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First-Principles Study of the Structural Phase Transition in Sn

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The dependence of phase transition temperature between the alpha and the beta phases of tin (Sn) on pressure is examined through first-principles calculation. The temperature and the pressure-dependent thermodynamic quantities, such as the entropy, the Helmholtz free energy, and the Gibbs free energy, are estimated by employing the phonon density of states, which is calculated by using the frozen phonon approximation. The calculated structural phase transition is compared with experiment.
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Journal of the Korean Physical Society, Vol. 56, No. 1, January 2010, pp. 494497
First-Principles Study of the Structural Phase Transition in Sn
Sung-Ho Naand Chul-Hong Park
Research Center for Dielectric and Advanced Matter Physics, Pusan National University, Pusan 609-735
(Received 2 April 2009, in ﬁnal form 9 July 2009)
The dependence of phase transition temperature between the αand the βphases of tin (Sn)
on pressure is examined through ﬁrst-principles calculation. The temperature and the pressure-
dependent thermodynamic quantities, such as the entropy, the Helmholtz free energy, and the
Gibbs free energy, are estimated by employing the phonon density of states, which is calculated
by using the frozen phonon approximation. The calculated structural phase transition is compared
with experiment.
PACS numbers: 63.20.Dj, 65.50.+m
Keywords: Tin, Phonon, Gibbs free energy
DOI: 10.3938/jkps.56.494
I. INTRODUCTION
The αphase and the βphase of tin are most commonly
observed (crystal structures described in Fig. 1). The
mass density of α-Sn is roughly 20 percent lower, and α-
Sn is slightly more stable at low temperatures below T
= 286 K, above which the β-Sn structure becomes more
stable [2,3]. α-Sn is a diamond structure semiconductor
with zero band gap while β-Sn has a tetragonal struc-
ture and is metallic. ‘Gray tin’ and ‘white tin’ are com-
mon names for the αphase and the βphase respectively,
but α-Sn can be metastable against β-Sn by applying
a pressure, and tin has been found to undergo succes-
sive phase transitions by further pressure increase [4,5].
The pressure-induced phase transitions of tin and other
group-IV elements have been investigated both theoret-
ically and experimentally [2–12]; however, little theoret-
ical study of the pressure-induced phase transition at
ﬁnite temperature exists.
First-principles calculations based on the density func-
tional theory have been powerful in analyzing the physi-
cal properties of matters, but are conﬁned to the ground
state. A quasi-harmonic approach based on informa-
tion on lattice vibration modes can be applied to inves-
tigate the ﬁnite-temperature state. Arbitrary pressure
and temperature conditions can be described by using
this approach with proper interpolation, and thermody-
namic properties, such as the entropy, the Helmholtz free
energy, and the Gibbs free energy can be determined. In
this study, we theoretically investigated the dependence
of the phase transition between α-Sn and β-Sn on both
E-mail: sunghona@kasi.re.kr; now at Korea Astronomy and Space
Science Institute, Daejeon 305-348
E-mail: cpark@pusan.ac.kr; Fax: +82-51-515-2390
the pressure and the temperature by utilizing the calcu-
lated eigenfrequencies of the lattice vibration.
II. CALCULATION AND DISCUSSION
We used the Vienna ab initio simulation package
(VASP) with the projector augmented wave (PAW)
schemes [13] and the generalized gradient approximation
(GGA) suggested by Perdew et al. [14]. The energy cut-
oﬀ was taken as 241 eV. The k-points were chosen by
using the Monkhorst-Pack method [15], and the k-point
meshes were set as (5, 5, 5) for the α-Sn and (10, 10, 10)
for the β-Sn. Many more k-points were necessary for the
β-Sn calculation due to its metallic character to assure
calculational convergence.
Fig. 1. (Color online) Crystal structures of two diﬀerent
states of tin : the αstate and the βstate. The α-Sn has a
diamond structure, while the β-Sn has a tetragonal structure
[1].
-494-
First-Principles Study of the Structural Phase Transition in Sn – Sung-Ho Na and Chul-Hong Park -495-
Table 1. Comparison of the lattice constant and the bulk
modulus of α-Sn and β-Sn with former calculations.
α-Sn β-Sn
[ lattice constant ] in A
our calculation 6.64 5.92
other calculations 6.381, 6.482, 6.5535.701, 5.922, 5.853
6.404, 6.4755.704, 5.735
experiment 6.49165.8206
[ bulk modulus ] in 109Pa
our calculation 47 57
other calculations 471, 42.12, 44.73611, 64.12, 54.43
51.24, 45.6560.54, 62.95
experiment (53)758.28
[αβtransition temperature
at zero pressure ] in K
our calculation 310
other calculation 311
experiment 2869
1-5 = Refs. 2–6, and 6-9 indicates Refs. 1, 16, 17, and 10.
Fig. 2. (Color online) Electronic density of states of the
α-Sn and the β-Sn.
The calculated results for the structural parameter are
shown in Table I, along with the experimental data and
the previously calculated results. The lattice constants
of the two phases of tin were calculated to be 6.64 A and
5.92 A, respectively, for α-Sn and β-Sn in the ground
state by minimizing the total energy. These are larger
by 2 percent than the experimental values [1]. The bulk
modulus of each phases were estimated as 47 GPa and
57 GPa. The bulk modulus experimental value for β-Sn
is between 55 and 59 GPa [17,18] and is in agreement
with our estimate, but experimental value for the α-Sn
bulk modulus are scarce.
The calculated electronic density of states and band
structure of α-Sn and β-Sn at the ground state are illus-
trated in Fig. 2 and Fig. 3, which show that α-Sn is a
semiconductor with zero band gap and that the β-Sn is
Fig. 3. (Color online) Electronic band structures of the
α-Sn and the β-Sn.
metallic. The Sn-3d orbital states are located far below
the Fermi level at about 21 or 20 eV for each of the
two phases, as shown in Fig. 3. In the semiconducting
α-phase, the bands of the sand the porbitals are split
around the Fermi level. This indicates that there is a
strong hybridization between the sand the porbitals in
α-Sn, which raises the energy level of the anti-bonding
states at the conduction band and lowers the the energy
level of the bonding states at the valence band. This
lowering of the electron-occupied bonding state can con-
tribute to stabilization of the α-phase relatively to the
β-phase.
We calculated the phonon frequency by solving the ω-~q
eigenvalue - eigenvector equation
det "1
pMiMj
Ckl
ij (~q)ω2#= 0,(1)
where the indices iand jare for the two atoms concerned
(displaced atom and target atom), and indices kand l
are for the directions of the small displacement of the
atom and of the direction of the resulting force on the
target atom, and ~q is the wave vector of the vibrational
mode. The interatomic force constant Ckl
ij is calculated
by placing a small displacement on the i-th atom in the
cell. Care should be taken to assign the amount of this
small displacement so that the quasi-harmonic approach
is valid; otherwise, too small or too large a displace-
ment would result in failure to ﬁnd the correct elastic
constants and eventually the correct phonon frequencies.
The choice of the cell size is often crucial. A larger cell
size is desirable for low-frequency acoustic phonon cal-
culation, but causes a quite longer calculation time. The
cell multiplicities in this study were taken as (3 ×3×3)
for α-Sn and (3 ×3×4) for β-Sn.
The calculated phonon dispersion relations for α-Sn
and β-Sn are shown in Fig. 4. The phonon densities of
the two states of Sn calculated by using uniform q-grid
points (10 ×10 ×10) in momentum space are shown in
Fig. 5. The top two ﬁgures in Fig. 4 correspond to
the phases under zero pressure, but the bottom ﬁgures
-496- Journal of the Korean Physical Society, Vol. 56, No. 1, January 2010
Fig. 4. (Color online) Calculated phonon dispersion curves
of the α-Sn and the β-Sn. (a) and (b) are for the α-Sn and
the β-Sn under zero pressure, and (c) and (d) correspond to
the high pressure states (α-Sn at 2.8 GPa, β-Sn at 2.9 GPa).
Fig. 5. (Color online) Phonon density of states of the α-Sn
and the β-Sn.
correspond to the high pressure state. For the case of
α-Sn, parts of the transverse acoustic phonon branch are
bent downward with increasing pressure. This indicates
phonon softening and has been discussed by Baroni et al.
[12]. We found that this tendency grows as the pressure
increases further. The overall vibrational frequencies of
the βphase phonon, especially for the optical modes,
are lower than or equal to those of the αphase. This
discrepancy is related with the larger entropy and the
lower free energy of β-Sn compared to α-Sn.
Fig. 6. (Color online) (a) Temperature dependence of the
entropy, heat capacity, and Helmholtz free energy of the β-Sn
at zero pressure, (b) Helmholtz free energies of the α-Sn and
the β-Sn at zero pressure.
Thermodynamic quantities, such as the partition func-
tion Q, the Helmholtz free energy F, the internal energy
U, the heat capacity Cv, and the entropy S can be cal-
culated by using [19–22]
Q=X
~q,n
exp ~ω(n+1
2)/kT ,(2)
F=kT lnQ =X
~q ~ω
2+kT ln (1 exp (~ω/kT ))
=X
~q
ln 2 sinh ~ω
2kT ,(3)
U=X
~q ~ω
2+~ω
exp(~ω/kT )1.(4)
The entropy can be expressed as S=∂F
∂T , and the heat
capacity can also be expressed as Cv=∂U
∂T . The calcu-
lated F, S, and Cv of β-Sn at zero pressure are shown
in Fig. 6, where the zero point energies were included.
The two Helmholtz free energy curves for α-Sn and β-
Sn cross at about 310 K, which is slightly higher than
the experimental estimate of 13oC for the phase transi-
tion under ambient pressure. Our estimate of the αβ
transition temperature as 310 K by using the GGA is
almost identical to the former estimate of 311 K from
the LDA calculation by Pavone et al. [2].
The pressure was estimated by adding two contribu-
tions - phonon contribution and electron contribution:
P=Pph +Pel =∂Fph
∂T +Fel
∂T . The free energy
F is the sum of the two parts, Fph and Fel.Fph is the
phonon contribution, i.e., lattice dynamical free energy,
which was calculated by using Eq. (3) above, and Fel
is the free energy of electronic contribution. The Gibbs
First-Principles Study of the Structural Phase Transition in Sn – Sung-Ho Na and Chul-Hong Park -497-
Fig. 7. (Color online) Calculated dependence of the phase
transition temperature between the α-Sn and the β-Sn on the
pressure is compared with an experimental estimate.
free energy G=F+P V is the relevant thermodynamic
quantity to determine the stable phase between neigh-
boring phases.
The discrete Gibbs free energy values acquired for
each of the two phases are interpolated. In Fig. 7,
the pressure-dependent phase transition temperature is
shown and compared with an experimental estimation by
Jayaraman et al. [10]. Jayaraman used the Clapeyron
equation to identify the slope dT/dP of the boundary.
Our curve is slightly shifted so that it would be recon-
ciled with the known transition temperature of 13 oC at
zero pressure. We estimate the phase transition pres-
sure at 0 K to be 0.3 GPa, which is in between the two
previous theoretical estimates of 0.2 GPa by Christensen
and Methfessel [4] and 0.8 GPa by Cheong and Chang
[5]. The diﬀerences between these theoretical estimates
should be ascribed to diﬀerent calculation schemes.
III. CONCLUSIONS
First-principles calculations of the thermodynamic
quantities for Sn were carried out in order to predict
the structural phase transition. The phonon dispersion
relations and the phonon density of states were calcu-
lated for two states of Sn. The Gibbs free energy and
other thermodynamic properties were calculated for α-
Sn and β-Sn by using phonon dispersion, by which the
phase transition boundary between the two phases of tin
was estimated.
ACKNOWLEDGMENTS
This study was supported by the Korea Research
Foundation funded by the Korean Government (KRF-
2006-005-J02804). The ﬁrst author thanks to his super-
visor for kind direction and teaching. He also thanks to
his wife for encouragement.
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The free energies of the α and β phases of tin are calculated in the harmonic approximation using density-functional theory and density-functional perturbation theory, within the local-density approximation. At T=0 K the free energy of the β phase lies ≈359cal/mole above that of the α structure. The narrower frequency range spanned by the vibrational band in the β phase makes its entropy larger at high temperature. As a consequence, the free energies of the two phases equal each other at a temperature of 38 °C, in close agreement with the observed transition temperature Tc≈13°C.
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The pressure-induced phase transitions in tin from the diamond (A4) to the white-tin (A5) and further to a simple face-centered-tetragonal structure continuously deforming upon higher compression towards a body-centered-cubic (A2) structure are calculated by means of a first-principles pseudopotential method without adjustable parameters. The critical compressions for the phase changes as well as the structural details of the tetragonal high-pressure modifications are predicted with good accuracy. For very high compression a transition to the hexagonal-close-packed (A3) structure is predicted.
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The structural and thermodynamic properties of two phases of tin are reported. Equilibrium properties of both phases obtained from an ab initio nonrelativistic calculation are in good agreement with experiment. From our calculation, we suggest that the measured bulk modulus for grey tin reported in the literature is in error. The grey-to-white-tin transition is discussed using the total internal energy and the free energy [E(T) and F(T)] curves as well as the valence charge distribution for each phase. Our method utilizes self-consistent pseudopotentials derived to identically reproduce results of all-electron calculations.
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The crystal dynamics of the low temperature semiconducting allotrope of tin, α-Sn, have been studied at 90°K by means of thermal neutron inelastic scattering. From the phonon frequency-wavevector dispersion relation along the [100], [110], and [111] symmetry axes a frequency distribution has been derived and used to calculate several thermodynamic properties. Limited measurements were made at 250°K to investigate the effect of the α → β transition at 286°K - no effect was detected.RésuméLes auteurs ont étudié la dynamique des cristaux de l'allotrope à basse température d'étain, α-Sn, à 90°K au moyen de la diffusion inélastique des neutrons lents. Ils ont utilisé les relations de dispersion (fréquence/vecteur d'onde) mesurées dans les directions [100], [110] et [111] pour établir une distribution de fréquence, qu'ils ont utilisée pour calculer plusieurs fonctions thermodynamiques. Ils ont fait des mesures limitées à 250°K pour examiner l'effet de la transition α → β à 286°K mais ils n'ont observé aucon effet.