Andreev Reflection in Fermi Arc Surface States of Weyl Semimetals

In 1929, Hermann Weyl proposed a new type of massless fermion with definite chirality Weyl (1929). After that, great efforts have been made in pursuing such elemental particles in high-energy particle physics Armitage et al. (2018), yet up until now, no candidate Weyl fermion has been reported Kajita (2016); McDonald (2016). In recent years instead, Weyl fermions are surprisingly found in an alternative form of quasi-particle excitations in a class of solid-state materials with conic band crossing, called Weyl semimetals (WSMs) Wan et al. (2011). The discovery of WSM opens up a new avenue for the study of relativistic Weyl fermion in condensed matter physics Murakami (2007); Burkov and Balents (2011); Weng et al. (2015); Huang et al. (2015a); Lv et al. (2015a); Xu et al. (2015a, b, c, 2016a); Deng et al. (2016); Yang et al. (2015); Huang et al. (2016); Tamai et al. (2016); Jiang et al. (2017); Belopolski et al. (2016); Lv et al. (2015b); Junwei et al. (2019). It provides an interesting platform for the experimental testing of predictions made by quantum field theory Adler (1969); Bell and Jackiw (1969); Nielsen and Ninomiya (1983) in terms of anomalous transport and optical properties in the condensed matter context Zyuzin and Burkov (2012); Aji (2012); Son and Spivak (2013); Chernodub et al. (2014); Zhou et al. (2013); Burkov (2015); Ma and Pesin (2015); Zhong et al. (2016); Spivak and Andreev (2016); Hirschberger et al. (2016); Huang et al. (2015b); Shekhar et al. (2015); Du et al. (2016); Wang et al. (2016a); Zhang et al. (2016); Chen and Franz (2016); Shuhei et al. (2014); Zhang et al. (2018); Breunig et al. (2019).

In addition to its high-energy counterpart, WSM also exhibits unique properties that exist only in the condensed matter context, especially the emergence of Fermi arc (FA) surface states on the sample boundaries Wan et al. (2011). In WSMs, Weyl points always appear in pairs with opposite topological charge (chirality) Nielsen and Ninomiya (1983), the FA spanning between each pair in the surface Brillouin zone Wan et al. (2011). Such disconnected FAs cannot be realized in any noninteracting 2D bulk states so that its emergence can serve as the hallmark of WSMs Huang et al. (2015a); Lv et al. (2015a); Xu et al. (2015a, b, c, 2016a); Deng et al. (2016); Yang et al. (2015); Huang et al. (2016); Tamai et al. (2016); Jiang et al. (2017); Belopolski et al. (2016); Lv et al. (2015b); Junwei et al. (2019). It was recently shown that the configurations of the FA are sensitive to the details of the sample boundary Morali et al. (2019); Yang et al. (2019); Ekahana et al. (2020), which opens the possibility for engineering FA and exploring its applications through surface modification. With this prospect, it is of great importance to extract and analyze clear signatures of the FA from measurement information in the presence of gapless bulk states. Surface transport measurements provide a natural solution to this goal because the setups contact directly to the sample boundaries Chen et al. (2018a, 2020).

In this work, we propose that Andreev reflection (AR) in a planar normal metal (N)-superconductor (S) junction on the WSM surface can provide a unique signature of the FA. The junction consists of two parallel N and S strip electrodes mediated by the topological surface states in between; see Fig. 1(a). For the unclosed FA, there generally exist two regions of different AR scenarios, referred to as I and II in Fig. 1(c). In region I, no normal reflection channel is available [Fig. 1(a)] so that there is a perfect AR with probability equal to unity within the energy gap [cf. Fig. 2(c)], in spite of the interface barrier. On the contrary, AR is strongly suppressed in region II by normal reflection at the boundary of the S [Fig. 1(b)] which results in a pair of resonant peaks of the AR probability at the gap edges [cf. Fig. 2(a)]. The proportion of electrons in two regions relies on the relative orientation between the WSM and the normal of the planar junction. This leads to a crossover between the double-peak and plateau structures in the conductance spectra by changing the orientation of the planar junction in different samples [Fig. 2(d)]. Two limiting cases occur as the electrons reside entirely in region I or II [Figs. 2(a) and 2(c)]. Remarkably, such a crossover can be greatly facilitated by imposing a magnetic field to a properly orientated planar junction such that the two AR regions coexist. The magnetic field drives electrons sliding along the FA and simultaneously opens up a transport channel in the bulk, i.e., the chiral Landau band, connected with the surface FA Potter et al. (2014). As a result, part of the electrons can switch between the two AR regions while the remaining ones penetrate the bulk with negligible contribution to the surface transport signature. In this way, the same crossover behavior of the conductance can be achieved along with a reduction of its magnitude. The AR spectra have the advantage that two scenarios can be clearly revealed by different shapes of the conductance rather than its magnitude, which provides a distinctive and robust signature of the FAs.

The rest of this paper is organized as follows. In Sec. II, we introduce the model and calculation details. In Sec. III, we show that a crossover of the conductance from the suppressed to perfect AR can be achieved by varying the orientation of the planar junction. We show that such an effect can be facilitated by the magnetic field in Sec. IV. Finally, some remarks are given in Sec. V.

We consider a WSM with four Weyl points, which can be captured by the effective two-band model as Chen et al. (2020)

where $v_{y}$ is the velocity in the $y$ direction, $k_{0,1}$ and $M_{1,2}$ are parameters, $\sigma_{x,y,z}$ are the Pauli matrices in the pseudo-spin space. The two bands are degenerate at four Weyl points $\bm{k}_{W}=({\pm}k_{1},0,{\pm}k_{0})$. The main results of this work are associated with the relative orientation between the FAs and the planar junction. We define $\theta$ as the azimuthal angle between the symmetry axis of the FA and the normal of the junction (set to the $x$ axis); see Fig. 1(c). Experimentally, this can be alternatively realized by fabricating various planar junctions along different directions. The FAs with azimuthal angle $\theta$ correspond to the rotated Hamiltonian $H_{W}(\bm{k})=H_{W}^{0}(U_{y}^{-1}\bm{k})$ with $U_{y}(\theta)=\left(\begin{smallmatrix}\cos\theta&0&-\sin\theta\\ 0&1&0\\ {\sin\theta}&0&{\cos\theta}\end{smallmatrix}\right)$ as the rotation operator around the $y$-axis Chen et al. (2013).

The configurations of the FAs are revealed by the spectra function $\mathcal{A}(E)=-(1/\pi)\text{Im}G^{R}(E)$, where $G^{R}(E)$ is the retarded Green’s function, under the open boundary condition in the $y$ direction. To yield curved FAs similar to those in real materials Koepernik et al. (2016); Belopolski et al. (2017); Haubold et al. (2017); Ekahana et al. (2020), an on-site potential is imposed to the top layer of the WSM lattice, which modifies the dispersion of the surface statesChen et al. (2020) through surface band bending effectBianchi et al. (2010); King et al. (2011); Benia et al. (2011) . The FAs for different azimuthal angle $\theta$ are shown in Figs. 2(a)-2(c).

To investigate the AR, we write the Hamiltonian for the whole system in Nambu space. The electron and hole components are decoupled in the WSM so that the Bogoliubov-de-Gennes Hamiltonian takes a diagonal form of $\mathcal{H}_{W}(\bm{k})=[H_{W}^{0}(\bm{k}),0;0,-H_{W}^{0*}(-\bm{k})]$. For a given $k_{z}$, the Hamiltonian $\mathcal{H}_{W}(\bm{k})$ defines a 2D slice in momentum space. Edge states induced by the non-trivial band topology of the bulk states emerge as the $k_{z}$ slice intersects the FAs under the open boundary condition Yang et al. (2011), which provide scattering channels for electrons (holes). Throughout the work, all the results are calculated using a lattice version of this model obtained by substituting $k_{i=x,y,z}\rightarrow a^{-1}\sin k_{i}a$ and $k_{i}^{2}\rightarrow 2a^{-2}(1-\cos k_{i}a)$, with $a$ the lattice constant of the fictitious cubic lattice (see Appendix A for details).

We then discuss the transport process of the 2D slices with different $k_{z}$. Generally, the FA can be divided into two parts, according to whether the normal reflection channels exist or not; see Fig. 1(c). In region I, only a single chiral edge state exists for each $k_{z}$ so that no normal reflection can occur; see Fig. 1(d). However, a hole channel is available for the AR, which corresponds to the electron-paired state with opposite $k_{z}$ as sketched by the blue dashed circle in Fig. 1(c). As a result, perfect AR with unity probability can be realized. Such a band structure for a given $k_{z}$ breaks particle-hole symmetry, which cannot be realized in any 2D system. It appears only as a subsystem of the whole 3D WSM, where two paired electrons are from opposite $k_{z}$ slices carrying zero net momentum. As all the $k_{z}$ channels are taken into account, the particle-hole symmetry retains. In region II, both normal reflection and AR channels exist as shown in Figs. 1(c) and 1(d), similar to a conventional metal. Given that the normal reflection generally occurs at the boundary of the S electrode due to the interface barrier or momentum mismatch, a suppression of AR is expected in region II, giving rise to two resonant peaks at the edges of the band gap.

We solve the transport problem through the surface planar junction as shown in Fig. 1(a). The N electrode deposited on top of the WSM is described by the effective Hamiltonian as $\mathcal{H}_{N}(\bm{k})=(C\bm{k}^{2}-\mu_{N})\tau_{z}$ with $C$ determined by the effective mass of the electron, $\mu_{N}$ the chemical potential, and $\tau_{x,y,z}$ the Pauli matrices in Nambu space. Similarly, the S bar is captured by $\mathcal{H}_{S}(\bm{k})=(C\bm{k}^{2}-\mu_{S})\tau_{z}+\Delta\tau_{x}$ with a different chemical potential $\mu_{S}$ and a finite $s$-wave pair potential $\Delta$. Due to the proximity effect, an effective superconducting gap $\Delta_{e}(k_{z})$ can be induced in the FA under the S electrode (see Appendix B for details). We assume that the size of both strip electrodes in the $z$ direction is much larger than the Fermi wavelength and their boundaries are smooth enough that the transverse momentum $k_{z}$ is approximately conserved during scattering. Then by taking $k_{z}$ as a parameter, the 3D system can be decomposed into a set of 2D slices labelled by $k_{z}$, thus simplifying the transport calculation.

We first study the transport properties of the 2D slices of the system labelled by $k_{z}$. For a hybridized square lattice, the Hamiltonian at a given $k_{z}$ is set as $\mathcal{H}_{W}(\bm{k})$, $\mathcal{H}_{N}(\bm{k})$, and $\mathcal{H}_{S}(\bm{k})$ for different parts. The coupling between the N (S) and the WSM is described by coupling strength $t_{N}$ ($t_{S}$) between two outmost lattice layers of contacting areas. The lattice Hamiltonian for calculation is elucidated in Appendix A. The thickness of the WSM and the N (S) electrode in the $y$ direction is 100 nm and 50 nm, respectively. The width of the hopping area between the WSM and the N electrode is $W=20$ nm, the separation between two electrodes is $L=40$ nm [cf. Fig. 1(b)] and the hopping between the WSM and the S electrode extends to infinity in the $x$ direction. An on-site potential of $6$ eV is introduced at the boundary line of the S electrode to simulate the interface barrier or the momentum mismatch in the planar junction. Both the WSM and N (S) electrodes connect to the leads extending to infinity in the $\pm x$ directions. In the energy scale smaller than $\Delta_{e}(k_{z})$, the current is dominated by the AR. For the two-dimensional lattice with fixed $k_{z}$, the scattering process can be described by

with $\psi^{in(out)}_{i,e(h)}$ represents the income (outgoing) wave amplitudes of electrons (holes) at the N electrode, $r^{ij}_{he}$ describes the scattering amplitude from electrons of channel $j$ to holes of channel $i$ at the same lead. Then the AR probability can be calculated by taking the trace of the electron-to-hole reflection matrix, $\text{A}_{k_{z}}(E)=\text{Tr}[\hat{r}_{he}^{\dagger}(E,k_{z})\hat{r}_{he}(E,k_{z})]=\sum\limits_{i,j}{|r_{he}^{ij}(E,k_{z}){|^{2}}}$, using KWANT Groth et al. (2014), which describes the AR process that an electron is incident from the N electrode and a hole is reflected back. We plot $\text{A}_{k_{z}}(E)$ in Figs. 2(a)-2(c) for different orientations of the FAs. In the limiting case of $\theta=0$ in Fig. 2(a), all $k_{z}$ is in region II, so that the AR is strongly suppressed, leaving only two resonant peaks at $E=\pm\Delta_{e}$. Note that $\Delta_{e}(k_{z})$ slightly varies with $k_{z}$, so that the positions of resonant peaks for different $k_{z}$ do not coincide. It stems from that the surface states labeled by $k_{z}$ possess different spreading in the $y$ direction, which determines the effective coupling between the surface states and the superconductor. Therefore, the proximity effect and the induced gap $\Delta_{e}(k_{z})$ varies with $k_{z}$; see Appendix B for details. In a general case, e.g., $\theta=0.15\pi$ in Fig. 2(b), region I and II coexist [cf. Fig. 1(c)]. As $k_{z}$ lies in region I, perfect AR occurs with nearly unity probability within $\Delta_{e}$ ; In stark contrast, for $k_{z}$ in region II, the AR probability exhibits double-spike behavior, which indicates a strong normal reflection. In the opposite limit, e.g., $\theta=0.25\pi$ in Fig. 2(c), all the electrons reside in region I, and so the AR probability exhibits a perfect AR plateau within $\Delta_{e}$. The AR probability is slightly smaller than unity which stems from that in our simulation, the incident channels of the FA are not fully occupied by the electrons injected from the N electrod. Nevertheless, the perfect AR can be manifested in the subgap plateau of its probability.

The crossover from suppressed to perfect AR can be probed by the conductance spectra. Imposing a bias voltage $V$ between the N and S electrodes drives a current $J$ in the S; see Fig. 1(a). The differential conductance $G=\partial J/\partial V$ at zero temperature can be obtained by summing the contributions from all the $k_{z}$ channels as

where the conductance $g_{k_{z}}$ for each $k_{z}$ channel is calculated by the Blonder-Tinkham-Klapwijk formula Blonder et al. (1982). $\text{N}_{k_{z}}$ is the number of incident channels below the Fermi energy in the N electrode and $\text{B}_{k_{z}}(E)=\text{Tr}[\hat{r}_{ee}^{\dagger}(E,k_{z})\hat{r}_{ee}(E,k_{z})]=\sum\limits_{i,j}{|r_{ee}^{ij}(E,k_{z}){|^{2}}}$ is the normal reflection probability.

The absolute value of conductance $G$ relies on sample details such as the length of the strip electrodes, which is not important to our main conclusion. We plot the normalized conductance $G/G_{0}$ with $G_{0}=G_{\Delta=0}$ in Fig. 2(d) for different $\theta$. The conductance spectra are contributed by all the $k_{z}$ channels, thus depending on the weight of two AR regions. In the limiting case of $\theta=0$, all the $k_{z}$ channels are in region II, giving rise to a double-peak structure in the conductance spectra, the conductance within the gap being strongly suppressed. As $\theta$ increases from zero, a portion of $k_{z}$ channels transfer from region II to I so that the AR probability with either double-spike or plateau shape exists for different $k_{z}$ channels [Fig. 2(b)]. Consequently, the conductance peaks are lowered accompanied by a rise of the conductance plateau within the gap. As $\theta$ exceeds the threshold $\tan^{-1}(k_{0}/k_{1})$, all the electrons reside in region I with perfect AR. Therefore, the conductance exhibits a plateau within the gap corresponding to perfect AR in all the $k_{z}$ channels. The crossover from the double-peak to plateau structure in the conductance spectra originates from the high anisotropy of FA configurations and thus provides a unique signature of the FA.

Such a crossover can be more easily observed by imposing a magnetic field $B$ in the $y$ direction. In this way, only a single sample with properly orientated planar junction is required. Here, we take $\theta=0.15\pi$. The incident electrons in the right-moving channels will slide along the FA by the Lorentz force, leading to a switching between two AR regions. Specifically, for $B<0$, a portion of electrons originally in region I are driven into region II, resulting in a transition from perfect to suppressed AR accordingly; see Fig. 3(a). Meanwhile, some of the electrons originally in region II are pushed into the chiral Landau bands of the bulk due to the surface-bulk connection at the Weyl points Potter et al. (2014); see Fig. 3(c). Those electrons cannot reach the S electrode so that they do not contribute to the current $J$ flowing into the S. On the other hand, some of the reflected electrons also enter the bulk [cf. Fig. 3(a)] and they do affect the conductance spectra via the competition with the AR process. In short, the magnetic field increases the proportion of the transport electrons in region II but reduces the total number of current carriers. This is well reflected in the conductance spectra of Fig. 3(e), where with increasing $B$, there are a more visible double-peak structure and a decrease of the conductance amplitude. As $B$ exceeds a critical value, all electrons reside in region II. If $B$ increases further to the saturation value $B_{c}=\hbar K_{z}/(eL)$, all electrons will transfer into the bulk and there will be no surface electron transport. Here $K_{z}$ is the span of the FA in the $k_{z}$ direction [Fig. 3(a)] and $L$ is the distance between the N and S electrodes [Fig, 1(b)]. For the parameters $K_{z}\simeq 0.71$ nm^-1 in Fig. 2(b) and $L=40$ nm, the saturated magnetic field is evaluated to be $B_{c}\simeq 11.7$ Tesla (see Appendix C for details).

Similarly, for $B>0$, electrons originally in region II are driven into region I [Fig. 3(b)], which induces a transition from the double-spike to plateau structure of the AR probability. Consequently, a plateau-like conductance spectrum gradually form with increasing $B$, accompanied by a reduction of its overall magnitude due to the transfer of electrons from region I into the bulk [Fig. 3(d)]; see Fig. 3(f). The response of the AR spectra to the magnetic field stems from the unique surface-bulk connection so that it provides another unambiguous evidence of the FA.

In the calculation, we adopt the Landau gauge $\bm{A}=({0,0,Bx})$ so that the Peierls substitution $\bm{k}\rightarrow-i\bm{\nabla}\pm e\bm{A}/\hbar$ (taking $e>0$) for both the electron and hole parts retains the $k_{z}$ conservation. The number of electrons that do not reach the S electrode is subtracted from $\text{N}_{k_{z}}$ in Eq. (3) by tracking their trajectory in the $-x$ direction on the bottom surface, and the number of electrons transferring to the bulk for the normal reflection is included in $\text{B}_{k_{z}}$ by tracking their trajectory in the $x$ direction on the bottom surface as well. To include the reduction of the conductance due to the magnetic field, all results in Figs. 3(e) and 3(f) are normalized by the same $G_{0}|_{B=0}$. For a strong magnetic field $B\simeq B_{c}$, all the incident electrons go into the bulk so that the current $J$ flowing into the S electrode is quenched.

Some remarks are made below about the experimental implementation of our proposal. The surface planar NS junction can be achieved by state-of-the-art fabrication techniques Li et al. (2020); Chen et al. (2018b); Ghatak et al. (2018). We considered the WSM with a pair of FAs here, which have been reported in NbIrTe₄ (TaIrTe₄) Koepernik et al. (2016); Belopolski et al. (2017); Haubold et al. (2017); Ekahana et al. (2020), WP₂ Yao et al. (2019), MoTe₂ Wang et al. (2016b), and YbMnBi₂ Borisenko et al. (2019). The main conclusion can be generalized to the situation with more Weyl nodes straightforwardly, as the main results stem from the anisotropy of the open FAs and the proximity effect between the superconductor and the FAs, which do not rely on the number of the Weyl nodes. For the multiple pairs of Weyl nodes, the present results still hold as long as the two regions of the Andreev reflection, without or with backscattering channels [region I and II in Fig. 1(c)], can be well separated in the reciprocal space denoted by $k_{z}$, then the switching between them can be achieved in the same way by tuning $\theta$ or $B$. The FA with a regular shape is beneficial to our proposal, in which the monotonic change of $k_{z}$ channels between two AR regions can be revealed visibly in the conductance spectra. This requires a big separation between Weyl points in momentum space Sun et al. (2015); Koepernik et al. (2016); Chang et al. (2016); Yao et al. (2019); Borisenko et al. (2019); Sie et al. (2019). In our calculations, for simplicity, zero chemical potential was taken in the WSM, where there is a vanishing density of bulk states. In real materials with finite density of bulk states, our main results remain unchanged as long as the FAs are well separated from the bulk states in the surface Brillouin zone. The presence of bulk states will only cause certain leakage of surface electrons, but does not change the current qualitative results. Finally, we focused on spin-degenerate FA in this work, and for the FA with fine spin textures Lv et al. (2015c); Xu et al. (2016b), the analysis of AR will be modified by including the spin degree of freedom.