Quantum Hall effect originated from helical edge states in ${\mathrm{Cd}}_3{\mathrm{As}}_2$

The recent experimental observations of the quantum Hall effect in three-dimensional (3D) topological semimetals have attracted great attention, but there are still debates on its origin. We systematically study the dependence of the quantum Hall effect in topological semimetals on the thickness, Fermi energy, and growth direction, taking into account the contributions from the Fermi-arc surface states, confinement-induced bulk subbands, and helical side-surface edge states. In particular, we focus on the intensively studied Dirac semimetal ${\mathrm{Cd}}_3{\mathrm{As}}_2$ and its slabs grown along experimentally accessible directions, including [001], [110], and [112]. We reveal an ignored mechanism from the Zeeman splitting of the helical edge states, which along with Fermi-arc 3D quantum Hall effect, may give a nonmonotonic dependence of the Hall conductance plateaus on the magnetic field in the most experimentally studied [112] direction slab. Our results will be insightful for exploring the quantum Hall effects beyond two dimensions.

Figure 1

The [001]-direction slab of the Dirac semimetal. (a) The helical edge states on the side surfaces in the $(x^{\prime },y^{\prime },z^{\prime })$ coordinates. The arrows denote the directions of propagation, and the red and blue colors distinguish opposite spin polarizations. (b) The confinement-induced energy gap ${\mathrm{\Delta }}_E$ (red solid line) and the spin Chern number $C_s$ (green dashed line) as functions of the slab thickness $L$. (c) The Hall conductance ${\sigma }_{\text{H}}$ (blue dots) and ${\sigma }_{\text{H}}^{\uparrow ,\downarrow }$ (blue and green lines) as functions of the magnetic field $1/B$ for $L=100$ nm. The red line correspond to ${\sigma }_{\text{H}}^{\uparrow }+{\sigma }_{\text{H}}^{\downarrow }$. (d) ${\sigma }_{\text{H}}$ as a function of $L$ and $1/B$. The Fermi energy $E_{\text{F}}$ is at the Weyl node $E_{\text{w}}$.

Figure 2

The [110]-direction slab of the Dirac semimetal, which consists of a Weyl semimetal and its time reversal. (a) The real-space correspondence of Fermi-arc surface states (red for top and blue for bottom) of the Weyl semimetal described by the upper block of $H$ in Eq. (1), for a slab with open boundary condition along the $y^{\prime }$ direction. $E_{\text{w}}$ marks the Weyl-node energy. (b) The projection of the Fermi arc on the $k_{x^{\prime }}-k_{z^{\prime }}$ plane. $S$ is the area of the Fermi loop. (c) The area $S$ of the Fermi surface of the Fermi-arc surface states in the $(k_{x^{\prime }},y^{\prime },k_{z^{\prime }})$ plane as a function of the slab thickness $L$. (d) The Hall conductance ${\sigma }_{\text{H}}$ as a function of $L$ and magnetic field $1/B$. The Fermi energy $E_{\text{F}}$ is at the Weyl node $E_{\text{w}}$.

Figure 3

The [112]-direction slab of the Dirac semimetal. (a) The area of the Fermi surface $S$ (solid) and spin Chern number $C_s$ (dotted) as functions of the thickness $L$. (b) The Hall conductance ${\sigma }_{\text{H}}$ as a function of $L$ and magnetic field $1/B$. (c) Schematic of where the Fermi energy $E_{\text{F}}$ and Weyl modes $E_{\text{w}}$ are in the bulk spectrum of the Dirac semimetal. (d)–(f) ${\sigma }_{\text{H}}$ as a function of (d) $(E_{\text{F}}-E_{\text{w}},1/B)$ and (e,f) $(L,1/B)$, respectively. Here we take (d) $L=50$ nm, (e) $E_{\text{F}}-E_{\text{w}}=5\text{meV}$, and (f) $E_{\text{F}}-E_{\text{w}}=-5\text{meV}$.

Figure 4

The dependence on $\theta$, the angle between the line connecting the Dirac nodes and $k_{x^{\prime }}-k_{z^{\prime }}$ plane. (a) Illustration of $\theta$. (b) The area of the Fermi surface of the Fermi-arc surface states $S$ (solid) and spin Chern number $C_s$ (dotted) as functions of $\theta$. (c,d) The Hall conductance ${\sigma }_{\text{H}}$ as a function of $(\theta ,1/B)$ and $(E_{\text{F}}-E_{\text{w}},1/B)$, respectively. The dashed line corresponds to the slab grown along the [112] direction. Here we take (c) $E_{\text{F}}-E_{\text{w}}=0$ and (d) $B=5$ T.

Figure 5

(a) Schematic illustration of the 3D device with a electrode attached on the left boundary, and the right lead is scanned by the STM tip as a 1D normal lead. (b) Schematic illustration of the 3D device with six electrodes on the top surface. (${\mathrm{c}}_1$)–(${\mathrm{c}}_3$) The spectra for the three different cases that possess the quantized Hall conductance. Here the blue dashed lines label the Fermi energy. (${\mathrm{d}}_1$)–(${\mathrm{d}}_3$) and (${\mathrm{e}}_1$)–(${\mathrm{e}}_3$) correspond to the numerically calculated two-terminal conductance ${\sigma }_{12}$ and the surface Hall conductance ${\sigma }_{x^{\prime }{z}^{\prime }}$ by using the devices shown in (a) and (b), respectively.

Figure 6

Numerically calculated spectrum of the Dirac semimetal slab with open boundary condition along the $y^{\prime }$ direction at $k_x^{\prime }=0$. The blue dashed lines label the Fermi level $E_{\text{F}}=E_{\text{w}}$. In (a), (b), and (c), $y^{\prime }$ correspond to [110], [001], and [112]-directions, respectively.

Figure 7

Numerically calculated spectrum of the Dirac semimetal slab with open boundary conditions along $y^{\prime }$ and $z^{\prime }$ directions. The blue dashed lines label the Fermi level $E_{\text{F}}=E_{\text{w}}$. In (a), (b), and (c), $y^{\prime }$ correspond to [110], [001], and [112]-directions, respectively.

Figure 8

Numerically calculated spectrum of the Dirac semimetal slab with open boundary conditions along the $y^{\prime }$ and $z^{\prime }$ directions with $B=5\text{T}$. The blue dashed lines label the Fermi level $E_{\text{F}}=E_{\text{w}}$. In (a), (b), and (c), $y^{\prime }$ correspond to [110], [001], and [112]-directions, respectively.

Figure 9

The Hall conductance ${\sigma }_{\text{H}}$ as a function of (a) the inverse of magnetic field $1/B$ at different tilting angles of the magnetic field, (b) disorder strength $U$ at different magnetic field strengths, and (c) the inverse of magnetic field $1/B$ at different temperatures. In (b), the error bars show the standard deviation averaged over 200 samples. The thicknesses of the [001]-direction slabs are 100 nm.

Figure 10

The Hall conductance ${\sigma }_{\text{H}}$ as a function of the inverse of magnetic field $1/B$ with the Fermi energy (a,d) $E_{\text{F}}=-0.004\text{eV}$, (b,e) $E_{\text{F}}=0$, and (c,f) $E_{\text{F}}=0.001\text{eV}$. The thicknesses of the [110]-direction slabs are 80 (red), 90 (blue), and 100 nm (yellow), respectively. The Zeeman terms are excluded and included in the upper and lower panels, respectively.

Figure 11

(a)–(c) Energy spectra of Weyl-semimetal films with open boundary conditions along the $y^{\prime }$ and $z^{\prime }$ directions and periodic boundary condition along the $x^{\prime }$ direction. Here, the sizes in the $y^{\prime }$ and $z^{\prime }$ directions are taken as $L_{y^{\prime }}=20a_{y^{\prime }}$ and $L_{z^{\prime }}=40a_{z^{\prime }}$. The red curves are the bulk bands and the blue dashed curves label the Fermi levels. The parameters are (a) $(\alpha ,\theta )=(0,0)$, $M=5$, $A=50$, $D_2=4$, $D_1=1$, $B_y=1/10$, and $k_w=1.5$ [13]; (b) $(\pi /2,0)$, $M=5$, $A=1$, $B_y=1/10$ and $D_1=D_2=k_w=0$; and (c) $(\pi /2,0)$, $M=5$, $A=1$, $D_1=D_2$, $B_y=0$, and $k_w=0.25$. (d)–(f) The wave function distributions along the $y^{\prime }$ direction. The red and blue dot lines correspond the red and blue points shown in (a)–(c). Here we define $|\mathrm{\Psi }(k_{x^{\prime }},y^{\prime }){|}^2={\sum }_{z^{\prime }}{\left|\psi (k_{x^{\prime }},y^{\prime },z^{\prime })\right|}^2$.