Isochoric, isobaric, and ultrafast conductivities of aluminum, lithium, and carbon in the warm dense matter regime

We study the conductivities $\sigma$ of (i) the equilibrium isochoric state ${\sigma }_{\mathrm{is}}$, (ii) the equilibrium isobaric state ${\sigma }_{\mathrm{ib}}$, and also the (iii) nonequilibrium ultrafast matter state ${\sigma }_{\mathrm{uf}}$ with the ion temperature $T_i$ less than the electron temperature $T_e$. Aluminum, lithium, and carbon are considered, being increasingly complex warm dense matter systems, with carbon having transient covalent bonds. First-principles calculations, i.e., neutral-pseudoatom (NPA) calculations and density-functional theory (DFT) with molecular-dynamics (MD) simulations, are compared where possible with experimental data to characterize ${\sigma }_{\mathrm{ic}}$, ${\sigma }_{\mathrm{ib}}$, and ${\sigma }_{\mathrm{uf}}$. The NPA ${\sigma }_{\mathrm{ib}}$ is closest to the available experimental data when compared to results from DFT with MD simulations, where simulations of about 64–125 atoms are typically used. The published conductivities for Li are reviewed and the value at a temperature of 4.5 eV is examined using supporting x-ray Thomson-scattering calculations. A physical picture of the variations of $\sigma$ with temperature and density applicable to these materials is given. The insensitivity of $\sigma$ to $T_e$ below 10 eV for carbon, compared to Al and Li, is clarified.

Figure 1

Static conductivities for Al from experiment and from DFT-MD and NPA calculations. The isobaric conductivity ${\sigma }_{\mathrm{ib}}$ is at densities $2.37\ge \rho \ge 1.65\mathrm{g}/{\mathrm{cm}}^3$ [cf. triangular region (a)]. The isochoric $[{\sigma }_{\mathrm{ic}}$, region (c)] and UFM $[{\sigma }_{\mathrm{uf}}$, region (b)] conductivities are for a density of $2.7\mathrm{g}/{\mathrm{cm}}^3$. Enlarged views of regions (a) and (c) are given in the Appendix. The blue closed diamond gives the conductivity of normal aluminum at its melting point (0.082 eV and $2.375\mathrm{g}/{\mathrm{cm}}^3$), viz., $\sigma =4.16\times {10}^6$ S/m from experiment (quoted in Ref. [34]) and $\sigma =4.09\times {10}^6$ S/m from the NPA calculation.

Figure 2

Isobaric conductivity of aluminum from near its melting point to about 0.4 eV, expanded from Fig. 1 and with linear scales. We compare the NPA calculation, Gathers' experiment, and our DFT with MD results. The experimental ${\sigma }_{\mathrm{ib}}$ of Al at its melting point (blue closed diamond) [34] $\rho =2.375\mathrm{g}/{\mathrm{cm}}^3$ is displayed and aligns with the NPA calculations for the Gathers data showing agreement between the NPA calculation and two independent experiments.

Figure 3

Ultrafast conductivity of Al at a density of $2.7\mathrm{g}/{\mathrm{cm}}^3$ obtained from the NPA-MHNC-Ziman formula for a solid initial state at 0.06 eV (blue curve with diamonds) and liquid initial state (black curve with triangles) just above the melting point 0.082 eV. The top curve was also displayed in Fig. 1 for comparison with ${\sigma }_{ib}$ and ${\sigma }_{ic}$.

Figure 4

(a) Static structure factor $S\left(k\right)$ of isochoric aluminum WDM at different temperatures; $S\left(k\right)$ at $2k_F$ changes by 65% from $T=0.1$ to $T=5$ eV. In ultrafast aluminum $S\left(k\right)$ remains fixed at the initial temperature, even when $T_e$ changes. (b) Evolution of $S\left(k\right)$ for isochoric Li at $0.542\mathrm{g}/{\mathrm{cm}}^3$ as a function of temperature. As $T$ increases, the peak broadens and shifts away from $2k_F$.

Figure 5

The NPA-MHNC $g\left(r\right)$ of isochoric Al at 1 eV, $\rho =2.70\mathrm{g}/{\mathrm{cm}}^3$, and $r_{\text{WS}}\simeq 3.0$ and isochoric Li at 1 eV, $\rho =0.542\mathrm{g}/{\mathrm{cm}}^3$, and $r_{\text{WS}}\simeq 3.3$ compared with the DFT-MD $g\left(r\right)$ obtained using abinit and 108 atoms in the simulation. At 1 eV, $T/E_F\sim 0.1$ and 0.2 for Al and Li, respectively, which correspond to partially degenerate WDM systems.

Figure 6

Isobaric (${\sigma }_{\mathrm{ib}}$), isochoric (${\sigma }_{\mathrm{ic}}$), and ultrafast (${\sigma }_{\mathrm{uf}}$) conductivities of Li at a density of $0.542\mathrm{g}/{\mathrm{cm}}^3$ obtained from the NPA-MHNC-Ziman formula. Isobaric experimental conductivities ${\sigma }_{\mathrm{ib}}$ are for $0.5\ge \rho \ge 0.4\mathrm{g}/{\mathrm{cm}}^3$. The DFT-MD-KG ${\sigma }_{\mathrm{ic}}$ value of Witte et al. at $0.6\mathrm{g}/{\mathrm{cm}}^3$ [43], and the corresponding NPA-MHNC-Ziman value for ${\sigma }_{\mathrm{ic}}$ are also shown.

Figure 7

Oak Ridge National Laboratory experimental data compared with the NPA-MHNC-Ziman results and the DFT-MD-KG conductivity of Kietzmann et al. [46] using the vasp code. Their 600 and 1000 K results have been slightly extrapolated to the low-density region covered by the experiments. The curve at 2000 K given by Kietzmann et al. is above the boiling point of Li and is not representative of the behavior of Li at 1500 K.

Figure 8

(a) Isochoric conductivity ${\sigma }_{\mathrm{ic}}$ and ultrafast conductivity ${\sigma }_{\mathrm{uf}}$ for carbon at $\rho =3.7\mathrm{g}/{\mathrm{cm}}^3$ from the NPA and DFT-MD methods [18] and isochoric conductivity from the NPA method for $\rho =2.0\mathrm{g}/{\mathrm{cm}}^3$. (b) Ion-ion $S\left(k\right)$ for several temperatures from the NPA method; note the nearly constant value of $S\left(k\right)$ at $2k_F$ (indicated by a vertical line).

Figure 9

Quantities relevant to XRTS calculated using the NPA-HNC method (this work) and the DFT-MD method (Ref. [43]) for lithium. Values for $k$ smaller than about 0.6 Å${}^{-1}$ are not available from the DFT-MD simulation due to the finite size of the DFT simulation cell. Here $q\left(k\right)$ is the Fourier transform of the free-electron density at the Li ion in the plasma, $f\left(k\right)$ is the bound-electron form factor, and $N\left(k\right)=f\left(k\right)+q\left(k\right)$. The ion feature $W\left(k\right)=N{\left(k\right)}^{2}S\left(k\right)$ involves the ion-ion structure factor $S\left(k\right)$. Experimental points are from Saiz et al. cited in the work of Witte et al. [43], Fig. 8.

Figure 10

The KG conductivity $\sigma \left(\omega \right)$ for Al, Li, and C. Note the slight non-Drude behavior of Li $\sigma \left(\omega \right)$ near 0.08 a.u. in (b). The carbon $\sigma \left(\omega \right)$ is highly non Drude-like, with the peak moving to higher energy as $T$ is lowered; no Drude form is shown for carbon.

Figure 11

Isochoric conductivity of aluminum from near its melting point to about 0.7 eV, expanded from Fig. 1, and now including the Witte et al. [42] calculation of the Al conductivity at 0.3 eV and $\rho =2.7\mathrm{g}/{\mathrm{cm}}^3$. Our DFT-MD data and those of Vlček et al. [35] are shown.