Electrical resistivity and thermal conductivity of liquid aluminum in the two-temperature state

The electrical resistivity and thermal conductivity of liquid aluminum in the two-temperature state is calculated by using the relaxation time approach and structural factor of ions obtained by molecular dynamics simulation. Resistivity witin the Ziman–Evans approach is also considered to be higher than in the approach with previously calculated conductivity via the relaxation time. Calculations based on the construction of the ion structural factor through the classical molecular dynamics and kinetic equation for electrons are more economical in terms of computing resources and give results close to the Kubo–Greenwood with the quantum molecular dynamics calculations.


Introduction
Electron heat conductivity is one of the most important intrinsic characteristics of the metal exposed to the ultrashort laser pulses. It significantly affects the rate of the heat propagation into the metal target and the thickness of the target heated layer. This in turn determines the total pattern of ablation of a laser irradiated metal target [1][2][3][4][5][6][7][8]. Resistivity and thermal conductivity at high temperatures are intensively investigated both by different aspects of Ziman or Kubo-Greenwood approach [9][10][11][12][13]. At the early stage of the laser action, when two-temperature state of a metal takes place, thermal conductivity depends on the electron and ion temperatures and density. Therefore kinetic coefficients of the two-temperature metals become important for the adequate description of the heat propagation in the laser target [14].
In [15][16][17][18][19][20][21][22][23][24][25][26], we have calculated thermal conductivity of some metals and their resistivity in the two-temperature state. In these works most attention was paid to the two-temperature state of solid metals while in the calculation of liquid state semiempirical approach was significantly used. Meanwhile, the intensities we are considering are such that the melting of the target material occurs [27,28].
In this work, we apply the relaxation time approach with the use of structure factor obtained by using the molecular dynamics with many-body interaction between ions to get kinetic coefficients of aluminum in the two-temperature state. Aluminum belongs to the so-called simple metals and being widely used as a laser target, is intensively investigated both experimentally and theoretically [9,10,12,29].

Resistivity
The collision integral in the kinetic equation for the electrons can be written in standard form [30]: where f k and f k ′ are the non-equilibrium distribution function of electrons involved in a collision in the volume V of the metal, w(k, k ′ ) is the transition probability per unit time of the electron from the state with wave vector k and the energy ε into the state with wave vector k ′ and the energy ε ′ . This probability in the Born approximation, taking into account the quasi-elastic scattering of the electron when considering electron-ion interactions, can be written as The matrix element of the interaction of electrons and ions of the metal U kk ′ assuming that the potential energy of the electron is the sum of the potential energies of interaction with individual ions with their concentration n i , at the positions R l can be represented as Here u(q) = u(q) = u(r)e −iqr dr is a Fourier transform of the pseudopotential u(r) of the individual ion, q = k ′ − k is the change of momentum of the electron in the scattering; is a structural factor of the isotropic liquid, depending on the momentum transfer q, ion concentration, as well as on the ion temperature T i and electron temperature T e in the nonequilibrium situation. Respectivelỹ Within the relaxation time approximation, which is an important feature of Ziman approach, with the relaxation time depending on the energy of the particle, the collision integral is replaced by with f ′ k = f k − f k0 and the equilibrium Fermi function f k0 . Calculations for the collisional plasma with arbitrary ionic charge based on the Fokker-Planck approach are made in [31]. At the relaxation time τ (ε) deviation from the equilibrium distribution function in the electric field E is f ′ k = (−∂f /∂ε) v k eEτ (ε). Substituting this expression into the collision integral in the form of and equating the two forms of the collision integral representation within the approximation of an isotropic effective mass m with the electron energy ε(k) = 2 k 2 /(2m) allows to find the electron relaxation time depending on the module k of its wave vector: Energy of electron is measured from the bottom of the conduction band. Taking into account the expressions forw(q) we get the relaxation time at electron-ion interactions With the wave vectors, taken only on the Fermi surface (their module is k F = (3π 2 zn i ) 1/3 at the number of conduction electrons per atom z = 3), conductivity in equilibrium situation (T e = T i = T ) can be obtained as Then the resistivity is This is Ziman formula.
To account for the strong excitation of electrons above the Fermi energy ε F and the appearence of a two-temperature situation at the interaction of femtosecond laser pulses with an aluminum target, we use the more general expression for the conductivity [30]: Substituting here the relaxation time from (1), we get At , and the expression for the conductivity (2) gives the Drude formula σ = ne 2 τ (k F )/m. In our two-temperature situation with the electron temperature T e , higher than the temperature of the ions T i , the expression for the −∂f /∂ε is not reduced to δ-function and is is a chemical potential of the electrons. We suppose the effective mass of the electrons to be constant. It can be obtained from the low temperature and zero pressure DFTcalculations of the electron spectra and density of states [32]. Fermi level at these conditions corresponds to ε F0 = 11.1 eV and effective mass can be obtained as with the ion concentration n i0 = 2.75 g/cm 3 . Thus calculated effective mass is 1.06 of the free electron mass. Then the chemical potential µ(T e , n) is the solution of equation dε.
It is convenient to introduce relative ion concentration: x = n i /n i0 . We choose pseudopotential u(r) necessary to calculate the electron relaxation time as a special case of Ashkroft pseudopotential with the screened Coulomb interaction u(r) = 0, r r 0 and u(r) = − ze 2 r exp(−r/λ), r > r 0 .
Here λ = λ(T e , x) is a screening length, r 0 is a core radius. The Fourier transform of the pseudopotential has a form u(q) = −4πze 2 r 2 0 exp(−r 0 /λ) cos y + r 0 sin y/(λy) where y = kr 0 . We calculate the inverse screening length of the Coulomb interactions in the Thomas-Fermi approximation as 1 λ(T e , x) = 4πe 2 ∂µ/∂n e .
Considering the asymptotic behavior of this expression at low and high temperatures, we use λ(T e , x) in the form Using the Fourier transform of the potential u(q) and the calculated structure factor S(q, T, T, n i ), the conductivity in the equilibrium case for the temperature T can be find from (3), and resistivity as the inverse of its value. The aluminum structure factor depends weakly on the electron temperature T e because of the low dependence of interatomic interactions on the electron temperature [33][34][35]. Considering it depends only on the ion temperature and taking into account that the Fourier transform of the pseudopotential depends upon the electron temperature, we obtain in a two-temperature situation, when the electron temperature is different from the ion: with Fourier transform of potential given by (4). The reverse of value (5) is the two-temperature resistivity: ρ(T e , T i , n i ) = 1/σ(T e , T i , n i ). Expression for the resistivity we used, obtained by applying (5) and (6), differs from the often used Ziman-Evans expression [36,37], which takes the form in the two-temperature state.

Thermal conductivity
Electron relaxation time from (1), depending on its wave vector, can now be used to find the electron thermal conductivity due to the scattering of electrons by ions. In the absence of electric current, it can be found from the Onsager coefficients [38] − ∂f s ∂ε dp, With the electron temperature increase the contribution of electron-electron collisions into the electronic thermal conductivity increases within the temperature range under consideration, not essentially exceeding half the Fermi energy of aluminum (up to 60 kK). Evaluating with the use of the Onsager coefficients the thermal conductivity due to electron-electron collisions κ ee (T e , n i ) by calculating the momentum-dependent electron-electron collision frequency ν ee (p) as in [26], we can now calculate the coefficient of electronic thermal conductivity as The calculated coefficient κ ee (T e , n i ) allows to calculatie the effective frequency of electronelectron collisions ν ee , included in the Drude formula for thermal conductivity κ ee = C v v 2 /ν ee and depending on the electron temperature and density. Isochoric electron heat capacity in the Drude formula is calculated as We introduce the variable t = 6k B T e /(ε F0 x 2/3 ) as in [19,26]. Then the squared velocity of electrons in Drude formula as Fermi velocity squared at low temperatures and Maxwell velocity squared at the high temperature limit can be represented as

Results
The structure factor S(k, T, n i ) in the equilibrium one-temperature situation is calculated within the molecular dynamics approach with the number of aluminum atoms equal to 13500 and 48688 (giving similar results) and the time step for the ion motion 1 fs. The many-body interatomic potential of the embedded atom model was taken as in [39,40]. Structure factor for several values of temperature T at the density 2.35 g/cm 3 which corresponds to the density in the liquid state in the melting is shown in figure 1. As it was mentioned above, due to the weak dependence of the force constants of aluminum on the electron temperature within its range under consideration, the structure factor S(k, T, n i ) can be used as the two-temperature structural factor S(k, T e , T i , n i ) with T i = T . The pseudopotential used has a free parameter r 0 which we choose as r 0 = 1.08 a.u. (1 a.u. = 0.529Å is the Bohr radius) to satisfy the well known value 0.24 µOhm m of the resistivity of liquid aluminum at the melting temperature.
In figure 2 results of calculation of the resistivity of liquid aluminum in the equilibrium state with T e = T i = T at the density 2.35 g/cm 3 are shown which are close to the data obtained in [12,41]. They are also very close to those obtained in [10].   Resistivity calculated by the use of the Ziman-Evans expression, is shown in figure 3 together with the approach used here, when the resistivity is the reverse of the primarily calculated conductuvity. Both approaches give the same results at low temperatures but differ significantly with the electron temperature increase. Similar discrepancy between the reverse conductivity approach based on the equations (5) and (6) and Ziman-Evans approach was found for the hydrogen plasma in [37].
Results obtained for the electron thermal conductivity of liquid aluminum in the equilibrium one-temperature state are shown in figure 4. Thermal conductivity increases with the electron temperature increase significantly slower when the electron-electron collisions are taken into account in addition to electron-ion collisions.
Thermal conductivity of liquid aluminum in the two-temperature state is presented in figure  5. Due to the growth of the electron-electron collision frequencies thermal conductivity is not growing strongly when the temperature of electrons increase. For the upper limit of electron temperature under consideration the excitation of L-electrons is negligibly small, and single electron energy band approach is applicable. Figure 6 presents the dependence of the resistivity on the electron temperature in the nonequilibrium two-temperature state of liquid aluminum for several values of the ion temperature.

Conclusion
We have considered kinetic coefficients of liquid aluminum by using the τ -approach of the kinetic equation. With the use of Ziman approximation applying the structure factor to relaxation time of electron (depending on the wavevector) the conductivity is calculated. Resistivity is then obtained as the reverse conductivity. Calculating the Onsager coefficients within the τ approach, the thermal conductivity has been found. Resistivity and thermal conductivity in the equilibrium one-temperature states agree well with the results of calculations based on the quantum molecular dynamics and the Kubo-Greenwood formalism. Resistivity and thermal conductivity in two-temperature states, which arise at the interaction of femtosecond laser pulses with metals, are calculated at temperature of electrons higher than that of ions.