Electronic band structure and magnetic susceptibility of Ge₁₋xSix solid solutions
The magnetic susceptibility (MS) of single crystal Ge₁-xSix (0 < x < 0.1) solid solutions was investigated. Considerable anomalous MS behavior was found. Theoretical estimations of paramagnetic and diamagnetic parts were made. Reconstruction of the band structure of these alloys results in pec...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2002
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| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Цитувати: | Electronic band structure and magnetic susceptibility of Ge₁₋xSix solid solutions / V.G. Deibuk, S.I. Shakhovtsova, V.A. Shenderovski, V.M. Tsmots // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2002. — Т. 5, № 1. — С. 5-8. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1195682017-06-08T03:06:39Z Electronic band structure and magnetic susceptibility of Ge₁₋xSix solid solutions Deibuk, V.G. Shakhovtsova, S.I. Shenderovski, V.A. Tsmots, V.M. The magnetic susceptibility (MS) of single crystal Ge₁-xSix (0 < x < 0.1) solid solutions was investigated. Considerable anomalous MS behavior was found. Theoretical estimations of paramagnetic and diamagnetic parts were made. Reconstruction of the band structure of these alloys results in peculiarities of MS as a function of the composition. 2002 Article Electronic band structure and magnetic susceptibility of Ge₁₋xSix solid solutions / V.G. Deibuk, S.I. Shakhovtsova, V.A. Shenderovski, V.M. Tsmots // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2002. — Т. 5, № 1. — С. 5-8. — Бібліогр.: 5 назв. — англ. 1560-8034 PACS: 78.20.-e http://dspace.nbuv.gov.ua/handle/123456789/119568 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The magnetic susceptibility (MS) of single crystal Ge₁-xSix (0 < x < 0.1) solid solutions was investigated. Considerable anomalous MS behavior was found. Theoretical estimations of paramagnetic and diamagnetic parts were made. Reconstruction of the band structure of these alloys results in peculiarities of MS as a function of the composition. |
| format |
Article |
| author |
Deibuk, V.G. Shakhovtsova, S.I. Shenderovski, V.A. Tsmots, V.M. |
| spellingShingle |
Deibuk, V.G. Shakhovtsova, S.I. Shenderovski, V.A. Tsmots, V.M. Electronic band structure and magnetic susceptibility of Ge₁₋xSix solid solutions Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Deibuk, V.G. Shakhovtsova, S.I. Shenderovski, V.A. Tsmots, V.M. |
| author_sort |
Deibuk, V.G. |
| title |
Electronic band structure and magnetic susceptibility of Ge₁₋xSix solid solutions |
| title_short |
Electronic band structure and magnetic susceptibility of Ge₁₋xSix solid solutions |
| title_full |
Electronic band structure and magnetic susceptibility of Ge₁₋xSix solid solutions |
| title_fullStr |
Electronic band structure and magnetic susceptibility of Ge₁₋xSix solid solutions |
| title_full_unstemmed |
Electronic band structure and magnetic susceptibility of Ge₁₋xSix solid solutions |
| title_sort |
electronic band structure and magnetic susceptibility of ge₁₋xsix solid solutions |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2002 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119568 |
| citation_txt |
Electronic band structure and magnetic susceptibility of Ge₁₋xSix solid solutions / V.G. Deibuk, S.I. Shakhovtsova, V.A. Shenderovski, V.M. Tsmots // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2002. — Т. 5, № 1. — С. 5-8. — Бібліогр.: 5 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT deibukvg electronicbandstructureandmagneticsusceptibilityofge1xsixsolidsolutions AT shakhovtsovasi electronicbandstructureandmagneticsusceptibilityofge1xsixsolidsolutions AT shenderovskiva electronicbandstructureandmagneticsusceptibilityofge1xsixsolidsolutions AT tsmotsvm electronicbandstructureandmagneticsusceptibilityofge1xsixsolidsolutions |
| first_indexed |
2025-07-08T16:11:21Z |
| last_indexed |
2025-07-08T16:11:21Z |
| _version_ |
1837095801271091200 |
| fulltext |
5© 2002, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2002. V. 5, N 1. P. 5-8.
PACS: 78.20.-e
Electronic band structure and magnetic susceptibility
of Ge1-xSix solid solutions
V.G. Deibuk1), S.I. Shakhovtsova2), V.A. Shenderovski2), V.M. Tsmots3)
1) Chernivtsi National University, 2 Kotsiubynskogo str., 58012 Chernivtsi, Ukraine
e-mail: vdei@chnu.cv.ua
2) Institute of Physics, NAS of Ukraine, 46 prospect Nauky, 03028 Kyiv, Ukraine
e-mail: schender@iop.kiev.ua
3) Drohobych State Pedagogical University, 3 Stryiska str., 82100 Drohobych, Ukraine
Abstract. The magnetic susceptibility (MS) of single crystal Ge1-xSix (0 ≤ x ≤ 0.1) solid solutions was
investigated. Considerable anomalous MS behavior was found. Theoretical estimations of paramagnetic
and diamagnetic parts were made. Reconstruction of the band structure of these alloys results in
peculiarities of MS as a function of the composition.
Keywords: band structure, magnetic susceptibility, Ge1-xSix, alloy.
Paper received 26.11.01; revised manuscript received 01.03.02; accepted for publication 05.03.02.
1. Introduction
The main consequence of alloying germanium with silicon
is the reconstruction of the band structure of Ge1�xSix alloys
(x is the molar fraction of Si) as a function of the Si
concentration. When x = 0.14, minima of the conduction
band of the types L1 and ∆1 are located at the same energy
level, whereas, when x > 0.14, inversion of the conduction
band of the type L1→∆1 occurs. The Ge1�xSix alloys are a
siutable object for the purpose of studying galvano-
magnetic properties of semiconductors with changing
dispersion law.
Magnetic susceptibility (MS) of the alloys allows to
receive reliable information about the nature of interatomic
interaction and parameters of chemical bonding. The
specificity of MS is that it does not depend on the scattering
mechanisms, but it is formed by features of the electronic
band structure. The doping Ge by silicon leads to the
dynamics of electronic bands, therefore to the change in
the topology of E(k) over Brillouin zone and as a rule in
concentration dependence of MS [1].
2. Experimental procedure
We investigated Ge1-xSix, doped by Sb up to concentration
1014�1015 cm-3. The highest content of the minority
component is 10 at. %. After X-ray microanalysis
measurements, irregularity of Si distribution over the sample
did not exceed 0.1%. The surface density of dislocations
was 103 cm�2 and it increases due to increase of Si compo-
sition up to 106 cm�2.
MS of Ge1-xSix alloys was investigated by the Faraday
method within the temperature range 77�300 K and at the
magnetic field 0.3�5 kOe. The samples with 1.5×1.5×10 mm3
dimensions were oriented along [100], [111], [110] direc-
tions. MS measurement error did not exceed 2%. In accu-
racy measurement limits the anisotropy of MS was not ob-
served. MS was not temperature and magnetic field
dependent. The experimental and theoretical results are
presented in the table (χexpt � the experimental MS at 77 K)
and the Fig. 1.
3. Theoretical estimations
Magnetic susceptibility of semiconductors consists
of Langevin�s diamagnetic χL, Van Vleck�s paramag-
netic χP terms and also the defects contribution χd(T):
)(TdpL χχχχ ++= (1)
Hamiltonian of the electron system in the external
magnetic field can be written as
( )∑∑ ++
−=
k
kk
k
kk ArotS
mc
e
UA
c
e
p
m
H
rrrr
2
2
1 (2)
where kp
r
and kS
r
are operators of momentum and spin
respectively, kA
r
� vector-potential of external magnetic field,
6 SQO, 5(1), 2002
V.G. Deibuk et al.: Electronic band structure and magnetic susceptibility...
χ,
c
m
/
m
ol
2
x
Fig. 1. Magnetic susceptibility variation χ(x) vs x for Ge1�xSix alloys
(1 � theory, 2 � experimental data at 77 K).
U � potential energy of electrostatic interaction of the elec-
trons and the summation runs over all electronic states of
the system. For the homogeneous magnetic field directed
along OZ axis the expression (2) transforms into the
following:
( ) ∑+++
+
∑ +
∂
∂−
∂
∂+∇−=
k
kzkkz SH
mc
e
Uyx
mc
e
H
k kxky
kykx
mci
e
zHkm
H
]
[
22
2
2
2
2
2
8
22
hh)
(3)
The multiplier zH
mc
e
2
in the second summand within
square brackets is the operator of z-component of angular
momentum.
∂
∂−
∂
∂=
k
k
k
kzk
x
y
y
x
i
l
h
(4)
Than the terms in (3), that linear depend on Hz, are:
( )∑ −=+−
k
zzzkzk MHSl
mc
e
H 2
2
(5)
zM is the operator of full electronic momentum. MS can
be defined as:
2
2
2
2
zz H
nHn
H
E
∂
∂
−=
∂
∂−=
)
χ (6)
For the calculation of (6), we used the perturbation
theory, supposing that the perturbation is interaction with
magnetic field. Than
1
)2(2)1( ... HHWHWHHH ozzo +=+++= (7)
( )22
2
2
2
1
8
kkzzz yx
mc
e
HMHH ++−= (8)
The energy of this system, taking into account the
second order terms over Hz, is equal ]
( ) −++−= ∑ nyxn
mc
e
HnMnH kk
k
z
o
zz
o
nn
22
2
2
2
8
εε
∑
−
−
'
)()(
'
2
2
'
n
o
n
o
n
o
z
z
nMn
H
εε
(9)
Substituting (9) into (6), the second term in energy gives
Langevin diamagnetic susceptibility and the third term is
Van Vleck paramagnetic susceptibility. In the case of tetra-
hedral covalent semiconductors, Langevin diamagnetic
susceptibility is:
2
222
24mc
dNem
L
γχ −= (10)
where d is the bond length, N � electronic density, γm �
scaling factor, that equal 1.12 for all semiconductors.
The paramagnetic contribution to MS can be presented [3]:
( )( ) ( )( )[ ]
)()(
1)(
)2(
2
2
3
2
kEkE
kEfkEfkM
kdP
nn
nn
nnocc
n n
B
p
rr
rrr
r
−
−
×
×=
′
′
′
′
∑∑ ∫
π
µ
χ
(11)
where )(kEn
r
is the energy of n-th band at k
r
point of
Brillouin zone (BZ); Mnn� � the oscillator strength; f � Fermi-
Dirac distribution function; P � means principal value
integration.
For the determination of Ge and Si band structure
)(kEn
r
, we used the tight-binding method [2]. As basic
functions we chose s-, px-, py-, pz- and s*- orbitals of anions
and cations, than the secular equation can be solved in
nearest neighbors approximation. We fit band structure in
the high symmetrical points of Brillouin zone to the most
V. G. Deibuk et al.: Electronic band structure and magnetic susceptibility...
7SQO, 5(1), 2002
recent experimental data. The two-center approximation is used
for expressions of interatomic matrix elements and only nearest-
neighbor interactions are taken into account. In our
investigations of the Ge1-xSix solid solution band structure,
we calculate the parameters of interactions in virtual crystal
approximation (VCA). Integral (11) over Brillouin zone was
calculated using thelinear tetrahedron method [4]. In this
method the 1/48-th part of BZ is divided by 256 identical
tetrahedrons.
Paramagnetic part of MS (11) was calculated at T =
= 0 K. We supposed also that the matrix elements of the
oscillator strength are constant within the tetrahedron that
corresponds to the Lehmann-Taut division. Than if we ex-
pand the energy difference into the Taylor series up to the
first order with respect to ( )okk
rr
− at the ok
r
point, we obtain
the following integrals
∫ −+
=′
)(
3
o
nn
kkBA
kd
PI rrr
r
(12)
where
( ) ( )kEkEA nn
rr
−= ′ (13)
[ ])()( kEkEB nnk
rrrr
−∇= ′ (14)
at okk
rr
= . Since the tetrahedrons have equal volumes, than
the integral (11) can be expressed analytically as [3]:
++Ω=′
4
3
3
2
3
4
2
2
2
2
4
1
1
2
1 lnlnln3
v
v
D
v
v
v
D
v
v
v
D
v
I nn (15)
where Ω is a tetrahedron volume, )()( inini kEkEv
rr
−= ′ .
Index i means the four tetrahedron apexes in which the
energies are defined. Besides that
( )( )( )
( )( )( )
( )( )( )1323433
1232422
2131411
vvvvvvD
vvvvvvD
vvvvvvD
−−−=
−−−=
−−−=
(16)
Paramagnetic susceptibility can be expressed in terms
of integrals Inn� introduced above
( ) ∑∑
′
′
′=
occ
n n
nn
nnB
p IM
2
3
2
2
2
π
µ
χ (17)
For the Ge1�xSix alloys the composition dependence of
MS is defined by the reconstruction of the electronic band
structure.
Matrix elements of the oscillator strength in the tight-
binding method can be estimated following Harrison [2]:
Table. Comparison of the theoretical calculations on mag-
netic susceptibility with experimental data for Ge1�xSix (at
77 K).
χ⋅106, cm3mol�1
x
�χexpt �χL χp �(χL + χp)
0 15.68 21.123 0.11866 21
0.01 12.99 21.108 0.08569 21.02
0.015 12.95 � � �
0.02 � 21.09 0.1066 20.98
0.03 � 21.076 0.11126 20.96
0.04 13.17 21.06 0.1054 20.95
0.05 � 21.045 0.0868 20.96
0.055 13.82 � � �
0.06 � 21.03 0.10347 20.93
0.07 � 21.01 0.12038 20.89
0.08 � 20.99 0.89 12.9
0.085 13.49 � � �
0.09 � 20.98 0.092 20.88
0.1 13.08 20.967 0.0917 20.87
( )
2
'
2
2
' '
2
n
x
n
EEm
M
nn
nn ∂
∂
−
∝
h (18)
From the quantum mechanical view, we can write
( )
''
2
' nxn
EEm
n
x
n nn
h
−
=
∂
∂
(19)
On the other hand,
( ) 2/11
32
' p
d
nxn αγ −=
where γ is the scaling factor; αp is the ionicity of the bond;
n , 'n and En, En are the eigenfunctions and eigenvalues
of the energy of antibonding and bonding states,
respectively. Then using (19), (20) and (18) we obtain:
( ) 2
2
'2
' '
2
nxn
EEm
M nn
nn
h
−
∝ ~10-2 (21)
Experimental (at 77 K) and theoretical results are
presented in the Table. The diamagnetic and paramagnetic
contributions to the MS were calculated using (10) and (17),
respectively. As seen from the Table, the small changes of
the alloy composition caused increasing of the
paramagnetic part of MS. In figure we plot the total MS
(1 � theory; 2 � experiment at 77 K).
As illustrated by the figure, the composition depend-
ence of MS has some peculiarities. They are obviously
connected with paramagnetic part of MS and caused by
8 SQO, 5(1), 2002
V.G. Deibuk et al.: Electronic band structure and magnetic susceptibility...
the changes in the topology of the electronic structure of
alloys with increasing Si concentration in solid solutions.
The difference between calculated and experimental
positions of singularities is due to insufficient precision of
the used version of the tight-binding method in calcula-
tions of )(kEn
r
, and also due to the linearity of integration
over BZ. In our theoretical calculations, we did not take
into account the impurity contribution to MS and influence
of the temperature, too.
However, it follows from our experimental investigations
[1] that the influence of the impurities and defects on MS is
strong. Moreover, the distribution of silicon atoms in Ge1�
xSix alloys is nonuniform. The scale of fluctuations of the
composition (geometric dimensions 50�100 Å and number
of silicon atoms per cluster 30�150) was estimated in [5].
The presence of these clusters causes the great local
distortions of the crystal lattice and the changes of the
electron density configuration.
References
1. V.A. Shenderovsky, I.N. Belokurova, B.R. Romanovsky,
V.M. Tsmots`, S.I. Shakhovtsova, M.M. Shwarts. Magnetic
susceptibility of Ge1�xSix solid solutions // Izvestiya AN SSSR
ser. Neorganicheskiye materialy 25(11), p. 1171�1174 (1989).
2. W. Harrison, Elecrtonic structure and properties of solids: Physic
of chemical bonds (in Russian), v.1, Mir, Moscow (1983).
3. J. Rath and A.J. Freeman. Generalized magnetic susceptibilities
in metals: Application of the analytic tetrahedron linear energy
method to Sc // Phys. Rev. B 11(6), pp.2109-2117 (1975).
4. G. Lehman and M. Taut. On the numerical calculation of the
density of states and related properties // Phys. Stat. Sol. 54(2),
pp.469-477 (1972).
5. S.I. Shakhovtsova, K.V. Shakhovtsov, L.I. Shpinar, I.I. Yakovets.
Scale of composition fluctuations in Ge1-xSix alloys (in Russian)
// Semiconductors, 27(6), p. 1035�1039 (1993).
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