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Coefficients of thermal expansion of thin metal films investigated using the statistical moment method

The coefficients of thermal expansion of thin metal film with face-centered cubic structure at zero pressure are investigated using the statistical moment method (SMM), including the anharmonic effects of thermal lattice vibration. The Helmholtz free energy, mean-square atomic displacement and linear thermal expansion coefficients are derived in closed analytic forms in terms of the power moments of the atomic displacement. | JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci. 2013 Vol. 58 No. 7 pp. 109-116 This paper is available online at http stdb.hnue.edu.vn COEFFICIENTS OF THERMAL EXPANSION OF THIN METAL FILMS INVESTIGATED USING THE STATISTICAL MOMENT METHOD Duong Dai Phuong1 Vu Van Hung2 and Nguyen Thi Hoa3 1 Tank Armour Officers Training School Tam Duong Vinh Phuc 2 Viet Nam Education Publishing House Hanoi 3 Hanoi University of Transport and Communications Abstract. The coefficients of thermal expansion of thin metal film with face-centered cubic structure at zero pressure are investigated using the statistical moment method SMM including the anharmonic effects of thermal lattice vibration. The Helmholtz free energy mean-square atomic displacement and linear thermal expansion coefficients are derived in closed analytic forms in terms of the power moments of the atomic displacement. Numerical calculations for Al Au and Ag thin films are found to be in good and reasonable agreement with other theoretical results and with the experimental data. Keywords Coefficients of thermal expansion statistical moment method anharmonicity thin film. 1. Introduction Materials in the form of thin film have come under the spotlight in recent years. They have been found to show physical chemical and mechanical properties that differ from corresponding bulk materials 3 4 . Properties limitations and advantages obtained on thin film geometry have been widely studied and they were found to depend on different factors structure size film thickness and different substrates 3 5 . Thin film is used in a vast range of applications which include microelectromechanical and nanoelectrome-chanical systems sensors and electronic textiles 1 . Metallic thin film displays a common geometry and presents enormous scientific interest mainly due to their attractive and novel properties for technological applications 1 . In the early twentieth century the theory of size-dependent effects in thin metal layers was