Ab initio modeling of nonequilibrium electron-ion dynamics of iron in the warm dense matter regime

The spatiotemporal electron and ion relaxation dynamics of iron induced by femtosecond laser pulses was studied using a one-dimensional two-temperature model (1D-TTM) where electron and ion temperature-dependent thermophysical parameters such as specific heat ($C$), electron-phonon coupling ($G$), and thermal conductivity ($K$) were calculated with ab initio density-functional-theory (DFT) simulations. Based on the simulated time evolutions of electron and ion temperature distributions [$T_e\left(x,t\right)$ and $T_i\left(x,t\right)$], the time evolution of x-ray absorption near-edge spectroscopy (XANES) was calculated and compared with experimental results reported by Fernandez-Pañella et al., where the slope of XANES spectrum at the onset of absorption ($s$) was used due to its excellent sensitivity to the electron temperature. Our results indicate that the ion temperature dependence on $G$ and $C$, which is largely neglected in the past studies, is very important for studying the nonequilibrium electron-ion relaxation dynamics of iron in warm dense matter (WDM) conditions. It is also shown that the $1/s$ behavior becomes very sensitive to the thermal gradient profile, in other words, to the values of $K$ in a TTM simulation, for target thickness of about two to four times the mean free path of conduction electrons. Our approach based on 1D-TTM and XANES simulations can be used to determine the optimal combination of target geometry and laser fluence for a given target material, which will enable us to tightly constrain the thermophysical parameters under electron-ion nonequilibrium WDM conditions.

Figure 1

Specific heat and electron-phonon coupling parameters calculated as a function of $T_e$ at different ion temperatures. The data of Lin et al. were calculated for a perfect bcc crystal at $T=0\mathrm{K}$, while the rest are ensemble averaged values based on ab initio MD simulations. All the solid lines are obtained based on EDOS self-consistently calculated as $T_e=T_i$, while dashed lines are obtained based on EDOS self-consistently calculated as $T_e=T_i+20000\mathrm{K}$ in order to assess the accuracy of rigid band approximation.

Figure 2

Thermal conductivity of iron calculated by the ab initio method described in the text (solid triangles for this work and solid squares for literature [28]) as well as experimental values for bcc solid (solid diamonds) [53] as well as for liquid (solid circle) [54]. Our simulation results in this figure are calculated for electron-ion equilibrium ($T_i=T_e$) conditions.

Figure 3

Ab initio thermal conductivity calculated for electron-ion nonequilibrium conditions ($T_e$ is different from $T_i$) and compared to equilibrium results. $T_i=2000\mathrm{K}$ and $T_e=20000\mathrm{K}$ (solid circle), $T_i=10000\mathrm{K}$ and $T_e=20000\mathrm{K}$ (solid square), $T_i=10000\mathrm{K}$ and $T_e=30000\mathrm{K}$ (solid square).

Figure 4

Time evolution of electron (solid lines) and ion (dashed lines) temperatures at selected delay times: $t=0.5,2.5,5.0,7.5$, and $10.0\mathrm{ps}$ calculated with several sets of thermophysical parameters (details in the text).

Figure 5

Electron (solid line) and ion (dashed line) temperature distributions (lower left), XANES at different target positions, $x=0,30$, and $60\mathrm{nm}$ (upper left), and the averaged XANES (right) calculated at the delay time, $t=2.5\mathrm{ps}$. The averaged XANES spectrum was obtained by integrating the XANES spectra calculated at the grid points spaced by 1.5 nm. The XANES spectrum at an arbitrary $\left(T_e,T_i\right)$ value was obtained using a linear interpolation of XANES spectra self-consistently calculated on the $\left(T_e,T_i\right)$ grid points. See text for more information about this procedure.

Figure 6

Time evolution of XANES calculated with ${{\left(G,C\right)}_{T_i=}}_{10000\mathrm{K}}$ and $K_{\mathrm{PC}}$. Convolution along the time axis using ${\sigma }_{\mathrm{FWHM}}=2.5\mathrm{ps}$ is performed in order to reflect the experimental time resolution of about 2.5 ps.

Figure 7

Simulated $1/s$ profiles for several sets of $\left(G,C,K\right)$ parameters compared with the experimental values.

Figure 8

Time evolution of electron and ion temperatures calculated for four different target thicknesses, $d=20$ (blue), 40 (green), 60 (orange), and 80 nm (magenta). Upper panels are for electron temperature profiles and lower panels are for ion temperature profiles. The delay times are $t=0.5,2.5,5.0$, and $7.5\mathrm{ps}$. The front side electron temperature reaches the maximum value at approximately $t=0.5\mathrm{ps}$. Note the ballistic range of electrons in these TTM simulations was set at $l=10\mathrm{nm}$.

Figure 9

$1/s$ calculated with $K_{\mathrm{PC}}$ (solid lines) and with $K_{\mathrm{PC}}\pm 20\%$ (dashed lines) in order to show the $K$ dependence of $1/s$. The pump laser fluences are chosen in such a way that the corresponding energy densities are equivalent to that of $d=60\mathrm{nm}$ (or the value used in Ref. [25]).