Meena Kumari,U.P.Verma

School of Studies in Physics,Jiwaji University,Gwalior 474011,India

Abstract In this paper,HoMg is studied in two different phases i.e.B1 and B2.The calculated lattice constants in B1 and B2 phases are 6.39°A and 3.78°A and corresponding energies are?25,648.49 Ry and?25,648.56 Ry,respectively.It is more stable in B2 phase(Pm-3m configuratio with no.221).The analysis of the obtained band structure and density of states shows metallic character of HoMg.Detailed information about the elastic properties is also presented in this paper.The calculated values of Cauchy pressure,G/B and Poisson ratio are?10.72,0.58 and 0.26,respectively which indicates that HoMg have brittle character.Calculations for the optical spectra such as the components of dielectric function,refractive index and other parameters of the optical properties are performed in the energy range of 0–20eV.For HoMg,the main peak of absorption spectra and energy loss function is located at 7.44eV and 11.76eV,respectively.Thermal parameters such as Gruneisen parameter,Debye temperature,Specifi heat etc.have been reported as a function of pressure and temperature.

Keywords:Rare-earth magnesium alloy;DFT;Elastic properties;Optical and thermal properties.

1.Introduction

Because of many unique properties of magnesium like good mechanical strength,low density,good heat dissipation and good electromagnetic properties.Therefore,it is considered as a potential material for technological applications[1].At high temperature Mg possesses low strength and poor corrosion performance which limits the wider application of Mg.Addition of Rare Earth(RE)Elements have significan effects at the high temperature.Magnesium alloys are stiffer than engineering plastics,more recyclable and less costly to produce.Because of these properties they have replaced engineering plastics in engineering,computer,aeronautical and automotive industry applications[2].These compounds also show high degree of biocompatibility[3–7].The rare earth Mg alloys can be composed with high strength and lower corrosion rates for engineering,automotive industry and biodegradable medical applications by understanding the Mg strengthening mechanism[8].Many researchers have contributed in this direction.For example:magnetic properties of some rare earth magnesium compounds have been reported by Aleonard et al.[9]and that of DyMg and ErMg by Belakhovsky et al.[10].The available information of some rare earth magnesium alloys have also been reported in[11–21].Detailed information about HoMg is still missing in the literature;therefore we wish to perform ab-initio calculations on structural,electronic,elastic,optical and thermal properties of HoMg.

2.Computational approach

Full-potential linear augmented plane wave(FP-LAPW)method within the density functional theory(DFT)as implemented in the WIEN2k[22]code has been applied to investigate the structural,electronic,elastic,optical and thermal properties of HoMg intermetallic compound in stable B2 phase.For the structural properties the exchange correlation potential has treated within local density approximation(LDA)[23],Wu–Cohen Generalized Gradient Approximation(WC-GGA)[24],Perdew–Burke–Ernzerhof(PBE)-GGA[25],Engel–Vosko generalized gradient approximation(EV-GGA)[26]and PBE Sol-GGA.For the rest of calculations PEBSol-GGA method has been used.To reach energy Eigen values convergence,the wave functions in the interstitial region were expanded in plane waves with a cutoff parameter Kmax?RMT=7,where RMTindicates the smallest atomic muffin-ti sphere radius and Kmaxprovides the magnitude of the largest K-vector in the plane wave expansion.The values of muffin-ti radii(RMT)for Ho and Mg were taken as 3.0 and 2.2 bohr,respectively.The wave functions inside the muffin tin spheres were expanded up to l=10,where l is the maximum value for the angular momentum quantum.The selfconsistent calculations are considered to be converged when the total energy of the system is stable within 10?4Ry.The integrals over the Brillouin zone are performed up to 37 kpoints in the irreducible Brillouin zone,using the Monkhorst–Pack[27]special k-points approach.The separation energy between the valence states from the core state has been set equal to?6.0 Ry.

Table 1 Calculated values of optimized crystal structure parameters i.e.lattice constant a(°A),volume V(bohr3),energy E0(Ry),the bulk modulus B0(GPa),and its firs pressure derivative B′0 for cubic HoMg in B1 and B2 phase using LDA,PBE-GGA,EV-GGA,PBE-Sol functional.

Table 2 The calculated elastic constants(in GPa unit)with Zener anisotropy factor(A),Poisson ratio(υ),Young’s modulus(Y)and isotropic shear modulus(G)in stable B2 phase.

3.Result and discussion

3.1.Structural properties

Firstly,we stabilized the crystal structure by minimizing the total energy as a function of unit cell volume to obtain the equilibrium lattice constant.HoMg alloy has been studied in two different phases NaCl(B1)and CsCl(B2).In both the phases atom Ho was placed at(0,0,0)and atom Mg at(0.5,0.5,0.5)in unit cell.Obtained optimization plots are shown in Fig.1.The calculated equilibrium parameters found using LDA,PBE-GGA,EV-GGA and Sol PBE-GGA methods are shown in Table 1,which also includes experimental data for comparison[28].Our reported results show that the ground state configuratio of HoMg in B2 phase which is in consistent with earlier result.In this phase,the obtained equilibrium lattice constant,obtained using PBE-GGASol,is 3.78°A,and corresponding energy,volume and bulk modulus,respectively,E0=?25,648.56 Ry,V0=365.36(a.u.3)and B0=31.53GPa as given in Table 1.and

3.2.Electronic properties

The electronic band structure and density of states(DOS)are obtained by using the optimized atomic positions at the equilibrium lattice constant.The obtained electronic band graph along the high symmetry direction in the firs Brillouin zone is shown in Fig.2.It is clear from the band profil that the valence and conduction bands overlap each other and there is no band gap at the Fermi level,which confirm the metallic nature of this intermetallic.The total density of states and partial density of states are shown in the Fig.3.In this figur f-states show the main contribution in PDOS plot at Fermi level.Non-zero value of total DOS at Fermi level confirm metallic character of HoMg.

3.3.Elastic properties

Fig.1.Total energy versus cell volume in B1(dotted line)and B2(solid line)phases of HoMg using local density approximation(LDA),Perdew–Burke–Ernzerhof(PBE),Engel Vosko(EV)and PBE-Sol generalized gradient approximation(GGA).

Table 3 The mass density(ρ)along with longitudinal(vl),transverse(vt)and average(vm)sound velocities in HoMg.

Fig.2.Band Profil of HoMg.

Taylor expansion of the total energy gives the information about the elastic constants i.e.the derivative of the energy as a function of a lattice strain.The elastic constants are a four-rank tensor,but these elastic constants are reduced to three independent ones:C11,C12,C44due to the symmetry for cubic crystal.The calculated values C11=36.64GPa,C12=28.96GPa and C44=39.72GPa are listed in Table 2.All elastic constants are larger than zero for HoMg which obey the Born’s criteria for the mechanical stability of material.To measure the elastic anisotropy of HoMg alloy,we have calculated Zener anisotropic factor define by A=2C44/(C11?C12).A=.The calculated value of A for HoMg is 10.34 which indicate that this material is completely elastically anisotropic because if A is smaller or greater than 1,the material is known as anisotropic otherwise isotropic.Cauchy pressure related to the brittle/ductile properties of materials by Johnson[29]and Pettif?r[30].Negative Cauchy pressure corresponds to brittle behavior while positive Cauchy pressure point out ductile characteristic.The present value of Cauchy pressure is?10.72.Therefore,HoMg show the brittle character.Bulk modulus and isotropic shear modulus can measure the hardness of a compound.If G/B＜0.5,the material behaves in a ductile manner,while G/B＞0.5,the material behaves in a brittle manner[31–33].In case of HoMg alloy our obtained value of G/B is 0.58 that indicates its behavior as brittle.The Poisson’s ratio plays a vital role for the manufacturing and technological applications.For covalent materials,the values of νare normally 0.1–0.25 and interatomic forces are non-central forces.Also,the lower and upper limits of’ν’are 0.25 and 0.5 for ionic crystals and the interatomic forces are central forces[34].The present case the value ofνis 0.26,therefore the ionic contribution is more dominant to atomic bonding for this material.The Poisson’s ratio also gives the information of brittle and ductile behavior of any solid.For a brittle material,the value Poisson ratio is ν＜0.33 whereas for a ductile metallic material,it is usually 0.33.In our case the value of obtained Poisson ratio 0.26 is smaller than 0.33,therefore,the material HoMg possesses the property of brittleness.=k The average sound velocity in HoMg is obtained using relation

Fig.3.Total and partial density of states of HoMg in B2 phase.The Fermi level is set to be 0eV.

where vland vtare the longitudinal and transverse elastic wave velocities,respectively.Their values are obtained from the following relations given in

In above equations B is the bulk modulus,G the shear modulus andρis the density of the material.The density of HoMg has been obtained using relationρ=M/V,where M and V are the unit cell mass and volume,respectively.The ovtained value of the density of HoMg and the average sound velocity in HoMg is given in Table 3.

3.4.Optical properties

3.4.1 Dielectric function

The optical properties of HoMg can be described by complex dielectric function ε(E)=ε1(E)+iε2(E).The real part of the dielectric function,ε1(E)represents the dispersion of the incident photons by the materials,while imaginary part,ε2(E)of the dielectric function is related to energy absorbed by the material.The real and imaginary part of the dielectric function is shown in Fig.4(a).From the analysis of the graph,it is observed that the static value of the real part of dielectric function isε1(0)=310.96.It becomes negative near E=0.6eV.Negative value of the real part of the dielectric function indicates that an incident plane wave will be perfectly reflecte by such a metal rather than absorbed.The imaginary part,ε2(E)of dielectric function decreases very fast up to 1eV and beyond 2eV it decreases very slowly and becomes zero near 9eV.The value of ε2(E)＞0 shows the range for optical absorption.

3.4.2.Refractive index

The refractive index n(E)and extinction coefficien k(E)are displayed in Fig.4(b).Obtained plots show that the n(E)nearly follow theε1(E)trend.The static value of n0(0)is 18.84.The value of refractive index is high in infrared region and gradually decreases in the visible and ultraviolet region.The extinction coefficien k(E)shows the maximum value in the energy range 0–1eV after that it starts decreasing.

Fig.4.Optical parameters of HoMg as function of the energy.(a)Components of dielectric functionε(E),(b)refractive index n(E)and extinction coeff cients k(E),(c)optical absorptionα(E),(d)optical conductivityσ(E),(e)energy loss function L(E)and(f)reflectvity(%)R(E).

3.4.3 Optical absorption

Optical absorption measures the attenuation percentage of light intensity when it travel one meter in a material.The absorption spectrum of HoMg is shown in Fig.4(c).Its highest peak(98.03 in arbitrary unit)appears at energy 7.44eV(166.65nm)in the absorption range 0–9eV.After that it decreases radically up to 15eV and then becomes almost constant.

3.4.4 Energy loss function

Energy Loss function L(E)is an important factor describing the energy loss of a fast electron traversing in a material.The prominent peak is found due to Plasmon resonance(a collective oscillation of valance electrons)and their corresponding frequency is named as plasma frequency.The energy loss spectrum is shown in Fig.4(d).For this compound,main peak is located at 11.76eV.This energy corresponds to～2.84Hz in ultra violet region.Above the plasma frequency,the material becomes transparent.

3.4.5 Optical conductivity

Optical conductivity increases the electrical conductivity because number of free charge carriers increases due to the photons absorption.Fig.4(e)shows the plot of the optical conductivity versus photon energy.High magnitude of optical conductivity shows very high electrical response of the material.HoMg highest optical conductivity near 0.72eV(1589.5409nm),after that it decreases rapidly up to 9eV.There is no photo conductivity,when energy is higher than 10eV.The property of optical conductivity makes the material more prominent for the electrical devices application.

Fig.5.Thermodynamical parameters of HoMg as functions of temperature T(in K)and pressure(in GPa).

3.4.6 Refl ectivity

The reflectvity spectrum is shown in Fig.4(f).The static value of reflectvity is about 81%.The reflectvity spectrum decreases to almost zero for photon energy higher than 16eV.[PARA]Static values of the real part of dielectric function,refractive index,extinction coefficien and reflectvity have been compiled in Table 4.

3.5.Thermal properties

To know the thermodynamic behavior of HoMg,we have used thequasi-harmonic Debye model[35].The thermal properties are determined in the temperature range from 0.0 to 1000K,where the quasi-harmonic model remains fully valid.The pressure effect was studied in the range of 0 to 30GPa.The obtained values of the tharmal parameters at zero pressure and different temoperatures are shown in Table 5.

3.5.1.Variation of volume

The variation of the volume with respect to the pressure at different temperature(T=0,300,600,900K)is shown in Fig.5(a).It is found that the change in the volume with respect to the pressure is non linear and its value decreases with increase in pressure.

3.5.2 Variation of bulk modulus

Variation of bulk modulus of a material indicates its compressibility.The present calculated value of bulk modulus is 29.74GPa at 0K and 0GPa while at 300K and 0GPa it is 28.27GPa.The pressure and temperature dependence of bulk modulus of HoMg is shown in Fig.5(b).The bulk modulus of this compound increases almost linearly with increasing pressure at a different temperature(T=0,500,1000K)which indicates that the cell volume undergoes a homologous behavior.

3.5.3 Thermal expansion coefficient

The thermal expansion coefficientα,is a function of temperature.It has been analyzed at hydrostatic pressure P=0,10,20 and 30GPa.Plots ofαversus temperature for different values of pressure are shown in Fig.5(c).From this figure it is clear that with the increase in temperature from T=0K to 200K,α(T)increases sharply.Beyond 200K,its gradient approaches a constant value between T=300 and 1000K.This shows that,at a fi ed temperature,thermal expansion decreases with increase in pressure.

3.5.4 Debye temperature

The Debye temperature gives the information about low and high temperature region of a crystal.We imagine that all modes have energy kBT when T＞θDand if T＜θDthen,we expects high-frequency modes to be frozen[36].Pressure dependent Debye temperature of HoMg at different temperature(T=0,500,1000K)is shown in Fig 5(d).At given temperature,Debye temperature increases with pressure.The calculated values of Debye temperature is shown in 5.At P=0GPa,T=0K and P=0GPa,T=500K the obtained values of Debye temperature of HoMg are 178.68K and 169.50K,respectively.

Fig.6.(a)Specifi heat at constant volume CV,(b)specifi heat at constant pressure CP,(c)Gruneisen parameter,(d)entropy variation at different pressure(GPa)of HoMg as a function temperature T(in K).

Table 4 Static values of dielectric constant,refractive index,extinction coeff cient and reflectvity of HoMg in B2 phase.

3.5.5.Specific heat

We can calculate specifi heat at constant volume(Cv)and specifi heat at constant pressure(Cp)using the following expression:

Heat capacity at constant pressure and volume versus temperature is plotted in Fig.6(a)and(b).For the entire range of temperature,especially above 200K Cp＞Cv,shows the thermodynamic stability of HoMg.At P=0GPa and T=300K,CPand Cvare 50.81 and 49.06J/mol K,respectively.

3.5.6.Gruneisen parameter

Gruneisen parameter is the measure of the vibrational anharmonicity.In the presence of anharmonicity,the phonon frequencies are volume dependent and this volume dependence of phonon is characterized by Gruneisen parameter.The pressure dependent Gruneisen parameter for HoMg at different temperature(T=0,500,1000K)is shown in Fig.6(c).

3.5.7 Entropy

The variation of entropy at high temperature is depicted in Fig.6(d).Microscopically,entropy is define as measure of disorder of a system.As the temperature of the material increases the particle vibrates more vigorously,causing an increase in entropy of the system.

4.Conclusions

We have performed ab-initio calculations of structural,electronic,elastic,optical and thermal properties of HoMg.It is observed that B2 phase is more stable.The calculated lattice constants,volume,energy,bulk modulus and the f rstorder pressure derivatives of bulk modulus in B2 phase are in good agreement with experimental results.The obtained band structure shows the metallic nature of HoMg.Our elastic constants calculation shows that HoMg is mechanically stabile but brittle and possess ionic nature.The optical parameters such asε(E),n(E),k(E)are also reported.Thestatic value of the real part of dielectric constant is 310.96 and its refractive index is obtained as 18.84.The quasi-harmonic Debye model is successfully applied to study the variations of the volume,bulk modulus,thermal expansion coefficient heat capacities,Debye temperature and entropy as a function of the pressure and temperature.

Table 5 Thermal parameters of HoMg at ambient conditions at temperature T=0,300,500 and 1000K and pressure P=0.

Acknowledgment

One of the authors(Meena Kumari)acknowledges the financia assistance provided by Jiwaji University,Gwalior,through sanction no.F/DEV/2015-16/208.