First-principles theory of the Dirac semimetal ${\mathrm{Cd}}_3{\mathrm{As}}_2$ under Zeeman magnetic field

Time-reversal broken Weyl semimetals have attracted much attention recently, but certain aspects of their behavior, including the evolution of their Fermi surface topology and anomalous Hall conductivity with Fermi-level position, have remained underexplored. A promising route to obtain such materials may be to start with a nonmagnetic Dirac semimetal and break time-reversal symmetry via magnetic doping or magnetic proximity. Here we explore this scenario in the case of the Dirac semimetal ${\mathrm{Cd}}_3{\mathrm{As}}_2$ based on first-principles density-functional calculations and subsequent low-energy modeling of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ in the presence of a Zeeman field applied along the symmetry axis. We clarify how each fourfold degenerate Dirac node splits into four Weyl nodes, two with chirality $\pm 1$ and two higher-order nodes with chirality $\pm 2$. Using a minimal $k·p$ model Hamiltonian whose parameters are fit to the first-principles calculations, we detail the evolution of the Fermi surfaces and their Chern numbers as the Fermi energy is scanned across the region of the Weyl nodes at fixed Zeeman field. We also compute the intrinsic anomalous Hall conductivity as a function of the Fermi-level position, finding a characteristic inverted-dome structure. ${\mathrm{Cd}}_3{\mathrm{As}}_2$ is especially well suited to such a study because of its high mobility, but the qualitative behavior revealed here should be applicable to other Dirac semimetals as well.

Figure 1

(a) Conventional unit cell of ${\mathrm{Cd}}_3{\mathrm{As}}_2$ with space group $I{4}_1/acd$. (b) Brillouin zone of the primitive unit cell with position of Dirac points. (c) Cd $s$ (red circles) and As $p$ (blue circles) character projected onto the energy manifold under nonmagnetic $\text{PBE}+\text{SOC}$ approximation. (d) $\text{PBE}+\text{SOC}$ band structure with linear Dirac crossing at the Fermi level.

Figure 2

(a) Nonmagnetic $\text{PBE}+\text{SOC}$ band structure of crossing Dirac bands along the $\mathrm{\Gamma }-Z$ direction. (b) Same but along $X^{\prime }-{\mathrm{\Gamma }}^{\prime }-X^{\prime }$ at $k_z=k_{\mathrm{D}}$, i.e., passing through the Dirac point. (c), (d) Same as (a), (b) but under a Zeeman field of $h=100\mathrm{T}$ along the [001] direction. Inset of (c) shows four Weyl nodes labeled as $w_1$ through $w_4$.

Figure 3

(a) Dispersion of the four Dirac-derived bands along the $k_z$ axis within the $k·p$ model. Bands are colored red, green, orange, and blue in order of increasing energy; Weyl points $w_1, w_2, w_3$, and $w_4$ are marked. (b) Electron pocket in third band (orange) and hole pocket in second band (green) for $E_F=0$, as indicated by horizontal orange and green lines in (a). Weyl point positions are shown by dots on the $k_z$ axis.

Figure 4

(a) Band structure showing several Fermi-level positions (dashed lines) as $E_F$ is tuned across the energy range of the Weyl nodes. (b) Contours of Fermi surfaces projected on the $(k_z,k_{\rho })$ plane corresponding to each Fermi-level position, for $G=10\mathrm{eV}{\AA }^2$, with $k_z$ and $k_{\rho }$ in units of ${10}^{-2}{\AA }^{-1}$. Chern numbers of electron and hole pockets are indicated.

Figure 5

(a)–(c) Band structure plotted versus $k_{\rho }$ for three different values of $k_z$ when $G$ is zero. (d)–(f) Berry curvature of the corresponding bands, color-coded accordingly.

Figure 6

(a)–(c) Band structure plotted versus $k_{\rho }$ for three different values of $k_z$ when $G$ is $10\mathrm{eV}{\AA }^2$. (d)–(f) Berry curvature of the corresponding bands, color coded accordingly.

Figure 7

Anomalous Hall conductivity ${\sigma }_{yx}^{\mathrm{AHC}}$ plotted versus Fermi level position for several values of the $G$ parameter. Energies of the Weyl nodes are indicated by the vertical dashed lines.