Field-Tunable One-Sided Higher-Order Topological Hinge States in Dirac Semimetals

Recently, higher-order topological matter and 3D quantum Hall effects have attracted a great amount of attention. The Fermi-arc mechanism of the 3D quantum Hall effect proposed to exist in Weyl semimetals is characterized by the one-sided hinge states, which do not exist in all the previous quantum Hall systems, and more importantly, pose a realistic example of the higher-order topological matter. The experimental effort so far is in the Dirac semimetal ${\mathrm{Cd}}_3{\mathrm{As}}_2$, where, however, time-reversal symmetry leads to hinge states on both sides of the top and bottom surfaces, instead of the aspired one-sided hinge states. We propose that under a tilted magnetic field, the hinge states in ${\mathrm{Cd}}_3{\mathrm{As}}_2$-like Dirac semimetals can be one sided, highly tunable by field direction and Fermi energy, and robust against weak disorder. Furthermore, we propose a scanning tunneling Hall measurement to detect the one-sided hinge states. Our results will be insightful for exploring not only the quantum Hall effects beyond two dimensions, but also other higher-order topological insulators in the future.

Figure 1

(a) Energy dispersions of the bulk (gray, $k_y=0$) and Fermi-arc surface states [red for top and blue for bottom in (b)] of a Weyl semimetal. (b) The Fermi-arc surface states on the top and bottom surfaces can form a 2D electron gas distributed in the 3D system to support a 3D quantum Hall effect under a magnetic field $\mathbf{B}$. The 3D quantum Hall effect is characterized by the one-sided hinge states (red and blue arrows) in (c), which do not exist in all the previous quantum Hall systems. In the experimentally accessible Dirac semimetals, however, time-reversal symmetry leads to a complete 2D electron gas on each surface (d) and hinge states on both sides (e), instead of the desired one-sided hinge states. We find that the hinge states in the Dirac semimetal can be one sided, highly tunable by a tilted magnetic field [from (e) to (c)] and Fermi energy [Fig. 2], giving a signature of the Fermi-arc 3D quantum Hall effect and higher-order topological states of matter. There are no hinge states along the $x$ direction because of the assumed translational symmetry.

Figure 2

(a₁) Energy spectrum of the Dirac semimetal ribbon in a magnetic field $B_y=15\mathrm{T}$ ($\theta =0$), with open (periodic) boundary conditions along the $y$ and $z$ ($x$) directions. The ribbon size is $n_{y}a=100\mathrm{nm}$ and $n_{z}a=200\mathrm{nm}$. The lattice constant $a=1\mathrm{nm}$. (a₂)–(a₅) The wave function distributions for the states marked by the horizontal dash lines in (a₁). Inside the Landau-level gap (at $E_F=-4.5\mathrm{meV}$), the hinge states lie on both sides, as schematically illustrated in (a₆). (a₇) The Fermi surface of the Dirac semimetal slab in (a₁), when considering only the Zeeman effect of the magnetic field. (a₈) The wave function distribution of the state marked by the star in (a₇). (b₁)–(b₈) The same as (a₁)–(a₈), except for $\theta =5°$ ($B_z=B_y\tan \theta$). The hinge states become one sided at $E_F=0$ and switch their positions at $E_F=-4.5\mathrm{meV}$. (c₁)–(c₇) The same as (a₁)–(a₇) except for $\theta =-5°$. The one-sided hinge states are swapped compared to those in (b₁)–(b₇). Red, blue, and purple represent top, bottom, and top-bottom surfaces, respectively.

Figure 3

The Landau bands of the Dirac semimetal [(a)–(c)] as functions of the wave vector $k_y$ at $B_y=15\mathrm{T}$ and [(d)–(f)] as functions of the magnetic strength $B_y$ at $k_y=0$. The color bars indicate the weights of $h_0(\mathbf{k})$ (red) and $h_0^{*}(-\mathbf{k})$ (blue) in Eq. (1). (a),(d) $\theta =0$ and the Zeeman coupling ${\mathrm{\Delta }}_y=0$. Only the energy spectra for $h_0(\mathbf{k})$ are shown as $h_0(\mathbf{k})$ and $h_0^{*}(-\mathbf{k})$ show the same behavior. (b),(e) $\theta =0$ and ${\mathrm{\Delta }}_y\ne 0$. (c),(f) $\theta =5°$ and the Zeeman energies ${\mathrm{\Delta }}_y\ne 0$ and ${\mathrm{\Delta }}_z\ne 0$. The orbital effect of $B_z$ is not included.

Figure 4

(a) The disorder-averaged conductance as a function of the disorder strength $U$ for different system sizes. The error bars show the standard deviation for 200 samples. $E_F=-4.5\mathrm{meV}$. (b) The disorder-averaged local density of states with system size $n_{x}a=200\mathrm{nm}$, $n_{y}a=n_{z}a=80\mathrm{nm}$. $U=20\mathrm{meV}$ and $E_F=-4.5\mathrm{meV}$. (c) Schematic of the scanning tunneling Hall measurements for probing the hinge states. Gray, green, and yellow correspond to the Dirac semimetal, in-plane Hall-bar electrodes, and scanning tunneling microscopy tip, respectively. (d₁), (d₂) The local density of states $\rho (n_y,z)$ as a function of $z$ at $E_F=-4.5\mathrm{meV}$ for $\theta =0$ and $\theta =5°$, respectively. (e) The total local density of states for the 20 outermost layers near the left ($L$) and right ($R$) hinges on the topmost layer $D_{L/R}$ as functions of the tilting angle $\theta$ with $E_F=-4.5\mathrm{meV}$. (d₃),(d₄),(f) The same as (d₁), (d₂), and (e), except for $E_F=4.5\mathrm{meV}$. (g) $\log (D_L/D_R)$ as a function of $E_F$ and $\theta$. In (c)–(g), the ribbon size is $n_{y}a=100\mathrm{nm}$ and $n_{z}a=200\mathrm{nm}$; the lattice constant $a=1\mathrm{nm}$.