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Theoretical Calculations for Structural, Elastic and Thermodynamic Properties of MgB2

under High Pressure

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2006 Commun. Theor. Phys. 45 558

(http://iopscience.iop.org/0253-6102/45/3/036)

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Commun. Theor. Phys. (Beijing, China) 45 (2006) pp. 558–564c© International Academic Publishers Vol. 45, No. 3, March 15, 2006

Theoretical Calculations for Structural, Elastic and Thermodynamic Properties of

MgB2 under High Pressure∗

WANG Hai-Yan,1,2 CHENG Yan,1,2 CHEN Xiang-Rong,1,2,3,† and GOU Qing-Quan1

1Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China

2College of Physical Science and Technology, Sichuan University, Chengdu 610064, China

3International Centre for Materials Physics, the Chinese Academy of Sciences, Shenyang 110016, China

(Received June 20, 2005)

Abstract We have investigated the structural and elastic properties of MgB2 under high pressures using the full-potential linearized muffin-tin orbital (FP-LMTO) scheme within the generalized gradient approximation correction(GGA) in the frame of density functional theory. The calculated pressure dependence of the normalized volume isin excellent agreement with the experimental results. At the same time the elastic constants and acoustic anisotropyas a function of applied pressure are presented. Through the quasi-harmonic Debye model, we also investigate thethermodynamic properties of MgB2.

PACS numbers: 71.15.Mb, 51.35.+a, 62.20.Dc, 74.70.AdKey words: elastic constants, thermodynamic properties, full-potential linearized muffin-tin orbital, MgB2

1 Introduction

The discovery of the superconductivity in MgB2 witha high superconducting temperature Tc of 39 K[1] has at-tracted considerable experimentalists as well as theoreti-cians, and has taken the whole materials science commu-nity by surprise. A great number of interesting results onMgB2 have been reported during the past several years,such as the boron isotope effect,[2,3] the superconductinggaps,[4,5] the thermodynamic and transport properties,[6]

the phonon density of states (PDOS),[7,8] and so on.Since MgB2 possesses a hexagonal crystal structure,

one thus expects anisotropy in the physical properties ofthis material. Rabindran et al.[9] noted that there are con-tradictory experimental results regarding the anisotropyin the elastic properties of MgB2, ranging from isotropicto highly anisotropic. They carried out the detailed elec-tronic structure studies on superconducting MgB2 to findthe large anisotropy in the elastic properties and polarizeddielectric tensor. From the high-resolution X-ray powderdiffraction, Vogt et al.[10] observed that MgB2 possessesnearly the isotropic mechanical behavior. However, theneutron diffraction measurements performed by Jorgensenet al.[11] showed large anisotropy in the thermal expansionand compressibility of MgB2.

Presently, the investigations of the elastic constantsof superconducting MgB2 under zero pressure have beenperformed successfully.[12−14] Since pressure is an impor-tant variable that can be used to tune physical properties,it should be attracting to us how to calculate the elastic

properties of MgB2 under high pressure. The question ofhow to calculate the elastic constants at arbitrary isotropicpressure requires special consideration. There are only afew papers[15−17] concerning this problem. Iitaka et al.[15]

computed the elastic properties of fcc solid argon at highpressure from 1 GPa up to 80 GPa. Jochym et al.[16]

obtained the elastic constants, equation of state and freeenergy of NaCl-, KOH-, and CsCl-type structures of AgBrbased on ab initio calculations. Gulseren et al.[17] investi-gated the elasticity of bcc Ta for the pressures up to 400GPa and temperatures up to 10 000 K. To our knowledge,there are no literatures reporting the elastic constants ofMgB2 under high pressure. Therefore, we here focus onthe structural, elastic and thermodynamic properties ofMgB2 under high pressure by using the full-potential lin-earized muffin-tin orbital (FP-LMTO) scheme[18] in theframe of density functional theory.[19,20]

2 Theoretical Method

2.1 Total Energy Electronic Structure Calcula-

tions

In the FP-LMTO method, we adopt the generalizedgradient approximation for exchange-correlation func-tional proposed by Perdew, Burke, and Ernzerhof.[21] Thebase geometry in this computational method consists ofmuffin-tin (MT) spheres and interstitial parts. The basisset is comprised of augmented linear muffin-tin orbitals.[22]

The MT sphere radii rMT are chosen to be 2.922 a.u. and1.687 a.u. for Mg and B, respectively. Inside the muffin-tin

∗The project supported by National Natural Science Foundation of China under Grant No. 60436010 and the Scientific Research

Foundation for Returned Overseas Chinese Scholars of the Ministry of Education under Grant No. 2004176-6-4†Corresponding author, E-mail: xrchen@126.com

No. 3 Theoretical Calculations for Structural, Elastic and Thermodynamic Properties of MgB2 under High Pressure 559

spheres, the basis functions, charge density, and potentialare expanded in symmetry adapted spherical harmonicfunctions together with a radial function. Fourier seriesare used in the interstitial regions. In the present calcula-tions, the K-space integration has been performed using6× 6× 6 k-points in the irreducible Brillouin zone (IBZ).The spherical-harmonic expansion of the charge density,potential, and basis functions is carried out up to the an-gular momentum component l = 6. The final convergenceis within 10−7 Ryd.

We carry out total energy electronic structure calcula-tions over a wide range of primitive cell volumes V , i.e.,from 0.8 V0 to 1.15V0, where V0 is the zero pressure equilib-rium primitive cell volume. No constraints are imposed onthe c/a ratio, i.e., both lattice parameters a and c are op-timized simultaneously. At each volume V , we determinethe corresponding equilibrium ratio c/a by performing thetotal energy electronic structure calculations on a series ofdifferent values of c/a. For a fixed value of c/a, a seriesof different values of a are set to calculate the total ener-gies E, which are minimized as a function of the c/a ratiofor a given V . Through these calculations, we can obtainthe equilibrium lattice parameters a and c as well as thecorresponding equilibrium ratio c/a for any pressure. Theenergy-volume (E-V ) curve can be obtained by fitting thecalculated E-V results to the Birch–Murnaghan EOS,[23]

in which the pressure-volume relationship expanded as

P = 3B0fE(1 + 2fE)5/2[1 +

32(B′ − 4)fE

+32

(B0B

′′ + (B′ − 4)(B′ − 3) +359

f2E

)], (1)

where fE = [(V0/V )2/3 − 1]/2, B, B′0(= dB0/dP ), and

B0 are hydrostatic bulk modulus, the pressure derivativeof the bulk modulus and the zero pressure bulk modulus,respectively. B′′ is given by[24]

B′′ =−1B0

[(3−B′)(4−B′) +

359

]. (2)

2.2 Elastic Constants

We calculate the elastic constants as the second deriva-tives of the internal energy with respect to the strain ten-sor. The theoretical method has been reported in detailin Ref. [25]. Only a brief description of the method ispresented as follows.

Let us consider a hexagonal crystal compressed by theisotropic pressure P to the density ρ1 = 1/V1 (where V1

is the distorted volume from the lattice distortion εij).Small homogeneous deformation of this crystal takes ev-ery Bravais lattice point ~R of the undistorted lattice to anew position ~R ′ in the strained lattice,

R′i =

∑j

(δij + εij)Rj . (3)

For a homogeneous strain, the parameters εij are simplyconstants, independent of ~R, where the subscripts i andj indicate the Cartesian components, and δij is the Kro-necker delta. Expanding the internal energy per unit massof the crystal, one obtains

E(ρ1, {ηmn}) = E(ρ1) +1ρ1

(∑ij

Tijηij

+12

∑ijkl

Cijklηijηkl + · · ·)

, (4)

where

ηij = εij +12

∑k

εikεkj , (5)

Tij = ρ1∂E(ρ1, {ηmn})

∂ηij

∣∣∣{ηmn=0}

, (6)

Cijkl = ρ1∂2E(ρ1, {ηmn})

∂ηij∂ηkl

∣∣∣{ηmn=0}

. (7)

Here Tij stands for the components of the stress tensorbefore deformation, Cijkl is the elastic constants of thecrystal at an arbitrary isotropic pressure P . It shouldbe pointed out that equation (4) represents the energy ofthe strained crystal in the case of full relaxation of atompositions within the distorted Bravais lattice.

To calculate the ab initio elastic constants, the defor-mation parameters are usually specified in the form

εij = sijγ + eijγ2 + · · · (8)

Then, equation (4) can be written as follows:

E(ρ1, {ηmn}) = E(ρ1) + Aγ +D

2γ2 + · · · , (9)

where

A = − P

ρ1

∑i

sii , (10)

D =1ρ1

∑ijkl

Cijklsijskl −2P

ρ1

∑ik

(eikδik +

s2ik

2

). (11)

The linear equation for the elastic constants Cijkl is then∑ijkl

Cijklsijskl = 2P∑ik

(eikδik +

s2ik

2

)+ ρ1

∂2E(ρ1, γ)∂γ2

∣∣∣γ=0

. (12)

We ought to note that equation (12) is equally valid forstrains with and without volume conservation. Obviously,for ambient conditions equation (12) is in agreement withthe ones used before.[12−14]

Using the symmetry properties of matrices ε and C,we can rewrite Eq. (12) as∑

αβ

ξαξβCαβsαsβ = 2P∑α

(2− ξα)eα + P∑α

ξαs2α

560 WANG Hai-Yan, CHENG Yan, CHEN Xiang-Rong, and GOU Qing-Quan Vol. 45

+ ρ1∂2E(ρ1, γ)

∂γ2

∣∣∣γ=0

. (13)

According to Eq. (13), pressure has an influence on thevalues of the elastic constants even when isochoric strainsare used.

2.3 Thermodynamic Properties

To investigate the thermodynamic properties, we hereapply the quasi-harmonic Debye model,[26] in which thenon-equilibrium Gibbs function G∗(V ;P, T ) takes theform of[26]

G∗(V ;P, T ) = E(V ) + PV + Avib(Θ(V );T ) , (14)

where Θ(V ) is the Debye temperature, and the vibrationalterm Avib can be written as[27,28]

Avib(Θ;T ) = nKT[98

ΘT

+3 ln(1− e−Θ/T )−D(Θ

T

)], (15)

where D(Θ/T ) represents the Debye integral, n is thenumber of atoms per formula unit. For an isotropic solid,Θ is expressed by[27]

Θ =h

K[6π2V 1/2n]1/3f(σ)

√BS

M, (16)

where M is the molecular mass per formula unit, BS isthe adiabatic bulk modulus, which can be approximatedby the static compressibility,[26]

BS ' B(V ) = V( d2E(V )

dV 2

). (17)

The Poisson ratio σ is taken as 0.25,[29] f(σ) is given byRefs. [30] and [31]. Therefore, the non-equilibrium Gibbsfunction G∗(V ;P, T ) as a function of (V ;P, T ) can be min-imized with respect to volume V(∂G∗(V ;P, T )

∂V

)P,T

= 0 . (18)

By solving Eq. (18), we get the thermal EOS. The isother-mal bulk modulus BT , the heat capacity Cv, and the ther-mal expansion coefficient α are given by[21]

BT (P, T ) = V(∂2G∗(V ;P, T )

∂V 2

)P,T

, (19)

CV = 3nk[4D

(ΘT

)− 3Θ/T

eΘ/T − 1

], (20)

α =γCV

BT V, (21)

where γ is the Gruneisen parameter.

3 Results and Discussions

3.1 Structural Behaviors

At zero pressure (i.e., Vn = 1.0), we find that the ra-tio c/a of about 1.138 is the most stable structure forhcp MgB2, and the equilibrium lattice parameters a and

c are about 3.093 A and 3.520 A, respectively, which areconsistent with experimental data[1,10] and other theoret-ical results.[9,12−14] The resulting pressure P dependenceof the normalized volume Vn of MgB2 obtained by fittingthe Birch–Murnaghan equation of state (EOS)[23] agreeswell with the experiment,[32] and seems to be better thanthose by Islam et al. from CRYSTAL 98.[33] The equilib-rium ratio c/a and the corresponding normalized latticeparameters a/a0, c/c0 and the normalized volume V/V0

as a function of the applied pressure are plotted in Fig. 1,where a0, c0, and V0 are their values at T = 0 and P = 0,respectively.

Fig. 1 Pressure dependence of the normalized latticeparameters, primitive cell volume, and c/a ratio of MgB2.

By fitting the calculated data to the second-order poly-nomial, we obtain their relationships at the temperatureT = 0 K,c

a= 1.138− 0.001 72P + 1.203 68× 10−5P 2 , (22)

a

a0= 1.000− 0.001 90P + 1.461 59× 10−5P 2 , (23)

c

c0= 1.000− 0.003 42P + 2.657 43× 10−5P 2 , (24)

V

V0= 1.000− 0.007 31P + 6.379 53× 10−5P 2 . (25)

These results are consistent with those measured using aMerrill–Bassett diamond anvil cell up to 8 GPa[10] andthose by Jorgensen et al.[11] through the neutron diffrac-tion measurements under low pressure (< 0.62 GPa). Itis shown that, as pressure increases, the equilibrium ra-tio c/a ratio decreases. The compression along c axisis much larger than that along the a axis, consistentwith the comparatively weaker (Mg-B) bonds that de-termine the c axis length. When pressure increases,the atoms in the interlayers become closer, and theirinteractions become stronger. Compared to K3C60, afairly soft material with weak intermolecular interactionswith d lnV/dP = −0.036 GPa−1,[34] MgB2 is a tightly-packed incompressible solid with volume compressibility,

No. 3 Theoretical Calculations for Structural, Elastic and Thermodynamic Properties of MgB2 under High Pressure 561

dlnV/dP = −0.0076 GPa−1. MgB2 is less compress-ible in the basal plane, in which the covalent B-B bondslie. The interlayer linear compressibility dlnc/dP =0.003 53 GPa−1 is about 1.8 times larger than the in-planevalue (dlna/dP = 0.001 96 GPa−1).

3.2 Elasticity

To determine the elastic constants of MgB2 at highpressure, we apply the five strains listed in Table 1.[25]

The atomic positions are optimized at all strains. Foreach strain, a number of infinitesimal parameter γ areemployed to calculate the total energy E, and the volumeV is not kept as a constant (i.e. volume-non-conserving)when the lattice is distorted by εij . All the obtained en-ergies are then fitted to a second-order polynomial in γ,where the second-order derivatives coefficients of the fit-ted E(ρ1, γ) with respect to γ are used to calculate theelastic constants of MgB2.

Table 1 The strains used to calculate the elastic constants of MgB2.

Strains Distortion (unlisted: εij = 0) ρ1∂2E(ρ1, γ)

∂γ2

∣∣∣γ=0

1 ε11 = ε33 = γ C11 + 2C13 + C33 − 2P

2 ε11 = −ε22 = γ 2(C11 − C12 − P )

3 ε11 = ε22 = γ 2(C11 + C12 − P )

4 ε13 = ε31 = γ 4C44 − 2P

5 ε33 = γ C33 − P

Table 2 The elastic constants Cij (in GPa) and the bulk modulus along a axis (Ba in GPa) and c axis (Bc inGPa) at different normalized volumes.

V/V0 P (GPa) C11 C12 C13 C33 C44 Ba Bc

This work 462 67 41 254 80 615.0 293.1

Exp. [35] 625 333

1.0 Ref. [12] 0.0 524 58 33 243 85 663 269

Ref. [13] 431 27 31 197 92 532 223

Ref. [14] 446 68 39 284 77

0.95 8.5 492.6 86.6 42.3 332.2 96.7 651.4 381.8

0.9 19.0 568.5 88.4 46.9 380.3 142.3 736.1 435.8

0.85 32.0 691.8 90.4 89.5 500.0 161 913.7 621.8

0.8 48.2 811 93.2 92.5 517.3 212.1 1061 626.6

The obtained compression dependencies of the elastic

constants at T = 0 are presented in Table 2, together

with the five elastic constants calculated by others at

V/V0 = 1. It is found that the five elastic constants in-

crease monotonously with applied pressure. C11 and C33

vary quickly as the pressure rises, and C44 becomes mod-

erate as well as C13. However, C12 is affected slightly

by applied pressure, comparatively. Our calculated elas-

tic constants at zero pressure agree with those by others.

Unfortunately, there are no experimental data to check

the elastic constants under high pressure.

To qualify the mechanical anisotropy of MgB2, one can

define bulk modulus values along the a axis (Ba) and the

c axis (Bc) as follows:

Ba = adp

da=

Λ2 + α

, (26)

Bc = cdp

dc=

Ba

α, (27)

Λ = 2(C11 + C12) + 4C13α + C33α2 , (28)

α =C11 + C12 − 2C13

C33 − C13. (29)

The bulk modulus Ba and Bc are also presented inTable 2. Obviously, Ba and Bc obtained here agree wellwith the experimental data (i.e. 625 GPa and 333 GPa)obtained from the hydrostatic high-pressure synchrotronXRD by Goncharov et al.[35] We find that the value ofBa/Bc has a trend of gradual decline as the value of V/V0

562 WANG Hai-Yan, CHENG Yan, CHEN Xiang-Rong, and GOU Qing-Quan Vol. 45

decreases under the applied pressure. It is suggested that,the mechanical behavior of MgB2 under zero pressure is ofhighly anisotropy. With the applied pressure increasing,the anisotropy becomes gradually weaken.

3.3 Anisotropy

It is known that the acoustic velocities are related tothe elastic constants by the Christoffel equation,

(Cijklnjnk −Mδil)ui = 0 , (30)

where Cijkl is the fourth rank tensor description of elasticconstants, n is the propagation, u the polarization vector,M = ρv2 is the modulus of propagation and v the velocity.The acoustic anisotropy can be described as

∆i =Mi[nx]Mi[100]

, (31)

where nx is the extremal propagation direction other than[100] and i is the index for the three types of elastic waves(one longitudinal and the two polarizations of the shearwave). Solving the Cristoffel equation (30) for a hexagonallattice, one can calculate the anisotropy of the compres-sional wave (P ) as

∆P =C33

C11. (32)

For the shear waves, the wave polarized perpendicular tothe basal plane (S1) and the one polarized in the basalplane (S2) have the anisotropies,

∆S1 =C11 + C33 − 2C13

4C44,

∆S2 =2C44

C11C12, (33)

while for S2 and P waves the extremum occurs along the caxis, for S1 it is at an angle of 45◦ from the c axis in the a-c plane. We note that an additional extremum may occurfor the compressional wave propagation at intermediatedirections depending on the elastic constants.

In Fig. 2, we illustrate the calculated pressure de-pendencies of the anisotropies of the compressional wave(∆P ), the shear waves ∆S1 (the wave polarized perpendic-ular to the basal plane S1), and ∆S2 (the one polarized inthe basal plane S2). It is found that ∆P and ∆S2 changeslightly in the wide range of applied pressure. ∆S1 variesgreatly from 0 GPa to 20 GPa, but it becomes moder-ate with the pressure rise, and is unsteady under pres-sure. These results may be understood by comparison toan hcp crystal interacting with central nearest-neighborforces (CNNF).[36] For this model the elastic anisotropy isindependent of the interatomic potential to lowest order inP/C11, hence the anisotropy is dependent on the symme-try of the crystal only. In our foregoing work, we find theratio c/a of MgB2 changes under different pressures, thatis, the structure is always varying with applied pressure.Therefore, the elastic anisotropy may be different withpressure. The relationship between the elastic anisotropy

and applied pressure also shows that ∆P , ∆S1, and ∆S2

have no remarkable changes under high pressure. Thesebehaviors may correspond to the bonding situations inMgB2, which are characterized as a strong covalent bond-ing in the boron layers, ionized magnesium atoms, and aweaker interlayer metallic bonding. Therefore, in a cer-tain range of pressures the elastic anisotropy tends to beinvariables.

Fig. 2 As a measure of anisotropy the elastic con-stant ratios C33/C11, (C11 + C33 − 2C13)/4C44, and2C44/(C11 −C12), which govern the compressional (∆P )and shear wave anisotropy (∆S1 and ∆S2, respectively)of MgB2, are shown as a function of pressure P .

Fig. 3 The volume compressions Vn of MgB2 up topressure 240 GPa at the temperatures of 300 K, 1000 K,and 2000 K, respectively.

3.4 Thermodynamic Properties

We illustrate the obtained isothermal Vn-P relation-ship curve in Fig. 3. It is shown that the volume is greatlyaffected by both the pressure and the temperature. Theeffect of increasing pressure on MgB2 is the same as de-creasing temperature of MgB2. On the other hand, as thepressure increases, the volume compression decreases at agiven temperature, and the volume compression at high

No. 3 Theoretical Calculations for Structural, Elastic and Thermodynamic Properties of MgB2 under High Pressure 563

temperature is less than that at lower temperature undera given pressure.

The obtained variations of the thermal expansion αwith temperatures and pressures are shown in Fig. 4. Wenote that at zero pressure α increases exponentially withT at low temperatures and gradually approaches a linearincrease at high temperatures. As the pressure increases,the increase of α with temperature becomes smaller, es-pecially at high temperature. At a given temperature, αdecreases drastically with the increase of pressure. Whenthe pressure is above 30 GPa, the thermal expansion αof 900 K is just a little larger than that of 600 K, whichmeans that the temperature dependence of α is very smallat high temperature and high pressure.

Fig. 4 The thermal expansion versus temperature andpressure for MgB2.

In Fig. 5, we show the heat capacity CV and the Debyetemperature Θ as a function of pressure P at the temper-atures of 300 K and 1000 K for MgB2. It is shown thatwhen the temperature is constant, the Debye temperatureΘ increases almost linearly with applied pressures, indi-cating the change of the vibration frequency of particlesin MgB2 under pressure. However, the heat capacity CV

decreases with the applied pressures, which is because theeffect of increasing pressure on MgB2 is the same as de-creasing temperature of MgB2.

4 ConclusionsThe elastic constants of hcp MgB2 at high pressure

are computed by the full-potential linearized muffin-tin or-

bital (FP-LMTO) scheme within the generalized gradientapproximation correction (GGA) in the frame of densityfunctional theory. We carry out total energy calculationsover a wide range of volumes from 0.8 V0 to 1.15 V0, andobtain the equilibrium ratio of the lattice parameters c/a

for a given volume, as well as the pressure dependenceof the normalized lattice parameters of MgB2. The ob-tained pressure dependence of the normalized volume isin excellent agreement with the experimental result.

Fig. 5 Variations of thermodynamic parameters X (X :Debye temperature or specific heat) with pressure P .They are normalized by (X − X0)/X0, where X andX0 are the Debye temperature or specific heat underany pressure P and zero pressure at the temperaturesof 300 K and 1000 K, respectively.

The elastic constants of MgB2 at high pressure from0 GPa to 50 GPa are also calculated. In order to qualifythe mechanical anisotropy of MgB2, we have investigatedthe bulk modulus values Ba and Bc, and the pressure de-pendence of acoustic anisotropy. An analysis of the calcu-lated parameters reveals the anisotropy of MgB2, that is,in a certain range of applied pressure, the anisotropy is re-markable, but beyond this range anisotropy tends to be in-variable. Finally, we investigate the thermodynamic prop-erties and obtain the isothermal Vn-P relationship curve,the relationships among the thermal expansion, tempera-ture and pressure, as well as the variations of Debye tem-perature, and specific heat with pressure.

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