3D quantum Hall effect of Fermi arcs in topological semimetals

TL;DR

This paper predicts a novel 3D quantum Hall effect in topological semimetals arising from Fermi arcs, enabled by Weyl node tunneling, with unique edge states and quantized Hall plateaus observable via gating.

Contribution

It introduces a mechanism for 3D quantum Hall effect in topological semimetals through Fermi arc tunneling, distinct from surface states in topological insulators.

Findings

Fermi arcs can form complete loops via Weyl node tunneling.

Hall conductivity shows transition from 1/B dependence to quantized plateaus.

The effect can be realized in materials like TaAs, Cd3As2, Na3Bi.

Abstract

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, Cd$_3$As$_2$, or Na$_3$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.

Equations

$$H=D_1{k}_y^2+D_2(k_x^2+k_z^2)+A(k_x{\sigma }_x+k_y{\sigma }_y)+M(k_w^2-k^2){\sigma }_z,$$

$$H_{\mathrm{arc}}=D_1{k}_w^2+v{k}_x+(D_2-D_1)(k_x^2+k_z^2),$$

$$k_x^2+k_z^2+2a{k}_x<k_w^2,$$

$${\sigma }_{\mathrm{H}}=\frac{e^2\hslash }{i{V}_{\mathrm{eff}}}\sum _{\delta ,{\delta }^{\prime }\ne \delta }\frac{⟨{\Psi }_{\delta }|v_x|{\Psi }_{{\delta }^{\prime }}⟩⟨{\Psi }_{{\delta }^{\prime }}|v_z|{\Psi }_{\delta }⟩[f(E_{{\delta }^{\prime }})-f(E_{\delta })]}{(E_{\delta }-E_{{\delta }^{\prime }})(E_{\delta }-E_{{\delta }^{\prime }}+i\Gamma )}.$$

$${\sigma }_{\mathrm{H}}^{\mathrm{arc}}=\frac{e^2}{h}\text{sgn}(R)\text{sgn}(eB)\lfloor \frac{S_{\mathrm{I}}/(2\pi {)}^2}{eB/h}+\frac{1}{2}\rfloor ,$$

$$H$$

Full text

Introduction - The discovery of the quantum Hall effect opens the door to the field of topological phases of matter Klitzing et al. (1980). In a strong magnetic field, the energy spectrum of a 2D electron gas evolves into Landau levels. The Landau levels deform at the sample edges and cut through the Fermi energy, forming 1D edge states protected by topology Thouless et al. (1982). Electrons can flow through the edge states in a dissipationless manner, giving rise to Hall conductance in units of $e^{2}/h$ that defines the quantum Hall effect. The quantum Hall effect can give transport signatures that distinguish different electron gases, such as the half-integer Hall conductance of the 2D massless Dirac fermions in graphene and topological surface states Klitzing et al. (1980); Novoselov et al. (2005); Zhang et al. (2005); Xu et al. (2014); Yoshimi et al. (2015). By contrast, in a 3D electron gas, the extra dimension along the magnetic field direction prevents the quantization of the Hall conductance. Therefore, the quantum Hall effect is usually observed in 2D systems.

In this Letter, we show that the quantum Hall effect is possible in a unique 3D system, specifically, in a topological semimetal, because of the Fermi arcs. The topological semimetal is a 3D topological state of matter Wan et al. (2011); Yang et al. (2011); Burkov and Balents (2011); Xu et al. (2011); Delplace et al. (2012); Jiang (2012); Young et al. (2012); Wang et al. (2012); Singh et al. (2012); Wang et al. (2013); Liu and Vanderbilt (2014); Bulmash et al. (2014), in which energy bands touch at discrete Weyl nodes [Fig 1 (a)]. It is equivalent to a 2D topological insulator for momenta ($k_{z}$ here) between the Weyl nodes, leading to the topologically protected states located at the surfaces [top and bottom in Fig. 1 (c)] parallel to the Weyl node separation direction. The protected states form the Fermi arcs on the Fermi surface [red curves in Figs. 1 (a) and 1(b)]. The Fermi arcs have been seen by ARPES Brahlek et al. (2012); Wu et al. (2013); Wang et al. (2012); Liu et al. (2014a); Wang et al. (2013); Xu et al. (2015a); Wang et al. (2013); Liu et al. (2014b); Neupane et al. (2014); Yi et al. (2014); Borisenko et al. (2014); Weng et al. (2015); Huang et al. (2015a); Lv et al. (2015); Xu et al. (2015b) and can induce novel quantum oscillations Potter et al. (2014); Moll et al. (2016). Topological phases of matter usually come with distinctive transports, making the transport signature of the Fermi arcs an intriguing topic Hosur (2012); Baum et al. (2015); Gorbar et al. (2016); Ominato and Koshino (2016); McCormick et al. (2017).

There are several issues for the Fermi arcs to exhibit the quantum Hall effect. First, the topological origin requires that the states of Fermi arcs occupy only a region between the Weyl nodes Zhang et al. (2016a) [Fig. 1(b)]. At one surface, the Fermi arcs cannot form a closed Fermi loop needed by Landau levels and the quantum Hall effect. We find that the Fermi arcs from opposite surfaces in a topological semimetal slab [Fig. 1(c)] can complete the needed closed Fermi loop [Fig. 1(d)]. Electrons can tunnel between the Fermi arcs at opposite surfaces via the Weyl nodes [Figs. 1(e)-1(g)]. Second, the quantum Hall effect solely from the Fermi arcs requires the bulk carriers to be depleted by tuning the Fermi energy to the Weyl nodes Ruan et al. (2016). Third, we find that the band anisotropy in the bulk Weyl fermions is necessary for the Fermi arcs to form a 2D electron gas. These properties in the quantum Hall effect can provide transport signatures for the Fermi arcs. Compared to the novel quantum oscillations Potter et al. (2014); Moll et al. (2016), the quantum Hall effect of the Fermi arcs contributes a quantum complement to the Fermi arc dominant electronic transports. The Weyl semimetals TaAs family Weng et al. (2015); Huang et al. (2015a); Lv et al. (2015); Xu et al. (2015b); Huang et al. (2015b); Yang et al. (2015); Shekhar et al. (2015); Zhang et al. (2016b) and the Dirac semimetals Cd3As2 and Na3Bi have extremely high mobilities He et al. (2014); Liang et al. (2015); Zhao et al. (2015); Narayanan et al. (2015); Xiong et al. (2015) required by the quantum Hall effect. Low carrier densities Li et al. (2015, 2016); Zhang et al. (2017) and gating Li et al. (2015) have also been achieved. We expect the quantum Hall effect of the Fermi arcs in slabs of the TaAs family, [110] or [1$\bar{1}$0] Cd3As2 Uchida et al. (2017), and [100] or [010] Na3Bi.

Minimal model - We will use a minimal model to illustrate the physics for the Fermi-arc quantum Hall effect. To preserve their topological properties, we need to derive the 2D effective model of the Fermi arcs from a 3D model of Weyl semimetal Shen (2012); Okugawa and Murakami (2014); Lu et al. (2015)

[TABLE]

where $(\sigma_{x},\sigma_{y},\sigma_{z})$ are the Pauli matrices, the wave vector $\mathbf{k}=(k_{x},k_{y},k_{z})$, and $D_{1}$, $D_{2}$, $A$, $M$, and $k_{w}$ are model parameters. We assume that $|M|>|D_{1}|$. The energy dispersion of the model is $E_{\pm}^{\mathbf{k}}=D_{1}k_{y}^{2}+D_{2}(k_{x}^{2}+k_{z}^{2})\pm[M^{2}(k_{w}^{2}-\mathbf{k}^{2})^{2}+A^{2}(k_{x}^{2}+k_{y}^{2})]^{1/2}$, with $\pm$ for the conduction and valence bands, respectively. The model hosts two Weyl nodes at $(0,0,\pm k_{w})$ having energy $E_{w}=D_{2}k_{w}^{2}$ [Fig. 1 (a)], and carries all of the topological semimetal properties Lu and Shen (2017). In contrast to the $k\cdot\sigma$ model, the Fermi arc states can be solved analytically from the model Zhang et al. (2016a).

Open Fermi arc at one surface - First, we show that the Fermi arc at a single surface of a Weyl semimetal cannot host the quantum Hall effect. We focus on the $y=L/2$ surface. By replacing $k_{y}$ with $-i\partial_{y}$ in $H$ and using open-boundary conditions, we can solve the wave function at $k_{x}=k_{z}=0$, and then project $H$ on the wave function to construct the effective model (see the procedure at Lu et al. (2010); Shan et al. (2010); Zhang et al. (2016a) and Sec. S1 of Sup ) for the Fermi arc

[TABLE]

where $v\equiv A\sqrt{M^{2}-D_{1}^{2}}/M$. If there is no anisotropic $D$ terms, the Fermi arc only disperses linearly with $k_{x}$; consequently, the Landau levels cannot be defined. Therefore, the anisotropic $D$ terms are necessary. Moreover, the electron gas of the Fermi arc is distinct from usual 2D electron gases because it is confined within a specific momentum space due to their topological nature Zhang et al. (2016a). For this model, the Fermi arc at the $y=L/2$ surface is confined in a region defined by the constraint

[TABLE]

where $a\equiv AD_{1}/2M\sqrt{M^{2}-D_{1}^{2}}$. This means that, the wave vectors of the Fermi arcs at the $y=L/2$ surface are only allowed within a circle of radius $\sqrt{k_{w}^{2}+a^{2}}$ centered at $(k_{x}$=$-a$, $k_{z}$=$0)$. The Fermi circle of $H_{\rm arc}$ at a given Fermi energy can only partially overlap with the constraint in Eq. (3), forming an “open” Fermi surface, as shown by the red solid curve in Figs. 1(a) and 1(b). Because of the open Fermi surface, electrons cannot undergo complete cyclotron motion in a perpendicular magnetic field. Thus, the 2D electron gas of Fermi arc at a single surface cannot form well-defined Landau levels required by the quantum Hall effect.

Fermi arc loop via “wormhole” tunneling - In contrast, the Fermi arcs at two opposite surfaces of a slab of Weyl semimetal, with the assistance of the Weyl nodes, can form a closed Fermi loop to support the quantum Hall effect. For a Weyl semimetal slab of thickness $L$, we consider two opposite surfaces at $y=\pm L/2$ [Fig. 1 (c)]. Similar to Eqs. (2) and (3), the model and constraint at the $y=-L/2$ surface are found as $H^{\prime}_{\rm arc}=D_{1}k_{w}^{2}-vk_{x}+(D_{2}-D_{1})(k_{x}^{2}+k_{z}^{2})$ and $k_{x}^{2}+k_{z}^{2}-2ak_{x}<k_{w}^{2}$, respectively. Figure 1 (d) shows the Fermi arcs at $E_{F}=E_{w}$ and constraints at the two surfaces. The Fermi arcs at opposite surfaces shift along opposite directions on the $k_{x}$ axis. The two open Fermi arcs [red and blue curves in Fig. 1 (d)] can form a Fermi loop well inside the overlapping constraint regions; thus, all states on this loop are allowed. We numerically calculate the energy spectrum for this slab by using the basis $\varphi_{n}(y)=\sqrt{2/L}\sin[n\pi(y/L+1/2)]$ (Sec. S2 of Sup ). Figure 2 (a) verifies the above picture for the Fermi loop formation. The energy band for the Fermi loop is marked as “I” (arc I). There is another band (marked as “II”), which appears below arc I at $k_{x}=\pm 0.1$ nm -1 but buried in the bulk valence bands. Moreover, the wave function on the Fermi loop can evolve from located at one surface [Figs. 1(e) and 1(g)] to spread out in the $y$ direction [Fig. 1 (f)] when moving from the Fermi arcs to the Weyl nodes. Therefore, the Weyl nodes act like “wormholes” that connect the top and bottom surfaces, and an electron can complete the cyclotron motion. Because the Weyl nodes are singularities in both energy and momentum, the wormhole tunneling can be infinite in both time and space, according to the uncertainty principle. In realistic materials, the tunneling distance is limited by the mean free path, which can be comparable to or longer than 100 nm in high-mobility topological semimetals Huang et al. (2015b); Yang et al. (2015); Shekhar et al. (2015); Zhang et al. (2016b); He et al. (2014); Liang et al. (2015); Zhao et al. (2015); Narayanan et al. (2015); Xiong et al. (2015), even up to 1 $\mu$m Moll et al. (2016), so the thickness in the calculation is chosen to be 100 nm. The loop formed by the Fermi arcs at opposite surfaces via the Weyl nodes can support a 3D quantum Hall effect. The wormhole effect has been addressed in different situations in topological insulators Rosenberg et al. (2010).

The Hall response - Now we demonstrate that arc I of the Weyl semimetal slab can host the quantum Hall effect. The Hall conductivity can be calculated from the Kubo formula (Sec. S3 of Sup )

[TABLE]

Here $|\Psi_{\delta}\rangle$ is the eigenstate of energy $E_{\delta}$ for $H$ in a $y$-direction magnetic field and with open boundaries at $y=\pm L/2$, $v_{x}$ and $v_{z}$ are the velocity operators, $f(x)$ is the Fermi distribution, $V_{\rm eff}$ is the volume of the slab or the area of the surfaces that host the Fermi arcs. $\sigma_{\rm H}$ has a dimension of $e^{2}/h$ in 2D and of $e^{2}/h$ over length in 3D. The sheet Hall conductivity for the slab can be defined as $\sigma_{\rm H}^{s}=\sigma_{\rm H}L$. We use the basis $|\phi_{\nu}(z)\rangle\otimes|\varphi_{n}(y)\rangle$ to find the eigenenergies for a slab in the $y$-direction magnetic field, where $\phi_{\nu}$ are the harmonic oscillator eigenfunctions. Figure 2 (b) shows the sheet Hall conductivity for the topological semimetal slab at Fermi energies far away from the Weyl nodes. $\sigma_{\rm H}^{s}$ follows the usual $1/B$ dependence. As the Fermi energy is shifted towards the Weyl nodes, the slope becomes smaller, indicating decreasing carrier density. Also, quantized plateaus of $\sigma_{\rm H}^{s}$ start to emerge as the Fermi energy approaches the Weyl nodes. When the Fermi energy crosses only arc I [Fig. 2(c)], $\sigma_{\rm H}^{s}$ shows well-formed quantized plateaus in units of $e^{2}/h$, indicating the quantum Hall effect of the Fermi arcs. Here disorder is included in the Kubo formula via the level broadening $\Gamma$. This treatment is capable of giving the quantization in graphene Gusynin and Sharapov (2005), which is massless in 2D. Because of the relation with the Chern number Thouless et al. (1982), the quantum Hall effect can be theoretically studied in the absence of disorder, as those in topological insulators Zyuzin and Burkov (2011); Zhang et al. (2014, 2015); Pertsova et al. (2016). To verify the numerical result in Fig. 2(c), we also calculate analytically the quantum Hall conductance from arc I (Sec. S4 of Sup ), by modeling arc I as an anisotropic parabolic band $H_{\rm arc_{I}}\approx D_{1}k_{w}^{2}+\hbar^{2}k_{x}^{2}/2m_{x}+\hbar^{2}k_{z}^{2}/2m_{z}$, with $m_{x}$ and $m_{z}$ being the effective masses. We can find the quantum Hall conductance of arc I in the clean limit $\Gamma\rightarrow 0$ Jain (2007); Laughlin (1981)

[TABLE]

where $\lfloor...\rfloor$ stands for rounding down, $R=D_{2}-D_{1}$ and the area of arc I in momentum space is $S_{\rm I}=2k_{w}^{2}(1+v^{2}/4R^{2}k_{w}^{2})\arctan(2|R|k_{w}/|v|)-|v|k_{w}/|R|$. Figure 2 (c) shows a good agreement between the analytic and numerical results on the Hall conductance and width of the plateaus.

Where are the edge states of the Fermi arcs? - Figures 1(h) and 1(i) show that the edge states of the Fermi arcs have a unique 3D spatial distribution. Figure 1 (h) shows the energies of the Landau levels in the $y$-direction field. The energies deform into edge states near $z_{0}=\pm 100$ nm. The green dot in Fig. 1(h) and the green curve in Fig. 1(i) show that the edge state near $z_{0}=100$ nm mainly distributes near the top surface at $y=50$ nm. By contrast, the edge state near $z_{0}=-100$ nm mainly distributes near the bottom surface (orange dot and curve). This unique 3D distribution of the edge states of the Fermi arcs can be probed by a combined measurement of in-plane transport and STM. Different from topological insulators Xu et al. (2014); Yoshimi et al. (2015), the Fermi-arc quantum Hall effect requires the collaboration of the two surfaces. Note that a 100-nm slab is still a 3D object. Therefore, the quantum Hall effect at Weyl nodes $E_{F}=E_{w}$ and the Fermi energy dependence can serve as transport signatures of the Fermi arcs. The above picture for the Fermi-arc quantum Hall effect can work for Weyl semimetals Weng et al. (2015); Huang et al. (2015a); Lv et al. (2015); Xu et al. (2015b); Huang et al. (2015b); Yang et al. (2015); Shekhar et al. (2015); Zhang et al. (2016b); Ruan et al. (2016).

Topological Dirac semimetals - Because of time-reversal symmetry, a single surface of the Dirac semimetal, such as Cd3As2 and Na3Bi, can support a complete Fermi loop required by the quantum Hall effect. The same-surface Fermi arc loop is not that robust and may get deformed Kargarian et al. (2016), and thus may show different characteristics (such as positions and widths of the Hall plateaus) compared to the two-surface Fermi arc loop. The spectrum and Fermi-arc Hall effect in Dirac semimetals can be studied (Secs. S5 and S6 of Sup ) by using the Hamiltonian Wang et al. (2012, 2013); Jeon et al. (2014)

[TABLE]

where $g_{s}$ and $g_{p}$ are the $g$ factors for the $s$ and $p$ bands Jeon et al. (2014), $k_{\pm}=k_{x}\pm ik_{y}$, $\varepsilon_{0}(\bm{k})=C_{0}+C_{1}k_{z}^{2}+C_{2}(k_{x}^{2}+k_{y}^{2})$, and $M(\bm{k})=M_{0}+M_{1}k_{z}^{2}+M_{2}(k_{x}^{2}+k_{y}^{2})$. The $x$, $y$, and $z$ axes in the Hamiltonian are defined along the [100], [010], and [001] crystallographic directions, respectively. The samples of Cd3As2 are usually cleaved or grown along [112] or [110] directions, which can be defined as the new $y^{\prime}$ axis for convenience, as shown in Fig. 3(a). For the [112] slab, the parameters Cano et al. (2017) yield that the Fermi arc bands are close to the bulk subbands [Fig. 3(c)], implying that the quantum Hall effect may exhibit a fourfold degeneracy. For the [110] slab, the quantum Hall effect may come from pure Fermi arc states [Fig. 3(d)]. Figure 3(e) shows that for the [110] Cd3As2 slab the odd plateaus are wider than the even plateaus, because the $g$ factor is large. This feature is robust when rotating the magnetic field. The Na3Bi samples cleaved along the [010] direction Xu et al. (2015a) can be used to probe the quantum Hall effect of the Fermi arcs [Figs. 3(f) and 3(g)]. The $C$ and $M$ terms in Eq. (6) secure the 2D Fermi arc on the (010) surface.

We thank helpful discussions with Faxian Xiu, Wang Yao, Hongming Weng, Xi Dai, Dapeng Yu, Yusheng Zhao, Wenqing Zhang, Jiaqing He, and Lang Chen. This work was supported by NBRPC (Grant No. 2015CB921102), Guangdong Innovative and Entrepreneurial Research Team Program (Grant No. 2016ZT06D348), the National Key R & D Program (Grant No. 2016YFA0301700), the National Natural Science Foundation of China (Grants No. 11534001, No. 11474005, and No. 11574127), and the Science, Technology and Innovation Commission of Shenzhen Municipality (Grant No. ZDSYS20170303165926217).