Carrier screening, transport, and relaxation in three-dimensional Dirac semimetals

A theory is developed for the density and temperature-dependent carrier conductivity in doped three-dimensional (3D) Dirac materials focusing on resistive scattering from screened Coulomb disorder due to random charged impurities (e.g., dopant ions and unintentional background impurities). The theory applies both in the undoped intrinsic (“high-temperature,” $T\gg T_F)$ and the doped extrinsic (“low-temperature,” $T\ll T_F)$ limit with analytical scaling properties for the carrier conductivity obtained in both regimes, where $T_F$ is the Fermi temperature corresponding to the doped free carrier density (electrons or holes). The scaling properties describing how the conductivity depends on the density and temperature can be used to establish the Dirac nature of 3D systems through transport measurements. We also consider the temperature-dependent conductivity limited by the acoustic phonon scattering in 3D Dirac materials. In addition, we theoretically calculate and compare the single-particle relaxation time ${\tau }_s$, defining the quantum level broadening, and the transport scattering time ${\tau }_t$, defining the conductivity, in the presence of screened charged impurity scattering. A critical quantitative analysis of the ${\tau }_t/{\tau }_s$ results for 3D Dirac materials in the presence of long-range screened Coulomb disorder is provided.

Figure 1

(a) Temperature-dependent polarizability $\Pi (q,T)$ of 3D Dirac semimetals as a function of wave vector for different temperatures and (b) temperature-dependent TF wave vector $q_{\mathrm{TF}}\left(T\right)$ as a function of temperature. Here, $D_F$ is the density of states of the 3D Dirac semimetals and $q_{\mathrm{TF}}\left(0\right)$ is the zero-temperature TF screening wave vector. The low- and high-temperature behaviors given in Eqs. (8) and (9), respectively, are also shown in (b). The TF screening wave vector $q_{\mathrm{TF}}\left(0\right)$ is related to the density of states at the Fermi energy through the relationship: $q_{\mathrm{TF}}\left(0\right)={(4\pi e^2{D}_F/\kappa )}^{1/2}$, and by definition $q_{\mathrm{TF}}^2=\Pi (q=0)4\pi e^2/\kappa$.

Figure 2

The density dependent conductivity calculated with the full RPA screening function for various temperatures. Here, $\alpha =1.2$ and a fixed impurity density $n_i={10}^{18}{\text{cm}}^{-3}$ are used.

Figure 3

The scaled conductivity $\sigma \left(T\right)/\sigma (T=0)$ for a strong screening parameter $q_0=q_{\mathrm{TF}}/2k_F=1.2$ (a) at low temperatures and (b) at high temperatures. Inset in (a) shows the same figures as (a) and (b) in the linear scale. The black line indicates the conductivity calculated with the full RPA screening. The red (blue) line indicates the conductivity calculated with temperature-dependent (independent) TF.

Figure 4

The same as Fig. 3 for a weak screening parameter $q_0=q_{\mathrm{TF}}/2k_F=0.17$.

Figure 5

(a) Calculated ratio of the transport scattering time to the single-particle scattering time ${\tau }_t/{\tau }_s$ as a function of $q_0$ for the screened charged impurity scattering. The solid (dashed) line represents the ratio for charged impurity with TF (RPA) screening effect. (b) The ratio as a function of $k_F/q_{\mathrm{TF}}$. Note that $q_0=\sqrt{2g/\pi }\sqrt{\alpha }$, where $\alpha =e^2/\kappa \hslash v_F$ is the dimensionless effective coupling constant. (c) and (d) show the scattering time ratio for the short-range impurity scattering as a function of $q_0$ and $k_F/q_{\mathrm{TF}}$, respectively. The horizontal solid lines in (c) and (d) indicate the ratio for the $\delta$-range unscreened short-range disorder. For comparison, we also show the results for screened short-range disorders within TF (thick solid lines) and RPA (dashed lines).