Solid State Communications 140 (2006) 538–543www.elsevier.com/locate/ssc

Ab initio calculation of the properties and pressure induced transition of Sn

Chun Yu∗, Junyan Liu, Hao Lu, Junmei Chen

School of Materials Science and Engineering, Shanghai Jiaotong University, Shanghai, 200030, PR China

Received 28 December 2005; received in revised form 7 March 2006; accepted 15 September 2006 by C.H.R. ThomsenAvailable online 6 October 2006

Abstract

In this paper, ab initio calculations of the electronic structure and pressure induced phase transition are performed for Sn by the densityfunctional theory, both local density approximation (LDA) and generalized gradient approximation (GGA) are chosen for the calculation ofexchange–correlation potential. Good agreement with previous experiments and calculations is found for the ground state properties of each phaseat atmospheric pressure. Both the band structure and density of states (DOS) of β-Sn are calculated and found to change greatly with the pressure.The transition sequence of α-Sn → β-Sn → bct (body-centered tetragonal) → bcc (body-centered cubic) → hcp (hexagonal close-packed)phase is obtained within generalized gradient approximation, and the conclusion is consistent with experiment. The transition pressures and atomvolumes of Sn are also calculated within generalized gradient approximation (GGA). The results show that the transition of α-Sn to β-Sn is at0.7 GPa, β-Sn to bct is at 12 GPa, and bct to bcc is at 45 GPa, bcc to hcp is at 160 GPa. These results are much closer to the experimental datathan those calculated within local density approximation (LDA).c© 2006 Elsevier Ltd. All rights reserved.

PACS: 64.70.Kb; 62.20.Dc; 71.15.Mb; 71.15.Dx

Keywords: A. Sn; D. Phase transition; E. Ab initio

1. Introduction

Tin occupies a special position within the carbon group ofthe periodic table; it is a very common and important element,archaeological evidence suggests that people have been usingtin for at least 5500 years. Two allotropes of tin occur nearroom temperature and atmospheric pressure. The first form oftin is called gray tin (α-Sn) and is stable at temperatures below13.2 ◦C [1]. At temperatures above 13.2 ◦C, gray tin slowlyturns into tin’s second form, white tin. White tin (β-Sn) hasa normal form of the metal and is very useful, for example,it can easily alloy with one or more of Pb, Cu, Ag, Zn, Sb,Bi etc, and the alloys are widely used in electronic packaging.At the same time, since the properties of tin are sensitive toexternal parameters like temperature and pressure, which israther unusual for an elemental solid, the study on tin is usefuland necessary.

∗ Corresponding address: School of Materials Science and Engineering,Shanghai Jiaotong University, 1954, Huashan road, 200030 Shanghai,PR China.

E-mail address: yuchun1980@sjtu.edu.cn (C. Yu).

0038-1098/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2006.09.026

Since tin is an element with many different properties andapplications, many experimental [2–9] and theoretical [10–17]works dealing with the structural and pressure inducedproperties of tin exist in the literature. The past researchesshow that the α-Sn has a diamond structure, and it is atypical covalently-bonded material but with a band gap equalto zero [7]. The transition of α-Sn to β-Sn is an entropy-drivenprocess, which is determined by the harmonic vibrationalproperties of both phases. The crystal structure of the β-Snis derived from a body-centered tetragonal Bravais lattice withtwo atoms in the basis at (0, 0, 0) and (a/2, 0, c/4). The atomicvolume of β-Sn is only 80% that of α-Sn. The β-Sn is stable forthe pressures up to 9.5 GPa. Above this pressure, β-Sn tends totransform to the body-centered tetragonal (bct) structure, thenit transforms finally to a body-centered cubic (bcc) structure atapproximately 50 GPa. And the bct and bcc structures co-existin the pressure range of 40–50 GPa. The bcc structure phaseremains stable up to a pressure of 120 GPa.

Specifically, Cheong [12], Christensen [13] and Andres [17]have published density functional theory (DFT) [18] calcula-tions of pressure induced phase transformations in tin withinthe local density approximation (LDA) [19]. Andres found that

C. Yu et al. / Solid State Communications 140 (2006) 538–543 539

the bcc structure will transform to hcp structure when the pres-sure is up to 163 GPa. All the theoretical predictions show apartial agreement between themselves and with experiment.

It is well know that most Sn (especially β-Sn) and Snalloys are not used under a very high pressure, so the studyof the properties of Sn, such as the structure, density of states,elastic constant near atmospheric pressure, is significant. Itis useful for predicting the properties of Sn alloys, Sn–Ag,Sn–Cu, Sn–Zn, Sn–Bi, etc., which are widely used in electronicpackaging nowadays. And also, the transition properties of Sninduced by pressure is of great interest, the existing data needbe verified and modified by different more precise methods.

In this paper, we calculate the property of Sn by DFT methodwith castep code [20], together with a comparison with previousexperimental and theoretical results to both study the propertiesof Sn and ascertain the expected accuracy of current DFT codes.

2. Methodology

The Cambridge Serial Total Energy Package (castep) wasdeveloped by Payne et al. [21]. It is a method of ‘totalenergy pseudopotential calculation’ bases on DFT in describingthe electron–electron interaction and on a pseudopotentialdescription of the electron–core interaction. It gives the sum ofelectronic energy of a large system, as well as its band structure.Transferability and robustness of the assumed pseudopotentialof each element are proven successful in reproducing thephysical properties such as lattice parameters of many metalsand compounds.

In this paper, the density functional calculations areperformed with castep code. Brillouin zone sampling isperformed using a Monkhorst–Pack [22] grid of special k point,and energy smearing of 0.1 eV is chosen for all the calculations.For the majority of this study, the generalized gradientapproximation (GGA) of Perdew and co-workers (PW91) [23,24] is chosen for the calculation of exchange–correlationenergy. The local density approximation (LDA) is also usedto compare the calculations. The ultrasoft pseudopotentialsscheme is used for this system. Before calculation, thek-point sampling and kinetic energy cutoff convergence for allsupercells are tested. With the results of the convergence tests,a kinetic energy cutoff of 450 eV and the Monkhorst–Packgrid of 10 × 10 × 10 are decided for most calculations.Structural relaxations are performed until forces on each atomare below 0.01 eV/A. The exchange–correlation functionsutilized in the atomic calculations for the pseudopotentials’generation are consistent with the functions used in the presentSn calculations. That is, pseudopotentials’ generation withinthe GGA are used only in GGA-based Sn calculations, andlikewise for the LDA.

In this paper, if not noted, the results are calculated atatmospheric pressure and zero temperature.

3. Results and discussions

3.1. Structural parameters

The equilibrium properties of α-Sn, β-Sn, bct, bcc, hcp andfcc at atmospheric pressure are listed in Table 1, and compared

Table 1Comparisons of the calculated properties, lattice constants (a), atom volume(V ), and bulk moduli (B) of various phases of Sn with previous theoretical andexperimental results at ambient pressure

a (A) c/a B (GPa) V (A3/atom)

α-Sn 6.417 45.2 33.03 LDA—this work6.452 46.7 33.57 GGA—this work6.483 42.6 34.06 Exp.—Ref. [25]

β-Sn 5.787 0.538 55.8 26.07 LDA—this work5.812 0.541 52.2 26.55 GGA—this work5.831 0.543 57.9 27.04 Exp.—Ref. [25]

bct 3.852 0.87 57.3 26.12 GGA—this work3.982 0.858 54.8 27.09 LDA—Ref. [17]3.891 0.90 52.0 26.50 LDA—Ref. [13]

bcc 3.747 62.5 26.30 GGA—this work3.733 52.8 26.01 Ref. [17]

hcp 3.282 1.635 53.2 24.91 GGA—this work3.322 1.653 51.3 26.24 Ref. [17]

fcc 4.721 53.4 26.31 GGA—this work4.735 51.4 26.54 Ref. [17]

with previous experiments and other calculated results. We cansee that, for both α and β phases, the calculated lattice constants(a) and atom volume (V ) are in good agreement with theexperimental values with the errors less than 1%. These resultsare more consistent with the experimental ones than previouscalculations. However, the bulk moduli (B) for α-Sn calculatedwithin LDA or GGA overestimates the experimental value by6.1% and 9.6%, respectively; by contrast, the bulk moduli forβ-Sn underestimates by 3.6% and 9.8%, respectively. For thelattice constants (a) or atom volume (V ), it can be seen that theresults calculated within GGA are better than those calculatedwithin LDA. But for bulk moduli (B), the results calculatedwithin GGA are worse than those calculated within LDA. Thesephenomena are also found in Andres’ work [17]. For bct, bcc,hcp and fcc phases, the calculated results of a, V and B arefound to be in good agreement with other calculations, this canbe understood from Table 1.

3.2. Band structure and density of states

There are few studies about the electronic properties of Snin the literature with DFT method. In this section, the bandstructure and density of states of α-Sn and β-Sn are analyzed toknow more about Sn theoretically.

The electronic structures are calculated in the region from−12 to 10 eV for α-Sn and −13 to 17 eV for β-Sn atatmospheric pressure. The band structures for α-Sn and β-Snare shown in Fig. 1(a), (c), and the corresponding total densityof states (TDOS) are presented in Fig. 1(b), (d), respectively.

For α-Sn, it can be seen from Fig. 1(a) that there isno conduction band or valence band across the Fermi level(indicated by the dashed red line in the graph, the energy atwhich is zero eV), and the calculated band gap energy for thediamond structure α-Sn phase is 0 eV (that is, the valence bandand conduction band intersect only at G or 0 point in the Fermilevel). This result indicates that the gray tin (α-Sn) is a zero

540 C. Yu et al. / Solid State Communications 140 (2006) 538–543

(a) Band structure for α-Sn. (b) TDOS (electrons/eV) for α-Sn.

(c) Band structure for β-Sn. (d) TDOS (electrons/eV) for β-Sn.

Fig. 1. Band structures and total density of states (TDOS). The dashed red line indicates the Fermi level at zero eV. The inset in (d) is the amplified part near theFermi level. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

band gap semiconductor [1], and the conclusion is consistentwith the experiment. For the β-Sn lattice, it can be seen fromFig. 1(c) that at atmospheric pressure there are two bands whichcross the Fermi level from the valence bands, and three bandsdown under the Fermi level from the conduction bands. Theenergy of the top of the valence bands is 1.448 eV, and that ofthe bottom of the conduction bands is −1.565 eV. Therefore,based on the calculations we can know that the β-Sn has anoverlap energy gap (indirect gap) [26], the energy is −3.013 eV.And this is one of the properties for the metal.

In order to study the effects of pressure on the electronicproperties of tin, the band structure and TDOS of β-Sn atdifferent pressures are calculated, and the results calculated ata pressure of 15 GPa are shown in Fig. 1(c), (d), indicated bydashed lines. It can be seen that both the valence bandwidth and

conduction bandwidth increase with pressure, the energy rangeincreases to about −15 to 19 eV at pressure of 15 GPa. Thenumber of bands across the Fermi level decreases from 5 to 4,but the indirect gap energy increases with pressure.

At the Fermi level, the TDOS for α-Sn is 0.784 electrons/eVand that for β-Sn is 2.183 electrons/eV at atmosphericpressure. But for β-Sn, we can see from the amplified partnear the Fermi level (the inset, as shown in Fig. 1(d)) that theTDOS at the Fermi level decreases greatly when the pressure isenhanced to 15 GPa, it is 1.665 electrons/eV.

From the analysis of the band structure and density ofstates of α-Sn and β-Sn, it is not difficult to deduce thatα-Sn is a semiconductor with zero band gap, and β-Sn is withmetallic property at ambient pressure, this is consistent with theexperimental result.

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Fig. 2. Partial density of states (PDOS). The dashed line indicates the Fermilevel at zero eV.

For clearly learning the electronic properties of Sn, thepartial density of states (PDOS) of α-Sn and β-Sn at normalpressure are calculated, the results are shown in Fig. 2. Thepseudo-atomic calculation is only performed for 5s and 5pstates in this work. From the PDOS graph, we can see thatthe contributions for the sum of DOS of α-Sn and β-Sn at theFermi level are mostly from p state. As far as the PDOS ofα-Sn are concerned (as shown in Fig. 2(a)), the 5s state of α-Snhas greater contribution to the valence bands zone from −12to −4.5 eV, while at the Fermi level the PDOS of 5s state is0 electrons/eV. And the 5p state contributes greatly to the restof the valence bands and all the conduction band zones but the5s state has little contribution. For β-Sn, we can know fromFig. 2(b) that the bigger contribution of the 5p state to the sumof the DOS is in the zone from −4.5 eV to 9.5 eV, and that thesum of DOS in the zone from −12 to −4.5 eV and 9.5 to 17 eVare mostly contributed by the 5s state. At the Fermi level, thePDOS of 5s and 5p states are about 0.44 and 1.95 electrons/eV,respectively. The peaks of the sum of DOS for α-Sn and β-Snare all near the Fermi level, but the former is twice that of thelatter.

3.3. Effects of pressure on the properties of β-Sn

Under high pressure, the lattice parameters will change, andthen the volume of atom will change, too, which finally affectsthe structural properties. In order to make clear the relationshipbetween the structural change and pressure, the volume changeof β-Sn is calculated within GGA. The results are listed inFig. 3 (the previous calculations are also listed for comparison),where, in the y-axis, V0 indicates the volume of the β-Sn atatmospheric pressure, 27.037 A3, and Vp indicates the volumeat different pressures. We can see from the figure that therelative volume decreases almost linearly with the pressure.But the c/a ratio changes little. The calculated results arecomparable with the earlier data.

Fig. 3. Relationship between relative volume changes and pressure ofβ-Sn �—calculated in this work; ©—others’ work [12]. V0—volume atatmospheric pressure, Vp—volume at high pressure.

Fig. 4. Relationship between DOS and pressure at the Fermi level.

Fig. 4 shows the changes of the DOS at the Fermi levelof β-Sn under different pressures. Density of states (DOS) isa property in condensed matter physics that quantifies howclosely packed the energy levels are in some physical systems.The value of DOS at the Fermi level reflects much information,such as the conductivity of the material [26], stable ability, etc.It can be seen from Fig. 4 that with the increase of pressure,the DOS at the Fermi level decreases greatly. At atmosphericpressure, the DOS is 2.183 electrons/eV, however the valuechanges to 1.902 and 1.665 electrons/eV when the pressureis up to 9.5 and 15 GPa, respectively. If there is no structuretransition, the value will decrease continuously.

The elastic constants of α-Sn and β-Sn are also calculatedat normal pressure to compare with the previous experimentsand calculations, all the results are listed in Table 2. We can seefrom the table that all elastic constants calculated in this workare in excellent agreement with the previous ones.

3.4. Phase transition calculation

In order to obtain the phase transition sequence and validatethe previous works, a number of structural optimizations and

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Table 2Elastic constants of α-Sn and β-Sn, all values in GPa

C11 C33 C12 C13 C44 C66

α-Sn 71.3 32.4 43.5 GGA67.6 31.0 44.0 Andres’ work [17]69.0 29.4 42.6 Exp. [27]

β-Sn 72.9 93.2 62.3 36.5 22.8 24.6 GGA73.7 95.3 61.9 32.8 23.2 24.7 Andres’ work [17]73.4 90.7 59.9 39.1 21.9 23.9 Exp. [27]

Fig. 5. Relation curves of energies vs pressures. P1, P2, P3, P4 represent thetransition pressures of the α-Sn to β-Sn, the β-Sn to bct, the bct to bcc, and thebcc to hcp, respectively.

energy calculations are performed for α-Sn, β-Sn, bct, bccand hcp phases at different pressures of 0–200 GPa. All thecalculations are within GGA to solve the exchange–correlationenergy, the cutoff energy is set to 450 eV. The unit cells of eachphase for these optimizations are completely unconstrainedexcept for the crystal symmetries. The main calculated resultsare presented in Fig. 5 and Table 3.

It is well known that, at the same conditions, lower theenergy, more stable the structure. Fig. 5 shows the total energyof each phase calculated for T = 0 as a function of pressure.Through comparing the energy of each phase at ambientpressure, the α-Sn is found with the least energy, so it is the

most stable structure at this condition. Then it is followed byβ-Sn, bct, bcc and hcp structure. According to the calculatedresults, we predict that the transition sequence will be α-Sn →

β-Sn → bct → bcc → hcp (this prediction will be confirmedin the following paragraph), that is in good agreement withexperimental and theoretical results.

To confirm the phase transition pressure at 0 K, we comparethe energy–pressure curve of each phase, and find that thereexists a pressure P for each couple (P1 for α-Sn and β-Sncouple, P2 for β-Sn and bct couple, P3 for bct and bcc couple,and P4 for bcc and hcp couple, as shown in Fig. 5), at which theenergies for the two phases of each couple are approximatelyequal. However, it can be seen from Fig. 5 that when thepressure is less than the value P , the energy (eV/atom) for theformer phase of each couple is less than that for the latter. Sounder this condition, the former phase is more stable than thelatter; and vice versa. Therefore, in addition to the calculationsof structural changes, we deduce that this pressure P is thephase transition pressure.

According to the above method, the energy and structurecalculations of the phases at different pressures indicate thatthe α → β transition is at P1, which is 0.7 GPa, and is ingood agreement with the results of Cheong and Chang [12],0.8 GPa, and Andres’ 0.75 GPa. The difference of the minimumenergies for the α-Sn and β-Sn phases is about 0.078 eV/atom,and it is only a little larger than the previous results of0.034 eV/atom [12] and 0.04 eV/atom [10]. Since it is wellknown that the α-Sn to β-Sn transition can take place at ambientpressure and below 13 ◦C, the Debye temperature for α-Snis higher than that for β-Sn. In addition, the entropy termincreases more rapidly for the β-Sn phase [10,12], so thepressure induced α-Sn to β-Sn transition is expected to occurat a lower pressure as the temperature increases. The volumecollapse for the transition is as big as 21.2%, that is comparableto the previous experimental and theoretical results as shown inTable 3.

The β → bct phase transition is located at 12 GPa. Thispressure is closer to the experimental value of 9.5 GPa, andit is a little less than Christensen’s calculation of 13 GPa [13]and Andres’ calculation of 15.9 GPa, the latter was calculatedusing SIESTA code within local density approximation (LDA).Although the volume decreases greatly in this transition (from0.822V0 to 0.795V0, as shown in Table 3), the c/a ratio of the β

phase changes very little at the pressure range from 0 to 20 GPa.On the contrary, the c/a ratio of the bct phase steadily increaseswith the pressure, and at 45 GPa, the c/a ratio jumps to 1. So thepressure of 45 GPa is expected as the bct to bcc phase transitionpressure, this is also in very good agreement with experimentalpredictions (40–50 GPa) and other calculations (Cheong andChang predicted a value of 46 GPa, and Andres’ calculationresult was 42.9 GPa). This transition volume is estimated to be0.68V0 or 18.07 A3/atom. It is also in good agreement with theobserved results.

There are many different opinions about the transitionfrom bcc to hcp phase. Cheong and Chang predicted thetransition pressure of 61 GPa and the transition volume of0.65V0 [12]. These results are underestimated for neglecting

C. Yu et al. / Solid State Communications 140 (2006) 538–543 543

Table 3

Transition pressure (in GPa) and volume (in A3/atom) for T = 0, and volume ratios Vt /V0, V0 indicates the calculated equilibrium volume of β-Sn phase,26.55 A3/atom

Vt (α-Sn) Vt (β-Sn) Vt (bct) Vt (bcc) Vt (hcp) P

33.12/1.250 26.07/0.982 0.7 GGA—this work21.82/0.822 21.16/0.795 12

18.07/0.681 18.06/0.680 4513.53/0.510 13.48/0.508 160

33.01 25.95 0.75 LDA—previous work [17]21.93 21.41 15.9

18.14 18.10 42.913.40 13.38 163

22.26 22.26 9.5 Exp. [2,3]17.71 17.71 40–50

the core–core interaction in their calculations. Christensen andMethfessel found the bcc phase to be more stable at least upto 200 GPa. Experimentally, the bcc phase is observed to bemore stable up to 120 GPa. Andres’ calculation predicted that abcc → hcp transition will generate at 163 GPa. In this paper,the calculations for the bcc and hcp structures are performedfrom 0 to 200 GPa, and the energy vs pressure curves are shownin Fig. 5(b), and P4 is determined to the transition pressure,160 GPa. The transition volume is 0.510V0 or 13.53 A3/atom,and the volume change between the two phases is very little.There is no exact experimental observation about the bcc →

hcp phase transition to validate the calculated results, but atleast, the results are consistent with Andres’ prediction.

There is still a lot of work needed to be done on Sn,especially β-Sn, such as, experimental and theoretical work tomake clear the phase transition behaviour of bcc to hcp andhcp to fcc, the study of the electronic structures, mechanicalproperties and other related important properties of Sn alloys.Those are the later works we will do.

4. Conclusions

In this work, the electronic structural, mechanical andphase transitional properties of Sn have been studied usinga self-consistent ab initio pseudopotential density functionaltheory method with castep code at zero temperature. Most ofthe results are much closer to the experimental observationsthan previous calculations. The main conclusions are listed asfollowing,

(1) The analysis of the electronic structures indicates that theα-Sn is a zero band gap semiconductor and β-Sn hasmetallic property, this is agreement with experiment.

(2) The TDOS of α-Sn and β-Sn at the Fermi level are0.784 and 2.183 electrons/eV at ambient pressure, and thevolume and TDOS of β-Sn decrease with the pressure.

(3) As pressure increases, successive phase transitions wasfound from α-Sn to β-Sn at 0.7 GPa, to bct at 12 GPa, to bcc

at 45 GPa and to hcp at 160 GPa. The transition sequenceα-Sn → β-Sn → bct → bcc → hcp is consistent withexperiment.

Acknowledgement

This work was supported by National Natural ScienceFoundation of China under Grant No. 50475021.

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