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International Journal of Modern Physics B

Vol. 28, No. 10 (2014) 1450057 (10 pages)
c World Scientific Publishing Company
DOI: 10.1142/S021797921450057X

FIRST-PRINCIPLES INVESTIGATION OF THE ELECTRONIC,

ELASTIC AND THERMODYNAMIC PROPERTIES OF
SUPERCONDUCTING MgB2
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HONGYING LU∗ , JIANPING LONG† , LIJUN YANG† and WEN HUANG‡

∗ School of Information Science & Technology, Chengdu University of Technology,
Chengdu 610059, P. R. China
† College of Materials and Chemistry & Chemical Engineering,

Chengdu University of Technology, Chengdu 610059, P. R. China

‡ College of Electronic Engineering,

Chongqing University of Post and Telecommunications,

Chongqing, 400065, P. R. China
∗ hylu@cdut.edu.cn

Received 6 November 2013

Accepted 26 December 2013
Published 24 January 2014

The electronic structure, elastic properties, Debye temperature and thermal conductivity
of MgB2 are investigated by using the first-principles density function theory within the
generalized gradient approximation (GGA). The calculated elastic constants indicate
that the MgB2 is mechanically stable. The shear modulus, Young’s modulus, Poisson’s
ratio, σ, the ratio B/G and universal anisotropy index are also calculated. Finally, the
averaged sound velocity, longitudinal sound velocity, transverse sound velocity, Debye
temperature and thermal conductivity are obtained.

Keywords: Electronic properties; elastic properties; thermodynamic properties; thermal

conductivity.

PACS numbers: 62.20.de, 74.70.Ad, 74.25.Bt

1. Introduction
MgB2 , the surprising discovery of its superconductivity transition temperature near
39 K, has aroused great scientific interests of many research groups around the
world, not only for its great potential applications but also it seems to be a BCS
type superconductor.1,2 Many related studies have been reported for understand-
ing the mechanics and superconductivity properties.3–6 Prassides et al.7 studied
the compressibility of MgB2 with applied pressure using synchrotron X-ray powder
diffraction techniques. Chen et al.8,9 investigated the structure and the equation
of state of compound MgB2 at high pressure using the density functional theory
(DFT). Islam et al.10,11 discussed the mechanical behavior of MgB2 under pressure

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with ab initio calculations. Zeng et al.12 studied the structural and superconducting
properties of MgB2 thin films. Masui et al.13 studied the pressure effect on transport
and superconducting properties of impurity substituted MgB2 single crystals. How-
ever, the mechanical behavior and superconductivity are observed in only some of
MgB2 , and various studies are currently directed to shed light on other properties,
including their elastic, mechanical, dielectric and thermodynamical properties.
In this work, we calculated the electronic structure, Born effective charge, di-
electric tensors, elastic properties, Debye temperature and thermal conductivity of
MgB2 by using the first-principles plane-wave pseudopotential (PWPP) method.
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The rest of the paper is organized as follows: in Sec. 2, we describe briefly the com-
putational methods used in this work; Sec. 3 contains our results and discussion,
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involving electronic structure, Born effective charge, dielectric tensors, elastic prop-
erties, Debye temperature and thermal conductivity of MgB2 ; finally, the conclusion
is given in Sec. 4.

2. Calculation Methods
The first-principles calculations were carried out by using the PWPP method within
the DFT, which was implemented in the CASTEP.14 As for the exchange and cor-
relation terms, the PBE for solids15 was used within the generalized gradient ap-
proximation (GGA).16 Using the PWPP method, 2p6 3s2 of Mg and 2s2 2p1 of B
were treated explicitly as valence electrons. In this study, we employed 600 eV as
the cutoff energy of plane-wave and a 9 × 9 × 9 Monkhorst–Pack k-point mesh
because it gives a sufficiently accurate energy for the MgB2 . The structural param-
eters of MgB2 were calculated by using the Brodyden–Fletcher–Goldfarb–Shanno
(BFGS)17–20 method.

3. Results and Discussions

3.1. Structure properties
MgB2 has a very simple crystal structure where the B atoms form graphite-like
sheets separated by hexagonal layers of Mg atoms. Its AlB2 -type structure with a
space group of P 6/mmm and lattice parameters a = 3.0849 Å, c = 3.5187 Å,21 as
shown in Fig. 1. The calculated equilibrium lattice parameters of MgB2 are summa-
rized in Table 1, together with the available experimental data for comparison. From
the Table 1, we can see that the calculated results are in a good agreement with
the experimental values and deviated from measured ones with 0.57% and 0.76%,
respectively. The reasons why there is a difference between theoretical values and
experimental data are as follow: (1) the theoretical values have been obtained at
0 K while experimental data at room temperature; (2) theoretical values that have
been obtained by GGA method are often higher than experimental data.

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First-Principles Investigation of the Electronic, Elastic and Thermodynamic Properties

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Fig. 1. Crystal structure and the first Brillouin zone of MgB2 .

Table 1. Calculated MgB2 structure parameters together with the experiment data.

Pearson Crystal Space

Compound symbol system group Lattice parameters (Å)

MgB2 hP 3 hexagonal P 6/mmm Cal. a= 3.06719 c = 3.5456

Ref. 21 a= 3.0849 c = 3.5187
Ref. 22 a= 3.08553 c = 3.52007
Ref. 23 a= 3.086 c = 3.524
Ref. 24 a= 3.0851 c = 3.5201

3.2. Electronic properties

We have calculated the energy band structure of MgB2 along high symmetry di-
rections as shown in Fig. 2. From Fig. 2, it is observed that MgB2 exhibit metallic
characters as there are no band gap at the Fermi level, the valence band (VB)
and conduction band (CB) overlap significantly at Fermi level. The calculated total
density of states (TDOS) and partial density of states (PDOS) of MgB2 in the
energy range between −13 eV and 20 eV are illustrated in Fig. 3. As it can be seen,
the VB are mainly composed of B–2s, 2p states hybridized with small amount of
Mg–2p, 3s states. The CB is mainly composed of Mg–2p, 3s states hybridized with
small amount of B–2s, 2p states.
The Mulliken charge populations analysis is a good method to reveal bonding
behavior of compound.25 The Mulliken charge populations of MgB2 were performed,
and the results of analysis are listed in Table 2. It can be easily seen that the
charge transfer from Mg to B is clear, and the transferred charge comes mainly
from the lost valence electrons on the 2p states of Mg atom after bonding. The
results from Mulliken charge populations confirm that there is an ionic nature for
Mg–B bonding.
The Born effective charge tensor quantifies the macroscopic electric response of
a crystal to the internal displacements of its atoms.26 The calculated Born effective

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Fig. 2. Calculated band structure of MgB2 .

Fig. 3. The TDOS and PDOS of MgB2 .

Table 2. Atomic Mulliken charge populations of MgB2 .

Element s p Total Charge (e)

B 1.11 2.75 3.86 −0.86

B 1.11 2.75 3.86 −0.86
Mg 2.25 6.57 8.82 1.72

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First-Principles Investigation of the Electronic, Elastic and Thermodynamic Properties

Table 3. Born effective charge and dielectric tensors.

Born effective charges tensors Dielectric tensors

   
−0.94496 0 0 20.77 0 0
B 0 −0.94496 0 Optical permittivity ε∞  0 20.77 0 
   

0 0 −1.34312 0 0 7.95
   
1.88992 0 0 25.83 0 0
Mg  0 1.88992 0 Static dielectric ε0  0 25.83 0
   
 
0 0 2.68625 0 0 15.97
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charges of MgB2 are given in Table 3. It is known that the Born effective charge
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can be strongly influenced by the local environment of each atom, and the form
and the number of independent elements in the Born effective charge tensors are
determined by the local atomic symmetry. The Born effective charge tensor of the
B and Mg atom is strongly anisotropic. The calculated optical permittivity ε∞
and static dielectric ε0 tensors are also given in Table 3. The dielectric tensors are
usually underestimated in GGA calculations due to the overestimation of the band
gap. From the Table 3, we can find that the MgB2 is isotropic in x–y plane, and is
strongly anisotropic in z-direction.

3.3. Elastic properties and mechanical stability

Elastic constants of crystals provide a link between mechanical and dynamical be-
haviors. Also, they give important information concerning the elastic response of
a crystal to an external pressure. For the hexagonal crystals, its five independent
elastic constants should satisfy the well-known Born stability criteria27
2
C11 > 0 , C33 > 0 , C44 > 0 , C11 − C12 > 0 , (C11 + C12 )C33 − 2C13 > 0.
(1)
The computed elastic constants at 0 K of MgB2 is shown in Table 4. Accord-
ing to the above criteria, it is clear that the MgB2 is mechanically stable at 0 K.
From the Table 4, it can be seen that the elastic constant C11 , which provides a
measure of rigidity against unidirectional deformation along a-axis, is larger than
elastic constant C33 , which provides an estimation of the elastic response of ma-
terial to a unidirectional pressure along c-axis direction, indicating that MgB2 is
incompressible under uniaxial stress along a-axis.

Table 4. The calculated elastic constants Cij (in GPa) of MgB2 alloys.

C11 C33 C44 C12 C13

Present work 351.7 244.3 42.9 112.2 49.8

Ref. 11 446 284 77 68 39
Ref. 28 438 264 80 43 33

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For a hexagonal structure, the computations of Voigt (GV ) and Reuss shear
modulus (GR ) and Voigt (BV ) and Reuss bulk modulus (BR ) are29
1 1
GV = (2C11 + C33 − C12 − 2C13 ) + (2C44 + C66 ) , (2)
15 5
 
2 1
BV = C11 + C12 + 2C13 + C33 , (3)
9 2
1
BR = , (4)
2(S11 + S33 ) + 2(S12 + 2S13 )
15
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GR = , (5)
4(2S11 + S33 ) − 4(S12 + 2S13 ) + 3(2S44 + S66 )
where the Sij are the elastic compliance constants.
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The elastic modulus of MgB2 can then be estimated by Hill’s average, BH =

(1/2)(BV + BR ) for bulk modulus and GH = (1/2)(GV + GR ) for shear modulus.
The Young’s modulus E and Poisson’s ratio σ can be computed by the following
equations,30 respectively:
9BG
E= , (6)
3B + G
3B − 2G
σ= . (7)
2(3B + G)
The ratio of bulk modulus to shear modulus of crystalline phases can predict the
brittle and ductile behavior of materials. If B/G > 1.75 the material will behave
in a ductile manner or else the material demonstrates brittleness.31 The values of
bulk and shear modulus B and G, Young’s modulus E, Poisson’s ratio σ and the
ratio B/G are given in Table 5. The obtained B/G ratio is 1.85, according to this
values MgB2 behave in a ductile manner.
The Poisson’s ratio σ provides more information about the characteristics of
the bonding forces than any other elastic constants.32 The obtained values of the
Poisson’s ratio (σ) are greater than 0.25, which indicates that the interatomic forces
of MgB2 is the central force.
Elastic anisotropy is one of the most important parameters for estimating the
mechanical properties of materials. The average Young’s modulus E, average shear
modulus G and average Poisson’s ratio σ on the (21̄1̄0), (011̄0) and (0001) planes
can be obtained using the following relationships33 :
1 1
E(21̄1̄0) = E(011̄0) = , E(0001) = , (8)
S11 S33

Table 5. Calculated bulk and shear modulus B and G (all in GPa), Young’s modulus E (all in
GPa), Poisson’s ratio σ and the ratio B/G of MgB2 .

BV BR BH GV GR GH B/G E σ

Present work 152.4 143.8 148.1 90.2 69.7 79.9 1.85 203.2 0.27
151 (Ref. 35) 172 (Ref. 36)

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First-Principles Investigation of the Electronic, Elastic and Thermodynamic Properties

Table 6. Calculated average Young’s modulus E, shear modulus G and Poisson’s ratio σ on the
(21̄1̄0), (011̄0) and (0001) planes.

Compounds E(21̄1̄0), E(011̄0) E(0001) G(21̄1̄0), G(011̄0) G(0001) σ(21̄1̄0), σ(011̄0) σ(0001)

MgB2 311.0 233.7 135.3 42.9 0.22 0.11

2 1
G(21̄1̄0) = G(011̄0) = , G(0001) = , (9)
S44 + 2S11 − 2S12 S44
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S12 + S13 S13

σ(21̄1̄0) = σ(011̄0) = − , σ(0001) = − . (10)
2S11 S33
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Table 6 is the calculated results for the Young’s modulus E, shear modulus G
and Poisson’s ratio σ on the (21̄1̄0), (011̄0) and (0001) planes. The calculated results
show that the anisotropy behavior of MgB2 is very significant due to the reason
that the Young’s modulus E and shear modulus G are the difference between the
prismatic planes (21̄1̄0), (011̄0) and the basal plane (0001). For hexagonal crystal,
compression anisotropy can be identified by Bc /Ba = (C11 + C12 − 2C13 )/(C33 −
C13 ), where Bc and Ba represent bulk modulus along c-axis and a-axis direction,
respectively. The calculated value of Bc /Ba is 1.87, it is concluded that MgB2
exhibits anisotropy elasticity. Most recently, Ranganathan and Ostoja–Starzewski34
introduced a concept of universal anisotropy index to measure the single crystal
elastic anisotropy. The universal anisotropy index:
GV BV
AU = 5 + −6. (11)
GR BR
where AU = 0 represents locally isotropic single crystals and AU > 0 denotes the
extent of single crystal anisotropy. The calculated value of AU is 1.53, suggesting
again its stronger anisotropy.

3.4. Thermodynamic properties

Dynamical properties were obtained from the linear response method, within den-
sity functional perturbation theory (DFTP). Unlike the previous calculations, the
calculations of phonons have been implemented using the norm-conserving pseudo-
potentials. Phonon calculations from DFTP can be used to evaluate the temper-
ature dependence of the entropy, free energy, enthalpy, heat capacity and Debye
temperature of a crystal in a quasi-harmonic approximation.
The variations of the entropy, enthalpy and free energy are shown in Fig. 4(a).
From Fig. 4(a), it is noted that the enthalpy and entropy increases rapidly when
the temperature increases while the free energy decreases when the temperature in-
creases. The contribution from the lattice vibrations to the specific heat capacity at
constant volume and Debye temperature (θD ) for MgB2 are presented in Fig. 4(b).
It is seen that when T < 450 K, Cv increases very rapidly with temperature; when

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Fig. 4. Enthalpy, free energy, entropy, constant volume heat capacity and temperature depen-
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dence of Debye temperature of MgB2 .

T > 450 K, Cv increases slowly with temperature and it moves to the Dulong–
Petit limit. It is also clearly seen from this figure that the Debye temperature θD
drops with the increasing temperature from 20 K to 101 K. However, in the tem-
perature range 101–1000 K, Debye temperature θD increases when the temperature
increases.
The Debye temperature is an important parameter to describe phenomena of
solid-state physics which are associated with lattice vibration, elastic constants,
specific heat and melting temperature. The Debye temperature (θD ) is not a strictly
determined parameter, various estimates may be obtained through well established
empirical or semi-empirical formulae. One of the semi-empirical formula can be used
to estimate the values of Debye temperature through elastic constants, averaged
sound velocity (vm ), longitudinal sound velocity (vl ) and transverse sound velocity
(vt ).37–41
   13
h 3n NA ρ
θD = vm , (12)
k 4π M
  − 13
1 2 1
vm = + 3 , (13)
3 vt3 vl
 12
4

B+ G
vl = 
 3 
 , (14)
1 ρ

  12
G
vt = , (15)
ρ
where h and k are Planck’s and Boltzmann’s constants; NA is Avogadro’s number;
ρ is the density; M is the molecular weight and n is the number of atoms in the
unit cell.

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First-Principles Investigation of the Electronic, Elastic and Thermodynamic Properties

Table 7. Calculated the density (ρ in g/cm3 ), transverse,

longitudinal, average sound velocity (vt , vl , vm in m/s), the
Debye temperatures (θD in K) and the minimum thermal
conductivity (Kmin in Wm−1 K−1 ) of MgB2 .

ρ vt vl vm θD Kmin

Present work 2.64 5503 9822 6125 857 1.87

Ref. 3 9650 880

The calculated values of vm , vl , vt and θD of MgB2 at 0 K are given in Table 7.

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The Debye frequency vD is given by vD = kB θD /h. On the basis of the Debye

temperature θD , the Debye frequency of MgB2 is 17.8 THz.
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Thermal conductivity K is the property of a material that indicates its ability

to conduct heat. However, in order to know if the material is a potential applica-
tion for thermal barrier coating, its thermal conductivity needs to be investigated.
Based on the Debye model, Clarke42 and Liu et al.43 suggested that the theoretical
minimum thermal conductivity can be calculated after replacing different atoms by
an equivalent atom with a mean atomic mass.
 − 23
M̄
Kmin = kB vm , (16)
ρ
where M̄ is the average atomic weight which is the molecular weight M divided by
the number of atoms in the unit cell. The calculated minimum thermal conductivity
of MgB2 is given in Table 7. Unfortunately, the experimental thermodynamic data
cannot be found. Therefore it is difficult to evaluate the magnitude of errors between
calculations and experiments.

4. Conclusions
In present work, the structural, elastic and thermodynamic properties of MgB2 have
been studied by means of DFT within the GGA. The most relevant conclusions are
summarized as follows:

(1) The calculated lattice parameters of MgB2 are in a good agreement with the
experimental values and deviated from measured ones with 0.57% and 0.76%,
respectively.
(2) The mechanical properties like shear modulus and Young’s modulus are
also calculated. From our results, we observe that MgB2 is mechanically
stable.
(3) The Poisson’s ratio σ and B/G ratio are calculated. According to these values,
we have revealed that the MgB2 behave in a ductile manner.
(4) The compression anisotropy Bc /Ba and the universal anisotropy index AU
are obtained. Based on our calculation, we can conclude that MgB2 exhibits
stronger anisotropy.

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(5) The averaged sound velocity (vm ), the longitudinal sound velocity (vl ), trans-
verse sound velocity (vt ), the Debye temperature (θD ) and thermal conductivity
are obtained.

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