First-Principle Calculations for Thermodynamic Properties of LiBC Under High Temperature

and High Pressure

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2006 Commun. Theor. Phys. 46 573

(http://iopscience.iop.org/0253-6102/46/3/034)

Download details:

IP Address: 139.184.30.133

The article was downloaded on 22/09/2012 at 15:27

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

Commun. Theor. Phys. (Beijing, China) 46 (2006) pp. 573–576c© International Academic Publishers Vol. 46, No. 3, September 15, 2006

First-Principle Calculations for Thermodynamic Properties of LiBC Under High

Temperature and High Pressure∗

LIU Zhong-Li, CHENG Yan,† TAN Ni-Na, and GOU Qing-Quan

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

(Received December 5, 2005)

Abstract The thermodynamic properties of LiBC are investigated by using the full-potential linearized muffin-tinorbital method (FP-LMTO) within the frame of density functional theory (DFT) and using the quasi-harmonic Debyemodel. The dependencies of the normalized lattice parameters a/a0 and c/c0, the ratio (c/a)/2, the normalized primitivevolume V/V0 on pressure and temperature are successfully obtained. It is found that the interlayer covalent interactions(Li-B bonds or Li-C bonds) are more sensitive to temperature and pressure than intralayer ones (B-C bonds), as givesrise to the extreme lattice anisotropy in the bulk hcp LiBC.

PACS numbers: 71.15.Mb, 65.40.-b, 62.50.+pKey words: thermal expansion, heat capacity, compressibility, LiBC, full-potential linearized muffin-tin orbital

Great interests have been stirred up in research of in-termetallic superconductor MgB2 experimentally and the-oretically since its critical high Tc = 39 K was discov-ered in 2001.[1] As a closely related intermetallic com-pound, the hole-doped LixBC has been predicted toachieve higher transition temperature of Tc ∼ 100 K(x = 0.5)[2] exceeding that of the isostructural intermetal-lic compound MgB2.[1] Up to now, a few investigationson LiBC, including lattice dynamics,[3,4] electronic andoptical properties,[5] vibrational properties,[6−8] and highpressure properties[9] have been made theoretically andexperimentally.

Hinks et al.[10] proposed that the electron-phonon cou-pling plays an important role in the superconductivity ofthe intermetallic compound. Electron-phonon couplingparameter λ increases monotonically with decreasing x toa maximum value of 1.4 for x = 0.125 and thus resultsin Tc ∼ 65 K.[4] The DC resistivity and AC dielectricmeasurements[5] show a strong anisotropy of the electri-cal transport properties in LiBC. The extreme thermalexpansion anisotropy in LiBC was observed by the high-resolution x-ray powder diffraction.[6]

In this letter, we focus on the thermodynamic proper-ties of LiBC under high temperature and high pressure,and apply the full-potential linearized muffin-tin orbital(FP-LMTO) scheme[11−14] in the frame of density func-tional theory[15,16] to calculate the lattice parameters andthermodynamic properties of LiBC. We adopt the gen-eralized gradient approximation for exchange-correlationfunctional proposed by Perdew, Burke, and Ernzerhof.[17]

The crystal space is partitioned into the non-overlapping(or slightly overlapping) muffin-tin spheres surroundingevery atom and the remaining interstitial region. The

basis set is comprised of augmented linearized muffin-tinorbitals.[18] The basis functions, charge density, and po-tential are expanded in symmetry-adapted spherical har-monic functions together with a radial function inside themuffin-tin spheres. The MT sphere radii RMT are chosento be 2.605 a.u., 1.504 a.u., and 1.504 a.u. for Li, B, andC atoms, respectively. We used Fourier series in the inter-stitial regions. The maximum angular momentum com-ponent for the spherical-harmonic expansion is lmax = 6.The K-space integration has been performed using 484 k-points mesh in the irreducible wedge of the Brillouin zone(BZ) for our structural properties. The self-consistent it-erations are continued until the total-energy differencesbetween two consecutive iterations are less than 10−5 Ryd.

The total energy electronic structure calculations arecarried out over a wide range of primitive cell volumes V ,i.e., from 0.8V0 to 1.15V0, where V0 is the zero pressureequilibrium primitive cell volume. No constraints are im-posed on the c/a ratio, that is, both lattice parametersa and c are optimized simultaneously. For each volumeV , we determine the corresponding equilibrium ratio c/a

by performing the total energy electronic structure calcu-lations on a series of different values of c/a. For a fixedvalue of c/a, a series of different values of a and c are takento calculate the total energies E, which are minimized asa function of the c/a ratio for a given V . Through thesecalculations, we can obtain the equilibrium lattice param-eters a and c as well as the corresponding equilibrium ratioc/a at any pressure and temperature.

The equilibrium lattice parameters obtained are listedin Table 1, together with the experimental data[4,10,19]

and the calculated results by Kobayashi et al.[9] OurGGA results are in better agreement with experimental

∗The project supported by National Natural Science Foundation of China under Grant No. 10576020†Corresponding author, E-mail: yan.cheng@126.com

574 LIU Zhong-Li, CHENG Yan, TAN Ni-Na, and GOU Qing-Quan Vol. 46

data than those obtained from the pseudopotential calcu-lation and the full-potential linearized augmented planewave calculation.[9] The bulk modulus B0 and the pres-sure derivative of bulk modulus B′

0 at zero pressure ob-tained from Birch–Murnaghan equation of state (EOS)[21]

are also shown in Table 1. Since there are no theoretical

or experimental data to check our calculated results, we

here present the lattice parameters, the bulk modulus B0,

and the pressure derivative of bulk modulus B′0 of hcp

structure MgB2 for comparison because the electron per

atom ratio for MgB2 is the same as that of LiBC.

Table 1 The calculated structural parameters compared with experiments and those by othersat P = 0.

c (A) a (A) c/a B0 (GPa) B′0

GGA (present) 6.993 2.756 2.537 180.1 3.66

Ref. [9] 6.87 2.73 2.52

Exp. [4] 7.053 2.749 2.566

Exp. [10] 7.05 2.75 2.56

Exp. [19] 7.058 2.752 2.565

Exp. [1, 20]a 3.086 3.524 1.141 151± 5 4 0.0aExperimental data for MgB2.

Fig. 1 The equilibrium ratio c/a and the correspond-ing normalized parameters a/a0, c/c0, and the normal-ized volume V/V0 as a function of the applied pressureat 0 K.

The obtained pressure dependence of the normalizedprimitive cell volume of LiBC is shown in Fig. 1, wherea0, c0, and V0 are the zero-pressure equilibrium latticeparameters and the primitive-cell volume, and P is thepressure in GPa. When the pressure increases from 0to 20 GPa, the c/a ratio decreases by 4.24% (2.60% forMgB2).[22] Compression along the c-axis is considerablylarger than along the a-axis in the basal plane, consistentwith the comparatively weaker Li-B bonds and Li-C bondsthat determine the interlayer distance together. LiBCis more compressible along the c-axis than in the basalplane. The interlayer linear compressibility, d ln c/d P =0.003 62 GPa−1 is about 3.45 times the basal plane value,d ln a/d P = 0.001 05 GPa−1 (for MgB2, d ln c/d P =0.003 53 GPa−1, d ln a/d P = 0.001 96 GPa−1).[22] The re-

sults indicate that the B-C bonds are stronger than the Li-B or Li-C bonds. LiBC is a tightly-packed incompressiblesolid with volume compressibility, d lnV/d P = −0.005 63GPa−1 (−0.0076 GPa−1 for MgB2)[22] in comparison to afairly soft material K3C60 with weak intermolecular inter-action with d lnV/d P = −0.036 GPa−1.[23]

Fig. 2 Ratios of (c/a)/2, a/a0, c/c0, and V/V0 versustemperature at zero pressure.

From the quasi-harmonic Debye model,[24] we investi-gate the temperature dependence of lattice constants al-lowing for temperature from 0 to 1000 K. At zero pres-sure, the equilibrium lattice parameters a/a0, c/c0, theratio (c/a)/2, and the normalized primitive cell volumeV/V0 as functions of temperature are plotted in Fig. 2.The obtained a, c, and c/a at various temperatures areillustrated in Fig. 2 and listed in Table 2 in comparisonto the experimental data.[6] The results agree well withexperiments. It is obvious from Figs. 1 and 2 that latticeconstant c is more sensitive to temperature and pressure

No. 3 First-Principle Calculations for Thermodynamic Properties of LiBC Under High Temperature and High Pressure 575

than a, consistent with the weaker Li-B bonds, and Li-Cbonds. To further investigate the thermal anisotropy, wecalculate the linear thermal expansion parameters αa andαc in the intralayer and interlayer directions, respectively.Through the following equation at zero pressure

α =(∆l/∆T )

l0=

1l0

∂l

∂T

∣∣∣∣P

, (1)

where l represents the lattice parameters a and c, we ob-

tain αa = 6.49 × 10−6 K−1 and αc = 2.78 × 10−5 K−1

at room temperature, consistent with the experimentaldata.[6] In Table 3, we list our results and those of closelyrelated intermetallic compound MgB2

[25] for comparison.It is found that the thermal expansion along the c-axis isabout four times that along the a-axis, and the interlayerbonding is especially weak in LiBC, indicating that the in-terlayer covalent interactions of Li-B and Li-C are weakerthan those of Mg-B in MgB2.

Table 2 The calculated a, c, and c/a values at various temperatures compared with the experi-mental data.

Temperature (K) 0 300 600 900 1200

a (A) Present work 2.756 2.762 2.768 2.777 2.788

Exp. [6] 2.746 2.750 2.755

c (A) Present work 6.993 7.058 7.120 7.181 7.228

Exp. [6] 7.042 7.059 7.157

c/a Present work 2.537 2.556 2.578 2.600 2.626

Exp. [6] 2.565 2.580 2.598

Table 3 The calculated linear thermal expansion parameters at room temperature.

Compound αa (10−6 K−1) αc (10−5 K−1)

LiBC Present wok 6.49 2.78

Fogg et al., [6]a 5.59 2.83

MgB2 [25] 5.40 1.14aObtained from the XRD experiment [6].

As an important fundamental parameter, the Debyetemperature is closely related to many physical propertiesof solids, such as elastic constants, specific heat, and melt-ing temperature. Quantum mechanical effects are very im-portant in understanding the thermodynamic propertiesbelow Debye temperature, but above Debye temperaturequantum effects can be neglected. Thermal expansion pa-rameters are of both practical and theoretical importance,and are essential for predicting the thermodynamic equa-tion of state together with the specific heat. Therefore,we have also considered the thermal expansion param-eters above Debye temperature at zero pressure. Fromquasi-harmonic Debye model,[24] we obtain Debye tem-perature ΘD = 818.01 K at T = 0 K and zero pressure.According to Eq. (1), we derive αa = 1.13×10−5 K−1 andαc = 2.59× 10−5 K−1.

In conclusion, we apply the FP-LMTO method to de-termine equilibrium structural parameters and the ther-modynamic properties of LiBC under high pressure. Thecalculated lattice constants agree well with the experi-mental data. From the quasi-harmonic Debye model, thedependencies of the normalized lattice parameters a/a0

and c/c0, the ratio (c/a)/2, the normalized primitive vol-ume V/V0 on pressure and temperature are successfullyobtained. The interlayer linear compressibility is aboutthree times the basal plane value under high pressure andthe thermal expansion coefficient along the c-axis is aboutfour times that along the a-axis. These results show thatthe interlayer covalent interactions (Li-B bonds or Li-Cbonds) are more sensitive to temperature and pressurethan intralayer ones (B-C bonds), resulting in the extremelattice anisotropy in LiBC.

References

[1] J. Nagamatsu, N. Nakagawa, T. Muranaka, et al., Nature(London) 410 (2001) 63

[2] H. Rosner, A. Kitaigorodsky, and W.E. Pickett, Phys.Rev. Lett. 88 (2002) 127001.

[3] B. Renker, H. Schober, P. Adelmann, et al., Phys. Rev.B 69 (2004) 52506.

576 LIU Zhong-Li, CHENG Yan, TAN Ni-Na, and GOU Qing-Quan Vol. 46

[4] J.K. Dewhurst, S. Sharma, C. Ambrosch-Draxl, and B.Johansson, Phys. Rev. B 68 (2003) 20504(R).

[5] A.V. Pronin, K. Pucher, P. Lunkenheimer, A. Krimmel,and A. Loidl, Phys. Rev. B 67 (2003) 132502.

[6] A.M. Fogg, P.R. Chalker, J. B. Claridge, et al., Phys. Rev.B 67 (2003) 245106.

[7] J.M. An, H. Rosner, S.Y. Savrasov, and W.E. Pickett,Physica B 328 (2003) 1.

[8] K.W. Lee and W.E. Pickett, Phys. Rev. B 68 (2003)085308.

[9] K. Kobayashi and M. Arai, Physica C 388-399 (2003)201.

[10] D.G. Hinks, H. Claus, and J.D. Jorgensen, Nature (Lon-don) 411 (2001) 457.

[11] J.M. Wills, O. Eriksson, M. Alouani, and D.L. Price, inElectronic Structure and Physical Properties of Solids, ed.H. Dreysse, Springer, Berlin (2000) p. 148.

[12] J.M. Wills and B.R. Cooper, Phys. Rev. B 36 (1987)3809.

[13] D.L. Price and B.R. Cooper, Phys. Rev. B 39 (1989) 4945.

[14] S.Y. Savrasov and D.Y. Savrasov, Phys. Rev. B 46 (1992)12181.

[15] P. Hohenberg and W. Kohn, Phys.Rev. B 136 (1964) 864.

[16] W. Kohn and L.J. Sham, Phys. Rev. A 140 (1965) 1133.

[17] J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. 77 (1996) 3865.

[18] O.K. Andersen, Phys. Rev. B 12 (1975) 3060.

[19] M. Worle, R. Nesper, G. Mair, M. Schwarz, H.G. von-Schnering, and Z. Anorg. Allg. Chem. 621 (1995) 1153.

[20] T. Vogt, G. Schneider, J.A. Hriljac, et al., Phys. Rev. B63 (2001) 220505(R).

[21] L. Fast, J.M. Wills, B. Johansson, and O. Eriksson, Phys.Rev. B 51 (1995) 17431.

[22] H.Y. Wang, X.R. Chen, W.J. Zhu, and Y. Cheng, Phys.Rev. B 72 (2005) 172502.

[23] O. Zhou, G.B.M. Vaughan, Q. Zhu, et al., Science 255(1992) 835.

[24] M.A. Blanco, E. Francisco, and V. Luana, Comput. Phys.Comm. 158 (2004) 57.

[25] J.D. Jorgensen, D.G. Hinks, and S. Short, Phys. Rev. B

63 (2001) 224522.