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Structural phase transition, electronic and lattice dynamical properties of half-Heusler compound CaAuBi
Accepted Manuscript Structural phase transition, electronic and lattice dynamical properties of Half-Heusler compound CaAuBi Deepika Shrivastava, Sankar P. Sanyal PII:
S0925-8388(18)30695-9
DOI:
10.1016/j.jallcom.2018.02.208
Reference:
JALCOM 45084
To appear in:
Journal of Alloys and Compounds
Received Date: 16 December 2017 Revised Date:
13 February 2018
Accepted Date: 15 February 2018
Please cite this article as: D. Shrivastava, S.P. Sanyal, Structural phase transition, electronic and lattice dynamical properties of Half-Heusler compound CaAuBi, Journal of Alloys and Compounds (2018), doi: 10.1016/j.jallcom.2018.02.208. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Structural Phase Transition, Electronic and Lattice Dynamical Properties of Half-Heusler Compound CaAuBi
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Deepika Shrivastava a,*, Sankar P. Sanyal a Department of Physics, Barkatullah University, Bhopal, 462026, India *
E-mail: [email protected]
Abstract:
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The structural phase transition, electronic and lattice dynamical properties of CaAuBi half-
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Heusler compound have been investigated using first-principles density functional theory within generalized gradient approximation (GGA). The comparison of enthalpies of cubic (MgAgAstype) and hexagonal (LiGaGe-type) structure as a function of pressure suggest the structural phase transition from cubic (MgAgAs-type) to hexagonal (LiGaGe-type) at 19.6 GPa, which agrees well with the experimental results. The calculated ground state properties are in good
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agreement with available experimental and theoretical data. The electronic and bonding properties reveal that CaAuBi compound has mixed metallic, ionic as well as covalent nature. The lattice dynamical study of CaAuBi in both phases cubic (MgAgAs-type) and hexagonal
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(LiGaGe-type) confirms that CaAuBi is stable in cubic structure at ambient pressure and
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transform to hexagonal structure at high pressure. The thermodynamic properties are also calculated.
Keywords: Half-Heusler compounds; Phase transition; Phonon properties; Thermodynamic properties. *
Corresponding author: Department of Physics, Barkatullah University, Bhopal, 462026, India
E-mail address: [email protected] Tel : + 91- 755-2517120, Fax : + 91- 755-2491823 Co authors: Condensed Matter Physics Laboratory, Department of Physics, Barkatullah University, Bhopal, M.P. 462-026, India Tel : + 91- 755-2517120, Fax : + 91-755-2491823, E-mail : [email protected]
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1. Introduction: An interesting class of ternary compounds having chemical formula XYZ named as half-Heusler
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(HH) alloys, crystallize in the face centered cubic MgAgAs-type structure (space group F4-3m; no.216) [1], where X is rare earth or transition metal, Y is transition metal and Z is a main group element [2-4]. This XYZ cubic structure can be understood by the filling of ‘X’ atom into a YZ zincblende structure [3]. The half-Heusler compounds have a narrow band gap at the Fermi level
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for one spin direction i.e. they possess metallic behavior, and are insulator or semiconductor in opposite spin orientation [5], suggesting that they can be used as spin valves, to increase gaint
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magneto-resistance (GMR) [6]. The ferromagnetic behavior in the half-Heusler compounds was first observed by de Groot et. al. [7]. These compounds have extensive potential for applications in the field of superconductivity, thermoelectric material and spintronics [2,8-10]. In spintronics they have the capability to use both charge and spin degree of freedom to achieve
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multifunctional electronic devices [11,12]. The half-Heusler compounds are also used in magnetic random access memories (MRAM) and magnetic sensors [12-14] because of spin polarization. The piezoelectric response and associated properties in half-Heusler compounds are
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predicted by Roy et. al. [15]. Half-Heusler compounds may also crystallize in several crystal structures. For instance REAuSn belongs to cubic structure (MgAgAs-type) when RE is smaller
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rare-earth element, while hexagonal structure (LiGaGe-type) is favorable when it contains larger rare-earth element. LiGaGe structure is considered as a hexagonal variant of half-Heusler XYZ structure type, where X atoms are filled into a hexagonal wurtzite structure [3]. In the recent past, several attempts have been made to understand the structural properties of half-Heusler alloy systems [16,17].
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In a recent study Xie et. al. [18] have reported the structural phase transition of ternary half-Heusler CaAuBi compound from cubic (MgAgAs-type) to hexagonal (LiGaGe-type) phase by synchrotron X-ray diffraction method at 18 GPa. These authors [18] have predicted that
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CaAuBi might undergo another structural phase transition to ZrBeSi-type hexagonal structure at further high pressure. In the present paper we first examine the stability of crystal structure of CaAuBi using density functional theory at ambient and if it has any possibility to undergo
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another structural phase transition. Another approach to establish stability of crystal structure is to calculate phonon dispersion curves. Therefore, it will be very interesting to examine the lattice
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dynamical properties of CaAuBi, although there has been no experimental or theoretical work reported on vibrational properties in literature so far. In the present paper we report for the first time a comprehensive ab-initio calculation of structural phase transition, electronic, vibrational and thermodynamic properties of CaAuBi half-Heusler compound.
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2. Computational method:
All the calculations reported in this paper have been performed within density functional theory (DFT) [19]. For phase transition, electronic structure and associated properties of CaAuBi, we
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have used the full-potential linearized augmented plane wave (FP-LAPW) method [20] with generalized gradient approximation (GGA) scheme of Perdew, Burke and Ernzerhof (PBE) [21]
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for exchange and correlation effects. The energy convergence is achieved by expanding the basis function up to RMT * Kmax = 7, where RMT is the smallest atomic radius in the unit cell and Kmax refers to the magnitude of the largest k vector in the plane wave expansion. The maximum value for partial waves inside the atomic sphere is lmax = 10, while the charge density is Fourier expanded up to Gmax = 12 (a.u.)-1. The self-consistent calculations are converged when the total energy of the system is stable within 10-4 Ry. The Monkhorst-Pack [22] k-points grid of
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10x10x10 is used to sample the Brillouin zone for electronic structure calculations. The phonon dispersion curves and phonon density of states are calculated using the plane wave pseudopotential method as implemented in PWSCF code [23] with density functional
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perturbation theory (DFPT) [24]. We have used the norm-conserving pseudopotential of Troullier and Martins type [25], for describing the interaction between the valence electron, nuclei and the core electrons. We have used smearing of Marzari-Vanderbilt [26] to select the
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occupation distribution. The convergence is achieved with a kinetic energy cut-off 80 Ry. which is sufficient to fully converge all properties. The force constants for phonon dispersion
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calculations are obtained using 4x4x4 q-mesh in the first Brillouin zone. The thermodynamic properties are calculated using quasi-harmonic approximation (QHA) [24].
3. Results and Discussion:
3.1 Structural and electronic properties:
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In order to acertain the stability of crystal structure of CaAuBi half-Heusler compound at ambient condition, we have assumed three crystallographic structures for this compound, namely cubic, MgAgAs-type (F4-3m, No. 216), hexagonal LiGaGe-type (P63mc, No. 186) and ZrBeSi-
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type (P63/mmc, No. 194) as shown in Figs. 1(a)-(c). The obtained optimal atomic positions are Ca-4a(0,0,0), Au-4c(1/4,1/4,1/4) and Bi-4b(1/2,1/2,1/2) for cubic (MgAgAs-type) phase, while
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Ca-2a(0,0,0) and Au and Bi are located at 2b(1/3,2/3,z) where z = 0.3104(Au)
and z =
0.6899(Bi) for hexagonal (LiGaGe-type) phase and Ca-2a(0,0,0), Au-2c(1/3,2/3,1/4) and Bi2d(1/3,2/3,3/4) for hexagonal (ZrBeSi-type) phase. As can be seen from Figs. 1(b) and 1(c), there is very little difference between the two hexagonal structures. Xie et. al. [18] have reported that at ambient conditions, CaAuBi is stable in cubic (MgAgAs-type) phase, and transforms to hexagonal (LiGaGe-type) phase at high pressure (around 18 GPa). They have also found in their
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XRD measurement that around 3 GPa the cubic phase of CaAuBi still remained and above 3 GPa a small peak of hexagonal phase appears, which increases with increasing pressure. Fig. 2(a) shows the calculated total energies as a function of volume in the three different phases (cubic
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(MgAgAs) and hexagonal (LiGaGe & ZrBeSi)) of CaAuBi half-Heusler compound. The calculated total energies are fitted to Birch-Murnaghan’s equation of state [27] to obtain the structural properties such as lattice constant (a0), bulk modulus (B) and pressure derivative of
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bulk modulus (B'). The calculated values of a0, B and B' in cubic (MgAgAs) phase are shown in Table 1 as cubic phase is most stable at ambient condition. It can be seen from Fig. 2(a) and (b)
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that there is a structural phase transition from cubic (MgAgAs-type) to hexagonal (LiGaGe-type) at around 19.6 GPa (present theoretical calculation) as the total energies in the two phases intercept. However we do not find any indication that there will be a hexagonal (LiGaGe-type) to hexagonal (ZrBeSi-type) transition at high pressure (or reduced volume). The calculated values
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of lattice constant, bulk modulus and its pressure derivative are in good agreement with available experimental and theoretical results [28, 18]. As can be seen from Fig. 2(a), we have found that under ambient condition CaAuBi crystallize in cubic (MgAgAs-type) structure and at high
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pressure the hexagonal (LiGaGe-type) phase would be favored. However, a cubic (MgAgAstype) to hexagonal (LiGaGe-type) structural transition can only be achieved at high pressure of
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around 19.6 GPa. We have also expressed the phase transition in appropriate manner, by plotting the variation of enthalpy with pressure as shown in Fig. 2(b), which explains that the enthalpy of both phases (cubic and hexagonal) are same at 19.6 GPa. The calculated value of phase transition pressure is in close agreement with the experimental value 18 GPa [18] and shown in Table 1. Fig. 2(c) shows the unit cell volumes of both the phases (cubic and hexagonal) monotonically decrease with increasing pressure which is in good agreement with experimental [18] results.
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The calculated electronic band structure and density of states (total and partial) of CaAuBi compound in its cubic phase are shown in Figs. 3(a) and (b). CaAuBi shows semimetallic character because there is no band gap at Fermi level and three bands cross the
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Fermi level. The lowest occupied bands in the energy range -6 to -4 eV are mainly due to the Au5d and Bi-6p states. The next group of bands (above the Fermi level) are predominantly due to the Ca-4s, Au-5d and Bi-6p states. As is seen from Fig. 3(b) in the energy range -6 to -4 eV, a
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strong peak appears at -5 eV which is mainly characterized by Au-5d and Bi-6p states with some contribution of Ca-4s states. In the energy range -3 to 4 eV there is a hybridization between Au-
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5d and Bi-6p states giving rise to a strong peak at -2 eV and near 3.0 eV. In the conduction region the hybridization between Au-5d and Bi-6p states leads to a covalent character of this material. As a consequence the cubic CaAuBi contains a mixture of metallic and covalent character. The density of states at Fermi level is 0.30 States/eV.
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We have plotted the Fermi surfaces (FS) of CaAuBi for the bands which are crossing the Fermi level and are shown in Figs. 4(a)-(d). The electronic nature of any metallic compounds can be understood by the Fermi surfaces. The measurement of energy distribution curve (EDC) for
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the different wave vectors in the Brilliouin zone is the appropriate way to find the Fermi surfaces
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and to ensure the exact location of wave vectors where the bands cross the Fermi level. All the Fermi surfaces contained the set of hole and electron sheets. As is seen in Fig. 3(a), there are three bands namely 20, 21 and 22 which cross the Fermi level and make Fermi surface (FS) sheets. The first two sheets for the bands 20 and 21 are mainly due to the hole pocket at Γ point and the third sheet is mainly due to the electron pocket along L → Γ point. The fourth sheet is the confluent FS sheet of these three sheets. The energy ranges for the bands 20, 21 and 22 are 0.22-0.44 Ry., 0.26-0.44 Ry. and 0.41-0.55 Ry. respectively.
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In order to identify the type of bonding between the different atoms in the cubic CaAuBi compound, we have calculated the charge density plots in (110) plane and is shown in Fig. 5. If the charge transfer between cation to anion and charge contours are isolated and spherical, the
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bonding is considered to be purely ionic, while the bonding is covalent when the charge density contours overlaps to each other. As is seen from Fig.5, the charge density distributions for the cation and anion in CaAuBi are spherical, which indicate that the bonding is ionic in nature. The
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metallic contribution is mainly due to the partially filled Bi-6p state. The covalent bonding comes from the mutually shared electron of Au-5d and Bi-6p states, which can be seen from
metallic and covalent bonding.
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partial density of states (see in Figure 3(b)). Therefore CaAuBi shows the mixing of ionic,
3.2 Phonon dispersion curves and thermodynamic properties:
The phonon dispersion curves and phonon density of states of CaAuBi in cubic (MgAgAs-type)
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phase are calculated at ambient pressure and shown in Figs. 6 (a)-(b). The primitive cell of cubic (MgAgAs-type) CaAuBi contains three atoms per unit cell, therefore, nine vibrational modes appear in the phonon dispersion curve in which three are acoustic branches and remaining six are
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optical ones. Figs. 6(a)-(b) show that, all the phonon modes are positive and the absence of any imaginary frequency confirms the stability of CaAuBi in cubic (MgAgAs-type) structure at
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ambient pressure. The metallic nature of CaAuBi in cubic phase is confirmed by the degeneracy of longitudinal optical and transverse optical mode at the zone centre (around 175 cm-1). It can be seen that there is no gap between longitudinal acoustic (LA) and transverse optical (TO) modes at low frequency regime (Fig. 6(a) (PDC) at X point) in the frequency range at about 75 cm-1. Thus a phonon can make a transition from the acoustic to optical mode without any momentum transfer. However there is a large energy gap between the two optical branches
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which can be ascribed to the mass difference between the constituent atom, Ca, Au and Bi. From Fig. 6(b), it can be noticed that, the phonon state in the acoustic region is dominantly created by the vibration of Au atoms due to their heavier mass with considerable contribution of Ca atoms.
Phonon DOS is found at around 175 cm-1.
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Ca and Bi atoms show the most contribution in the optical phonon states. The highest peak in
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Further to understand the phase transition of CaAuBi HH compound from Cubic (MgAsAs-type) to hexagonal (LiGage-type) structure in terms of lattice dynamics, we have
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calculated the phonon dispersion curves of CaAuBi in hexagonal structure at zero pressure and around the transition pressure (25 GPa) and are shown in Figs. 7 (a)-(b). It is clear from Fig. 7(a), that the transverse acoustic (TA) phonon branch along Γ-K and Γ-M directions becomes negative indicating that the hexagonal phase of CaAuBi is unstable at zero pressure. However, no negative frequencies are found and all the phonon branches are positive at 25 GPa in hexagonal
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phase of CaAuBi (Fig. 7(b)). This stability indicates that the hexagonal phase of CaAuBi HH compound is stable at high pressure (25 GPa). Therefore, the high pressure stability of hexagonal (LiGaGe) phase phonon is responsible for cubic (MgAgAs phase) to hexagonal (LiGaGe phase)
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phase transition in CaAuBi. There is no experimental or theoretical result on the lattice dynamics
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of CaAuBi in the literature.
Figs. 8(a)-(e) show the variation of specific heat at constant volume (CV), entropy (S),
internal energy (E), vibrational free energy (F) and Debye temperature (θD) with temperature for CaAuBi compound in cubic phase. It is seen from Fig. 8(a) that CV increases rapidly at low temperature, and then becomes constant at high temperature. CV versus temperature plot is obeys the Dulong & Petit’s rule, CV = 9NkB at high temperature, here N is Avogadro’s constants. Fig. 8(b) shows that, when the temperature increase the entropy increases smoothly as well. This
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behavior is clear because the phonon frequency should increase with temperature. Fig. 8(c) and (d) show the behavior of internal energy (E) and vibrational free energy (F) respectively. The internal energy increase and vibrational free energy decrease with temperature. Here we have
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noticed that at zero temperature the value of internal energy and vibrational free energy are equal and do not vanish due to zero point vibration. The temperature variation of Debye temperature
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(θD) for CaAuBi is shown in Figure 8(e) and the value of θD is 170 K at 0 K.
4. Conclusion:
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In conclusion, by applying the ab-initio density functional theory, we have carried out a detailed exploration of structural phase transition, electronic, lattice dynamical and thermodynamical properties for CaAuBi half-Heusler compound. The calculated structural phase transition pressure and ground state properties are in good agreement with available experimental and theoretical results. Metallic nature is confirmed by the calculated band structure, total and partial
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density of states. From the charge density plots, it is seen that the CaAuBi having the mixture of ionic, metallic and covalent bonding. The first time calculated phonon dispersion curves show
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that the CaAuBi compound is dynamically stable in cubic (MgAgAs-type) phase at zero pressure and hexagonal (LiGaGe-type) phase at high pressure and it is unstable in hexagonal (LiGaGe-
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type) phase at zero pressure which suggested that the structural phase transition from cubic (MgAgAs-type) to hexagonal (LiGaGe-type) phase occurs at around 25 GPa. By the calculated thermodynamics properties, we have got the usual thermodynamic behavior of CaAuBi compound.
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Acknowledgement The authors are thankful to Prof. P. K. Jha and Mr. Narayan Som for useful discussion. DS is
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thankful to UGC, New Delhi for granting a UGC-BSR fellowship (Grant number F.25-1/2013-
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14(BSR)/7-19/2007(BSR)).
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Table 1: Calculated lattice parameter (a0), bulk modulus (B), pressure derivative of bulk modulus (Bˈ), phase transition pressure (PT) and volume collapse of CaAuBi half-Heusler compound in cubic (MgAgAs-type) phase.
6.9606
B (GPa) 53.47
Pre. (Full-potential)
B' 4.72
(Pseudopotential)
6.970
50
5.0
Exp.
6.851a
-
-
Theo.
6.90b
60.8b
5.0b
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Pre. = Present a Ref. [28], bRef. [18],
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PT (GPa) 19.6
∆V/V0 % 4.9%
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a0(Å)
-
-
18b
-
-
-
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CaAuBi
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Fig. 1 Crystal structure of CaAuBi half-Heusler compound in (a) cubic (MgAgAs-type), hexagonal- (b) (LiGaGe-type) and (c) (ZrBeSi-type) phases.
Fig. 2 Variation of (a) total energy with volume in cubic (MgAgAs-type) and hexagonal (LiGaGe-type and ZrBeSi-type) phase, (b) enthalpy with pressure in cubic (MgAgAstype) and hexagonal (LiGaGe-type) phase (c) equation of state with experimental results ( ) for cubic (MgAgAs-type) and ( ) for hexagonal (LiGaGe-type) taken from ref [18] for CaAuBi half-Heusler compound.
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Fig. 4
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Fig. 3 (a) Electronic band structure and (b) Density of states (total and partial) in cubic (MgAgAs-type) phase at ambient pressure for CaAuBi half-Heusler compound.
Fermi surfaces (a)-(d) in cubic (MgAgAs-type) phase at ambient pressure for CaAuBi half-Heusler compound
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Charge density plots in (110) plane in cubic (MgAgAs-type) phase at ambient pressure for CaAuBi half-Heusler compound.
Fig. 6
(a) Phonon dispersion curves and (b) Phonon density of states in cubic (MgAgAs-type) phase for CaAuBi half-Heusler compound at ambient pressure.
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Fig. 5
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Fig. 7 Phonon dispersion curves in hexagonal (LiGaGe-type) phase for CaAuBi half-Heusler compound (a) 0 GPa and (b) 25 GPa.
Fig. 8 (a) Specific heat, (b) Entropy, (c) Internal energy, (d) Vibrational free energy and (e) Debye temperature in cubic (MgAgAs-type) phase for CaAuBi half-Heusler compound as a function of temperature.
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Research Highlights
The structural phase transition calculations have been performed for CaAuBi.
Phase transition occurs from cubic (MgAgAs) to hexagonal (LiGaGe) at 19.6 GPa.
CaAuBi shows the mixing of ionic, metallic and covalent bonding.
Phonon dispersion curves show that CaAuBi is stable in Cubic phase at 0 GPa.
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S0925-8388(18)30695-9
DOI:
10.1016/j.jallcom.2018.02.208
Reference:
JALCOM 45084
To appear in:
Journal of Alloys and Compounds
Received Date: 16 December 2017 Revised Date:
13 February 2018
Accepted Date: 15 February 2018
Please cite this article as: D. Shrivastava, S.P. Sanyal, Structural phase transition, electronic and lattice dynamical properties of Half-Heusler compound CaAuBi, Journal of Alloys and Compounds (2018), doi: 10.1016/j.jallcom.2018.02.208. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Structural Phase Transition, Electronic and Lattice Dynamical Properties of Half-Heusler Compound CaAuBi
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Deepika Shrivastava a,*, Sankar P. Sanyal a Department of Physics, Barkatullah University, Bhopal, 462026, India *
E-mail: [email protected]
Abstract:
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The structural phase transition, electronic and lattice dynamical properties of CaAuBi half-
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Heusler compound have been investigated using first-principles density functional theory within generalized gradient approximation (GGA). The comparison of enthalpies of cubic (MgAgAstype) and hexagonal (LiGaGe-type) structure as a function of pressure suggest the structural phase transition from cubic (MgAgAs-type) to hexagonal (LiGaGe-type) at 19.6 GPa, which agrees well with the experimental results. The calculated ground state properties are in good
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agreement with available experimental and theoretical data. The electronic and bonding properties reveal that CaAuBi compound has mixed metallic, ionic as well as covalent nature. The lattice dynamical study of CaAuBi in both phases cubic (MgAgAs-type) and hexagonal
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(LiGaGe-type) confirms that CaAuBi is stable in cubic structure at ambient pressure and
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transform to hexagonal structure at high pressure. The thermodynamic properties are also calculated.
Keywords: Half-Heusler compounds; Phase transition; Phonon properties; Thermodynamic properties. *
Corresponding author: Department of Physics, Barkatullah University, Bhopal, 462026, India
E-mail address: [email protected] Tel : + 91- 755-2517120, Fax : + 91- 755-2491823 Co authors: Condensed Matter Physics Laboratory, Department of Physics, Barkatullah University, Bhopal, M.P. 462-026, India Tel : + 91- 755-2517120, Fax : + 91-755-2491823, E-mail : [email protected]
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1. Introduction: An interesting class of ternary compounds having chemical formula XYZ named as half-Heusler
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(HH) alloys, crystallize in the face centered cubic MgAgAs-type structure (space group F4-3m; no.216) [1], where X is rare earth or transition metal, Y is transition metal and Z is a main group element [2-4]. This XYZ cubic structure can be understood by the filling of ‘X’ atom into a YZ zincblende structure [3]. The half-Heusler compounds have a narrow band gap at the Fermi level
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for one spin direction i.e. they possess metallic behavior, and are insulator or semiconductor in opposite spin orientation [5], suggesting that they can be used as spin valves, to increase gaint
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magneto-resistance (GMR) [6]. The ferromagnetic behavior in the half-Heusler compounds was first observed by de Groot et. al. [7]. These compounds have extensive potential for applications in the field of superconductivity, thermoelectric material and spintronics [2,8-10]. In spintronics they have the capability to use both charge and spin degree of freedom to achieve
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multifunctional electronic devices [11,12]. The half-Heusler compounds are also used in magnetic random access memories (MRAM) and magnetic sensors [12-14] because of spin polarization. The piezoelectric response and associated properties in half-Heusler compounds are
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predicted by Roy et. al. [15]. Half-Heusler compounds may also crystallize in several crystal structures. For instance REAuSn belongs to cubic structure (MgAgAs-type) when RE is smaller
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rare-earth element, while hexagonal structure (LiGaGe-type) is favorable when it contains larger rare-earth element. LiGaGe structure is considered as a hexagonal variant of half-Heusler XYZ structure type, where X atoms are filled into a hexagonal wurtzite structure [3]. In the recent past, several attempts have been made to understand the structural properties of half-Heusler alloy systems [16,17].
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In a recent study Xie et. al. [18] have reported the structural phase transition of ternary half-Heusler CaAuBi compound from cubic (MgAgAs-type) to hexagonal (LiGaGe-type) phase by synchrotron X-ray diffraction method at 18 GPa. These authors [18] have predicted that
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CaAuBi might undergo another structural phase transition to ZrBeSi-type hexagonal structure at further high pressure. In the present paper we first examine the stability of crystal structure of CaAuBi using density functional theory at ambient and if it has any possibility to undergo
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another structural phase transition. Another approach to establish stability of crystal structure is to calculate phonon dispersion curves. Therefore, it will be very interesting to examine the lattice
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dynamical properties of CaAuBi, although there has been no experimental or theoretical work reported on vibrational properties in literature so far. In the present paper we report for the first time a comprehensive ab-initio calculation of structural phase transition, electronic, vibrational and thermodynamic properties of CaAuBi half-Heusler compound.
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2. Computational method:
All the calculations reported in this paper have been performed within density functional theory (DFT) [19]. For phase transition, electronic structure and associated properties of CaAuBi, we
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have used the full-potential linearized augmented plane wave (FP-LAPW) method [20] with generalized gradient approximation (GGA) scheme of Perdew, Burke and Ernzerhof (PBE) [21]
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for exchange and correlation effects. The energy convergence is achieved by expanding the basis function up to RMT * Kmax = 7, where RMT is the smallest atomic radius in the unit cell and Kmax refers to the magnitude of the largest k vector in the plane wave expansion. The maximum value for partial waves inside the atomic sphere is lmax = 10, while the charge density is Fourier expanded up to Gmax = 12 (a.u.)-1. The self-consistent calculations are converged when the total energy of the system is stable within 10-4 Ry. The Monkhorst-Pack [22] k-points grid of
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10x10x10 is used to sample the Brillouin zone for electronic structure calculations. The phonon dispersion curves and phonon density of states are calculated using the plane wave pseudopotential method as implemented in PWSCF code [23] with density functional
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perturbation theory (DFPT) [24]. We have used the norm-conserving pseudopotential of Troullier and Martins type [25], for describing the interaction between the valence electron, nuclei and the core electrons. We have used smearing of Marzari-Vanderbilt [26] to select the
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occupation distribution. The convergence is achieved with a kinetic energy cut-off 80 Ry. which is sufficient to fully converge all properties. The force constants for phonon dispersion
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calculations are obtained using 4x4x4 q-mesh in the first Brillouin zone. The thermodynamic properties are calculated using quasi-harmonic approximation (QHA) [24].
3. Results and Discussion:
3.1 Structural and electronic properties:
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In order to acertain the stability of crystal structure of CaAuBi half-Heusler compound at ambient condition, we have assumed three crystallographic structures for this compound, namely cubic, MgAgAs-type (F4-3m, No. 216), hexagonal LiGaGe-type (P63mc, No. 186) and ZrBeSi-
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type (P63/mmc, No. 194) as shown in Figs. 1(a)-(c). The obtained optimal atomic positions are Ca-4a(0,0,0), Au-4c(1/4,1/4,1/4) and Bi-4b(1/2,1/2,1/2) for cubic (MgAgAs-type) phase, while
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Ca-2a(0,0,0) and Au and Bi are located at 2b(1/3,2/3,z) where z = 0.3104(Au)
and z =
0.6899(Bi) for hexagonal (LiGaGe-type) phase and Ca-2a(0,0,0), Au-2c(1/3,2/3,1/4) and Bi2d(1/3,2/3,3/4) for hexagonal (ZrBeSi-type) phase. As can be seen from Figs. 1(b) and 1(c), there is very little difference between the two hexagonal structures. Xie et. al. [18] have reported that at ambient conditions, CaAuBi is stable in cubic (MgAgAs-type) phase, and transforms to hexagonal (LiGaGe-type) phase at high pressure (around 18 GPa). They have also found in their
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XRD measurement that around 3 GPa the cubic phase of CaAuBi still remained and above 3 GPa a small peak of hexagonal phase appears, which increases with increasing pressure. Fig. 2(a) shows the calculated total energies as a function of volume in the three different phases (cubic
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(MgAgAs) and hexagonal (LiGaGe & ZrBeSi)) of CaAuBi half-Heusler compound. The calculated total energies are fitted to Birch-Murnaghan’s equation of state [27] to obtain the structural properties such as lattice constant (a0), bulk modulus (B) and pressure derivative of
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bulk modulus (B'). The calculated values of a0, B and B' in cubic (MgAgAs) phase are shown in Table 1 as cubic phase is most stable at ambient condition. It can be seen from Fig. 2(a) and (b)
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that there is a structural phase transition from cubic (MgAgAs-type) to hexagonal (LiGaGe-type) at around 19.6 GPa (present theoretical calculation) as the total energies in the two phases intercept. However we do not find any indication that there will be a hexagonal (LiGaGe-type) to hexagonal (ZrBeSi-type) transition at high pressure (or reduced volume). The calculated values
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of lattice constant, bulk modulus and its pressure derivative are in good agreement with available experimental and theoretical results [28, 18]. As can be seen from Fig. 2(a), we have found that under ambient condition CaAuBi crystallize in cubic (MgAgAs-type) structure and at high
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pressure the hexagonal (LiGaGe-type) phase would be favored. However, a cubic (MgAgAstype) to hexagonal (LiGaGe-type) structural transition can only be achieved at high pressure of
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around 19.6 GPa. We have also expressed the phase transition in appropriate manner, by plotting the variation of enthalpy with pressure as shown in Fig. 2(b), which explains that the enthalpy of both phases (cubic and hexagonal) are same at 19.6 GPa. The calculated value of phase transition pressure is in close agreement with the experimental value 18 GPa [18] and shown in Table 1. Fig. 2(c) shows the unit cell volumes of both the phases (cubic and hexagonal) monotonically decrease with increasing pressure which is in good agreement with experimental [18] results.
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The calculated electronic band structure and density of states (total and partial) of CaAuBi compound in its cubic phase are shown in Figs. 3(a) and (b). CaAuBi shows semimetallic character because there is no band gap at Fermi level and three bands cross the
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Fermi level. The lowest occupied bands in the energy range -6 to -4 eV are mainly due to the Au5d and Bi-6p states. The next group of bands (above the Fermi level) are predominantly due to the Ca-4s, Au-5d and Bi-6p states. As is seen from Fig. 3(b) in the energy range -6 to -4 eV, a
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strong peak appears at -5 eV which is mainly characterized by Au-5d and Bi-6p states with some contribution of Ca-4s states. In the energy range -3 to 4 eV there is a hybridization between Au-
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5d and Bi-6p states giving rise to a strong peak at -2 eV and near 3.0 eV. In the conduction region the hybridization between Au-5d and Bi-6p states leads to a covalent character of this material. As a consequence the cubic CaAuBi contains a mixture of metallic and covalent character. The density of states at Fermi level is 0.30 States/eV.
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We have plotted the Fermi surfaces (FS) of CaAuBi for the bands which are crossing the Fermi level and are shown in Figs. 4(a)-(d). The electronic nature of any metallic compounds can be understood by the Fermi surfaces. The measurement of energy distribution curve (EDC) for
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the different wave vectors in the Brilliouin zone is the appropriate way to find the Fermi surfaces
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and to ensure the exact location of wave vectors where the bands cross the Fermi level. All the Fermi surfaces contained the set of hole and electron sheets. As is seen in Fig. 3(a), there are three bands namely 20, 21 and 22 which cross the Fermi level and make Fermi surface (FS) sheets. The first two sheets for the bands 20 and 21 are mainly due to the hole pocket at Γ point and the third sheet is mainly due to the electron pocket along L → Γ point. The fourth sheet is the confluent FS sheet of these three sheets. The energy ranges for the bands 20, 21 and 22 are 0.22-0.44 Ry., 0.26-0.44 Ry. and 0.41-0.55 Ry. respectively.
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In order to identify the type of bonding between the different atoms in the cubic CaAuBi compound, we have calculated the charge density plots in (110) plane and is shown in Fig. 5. If the charge transfer between cation to anion and charge contours are isolated and spherical, the
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bonding is considered to be purely ionic, while the bonding is covalent when the charge density contours overlaps to each other. As is seen from Fig.5, the charge density distributions for the cation and anion in CaAuBi are spherical, which indicate that the bonding is ionic in nature. The
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metallic contribution is mainly due to the partially filled Bi-6p state. The covalent bonding comes from the mutually shared electron of Au-5d and Bi-6p states, which can be seen from
metallic and covalent bonding.
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partial density of states (see in Figure 3(b)). Therefore CaAuBi shows the mixing of ionic,
3.2 Phonon dispersion curves and thermodynamic properties:
The phonon dispersion curves and phonon density of states of CaAuBi in cubic (MgAgAs-type)
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phase are calculated at ambient pressure and shown in Figs. 6 (a)-(b). The primitive cell of cubic (MgAgAs-type) CaAuBi contains three atoms per unit cell, therefore, nine vibrational modes appear in the phonon dispersion curve in which three are acoustic branches and remaining six are
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optical ones. Figs. 6(a)-(b) show that, all the phonon modes are positive and the absence of any imaginary frequency confirms the stability of CaAuBi in cubic (MgAgAs-type) structure at
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ambient pressure. The metallic nature of CaAuBi in cubic phase is confirmed by the degeneracy of longitudinal optical and transverse optical mode at the zone centre (around 175 cm-1). It can be seen that there is no gap between longitudinal acoustic (LA) and transverse optical (TO) modes at low frequency regime (Fig. 6(a) (PDC) at X point) in the frequency range at about 75 cm-1. Thus a phonon can make a transition from the acoustic to optical mode without any momentum transfer. However there is a large energy gap between the two optical branches
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which can be ascribed to the mass difference between the constituent atom, Ca, Au and Bi. From Fig. 6(b), it can be noticed that, the phonon state in the acoustic region is dominantly created by the vibration of Au atoms due to their heavier mass with considerable contribution of Ca atoms.
Phonon DOS is found at around 175 cm-1.
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Ca and Bi atoms show the most contribution in the optical phonon states. The highest peak in
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Further to understand the phase transition of CaAuBi HH compound from Cubic (MgAsAs-type) to hexagonal (LiGage-type) structure in terms of lattice dynamics, we have
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calculated the phonon dispersion curves of CaAuBi in hexagonal structure at zero pressure and around the transition pressure (25 GPa) and are shown in Figs. 7 (a)-(b). It is clear from Fig. 7(a), that the transverse acoustic (TA) phonon branch along Γ-K and Γ-M directions becomes negative indicating that the hexagonal phase of CaAuBi is unstable at zero pressure. However, no negative frequencies are found and all the phonon branches are positive at 25 GPa in hexagonal
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phase of CaAuBi (Fig. 7(b)). This stability indicates that the hexagonal phase of CaAuBi HH compound is stable at high pressure (25 GPa). Therefore, the high pressure stability of hexagonal (LiGaGe) phase phonon is responsible for cubic (MgAgAs phase) to hexagonal (LiGaGe phase)
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phase transition in CaAuBi. There is no experimental or theoretical result on the lattice dynamics
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of CaAuBi in the literature.
Figs. 8(a)-(e) show the variation of specific heat at constant volume (CV), entropy (S),
internal energy (E), vibrational free energy (F) and Debye temperature (θD) with temperature for CaAuBi compound in cubic phase. It is seen from Fig. 8(a) that CV increases rapidly at low temperature, and then becomes constant at high temperature. CV versus temperature plot is obeys the Dulong & Petit’s rule, CV = 9NkB at high temperature, here N is Avogadro’s constants. Fig. 8(b) shows that, when the temperature increase the entropy increases smoothly as well. This
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behavior is clear because the phonon frequency should increase with temperature. Fig. 8(c) and (d) show the behavior of internal energy (E) and vibrational free energy (F) respectively. The internal energy increase and vibrational free energy decrease with temperature. Here we have
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noticed that at zero temperature the value of internal energy and vibrational free energy are equal and do not vanish due to zero point vibration. The temperature variation of Debye temperature
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(θD) for CaAuBi is shown in Figure 8(e) and the value of θD is 170 K at 0 K.
4. Conclusion:
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In conclusion, by applying the ab-initio density functional theory, we have carried out a detailed exploration of structural phase transition, electronic, lattice dynamical and thermodynamical properties for CaAuBi half-Heusler compound. The calculated structural phase transition pressure and ground state properties are in good agreement with available experimental and theoretical results. Metallic nature is confirmed by the calculated band structure, total and partial
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density of states. From the charge density plots, it is seen that the CaAuBi having the mixture of ionic, metallic and covalent bonding. The first time calculated phonon dispersion curves show
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that the CaAuBi compound is dynamically stable in cubic (MgAgAs-type) phase at zero pressure and hexagonal (LiGaGe-type) phase at high pressure and it is unstable in hexagonal (LiGaGe-
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type) phase at zero pressure which suggested that the structural phase transition from cubic (MgAgAs-type) to hexagonal (LiGaGe-type) phase occurs at around 25 GPa. By the calculated thermodynamics properties, we have got the usual thermodynamic behavior of CaAuBi compound.
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Acknowledgement The authors are thankful to Prof. P. K. Jha and Mr. Narayan Som for useful discussion. DS is
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thankful to UGC, New Delhi for granting a UGC-BSR fellowship (Grant number F.25-1/2013-
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14(BSR)/7-19/2007(BSR)).
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Table 1: Calculated lattice parameter (a0), bulk modulus (B), pressure derivative of bulk modulus (Bˈ), phase transition pressure (PT) and volume collapse of CaAuBi half-Heusler compound in cubic (MgAgAs-type) phase.
6.9606
B (GPa) 53.47
Pre. (Full-potential)
B' 4.72
(Pseudopotential)
6.970
50
5.0
Exp.
6.851a
-
-
Theo.
6.90b
60.8b
5.0b
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Pre. = Present a Ref. [28], bRef. [18],
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PT (GPa) 19.6
∆V/V0 % 4.9%
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a0(Å)
-
-
18b
-
-
-
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CaAuBi
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Fig. 1 Crystal structure of CaAuBi half-Heusler compound in (a) cubic (MgAgAs-type), hexagonal- (b) (LiGaGe-type) and (c) (ZrBeSi-type) phases.
Fig. 2 Variation of (a) total energy with volume in cubic (MgAgAs-type) and hexagonal (LiGaGe-type and ZrBeSi-type) phase, (b) enthalpy with pressure in cubic (MgAgAstype) and hexagonal (LiGaGe-type) phase (c) equation of state with experimental results ( ) for cubic (MgAgAs-type) and ( ) for hexagonal (LiGaGe-type) taken from ref [18] for CaAuBi half-Heusler compound.
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Fig. 4
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Fig. 3 (a) Electronic band structure and (b) Density of states (total and partial) in cubic (MgAgAs-type) phase at ambient pressure for CaAuBi half-Heusler compound.
Fermi surfaces (a)-(d) in cubic (MgAgAs-type) phase at ambient pressure for CaAuBi half-Heusler compound
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Charge density plots in (110) plane in cubic (MgAgAs-type) phase at ambient pressure for CaAuBi half-Heusler compound.
Fig. 6
(a) Phonon dispersion curves and (b) Phonon density of states in cubic (MgAgAs-type) phase for CaAuBi half-Heusler compound at ambient pressure.
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Fig. 7 Phonon dispersion curves in hexagonal (LiGaGe-type) phase for CaAuBi half-Heusler compound (a) 0 GPa and (b) 25 GPa.
Fig. 8 (a) Specific heat, (b) Entropy, (c) Internal energy, (d) Vibrational free energy and (e) Debye temperature in cubic (MgAgAs-type) phase for CaAuBi half-Heusler compound as a function of temperature.
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Research Highlights
The structural phase transition calculations have been performed for CaAuBi.
Phase transition occurs from cubic (MgAgAs) to hexagonal (LiGaGe) at 19.6 GPa.
CaAuBi shows the mixing of ionic, metallic and covalent bonding.
Phonon dispersion curves show that CaAuBi is stable in Cubic phase at 0 GPa.
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