Lattice dynamics and chemical bonding in Sb2Te3 from first ... · Pressure effects on the lattice...

J. Chem. Phys. 142, 174702 (2015); https://doi.org/10.1063/1.4919683 142, 174702

© 2015 AIP Publishing LLC.

Lattice dynamics and chemical bonding inSb2Te3 from first-principles calculations

Cite as: J. Chem. Phys. 142, 174702 (2015); https://doi.org/10.1063/1.4919683Submitted: 13 November 2014 . Accepted: 22 April 2015 . Published Online: 05 May 2015

Bao-Tian Wang, Petros Souvatzis, Olle Eriksson, and Ping Zhang

ARTICLES YOU MAY BE INTERESTED IN

Phonon spectrum and bonding properties of Bi2Se3: Role of strong spin-orbit interaction

Applied Physics Letters 100, 082109 (2012); https://doi.org/10.1063/1.3689759

Dynamics of all the Raman-active coherent phonons in Sb2Te3 revealed via transient

reflectivityJournal of Applied Physics 117, 143102 (2015); https://doi.org/10.1063/1.4917384

Electronic topological transition and semiconductor-to-metal conversion of Bi2Te3 under

high pressureApplied Physics Letters 103, 052102 (2013); https://doi.org/10.1063/1.4816758

https://images.scitation.org/redirect.spark?MID=176720&plid=1007006&setID=378408&channelID=0&CID=326229&banID=519800491&PID=0&textadID=0&tc=1&type=tclick&mt=1&hc=9f34f1cb620828b56eaf9bdebc7286d6c339bcff&location=https://doi.org/10.1063/1.4919683https://doi.org/10.1063/1.4919683https://aip.scitation.org/author/Wang%2C+Bao-Tianhttps://aip.scitation.org/author/Souvatzis%2C+Petroshttps://aip.scitation.org/author/Eriksson%2C+Ollehttps://aip.scitation.org/author/Zhang%2C+Pinghttps://doi.org/10.1063/1.4919683https://aip.scitation.org/action/showCitFormats?type=show&doi=10.1063/1.4919683http://crossmark.crossref.org/dialog/?doi=10.1063%2F1.4919683&domain=aip.scitation.org&date_stamp=2015-05-05https://aip.scitation.org/doi/10.1063/1.3689759https://doi.org/10.1063/1.3689759https://aip.scitation.org/doi/10.1063/1.4917384https://aip.scitation.org/doi/10.1063/1.4917384https://doi.org/10.1063/1.4917384https://aip.scitation.org/doi/10.1063/1.4816758https://aip.scitation.org/doi/10.1063/1.4816758https://doi.org/10.1063/1.4816758

THE JOURNAL OF CHEMICAL PHYSICS 142, 174702 (2015)

Lattice dynamics and chemical bonding in Sb2Te3from first-principles calculations

Bao-Tian Wang,1,2,a) Petros Souvatzis,2 Olle Eriksson,2,b) and Ping Zhang31Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan 030006, China2Department of Physics and Astronomy, Division of Materials Theory, Uppsala University, P.O. Box 516,SE-75120 Uppsala, Sweden3LCP, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

(Received 13 November 2014; accepted 22 April 2015; published online 5 May 2015)

Pressure effects on the lattice dynamics and the chemical bonding of the three-dimensional topo-logical insulator, Sb2Te3, have been studied from a first-principles perspective in its rhombohedralphase. Where it is possible to compare, theory agrees with most of the measured phonon dispersions.We find that the inclusion of relativistic effects, in terms of the spin-orbit interaction, affects thevibrational features to some extend and creates large fluctuations on phonon density of state inhigh frequency zone. By investigations of structure and electronic structure, we analyze in detailthe semiconductor to metal transition at ∼2 GPa followed by an electronic topological transition at apressure of ∼4.25 GPa. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4919683]

I. INTRODUCTION

Topological insulators (TIs) are novel materials that havea bulk energy gap that coexists with gapless Dirac fermionstates on the surface, due to the effects of spin-orbit interaction(SOI).1–5 Among the three-dimensional (3D) TIs discovered,the group V chalcogenides Bi2Se3, Bi2Te3, and Sb2Te3 havespurred tremendous interests by virtue of offering physicalproperties suitable for future spintronic devices, quantumcomputing, and photonics.6–9 For example, Zhang et al. haveconfirmed pressure-induced superconductivity in Bi2Te3 byresistance measurements;10 Jiang et al. have experimentallyfound that the Fermi level of Sb2Te3 can be tuned over theentire range of the bulk band gap by regulating intrinsicdefects and substrate transfer doping.11 Furthermore, thesecompounds are also well known for harboring substantialthermoelectric effects at ambient conditions.12

The study of physical properties in these materials undercompression is receiving increasing attention especially dueto continuous experimental and theoretical developments.13–22

The high-pressure phases II, III, and IV of Bi2Te3 were recog-nized by x-ray diffraction experiments with the additionalhelp from the particle swarm optimization algorithm.13–15 Thesame type of phases was observed in Bi2Se3 and Sb2Te3 bythe experiment group of Vilaplana et al.16,17 Here, the phasetransition from the α phase I to the β phase II was between therhombohedral R3̄m crystal structure to the monoclinic C2/mstructure at 10 GPa for Bi2Se3, 7.4 GPa for Bi2Te3, and 7.7 GPafor Sb2Te3.

In a recent work,23 the phonon dispersions of Bi2Se3 andBi2Te3 at zero pressure were calculated from first-principlestheory. In the present study, however, we take a step further andfocus on Sb2Te3 in order to exemplify how pressure influencesthe dynamical stability and chemical bonding of a typical

a)E-mail: [email protected])E-mail: [email protected]

3D TI. Our present calculations are only concerned withthe low-pressure α phase. From previous experimental work,the phonon dispersion along Γ–Z direction in the Brillouinzone (BZ)24,25 and the pressure dependence of Raman-activemodes17 have been obtained for Sb2Te3. However, the fulldispersions of the phonons along different directions in recip-rocal space have not yet been either calculated nor measured,and this is the main topic of the present investigation. The effectof pressure on the phonon spectrum and chemical bonding hasalso not been studied before and is presented here.

II. METHODS

All calculations were performed by first-principles den-sity functional theory (DFT). Here, the frozen-core projectoraugmented wave (PAW) method was used, as is implementedin the Vienna ab initio simulation package (VASP),26 mainlywithin the context of the local density approximation (LDA).27

We also have used the Perdew, Burke, and Ernzerhof (PBE)28

form of the generalized gradient approximation. Furthermore,since PBE nor LDA hardly can treat the weak van der Waals(vdW) type interaction in this layered material, we also usedthe density functional theory approach including dispersioncorrections (DFT-D2) of Grimme29 to correct the description.Relativistic effects are included in the calculations in terms ofthe SOI. To obtain an accurate total energy, a cutoff energy of300 eV is used for the plane-wave basis-set with a 10 × 10 × 10k-point mesh in the BZ for the rhombohedral crystal structure.All atoms were fully relaxed until the Hellmann-Feynmanforces became less than 0.001 eV/Å. The Sb 5s25p3 and theTe 5s25p4 orbitals are included as valence electrons. In orderto study the dynamical stability of this typical 3D TI underfinite pressure, phonon spectrum calculations were carriedout by using the direct force method.30,31 In these phononcalculations, 3 × 4 × 4 rhombohedral supercells were usedtogether with a 2 × 2 × 2 k-point mesh. An amplitude of thedirect force displacements around 0.04 Å was used.

0021-9606/2015/142(17)/174702/6/$30.00 142, 174702-1 © 2015 AIP Publishing LLC

http://dx.doi.org/10.1063/1.4919683http://dx.doi.org/10.1063/1.4919683http://dx.doi.org/10.1063/1.4919683http://dx.doi.org/10.1063/1.4919683http://dx.doi.org/10.1063/1.4919683http://dx.doi.org/10.1063/1.4919683http://dx.doi.org/10.1063/1.4919683http://dx.doi.org/10.1063/1.4919683http://dx.doi.org/10.1063/1.4919683http://dx.doi.org/10.1063/1.4919683mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://crossmark.crossref.org/dialog/?doi=10.1063/1.4919683&domain=pdf&date_stamp=2015-05-05

174702-2 Wang et al. J. Chem. Phys. 142, 174702 (2015)

TABLE I. Optimized structural parameters and insulating band gap (Eg ) for Sb2Te3, together with experimentalvalues.

Latticeparameters (Å) Bond length (Å) Band gap (eV)

Method a c µ ν Te1− Sb Te2− Sb Te1− Te1 Eg

LDA 4.254 29.460 0.3992 0.2096 2.989 3.130 3.528 0.00LDA+SOI 4.252 29.450 0.3993 0.2094 2.991 3.131 3.515 0.05PBE+SOI 4.342 31.301 0.3969 0.2152 3.034 3.201 3.937 0.12DFT-D2 4.246 30.957 3.970 0.2140 2.996 3.145 3.822 0.20DFT-D2 (SOI) 4.248 30.970 0.3971 0.2138 2.999 3.148 3.815 0.14Expt. 4.264a 30.458a 0.3988a 0.2128a 2.979a 3.168a 3.736a 0.3b

aReference 32.bReferences 11 and 24.

III. RESULTSA. Structure and electronic structure

The rhombohedral crystal structure of Sb2Te3 belongsto space group R3̄m, which has the Sb atoms situated at2c(µ, µ, µ), the type 1 Te atoms (Te1) at 2c(ν, ν, ν), and type2 Te atoms (Te2) at 1a(0, 0, 0) Wyckoff positions (seeTable I). The theoretical equilibrium structural parameters andinsulating band gap (Eg) within LDA, LDA+SOI, PBE+SOI,DFT-D2, and DFT-D2 with SOI are listed in Table I, togetherwith the corresponding experimental values.11,24,32 Clearly,one can find that the PBE+SOI overestimates the latticeconstant a of about 2% compared with experiment. Afterincluding vdW modification, the DFT-D2 gives a value of athat is almost identical with the experimental lattice constants.For the lattice constant c, LDA or LDA+SOI underestimatesits value with 3.3%, PBE+SOI overestimates it with 2.8%,and DFT-D2 with or without SOI overestimates it with 1.7%.SOI effects on the structural parameters are not as important as

the vdW correction. Including vdW interaction results in thebest agreement between theory and experiment, as far asthe lattice constants are concerned. However, SOI effects onthe electronic structure are more prominent than the vdWcorrection (see Fig. 1). While the LDA predicts metallic bulkcharacter, the LDA+SOI opens the band gap and inverts theoccupation character of the valence-band maximum (VBM)and conduction-band minimum (CBM) states, reflecting thetopological nature of this material. Direct semiconductorfeatures described by DFT-D2 are modified by LDA+SOI andDFT-D2 with SOI to features of an indirect semiconductor,with the VBM located halfway in the Z–F direction andthe CBM in the Γ–Z direction. However, as shown inTable I, the underestimation of Eg compared with theexperimental value of around 0.3 eV11,24 is clear. This factis due to the limitation of the approximations used in theexchange-correlation functional. LDA and PBE are knownto underestimate the band gaps and might change the bandinversion energy ∆i = εs − εp,d of the TI transition.33

FIG. 1. Band structure for Sb2Te3 at0 GPa calculated by (a) LDA, (b)LDA+SOI, (c) DFT-D2, (d) PBE+SOI,and DFT-D2 with SOI. The Fermi en-ergy level is set at zero.


FIG. 2. (a) Calculated phonon dispersion curves and phonon DOS for Sb2Te3 at 0 GPa. Calculation formalisms are indicated in the legends. The full drawncurves in the phonon DOS correspond to our present calculations. Experimental phonon dispersion by neutrons as well as phonon DOS by neutrons and nuclearresonant inelastic scattering (NIS) is taken from Refs. 24, 34, and 35. (b) Pressure dependence of the phonon frequencies at Γ point obtained from LDA+SOIcalculations. Recent Raman scattering values17 are also presented for comparison. (c) BZ for Sb2Te3 with space group R3̄m.

B. Phonons

We show in Fig. 2(a) the phonon dispersion alongU–Γ–Z–F–Γ–L directions and the phonon density of state(DOS) of Sb2Te3 at 0 GPa by LDA and LDA+SOI, togetherwith experimental phonon frequencies24 along the Γ–Zsymmetry line and phonon DOS34,35 for comparison. Pressuredependence of the phonon frequencies at Γ point obtained fromLDA+SOI calculations is presented in Fig. 2(b), in comparingwith recent Raman scattering values.17 High symmetry pointsin the BZ are illustrated in Fig. 2(c).

The R3̄m structure possesses the point group symmetryof D3d. By analyzing the sites symmetry, we list in Table IIthe Wyckoff positions of the Sb, Te1, and Te2 atoms andthe corresponding vibration modes. Here, one Eu and oneA2u modes are acoustic. Due to the fact that E and A modesrepresent displacement in the a − b plane and along the caxis, respectively, all the five vibration E modes along Γ–Zdirection are double degenerate.

At 0 GPa, Fig. 2(a) displays a good agreement betweenthe calculated phonon dispersion and the correspondingexperimental frequencies along Γ–Z . Also in Fig. 2(a), it can

TABLE II. Symmetry analysis and phonon frequencies at the Γ point forSb2Te3, together with experimental values.

Optical modes

Atom Wyckoff position Raman active Infrared active

Sb 2c A1g +Eg A2u+EuTe1 2c A1g +Eg A2u+EuTe2 1a A2u+Eu

Modes LDA+SOI Expt.a

Symmetry Atoms (THz) (THz)E1g Sb,Te1 1.56 1.41A11g Sb,Te1 2.18 2.17E2u Sb,Te1,Te2 2.21 1.95E3u Te1,Te2 2.97 2.30A22u Te1,Te2 3.26 3.18E2g Sb,Te1 3.50 3.33A32u Sb,Te1,Te2 4.40A21g Sb,Te1 4.95 4.98

aReference 24.

be observed that the calculated phonon DOSs and the DOSsfrom inelastic neutron scattering34 as well as nuclear resonantinelastic scattering (NIS) data35 show a large overlap. Wenote that a theoretical work,25 where the Plane-Wave Self-Consistent Field code (PWSCF) was used, also reproducedthe experimental phonon frequencies along Γ–Z direction.Furthermore, different with the previous study of Bi2Se3 andBi2Te323 where the two lowest acoustic branches obtainedby LDA+SOI exhibit imaginary frequencies along the Γ–Fand/or Z–F directions, our present phonon spectra of Sb2Te3by LDA and LDA+SOI are all positive. Actually, when weusing the same supercell of 3 × 3 × 3 as previous study,23 eitherwithin LDA+SOI or DFT-D2 with SOI, imaginary frequenciesalong Γ–U, Γ–F, and Z–F directions were observed. Onlyafter increasing the supercell to 3 × 4 × 4, the imaginaryfrequencies along the two lowest acoustic branches disappear.The unphysical errors of our previous study are mainly due tothe limitation of the computational resource. Inclusion of SOIand increasing size of the supercell result in great requirementof the computer memory which was not satisfied at thatmoment. Anyway, our present results indicate that inclusion ofSOI, although not changes phonon spectra and phonon DOSsthat much, does impact them in some places. The acousticEu mode shows uplift along the Γ–F direction and also nearthe U , F, and L points. This makes the phonon DOS near1.5 THz smoother and closer to experimental observations34,35

than pure LDA. In range of 3.8-5.4 THz, the phonon DOSby LDA+SOI exhibits larger fluctuations than that by LDA.While the result by LDA shows a good agreement with theold neutron data, the phonon DOS curve by LDA+SOI isconsistent with the new NIS results which also show largefluctuations in the high frequency zone.

By employing the LDA+SOI, we study carefully thepressure effects on the phonon spectra of this typical 3D TI.Under pressure up to 7 GPa, no unstable phonon branches havebeen observed. Thus, the rhombohedral phase in Sb2Te3 can beexpected to remain dynamically stable at pressures exceeding7 GPa. Similar with studies in other systems,36,37 pressure onlypushes the vibrational spectra even harder. In Fig. 2(b), weshow the pressure dependence of the phonon frequencies at theΓ point. We choose to perform this investigation with a LDAfunctional, due to the computational efficiency over DFT-D2.


These two methods give similar results and motivating themore efficient methodology. Good agreement with recentRaman scattering data17 of A11g , E

2g , and A

21g modes can be

found. This guarantees the correctness of our calculationresults. Gomis et al. have revealed that both frequency andlinewidth of their observed Raman modes show differentbehaviors above 3.5 GPa,17 where experimentally the effectsof an electronic topological transition (ETT) are observed.

C. Topology of Fermi surface

Our calculations show that a semiconductor to metal tran-sition occurs at around 2 GPa and an ETT happens just above4 GPa. Here, the ETT has been suggested experimentally17,38

in Sb2Te3 by the change in the compressibility. However, itis possible that the observed change in compressibility iscaused by the semiconductor to metal transition. Nevertheless,we now proceed with a detailed analysis of the pressuredependence of the ETT.

In Fig. 3, the calculated Fermi surfaces of Sb2Te3 at 3 GPaand 7 GPa are displayed. At the higher pressure the topologyhas drastically changed as compared to the lower pressureFermi surface, with the emergence of ellipsoidal surfacesradiating out from the Γ-point [blue surface sheets in Fig. 3(b)]and extra blobs (green surface sheets in Fig. 3) appearingat the Brillouin zone boundaries. The ETT represented bythe occurrence of ellipsoidal surfaces will give rise to VanHove singularity in the electronic density of states N(E)∼ |E − Ec |, close to the critical energy Ec where the

topological transition occurs.39 Furthermore, close to the VanHove singularity, the band contribution to the compressibility,

FIG. 3. The Fermi surface of Sb2Te3 calculated with LDA+SOI at (a) 3 GPaand at (b) 7 GPa. The Fermi surface is displayed in the plane orthogonal tothe Γ–Z high symmetry direction.

B, will be dominated by B ∼ |EF − Ec |−1/2, which is singularwhenever the Fermi energy, EF, equals the critical energy andwill thus induce drastic changes in the compressibility.

By shifting the Fermi level continuously until the topol-ogy of the Fermi Surface changed by the appearance of theabove mentioned ellipsoidal surfaces, the energy differenceEF − Ec was estimated to 35 meV at 3 GPa and to −77 meVat 7 GPa. Furthermore, by assuming that EF − Ec dependslinearly upon pressure, we were able to estimate the pressureat which the ETT will occur at ∼4.25 GPa.

Similar ETTs have been observed in Bi2Se3 around5 GPa16 and Bi2Te3 around 3.2 GPa.10 Furthermore, in Bi2Te3at pressures between 3 and 6 GPa, a superconductive transitionwith a transition temperature, Tc, of ∼3 K has been observed.10Similar phenomena have also been reported for Sb2Te320 at itsETT pressures.

IV. DISCUSSION

In order to explore the effects of the pressure-induced ETTin Sb2Te3 in detail, we show in Fig. 4 the pressure dependencesof structural parameters, bond lengths, line charge density atthe corresponding bond points (CDb), electron transfer fromthe Sb atoms to the Te atoms, and the value of the energydifference (Ed) between the bottom of the conduction and thetop of the valence band. Here, positive value of Ed standsfor the indirect energy gap (Eg). In Fig. 4(e), the number ofelectrons of each atom is obtained through Bader analysis.40

When comparing with the experimental structural param-eters,38,41 a good agreement is found for the calculated latticeconstant a while the c lattice parameter is slightly under-estimated by the present calculations. However, the pressuredependency of the calculated lattice parameters is in qualita-tive agreement with the experimentally observed parameters.For instance, the experimentally observed minimum in thec/a parameter at about 2-4 GPa is also found in the presentlycalculated c/a ratio.

As shown in Figs. 4(c)-4(e), when viewed separatelyat around 3.5 GPa, the bond lengths and electron transferfrom Sb atom to Te decrease while the CDb increases nearlylinearly with elevated pressure. The decreasing or increasingrates, obtained by performing first-order linear fitting onthese curves, are reminiscent of the pressure dependenttransition characteristic of an ETT at around 3.5 GPa. Thesecharacteristics are not evident for the pressure dependency ofthe Te1–Sb and Te2–Sb bonds due to their covalent bondingcharacter. But, for the Te1–Te1 bond, the characteristics of apressure-induced transition can be clearly seen with a slowdown about 41% and 26% for decreasing rate of bond lengthand increasing rate of CDb, respectively. Furthermore, theeffect of pressure upon the Te1–Te1 bond is more prominentthan on the Te1–Sb and Te2–Sb bonds in the whole pressurezone, which is mainly due to relative strong compression effecton the weak vdW type bonding of Te1–Te1. Specifically, thedecreasing rate of Te1–Te1 bond length is about 4 times thatof the other two types of bonds before 3.5 GPa, and a factor of2 larger after 3.5 GPa; the increasing rate of CDb for Te1–Te1bond is about 2 times that of the of Te1–Sb and Te2–Sb bondsin the whole pressure zone considered. Transition at around


FIG. 4. Pressure dependences of (a) lattice constants, (b) c/a ratio, (c) bond lengths, (d) charge density values at bond points (CDb), (e) electrons transfer fromSb atom to Te, and (f) the energy difference (Ed) between the bottom of the conduction band and the top of the valence band obtained by LDA+SOI scheme.The solid lines are first-order linear fittings. Experimental lattice constants and c/a ratio are taken from Refs. 38 and 41.

3.5 GPa for electrons transfer from the Sb atoms to the Teatoms is also evident with a slowdown of the decreasing rateabout 31%. Therefore, the covalent and ionic bonding natureof this typical 3D TI also occurs transition at around 3-4 GPa.This kind of transition behavior supplies more theoreticalunderstanding of the semiconductor to metal transition as wellas the ETT.

The pressure dependence of Ed [see Fig. 4(f)] obtainedfrom LDA+SOI clearly shows a transition from semiconductorto metal at around 2 GPa. This transition coincides well withthe emergence of superconductivity in this material.20 Besides,the slope of the pressure dependent Ed shows a change ataround 3 GPa, which should also possess potential correlationwith aforementioned ETT as well as the changes in slope ofthe Te1–Te1 distance and charge transfer at 3-4 GPa.

To investigate the pressure effect of electronic structuresin more detail, we compare in Fig. 5 the band structures ofSb2Te3 at 0 and 7 GPa calculated from LDA+SOI. At 7 GPa,curves near CBM, F, and L points have already passed acrossthe Fermi level which clearly indicates the semiconductor tometal transition. However, the gap at the Γ point (EΓg ) showsno evident change upon compression. Recent experimentalwork found that the EΓg of Bi2Se3 increases with pressure.

42

FIG. 5. Band structures for Sb2Te3 at 0 GPa (dark line) and 7 GPa (red line)calculated with LDA+SOI. The Fermi energy level is set at zero.

Since DFT studies of Bi2Se342 and Bi2Te3 have not beenable to find a semiconductor to metal transitions and theexperiments10,18,20,22 exist controversy, further experimentaland theoretical studies are needed before any general conclu-sions can be drawn about how pressure affects the semicon-ductor to conductor transitions in the group V chalcogenides.

V. CONCLUSION

In summary, through the investigation of the latticedynamics of one of the typical 3D, TI Sb2Te3, we have foundthat the SOI has some effects on the phonon spectra in thegroup V chalcogenides, although not that visible. Concerningthe use of different functionals, we find that the inclusion ofvdW interaction gives slightly improved lattice parameterswhen compared to experiment. The calculated pressure-induced behavior of the phonon frequencies at the Γ pointis in an excellent agreement with experiment. Furthermore,the different pressure dependencies of phonon frequencies,the c/a ratio, bond lengths, and electron transfer togetherwith our Fermi surface calculations support the experimentalsuggestion of an ETT.17,38

ACKNOWLEDGMENTS

This work was supported by NSFC under Grant Nos.11104170, 51071032, and 11074155. O.E. acknowledgessupport from the Swedish research council, the KAW foun-dation, ESSENCE, STANDUPP, and the ERC (Project No.247062–ASD). Calculations done via allocation of Swedishsupercomputer resources granted by SNIC.

1D. Hsieh, Y. Xia, D. Qian, L. Wray, F. Meier, J. H. Dil, J. Osterwalder, L.Patthey, A. V. Fedorov, H. Lin et al., Phys. Rev. Lett. 103, 146401 (2009).

2X. L. Qi and S. C. Zhang, Phys. Today 63(1), 33 (2010).3J. Moore, Nature 464, 194 (2010).4M. Hasan and C. Kane, Rev. Mod. Phys. 82, 3045 (2010).5X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).6H. J. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, Nat. Phys.5, 438 (2009).

http://dx.doi.org/10.1103/PhysRevLett.103.146401http://dx.doi.org/10.1063/1.3293411http://dx.doi.org/10.1038/nature08916http://dx.doi.org/10.1103/RevModPhys.82.3045http://dx.doi.org/10.1103/RevModPhys.83.1057http://dx.doi.org/10.1038/nphys1270


7Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y.S. Hor, R. J. Cava, and M. Z. Hasan, Nat. Phys. 5, 398 (2009).

8D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L.Patthey, J. G. Checkelsky, N. P. Ong et al., Nature 460, 1101 (2009).

9Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J.Zhang, D. H. Lu, X. Dai, Z. Fang et al., Science 325, 178 (2009).

10J. L. Zhang, S. J. Zhang, H. M. Weng, W. Zhang, L. X. Yang, Q. Q. Liu, S.M. Feng, X. C. Wang, R. C. Yu, L. Z. Cao, L. Wang, W. G. Yang, H. Z. Liu,W. Y. Zhao, S. C. Zhang, X. Dai, Z. Fang, and C. Q. Jin, Proc. Natl. Acad.Sci. 108, 24 (2011).

11Y. Jiang, Y. Y. Sun, M. Chen, Y. Wang, Z. Li, C. Song, K. He, L. Wang, X.Chen, Q.-K. Xue, X. Ma, and S. B. Zhang, Phys. Rev. Lett. 108, 066809(2012).

12G. J. Snyder and E. S. Tobere, Nat. Mater. 7, 105 (2008).13M. Einaga, F. Ishikawa, A. Ohmura, A. Nakayama, Y. Yamada, and S.

Nakano, Phys. Rev. B 83, 092102 (2011).14L. Zhu, H. Wang, Y. Wang, J. Lv, Y. Ma, Q. Cui, Y. Ma, and G. Zou, Phys.

Rev. Lett. 106, 145501 (2011).15R. Vilaplana, O. Gomis, F. J. Manjón, A. Segura, E. Pérez-González, P.

Rodríguez-Hernández, A. Muñoz, J. González, V. Marín-Borrás, V. Muñoz-Sanjose, C. Drasar, and V. Kucek, Phys. Rev. B 84, 104112 (2011).

16R. Vilaplana, D. Santamaría-Pérez, O. Gomis, F. J. Manjón, J. González, A.Segura, A. Muñoz, P. Rodríguez-Hernández, E. Pérez-González, V. Marín-Borrás, V. Muñoz-Sanjose, C. Drasar, and V. Kucek, Phys. Rev. B 84, 184110(2011).

17O. Gomis, R. Vilaplana, F. J. Manjón, P. Rodríguez-Hernández, E. Pérez-González, A. Muñoz, V. Kucek, and C. Drasar, Phys. Rev. B 84, 174305(2011).

18J. Zhang, C. Liu, X. Zhang, F. Ke, Y. Han, G. Peng, Y. Ma, and C. Gao, Appl.Phys. Lett. 103, 052102 (2013).

19K. Kirshenbaum, P. S. Syers, A. P. Hope, N. P. Butch, J. R. Jeffries, S. T.Weir, J. J. Hamlin, M. B. Maple, Y. K. Vohra, and J. Paglione, Phys. Rev.Lett. 111, 087001 (2013).

20J. Zhu, J. L. Zhang, P. P. Kong, S. L. Zhang, X. H. Yu, J. L. Zhu, Q. Q. Liu,X. Li, R. C. Yu, R. Ahuja, W. G. Yang, G. Y. Shen, H. K. Mao, H. M. Weng,X. Dai, Z. Fang, Y. S. Zhao, and C. Q. Jin, Sci. Rep. 3, 2016 (2013).

21J. Zhang, Y. Han, C. Liu, X. Zhang, F. Ke, G. Peng, Y. Ma, Y. Ma, and C.Gao, Appl. Phys. Lett. 105, 062102 (2014).

22K. Matsubayashi, T. Terai, J. S. Zhou, and Y. Uwatoko, Phys. Rev. B 90,125126 (2014).

23B. T. Wang and P. Zhang, Appl. Phys. Lett. 100, 082109 (2012).24Landolt-Börnstein, “Antimony telluride (Sb2Te3) phonon dispersion,

phonon frequencies,” in Numerical Data and Functional Relationships inScience and Technology (New Series, Group III (Condensed Matter) Vol.41 (Semiconductors), Subvolume C (Non Tetrahedrally Bonded Elementsand Binary Compounds I)), edited by O. Madelung, M. Schulz, and H.Weiss (Springer, New York, 1983).

25G. C. Sosso, S. Caravati, and M. Bernasconi, J. Phys.: Condens. Matter 21,095410 (2009).

26G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).27W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).28J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).29S. Grimme, J. Comput. Chem. 27, 1787 (2006).30W. Frank, C. Elsasser, and M. Fahnle, Phys. Rev. Lett. 74, 1791 (1995).31K. Parlinski, Z.-Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997).32T. L. Anderson and H. B. Krause, Acta Crystallogr. B 30, 1307 (1974).33J. Vidal, X. Zhang, L. Yu, J. W. Luo, and A. Zunger, Phys. Rev. B 84, 041109

(2011).34H. Rauh, R. Geick, H. Köhler, N. Nücker, and N. Lehner, J. Phys. C: Solid

State Phys. 14, 2705 (1981).35D. Bessas, I. Sergueev, H.-C. Wille, J. Perßon, D. Ebling, and R. P. Hermann,

Phys. Rev. B 86, 224301 (2012).36B. T. Wang, P. Zhang, H. Y. Liu, W. D. Li, and P. Zhang, J. Appl. Phys. 109,

063514 (2011).37E. G. Maksimov, S. Q. Wang, M. V. Magnitskaya, and S. V. Ebert, Super-

cond. Sci. Tech. 22, 075004 (2009).38N. Sakai, T. Kajiwara, K. Takemura, S. Minomura, and Y. Fujii, Solid State

Commun. 40, 1045 (1981).39I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960).40W. Tang, E. Sanville, and G. Henkelman, J. Phys.: Condens. Matter 21,

084204 (2009).41S. M. Souza, C. M. Poffo, D. M. Trichês, J. C. de Lima, T. A. Grandi, A.

Polian, and M. Gauthier, Physica B 407, 3781-3789 (2012).42A. Segura, V. Panchal, J. F. Sánchez-Royo, V. Marín-Borrás, V. Muñoz-

Sanjose, P. Rodríguez-Hernández, A. Muñoz, E. Pérez-González, F. J. Man-jón, and J. González, Phys. Rev. B 85, 195139 (2012).
http://dx.doi.org/10.1038/nphys1274http://dx.doi.org/10.1038/nature08234http://dx.doi.org/10.1126/science.1173034http://dx.doi.org/10.1073/pnas.1014085108http://dx.doi.org/10.1073/pnas.1014085108http://dx.doi.org/10.1103/PhysRevLett.108.066809http://dx.doi.org/10.1038/nmat2090http://dx.doi.org/10.1103/PhysRevB.83.092102http://dx.doi.org/10.1103/PhysRevLett.106.145501http://dx.doi.org/10.1103/PhysRevLett.106.145501http://dx.doi.org/10.1103/PhysRevB.84.104112http://dx.doi.org/10.1103/PhysRevB.84.184110http://dx.doi.org/10.1103/PhysRevB.84.174305http://dx.doi.org/10.1063/1.4816758http://dx.doi.org/10.1063/1.4816758http://dx.doi.org/10.1103/PhysRevLett.111.087001http://dx.doi.org/10.1103/PhysRevLett.111.087001http://dx.doi.org/10.1038/srep02016http://dx.doi.org/10.1063/1.4892661http://dx.doi.org/10.1103/PhysRevB.90.125126http://dx.doi.org/10.1063/1.3689759http://dx.doi.org/10.1088/0953-8984/21/9/095410http://dx.doi.org/10.1103/PhysRevB.54.11169http://dx.doi.org/10.1103/PhysRev.140.A1133http://dx.doi.org/10.1103/PhysRevLett.77.3865http://dx.doi.org/10.1002/jcc.20495http://dx.doi.org/10.1103/PhysRevLett.74.1791http://dx.doi.org/10.1103/PhysRevLett.78.4063http://dx.doi.org/10.1107/S0567740874004729http://dx.doi.org/10.1103/PhysRevB.84.041109http://dx.doi.org/10.1088/0022-3719/14/20/009http://dx.doi.org/10.1088/0022-3719/14/20/009http://dx.doi.org/10.1103/PhysRevB.86.224301http://dx.doi.org/10.1063/1.3556753http://dx.doi.org/10.1088/0953-2048/22/7/075004http://dx.doi.org/10.1088/0953-2048/22/7/075004http://dx.doi.org/10.1016/0038-1098(81)90248-9http://dx.doi.org/10.1016/0038-1098(81)90248-9http://dx.doi.org/10.1088/0953-8984/21/8/084204http://dx.doi.org/10.1016/j.physb.2012.05.061http://dx.doi.org/10.1103/PhysRevB.85.195139

Lattice dynamics and chemical bonding in Sb2Te3 from first ... · Pressure effects on the lattice...

Documents

Transcript of Lattice dynamics and chemical bonding in Sb2Te3 from first ... · Pressure effects on the lattice...