# Two-temperature models

During laser-metal interaction, the laser energy is first deposited into electrons on the metal surface, where two competing processes occur (Hohlfeld, 2000). One is ballistic motion of the excited electrons into deeper parts of the metal with velocity close to the Fermi velocity (~106 m/s). Another is collision between the excited electrons and electrons around the Fermi level – an electron temperature is defined upon establishment of equilibrium among hot electrons. These hot electrons are then diffused into deeper parts of the electron gas at a speed (<104 m/s) much lower than that of the ballistic motion. Meanwhile, the hot electrons are cooled by transferring their energy to the lattice through electron-phonon coupling. The nonequilibrium between electrons and lattice has been observed experimentally (Eesley, 1986; Elsayed-Ali et al., 1987) and can be described by the two-temperature model, which was originally proposed by Anisimov et al. (Anisimov et al., 1974) and rigorously derived by Qiu and Tien (1993) from the Boltzmann transport equation.

## Parabolic Two-Step Model

Assuming heat conduction in the electron can be described by Fourier’s law and neglecting heat conduction in the lattice, the energy equations of the free electrons and lattices (phonons) are

${C_e}\frac{{\partial {T_e}}}{{\partial t}} = \nabla \cdot \left( {{k_e}\nabla {T_e}} \right) - G\left( {{T_e} - {T_l}} \right) + q''' \qquad \qquad(1)$
${C_l}\frac{{\partial {T_l}}}{{\partial t}} = G\left( {{T_e} - {T_l}} \right) \qquad \qquad(2)$

where the volumetric heat capacity of lattice is Cl = ρcp, and the volumetric heat capacity of electrons is

${C_e} = \frac{{{\pi ^2}{n_e}{k_B}}}{{2{\mu _F}}}{T_e} = {B_e}{T_e} \qquad \qquad(3)$

where ne is the number density of electrons, kB is the Boltzmann constant, and F is Fermi energy. Equation (3) indicates that the volumetric heat capacity of the electron is proportional to the electron temperature. It should be noted that the volumetric heat capacity of electrons is much less than that of the lattice even at very high electron temperature. At nonequilibrium condition, thermal conductivity of the electrons depends on the temperatures of both electrons and lattice, i.e.,

${k_e} = {k_{eq}}\left( {\frac{{{T_e}}}{{{T_l}}}} \right) \qquad \qquad(4)$

where keq(T) is the thermal conductivity of the electron when the electrons and lattice are in thermal equilibrium. The electron-lattice coupling factor, G, is to account for the rate of energy exchange between electrons and phonons and it can be estimated by

$G = \frac{9}{{16}}\frac{{{n_e}k_B^2T_D^2{v_F}}}{{\Lambda ({T_l}){T_l}{\mu _F}}} \qquad \qquad(5)$

where TD is Debye temperature, vF is Fermi velocity, and Λ is the electron mean free path. Neglecting conduction in the lattice is justified by the fact that the thermal conductivity of the lattice is two orders of magnitude smaller than that of the free electrons (Klemens and Williams, 1986). The heat conduction model represented by eqs. (1) and (2) is referred to as a parabolic two-step model because Fourier’s law was used to describe heat transfer in the electron gas.

Assuming all properties of electrons and lattice are independent from temperatures, one can obtain a single energy equation for lattice temperature by combining eqs. (1) and (2). Solving for Te from eq. (2) yields

${T_e} = {T_l} + \frac{{{C_l}}}{G}\frac{{\partial {T_l}}}{{\partial t}} \qquad \qquad(6)$

Substituting eq. (6) into eq. (1), we have

${C_e}\frac{\partial }{{\partial t}}\left( {{T_l} + \frac{{{C_l}}}{G}\frac{{\partial {T_l}}}{{\partial t}}} \right) = {k_e}{\nabla ^2}\left( {{T_l} + \frac{{{C_l}}}{G}\frac{{\partial {T_l}}}{{\partial t}}} \right) - G\left( {\frac{{{C_l}}}{G}\frac{{\partial {T_l}}}{{\partial t}}} \right) + q''' \qquad \qquad(7)$

which can be rearranged as

$\frac{{{C_e} + {C_l}}}{{{k_e}}}\frac{{\partial T}}{{\partial t}} + \frac{{{C_e}{C_l}}}{{G{k_e}}}\frac{{{\partial ^2}T}}{{\partial {t^2}}} = {\nabla ^2}{T_l} + \frac{{{C_l}}}{G}\frac{\partial }{{\partial t}}({\nabla ^2}T) + \frac{{q'''}}{{{k_e}}} \qquad \qquad(8)$

where the subscript l for lattice has been dropped. Comparing eq. (8) with the energy equation for the from dual-phase lag model:

$\frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} + \frac{{{\tau _q}}}{\alpha }\frac{{{\partial ^2}T}}{{\partial {t^2}}} = {\nabla ^2}T + {\tau _T}\frac{\partial }{{\partial t}}({\nabla ^2}T) + \frac{1}{k}\left( {q''' + {\tau _q}\frac{{\partial q'''}}{{\partial t}}} \right)$

it is apparent that they have almost identical form except the partial derivative of heat source with respective to time is not present in eq. (8). The thermophysical properties in the dual-phase lag model are related to the properties appearing in the two-temperature model by

$k = {k_e},{\rm{ }}\alpha = \frac{{{k_e}}}{{{C_e} + {C_l}}},{\rm{ }}{\tau _T} = \frac{{{C_l}}}{G},{\rm{ }}{\tau _q} = \frac{{{C_e}{C_l}}}{{G({C_e} + {C_l})}} \qquad \qquad(9)$

The ratio of two phase-lag times is

$\frac{{{\tau _T}}}{{{\tau _q}}} = \frac{{{C_e} + {C_l}}}{{{C_e}}} = 1 + \frac{{{C_l}}}{{{C_e}}} \qquad \qquad(10)$

which indicates that τT is always greater than τq.

## Hyperbolic Two-Step Model

If we consider the hyperbolic effect on the conduction in the electron gas, the energy equation for the electron gas is

${C_e}\frac{{\partial {T_e}}}{{\partial t}} = - \nabla \cdot {\mathbf{q''}} - G\left( {{T_e} - {T_l}} \right) + q''' \qquad \qquad(11)$

where

${\mathbf{q''}} + {\tau _e}\frac{{\partial {\mathbf{q''}}}}{{\partial t}} = - {k_e}\nabla {T_e} \qquad \qquad(12)$

while the energy equation for the lattice is still eq. (2). Equations (11) and (12) can be combined to yield

$\begin{array}{l} {C_e}\frac{{\partial {T_e}}}{{\partial t}} + {C_e}{\tau _e}\frac{{{\partial ^2}{T_e}}}{{\partial {t^2}}} = \nabla \cdot ({k_e}\nabla {T_e}) \\ - G\left( {{T_e} - {T_l}} \right) - {\tau _e}\frac{\partial }{{\partial t}}\left[ {G\left( {{T_e} - {T_l}} \right)} \right] + q''' + {\tau _e}\frac{{\partial q'''}}{{\partial t}} \\ \end{array} \qquad \qquad(13)$

The conduction model represented by eqs. (13) and (2) is referred to as a hyperbolic two-step model. Qiu and Tien (1993) simulated picosecond laser-metal interaction and concluded that the parabolic two-step model predicted the general temperature response but it failed to predict the finite speed of energy propagation. Therefore, the hyperbolic two-step model can provide better accuracy for ultrafast laser-metal interaction.

## Dual-Parabolic Two-Step Model

The contribution of heat conduction in phonons was neglected in the above two models. If it is assumed that the heat conduction in the phonons can be modeled using the classical Fourier’s law, the energy equations of the lattices (phonons) are

${C_l}\frac{{\partial {T_l}}}{{\partial t}} = \nabla \cdot \left( {{k_l}\nabla {T_l}} \right) + G\left( {{T_e} - {T_l}} \right) \qquad \qquad(14)$

The bulk thermal conductivity of metal measured at equilibrium, keq, is the sum of electron thermal conductivity, ke, and the lattice thermal conductivity, kl. Since the mechanism for heat conduction in metal is diffusion of free electron, ke is usually dominant. For gold, ke is 99% of keq, while kl only contributes to 1% of keq (Klemens and Williams, 1986). Since both eqs. (1) and (14) are parabolic, this model is referred to as dual-parabolic two-step model. For the case that phonon temperature gradient is significant, inclusion of conduction in the phonons is essential.

## Dual-Hyperbolic Two-Step Model

For the case that heat conduction in both electrons and phonons needs to be considered using hyperbolic model, the energy equation for the lattice becomes

${C_l}\frac{{\partial {T_l}}}{{\partial t}} = - \nabla \cdot {{\mathbf{q''}}_l} + G\left( {{T_e} - {T_l}} \right) \qquad \qquad(15)$

where

${{\mathbf{q''}}_l} + {\tau _l}\frac{{\partial {{{\mathbf{q''}}}_l}}}{{\partial t}} = - {k_l}\nabla {T_l} \qquad \qquad(16)$

Combining (11) and (12) to eliminate ${{\mathbf{q''}}_l}$ yield

${C_l}\frac{{\partial {T_l}}}{{\partial t}} + {C_l}{\tau _l}\frac{{{\partial ^2}{T_l}}}{{\partial {t^2}}} = \nabla \cdot ({k_l}\nabla {T_l}) + G\left( {{T_e} - {T_l}} \right) + {\tau _l}\frac{\partial }{{\partial t}}\left[ {G\left( {{T_e} - {T_l}} \right)} \right] \qquad \qquad(17)$

Equation (17) together with eq. (13) become governing equations for the dual-hyperbolic two-step model. Chen and Beraun (2001) applied the dual-hyperbolic two-step model to simulate ultrashort laser pulse interactions with metal film. They found that the electron temperatures obtained from the dual-hyperbolic model and the hyperbolic model are very close. However, the lattice temperatures obtained from the two models differ significantly.

## References

Anisimov, S. I., Kapeliovich, B. L., Perel’man, T. L., 1974, “Electron Emission from Metal Surface Exposed to Ultrashort Laser Pulses,” Sov. Phys. JETP, Vol. 39, pp. 375-377.

Eesley, G. L., 1986, “Generation of Non-equilibrium Electron and Lattice Temperatures in Copper by Picosecond Laser Pulses,” Physical Review B, Vol. 33, pp. 2144-2155.

Elsayed-Ali, H. E., Norris, T. B., Pessot, M. A., and Mourou, G. A., 1987, “Time-Resolved Observation of Electron-Phonon Relaxation in Copper,” Physical Review Letters, Vol. 58, pp. 1212-1215.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Hohlfeld, J., Wellershoff, S. S., Gudde, J., Conrad, U., Jahnke, V., and Matthias, E., 2000, “Electron and Lattice Dynamics Following Optical Excitation of Metals,” Chemical Physics, Vol. 251, pp. 237-258.

Klemens, P. G., and Williams, R. K., 1986, “Thermal Conductivity of Metals and Alloys,” Int. Metals Review, 31, pp. 197-215.

Chen, J. K., and Beraun, J. E., 2001, “Numerical Study of Ultrashort Laser Pulse Interactions with Metal Films,” Numerical Heat Transfer A: Applications, Vol. 40, pp. 1-20.

Qiu, T.Q., and Tien, C.L., 1993, “Heat Transfer Mechanism During Short-Pulsed Laser Heating of Metals,” ASME Journal of Heat Transfer, Vol. 115, pp. 835-841.