects with high prospects of using
x
x
1
Ge
Si
-
as thermoelectric material, the Seebeck effect of whisker semiconductors is investigated insufficiently. It is related with the fact that it is very difficult to calculate contribution of mutual dragging the quasi-particles in thermoelectric effects [3]. Difficulties exist also in measurements of the Hall effect in whisker samples. This forces to approach more correctly to measurements of thermal e.m.f. and, all the more, to calculation of the integral of collisions in the specific case. All this will make it possible to reliably determine the temperature dependence of carrier concentrations in the whisker materials and to determine the performance characteristics of sensors. In the case of bulk
Si
-
p
samples, measurements of the Hall effect [4] and determination of the thermal e.m.f. in these materials [5] were carried out reliably. The influence of the effects of carrier dragging by phonons, their scattering by phonons and at the boundaries of surface, and their influence on the measured temperature dependence of thermal e.m.f. were investigated.
2. Experimental details and results
Investigated in this work were
SiGe
-
p
whisker samples with the specific resistance ρ = 0.018 – 0.03 Ohm·cm grown by the method of gas transport reactions in the closed bromide system from the source materials Si and Ge. The composition of solid solutions was controlled using the method of microprobe analysis and differed within the limits no more than 3%. The crystals for investigations with the length of 10…20 mm and with the transverse sizes of approximately 20 to 30 μm were selected. Our measurements of Seebeck’s coefficient was carried out using the method of the temperature gradient in whisker crystals created with the aid of current transmission through the part of the sample with the length ~0.3 to 0.5 mm. The temperature of cold end was measured by
Au
Pt
-
thermocouple, while the temperature of the heated part of the sample was determined from the dependence R(T) [6]. The accuracy of absolute thermal e.m.f. determination when measuring the e.m.f. on Pt contacts proved to be ~3%. Measurements of the conductivity with the accuracy 1% and thermal e.m.f. 3% were performed within the temperature interval 5 to 400 K with the gradient 1…3 K. The results of the carried out measurements and calculations have been depicted in Figs 1 to 5. Shown in Figs 1 and 2 are the experimental temperature dependences for the thermal e.m.f. in bulk
Si
-
p
samples and whisker samples of
SiGe
-
p
. Also shown are the computed values corresponding to them and obtained with account of interaction between carriers and phonons, determined by the integral of scattering. Represented in Figs 3 and 4 are the temperature dependences of the hole concentration in the bulk and whisker samples, obtained by determining the Fermi integral of ½ degree from the temperature dependences of the thermal e.m.f., multiplied by the integral of scattering. Shown in Fig. 5 are the dependences of the hole mobility on temperature for two whisker
SiGe
-
p
samples.
3. Theoretical analysis
The differential thermal e.m.f. (() for semiconductor with one type of charge carriers is defined as:
÷
÷
ø
ö
ç
ç
è
æ
h
-
±
=
a
*
kT
Q
e
k
,
kT
F
=
h
,
where the sign coincides with the sign of charge carrier, η is the reduced Fermi level, k – Boltzmann constant, T – absolute temperature, Q* – transfer energy:
(
)
(
)
(
)
h
h
+
=
+
*
r
r
F
F
r
kT
Q
1
2
.
The value r is determined by a carrier scattering mechanism. At temperatures higher than the Debye one r = 1. Then:
(
)
(
)
(
)
ú
û
ù
ê
ë
é
h
-
h
h
+
=
a
+
r
r
F
F
r
e
k
1
2
m
,
where Fr+1(η) and Fr(η) are the Fermi integrals of r degree. In the case
1
-
>
h
, the Fermi integrals are equal:
p
÷
÷
ø
ö
ç
ç
è
æ
h
p
+
h
=
-
2
8
3
2
2
1
2
2
3
2
1
F
;
÷
÷
ø
ö
ç
ç
è
æ
p
+
h
=
3
2
1
2
2
1
F
;
(
)
2
1
3
1
2
3
2
h
p
+
h
=
F
.
But within the interval
-¥
>
h
>
-
1
for the integral F1/2, it is correct to use the approximation:
(
)
h
÷
÷
ø
ö
ç
ç
è
æ
p
=
exp
4
2
1
F
.
If scattering is determined by the only one mechanism of scattering, then
ú
û
ù
ê
ë
é
+
+
=
a
n
N
p
e
k
V
ln
2
5
,
where p is the exponent of the energy dependence for the carrier relaxation time, n – carrier concentration in the sample, NV – density of states in the valence band. Then, it is possible to construct a matrix and describe the dependence F1/2 on the absolute thermal e.m.f. (Table), as the thermal e.m.f. measurements were carried out using Pt contacts.
Knowing F1/2, we can obtain the concentration of carriers in a sample:
2
1
2
F
N
n
V
p
=
.
According to the condition of the electroneutrality of the extrinsic semiconductor, the concentration of holes in dependence on temperature is equal:
(
)
(
)
(
)
(
)
,
exp
2
1
exp
4
1
exp
2
/
1
2
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
-
ú
û
ù
÷
÷
ø
ö
ç
ç
è
æ
+
+
ê
ê
ë
é
×
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
=
d
a
V
a
a
V
d
a
V
N
kT
E
T
N
kT
E
N
T
N
N
kT
E
T
N
T
n
b
b
b
(
)
,
01
.
0
exp
5
1
1
300
10
5
.
2
2
3
2
3
19
÷
÷
ø
ö
ç
ç
è
æ
÷
ø
ö
ç
è
æ
-
+
´
´
÷
ø
ö
ç
è
æ
×
×
×
=
kT
T
m
T
N
p
V
(1)
where β is the degree of degeneracy of the boron acceptor level (1/2); Na, Ea – concentration value and energy position of boron levels in
Si
-
p
.
Within the region of mixed scattering, the high level of doping leads to the fact that the best adjustment to the temperature dependence of the carrier concentration is obtained with r = 1. With carrier scattering by acoustic phonons
2
/
1
-
=
r
, and by ionized impurities r = 3/2. Interaction of carriers with phonons is determined by the integral of collisions:
p
ep
e
ep
ep
S
S
S
+
=
.
The integral
e
ep
S
describes collisions with the equilibrium phonons. The term
p
ep
S
considers the non-equilibrium of phonons, and it is critical for drag of carriers by phonons. Scattering the holes by phonons gives not only mobility but also thermal e.m.f. to the dependence ( (
3
.
2
-
T
. Within the range of low temperatures, thermal e.m.f. is proportional to the specific heat capacity per unit of volume. Since the specific heat capacity at temperatures considerably smaller than the Debye temperature is proportional to Т3, then ((Т) will also be proportional to Т3. Therefore, we will define the integral of collisions as:
32
.
2
3
-
×
+
×
+
=
T
h
T
g
d
l
S
ep
.
(2)
The contribution from the phonon drag to the thermal e.m.f. of crystal should contain the parameter of l /d, where d is the transverse size of a sample, l – path length of carriers. At very low temperatures, phonon scattering on the crystal boundaries is the factor that limits the drag effect. On the other hand, in the near-surface regions of crystal, there are gradients of non-equilibrium carrier concentrations that collapse to the bulk ones at the distances of the order of diffusion length, which from the viewpoint of phonon drag should be treated as the concentration de-compensation. Phonon-phonon scattering also decreases the effectiveness of the drag effect. For this very reason, the carrier drag by phonons is negligibly small and in the experiments with thermal e.m.f. measurements is weakly pronounced near room temperatures, and at low temperatures the thermal e.m.f. is reduced with decreasing the crystal thickness.
Table. Technique of account of
2
/
1
F
dependence on (.
F1/2 :=
8.8442
( :=
100.6720
7.4100
110.9355
6.5115
118.9494
5.4580
130.3624
4.4876
143.7049
3.6070
159.3120
2.8237
177.5778
2.1449
198.8875
1.5756
223.5701
1.1173
251.7649
0.7652
283.3738
0.5075
318.0719
0.3278
355.3125
0.2074
394.4944
0.1293
435.0507
0.0798
476.5373
0.0489
518.6230
0.0299
561.0935
0.0182
603.8098
0.00674
775.5611
0.00034
1034.0815
8·10–7
1551.1223
4. Discussion
For p-type silicon samples, the temperature dependence of the hole concentration [4] and thermal e.m.f. [5] were experimentally obtained. After describing the carrier concentration in
Si
-
p
depending on temperature, Fig. 3, and using the equation of electroneutrality (1), we obtained the values of the ionization energy (EV + 0.034 eV) and boron atom concentration (
3
18
B
cm
10
5
-
×
=
=
a
N
N
) in the forbidden band of silicon. Reduction in the ionization energy of boron in silicon in comparison with the value (EV+0.046 eV) accepted attests to the fact that the impurity band is formed with this level of doping in the forbidden band of silicon. In this case, the ionization energy of the doping impurity is reduced. Formation of the impurity band is confirmed by a low value of the carrier mobility and by the fact that the thermal e.m.f. at T < 20 K reverses the sign to the opposite one [4, 5]. The integral of collisions is calculated by selecting the coefficients in the expression (2) in such manner that the determined values of the Fermi integral of degree ½ from its functional dependence on the thermal e.m.f. (F1/2(()) make it possible to reliably determine the concentration of holes (Fig. 3), which coincides with the experimental temperature dependence [4]. The values of the coefficients: l /d = 0.17; g = 1.0(10–8; h = 3.7·103 have such dimensionality that the integral of collisions is dimensionless. The effect of the integral of collisions on the experimentally determined thermal e.m.f. is shown in Fig. 1.
Fig. 3. Temperature dependences of the hole concentration in the silicon sample doped with boron: ( – taken from the work [4]; solid line – calculation of p(T) according to the equation (1); □ – р(Т) ( NV(T)·F1/2.
Fig. 4. Temperature dependences of the hole concentration in the p–SiGe whisker samples doped with boron: solid line – calculation of р(Т) according to Eq.(1); (, □ – р(Т) ( NV(T)·F1/2.
Fig. 5. Temperature dependences of the hole drift mobility in the boron doped p–SiGe whisker samples with the specific resistances: 0.03 Ohm·cm (1), 0.018 Ohm·cm (2).
Unfortunately, measurements of the Hall-effect in
SiGe
-
p
whisker samples are not currently possible. Therefore, after being certified in the reliability of the function F1/2((), and after determining the coefficients in the integral of collisions: l /d = 0.21;
8
10
5
.
3
-
×
=
g
; h = 6.5·103, the temperature dependence of the hole concentration in
SiGe
-
p
(Fig. 4) was calculated. The influence of the integral of collisions on the experimentally measured thermal e.m.f. is shown in Fig. 2. The temperature dependence of the calculated concentration of holes was described using the equation (1). With the boron atom concentration
3
18
cm
10
5
-
×
=
a
N
and concentration of the shallow compensating centers
3
15
cm
10
5
-
×
=
d
N
, the ionization energy of boron atoms is equal (EV + 0.038 eV). An increase of the ionization energy of boron atoms in
SiGe
-
p
whisker samples indicates the smaller energy extent of impurity band in the whisker samples in comparison with that in bulk
Si
-
p
samples. In the whisker samples, in comparison with the bulk ones, the coefficients in the integral of collisions have increased. This only can mean that in the whisker samples, as compared with the bulk
Si
-
p
ones, the lower value of the maximum thermal e.m.f. due to the lower cross-section of samples is observed. An increase of g in the whisker samples does testify that phonons transfer high energy to charge carriers, and thus they strengthen the phonon drag. The increase of h in the whisker samples testifies about the decrease of carrier scattering by phonons. Actually, in the whisker
SiGe
-
p
samples, in comparison with bulk
Si
-
p
, the mobility of holes is increased, which indicates the decrease of charge carrier scattering by phonons in the whisker samples (Fig. 5).
5. Conclusion
The concentration of boron atoms and mobility of charge carriers in the whisker p – SiGe samples have been determined. It is shown that on the average the mobility of holes is ~1.5 times higher in the whisker p – SiGe samples in comparison with the bulk p – Si ones. Using selection of coefficients in the integral of collisions, values of the thermal e.m.f. corresponding to the charge carrier concentration have been calculated. The great magnitude of the integral of collisions in the whisker samples, in comparison with that in the bulk
Si
-
p
samples, is indicative of a decrease in carrier scattering by phonons and increase in the phonon drag in
SiGe
-
p
whisker samples.
References
1. V.G. Dubrovsky, G.E. Tsyrlin, V.M. Ustinov, Semiconductor whisker nanocrystals: synthesis, properties, application // Fizika tekhnika i poluprovodnikov, 43(12), p. 1585-1628 (2009), in Russian.
2. Ya.S. Budzhak, S.S. Varshava, I.P. Ostrovskii, Theoretical and experimental base of Seebeck elements design by use of semiconductor whiskers // 8th Intern. Symp. on Temperature and Thermal Measurements in Industry and Science: Proceedings. Berlin, Germany, p. 121-126 (2001).
3. Yu.V. Ivanov, V.K. Zaitsev, M.I. Fedorov, Contribution of non-equilibrium optical phonons to the Peltier and Seebeck effects in polar
semiconductors // Fizika tverdogo tela, 40(7), p. 1209-1215 (1998), in Russian.
4. F.J. Morin and J.P. Maita, Electrical properties of silicon containing arsenic and boron // Phys. Rev. 96(1), p. 28-35 (1954).
5. T.H. Geballe and G.W. Hull, Seebeck effect in silicon // Phys. Rev. 98(4), p. 940-947 (1955).
6. A.O. Druzhinin, I.P. Ostrovskii, N.S. Liakh, S.M. Matviyenko, The way to determine the coefficient of thermal e.m.f. for whiskers crystals // Declaration Patent for Useful Model №3531 from 15.11.2004. Certificate №2004042607 from 06.04.2004. Bulletin №11 from 15.11.2004 (in Ukrainian).
�
Fig. 1. Temperature dependences of the thermal e.m.f. for p-type silicon doped with boron in the samples with the sizes 0.12×0.16×1.1 cm [5]: 1 – experiment; 2 – with account of phonon drag and carrier scattering by phonons; 3 – ( ~ T–2.32.
�
Fig. 2. Temperature dependences of the thermal e.m.f. for �p–SiGe doped with boron in the samples with the sizes: d = 40 μm, l = 1 cm: 1 – experiment; 2 – with account of phonon drag and carrier scattering by phonons; 3 – ( ( 7.5(10–8 T–2.32.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
456
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