Thermal behaviour of metal films ± a hyperbolic two-step model

A perturbation technique is proposed for solution of the generalized equations governing the thermal behaviour of thin metal ®lms described by a hyperbolic two-step model. The generalized equations of this model contains diffusion terms in both the electron and lattice energy equations and assumes that incident laser radiation is absorbed by both the electron gas and solid lattice to account for the thermal behaviour of semiconducting and impure metals. A perturbation technique is utilized to eliminate the coupling between the electron and phonon energy equations when the normalized temperature difference between electrons and phonons is a small quantity, which is true in materials exhibit high coupling factors. List of symbols C heat capacity Jmÿ3 Kÿ1 CR heat capacity ratio, Ce=Cl G electron±phonon coupling factor, Wmÿ3 Kÿ1 G Green's function h Planck constant, J s kB Boltzmann constant, J K ÿ1 K thermal conductivity Wmÿ1 Kÿ1 KR thermal conductivity ratio, Ke=Kl L ®lm thickness, m m effective mass of electrons, kg ne electron number density per unit volume 1/m 3 na atomic density per unit volume, 1/m 3 N number of atoms in the solid Pe dimensionless heat source in the electron gas, SeL =TiKe Pl dimensionless heat source in the lattice, SlL CR=TiKe q conduction heat ̄ux, Wmÿ2 Q dimensionless conduction heat ̄ux, qL=TiKe Se volumetric heat source in the electron gas, Wm ÿ3 Sl volumetric heat source in the lattice, Wm ÿ3 t time, s T temperature, K TD Debye temperature, K Ti initial temperature of both lattice and electron gas, K vs speed of sound, m s ÿ1 x spatial coordinate, m X dimensionless spatial coordinate, x=L Greek symbols d Dirac's delta function D difference function dimensionless small parameter h dimensionless temperature, …T ÿ Ti†=Ti s dimensionless time, tKe=…LCe† Subscripts e electron i initial l lattice