3D Quantum Hall Effect of Fermi Arcs in Topological Semimetals

The quantum Hall effect is usually observed in 2D systems. We show that the Fermi arcs can give rise to a distinctive 3D quantum Hall effect in topological semimetals. Because of the topological constraint, the Fermi arc at a single surface has an open Fermi surface, which cannot host the quantum Hall effect. Via a “wormhole” tunneling assisted by the Weyl nodes, the Fermi arcs at opposite surfaces can form a complete Fermi loop and support the quantum Hall effect. The edge states of the Fermi arcs show a unique 3D distribution, giving an example of ($d-2$)-dimensional boundary states. This is distinctly different from the surface-state quantum Hall effect from a single surface of topological insulator. As the Fermi energy sweeps through the Weyl nodes, the sheet Hall conductivity evolves from the $1/B$ dependence to quantized plateaus at the Weyl nodes. This behavior can be realized by tuning gate voltages in a slab of topological semimetal, such as the TaAs family, ${\mathrm{Cd}}_3{\mathrm{As}}_2$, or ${\mathrm{Na}}_3\mathrm{Bi}$. This work will be instructive not only for searching transport signatures of the Fermi arcs but also for exploring novel electron gases in other topological phases of matter.

Figure 1

(a) The energy dispersions for the Fermi arc (at $y=L/2$) and bulk states in a topological Weyl semimetal. $k_{||}$ stands for $(k_x,k_y)$ for the bulk and $k_x$ for the arc, respectively. (b) The Fermi arc at $y=L/2$ and $E_F=E_w$ on the $k_z-k_x$ plane. The shadow defines the “constraint” region where the Fermi arcs can exist. (c) A slab of topological semimetal of thickness $L$ and width $W$. (d) The Fermi arcs at $E_F=E_w$ (solid) and constraints (shadow) at the $y=L/2$ (red) and $-L/2$ (blue) surfaces of the slab. [(e)–(g)] The wave function distributions at $k_z=0$ along the $y$ axis, at the blue (bottom arc, $k_x<0$), black (Weyl nodes), and red (top arc, $k_x>0$) dots in (d). (h) Landau levels of the Fermi arcs at $B=5\mathrm{T}$ vs the guiding center $z_0$. (i) The wave function distributions along the $y$ axis for the edge states of the Fermi arcs marked by the green and orange dots in (h). $L=100\mathrm{nm}$, $W=200\mathrm{nm}$, and other parameters can be found in Fig. 2.

Figure 2

(a) In a topological semimetal slab, the numerically calculated energy spectrum (pink) for the bulk states and Fermi arcs at $k_x=0$ (left) and $k_x=\pm 0.1{\mathrm{nm}}^{-1}$ (right). The blue curves are the Fermi arc bands plotted using $H_{\mathrm{arc}}$ and $H_{\mathrm{arc}}^{\prime }$. (b) The sheet Hall conductivity when the Fermi energy $E_F$ crosses the bulk states for $\mathrm{\Gamma }\to 0$. $\mathrm{\Gamma }$ is the disorder-induced level broadening. Recent experiments show that gating can tune carriers from $n$- to $p$-type in 100-nm-thick devices of topological semimetal [49]. (c) The sheet Hall conductivity ${\sigma }_{\mathrm{H}}^s$ at $E_F=E_w$, where the Fermi energy crosses only arc I. The right inset shows the analytic Hall conductance ${\sigma }_{\text{H}}^{\text{arc}}$ in Eq. (5). In the presence of a residual detuning from the Weyl nodes, the bulk states also contribute to ${\sigma }_{\mathrm{H}}^s$. Unlike that from the Fermi arcs, the contribution from the bulk states may change with the slab thickness. The left inset shows the width of the Hall plateaus in the clean limit as a function of $D_1$ for different $k_w$. The dots and lines are the numerical and analytic results, respectively. The parameters are $M=5\mathrm{eV}{\mathrm{nm}}^2$, $A=0.5\mathrm{eV}\mathrm{nm}$, and $D_2=3\mathrm{eV}{\mathrm{nm}}^2$, $D_1=2\mathrm{eV}{\mathrm{nm}}^2$, $k_w=0.3{\mathrm{nm}}^{-1}$, and $L=100\mathrm{nm}$.

Figure 3

(a) The crystallographic directions and coordinates system in the model of Dirac semimetal Eq. (6). Slabs grown along the [112] and [110] directions corresponds to $(\alpha ,\theta )=(-\pi /4,{\tan }^{-1}\sqrt{2})$ and $(-\pi /4,0)$, respectively. (b) $y^{\prime }$ is the slab growth direction. The Hall conductance is defined in the $x^{\prime }-z^{\prime }$ plane. The magnetic field can be rotated from the $y^{\prime }$ to the $x^{\prime }$ direction by the angle $\xi$. [(c),(d),(f)] For a slab of 100 nm, the energy spectrum at $k_x=0.01{\mathrm{nm}}^{-1}$ and the wave function distribution along the $y^{\prime }$ direction for the states marked by the red points. (e) The sheet Hall conductivity at the Dirac node $E_w=C_0-C_1{M}_0/M_1$ for the ${\mathrm{Cd}}_3{\mathrm{As}}_2$ [110] slab at different $\xi$. (g) The same as (e) but for the [010] ${\mathrm{Na}}_3\mathrm{Bi}$ slab. The parameters for ${\mathrm{Cd}}_3{\mathrm{As}}_2$ are $C_0=-0.0145\mathrm{eV}$, $C_1=10.59\mathrm{eV}{\AA }^2$, $C_2=11.5\mathrm{eV}{\AA }^2$, $M_0=0.0205\mathrm{eV}$, $M_1=-18.77\mathrm{eV}{\AA }^2$, $M_2=-13.5\mathrm{eV}{\AA }^2$, $A=0.889\mathrm{eV}\AA$ [70], $g_s=18.6$, and $g_p=2$ [69]. The parameters for ${\mathrm{Na}}_3\mathrm{Bi}$ are $C_0=-0.06382\mathrm{eV}$, $C_1=8.7536\mathrm{eV}{\AA }^2$, $C_2=-8.4008\mathrm{eV}{\AA }^2$, $M_0=0.08686\mathrm{eV}$, $M_1=-10.6424\mathrm{eV}{\AA }^2$, $M_2=-10.361\mathrm{eV}{\AA }^2$, $A=2.4598\mathrm{eV}\AA$ [70] $g_s=20$, and $g_p=20$ [48]. $\mathrm{\Gamma }=1\mathrm{K}$ in panels (e) and (g).