Phase diagrams of Si₁-xGex solid solution: a theoretical approach
In this work, we have used the pseudo-alloy atom model and higher-order perturbation theory based on pseudopotential approach to investigate phase diagram at different temperatures for Si₁–xGex solid solution system where x is the arbitrary (atomic) concentration of the second constituting elemen...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2012
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| Цитувати: | Phase diagrams of Si₁-xGex solid solution: a theoretical approach / A.R. Jivani, A.R. Jani // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 1. — С. 17-20. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1182682017-05-30T03:02:00Z Phase diagrams of Si₁-xGex solid solution: a theoretical approach Jivani, A.R. Jani, A.R. In this work, we have used the pseudo-alloy atom model and higher-order perturbation theory based on pseudopotential approach to investigate phase diagram at different temperatures for Si₁–xGex solid solution system where x is the arbitrary (atomic) concentration of the second constituting element. We have also investigated the phase diagram near the melting temperature as well as at low temperatures and compared with the available experimental results. Our calculated phase diagram near the melting point agrees well with the experimental data. 2012 Article Phase diagrams of Si₁-xGex solid solution: a theoretical approach / A.R. Jivani, A.R. Jani // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 1. — С. 17-20. — Бібліогр.: 14 назв. — англ. 1560-8034 PACS 64.70.kg, 64.75.Nx, 71.15.Nc, Dx http://dspace.nbuv.gov.ua/handle/123456789/118268 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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In this work, we have used the pseudo-alloy atom model and higher-order
perturbation theory based on pseudopotential approach to investigate phase diagram at
different temperatures for Si₁–xGex solid solution system where x is the arbitrary (atomic)
concentration of the second constituting element. We have also investigated the phase
diagram near the melting temperature as well as at low temperatures and compared with
the available experimental results. Our calculated phase diagram near the melting point
agrees well with the experimental data. |
| format |
Article |
| author |
Jivani, A.R. Jani, A.R. |
| spellingShingle |
Jivani, A.R. Jani, A.R. Phase diagrams of Si₁-xGex solid solution: a theoretical approach Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Jivani, A.R. Jani, A.R. |
| author_sort |
Jivani, A.R. |
| title |
Phase diagrams of Si₁-xGex solid solution: a theoretical approach |
| title_short |
Phase diagrams of Si₁-xGex solid solution: a theoretical approach |
| title_full |
Phase diagrams of Si₁-xGex solid solution: a theoretical approach |
| title_fullStr |
Phase diagrams of Si₁-xGex solid solution: a theoretical approach |
| title_full_unstemmed |
Phase diagrams of Si₁-xGex solid solution: a theoretical approach |
| title_sort |
phase diagrams of si₁-xgex solid solution: a theoretical approach |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/118268 |
| citation_txt |
Phase diagrams of Si₁-xGex solid solution: a theoretical approach / A.R. Jivani, A.R. Jani // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 1. — С. 17-20. — Бібліогр.: 14 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT jivaniar phasediagramsofsi1xgexsolidsolutionatheoreticalapproach AT janiar phasediagramsofsi1xgexsolidsolutionatheoreticalapproach |
| first_indexed |
2025-07-08T13:39:19Z |
| last_indexed |
2025-07-08T13:39:19Z |
| _version_ |
1837086237985341440 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 1. P. 17-20.
PACS 64.70.kg, 64.75.Nx, 71.15.Nc, Dx
Phase diagrams of Si1-xGex solid solution: a theoretical approach
A.R. Jivani1 and A.R. Jani2
1V P and R P T P Science College, Vallabh Vidyanagar-388120, Gujarat, India
2Department of Physics, Sardar Patel University, Vallabh Vidyanagar-388120, Gujarat, India
E-mail: jivani_a_r@yahoo.com
Phone: (office) +91-2692 – 230011; cell phone +91-9909012156; fax: +91-2692 235207
Abstract. In this work, we have used the pseudo-alloy atom model and higher-order
perturbation theory based on pseudopotential approach to investigate phase diagram at
different temperatures for Si1–xGex solid solution system where x is the arbitrary (atomic)
concentration of the second constituting element. We have also investigated the phase
diagram near the melting temperature as well as at low temperatures and compared with
the available experimental results. Our calculated phase diagram near the melting point
agrees well with the experimental data.
Keywords: solid solution, pseudopotential method, phase diagram.
Manuscript received 14.04.11; revised manuscript received 29.11.11; accepted for
publication 26.01.12; published online 29.02.12.
1. Introduction
Group IV solid solutions such as SiGe, GeSn etc. exhibit
unique optoelectronic properties. The Group IV
semiconductor alloys have wide (direct) bandgap
materials, and hence they are used in fabrication of
optoelectronic devices. These systems also improve the
transport and optical properties as compared to its
constituting atoms [1–5]. Therefore, the study of various
physical properties of system is very important
both for experimentalists and theoreticians. The
knowledge of phase diagrams for alloy systems is
essential to understand behavior of alloys. In spite of this
importance, only very little information about phase
diagrams at and around the melting temperature are
available for Si – Ge system.
GeSi −
Previously, we have successfully employed our own
proposed potential [6–10] to investigate a large number of
physical properties of various semiconductor systems. We
fruitfully reported the total energy, bulk modulus and heat
of formation of Si – Ge system [9] using our potential and
hence this prompted us to make these calculations of
phase diagrams at various temperatures with using our
proposed pseudopotential [6–10].
2. Simulation details
2.1. Total energy calculations
For a covalent crystal with the diamond structure, if
being based on the higher-order perturbation theory, the
total energy per atom of the crystal [6–10] is given by
cov210 EEEEEE i ++++= . (1)
In Eq. (1), Ei is energy contribution due to ion-ion
interactions, usually known as the Madelung energy, E0
is the energy of the homogeneous electron gas which is a
sum of the kinetic energy, exchange energy and
correlation energy, E1 is the first-order energy of the
valence electron due to the pseudopotential, E2 is the
band structure energy, and Ecov is the covalent-correction
term. The covalent term is approximately a sum of third
and fourth-order perturbation energies, which is essential
while we study electronic, elastic and vibrational
properties of semiconducting materials [6–10].
2.2. Pseudo-alloy atom model
In these investigations of phase diagrams for solid
solutions, we have the used pseudo-alloy atom model
(hereafter referred to as PAA) [11]. In this model, a
hypothetical monatomic periodic lattice is replaced by
the disordered alloy. The screened potential form factor
Ws
PAA(q) of an electron with a single PAA in the case of
solid solution is given by
( )
( ) ( )
( )
,sin1
2
sin
εΩ
π12)(
2
3
2
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+−×
×−=
PAA
c
PAA
c
PAA
c
PAA
c
PAA
c
PAA
PAA
S
qR
qR
Rq
qR
qRq
eZqW
(2)
where ZPAA = ZSi = ZGe = 4 is ion valency, e is the
electronic charge, Ω is the atomic volume, q is the
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
17
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 1. P. 17-20.
wave vector, and ε(q) is the static Hartree dielectric
function. The parameter of the potential at various
concentrations x is determined by
( ) ( ) .1 GeSi
cc
PAA
c xRRxxR +−= (3)
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
)
]
)
)
The total energy EPAA(x) of the solid
solution is calculated using the higher-order perturbation
theory [ ] and with use of the above average
screened potential (Eq. (2)) and potential parameter (as
defined in Eq. (3)). In this work, we have used Sarkar
et al. [12] local-field correction function to consider
exchange and correlation effects on dielectric function.
GeSi −
106 −
2.3. Heat of formation
The heat of formation ΔE(x) for solid solutions is
defined as the energy difference between the energy of
solid solution EPAA(x) and the energy of phase mixture
Emix(x) which is given by
ΔE(x) = EPAA(x) – Emix(x). (4)
In the equation (4), the total energy of phase
mixture of solid solutions for is given by GeSi −
( ) ( ) .1 GeSi
mix xEExxE +−= (5)
In the equation (5), ESi and EGe are the total energy
per atom of Si and Ge calculated with the equation (1).
2.4. Phase diagram at low temperatures
In this work, we have investigated the phase diagram at
low temperatures as well as at melting temperatures.
The Helmhöltz free energy for the solid
solution at the temperature T (in K) is given by [13]
( TxFS ,
( ) ( )
( ) ( ) ( )[ .1ln1ln
,,
xxxxkT
TxETxFS
−−++
+=
. (6)
Then, the Helmhöltz free energy of formation
for the solid solution [13] is equal to difference
between and the energy of phase mixture
which is given by
( TxFS ,′
( TxFS ,
( )xEmix
( ) ( ) ( ) =−=′ xETxFTxF SS mix,,
( ) ( ) ( ) ( )[ ]xxxxkTxE −−++Δ= 1ln1ln . (7)
Here, in deriving the equation (7), the vibrational
contribution to the internal energy and the thermal
entropy were assumed to be independent of the alloy
composition and the configurational entropy was taken
independent of the random distribution.
2.5. Phase diagram near the melting temperature
For solid solution, the Helmhöltz free energy ( )TxF ,1
for the liquid phase at the temperature T (in K) is given
by [13]
( ) ( )( )
( ) ( )
( ) ( ) ( ) ( )[ ] .1ln1ln
1,
1
GeGe
SiSi
xxxxkTxE
TxSLEx
LExTxF
m
L
−−++Δ+
+Δ−++
++−=
(8)
Similar to a solid phase, we can write the
Helmhöltz free energy of formation for the
liquid phase as given by
( TxF ,′ )
( ) ( ) ( ) =−=′ TxETxFTxF LL ,,, mix
( ) ( ) +Δ−+−= TxSLxLx m
GeSi1 (9)
( ) ( ) ( ) ( )[ ]xxxxkTxE −−++Δ+ 1ln1ln1 .
We have also taken into consideration all the
assumptions as for the solid phase and we have also
taken the heat of solution for the liquid phase
( ) 01 =Δ xE .
In order to perform numerical calculations of the
phase diagrams near the melting point, we also assumed
that the entropy change in the course of melting ( )xSmΔ
is
( ) ( )
Ge
Ge
Si
Si
1
mm
m
T
Lx
T
LxxS +−=Δ , (10)
where Tm and L are the melting points and heat of fusion,
respectively.
3. Results and discussion
3.1. Phase diagram at low temperatures
The presently investigated for system
as a function of the concentration (x) is shown for three
typical temperatures in Fig. 1.
( TxFS ,′ ) GeSi −
Fig. 1. The Helmhöltz free energy of formation ( )TxFS ,′ for
Si – Ge solid solution at 50, 200 and 250 K.
The Helmhöltz free energy of formation ( )TxFS ,′
for solid solutions is found to be positive as well as
negative depending on temperature. Depending upon the
18
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 1. P. 17-20.
sign, the physical interpretation may be formulated like
this.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Fig. 2. The calculated phase diagram for Si – Ge system at low
temperatures.
The Helmhöltz free energy of formation ( )TxFS ,′
for at 50 K is positive over the whole x region
as shown in Fig. 1. This suggests that the
GeSi −
GeSi −
system is insoluble at this temperature. The Helmhöltz
free energy of formation ( )TxFS ,′ for at 200 K
is found to be positive as well as negative. This curve
gives the values of concentration and temperature at
which the phase will change. The negative value of the
Helmhöltz free energy of formation
GeSi −
( )TxFS ,′ for
at 250 K over the whole x range indicates that
system forms a stable solid solution.
GeSi −
GeSi −
Fig. 2 shows the calculated solubility limit of
system. The system exists as phase
mixture with 10 to 90% concentration of Ge. There
exists a wide region of phase mixture at low
temperatures.
GeSi − GeSi −
3.2. Phase diagrams near the melting temperature
In Figs 3a to 3c, the free energy curves ( ( )TxFS ,′ and
) are shown for three typical temperatures,
namely: 1200, 1450 and 1700 K for solid
solution.
( TxFL ,′ )
)
GeSi −
It is seen from Fig. 3a that two curves coincide at
x = 1 and 1200 K. The free energy of solid phase is
negative throughout the x-region. Thus, system
at 1200 K forms a solid phase. The free energy of a
liquid phase is initially positive up to x = 0.54 and then
after it becomes negative.
GeSi −
The free energy curves and ( TxFS ,′ ( )TxFL ,′ for
solid and liquid phases crosses at 1450 K as shown in
Fig. 3b. Hence, this graph gives information about phase
boundaries of these two phases.
In Fig. 3c, two curves and ( TxFS ,′ ) ( )TxFL ,′
coincide at x = 0 and 1700 K temperature. The free
energy of the solid phase as well as liquid phase is
negative throughout the x-region. The value of ( )TxFS ,′
is more negative than at a particular
concentration, and this indicates that system
transforms to the liquid state.
( TxFL ,′ )
GeSi −
The calculated phase diagram near the melting
point is shown in Fig. 4 for solid solution
alongwith available experimental data [14]. From this
figure, it is clear that the agreement of our calculation
with the experimental data [14] is good.
GeSi −
a
b
c
Fig. 3. The free energy curves for solid and liquid phases for
Si – Ge solid solution at 1200 (a), 1450 (b), and 1700 K (c).
19
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 1. P. 17-20.
Fig. 4. The calculated phase diagrams near the melting points
for Si – Ge solid solution.
4. Conclusions
Thus, the phase diagram at various temperatures of
solid solution are studied employing the present
formulation of the model potential [ ] with the
higher-order perturbation theory and the pseudo-alloy
atom model [ ]. From the study of phase diagrams
for the system at low temperatures, we can
conclude that there exists a wide region of phase mixture
at low temperatures. It is also evident from this study
that the calculated phase diagram near the melting point
for solid solution agrees well with the available
experimental data [14]. Thus, our proposed potential
[ ] is efficient to study various properties of
solid solution, and hence this shows the
strength of the potential.
GeSi −
106 −
106 −
GeSi −
GeSi −
106 −
GeSi −
Acknowledgement
A.R. Jivani thanks University Grants Commission, New
Delhi, India, for financial support (Grant No. 47-
625/08(WRO)) to carry out this research work.
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© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
20
http://www.doitpoms.ac.uk/tlplib/phase-diagrams
2.3. Heat of formation
Fig. 2. The calculated phase diagram for Si – Ge system at low temperatures.
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