Structural stabilities, electronic, elastic and optical properties of SrTe under pressure: A first-principles study

Physica B 406 (2011) 181–186

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Structural stabilities, electronic, elastic and optical properties of SrTe under pressure: A first-principles study Liwei Shi , Yifeng Duan, Xianqing Yang, Lixia Qin Department of Physics, School of Sciences, China University of Mining and Technology, Xuzhou 221116, China

a r t i c l e in f o

abstract

Article history: Received 11 March 2010 Received in revised form 27 August 2010 Accepted 14 October 2010

The structural, electronic, elastic and optical properties as well as phase transition under pressure of SrTe have been systematically investigated by first-principles pesudopotential calculations. Five possible phases of SrTe have been considered. Our results show that SrTe undergoes a phase transition from NaCl-type (B1) to CsCl-type (B2) structure at 10.9 GPa with a volume collapse of 9.43%, and no further transition is found. We find that SrTe prefer h-MgO instead of wurtzite (B4) structure for its metastable phase because that the ionic compound prefers a high coordination. The elastic moduli, energy band structures, real and imaginary parts of the dielectric functions have been calculated for all considered phases, and we find that a smaller energy gap yields a larger high-frequency dielectric constant. Our calculated results are discussed and compared with the available experimental and theoretical data. & 2010 Elsevier B.V. All rights reserved.

Keywords: First-principles Metastable phase Phase transition Elastic property Electronic property Optical property

1. Introduction Strontium telluride, SrTe, like other alkaline-earth tellurides has recently attracted significant attention of the researchers because of its potential technological applications ranging from catalysis to microelectronics and optoelectronics [1–4]. SrTe is an important closed-shell ionic compound, which crystallizes in NaCl-type (B1) structure at ambient conditions. However, high pressure X-ray diffraction experiment has shown that on applying pressure up to about (1271) GPa, SrTe undergoes a structural phase transition to CsCl-type (B2) structure with a relative volume collapse of (11.170.7)% [5]. Several experimental and quite a few ab initio theoretical studies involving method of full potential linear augmented plane waves (FP-LAPW), pseudopotential plane wave (PP-PW), two body interionic potential approach and other calculations within the density functional theory (DFT) have been performed to study the structural stabilities, electronic energy band structures and optical properties of strontium chalcogenides over the last two decades [6–19]. Only B1 and B2 phases of SrTe, however, had been considered in most former theoretical studies, which we thought that it was inadequate. Moreover, so far little theoretical work has been done to study the elastic constants and related properties, as well as band structures and optical properties of SrTe, especially for its metastable phases.

 Corresponding author. Tel.: +86 516 83591580.

E-mail address: [email protected] (L. Shi). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.10.038

Being an intermediate structure between wurtzite (B4) and B1 phase, the h-MgO structure has been systematically studied for theoretical cases [20–23]. The B4 structure is characterized by three parameters: the lattice constant a, the c/a ratio and the internal parameter u, which fixes the relative position of the two hexagonal close-packed sublattices. For the ideal B4 structure, c/a¼ (8/3)1/2, and u¼0.375 [24], whereas for the h-MgO structure, c=a r 1:2, and u ¼0.5, and the space group changes from P63mc to P63/mmc [21]. The phase transformation from B4 to h-MgO structure could be a consequence of an applied external uniaxial compression along the c-axis. The coordination of h-MgO increases from fourfold to fivefold, resulting in higher symmetry. Since the ionic compounds prefer a high coordination and given that the ionicity of the group-IIA tellurides increases as the cation atomic number increases, the questions are as follows: (I) Being an important closed-shell ionic compound, does SrTe prefer h-MgO or B4 structure for its metastable phase? (II) How does the atomic configuration influence the elastic, electronic and optical properties of SrTe? With these motivations mentioned above, in this work we have employed the first-principles plane-wave pesudopotential method to investigate structural stabilities, electronic, elastic and optical properties of SrTe under hydrostatic pressure considering some potential phases including B1, B2, zincblende (B3), B4 and NiAs-type (B8) structures, which other group-II tellurides possess at ambient or high pressure conditions. In Section 2, we describe the computational methods used in this work, then we report and discuss the main results of this work in Section 3. Finally, a brief summary will be given in Section 4.

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2. Computational methods Our calculations are performed within the local density approximation (LDA) to density functional theory (DFT), as implemented in the plane-wave pseudopotential ABINIT package [25]. To obtain good convergence, the Brillouin zone integration is performed with 8  8  8 k-meshpoints. The energy cutoff of plane-wave basis is set to 70 Ry. The norm-conserving pseudopotentials are generated using the OPIUM program [26], which are rigorously tested against the all-electron FP-LAPW method [27]. The Sr 4s24p65s2 and Te 5s25p4 orbitals are explicitly included as valence electron states and all lower-lying states as part of the core. The relative stability of different structural phases is examined by comparing the Gibbs free energy value, which can be obtained from G ¼ Etot þPV þ TS

ð1Þ

Since in the theoretical calculations, we only consider the zerotemperature limit, T¼ 0 K, the Gibbs free energy equals the enthalpy: H ¼ Etot þPV

ð2Þ

3. Results and discussions 3.1. Structural parameters and phase transition To start with the structural properties, the total energy is calculated as a function of the unit-cell volume around the equilibrium cell volume V0 for all the considered phases, in which the cell shape and internal parameters are fully relaxed. By fitting the Murnaghan equation of state (EOS) [28], we obtain the equilibrium lattice constants a, the isothermal bulk modulus B0 and its pressure coefficient B0u ¼ ðdB0 =dpÞp ¼ 0 , as well as the lowest total energy per cation–anion pair Etot. The calculated structural parameters for all five phases of SrTe at ambient pressure are listed in Table 1. Note that the equilibrium lattice constants for both B1 and B2 phases are in good agreement with other theoretical values [7,8,10–12], but are slightly smaller than the available experimental data [5] due to the LDA underestimation of volume, suggesting that the experimental volumes correspond to the Table 1 Calculated lattice constants a, bulk modulus B0 and its pressure derivative B0u , as well as lowest total energy Etot for SrTe polymorphs at ambient conditions. Phases

˚ a (A)

B0 (GPa)

B0u

Etot (meV)

Refs.

B1

6.531 6.659 7 0.006a 6.661b, 6.76d 6.48c, 6.64e

39.1 39.5 7 3a 37.6b, 36.0d 44c, 39.2e

4.4 5a 3.23b 3.927c

0

Present Exp. Theory

B2

3.963 3.990a 4.051e, 4.123c 3.95d, 4.009b

43.9

4.2

423

31.5e, 35c 45d, 45.6b

3.56c 3.64b

Present Exp. Theory

B3

7.286

24.8

2.4

775

Present

h-MgO

5.547 (c/a ¼1.167, u¼ 0.500)

30.2

3.8

402

Present

B8

4.540 (c/a ¼1.744, u¼ 0.250)

39.2

4.4

125

Present

The c/a ratio and internal parameter u for hexagonal h-MgO and B8 phases are given in the brackets. The Etot of B1 phase is set to zero as reference. Comparison is made with the available experimental and theoretical data. a

Ref. [5]. b Ref. [7]. c Ref. [10]. d Ref. [8]. e Ref. [11].

Fig. 1. (a) Enthalpy differences versus pressure for all considered structures of SrTe with respect to B1 phase; (b) Pressure–volume relation of SrTe in the most stable structural phases.

negative hydrostatic pressure. The calculated bulk moduli B0 and its pressure coefficient B0u are in consistent well with available experimental [5] and other theoretical values [7,8,10,11]. The pressure derivative B0u is almost independent of the crystal structure, with an estimated error of less than 0.5. Note that the E(V) for h-MgO and B8 phases are fully relaxed values with respect to c/a and u. By comparing Etot for all five phases, we find that B1 phase, whose Etot is set to zero as reference, is the ground-state configuration of SrTe at ambient conditions. Since the ionic compound prefers a high coordination and strontium has higher ionicity, SrTe display h-MgO structure (c/a¼ 1.167, u¼0.5) instead of wurtzite structure for its metastable phase. It should be noted that the initial data (the c/a ratio and the internal parameter pffiffiffiffiffiffiffiffiffiu) prepared for full relaxation of SrTe satisfy the conditions c=a ¼ 8=3 and u¼3/8 of ideal wurtzite. The calculated relative enthalpy values as a function of applied hydrostatic pressure to SrTe solids for B1, B2, B3, h-MgO, and B8 phases are shown in Fig. 1(a). The transition pressure (PT) is at the cross-points of the H(P) curves for these phases. As shown in Fig. 1(a), there exists a phase transition from B1 phase to B2 phase. Other structures such as B3, h-MgO and B8 structures are unstable as high pressure phase in our calculated pressure range. Fig. 1(b) shows the pressure–volume relation of SrTe in the most stable structural phases. It can be seen that there is a reduction in volume accompanying the B1-to-B2 transition. As shown in Table 2, the calculated transition pressure PT is about 10.9 GPa, with a volume collapse DV=VT1 value of about 9.43%, which is in good agreement with the available experimental data [5] and better than other theoretical reports [6–8,10]. 3.2. Elastic properties The mechanical stability conditions under ambient pressure can be expressed in terms of elastic constants, for the cubic B1, B2 and B3 structures as: c44 40,

c11 4jc12 j,

c11 þ2c12 40

ð3Þ

for the hexagonal h-MgO and B8 structures as c44 40,

c11 4jc12 j,

2 c33 ðc11 þc12 Þ 4 2c13

ð4Þ

The calculated elastic constants at equilibrium lattice constants of five polymorphs for SrTe given in Table 3 satisfy these stability conditions. For the ground-state B1 phase of SrTe, the elastic constants are calculated at optimized and experimental structures, respectively. It is evident that the calculated values are strongly

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183

Table 2 Calculated transition pressure PT, relative volume VT1/V01, as well as volume collapse DV=VT1 for the B1-to-B2 phase transition in SrTe. PT (GPa)

VT1/V01

V02/V01

DV=VT1 ð%Þ

Refs.

10.9 12 7 1a 12.9b, 13.24c 11.2d, 10.3e

0.831 0.828a

0.894 0.876a

Present Exp. Theory

0.8d

0.910d

9.43 11.1 7 0.7a 7.30b, 8.92c 8.0d, 7.30e

V02/V01 is the hypothetical zero-pressure volume of the B2 phase. Comparison is made with the available experimental and theoretical data. a

Ref. [5]. Ref. [6]. c Ref. [7]. d Ref. [8]. e Ref. [10]. b

Table 3 Calculated elastic constants cij (in GPa) and corresponding bulk modulus B0(1) for SrTe polymorphs at ambient conditions, along with the available theoretical data for comparison. Phases

c11

c12

B1

98.4 74.5 73.04a 101.5b 52.1c

B2

c13

c33

c44

B0(1)

Refs.

10.2 8.6 16.27a 7.8b 32.8c

15.6 15.4 16.78a 44.8b 6.14c

39.6 30.6

COS CES Theory

114.2 108.9 99.7b 60.6c

8.8 6.2 3.3b 37.1c

6.8 5.1 41.1b 8.28c

43.6 40.4

COS CES Theory

B3

26.0

24.7

3.3

25.1

COS

h-MgO

44.5

30.6

8.0

87.3

11.4

29.6

COS

B8

60.5

28.4

24.6

80.5

29.8

39.2

COS

COS and CES denotes the results calculated at optimized and experimental structure, respectively. a b c

Ref. [6]. Ref. [10]. Ref. [11].

c11 þ2c12 3

ð5Þ

For h-MgO and B8 phases [29–31], B0ð1Þ ¼

2 ðc11 þ c12 Þc33 2c13 c11 þc12 þ 2c33 4c13

shear as the phase transition is approached. The pressure derivative @c44 =@P is 0.265, which is in well-consistent with previously reported theoretical value of 0.253 calculated using two body interionic potential approach with modified ionic charge [6]. The shear wave modulus G and Young’s modulus E for the B1 phase of SrTe can be estimated by G¼

c11 c12 2

ð7Þ

E¼

ðc11 c12 Þðc11 þ2c12 Þ c11 þ c12

ð8Þ

The calculated pressure dependencies of bulk, shear and Young’s moduli for B1 phase of SrTe are illustrated in Fig. 2(b). All B0(1), G and E increase almost linearly with pressure. This material is hard to be broken at least in the calculated pressure range because the Young’s modulus is much higher than the bulk modulus. 3.3. Electronic band structures

affected by the relaxation of c/a and u. The elastic constants calculated at optimized structures are larger than those calculated at experimental structures due to the LDA underestimation of volume. Our calculated results are compared with other theoretical data [6,10,11] but only for B1 and B2 phases. So far no theoretical or experimental data are available for B3, h-MgO and B8 phases of SrTe. To check the accuracy of our results, we calculate the isothermal bulk moduli B0(1) using the corresponding elastic constants. For B1, B2 and B3 structures [6,29], B0ð1Þ ¼

Fig. 2. Calculated (a) elastic constants cij and (b) bulk modulus B0(1), shear wave modulus G and Young’s modulus E versus pressure for B1 phase of SrTe.

ð6Þ

The bulk moduli B0(1) and B0 obtained by fitting E(V) to the EOS listed in Table 1 are in very good agreement, indicating the computational accuracy of elastic constants and bulk moduli. Fig. 2(a) shows the variation of elastic constants with pressure for B1 phase of SrTe, which can reveal many important features of the short range forces at high pressure. Both c11 and c12 increase monotonically with pressure, and the slope of c11 is greatly larger than that of c12. However, c44 decreases monotonically with the increase of applied pressure, showing the reduction in resistance to

Knowledge of the energy band structures provides valuable information regarding its potential utility in fabricating electronic devices. Since SrTe is an promising material for application in solid state devices, accurate knowledge about its band structure in all the possible phases becomes essential. The band structures, total density of states (TDOS) and partial density of states (PDOS) for all possible phases, B1, B2, B3, h-MgO and B8 of SrTe calculated at the equilibrium volumes within DFT-LDA are illustrated in Fig. 3(a)–(e), which clearly indicates energy band structure strongly dependent of the atomic arrangement pattern. The variation in the energy band is associated with the change of symmetry at the structural transformation, as the tertrahedrally coordinated B3 phase and fivefold coordinated h-MgO phase possess no inversion symmetry center in contrast to the cubic B1 and B2 phases. As shown in Fig. 3, the valence-band maximum (VBM) occurs at the G point in B1, h-MgO and B8 phases, while it occurs at the M point in B2 and the X point in B3 phase. The conduction-band minimum (CBM) occurs at the M point in B2 and B8 phases, while it occurs at the X, G and K points for B1, B3 and h-MgO phases, respectively. As a consequence, the band gap is direct only in B2 phase and indirect in all other phases of SrTe at ambient conditions. The values of fundamental energy gap (Eg) and the energy gap at the G point (EG g ) for all the possible phases of SrTe are summarized in Table 4, along with the existing experimental data [32] for comparison. It is well-known that in the energy band structure

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Fig. 3. Band structures and density of states for SrTe polymorphs under ambient conditions calculated within the DFT-LDA framework: (a) B1, (b) B2, (c) B3, (d) h-MgO and (e) B8. The shaded region indicates the fundamental gap.

calculations within DFT, both LDA and GGA usually underestimate the energy band gap [33]. The calculated value (1.458 eV) of indirect (GX) band gap for most-studied B1 phase of SrTe is underestimated as compared to the experimental value of 2.9–3.4 eV. The direct (M–M) band gap for B2 phase of SrTe is 0.199 eV. From the PDOS one can identify the angular momentum character of various structures. In all phases of SrTe, two band

groups originate from tellurium atoms in the valence band. The lower weakly dispersion bands is totally dominated by Te 5s states, which are separated by a large interanionic valence gap from Te 5p bands of the uppermost valence bands. The lower part of conduction band is dominated mainly by Sr 4d states, with some contributions of Sr 5s states. In case of B2 phase, one can note the broadening of the valence band due to Te 5p states resulting

L. Shi et al. / Physica B 406 (2011) 181–186

185

Table 4 Calculated fundamental energy gap (Eg) and gap at the G point (EG g ) of SrTe polymorphs at ambient conditions, along with the available experimental data for comparison.

Eg(eV) Exp. EG g ðeVÞ a

B1

B2

B3

h-MgO

B8

1:458ðG2XÞ 2.9–3.4a 2:855ðG2GÞ

0.199(M–M)

2:634ðX2GÞ

1:873ðG2KÞ

1:601ðG2MÞ

1:332ðG2GÞ

2:837ðG2GÞ

2:101ðG2GÞ

1:825ðG2GÞ

Ref. [32].

from structural change under pressure which is from a relative open to more dense atomic arrangement. This maybe a possible reason for change of the nature of the band gap in this phase. 3.4. Optical properties It is known that the dielectric function not only pertains to the electronic linear response, but also mainly determines the optical properties. In the framework of the linear-response theory [34], the imaginary part of the dielectric function e2 can be written as !Z X Ve2 h3 dk j/knj@=@xjkunuSj2 e2 ðEÞ ¼ 8p3 m2 E2 fFD ðEkn Þð1fFD ðEknu ÞÞdðEkn Eknu EÞ

ð9Þ

where h is the Planck constant, E is the energy of an incident photon and fFD(Ekn) is the Fermi-Dirac function of the electron occupied as the jknS state. Other notations have their usual meanings such as electron mass m and charge e, and so on. The real part e1 ðoÞ of the dielectric function is related to its imaginary part e2 ðoÞ by the ¨ Kramers–Kronig formula. Fig. 4 demonstrates the influence of the crystallographic structure on the dielectric function of SrTe, which is calculated using the optical package in the WIEN2K code [35]. The calculated imaginary e2 ðoÞ and real part e1 ðoÞ of the dielectric functions in the energy range from 0 to 25 eV for all the five possible phases are shown in Fig. 4. As one can see, the change of the coordination of the atoms significantly affects the dielectric function, and the best noticeable is the imaginary part e2 ðoÞ, which is proportional to the summation over all distinct optical dipole–dipole transition probabilities between interbands. The first critical point of the e2 ðoÞ occurs at about 2.06, 1.20, 2.69, 2.48 and 2.15 eV for B1, B2, B3, h-MgO and B8 phases, respectively, and is attributed to the threshold for the direct optical transitions between the valenceband maximum and the first conduction-band minimum for the respective phase. Note that because of the restriction of FermiDirac function on electron transition between interbands, the dielectric function reflects the joint feature of electronic bands [36,37]. In the infrared region, the e2 ðoÞ of the dielectric functions of B1, B2, h-MgO and B8 structures display multi-peak characteristics, whereas those of B3 phase mainly show nearly a single-peak absorption. As this possesses cubic zincblend structure with the lowest crystallographic symmetry among the five phases, and some electronic exist states are forbidden to optical dipole–dipole transitions. The main features of the e1 ðoÞ curves are: the occurrence of strong peaks in the energy range 1.4–4.1 eV, then a rather steep decrease between 3.5 and 4.8 eV for B2 and B3 phases, while for B1, h-MgO and B8 phases the decrease is over a broader range between 2.7 and 6.0 eV, after which it becomes negative, then a minimum followed by a slow increase towards zero at higher energies. The calculated high-frequency electronic dielectric constant e1 is 7.70, 9.22, 6.54, 5.18 and 6.74 for B1, B2, B3, h-MgO and B8 phases,

Fig. 4. Imaginary and real parts of the frequency-dependent dielectric functions for SrTe polymorphs under ambient conditions: (a) B1, (b) B2, (c) B3, (d) h-MgO and (e) B8 as calculated within the independent-particle approximation using Kohn– Sham eigenstates and eigenvalues from DFT-LDA. In the case of h-MgO and B8, the tensor components exx ¼ eyy (solid line) as well as the zz-component ezz (dotted line) is presented.

respectively, indicating that a smaller energy gap yields a larger e1 value. Note that for h-MgO and B8 phases, e1 ¼ ðexx þ eyy þ ezz Þ=3. Fig. 5 demonstrates the pressure dependence of fundamental energy gap Eg and high-frequency electronic dielectric constant e1 for B1 phase of SrTe. The energy gap decreases linearly, while highfrequency dielectric constant increases linearly with increasing pressure. This is attributed to band widening by strengthened overlap interaction of the wave function with the shrinking atomic distance under high pressure. Despite that so far no experimental results are available, the values of e1 calculated from LDA are expected to be larger than the experimental data due to the underestimation of band gap within the DFT.

4. Summary We have systematically investigated the structural, electronic, elastic and optical properties of all possible phases of SrTe using first-principles plane-wave pesudopotential calculations. Our calculations show that SrTe undergoes a phase transition from B1 to B2 structure at about 10.9 GPa with a volume collapse of 9.43%, and no further transitions are found up to 50 GPa. Since the ionic compound prefers a high coordination, SrTe prefers h-MgO instead of B4 structure for its metastable phase. The real and imaginary parts of the dielectric

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Fig. 5. The calculated pressure dependence of fundamental energy gap Eg and highfrequency electronic dielectric constant e1 for B1 phase of SrTe.

functions have been calculated for all five possible phases of SrTe, and we find that a smaller energy gap yields a larger high-frequency dielectric constant. Moreover, the influence of hydrostatic pressure on elastic moduli and energy gaps of B1 phases has also been investigated. Our calculated results are in reasonable agreement with some available experimental and theoretical data, and are expected to be useful when SrTe material is considered for related applications.

Acknowledgments The work is supported by the National Natural Science Foundation of China under Grant no. 10947119 and the Youth Science Funds of China University of Mining and Technology under Grant nos. 2009A040 and 2009A048. Finally, many thanks for Editor and Reviewers’ positive comments and useful suggestions. References [1] Y. Nakanishi, T. Ito, Y. Hatanaka, G. Shimaoka, Appl. Surf. Sci. 66 (1992) 515. [2] L. Abtin, G. Springholz, Appl. Phys. Lett. 93 (2008) 163102.

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