Two –Temperature Model (Chap 7.1.3) 6 Dec 2011 Injoo Hwang

Contents 1. Two – Temperature Model 1 2. Conclustion 2

Two-Temperature Model A pair of coupled nonlinear equations governing the effective temperatures of electrons and phonons (by S.I. Anisimov et al. 1974) Assumed in the Two – Temperature Model Electron System Phonon System Mutually Equilibrium Equilibrium Non-Equilibrium

Two-Temperature Model : Electron System : Phonon System : the volumetric heat capacity : the electron – phonon coupling constant : the source term ( during the laser pulse ) : the thermal conductivity ( ∵ heat conduction by phonons is neglected )

Two-Temperature Model Examining parameters - The volumetric heat capacity - The coupling constant or - The thermal conductivity G is independent of temperature and proportional to the square of the speed of sound in the metal

Two-Temperature Model The associated electron and phonon temperatures near the surface Ultrafast thermoreflectance experiments Pump beam Thermoreflec-tance signal Probe beam Specimen The electron temperature rises quickly during the pulse and begins to decrease afterward The lattice temperature gradually increases until the electron and lattice systems reach a thermal equilibrium Measuring the effective electron temperature by no contact thermometer which performs the femtosecond or picosecond thermoreflectrance technique

Two-Temperature Model Differential equations for the electron temperature

Two-Temperature Model Differential equations for the phonon temperature Identical to the lagging heat equations Following the general trends τq : Thermalization time (thermal time constant for the electron system to reach an equilibrium with the phonon system)

Two-Temperature Model - For noble metals at room temperature - Relaxation time : τ ~ 30 to 40 fs Thermalization time : τq ~ 0.5 to 0.8 ps Retatdation time : τT ~ 60 to 90 ps The two-temperature model cannot be applied to t < τ ( the limitation of Fourier’s law ) - Difficult issues - The processes below 20 fs (relaxation time of Cr is about 3 fs) Electron – electron inelastic scattering Thermionic emission, ionization, phase transformation, chemical reaction The reduced pulse width include widened frequency spectrum Increased pulse intensity, decreased pulse energy

Conclusion Two – Temperature Model Examine parameters (their own local equilibrium not in mutual equilibrium) Examine parameters (the volumertic heat capacity, the thermal conductivity, the coupling constant) The behavior of the electron and phonon temperatures Non contact thermometer ( thermoreflctance technique known as pump-and-probe method) Derive PDE for the electron or phonon temperature

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