Thermodynamic Properties of Copper in a Wide Range of Pressure and Temperature Within the Quasi-Harmonic Approximation
Tóm tắt
A first-principles calculations have been performed based on density functional theory to study the thermal properties of copper. Calculations have been performed using the pseudo-potential method within the generalized gradient approximation (GGA) and local density approximation. Thermodynamic properties including the bulk modulus, thermal expansion coefficient, and heat capacities at constant volume and constant pressure were calculated as a function of pressure and temperature using three different models based on the quasi-harmonic approximation: the Debye–Slater model, the Debye–Grüneisen model, and the full quasi-harmonic model (that requires the phonon density of states at each calculated volume). Also, empirical energy corrections are applied to the results of the three models. The electronic contributions to the specific heat are calculated and discussed, and it was found that they become important at high temperatures. The calculated values are in good agreement with experimental results. It is found that the full quasi-harmonic model with the GGA approximation provides more accurate estimates in comparison with the other models.
Tài liệu tham khảo
R.K. Kirby, T.A. Hahn, in NBS Certificate Standard Reference Material 736, Copper—Thermal Expansion (National Bureau of Standards, Gaithersburg, Washington, DC 1969)
T.A. Hahn, J. Appl. Phys. 41, 5096 (1970)
S.J. Bennett, J. Phys. D 11, 777 (1978)
J.L. Verhaeghe, G.G. Robbrecht, E. Van Outryve, J. Phys. 23, 109 (1962)
A.F. Pojur, B. Yates, J. Phys. E 6, 63 (1973)
R.O. Simmons, R.W. Balluffi, Phys. Rev. 129, 1533 (1963)
F.C. Nix, D. MacNair, Phys. Rev. 60, 597 (1941)
O. Prakash, A. Rao, P.N. Dheer, Pramana 39, 655 (1992)
D. Bijl, H. Pullan, Physica 21, 285 (1955)
R.F. Cooper, B. Yates, Cryogenics 10, 442 (1970)
T. Rubin, H.W. Altman, H.L. Johnston, J. Am. Chem. Soc. 76, 5289 (1954)
G.V. Bunton, S. Weintroub, Cryogenics 8, 354 (1968)
G.K. White, J.G. Collins, J. Low Temp. Phys. 7, 43 (1972)
K.O. McLean, C.A. Swenson, C.R. Case, J. Low Temp. Phys. 7, 76 (1972)
F.N.D.D. Perierat, G.M. Graham, J. Low Temp. Phys. 9, 203 (1972)
W.C. Overton, J. Gaffney, Phys. Rev. 98, 969 (1955)
G.K. White, S.J. Collocott, J. Phys. Chem. Ref. Data 13, 1251 (1984)
D.L. Martin, Can. J. Phys. 38, 17 (1960)
R.A. Robie, B.S. Hemingway, W.H. Wilson, J. Res. US Geol. Surv. 4, 631 (1976)
F. Jona, P.M. Marcus, Phys. Rev. B 63, 094113 (2001)
F. Jona, X.Z. Ji, P.M. Marcus, Phys. Rev. B 68, 172101 (2003)
C. Bercegeay, S. Bernard, Phys. Rev. B 72, 214101 (2005)
Z. Tang, M. Hasegawa, Y. Nagai, M. Saito, Phys. Rev. B 65, 195108 (2002)
Y.B. Liu, X.S. Li, Y.L. Feng, Y.L. Cui, X. Han, Physica B 394, 14 (2007)
J.C. Slater, Introduction to Chemical Physics (McGraw-Hill, New York, 1939)
V.L. Moruzzi, J.F. Janak, K. Schwarz, Phys. Rev. B 37, 790 (1988)
A. Otero-de-la-Roza, D. Abbasi-P’erez, V. Lauña, Comput. Phys. Commun. 182, 2232 (2011)
Y. Lu, P. Zhang, Comput. Mater. Sci. 82, 5 (2014)
X. Zhang, C. Ying, S. Quan, G. Shi, Z. Li, Comput. Mater. Sci. 58, 12 (2012)
P. Carrier, R. Wentzcovitch, J. Tsuchiya, Phys. Rev. B 76, 064116 (2007)
S. Laref, A. Laref, Comput. Mater. Sci. 51, 135 (2012)
A. Togo, L. Chaput, I. Tanaka, G. Hug, Phys. Rev. B 81, 174301 (2010)
J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46, 6671 (1992)
J.P. Perdew, A. Zunger, Phys. Rev. B 23, 5048 (1981)
P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G.L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A.P. Seitsonen, A. Smogunov, P. Umari, R.M. Wentzcovitch, J. Phys. Condens. Matter. 21, 395502 (2009)
D. Vanderbilt, Phys. Rev. B 41, 7892 (1990)
J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996)
H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13, 5188 (1976)
M. Methfessel, A.T. Paxton, Phys. Rev. B 40, 3616 (1989)
A. Abdollahi, Physica B 410, 57 (2013)
Y. Wang, Z.K. Liu, L.Q. Chen, Acta Mater. 52, 2665 (2004)
F. Birch, Phys. Rev. 71, 809 (1947)
Z. Guo, C.S.S.R. Kumar, L.L. Henry, E.E. Doomes, J. Hormes, E.J. Podlaha, J. Electrochem. Soc. 152, D1 (2005)
C. Kittel, Introduction to Solid State Physics, 5th edn. (Wiley, New York, 1976)