Conductance oscillation in surface junctions of Weyl semimetals

Fermi arc surface states, the manifestation of the bulk-edge correspondence in Weyl semimetals, have attracted much research interest. In contrast to the conventional Fermi loop, the disconnected Fermi arcs provide an exotic two-dimensional (2D) system for exploration of novel physical effects on the surface of Weyl semimetals. Here, we propose that visible conductance oscillation can be achieved in planar junctions fabricated on the surface of a Weyl semimetal with a pair of Fermi arcs. It is shown that Fabry-Pérot-type interference inside the 2D junction can generate conductance oscillation with its visibility strongly relying on the shape of the Fermi arcs and their orientation relative to the strip electrodes, the latter clearly revealing the anisotropy of the Fermi arcs. Moreover, we show that the visibility of the oscillating pattern can be significantly enhanced by a magnetic field perpendicular to the surface taking advantage of the bulk-surface connected Weyl orbits. Our work offers an effective way to identify Fermi arc surface states through transport measurement and predicts the surface of Weyl semimetals as a novel platform for the implementation of 2D conductance oscillations.

Figure 1

(a) Schematic illustration of the planar normal metal (N)-FA-normal metal (N) on the top surface of a WSM slab and scattering of particles at the interface. The trajectories of electrons (solid circles) are sketched as solid lines, where the colors of them denote that they belong to different $k_z$ regions. (b) Two regions $\mathrm{I}$ and $\mathrm{II}$ for left incident electrons are defined by the transverse momentum $k_z$. (c) The corresponding band structures for a fixed $k_z$ of the two regions. The red lines denote the FA top surface states and black arrows indicate the moving directions.

Figure 2

Conductance oscillations for different FAs given by Eq. (3). Top rows illustrate the FAs with different curvature and length, where the relevant parameters are (a) $k_0=0.6{\mathrm{nm}}^{-1}$, $d=1.5\mathrm{eV}{\mathrm{nm}}^2$; (b) $k_0=0.6{\mathrm{nm}}^{-1}$, $d=0.3\mathrm{eV}{\mathrm{nm}}^2$; and (c) $k_0=0.4{\mathrm{nm}}^{-1}$, $d=1.5\mathrm{eV}{\mathrm{nm}}^2$. [(d)–(f)] Middle rows illustrate the corresponding analytical results of the transmission coefficient $\tilde{T}(\theta =0)$ as a function of $\varepsilon$ and $k_z$. [(g)–(i)] Bottom rows exhibit the corresponding conductance $\sigma$ after integrates over $k_z$. The other parameters are set as $a=1\mathrm{nm}$, $R_1=R_2=1$, $M_1=1.25\mathrm{eV}{\mathrm{nm}}^2$, and $k_1=1.2{\mathrm{nm}}^{-1}$.

Figure 3

FAs and conductance oscillations for different WSM given by Eq. (13), with $V$ being the on-site potential on the surface layer. Top rows illustrate the FA spectra with different curvature and length, where the relevant parameters are (a) $k_0=0.6{\mathrm{nm}}^{-1}$, $V=1.0\mathrm{eV}$; (b) $k_0=0.6{\mathrm{nm}}^{-1}$, $V=0.2\mathrm{eV}$; and (c) $k_0=0.4{\mathrm{nm}}^{-1}$, $V=1.0\mathrm{eV}$. [(d)–(f)] Middle rows illustrate the corresponding numerical results of the transmission coefficient $T$ as a function of $\varepsilon$ and $k_z$. [(g)–(i)] Bottom rows exhibit the corresponding conductance $\sigma$ after integrates over $k_z$. The other parameters are $a=1\mathrm{nm}$, $U_1=U_2=1.0\mathrm{eV}$, $M_1=M_2=1.25\mathrm{eV}{\mathrm{nm}}^2$, $v_y=0.66\mathrm{eV}\mathrm{nm}$, $k_1=1.2{\mathrm{nm}}^{-1}$, $C=0.5\mathrm{eV}{\mathrm{nm}}^2$, ${\mu }_N=1.0\mathrm{eV}$, and $t_N=0.5\mathrm{eV}{\mathrm{nm}}^2$.

Figure 4

Conductance oscillations in disordered WSMs. Top rows illustrate the transmission coefficient $T$ as a function of $\varepsilon$ and $k_z$, where the disorder strengths are (a) 0, (b) 0.1, and (c) 0.2 eV. Bottom rows exhibit the corresponding conductance $\sigma$ after integrates over $k_z$. Other parameters are the same as those in Fig. 3.

Figure 5

(a) The transmission coefficient $T$ as a function of $\varepsilon$ and $k_z$ when the Fermi surface of the WSM is set to $0.2\mathrm{eV}$. (b) The corresponding differential conductance after summing over the transverse channels $k_z$. The band structures of the $k_z$ channels with (c) $k_z=0{\mathrm{nm}}^{-1}$, (d) $k_z=0.3{\mathrm{nm}}^{-1}$, and (e) $k_z=0.6{\mathrm{nm}}^{-1}$. The spatial current distributions (f) $J_1$, (g) $J_2$, and (h) $J_3$ for various $k_z$ slice and a finite Fermi energy $0.1\mathrm{eV}$ marked in panel (a). The other parameters are the same as those in Fig. 3.

Figure 6

Conductance oscillations in WSM with FAs of different orientations. [(a)–(c)] Upper panel: FA spectra for different azimuthal angles $\theta$. [(d)–(f)] Middle panel: Corresponding transmission probability as a function of $\varepsilon$ and $k_z$. [(g)–(i)] Lower panel: Corresponding differential conductance after summing over all the transverse channels $k_z$. The other parameters are the same as those in Fig. 3.

Figure 7

Conductance oscillations in a WSM under a magnetic field perpendicular to the surface. (a) FA spectrum for azimuthal angle $\theta =0$, and electrons slide along the FA driven by the Lorentz force. (b) Trajectories of electrons in real space corresponding to the left panel. (c) Resolution of the conductance as a function of the magnetic field, where $B_C$ is the saturated magnetic field. The transmission probability as a function of $\varepsilon$ and $k_z$ with (d) $B=0$, (e) $B=0.4B_C$, and (f) $B=0.8B_C$. [(g)–(i)] The corresponding conductance after summing over all the transverse channels $k_z$ for different magnetic fields in the $\widehat{y}$ direction. The other parameters are the same for those in Fig. 3.

Figure 8

The current distributions (a) $J_1$ and (b) $J_2$ for a given $k_z$ and $\varepsilon$ marked in Fig. 7.