- Email: [email protected]
High-pressure lattice dynamics and thermodynamic properties of zinc-blende BN from first-principles calculation
Physics Letters A 373 (2009) 2082–2086
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
High-pressure lattice dynamics and thermodynamic properties of zinc-blende BN from first-principles calculation HuanYou Wang a,b,∗ , Hui Xu a , XianChun Wang c , ChunZhi Jiang b a b c
Department of Physics, Central South University, Changsha 410083, China Department of Physics, Xiangnan University, Chenzhou 423000, China College of Physics and Electronic Science, Hunan University of Arts and Science, Changde 415000, China
a r t i c l e
i n f o
Article history: Received 6 January 2009 Received in revised form 26 March 2009 Accepted 10 April 2009 Available online 18 April 2009 Communicated by R. Wu PACS: 65.40.De 62.20.Dc 63.20.-e
a b s t r a c t The density function perturbation theory (DFPT) is employed to study the lattice dynamics and thermodynamic properties (with quasiharmonic approximation) of zinc-blende BN. First we discuss the structural properties and compare the phonon spectrum with available Raman scattering experiments. Thereafter using the calculated phonon dispersions we obtain the PTV equation of state from the free energy. Our results for the above properties are generally speaking in good agreement with experiments and with similar theoretical calculations. Owing to the anharmonic effect at high temperature, the calculated linear thermal expansion coefficients (CTE) are low to experimental data. © 2009 Elsevier B.V. All rights reserved.
Keywords: First-principle Lattice dynamics Heat expansion Pressure
1. Introduction Recently, the boron nitride has attracted both scientific and technological attention. Due to its fascinating mechanical properties, such as high hardness, high melting point, high thermal conductivity, large bulk moduli, etc., making it useful for protective coating. Furthermore, it is also attractive for use in field effect transistors intended to operate at high power and/or temperature. Current activities in optoelectronic devices have led to significant interest in studies of electronic, dynamical and thermal properties. In these important properties, thermal properties are one of the most basic properties of any material. Thermal expansion is connected not only with thermal properties (thermal conductivity, specific heat, etc.) but also it influences many other properties, such as the temperature (T) variation of the energy band gap. Moreover, the knowledge of the thermal expansion coefficient is especially important for epitaxial growth. Following above reality, several experimental and theoretical works have been carried out on the structural, vibrational and
*
Corresponding author at: Department of Physics, Central South University, Changsha 410083, China. E-mail address: [email protected] (H. Wang). 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.04.030
thermal properties. Experimentally zone-centre phonon modes have been determined by G.L. Doll [1] by first-order Raman scattering technique. Very recently, zone-edge phonon modes of this material have also been measured by S. Reich [2] by using secondorder Raman scattering technique. Vladimir L. Solozhenko [3] have measured the pressure–volume data at room temperature up to 66 GPa. F. Datchi [4] have measured the equation of state of cubic boron nitride by X-ray diffraction up to 160 GPa at 295 K and 80 GPa in the range 500–900 K. The measurements of thermal expansion coefficient were also performed by A.S. Glen [5] and F.G. Alexander [6], respectively. Several theoretical calculations have been performed to study the dynamical properties [7–11] of zinc-blende BN recently. These methods include valence-force model, rigid-ion model and ab initio calculation, etc. The calculation of thermal properties is rare. In the 1990s, another approach had been made possible by the achievement of DFPT [12,13] which allowed exact calculation of vibrational frequencies in every point of the Brillouin zone. The vibrational free energy can be obtained using the quasiharmonic approximation (QHA) which include anharmonic effects through the explicit volume dependence of the vibrational frequencies. In the Letter we apply DFPT within QHA to the study of the structural, vibrational and thermal properties of zinc-blende BN at high pressures.
H. Wang et al. / Physics Letters A 373 (2009) 2082–2086
2083
2. Theory and computational details 2.1. Theory The knowledge of entire phonon spectrum of a given system enables the calculation of its thermodynamical properties and the relative stability of its different phases as functions of T. The thermodynamical properties are usually determined by the appropriate thermodynamical potential relevant to the given ensemble. The relevant potential, e.g. Helmholtz free energy (F) can be written as F (a, T ) = E tot (a) + K B T
ln 2 sinh
qλ
h¯ ωqλ (a) 2K B T
,
(1)
the equilibrium structure parameters a = (a1 , a2 , . . .) of a crystal at any T are obtained by minimizing F of a system. Where E tot (a) is the ground state (T = 0 K) total energy of the crystal. The next term is vibrational free energy. The electronic entropy contribution vanishes for semiconductor, and thus, it is not include in Eq. (1). Even for metal this contribution is usually neglected, although it is easy to calculate. ωqλ (a) of Eq. (1) is the phonon frequency of the qth phonon mode with branch index λ. The QHA accounts only partially for the effects of anharmonicity. However, QHA is found to be a very good approximation at temperatures not too high. For given T and V , the equilibrium state of a crystal is determined by minimizing F with respect of all possible degrees of freedom. The equation of state ( P versus V ) of the system is obtained by equating P to minus the derivative of F with respect to V at constant T ,
P =−
∂F ∂V
Fig. 1. Calculated total energies per unit cell versus volume.
(2)
. T
Thermal expansion is obtained directly from the equation of state, and the volume thermal expansion coefficient is defined as
αv =
1 V
∂V ∂T
.
(3)
P
The linear thermal expansion coefficients for cubic crystal is given as 1 αa = α v . (4) 3
Fig. 2. Pressure–volume data for zinc-blend BN (solid line) at 0 K. Circles, diamonds and asterisks are experimental data from Refs. [3,4,6], respectively.
2.2. Computational details The lattice dynamics and thermodynamic properties of zincblende BN were performed using the ABINIT codes [14,15]. We use a first-principles pseudopotential method based on DFPT with wave function represented in a plane-wave basis set. A review of the method (and of the algorithm used for the convergence of electronic density and atomic positions) can be found in Ref. [16]. The static energies were computed using density function theory (DFT), and phonon frequencies using DFPT. The interactions between the ions and valence electrons were described using norm-conserving local density approximation (LDA) pseudopotentials which are generated in the scheme of Troullier–Martins [17]. Brillouin-zone integrations were performed using 12×12×12 k-point mesh and phonon frequencies were computed on a 8×8×8 q-point mesh. Plane-wave basis set with a cutoff of 40 Hartree was used. These calculating parameters are chosen to guarantee the total energy error in 0.01 mHartree. 3. Results and analysis 3.1. Structural properties Firstly we determine the equilibrium lattice constant of the ground state of the zinc-blende BN by calculating the total en-
Table 1 Comparison of calculated and measured structural properties of zinc-blende BN: lattice constant a (Å), bulk module B (Mbar) and its pressure derivative B .
Calc Expt (Ref. [4]) Expt (Ref. [18]) Expt (Ref. [19]) Calc (Ref. [20]) Calc (Ref. [21])
a
B
3.590 3.612 3.615 3.6175 3.593 3.58
3.98 3.95 3.69 3.87 3.95 3.80
B 3.6 3.54 4.0 3.06 3.65 3.56
ergy per primitive unit cell as a function of volume. The Birch– Murnaghan’s equation of state (EOS) is then used to fit the calculated energy–volume data. Fig. 1 shows the total energy per unit cell as a function of volume for BN. The obtained structural parameters are compared in Table 1 with the available experimental data and theoretical values. The calculated lattice constant at ground sate is in good agreement with experimental data of E. Knittle [18], F.G. Alexander [19] and F. Datchi [4]. The calculated static bulk moduli is overestimated by about 2% and 8% compared with experimental data [4,18,19]. The present determined value of the bulk modulus is in good agreement with the theoretical results of A. Janotti [20], but higher than result of H.M. Tütüncü [21]. Fig. 2 shows pres-
2084
H. Wang et al. / Physics Letters A 373 (2009) 2082–2086
sure vs volume for zinc-blende BN. Our conclusion is similar to those found in Refs. [3,4,6]. The experimental data in reference are almost equal, and our values are lightly low to those experimental data. Firstly, the underestimate of lattice constants are caused by selection of pseudopotential and the LDA itself. On the other hand, the experimental data are achieved at room temperature and our values neglect thermal expansion. Meanwhile, one should also notice that the experimental values of bulk moduli are somehow uncertain due to the difficult of growing high-quality single crystal. 3.2. Interatomic force constant The interatomic force constants (IFCs) describing the atomic interactions in a crystalline solid are defined in real space as [22] C kα ,k β (a, b) =
∂2 E ∂ τkaα ∂ τkb β
mensions are four times larger than the primitive zinc-blende cell, thus containing 128 atoms. Generally, the decay of local interaction for B–B, B–N, and N–N are faster than those of total interaction. At the forth neighbor the local part go to zero for each species pair, but only at the eighth neighbor go to zero for total interaction. The local interatomic force constants drop off really rapidly, the first-neighbor force constants are over 5 times as large as any other force constants. These large first-neighbor interactions are mainly due to the spatial extent of the p-electron wave function around each atom. Since BN is a polar material, when an atom is displaced from its original position it creates a dipole. The macroscopic electric field, caused by the long-range character of the Coulomb forces, contributes to the longitudinal optical phonons in the long-wavelength (q → 0) limit. 3.3. Vibrational properties
.
(5)
Here, τkaα is the displacement vector of kth atom in the ath primitive cell (with translation vector R a ) along α axis. E is the Born– Oppenheimer (BO) total energy surface of the system (electrons plus clamped ions). IFCs offer a convenient way of storing the information in the dynamical matrix, so an adequate description of the ions motion in DFPT is really necessary. The IFC can be decomposed into an electrostatic (Ewald) contribution, which is long ranged, and a “local” contribution which can be attributed covalent bonding. The behavior of the total IFC and of the local contribution as a function of interneighbour distance is shown in Fig. 3. Dynamical matrices have been calculated on a (8 × 8 × 8) reciprocal space fcc grid. Fourier deconvolution on its mesh yields real-space interatomic force constants up to the ninth shell of neighbor. This procedure is equivalent to calculating real-space force constants using an fcc supercell whose linear di-
The calculated phonon dispersion curve at P = 0 GPa and P = 30 GPa are displayed in Fig. 4. The vibrational frequencies of BN were determined at several volumes within the linear response framework. There is no gap between the acoustical and optical phonon branches. This overlap is caused by the almost identical masses of B and N atoms. Our result at P = 0 GPa are in good agreement with experimental data [2] at ambient condition. In particular, the transverse optic (TO) and longitudinal optic (LO) phonon modes at the zone center are found to be 1075 cm−1 and 1310 cm−1 . These phonon modes have been reported with frequencies of 1055 cm−1 and 1305 cm−1 in the recent Raman scattering study [2]. In addition to this agreement, the calculated TO (1030 cm−1 ) and LO (1160 cm−1 ) at X point are close to the experimental value of 900 cm−1 and 1135 cm−1 [2]. The pressure dependence of LO–TO splitting is shown in Fig. 5. In a polar lattice, the splitting of the optical phonon modes is determined by two parameter, Born’s dynamical effective charge
Fig. 3. Real-space interatomic force constants.
H. Wang et al. / Physics Letters A 373 (2009) 2082–2086
Fig. 4. Phonon spectra at zero (solid line) and 30 GPa (dotted line). Diamonds are experimental data from Ref. [2].
2085
Fig. 6. The calculated effective charge vs pressure.
Fig. 5. Pressure dependence of the zone-center optical frequencies (upper panel). The difference of the optical phonon frequencies are plotted in the lower panel.
of the lattice ions and the screening of the Coulomb interaction, which depends on the electronic part of the dielectric constant in the phonon frequency regime. In this work, the phonon frequencies shift to higher energies and LO–TO splitting increases with pressure. The change of the dynamical effective charge under pressure can be determined from the frequencies of the optical phonon using equation [23] Z ∗ 2 = ε0 ε∞ V μ
2 2 ωLO − ωTO ,
(6)
where ε0 is the vacuum permittivity, μ is the reduced mass of an anion–cation pair. V is the available volume per pair, and ω is angular mode frequency. The calculated effective charge vs pressure is shown in Fig. 6. The effective charge is only extremely small decrease under pressure. This indicates that increase of the LO–TO splitting at high pressure is mainly due to the reduction of dielectric constant, which is a signature of strong covalent bonding and the related overall increase of direct optical gaps with pressure. 3.4. Thermodynamic properties The P –V equation of state (EOS) isotherms of BN are depicted in Fig. 7. The shown EOS are determined by fitting the calculated Helmholtz free energy at certain T , given by Eq. (3), to Birch– Murnaghan’s EOS. Fig. 8 shows the pressure as a function of the
Fig. 7. Calculated equation of state of BN for different temperatures. The inset shows the equilibrium volume vs T .
volume for several temperatures. The intersection between the curve and the P = 0 GPa line gives the equilibrium volume V 0 for each temperature. In the inset, we show also volume V as a function of T . We can see that the room temperature value of lattice constant is in better agreement with the experimental one than T = 0 K value. Fig. 8 shows the linear thermal expansion coefficients at several different pressure. The available experimental data from A.S. Glen [5], F. Datchi [4] and F.G. Alexander [19] are also shown in same figure. We see that the effect of the pressure is to reduce the thermal expansion as observed in other materials. The calculated results are good agreement with experimental data of Ref. [5] for at T < 700 K, but the discrepancy is large above this T . Compared with fitting curves of Ref. [19] from experimental data, our calculated values at T < 900 K are good agreement with fitting curves of Ref. [19], but above 900 K, our results are low to fitting curves of Ref. [19]. Our calculated values are lower than experimental data of Ref. [4]. At high temperature, the calculated CTEs have large errors compared with experimental data, and increase with temperature. It is instructive to verify how well the calculated and experimental
2086
H. Wang et al. / Physics Letters A 373 (2009) 2082–2086
0–1600 K, respectively. The calculations were performed employing density functional perturbation theory with the local-density approximation. We calculated the linear thermal expansion coefficient in the quasiharmonic approximation. Our calculated structural parameters, phonon spectra, linear thermal expansion coefficients are generally speaking in good agreement with the available experimental data and other theoretical values. In this work, the calculated phonon frequencies shift to higher energies and LO– TO splitting increase with pressure. The increases of the LO–TO splitting with pressure are mainly caused by the reduction of dielectric constant. The effect of pressure is to reduce the expansion. Although our results are good agreement with experimental linear thermal expansion coefficients, the errors increase with temperature at high temperature. With the increase of temperature, the contribution of lattice anharmonicity can not be neglected. The phonon softing at high temperature can also increase errors. Hence, the QHA becomes increasing less adequate at high temperature, however anharmonic effects decrease with increasing pressure. Fig. 8. Calculated linear thermal expansion coefficients of BN as a function of temperature, at different pressure (solid line). Circles: experimental data of Ref. [5] ( P = 0 GPa). Diamonds and asterisks are experimental data of Ref. [4] (Diamonds: P = 0 GPa, asterisks: P = 20 GPa). Dotted lines: linear thermal expansion coefficient of Ref. [19]. Dotted lines are fits to the experimental data at P = 54 GPa.
CTEs values following the theoretical dependence on other thermodynamical material properties:
α=
γ CV VB
,
(7)
where Grüneisen parameter γ = −d ln ω/d ln V . At high temperature, with the increase of temperature, the contribution of lattice anharmonicity cannot be neglected. Recent measurements of the phonon density of state on GaN [24] and PbS [25] have shown an unusually large thermal softing of phonon frequencies at high temperature. The phonon softing at high temperature comes from the weakening of force constant in a harmonic potential. Another cause of large errors at high temperature is that the LDA underestimates lattice parameters and overestimates bulk modulus, then underestimates thermal expansion. In the QHA, phonon frequencies depend on V , while intrinsic anharmonic effects arising from phonon–phonon interaction are neglected. These effect become more important at elevated T , hence, QHA becomes increasingly less adequate at high temperatures. However, anharmonic effects decrease with increasing pressure. Indeed, the leading harmonic term in the expansion of the potential energy in terms of atomic displacement is expected to become increasing predominant by decreasing V . 4. Conclusion We presented the results of the pressure dependence of lattice dynamics, thermodynamic properties of zinc-blende BN. The considered pressure and temperature ranges are 0–70 GPa and
Acknowledgements We wish to acknowledge the support of the National Natural Science Foundation No. 50271085 and the Hunan Provincial Natural Science Foundation of China No. 05JJ40135. We thank Yao-zhuang Nie for useful discussion. References [1] G.L. Doll, in: Properties of Group III Nitrides, INSPEC, IEE, London, 1994. [2] S. Reich, A.C. Ferrari, R. Arenal, A. Loiseau, I. Bello, J. Robertson, Phys. Rev. B 71 (2005) 205201. [3] V.L. Solozhenko, D. Häusermann, M. Mezouar, Appl. Phys. Lett. 72 (1998) 1691. [4] F. Datchi, A. Dewaele, Y. Le Godec, P. Loubeyre, Phys. Rev. B 75 (2007) 214104. [5] A.S. Glen, S.F. Bartram, J. Appl. Phys. 46 (1975) 89. [6] F.G. Alexander, J.C. Crowhurst, J.K. Dewhurst, Phys. Rev. B 75 (2007) 224114. [7] Y. Zhong, H. Sun, C. Chen, Phys. Rev. B 73 (2006) 144115. [8] R. Pässler, J. Appl. Phys. 101 (2007) 093513. [9] J. Cai, N. Chen, Phys. Rev. B 75 (2007) 134109. [10] N. Ohba, K. Miwa, N. Nagasako, A. Fukumoto, Phys. Rev. B 63 (2001) 115207. [11] H.M. Tütüncü, G.P. Srivastava, Phys. Rev. B 62 (2000) 5028. [12] S. Baroni, P. Giannozzi, A. Testa, Phys. Rev. Lett. 58 (1987) 1861. [13] P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B 43 (1991) 7231. [14] X. Gonze, http://www.abinit.org. [15] X. Gonze, et al., Comput. Mater. Sci. 25 (2002) 478. [16] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Jonannopoulos, Rev. Mod. Phys. 64 (1992) 1045. [17] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [18] E. Knittle, R.M. Wentzcovitsch, R. Jeanloz, M.L. Cohen, Nature (London) 337 (1989) 349. [19] A.F. Goncharov, et al., Phys. Rev. B 75 (2007) 224114. [20] A. Janotti, S.-H. Wei, Phys. Rev. B 64 (2001) 174107. [21] H.M. Tütüncü, S. Baci, G.P. Srivastava, A.T. Albudak, G. Ugur, Phys. Rev. B 71 (2005) 195309. [22] P. Giannozzi, S. de Gironcoli, Phys. Rev. B 43 (1990) 7231. [23] A.R. Goni, H. Siegle, K. Syassen, Phys. Rev. B 64 (2001) 035205. [24] R.K. Kremer, M. Cardona, E. Schmitt, Phys. Rev. B 72 (2005) 075209. [25] M. Cardona, R.K. Kremer, R. Lauck, G. Siegle, Phys. Rev. B 76 (2007) 075211.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
High-pressure lattice dynamics and thermodynamic properties of zinc-blende BN from first-principles calculation HuanYou Wang a,b,∗ , Hui Xu a , XianChun Wang c , ChunZhi Jiang b a b c
Department of Physics, Central South University, Changsha 410083, China Department of Physics, Xiangnan University, Chenzhou 423000, China College of Physics and Electronic Science, Hunan University of Arts and Science, Changde 415000, China
a r t i c l e
i n f o
Article history: Received 6 January 2009 Received in revised form 26 March 2009 Accepted 10 April 2009 Available online 18 April 2009 Communicated by R. Wu PACS: 65.40.De 62.20.Dc 63.20.-e
a b s t r a c t The density function perturbation theory (DFPT) is employed to study the lattice dynamics and thermodynamic properties (with quasiharmonic approximation) of zinc-blende BN. First we discuss the structural properties and compare the phonon spectrum with available Raman scattering experiments. Thereafter using the calculated phonon dispersions we obtain the PTV equation of state from the free energy. Our results for the above properties are generally speaking in good agreement with experiments and with similar theoretical calculations. Owing to the anharmonic effect at high temperature, the calculated linear thermal expansion coefficients (CTE) are low to experimental data. © 2009 Elsevier B.V. All rights reserved.
Keywords: First-principle Lattice dynamics Heat expansion Pressure
1. Introduction Recently, the boron nitride has attracted both scientific and technological attention. Due to its fascinating mechanical properties, such as high hardness, high melting point, high thermal conductivity, large bulk moduli, etc., making it useful for protective coating. Furthermore, it is also attractive for use in field effect transistors intended to operate at high power and/or temperature. Current activities in optoelectronic devices have led to significant interest in studies of electronic, dynamical and thermal properties. In these important properties, thermal properties are one of the most basic properties of any material. Thermal expansion is connected not only with thermal properties (thermal conductivity, specific heat, etc.) but also it influences many other properties, such as the temperature (T) variation of the energy band gap. Moreover, the knowledge of the thermal expansion coefficient is especially important for epitaxial growth. Following above reality, several experimental and theoretical works have been carried out on the structural, vibrational and
*
Corresponding author at: Department of Physics, Central South University, Changsha 410083, China. E-mail address: [email protected] (H. Wang). 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.04.030
thermal properties. Experimentally zone-centre phonon modes have been determined by G.L. Doll [1] by first-order Raman scattering technique. Very recently, zone-edge phonon modes of this material have also been measured by S. Reich [2] by using secondorder Raman scattering technique. Vladimir L. Solozhenko [3] have measured the pressure–volume data at room temperature up to 66 GPa. F. Datchi [4] have measured the equation of state of cubic boron nitride by X-ray diffraction up to 160 GPa at 295 K and 80 GPa in the range 500–900 K. The measurements of thermal expansion coefficient were also performed by A.S. Glen [5] and F.G. Alexander [6], respectively. Several theoretical calculations have been performed to study the dynamical properties [7–11] of zinc-blende BN recently. These methods include valence-force model, rigid-ion model and ab initio calculation, etc. The calculation of thermal properties is rare. In the 1990s, another approach had been made possible by the achievement of DFPT [12,13] which allowed exact calculation of vibrational frequencies in every point of the Brillouin zone. The vibrational free energy can be obtained using the quasiharmonic approximation (QHA) which include anharmonic effects through the explicit volume dependence of the vibrational frequencies. In the Letter we apply DFPT within QHA to the study of the structural, vibrational and thermal properties of zinc-blende BN at high pressures.
H. Wang et al. / Physics Letters A 373 (2009) 2082–2086
2083
2. Theory and computational details 2.1. Theory The knowledge of entire phonon spectrum of a given system enables the calculation of its thermodynamical properties and the relative stability of its different phases as functions of T. The thermodynamical properties are usually determined by the appropriate thermodynamical potential relevant to the given ensemble. The relevant potential, e.g. Helmholtz free energy (F) can be written as F (a, T ) = E tot (a) + K B T
ln 2 sinh
qλ
h¯ ωqλ (a) 2K B T
,
(1)
the equilibrium structure parameters a = (a1 , a2 , . . .) of a crystal at any T are obtained by minimizing F of a system. Where E tot (a) is the ground state (T = 0 K) total energy of the crystal. The next term is vibrational free energy. The electronic entropy contribution vanishes for semiconductor, and thus, it is not include in Eq. (1). Even for metal this contribution is usually neglected, although it is easy to calculate. ωqλ (a) of Eq. (1) is the phonon frequency of the qth phonon mode with branch index λ. The QHA accounts only partially for the effects of anharmonicity. However, QHA is found to be a very good approximation at temperatures not too high. For given T and V , the equilibrium state of a crystal is determined by minimizing F with respect of all possible degrees of freedom. The equation of state ( P versus V ) of the system is obtained by equating P to minus the derivative of F with respect to V at constant T ,
P =−
∂F ∂V
Fig. 1. Calculated total energies per unit cell versus volume.
(2)
. T
Thermal expansion is obtained directly from the equation of state, and the volume thermal expansion coefficient is defined as
αv =
1 V
∂V ∂T
.
(3)
P
The linear thermal expansion coefficients for cubic crystal is given as 1 αa = α v . (4) 3
Fig. 2. Pressure–volume data for zinc-blend BN (solid line) at 0 K. Circles, diamonds and asterisks are experimental data from Refs. [3,4,6], respectively.
2.2. Computational details The lattice dynamics and thermodynamic properties of zincblende BN were performed using the ABINIT codes [14,15]. We use a first-principles pseudopotential method based on DFPT with wave function represented in a plane-wave basis set. A review of the method (and of the algorithm used for the convergence of electronic density and atomic positions) can be found in Ref. [16]. The static energies were computed using density function theory (DFT), and phonon frequencies using DFPT. The interactions between the ions and valence electrons were described using norm-conserving local density approximation (LDA) pseudopotentials which are generated in the scheme of Troullier–Martins [17]. Brillouin-zone integrations were performed using 12×12×12 k-point mesh and phonon frequencies were computed on a 8×8×8 q-point mesh. Plane-wave basis set with a cutoff of 40 Hartree was used. These calculating parameters are chosen to guarantee the total energy error in 0.01 mHartree. 3. Results and analysis 3.1. Structural properties Firstly we determine the equilibrium lattice constant of the ground state of the zinc-blende BN by calculating the total en-
Table 1 Comparison of calculated and measured structural properties of zinc-blende BN: lattice constant a (Å), bulk module B (Mbar) and its pressure derivative B .
Calc Expt (Ref. [4]) Expt (Ref. [18]) Expt (Ref. [19]) Calc (Ref. [20]) Calc (Ref. [21])
a
B
3.590 3.612 3.615 3.6175 3.593 3.58
3.98 3.95 3.69 3.87 3.95 3.80
B 3.6 3.54 4.0 3.06 3.65 3.56
ergy per primitive unit cell as a function of volume. The Birch– Murnaghan’s equation of state (EOS) is then used to fit the calculated energy–volume data. Fig. 1 shows the total energy per unit cell as a function of volume for BN. The obtained structural parameters are compared in Table 1 with the available experimental data and theoretical values. The calculated lattice constant at ground sate is in good agreement with experimental data of E. Knittle [18], F.G. Alexander [19] and F. Datchi [4]. The calculated static bulk moduli is overestimated by about 2% and 8% compared with experimental data [4,18,19]. The present determined value of the bulk modulus is in good agreement with the theoretical results of A. Janotti [20], but higher than result of H.M. Tütüncü [21]. Fig. 2 shows pres-
2084
H. Wang et al. / Physics Letters A 373 (2009) 2082–2086
sure vs volume for zinc-blende BN. Our conclusion is similar to those found in Refs. [3,4,6]. The experimental data in reference are almost equal, and our values are lightly low to those experimental data. Firstly, the underestimate of lattice constants are caused by selection of pseudopotential and the LDA itself. On the other hand, the experimental data are achieved at room temperature and our values neglect thermal expansion. Meanwhile, one should also notice that the experimental values of bulk moduli are somehow uncertain due to the difficult of growing high-quality single crystal. 3.2. Interatomic force constant The interatomic force constants (IFCs) describing the atomic interactions in a crystalline solid are defined in real space as [22] C kα ,k β (a, b) =
∂2 E ∂ τkaα ∂ τkb β
mensions are four times larger than the primitive zinc-blende cell, thus containing 128 atoms. Generally, the decay of local interaction for B–B, B–N, and N–N are faster than those of total interaction. At the forth neighbor the local part go to zero for each species pair, but only at the eighth neighbor go to zero for total interaction. The local interatomic force constants drop off really rapidly, the first-neighbor force constants are over 5 times as large as any other force constants. These large first-neighbor interactions are mainly due to the spatial extent of the p-electron wave function around each atom. Since BN is a polar material, when an atom is displaced from its original position it creates a dipole. The macroscopic electric field, caused by the long-range character of the Coulomb forces, contributes to the longitudinal optical phonons in the long-wavelength (q → 0) limit. 3.3. Vibrational properties
.
(5)
Here, τkaα is the displacement vector of kth atom in the ath primitive cell (with translation vector R a ) along α axis. E is the Born– Oppenheimer (BO) total energy surface of the system (electrons plus clamped ions). IFCs offer a convenient way of storing the information in the dynamical matrix, so an adequate description of the ions motion in DFPT is really necessary. The IFC can be decomposed into an electrostatic (Ewald) contribution, which is long ranged, and a “local” contribution which can be attributed covalent bonding. The behavior of the total IFC and of the local contribution as a function of interneighbour distance is shown in Fig. 3. Dynamical matrices have been calculated on a (8 × 8 × 8) reciprocal space fcc grid. Fourier deconvolution on its mesh yields real-space interatomic force constants up to the ninth shell of neighbor. This procedure is equivalent to calculating real-space force constants using an fcc supercell whose linear di-
The calculated phonon dispersion curve at P = 0 GPa and P = 30 GPa are displayed in Fig. 4. The vibrational frequencies of BN were determined at several volumes within the linear response framework. There is no gap between the acoustical and optical phonon branches. This overlap is caused by the almost identical masses of B and N atoms. Our result at P = 0 GPa are in good agreement with experimental data [2] at ambient condition. In particular, the transverse optic (TO) and longitudinal optic (LO) phonon modes at the zone center are found to be 1075 cm−1 and 1310 cm−1 . These phonon modes have been reported with frequencies of 1055 cm−1 and 1305 cm−1 in the recent Raman scattering study [2]. In addition to this agreement, the calculated TO (1030 cm−1 ) and LO (1160 cm−1 ) at X point are close to the experimental value of 900 cm−1 and 1135 cm−1 [2]. The pressure dependence of LO–TO splitting is shown in Fig. 5. In a polar lattice, the splitting of the optical phonon modes is determined by two parameter, Born’s dynamical effective charge
Fig. 3. Real-space interatomic force constants.
H. Wang et al. / Physics Letters A 373 (2009) 2082–2086
Fig. 4. Phonon spectra at zero (solid line) and 30 GPa (dotted line). Diamonds are experimental data from Ref. [2].
2085
Fig. 6. The calculated effective charge vs pressure.
Fig. 5. Pressure dependence of the zone-center optical frequencies (upper panel). The difference of the optical phonon frequencies are plotted in the lower panel.
of the lattice ions and the screening of the Coulomb interaction, which depends on the electronic part of the dielectric constant in the phonon frequency regime. In this work, the phonon frequencies shift to higher energies and LO–TO splitting increases with pressure. The change of the dynamical effective charge under pressure can be determined from the frequencies of the optical phonon using equation [23] Z ∗ 2 = ε0 ε∞ V μ
2 2 ωLO − ωTO ,
(6)
where ε0 is the vacuum permittivity, μ is the reduced mass of an anion–cation pair. V is the available volume per pair, and ω is angular mode frequency. The calculated effective charge vs pressure is shown in Fig. 6. The effective charge is only extremely small decrease under pressure. This indicates that increase of the LO–TO splitting at high pressure is mainly due to the reduction of dielectric constant, which is a signature of strong covalent bonding and the related overall increase of direct optical gaps with pressure. 3.4. Thermodynamic properties The P –V equation of state (EOS) isotherms of BN are depicted in Fig. 7. The shown EOS are determined by fitting the calculated Helmholtz free energy at certain T , given by Eq. (3), to Birch– Murnaghan’s EOS. Fig. 8 shows the pressure as a function of the
Fig. 7. Calculated equation of state of BN for different temperatures. The inset shows the equilibrium volume vs T .
volume for several temperatures. The intersection between the curve and the P = 0 GPa line gives the equilibrium volume V 0 for each temperature. In the inset, we show also volume V as a function of T . We can see that the room temperature value of lattice constant is in better agreement with the experimental one than T = 0 K value. Fig. 8 shows the linear thermal expansion coefficients at several different pressure. The available experimental data from A.S. Glen [5], F. Datchi [4] and F.G. Alexander [19] are also shown in same figure. We see that the effect of the pressure is to reduce the thermal expansion as observed in other materials. The calculated results are good agreement with experimental data of Ref. [5] for at T < 700 K, but the discrepancy is large above this T . Compared with fitting curves of Ref. [19] from experimental data, our calculated values at T < 900 K are good agreement with fitting curves of Ref. [19], but above 900 K, our results are low to fitting curves of Ref. [19]. Our calculated values are lower than experimental data of Ref. [4]. At high temperature, the calculated CTEs have large errors compared with experimental data, and increase with temperature. It is instructive to verify how well the calculated and experimental
2086
H. Wang et al. / Physics Letters A 373 (2009) 2082–2086
0–1600 K, respectively. The calculations were performed employing density functional perturbation theory with the local-density approximation. We calculated the linear thermal expansion coefficient in the quasiharmonic approximation. Our calculated structural parameters, phonon spectra, linear thermal expansion coefficients are generally speaking in good agreement with the available experimental data and other theoretical values. In this work, the calculated phonon frequencies shift to higher energies and LO– TO splitting increase with pressure. The increases of the LO–TO splitting with pressure are mainly caused by the reduction of dielectric constant. The effect of pressure is to reduce the expansion. Although our results are good agreement with experimental linear thermal expansion coefficients, the errors increase with temperature at high temperature. With the increase of temperature, the contribution of lattice anharmonicity can not be neglected. The phonon softing at high temperature can also increase errors. Hence, the QHA becomes increasing less adequate at high temperature, however anharmonic effects decrease with increasing pressure. Fig. 8. Calculated linear thermal expansion coefficients of BN as a function of temperature, at different pressure (solid line). Circles: experimental data of Ref. [5] ( P = 0 GPa). Diamonds and asterisks are experimental data of Ref. [4] (Diamonds: P = 0 GPa, asterisks: P = 20 GPa). Dotted lines: linear thermal expansion coefficient of Ref. [19]. Dotted lines are fits to the experimental data at P = 54 GPa.
CTEs values following the theoretical dependence on other thermodynamical material properties:
α=
γ CV VB
,
(7)
where Grüneisen parameter γ = −d ln ω/d ln V . At high temperature, with the increase of temperature, the contribution of lattice anharmonicity cannot be neglected. Recent measurements of the phonon density of state on GaN [24] and PbS [25] have shown an unusually large thermal softing of phonon frequencies at high temperature. The phonon softing at high temperature comes from the weakening of force constant in a harmonic potential. Another cause of large errors at high temperature is that the LDA underestimates lattice parameters and overestimates bulk modulus, then underestimates thermal expansion. In the QHA, phonon frequencies depend on V , while intrinsic anharmonic effects arising from phonon–phonon interaction are neglected. These effect become more important at elevated T , hence, QHA becomes increasingly less adequate at high temperatures. However, anharmonic effects decrease with increasing pressure. Indeed, the leading harmonic term in the expansion of the potential energy in terms of atomic displacement is expected to become increasing predominant by decreasing V . 4. Conclusion We presented the results of the pressure dependence of lattice dynamics, thermodynamic properties of zinc-blende BN. The considered pressure and temperature ranges are 0–70 GPa and
Acknowledgements We wish to acknowledge the support of the National Natural Science Foundation No. 50271085 and the Hunan Provincial Natural Science Foundation of China No. 05JJ40135. We thank Yao-zhuang Nie for useful discussion. References [1] G.L. Doll, in: Properties of Group III Nitrides, INSPEC, IEE, London, 1994. [2] S. Reich, A.C. Ferrari, R. Arenal, A. Loiseau, I. Bello, J. Robertson, Phys. Rev. B 71 (2005) 205201. [3] V.L. Solozhenko, D. Häusermann, M. Mezouar, Appl. Phys. Lett. 72 (1998) 1691. [4] F. Datchi, A. Dewaele, Y. Le Godec, P. Loubeyre, Phys. Rev. B 75 (2007) 214104. [5] A.S. Glen, S.F. Bartram, J. Appl. Phys. 46 (1975) 89. [6] F.G. Alexander, J.C. Crowhurst, J.K. Dewhurst, Phys. Rev. B 75 (2007) 224114. [7] Y. Zhong, H. Sun, C. Chen, Phys. Rev. B 73 (2006) 144115. [8] R. Pässler, J. Appl. Phys. 101 (2007) 093513. [9] J. Cai, N. Chen, Phys. Rev. B 75 (2007) 134109. [10] N. Ohba, K. Miwa, N. Nagasako, A. Fukumoto, Phys. Rev. B 63 (2001) 115207. [11] H.M. Tütüncü, G.P. Srivastava, Phys. Rev. B 62 (2000) 5028. [12] S. Baroni, P. Giannozzi, A. Testa, Phys. Rev. Lett. 58 (1987) 1861. [13] P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B 43 (1991) 7231. [14] X. Gonze, http://www.abinit.org. [15] X. Gonze, et al., Comput. Mater. Sci. 25 (2002) 478. [16] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Jonannopoulos, Rev. Mod. Phys. 64 (1992) 1045. [17] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [18] E. Knittle, R.M. Wentzcovitsch, R. Jeanloz, M.L. Cohen, Nature (London) 337 (1989) 349. [19] A.F. Goncharov, et al., Phys. Rev. B 75 (2007) 224114. [20] A. Janotti, S.-H. Wei, Phys. Rev. B 64 (2001) 174107. [21] H.M. Tütüncü, S. Baci, G.P. Srivastava, A.T. Albudak, G. Ugur, Phys. Rev. B 71 (2005) 195309. [22] P. Giannozzi, S. de Gironcoli, Phys. Rev. B 43 (1990) 7231. [23] A.R. Goni, H. Siegle, K. Syassen, Phys. Rev. B 64 (2001) 035205. [24] R.K. Kremer, M. Cardona, E. Schmitt, Phys. Rev. B 72 (2005) 075209. [25] M. Cardona, R.K. Kremer, R. Lauck, G. Siegle, Phys. Rev. B 76 (2007) 075211.