Quantum and transport mobilities of a ${\mathrm{Na}}_3\mathrm{Bi}$-based three-dimensional Dirac system

The electronic and transport properties of a three-dimensional (3D) Dirac system are investigated theoretically, which is motivated by recent experimental measurements on quantum and transport mobilities in the 3D Dirac semimetal ${\mathrm{Na}}_3\mathrm{Bi}$ by J. Xiong et al. [Science 350, 413 (2015); Europhys. Lett. 114, 27002 (2016)]. The electron Hamiltonian is taken from a simplified $\mathbf{k}·\mathbf{p}$ approach. From the obtained electronic band structure and the Fermi energy, we explain why the anomalous effect induced by the chiral anomaly and the Berry curvature in the energy band can be observed experimentally in magnetotransport coefficients in both low- and high-density samples. Moreover, the quantum and transport mobilities are calculated on the basis of the momentum-balance equation derived from a semiclassical Boltzmann equation with the electron-impurity interaction. The quantum and transport mobilities obtained from this study agree both qualitatively and quantitatively with those measured experimentally. We also examine the electron mobilities along different crystal directions in ${\mathrm{Na}}_3\mathrm{Bi}$ and find them largely anisotropic. The theoretical findings from this work can be helpful in gaining an in-depth understanding of the experimental results and of the basic electronic and transport properties of newly developed 3D Dirac systems.

Figure 1

The energy spectrum of the conduction band as a function of $k$ and $k_z$ for different values of $B_3$ as indicated (left panel). Here $B_3\ne 0$ corresponds to the Hamiltonian given by Eq. (1) and $B_3=0$ to that given by Eq. (2). The pink line refers to Fermi energy $E_F=37\mathrm{meV}$ calculated by taking $N_e=4\times {10}^{18}{\mathrm{cm}}^{-3}$ and $T=4.2\mathrm{K}$. The right panel shows the result for $k=0$, which also leads to $B_3=0$.

Figure 2

The Fermi energy $E_F$ in a 3D Dirac system as a function of electron density $N_e$ at different temperatures as indicated. Here we also show the result for an ideal 3D electron gas (I3DEG) at $T=0$. The results for ${\mathrm{Na}}_3\mathrm{Bi}$ 3D Dirac system at $T=0\mathrm{K}$, 4.2 K, and 10 K coincide roughly. The inset shows the electronic DOS for a 3D Dirac system and for an I3DEG, with $D_0=C_0^2/A^3$ for 3D Dirac system and $D_0={m^{*}}^{3/2}\sqrt{2|C_0|}/\pi {\hslash }^3$ for an I3DEG.

Figure 3

Inverse screening length $K_s$ as a function of electron density $N_e$ at different temperatures. The results for $T=0\mathrm{K}$, 4.2 K, and 10 K coincide roughly.

Figure 4

The quantum ${\mu }_q$ and transport ${\mu }_t$ mobilities as a function of electron density at different impurity concentrations $N_i=5.0\times {10}^{18}{\mathrm{cm}}^{-3}$ (black curves) and $N_i=1.2\times {10}^{19}{\mathrm{cm}}^{-3}$ (red curves). The black and red curves are obtained from the present theoretical calculation and the symbols are experimental results [17] with corresponding sample numbers. Note that the experimental result of the quantum lifetime for sample B2 was shown in the caption of Fig. 2 in Ref. [17].

Figure 5

The transport mobility along the $x$ direction, ${\mu }_{xx}$ (solid curves), and the $z$ direction, ${\mu }_{zz}$ (dashed curves), as a function of electron density $N_e$ for a fixed impurity concentration $N_i=5.0\times {10}^{18}{\mathrm{cm}}^{-3}$.