_{See discussions stats and author profiles for this publication at httpswwwresearchgatenetpublication268284024}

_{Electronic Structure Mechanics and Thermodynamics of}

_{ZrBltSUBgt12ltSUBgt Under Pressure}

_{Article in Journal of Applied Statistics middot October 2014}

_{DOI 101166sam20141994}

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_{Bao-Tian Wang}

_{Chinese Academy of Sciences}

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_{Wei-Dong Li}

_{Shanxi University}

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_ARTICLE

_{Copyright copy 2014 by American Scientific Publishers}

_{All rights reserved}

_{Printed in the United States of America}

_{Science of Advanced MaterialsVol 6 pp 2281ndash2285 2014}

_{(wwwaspbscomsam)}

_{Electronic Structure Mechanics andThermodynamics of ZrB12 Under PressureWenxue Zhang Bao-Tian Wanglowast Xinlin Cui Li Li and Wei-Dong Li}

_{Institute of Theoretical Physics and Department of Physics Shanxi University Taiyuan 030006Peoplersquos Republic of China}

_ABSTRACT

_{The effects of pressure on the electronic mechanical and thermodynamical properties of ZrB12 have beenstudied from a first-principles perspective Due to the good mechanical and dynamical stabilities of this mate-rial our results show that the pressure effects on the structure and electronic structure are limited Howeverthe mechanical parameters such as the elastic constants elastic moduli and elastic wave velocities and thethermodynamical properties of the Debye temperature and melting points are promptly enhanced by increasingpressure Under pressure the density of states at the Fermi level NEF decreases while the low frequencyvibration of Zr atoms increases in the energy level which indicates a negative pressure effect on the super-conductivity of ZrB12 According to phonon dispersions we have calculated the phonon free energy phononentropy and the specific heat at constant volume under different pressure and temperature conditions}

_{KEYWORDS ZrB12 Elastic Constant Phonon First-Principles Pressure}

_{1 INTRODUCTIONAmong dodecaborides ZrB12 exhibits the highest super-conducting transition temperature of Tc sim 6 K1 andhas attracted much investigation interest leading toexperiments2ndash8 and computational simulations8ndash11 Theband-structure calculations of YB12 and ZrB12 by the full-potential linearized muffin-tin orbital (FLMTO) calcula-tions indicated that an increase in the contribution ofthe transition metal 4d states to the density of states atthe Fermi level NEF is responsible for the increase ofthe superconducting transition temperature Tc (from 47 Kfor YB12 to 58 K for ZrB12)}

_{9 After calculations Teyssieret al have found that the isotropic Fermi surface (FS)of ZrB12 is composed by an electron character near theX point and a hole character near the point in theBrillouin zone (BZ)4 One recent combination work of thede Haas-van Alphen effect experiment and the FLMTOcalculation showed unusually large electronndashphonon inter-action on the neck and box sections of the FS on the BZboundaries6 All these works illustrated that the main con-tribution to the NEF comes from the Zr 4d and B 2pstates11}

_{It is well known that the lattice dynamic prop-erties connect tightly with the superconductivity ofmaterials12 In the study of those properties of ZrB12 many}

_{lowastAuthor to whom correspondence should be addressedEmail wbt11129sxueducnReceived 18 July 2013Accepted 9 November 2013}

_{investigations have been performed Through a detailedmeasurement of the thermal and transport properties ofZrB12}

_{2 the lattice vibration modes involving Zr atomswere experimentally found to play a critical role in theelectronndashphonon coupling responsible for superconductiv-ity Their specific heat and resistivity results unveiled a15 meV mode in the low-energy phonon density of states(PhDOS) More recently Rybina et al determined thequasilocal Zr vibration modes to be about 175 meV bya lattice dynamics study from ab initio calculations andinelastic neutron scattering8 Based on density-functionaltheory (DFT) in our previous work11 we have also con-firmed that the vibration modes of the Zr atoms exhibit themain peak at about 169 meV in the low frequency zoneIn the present study however we make a step further}

_{and focus on the effects of pressure on the electronicmechanic lattice dynamic and thermodynamic propertiesof ZrB12 We found that the NEF decreases upon com-pression and the low frequency vibration of Zr atomsincreases in the energy level These observations clearlyindicate that the superconducting transition temperature Tcof ZrB12 decreases with pressure}

_{2 COMPUTATIONAL METHODSFirst-principles DFT calculations on the basis of the pro-jected augmented wave (PAW) method of Bloumlchl13 are per-formed within the Vienna ab initio simulation package14}

_{The exchange and correlation effects are described by thePerdew Burke and Ernzerhof15 form of the generalized}

_{Sci Adv Mater 2014 Vol 6 No 10 1947-2935201462281005 doi101166sam20141994 2281}

_{Electronic Structure Mechanics and Thermodynamics of ZrB12 Under Pressure Zhang et al}

_ARTICLE

_{Fig 1 The total energy (E)ndashvolume (V ) and the pressure (P )ndashV rela-tions of ZrB12 The black solid circles are the first-round calculated EThen we do polynomial fitting to these data and obtain the solid blackline Using the polynomial fitted EndashV we can deduce the P by the rela-tion P =minusEV Our used PndashV points are indicated by the red hollowcircles}

_{gradient approximation For the plane-wave set a cutoffenergy of 500 eV is used The -centered k point-meshin the full wedge of the BZ is sampled by a 6times 6times 6grid according to the Monkhorst-Pack (MP)16 in the face-centered cubic (fcc) unit cell (52-atoms cell) All atoms arefully relaxed until the Hellmann-Feynman forces becomeless than 0001 eVAring The Zr 4s24p64d35s1 and the B2s22p1 orbitals are explicitly included as valence electronsTo avoid the Pulay stress problem the geometry opti-}

_{mization at each volume is performed at fixed volumerather than constant pressure As shown in Figure 1 wecalculate the total energy (E) at a series of volumes (V )in the first-round calculations Then we conduct seventhorder polynomial fitting to these data and obtain the EndashVrelation Using this EndashV relation we can deduce the PndashVrelation by the thermodynamic derivative P = minusEV Utilizing this PndashV relation we perform the second-roundcalculations at a series of volumes corresponding to someinteger pressures Note that these pressure values deducedby fitting the EndashV relation are different from the pressure}

_{Table I Calculated lattice constant (a) and volume of the unit cell (Vuc) elastic constants (C11 C12 and C44) bulk modulus (B) shear modulus (G)Youngrsquos modulus (E) Poissonrsquos ratio () density () as well as transverse (t) longitudinal (l) and average (m) sound velocities derived frompolycrystalline bulk and shear modulus of ZrB12 at selected pressures}

_{Pressure (GPa) a (Aring) Vuc (Aring3) C11 (GPa) C12 (GPa) C44 (GPa) B (GPa) G (GPa) E (GPa) (gcm3) t (kms) l (kms) m (kms)}

_{0 7410 40687 4676 1183 2695 2347 2265 5141 0135 3607 7924 12198 86905 7358 39844 4955 1262 2885 2493 2412 5472 0134 3684 8093 12450 887510 7311 39084 5232 1339 3039 2636 2542 5772 0135 3755 8228 12668 902415 7268 38390 5494 1434 3186 2788 2659 6053 0138 3823 8340 12870 914920 7227 37740 5755 1484 3328 2908 2786 6334 0137 3889 8464 13049 928425 7188 37137 6016 1557 3472 3043 2907 6615 0138 3952 8576 13232 940930 7151 36572 6276 1629 3615 3178 3028 6894 0138 4013 8686 13409 953035 7117 36043 6528 1699 3753 3309 3144 7164 0139 4072 8788 13573 964240 7084 35548 6774 1768 3888 3437 3258 7428 0140 4129 8884 13728 974845 7053 35080 7017 1836 4020 3563 3371 7688 0140 4184 8976 13878 985050 7023 34640 7257 1904 4151 3688 3482 7945 0141 4237 9065 14021 9947}

_{values directly computed from DFT with the Hellmann-Feynman theoremThe elastic constants are calculated by applying stress}

_{tensors with various small strains onto the equilibriumstructures The strain amplitude is varied in steps of0006 from = minus0036 to 0036 After obtaining elasticconstants the polycrystalline bulk modulus B and shearmodulus G are calculated from the Voigt (Reuss) boundson the bulk modulus BV BR) and shear modulus GV GR)by the Voigt-Reuss-Hill (VRH) approximations17}

_{B = BV = BR = C11+2C123}

_{GV = C11minusC12+3C445}

_{GR = 5lowast C11minusC12C444C44+3C11minusC12}

_{G= GV +GR2 (1)}

_{The Youngrsquos modulus E and Poissonrsquos ratio are cal-culated through E = 9BG3B + G and = 3B minus2G23B+G The transverse (t) longitudinal (l)and average (m) sound velocities as well as the Debyetemperature (13D) are derived from polycrystalline bulk andshear modulus For a more detailed overview of the com-putational details we redirect the reader to Ref [12]After obtaining the Debye temperature at a pressure}

_{range from ambient condition to 50 GPa we perform themelting curve calculations based on the Lindemann melt-ing model18 This model is based on the harmonic approx-imation predicting that melting will occur when the ratioof the root mean square (rms) atomic displacement to themean interatomic distance reaches a certain value (gen-erally about 18) It can be expressed as Tm = CV 2313Dwhere Tm is the melting point C is a constant and V isatomic volumePhonon frequency at selected pressures is obtained by}

_{using the supercell approach within the FROPHO code19}

_{To reach high accuracy a 3times 3times 3 MP k-point mesh isutilized in the BZ integration for a 3times3times3 rhombohedralZrB12 supercell}

_{2282 Sci Adv Mater 6 2281ndash2285 2014}

_{Zhang et al Electronic Structure Mechanics and Thermodynamics of ZrB12 Under Pressure}

_ARTICLE

_{3 RESULTS AND DISCUSSIONIn ambient conditions ZrB12 crystallizes in a fcc struc-ture with space group Fm3m (No 225) with Zr in 4a(00 0) and B in 48e(05 y y) Wyckoff positions in whichthe Zr atoms and cuboctahedral B12 cluster are arrangedin an NaCl type structure The present optimized latticeconstant a and volume of the unit cell (Vuc) at selectedpressures in the pressure range of 0ndash50 GPa for ZrB12 arepresented in Table I As having been clearly indicated inour previous study11 our calculated results of equilibriumstate properties are in good agreement with experimentaldata3 This supplies the safeguard for our extending studyof ZrB12 upon compression At pressures of 0 25 and50 GPa the boron position parameters y are optimized tobe 01695 01696 and 01697 respectively This indicatesthat the pressure effect on the structure is limited}

_{To study the pressure effect on the electronic structure weplot the calculated total and partial density of states (DOSs)of ZrB12 at 0 25 and 50 GPa in Figure 2 At ambient condi-tion the main features of the orbital occupation are consis-tent with previous FLMTO calculations49 and the experi-mental X-ray photoemission spectra measurements7 Samewith previous theoretical calculations49 we also find thatthe main contribution to the NEF is from the B-2p stateUnder pressure the mainly occupied states of the B-spextend to deeper energy level while the NEF decreases alittle (see Fig 3) The decrement of NEF from zero pres-sure to 50 GPa is about 133 This kind of decrement ofNEF has been previously observed by Escamilla et al10}

_{in a pressure range of 0ndash104 GPa Due to the critical roleof the NEF to the superconducting transition tempera-ture Tc}

_{1220 we can speculate a negative pressure effecton the superconductivity of ZrB12}

_{1021 In spite of thisfact we still can say that the pressure effect on the elec-tronic structure is also limited which supports the goodmechanical and dynamical stabilities of this materials11}

_{Fig 2 Total and partial densities of states for ZrB12 at (a) 0 GPa(b) 25 GPa and (c) 50 GPa The Fermi energy level is set at zero}

_{Fig 3 Ratio of the NEF at finite pressures to that at 0 GPa for ZrB12The NEF at 0 GPa is about 085 stateeVfu}

_{In the following we will see that the pressure will enhancethe mechanical and dynamical propertiesIn our previous study11 we have shown that our calcu-}

_{lated elastic constants (C11 C12 and C44) elastic moduli(B G and E) Poissonrsquos ratio () density () transversesound velocities (t) longitudinal sound velocities (l)average sound velocities (m) and Debye temperature (13D)of ZrB12 at zero pressure are wholly consistent with corre-sponding results directly read or indirectly deduced fromexperiments3 In the present study we focus on the pres-sure effects on these mechanical andor thermodynami-cal properties We present our calculated data in pressurerange of 0ndash50 GPa in Table I and in Figure 4 Clearly pres-sure induced enhancements of the mechanical and thermo-dynamical properties can be seen This kind of pressureeffect has also been found by Korozlu et al22 in study ofMB12 (M= Zr Hf Y Lu) and by Wang et al in study ofthe element Zr and the TiZr alloy1223 As already shownin our previous study11 ZrB12 possesses almost the samemechanical properties of the bulk modulus B shear modu-lus G Youngrsquos modulus E and Poissonrsquos ratio with thatof ZrB2 but prominently lower density This point makesZrB12 more promising than ZrB2 in real engineered appli-cations The enhancement of the mechanical propertiesunder pressure for this material would be responsible forgood application in extreme andor complex conditions}

_{Fig 4 The melting temperature and the Debye temperature of ZrB12 asa function of pressure The solid lines are the polynomial fitting resultsof our data}

_{Sci Adv Mater 6 2281ndash2285 2014 2283}

_{Electronic Structure Mechanics and Thermodynamics of ZrB12 Under Pressure Zhang et al}

_ARTICLE

_{As shown in Table I one can find that the most slug-gish one upon compression is the Poissonrsquos ratio It onlyincreases 44 from zero pressure to 50 GPa The increaseproportions of C11 C12 C44 B G E t l m and 13Dare 552 610 540 571 537 545 175144 150 145 and 208 respectively Actuallythe pressure induced behaviors of the mechanical and ther-modynamical properties are mainly controlled by the elas-tic constants and the density (see method part) Due tothe fact that the elastic constants increase almost linearlywith pressure we can do first-order polynomial fitting Theincreasing rates of C11 C12 and C44 are then obtained tobe 5162 1442 and 2912 Here the increasing rates Rij}

_{of the elastic constants Cij are defined as Rij = CijP It is clear that the R11 is prominently larger than R12 aswell as R44}

_{22 However the specific values of the elasticconstants and also the increasing rates do possess somedifferences between our results and the data calculated byKorozlu et al22 While they performed their calculationsby using the plane wave pseudopotential we use here thePAW basis pseudopotential The PAW method is more use-ful when both localized and delocalized valence states areimportant which is just the case of our present study ofthe transition metal Zr based systemThe melting curves of materials have great scientific and}

_{technological interest We can easily deduce the meltingcurve of ZrB12 by using the Lindemann criterion from ourcalculated Debye temperature The Lindemann criterion isa single-parameter model and the free constant C can becalculated to equal to 00329 from a single point (Tm =2353 K at 0 GPa for ZrB12 by experiment24 as well as V =40687 Aring3 and 13D = 13029 K by the present work) Weshow the melting curves in Figure 4 It can be seen thatthe melting temperature of ZrB12 increase by about 200 Kfrom ambient pressure to 50 GPa But the increase rate isslowed down by pressure These pressure effects need moreexperiments to clarifyIn our previous study we have investigated the phonon}

_{spectrum of ZrB2 and ZrB12 at ambient pressure11 Goodagreement between our calculations and the experimentshas been found In the present study we extend our studyunder high pressures We present in Figure 5 the calcu-lated phonon spectrum and the corresponding PhDOS attwo typical pressures (25 and 50 GPa) It can be seenthat the gap between the acoustical branches and the opti-cal branches observed at ambient condition in our previ-ous study11 vanishes through compressing of the cell tosome extend At ambient condition the first main peakof the PhDOS lies at about 404 THz which stands forabout 169 meV in the energy space11 This observationis well consistent with experiments2 where they unveileda 15 meV mode of the PhDOS and the spectral electronndashphonon scattering function by measuring the specific heatand resistivity curves According to their analysis the lowfrequency vibration of the loosely bound Zr atoms is the}

_{Fig 5 Phonon dispersion of ZrB12 at (a) 25 GPa and (b) 50 GPa}

_{main contribution to the superconductivity of ZrB12 Thisgives us clear guidance in analyzing of our data From ourpresent data of the PhDOS the low frequency vibrationof the Zr atoms moves slightly higher in the frequencyspace The energy level of the first peak in the PhDOS isincreased from 169 meV at ambient condition to about237 meV at 50 GPa This observation clearly predictsthat the superconducting transition temperature Tc of ZrB12}

_{decreases with pressure thus supplies direct theoreticalcertification of the negative pressure effect on the super-conductivity of ZrB12}

₁₀₂₁

_{Thermodynamic properties can be determined throughphonon calculations using the quasiharmonic approxima-tion (QHA)2526 Under QHA the phonon contributionto the free energy FphT V and the phonon entropySphT V can be calculated by}

_{FphT V = kBTint}

_{0g ln}

_{[2 sinh}

₍

_2kBT

_{)]d (2)}

_and

_{SphT V =int}

_0gd

₍

_2T

_)[coth

₍

_2kBT

_)minus1

_]

_minuskB

_int

_{0gd ln}

_{[1minus exp}

_(minus

_kBT

_)](3)

_{where represents the phonon frequencies and g is thePhDOS Besides the specific heat at constant volume CV}

_{can be directly calculated through}

_{CV =kB}

_int

_0g

₍

_kBT

_{)2 expkBT expkBT minus12}

_{d (4)}

_{We plot the calculated temperature dependences of theFph Sph and CV of ZrB12 at three selected pressures in}

_{2284 Sci Adv Mater 6 2281ndash2285 2014}

_{Zhang et al Electronic Structure Mechanics and Thermodynamics of ZrB12 Under Pressure}

_ARTICLE

_{Fig 6 Temperature dependences of the (a) phonon free energy (Fph) (b) entropy (Sph) and (c) heat capacity at constant volume (CV ) of ZrB12 atthree selected pressures Results at zero pressure are obtained by using the PhDOS data in Ref [11] For comparison experimental CV data fromRef [2] are presented in the inset}

_{Figure 6 Results at zero pressure are obtained by using thePhDOS data in Ref [11] As shown in Figure 6(a) our cal-culated zero point energies are 112 124 and 133 kJmolrespectively under pressures of 0 25 and 50 GPa Thisillustrates that the zero point energy also increases withpressure As shown in Figure 6(a) the phonon free energydecreases gradually with increasing temperature but iswholly increased by applying compression in all tempera-ture we considered For the phonon entropy SphT V wesee that it increases rapidly with temperature in the lowtemperature domain Over about 800 K this increasingbehavior becomes slow down Through applying externalpressure to this material the phonon entropy SphT V decreases in the whole temperature domain For the spe-cific heat at constant volume CV it shows a sharp increaseup to sim 400 K and at high temperature is close to a con-stant which is the so-called Dulong-Petit limit27 Our cal-culated results in the low temperature domain are in goodagreement with available experimental data by Lortz et al2}

_{Under pressure the CV changes a little in temperature zoneof 200ndash1200 K Given the importance of this material insuperconductor field we expect these results to be usefulfor further theoretical and experimental investigations}

_{4 CONCLUSIONSIn summary we have calculated the structural parame-ters elastic constants elastic moduli elastic wave veloci-ties Debye temperature melting points phonon spectrumphonon free energy phonon entropy and specific heat ofZrB12 under a pressure range of 0ndash50 GPa Great enhance-ments of the elastic constants (C11 C12 and C44) and elas-tic muduli (B G and E) by external pressure have beenfound Our results show that the NEF decreases almostlinearly upon compression and the low frequency vibrationof the loosely bonded Zr atoms increases in the energylevel From these investigations we have supplied directtheoretical certifications of the negative pressure effect onthe superconductivity of ZrB12}

_{Acknowledgments This work was supported by NSFCunder Grant Nos 11104170 11074155 11374197 and11302121}

_{References and Notes1 B T Matthias T H Geballe K Andres E Corenzwit G W Hull}

_{and J P Maita Science 159 530 (1968)2 R Lortz Y Wang S Abe C Meingast Y B Paderno V Filippov}

_{and A Junod Phys Rev B 72 024547 (2005)3 G E Grechnev A E Baranovskiy V D Fil T V Ignatova I G}

_{Kolobov A V Logosha N Yu Shisevalova V B Filippov andO Eriksson Low Temp Phys 34 921 (2008)}

_{4 J Teyssier A B Kuzmenko D van der Marel F Marsiglio A BLiashchenko N Shitsevalova and V Filippov Phys Rev B 75134503 (2007)}

_{5 J Teyssier R Lortz A Petrovic D van der Marel V Filippov andN Shitsevalova Phys Rev B 78 134504 (2008)}

_{6 V A Gasparov I Sheikin F Levy J Teyssier and G Santi PhysRev Lett 101 097006 (2008)}

_{7 L Huerta A Duran R Falconi M Flores and R EscamillaPhysca C 470 456 (2010)}

_{8 A V Rybina K S Nemkovski P A Alekseev J M MignotE S Clementyev M Johnson L Capogna A V Dukhnenko A BLyashenko and V B Filippov Phys Rev B 82 024302 (2010)}

_{9 I R Shein and A L Ivanovskii Phys Solid State 45 1429 (2003)10 R Escamilla M Romero and F Morales Solid State Commun 152}

_{249 (2012)11 B T Wang W Zhang and W D Li Sci Adv Mater 5 1916 (2013)12 B T Wang P Zhang H Y Liu W D Li and P Zhang J Appl}

_{Phys 109 063514 (2011)13 P E Bloumlchl Phys Rev B 50 17953 (1994)14 G Kresse and J Furthmuumlller Phys Rev B 54 11169 (1996)15 J P Perdew K Burke and M Ernzerhof Phys Rev Lett 77 3865}

_{(1996)16 H J Monkhorst and J D Pack Phys Rev B 13 5188 (1972)17 R Hill Phys Phys Soc London 65 349 (1952)18 F R Lindemann Phys Z 11 609 (1910)19 A Togo F Oba and I Tanaka Phys Rev B 78 134106 (2008)20 W L McMillan Phys Rev 167 331 (1968)21 R Khasanov D Di Castro M Belogolovskii Yu Paderno}

_{V Filippov R Bruumltsch and H Keller Phys Rev B 72 224509(2005)}

_{22 N Korozlu K Colakoglu E Deligoz and S Aydin J AlloyCompd 546 157 (2013)}

_{23 B T Wang W D Li and P Zhang J Nucl Mater 420 501 (2012)24 B Post and F W Glaser Trans AIME 194 631 (1952)25 A Siegel K Parlinski and U D Wdowik Phys Rev B 74 104116}

_{(2006)26 P Zhang B-T Wang and X-G Zhao Phys Rev B 82 144110}

_{(2010)27 C Kittel Introduction to Solid State Physics 7th edn Wiley}

_{New York (1996)}

_{Sci Adv Mater 6 2281ndash2285 2014 2285}

_{View publication statsView publication stats}

_{Zhang et al Electronic Structure Mechanics and Thermodynamics of ZrB12 Under Pressure}

_ARTICLE

_{3 RESULTS AND DISCUSSIONIn ambient conditions ZrB12 crystallizes in a fcc struc-ture with space group Fm3m (No 225) with Zr in 4a(00 0) and B in 48e(05 y y) Wyckoff positions in whichthe Zr atoms and cuboctahedral B12 cluster are arrangedin an NaCl type structure The present optimized latticeconstant a and volume of the unit cell (Vuc) at selectedpressures in the pressure range of 0ndash50 GPa for ZrB12 arepresented in Table I As having been clearly indicated inour previous study11 our calculated results of equilibriumstate properties are in good agreement with experimentaldata3 This supplies the safeguard for our extending studyof ZrB12 upon compression At pressures of 0 25 and50 GPa the boron position parameters y are optimized tobe 01695 01696 and 01697 respectively This indicatesthat the pressure effect on the structure is limited}

_{To study the pressure effect on the electronic structure weplot the calculated total and partial density of states (DOSs)of ZrB12 at 0 25 and 50 GPa in Figure 2 At ambient condi-tion the main features of the orbital occupation are consis-tent with previous FLMTO calculations49 and the experi-mental X-ray photoemission spectra measurements7 Samewith previous theoretical calculations49 we also find thatthe main contribution to the NEF is from the B-2p stateUnder pressure the mainly occupied states of the B-spextend to deeper energy level while the NEF decreases alittle (see Fig 3) The decrement of NEF from zero pres-sure to 50 GPa is about 133 This kind of decrement ofNEF has been previously observed by Escamilla et al10}

_{in a pressure range of 0ndash104 GPa Due to the critical roleof the NEF to the superconducting transition tempera-ture Tc}

_{1220 we can speculate a negative pressure effecton the superconductivity of ZrB12}

_{1021 In spite of thisfact we still can say that the pressure effect on the elec-tronic structure is also limited which supports the goodmechanical and dynamical stabilities of this materials11}

_{Fig 2 Total and partial densities of states for ZrB12 at (a) 0 GPa(b) 25 GPa and (c) 50 GPa The Fermi energy level is set at zero}

_{Fig 3 Ratio of the NEF at finite pressures to that at 0 GPa for ZrB12The NEF at 0 GPa is about 085 stateeVfu}

_{In the following we will see that the pressure will enhancethe mechanical and dynamical propertiesIn our previous study11 we have shown that our calcu-}

_{lated elastic constants (C11 C12 and C44) elastic moduli(B G and E) Poissonrsquos ratio () density () transversesound velocities (t) longitudinal sound velocities (l)average sound velocities (m) and Debye temperature (13D)of ZrB12 at zero pressure are wholly consistent with corre-sponding results directly read or indirectly deduced fromexperiments3 In the present study we focus on the pres-sure effects on these mechanical andor thermodynami-cal properties We present our calculated data in pressurerange of 0ndash50 GPa in Table I and in Figure 4 Clearly pres-sure induced enhancements of the mechanical and thermo-dynamical properties can be seen This kind of pressureeffect has also been found by Korozlu et al22 in study ofMB12 (M= Zr Hf Y Lu) and by Wang et al in study ofthe element Zr and the TiZr alloy1223 As already shownin our previous study11 ZrB12 possesses almost the samemechanical properties of the bulk modulus B shear modu-lus G Youngrsquos modulus E and Poissonrsquos ratio with thatof ZrB2 but prominently lower density This point makesZrB12 more promising than ZrB2 in real engineered appli-cations The enhancement of the mechanical propertiesunder pressure for this material would be responsible forgood application in extreme andor complex conditions}

_{Fig 4 The melting temperature and the Debye temperature of ZrB12 asa function of pressure The solid lines are the polynomial fitting resultsof our data}

_{Sci Adv Mater 6 2281ndash2285 2014 2283}

_{Electronic Structure Mechanics and Thermodynamics of ZrB12 Under Pressure Zhang et al}

_ARTICLE

_{As shown in Table I one can find that the most slug-gish one upon compression is the Poissonrsquos ratio It onlyincreases 44 from zero pressure to 50 GPa The increaseproportions of C11 C12 C44 B G E t l m and 13Dare 552 610 540 571 537 545 175144 150 145 and 208 respectively Actuallythe pressure induced behaviors of the mechanical and ther-modynamical properties are mainly controlled by the elas-tic constants and the density (see method part) Due tothe fact that the elastic constants increase almost linearlywith pressure we can do first-order polynomial fitting Theincreasing rates of C11 C12 and C44 are then obtained tobe 5162 1442 and 2912 Here the increasing rates Rij}

_{of the elastic constants Cij are defined as Rij = CijP It is clear that the R11 is prominently larger than R12 aswell as R44}

_{22 However the specific values of the elasticconstants and also the increasing rates do possess somedifferences between our results and the data calculated byKorozlu et al22 While they performed their calculationsby using the plane wave pseudopotential we use here thePAW basis pseudopotential The PAW method is more use-ful when both localized and delocalized valence states areimportant which is just the case of our present study ofthe transition metal Zr based systemThe melting curves of materials have great scientific and}

_{technological interest We can easily deduce the meltingcurve of ZrB12 by using the Lindemann criterion from ourcalculated Debye temperature The Lindemann criterion isa single-parameter model and the free constant C can becalculated to equal to 00329 from a single point (Tm =2353 K at 0 GPa for ZrB12 by experiment24 as well as V =40687 Aring3 and 13D = 13029 K by the present work) Weshow the melting curves in Figure 4 It can be seen thatthe melting temperature of ZrB12 increase by about 200 Kfrom ambient pressure to 50 GPa But the increase rate isslowed down by pressure These pressure effects need moreexperiments to clarifyIn our previous study we have investigated the phonon}

_{spectrum of ZrB2 and ZrB12 at ambient pressure11 Goodagreement between our calculations and the experimentshas been found In the present study we extend our studyunder high pressures We present in Figure 5 the calcu-lated phonon spectrum and the corresponding PhDOS attwo typical pressures (25 and 50 GPa) It can be seenthat the gap between the acoustical branches and the opti-cal branches observed at ambient condition in our previ-ous study11 vanishes through compressing of the cell tosome extend At ambient condition the first main peakof the PhDOS lies at about 404 THz which stands forabout 169 meV in the energy space11 This observationis well consistent with experiments2 where they unveileda 15 meV mode of the PhDOS and the spectral electronndashphonon scattering function by measuring the specific heatand resistivity curves According to their analysis the lowfrequency vibration of the loosely bound Zr atoms is the}

_{Fig 5 Phonon dispersion of ZrB12 at (a) 25 GPa and (b) 50 GPa}

_{main contribution to the superconductivity of ZrB12 Thisgives us clear guidance in analyzing of our data From ourpresent data of the PhDOS the low frequency vibrationof the Zr atoms moves slightly higher in the frequencyspace The energy level of the first peak in the PhDOS isincreased from 169 meV at ambient condition to about237 meV at 50 GPa This observation clearly predictsthat the superconducting transition temperature Tc of ZrB12}

_{decreases with pressure thus supplies direct theoreticalcertification of the negative pressure effect on the super-conductivity of ZrB12}

₁₀₂₁

_{Thermodynamic properties can be determined throughphonon calculations using the quasiharmonic approxima-tion (QHA)2526 Under QHA the phonon contributionto the free energy FphT V and the phonon entropySphT V can be calculated by}

_{FphT V = kBTint}

_{0g ln}

_{[2 sinh}

₍

_2kBT

_{)]d (2)}

_and

_{SphT V =int}

_0gd

₍

_2T

_)[coth

₍

_2kBT

_)minus1

_]

_minuskB

_int

_{0gd ln}

_{[1minus exp}

_(minus

_kBT

_)](3)

_{where represents the phonon frequencies and g is thePhDOS Besides the specific heat at constant volume CV}

_{can be directly calculated through}

_{CV =kB}

_int

_0g

₍

_kBT

_{)2 expkBT expkBT minus12}

_{d (4)}

_{We plot the calculated temperature dependences of theFph Sph and CV of ZrB12 at three selected pressures in}

_{2284 Sci Adv Mater 6 2281ndash2285 2014}

_{Zhang et al Electronic Structure Mechanics and Thermodynamics of ZrB12 Under Pressure}

_ARTICLE

_{Fig 6 Temperature dependences of the (a) phonon free energy (Fph) (b) entropy (Sph) and (c) heat capacity at constant volume (CV ) of ZrB12 atthree selected pressures Results at zero pressure are obtained by using the PhDOS data in Ref [11] For comparison experimental CV data fromRef [2] are presented in the inset}

_{Figure 6 Results at zero pressure are obtained by using thePhDOS data in Ref [11] As shown in Figure 6(a) our cal-culated zero point energies are 112 124 and 133 kJmolrespectively under pressures of 0 25 and 50 GPa Thisillustrates that the zero point energy also increases withpressure As shown in Figure 6(a) the phonon free energydecreases gradually with increasing temperature but iswholly increased by applying compression in all tempera-ture we considered For the phonon entropy SphT V wesee that it increases rapidly with temperature in the lowtemperature domain Over about 800 K this increasingbehavior becomes slow down Through applying externalpressure to this material the phonon entropy SphT V decreases in the whole temperature domain For the spe-cific heat at constant volume CV it shows a sharp increaseup to sim 400 K and at high temperature is close to a con-stant which is the so-called Dulong-Petit limit27 Our cal-culated results in the low temperature domain are in goodagreement with available experimental data by Lortz et al2}

_{Under pressure the CV changes a little in temperature zoneof 200ndash1200 K Given the importance of this material insuperconductor field we expect these results to be usefulfor further theoretical and experimental investigations}

_{4 CONCLUSIONSIn summary we have calculated the structural parame-ters elastic constants elastic moduli elastic wave veloci-ties Debye temperature melting points phonon spectrumphonon free energy phonon entropy and specific heat ofZrB12 under a pressure range of 0ndash50 GPa Great enhance-ments of the elastic constants (C11 C12 and C44) and elas-tic muduli (B G and E) by external pressure have beenfound Our results show that the NEF decreases almostlinearly upon compression and the low frequency vibrationof the loosely bonded Zr atoms increases in the energylevel From these investigations we have supplied directtheoretical certifications of the negative pressure effect onthe superconductivity of ZrB12}

_{Acknowledgments This work was supported by NSFCunder Grant Nos 11104170 11074155 11374197 and11302121}

_{References and Notes1 B T Matthias T H Geballe K Andres E Corenzwit G W Hull}

_{and J P Maita Science 159 530 (1968)2 R Lortz Y Wang S Abe C Meingast Y B Paderno V Filippov}

_{and A Junod Phys Rev B 72 024547 (2005)3 G E Grechnev A E Baranovskiy V D Fil T V Ignatova I G}

_{Kolobov A V Logosha N Yu Shisevalova V B Filippov andO Eriksson Low Temp Phys 34 921 (2008)}

_{4 J Teyssier A B Kuzmenko D van der Marel F Marsiglio A BLiashchenko N Shitsevalova and V Filippov Phys Rev B 75134503 (2007)}

_{5 J Teyssier R Lortz A Petrovic D van der Marel V Filippov andN Shitsevalova Phys Rev B 78 134504 (2008)}

_{6 V A Gasparov I Sheikin F Levy J Teyssier and G Santi PhysRev Lett 101 097006 (2008)}

_{7 L Huerta A Duran R Falconi M Flores and R EscamillaPhysca C 470 456 (2010)}

_{8 A V Rybina K S Nemkovski P A Alekseev J M MignotE S Clementyev M Johnson L Capogna A V Dukhnenko A BLyashenko and V B Filippov Phys Rev B 82 024302 (2010)}

_{9 I R Shein and A L Ivanovskii Phys Solid State 45 1429 (2003)10 R Escamilla M Romero and F Morales Solid State Commun 152}

_{249 (2012)11 B T Wang W Zhang and W D Li Sci Adv Mater 5 1916 (2013)12 B T Wang P Zhang H Y Liu W D Li and P Zhang J Appl}

_{Phys 109 063514 (2011)13 P E Bloumlchl Phys Rev B 50 17953 (1994)14 G Kresse and J Furthmuumlller Phys Rev B 54 11169 (1996)15 J P Perdew K Burke and M Ernzerhof Phys Rev Lett 77 3865}

_{(1996)16 H J Monkhorst and J D Pack Phys Rev B 13 5188 (1972)17 R Hill Phys Phys Soc London 65 349 (1952)18 F R Lindemann Phys Z 11 609 (1910)19 A Togo F Oba and I Tanaka Phys Rev B 78 134106 (2008)20 W L McMillan Phys Rev 167 331 (1968)21 R Khasanov D Di Castro M Belogolovskii Yu Paderno}

_{V Filippov R Bruumltsch and H Keller Phys Rev B 72 224509(2005)}

_{22 N Korozlu K Colakoglu E Deligoz and S Aydin J AlloyCompd 546 157 (2013)}

_{23 B T Wang W D Li and P Zhang J Nucl Mater 420 501 (2012)24 B Post and F W Glaser Trans AIME 194 631 (1952)25 A Siegel K Parlinski and U D Wdowik Phys Rev B 74 104116}

_{(2006)26 P Zhang B-T Wang and X-G Zhao Phys Rev B 82 144110}

_{(2010)27 C Kittel Introduction to Solid State Physics 7th edn Wiley}

_{New York (1996)}

_{Sci Adv Mater 6 2281ndash2285 2014 2285}

_{View publication statsView publication stats}

_{Electronic Structure Mechanics and Thermodynamics of ZrB12 Under Pressure Zhang et al}

_ARTICLE

_{As shown in Table I one can find that the most slug-gish one upon compression is the Poissonrsquos ratio It onlyincreases 44 from zero pressure to 50 GPa The increaseproportions of C11 C12 C44 B G E t l m and 13Dare 552 610 540 571 537 545 175144 150 145 and 208 respectively Actuallythe pressure induced behaviors of the mechanical and ther-modynamical properties are mainly controlled by the elas-tic constants and the density (see method part) Due tothe fact that the elastic constants increase almost linearlywith pressure we can do first-order polynomial fitting Theincreasing rates of C11 C12 and C44 are then obtained tobe 5162 1442 and 2912 Here the increasing rates Rij}

_{of the elastic constants Cij are defined as Rij = CijP It is clear that the R11 is prominently larger than R12 aswell as R44}

_{22 However the specific values of the elasticconstants and also the increasing rates do possess somedifferences between our results and the data calculated byKorozlu et al22 While they performed their calculationsby using the plane wave pseudopotential we use here thePAW basis pseudopotential The PAW method is more use-ful when both localized and delocalized valence states areimportant which is just the case of our present study ofthe transition metal Zr based systemThe melting curves of materials have great scientific and}

_{technological interest We can easily deduce the meltingcurve of ZrB12 by using the Lindemann criterion from ourcalculated Debye temperature The Lindemann criterion isa single-parameter model and the free constant C can becalculated to equal to 00329 from a single point (Tm =2353 K at 0 GPa for ZrB12 by experiment24 as well as V =40687 Aring3 and 13D = 13029 K by the present work) Weshow the melting curves in Figure 4 It can be seen thatthe melting temperature of ZrB12 increase by about 200 Kfrom ambient pressure to 50 GPa But the increase rate isslowed down by pressure These pressure effects need moreexperiments to clarifyIn our previous study we have investigated the phonon}

_{spectrum of ZrB2 and ZrB12 at ambient pressure11 Goodagreement between our calculations and the experimentshas been found In the present study we extend our studyunder high pressures We present in Figure 5 the calcu-lated phonon spectrum and the corresponding PhDOS attwo typical pressures (25 and 50 GPa) It can be seenthat the gap between the acoustical branches and the opti-cal branches observed at ambient condition in our previ-ous study11 vanishes through compressing of the cell tosome extend At ambient condition the first main peakof the PhDOS lies at about 404 THz which stands forabout 169 meV in the energy space11 This observationis well consistent with experiments2 where they unveileda 15 meV mode of the PhDOS and the spectral electronndashphonon scattering function by measuring the specific heatand resistivity curves According to their analysis the lowfrequency vibration of the loosely bound Zr atoms is the}

_{Fig 5 Phonon dispersion of ZrB12 at (a) 25 GPa and (b) 50 GPa}

_{main contribution to the superconductivity of ZrB12 Thisgives us clear guidance in analyzing of our data From ourpresent data of the PhDOS the low frequency vibrationof the Zr atoms moves slightly higher in the frequencyspace The energy level of the first peak in the PhDOS isincreased from 169 meV at ambient condition to about237 meV at 50 GPa This observation clearly predictsthat the superconducting transition temperature Tc of ZrB12}

_{decreases with pressure thus supplies direct theoreticalcertification of the negative pressure effect on the super-conductivity of ZrB12}

₁₀₂₁

_{Thermodynamic properties can be determined throughphonon calculations using the quasiharmonic approxima-tion (QHA)2526 Under QHA the phonon contributionto the free energy FphT V and the phonon entropySphT V can be calculated by}

_{FphT V = kBTint}

_{0g ln}

_{[2 sinh}

₍

_2kBT

_{)]d (2)}

_and

_{SphT V =int}

_0gd

₍

_2T

_)[coth

₍

_2kBT

_)minus1

_]

_minuskB

_int

_{0gd ln}

_{[1minus exp}

_(minus

_kBT

_)](3)

_{where represents the phonon frequencies and g is thePhDOS Besides the specific heat at constant volume CV}

_{can be directly calculated through}

_{CV =kB}

_int

_0g

₍

_kBT

_{)2 expkBT expkBT minus12}

_{d (4)}

_{We plot the calculated temperature dependences of theFph Sph and CV of ZrB12 at three selected pressures in}

_{2284 Sci Adv Mater 6 2281ndash2285 2014}

_{Zhang et al Electronic Structure Mechanics and Thermodynamics of ZrB12 Under Pressure}

_ARTICLE

_{Fig 6 Temperature dependences of the (a) phonon free energy (Fph) (b) entropy (Sph) and (c) heat capacity at constant volume (CV ) of ZrB12 atthree selected pressures Results at zero pressure are obtained by using the PhDOS data in Ref [11] For comparison experimental CV data fromRef [2] are presented in the inset}

_{Figure 6 Results at zero pressure are obtained by using thePhDOS data in Ref [11] As shown in Figure 6(a) our cal-culated zero point energies are 112 124 and 133 kJmolrespectively under pressures of 0 25 and 50 GPa Thisillustrates that the zero point energy also increases withpressure As shown in Figure 6(a) the phonon free energydecreases gradually with increasing temperature but iswholly increased by applying compression in all tempera-ture we considered For the phonon entropy SphT V wesee that it increases rapidly with temperature in the lowtemperature domain Over about 800 K this increasingbehavior becomes slow down Through applying externalpressure to this material the phonon entropy SphT V decreases in the whole temperature domain For the spe-cific heat at constant volume CV it shows a sharp increaseup to sim 400 K and at high temperature is close to a con-stant which is the so-called Dulong-Petit limit27 Our cal-culated results in the low temperature domain are in goodagreement with available experimental data by Lortz et al2}

_{Under pressure the CV changes a little in temperature zoneof 200ndash1200 K Given the importance of this material insuperconductor field we expect these results to be usefulfor further theoretical and experimental investigations}

_{4 CONCLUSIONSIn summary we have calculated the structural parame-ters elastic constants elastic moduli elastic wave veloci-ties Debye temperature melting points phonon spectrumphonon free energy phonon entropy and specific heat ofZrB12 under a pressure range of 0ndash50 GPa Great enhance-ments of the elastic constants (C11 C12 and C44) and elas-tic muduli (B G and E) by external pressure have beenfound Our results show that the NEF decreases almostlinearly upon compression and the low frequency vibrationof the loosely bonded Zr atoms increases in the energylevel From these investigations we have supplied directtheoretical certifications of the negative pressure effect onthe superconductivity of ZrB12}

_{Acknowledgments This work was supported by NSFCunder Grant Nos 11104170 11074155 11374197 and11302121}

_{References and Notes1 B T Matthias T H Geballe K Andres E Corenzwit G W Hull}

_{and J P Maita Science 159 530 (1968)2 R Lortz Y Wang S Abe C Meingast Y B Paderno V Filippov}

_{and A Junod Phys Rev B 72 024547 (2005)3 G E Grechnev A E Baranovskiy V D Fil T V Ignatova I G}

_{Kolobov A V Logosha N Yu Shisevalova V B Filippov andO Eriksson Low Temp Phys 34 921 (2008)}

_{4 J Teyssier A B Kuzmenko D van der Marel F Marsiglio A BLiashchenko N Shitsevalova and V Filippov Phys Rev B 75134503 (2007)}

_{5 J Teyssier R Lortz A Petrovic D van der Marel V Filippov andN Shitsevalova Phys Rev B 78 134504 (2008)}

_{6 V A Gasparov I Sheikin F Levy J Teyssier and G Santi PhysRev Lett 101 097006 (2008)}

_{7 L Huerta A Duran R Falconi M Flores and R EscamillaPhysca C 470 456 (2010)}

_{8 A V Rybina K S Nemkovski P A Alekseev J M MignotE S Clementyev M Johnson L Capogna A V Dukhnenko A BLyashenko and V B Filippov Phys Rev B 82 024302 (2010)}

_{9 I R Shein and A L Ivanovskii Phys Solid State 45 1429 (2003)10 R Escamilla M Romero and F Morales Solid State Commun 152}

_{249 (2012)11 B T Wang W Zhang and W D Li Sci Adv Mater 5 1916 (2013)12 B T Wang P Zhang H Y Liu W D Li and P Zhang J Appl}

_{Phys 109 063514 (2011)13 P E Bloumlchl Phys Rev B 50 17953 (1994)14 G Kresse and J Furthmuumlller Phys Rev B 54 11169 (1996)15 J P Perdew K Burke and M Ernzerhof Phys Rev Lett 77 3865}

_{(1996)16 H J Monkhorst and J D Pack Phys Rev B 13 5188 (1972)17 R Hill Phys Phys Soc London 65 349 (1952)18 F R Lindemann Phys Z 11 609 (1910)19 A Togo F Oba and I Tanaka Phys Rev B 78 134106 (2008)20 W L McMillan Phys Rev 167 331 (1968)21 R Khasanov D Di Castro M Belogolovskii Yu Paderno}

_{V Filippov R Bruumltsch and H Keller Phys Rev B 72 224509(2005)}

_{22 N Korozlu K Colakoglu E Deligoz and S Aydin J AlloyCompd 546 157 (2013)}

_{23 B T Wang W D Li and P Zhang J Nucl Mater 420 501 (2012)24 B Post and F W Glaser Trans AIME 194 631 (1952)25 A Siegel K Parlinski and U D Wdowik Phys Rev B 74 104116}

_{(2006)26 P Zhang B-T Wang and X-G Zhao Phys Rev B 82 144110}

_{(2010)27 C Kittel Introduction to Solid State Physics 7th edn Wiley}

_{New York (1996)}

_{Sci Adv Mater 6 2281ndash2285 2014 2285}

_{View publication statsView publication stats}

_{Zhang et al Electronic Structure Mechanics and Thermodynamics of ZrB12 Under Pressure}

_ARTICLE

_{Fig 6 Temperature dependences of the (a) phonon free energy (Fph) (b) entropy (Sph) and (c) heat capacity at constant volume (CV ) of ZrB12 atthree selected pressures Results at zero pressure are obtained by using the PhDOS data in Ref [11] For comparison experimental CV data fromRef [2] are presented in the inset}

_{Figure 6 Results at zero pressure are obtained by using thePhDOS data in Ref [11] As shown in Figure 6(a) our cal-culated zero point energies are 112 124 and 133 kJmolrespectively under pressures of 0 25 and 50 GPa Thisillustrates that the zero point energy also increases withpressure As shown in Figure 6(a) the phonon free energydecreases gradually with increasing temperature but iswholly increased by applying compression in all tempera-ture we considered For the phonon entropy SphT V wesee that it increases rapidly with temperature in the lowtemperature domain Over about 800 K this increasingbehavior becomes slow down Through applying externalpressure to this material the phonon entropy SphT V decreases in the whole temperature domain For the spe-cific heat at constant volume CV it shows a sharp increaseup to sim 400 K and at high temperature is close to a con-stant which is the so-called Dulong-Petit limit27 Our cal-culated results in the low temperature domain are in goodagreement with available experimental data by Lortz et al2}

_{Under pressure the CV changes a little in temperature zoneof 200ndash1200 K Given the importance of this material insuperconductor field we expect these results to be usefulfor further theoretical and experimental investigations}

_{4 CONCLUSIONSIn summary we have calculated the structural parame-ters elastic constants elastic moduli elastic wave veloci-ties Debye temperature melting points phonon spectrumphonon free energy phonon entropy and specific heat ofZrB12 under a pressure range of 0ndash50 GPa Great enhance-ments of the elastic constants (C11 C12 and C44) and elas-tic muduli (B G and E) by external pressure have beenfound Our results show that the NEF decreases almostlinearly upon compression and the low frequency vibrationof the loosely bonded Zr atoms increases in the energylevel From these investigations we have supplied directtheoretical certifications of the negative pressure effect onthe superconductivity of ZrB12}

_{Acknowledgments This work was supported by NSFCunder Grant Nos 11104170 11074155 11374197 and11302121}

_{References and Notes1 B T Matthias T H Geballe K Andres E Corenzwit G W Hull}

_{and J P Maita Science 159 530 (1968)2 R Lortz Y Wang S Abe C Meingast Y B Paderno V Filippov}

_{and A Junod Phys Rev B 72 024547 (2005)3 G E Grechnev A E Baranovskiy V D Fil T V Ignatova I G}

_{Kolobov A V Logosha N Yu Shisevalova V B Filippov andO Eriksson Low Temp Phys 34 921 (2008)}

_{4 J Teyssier A B Kuzmenko D van der Marel F Marsiglio A BLiashchenko N Shitsevalova and V Filippov Phys Rev B 75134503 (2007)}

_{5 J Teyssier R Lortz A Petrovic D van der Marel V Filippov andN Shitsevalova Phys Rev B 78 134504 (2008)}

_{6 V A Gasparov I Sheikin F Levy J Teyssier and G Santi PhysRev Lett 101 097006 (2008)}

_{7 L Huerta A Duran R Falconi M Flores and R EscamillaPhysca C 470 456 (2010)}

_{8 A V Rybina K S Nemkovski P A Alekseev J M MignotE S Clementyev M Johnson L Capogna A V Dukhnenko A BLyashenko and V B Filippov Phys Rev B 82 024302 (2010)}

_{9 I R Shein and A L Ivanovskii Phys Solid State 45 1429 (2003)10 R Escamilla M Romero and F Morales Solid State Commun 152}

_{249 (2012)11 B T Wang W Zhang and W D Li Sci Adv Mater 5 1916 (2013)12 B T Wang P Zhang H Y Liu W D Li and P Zhang J Appl}

_{Phys 109 063514 (2011)13 P E Bloumlchl Phys Rev B 50 17953 (1994)14 G Kresse and J Furthmuumlller Phys Rev B 54 11169 (1996)15 J P Perdew K Burke and M Ernzerhof Phys Rev Lett 77 3865}

_{(1996)16 H J Monkhorst and J D Pack Phys Rev B 13 5188 (1972)17 R Hill Phys Phys Soc London 65 349 (1952)18 F R Lindemann Phys Z 11 609 (1910)19 A Togo F Oba and I Tanaka Phys Rev B 78 134106 (2008)20 W L McMillan Phys Rev 167 331 (1968)21 R Khasanov D Di Castro M Belogolovskii Yu Paderno}

_{V Filippov R Bruumltsch and H Keller Phys Rev B 72 224509(2005)}

_{22 N Korozlu K Colakoglu E Deligoz and S Aydin J AlloyCompd 546 157 (2013)}

_{23 B T Wang W D Li and P Zhang J Nucl Mater 420 501 (2012)24 B Post and F W Glaser Trans AIME 194 631 (1952)25 A Siegel K Parlinski and U D Wdowik Phys Rev B 74 104116}

_{(2006)26 P Zhang B-T Wang and X-G Zhao Phys Rev B 82 144110}

_{(2010)27 C Kittel Introduction to Solid State Physics 7th edn Wiley}

_{New York (1996)}

_{Sci Adv Mater 6 2281ndash2285 2014 2285}

_{View publication statsView publication stats}