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Journal of the Korean Physical Society, Vol. 56, No. 1, January 2010, pp. 494∼497

First-Principles Study of the Structural Phase Transition in Sn

Sung-Ho Na∗and 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

[α⇔βtransition 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 α⇔β

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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