A New Phase of GaN

Fan, Qingyang; Chai, Changchun; Wei, Qun; Yang, Jionghao; Zhou, Peikun; Zhang, Dongyun; Yang, Yintang

doi:https://doi.org/10.1155/2016/8612892

Journal of Chemistry

On this page

Abstract Results and Discussion Conclusions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2016 | Article ID 8612892 | https://doi.org/10.1155/2016/8612892

A New Phase of GaN

Qingyang Fan,¹Changchun Chai,¹Qun Wei,²Jionghao Yang,³Peikun Zhou,⁴Dongyun Zhang,⁵and Yintang Yang¹

Academic Editor: Zhongfang Chen

Received24 Dec 2015

Revised04 Mar 2016

Accepted13 Mar 2016

Published14 Apr 2016

Abstract

The structural, mechanical, and electronic properties of the orthorhombic GaN (Pnma-GaN) are investigated at ambient pressure by using first-principles calculations method with the ultrasoft pseudopotential scheme. The elastic constants and phonon calculations reveal Pnma-GaN is mechanically and dynamically stable at ambient pressure. The calculated Young modulus of Pnma-GaN is 170 GPa, which is the three-fifths of wurtzite-GaN. Electronic structure study shows that Pnma-GaN is a direct semiconductor with band gap of 1.847 eV. The anisotropic calculation shows that wurtzite-GaN has a smaller elastic anisotropy than that of Pnma-GaN in Young’s modulus. In addition, when the composition of aluminum increases from 0 to 0.063 in the alloy, the band gap decreases initially and increases afterward for Pnma-Ga_1−xAl_xN, while, for wurtzite-Ga_1−xAl_xN, the band gap increases with the increasing composition x. Due to the structural porous feature, Pnma-GaN can also be expected to be a good hydrogen storage material.

1. Introductions

A complete acquisition of the physical properties of materials from the macro to the atomic scale is compulsory for device applications. Consequently, theoretical and experimental efforts are beneficial to the study of major properties of main materials and to the search for new materials. Theoretical studies are built on systematic models and computer simulations. One of the most precise microscopic theories in computational materials science is density functional theory (DFT) [1], which sufficiently defines the ground state physical properties of electronic system. It is well known that the GaN single crystal can be considered as the best candidate for wide-gap UV detectors and photodiodes [2, 3]. One of the best ways to improve their technology parameters in our opinion consists of the synthesis of an appropriate solid alloy to continuously vary their band energy gaps creating a continuous set of solid alloy. There have been many published articles of electronic and optical properties in GaN single crystal. The first band energy structure for wurtzite Ga_1−xAl_xN (x = 0.25, 0.50, and 0.75) has been calculated by Rubio et al. [4]. Then, Malachowski et al. [5] have calculated the band energy structure of Ga_1−xAl_xN (x = 0.15 and 0.35) solid alloys. The band structure of the Ga_1−xAl_xN (x = 0.15, 0.52, and 0.65) solid crystalline alloys was calculated by Kityk [6] first using another approach of calculations. This approach includes the use of the orthogonalisation procedure for the plane-wave basis set with respect to the core-like states and the applications of the PA screening procedure; and they found that this approach can be extended to different binary wide-gap semiconductors solid alloys during the estimation of the band energy structure. In [7], a good agreement was predicted with experimental results for the minimum of band gaps in the wurtzite-like Ga_1−xAl_xN structure. For the zinc-blend structure, it was evaluated that GaN will have a direct band gap of 3.1 eV at Γ point in good agreement with recent reported experiments on cubic GaN crystals (3.35 eV) [8]. Miwa and Fukumoto [9] calculated the GaN with wurtzite-like structure and zinc-blend structure using the first-principle pseudopotential calculations. Similar results were concluded in [10, 11].

Recently, we have found that Pnma-GaN is mechanically and dynamically stable at ambient pressure via elastic constants and phonon calculations. Previous work has not systematically investigated the structural and mechanical properties of the new Pnma phase of GaN and their Ga_1−xAl_xN alloys. In this work, we have first detailed the equilibrium crystal structure, mechanical properties, and electronic properties of GaN based on DFT.

2. Computational Methods

The self-consistent calculations were performed using density functional theory (DFT) [1, 12] within the Perdew-Burke-Ernzerhof (PBE) exchange-correlation and employing the generalized gradient approximation (GGA) [13] and the local density approximation (LDA) [14, 15] as implemented in the Cambridge Serial Total Energy Package (CASTEP) code [16]. A plane-wave basis set with energy cut-off of 380 eV was used for Pnma-GaN and their Ga_1−xAl_xN alloys in these calculations. For wurtzite-GaN, we used a plane-wave basis set with energy cut-off 400 eV, which is used with GGA and LDA. The Monkhorst-Pack [17] k-point separation with a grid of 0.025 Å⁻¹ along the three axes in the reciprocal space was used in all calculations. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) [18] minimization scheme was used in geometry optimization. The total energy convergence tests showed that convergence to less than 1 meV/atom was achieved with the above calculation parameters. All energy on atoms converged to less than 1 meV/atom for the geometry relaxation; and all the components of the Hellmann-Feynman stress tensor were reduced to the order of 0.01 GPa at the given pressure. The elastic constants were determined by applying a set of given homogeneous strains with a finite value (0.1%) to the equilibrium lattice and calculating the resulting stress according to optimizing the structure under the strain. The bulk modulus, shear modulus, and Young’s modulus were estimated by employing Voigt-Reuss-Hill approximation [19–21]. The phonons spectra of Pnma-GaN are using the linear response approach, or the so-called density functional perturbation theory (DFPT) is one of the most popular methods of ab initio calculation of lattice dynamics [22].

3. Results and Discussion

The calculated lattice parameters of Pnma-GaN are presented in Table 1. Similarly, it is recognized within DFT that the lattice parameters of Pnma-GaN and wurtzite-GaN calculations are commonly underestimated for both GGA and LDA, while the LDA values are close to the experimental values for wurtzite-GaN, so LDA method can obtain more accurate results. The calculated lattice constants of Pnma-GaN are a = 5.900 Å, b = 3.232 Å, and c = 5.374 Å with LDA. The crystal structure of Pnma-GaN is shown in Figure 1(a). There are eight atoms in conventional cell, including four Ga atoms and four N atoms. There is an equivalent position for Ga or N atoms; that is to say, the four Ga atoms or N atoms are equivalent. Hence, it is reasonable that Al atom can replace any Ga atom. The Ga atoms occupy the crystallographic 4c sites in a conventional cell, which is (−0.1673, 0.2500, and 0.4008); and the N atoms occupy the crystallographic 4c sites in a conventional cell, which is (0.1731, 0.2500, and 0.4016), respectively. The crystal structure of Ga_1−xAl_xN (x = 0.031, 0.063, 0.094, and 0.125) is shown in Figures 1(b), 1(c), 1(d), and 1(e), respectively. In Ga_1−xAl_xN alloys, aluminum atoms substituted position is the lowest energy substitution method. In addition, wurtzite-GaN results were calculated using the same method; together with previous experimental results obtained other calculated results are presented in Table 1. The calculated lattice constants wurtzite-GaN (space group: P6₃mc) are a = b = 3.156 Å (3.225 Å), c = 5.148 Å (5.257 Å), and internal parameter u = 0.376 (0.375) with LDA (GGA), respectively. In addition, the experimental results of wurtzite-GaN are a = b = 3.190 Å, c = 5.189 Å [23], and u = 0.376. The calculated lattice parameters of wurtzite-GaN are in good agreement with the reported calculated results of other theoretical results [23–26], while larger deviations from experimental results are found in both a and c directions 1.10% and 3.11% with GGA, respectively, but smaller deviations are found in both a and c directions 1.07% and 0.79% with LDA, respectively. Due to the structural porous feature (see Figures 1(b)–1(e)), Pnma-GaN can also be expected to be a good hydrogen storage material. At ambient pressure, the enthalpy of Pnma-GaN is slightly greater 0.264 eV per formula unit than that of wurtzite-GaN.

(a) GaN

(b) Ga0.969Al0.031N (2 × 1 × 2)

(c) Ga0.906Al0.063N (2 × 1 × 2)

(d) Ga0.937Al0.094N (2 × 1 × 2)

(e) Ga0.875Al0.125N (2 × 1 × 2)

The elastic properties of a solid are important. They are not only closely related to various fundamental solid-state phenomena such as interatomic bonding, equations of state, and phonon spectra, but also linked thermodynamically with specific heat, thermal expansion, Debye temperature, and Grüneisen parameter. Most importantly, knowledge of the elastic constants is essential for many practical applications related to the mechanical properties of a solid. The criteria for mechanical stability of orthorhombic phase are given by [27]:Obviously, the elastic constants of Pnma-GaN satisfy the generalized elastic stability criteria for a stable orthorhombic symmetry. To ensure the stability of Pnma-GaN, the phonon spectra were calculated at ambient conditions (see Figure 2(a)). There is no imaginary frequency, which means Pnma-GaN can be stable at ambient conditions. Calculated single crystal elastic constants , bulk modulus B, shear modulus G, and Young’s modulus for Pnma and wurtzite phases of GaN are listed in Table 2. The calculated elastic constants of wurtzite-GaN are in excellent agreement with the reported calculated results of other theoretical results [28, 35]. The calculated related enthalpies of Pnma-GaN, wurtzite-GaN, zinc blende-GaN, and NaCl-GaN are shown in Figure 2(b). All of them are smaller than the enthalpy of α-Ga and α-N₂. That is to say, they can be synthesized in the experiment.

(a)

(b)

The Debye temperature () is a fundamental physical property and correlates with many physical properties of solids, such as specific heat and the thermal coefficient [36]. It is used to distinguish between high and low temperature regions for a solid. If the temperature , we expect all modes to have the energy of , and if , one expects the high-frequency modes to be frozen; that is, the vibrational excitations arise solely from the acoustic vibrations. The standard formula to evaluate the Debye temperature is given by [37]:where is Planck’s constant, is Boltzmann’s constant, is the number of atoms per formula unit, and , , and represent Avogadro’s number, molar mass, and density, respectively. The average sound velocity is described by the following expression:The transverse and longitudinal elastic wave velocity, and , can be achieved from Navier’s equation as follows:where and are bulk modulus and shear modulus, respectively. The calculated Debye temperature K is evaluated from (2). The Debye temperature of Pnma-GaN is slightly smaller than that of wurtzite-GaN (600 K).

It is well known that the anisotropy of elasticity is an important implication in engineering science and crystal physics. The directional dependence of Young’s modulus and the projection in , , and planes for Pnma-GaN are illustrated in Figures 3(a) and 3(b). The maximum value of = 221 GPa and the minimal of = 106 GPa. The average value of all directions is 176 GPa, which is very close to CASTEP value (170 GPa; see Table 2). The ratio / = 2.09 shows Pnma-GaN exhibits a large elastic anisotropy. The directional dependence of Young’s modulus and the projection in , , and planes for wurtzite-GaN are shown in Figures 3(c) and 3(d). The maximal value of = 301 GPa and the minimal value of = 235 GPa. Its CASTEP value is 257 GPa, and this value is smaller than the average value of all directions by 1.15% (260 GPa; see Table 2). The ratio of / is 1.28; it shows that wurtzite-GaN exhibits a smaller elastic anisotropy than that of Pnma-GaN.

(a) Pnma

(b) Pnma

(c) P63mc

(d) P63mc

The electronic band structure of Pnma-GaN is shown in Figure 4(a). The dashed line represents the Fermi level (). The coordinate of high symmetry points in Brillouin zone for Pnma-GaN is , , , , , , , and . From the electronic band structure, it is shown that Pnma-GaN is a direct semiconductor and its band gap is 1.567 eV. It is known that the calculated band gap with DFT is usually underestimated by 30–50%, or even more; the true band gap must be larger than the calculated results. The valence band maximum and conduction band minimum are both at point. The total density of states and the partial wave density of states projected onto different atoms at ambient pressure are also shown in Figure 4. The main features of Pnma-GaN can be illustrated as follows: (i) the part present in the energy part (2.5 to 12.5 eV) of the DOS is mainly due to contributions from the electrons and electrons of Ga atom, and the contribution of the N atom orbital is the smallest; (ii) the states from −7.5 eV to Fermi energy mainly originate from electrons of N atom and with slight contributions from electrons of N atom; (iii) the peak present in the lower energy part (−17.5 to −10 eV) of the DOS is mainly due to contributions from the electrons of N, and the contribution from N- is very small.

(a)

(b)

The calculated band structures of Ga_1−xAl_xN (x = 0.031, 0.063, 0.094, and 0.125) for Pnma and wurtzite phases are plotted in Figures 5(a)–5(d) and 5(e)–5(h), respectively. The coordinate of high symmetry points in Brillouin zone for Pnma-Ga_1−xAl_xN is , , , and . The dashed line also represents the Fermi level (). Calculating the band structures of Ga_1−xAl_xN, aluminum atoms substituted position is the lowest energy substitution method. The values of band gap as a function of concentration for Pnma-Ga_1−xAl_xN and wurtzite-Ga_1−xAl_xN are listed in Table 3. Similarly, it is recognized within DFT that the energy band gap of electronic structure calculations is commonly underestimated for both GGA and LDA, while the LDA values are close to the experimental values for wurtzite-GaN, so we show the band structures of Pnma-Ga_1−xAl_xN and wurtzite-Ga_1−xAl_xN in Figure 5 with LDA. This is generally due to the circumstance that they have sample forms that are not adequately flexible to precisely reproduce both the exchange-correlation energy and equivalent potential [31]. The electronic energy band structure of Ga_0.969Al_0.031N, Ga_0.937Al_0.063N, Ga_0.906Al_0.094N, and Ga_0.875Al_0.125N for Pnma phase shows that they are direct semiconductors and its band gap is 1.740, 1.640, 1.710, and 1.741 eV, respectively. Furthermore, the electronic energy band structure of Ga_0.969Al_0.031N, Ga_0.937Al_0.063N, Ga_0.906Al_0.094N, and Ga_0.875Al_0.125N for wurtzite phase shows that they are direct semiconductors and its band gap is 2.217, 2.327, 2.489, and 2.648 eV, respectively. With the same composition x, the band gap of Ga_1−xAl_xN alloy for Pnma phase is smaller than that of wurtzite phase. With the increasing composition x, the band gap decreases initially and increases afterward for Pnma-Ga_1−xAl_xN, while, for wurtzite-Ga_1−xAl_xN, the band gap increases with the composition x increasing.

(a) Ga0.969Al0.031N band structure (Pnma)

(b) Ga0.937Al0.063N band structure (Pnma)

(c) Ga0.906Al0.094N band structure (Pnma)

(d) Ga0.875Al0.125N band structure (Pnma)

(e) Ga0.969Al0.031N band structure (P63mc)

(f) Ga0.937Al0.063N band structure (P63mc)

(g) Ga0.906Al0.094N band structure (P63mc)

(h) Ga0.875Al0.125N band structure (P63mc)

4. Conclusions

This study reports a detailed investigation of the structural and electronic properties of Pnma-GaN/Ga_1−xAl_xN semiconductor alloy using the DFT within the ultrasoft pseudopotential scheme in the frame of LDA and GGA. The elastic constants and phonon calculations reveal Pnma-GaN is mechanically and dynamically stable at ambient pressure and its Debye temperature is 510.55 K. Pnma-GaN is a direct semiconductor and its band gap is 1.847 eV within LDA at ambient pressure. The elastic anisotropy properties show that Pnma-GaN exhibits a large anisotropy in its Young’s modulus. When the composition of aluminum increases from 0 to 0.125 in the alloys, the band gap decreases initially and increases afterwards for Pnma-Ga_1−xAl_xN, while, for wurtzite-Ga_1−xAl_xN, the band gap increases as the composition x increases. Due to the structural porous feature, Pnma-GaN can also be expected to be a good hydrogen storage material.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the Natural Science Foundation of China (no. 61474089) and Open Fund of Key Laboratory of Complex Electromagnetic Environment Science and Technology, China Academy of Engineering Physics (no. 2015-0214. XY.K).

References

W. Kohn and L. J. Sham, “Self-consistent equations including exchange and correlation effects,” Physical Review A, vol. 140, pp. A1133–A1138, 1965.
View at: Google Scholar | MathSciNet
E. Przeździecka, K. Gościński, M. Stachowicz et al., “Spectrum selective UV detectors from an p-ZnO:As/n-GaN diodes grown by Molecular Beam Epitaxy,” Sensors and Actuators A: Physical, vol. 195, pp. 27–31, 2014.
View at: Publisher Site | Google Scholar
X. D. Wang, W. D. Hu, M. Pan et al., “Study of gain and photoresponse characteristics for back-illuminated separate absorption and multiplication GaN avalanche photodiodes,” Journal of Applied Physics, vol. 115, Article ID 013103, 2014.
View at: Publisher Site | Google Scholar
A. Rubio, J. L. Corkill, and M. L. Cohen, “Quasiparticle band structures of short-period superlattices and ordered alloys of AlN and GaN,” Physical Review B, vol. 49, no. 3, pp. 1952–1956, 1994.
View at: Publisher Site | Google Scholar
M. Malachowski, I. R. Kityk, and B. Sahraoui, “Electronic structure and optical response in Ga_xAl_1−xN solid alloys,” Physics Letters A, vol. 242, no. 6, pp. 337–342, 1998.
View at: Publisher Site | Google Scholar
I. Kityk, “Band structure of wurtzite-like Ga_xAl_1−xN solid state alloys,” Materials Letters, vol. 57, no. 12, pp. 1798–1804, 2003.
View at: Publisher Site | Google Scholar
M. Malachowski, I. R. Kityk, and B. Sahraoui, “Band structure and optical response in Ga_xAl_1-xN,” Physica Status Solidi B, vol. 207, no. 2, pp. 405–418, 1998.
View at: Publisher Site | Google Scholar
H. Okumura, S. Yoshida, and T. Okahisa, “Optical properties near the band gap on hexagonal and cubic GaN,” Applied Physics Letters, vol. 64, no. 22, pp. 2997–2999, 1994.
View at: Publisher Site | Google Scholar
K. Miwa and A. Fukumoto, “First-principles calculation of the structural, electronic, and vibrational properties of gallium nitride and aluminum nitride,” Physical Review B, vol. 48, no. 11, pp. 7897–7902, 1993.
View at: Publisher Site | Google Scholar
J. L. P. Hughes, Y. Wang, and J. E. Sipe, “Calculation of linear and second-order optical response in wurtzite GaN and AlN,” Physical Review B, vol. 55, no. 20, pp. 13630–13640, 1997.
View at: Publisher Site | Google Scholar
Y. Xu and W. Y. Ching, “Electronic, optical, and structural properties of some wurtzite crystals,” Physical Review B, vol. 48, article 4335, 1993.
View at: Publisher Site | Google Scholar
P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Physical Review Letters, vol. 136, no. 3, p. B864, 1964.
View at: Publisher Site | Google Scholar
J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Physical Review Letters, vol. 77, no. 18, pp. 3865–3868, 1996.
View at: Publisher Site | Google Scholar
J. P. Perdew and A. Zunger, “Self-interaction correction to density-functional approximations for many-electron systems,” Physical Review B, vol. 23, no. 10, pp. 5048–5079, 1981.
View at: Publisher Site | Google Scholar
D. M. Ceperley and B. J. Alder, “Ground state of the electron gas by a stochastic method,” Physical Review Letters, vol. 45, no. 7, pp. 566–569, 1980.
View at: Publisher Site | Google Scholar
S. J. Clark, M. D. Segall, C. J. Pickard et al., “First principles methods using CASTEP,” Zeitschrift für Kristallographie, vol. 220, no. 5-6, pp. 567–570, 2005.
View at: Publisher Site | Google Scholar
H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integrations,” Physical Review B: Solid State, vol. 13, no. 12, pp. 5188–5192, 1976.
View at: Publisher Site | Google Scholar | MathSciNet
B. G. Pfrommer, M. Côté, S. G. Louie, and M. L. Cohen, “Relaxation of crystals with the quasi-newton method,” Journal of Computational Physics, vol. 131, no. 1, pp. 233–240, 1997.
View at: Publisher Site | Google Scholar
W. Voigt, Lehrburch der Kristallphysik, Teubner, Leipzig, Germany, 1928.
A. Reuss, “Berechnung der fließgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle,” Zeitschrift für Angewandte Mathematik und Mechanik, vol. 9, no. 1, pp. 49–58, 1929.
View at: Publisher Site | Google Scholar
R. Hill, “The elastic behaviour of a crystalline aggregate,” Proceedings of the Physical Society, Section A, vol. 65, no. 5, pp. 349–354, 1952.
View at: Publisher Site | Google Scholar
S. Baroni, S. de Gironcoli, A. dal Corso, and P. Giannozzi, “Phonons and related crystal properties from density-functional perturbation theory,” Reviews of Modern Physics, vol. 73, no. 2, pp. 515–562, 2001.
View at: Publisher Site | Google Scholar
H. Schulz and K. H. Thiemann, “Crystal structure refinement of AlN and GaN,” Solid State Communications, vol. 23, no. 11, pp. 815–819, 1977.
View at: Publisher Site | Google Scholar
Y. Duan, G. Tang, L. Qin, and L. Shi, “Elasticity and polarization of ${Ga}_{x} {Al}_{1 - x} N$ alloys subjected to uniaxial and biaxial compression,” The European Physical Journal B, vol. 66, no. 2, pp. 211–215, 2008.
View at: Publisher Site | Google Scholar
F. Bernardini, V. Fiorentini, and D. Vanderbilt, “Spontaneous polarization and piezoelectric constants of III-V nitrides,” Physical Review B—Condensed Matter and Materials Physics, vol. 56, no. 16, pp. R10024–R10027, 1997.
View at: Publisher Site | Google Scholar
J. M. Wagner and F. Bechstedt, “Properties of strained wurtzite GaN and AlN: Ab initio studies,” Physical Review B, vol. 66, no. 11, Article ID 115202, 2002.
View at: Publisher Site | Google Scholar
J. F. Nye, Physical Properties of Crystals, Oxford University Press, 1985.
O. Madelung, Semiconductors: Data Handbook, Springer, Berlin, Germany, 2004.
View at: Publisher Site
T. Deguchi, D. Ichiryu, K. Toshikawa et al., “Structural and vibrational properties of GaN,” Journal of Applied Physics, vol. 86, no. 4, pp. 1860–1866, 1999.
View at: Publisher Site | Google Scholar
M. Trivedi and K. Shenai, “Performance evaluation of high-power wide band-gap semiconductor rectifiers,” Journal of Applied Physics, vol. 85, no. 9, pp. 6889–6897, 1999.
View at: Publisher Site | Google Scholar
S. Strite, “GaN, AlN, and InN: a review,” Journal of Vacuum Science & Technology B, vol. 10, article 1237, 1992.
View at: Publisher Site | Google Scholar
H. Morkoç, S. Strite, G. B. Gao, M. E. Lin, B. Sverdlov, and M. Burns, “Large-band-gap SiC, III-V nitride, and II-VI ZnSe-based semiconductor device technologies,” Journal of Applied Physics, vol. 76, no. 3, pp. 1363–1398, 1994.
View at: Publisher Site | Google Scholar
V. Bougrov, M. E. Levinshtein, S. L. Rumyantsev, and A. Zubrilov, Properties of Advanced Semiconductor Materials GaN, AlN, InN, BN, SiC, SiGe, John Wiley & Sons, New York, NY, USA, 2001.
T. P. Chow and M. Ghezzo, “SiC power devices,” in III-Nitride, SiC, and Diamond Materials for Electronic, D. K. Gaskill, C. D. Brandt, and R. J. Nemanich, Eds., vol. 423 of MRS Symposium Proceedings, pp. 69–73, Materials Research Society, Pittsburgh, Pa, USA, 1996.
View at: Publisher Site | Google Scholar
S. P. Wan, J. B. Xia, and K. Chang, “Effects of piezoelectricity and spontaneous polarization on electronic and optical properties of wurtzite III-V nitride quantum wells,” Journal of Applied Physics, vol. 90, no. 12, article 6210, 2001.
View at: Publisher Site | Google Scholar
Y.-J. Hao, X.-R. Chen, H.-L. Cui, and Y.-L. Bai, “First-principles calculations of elastic constants of c-BN,” Physica B: Condensed Matter, vol. 382, no. 1-2, pp. 118–122, 2006.
View at: Publisher Site | Google Scholar
O. L. Anderson, “A simplified method for calculating the debye temperature from elastic constants,” Journal of Physics and Chemistry of Solids, vol. 24, no. 7, pp. 909–917, 1963.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2016 Qingyang Fan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

1854

Downloads

1120

Citations