## 1 Introduction

Weyl semimetals (WSMs) are the topological phases with broken time-reversal or space-inversion symmetry, whose electronic structure is composed of pairs of Weyl nodes with opposite chirality [1], [2]. They are a prototypical representative of the gapless topological materials, and have been experimentally discovered in three-dimensional condensed matters including MoTe_{2}, WTe_{2}, NbAs, TaP, TaAs, and so forth [3], [4], [5], [6]. WSMs are privileged for many intriguing topographies such as anomalous Hall effect [1], [7], surface states with Fermi arcs [4], [5], large second harmonic generation [2], [8], and circular photogalvanic effect [9], [10], etc. The Weyl nodes in WSMs can be anisotropic and tilted away from the vertical axis [11]. Accordingly, WSMs are classified into type-I (partially tilted) phase with vanishing density of states (DOS) at the Weyl point, and type-II (overtilted) phase with a finite DOS at the Fermi energy and an electron and a hole pocket on either side of the Weyl point [12], [13]. To date, ideal type-I Weyl points with symmetric cone spectra have been ascertained in available semimetals (e.g. NbAs, TaP, TaAs) and also in artificial photonic crystal structures [4], [5], [6], [14], [15], [16]. In contrast, it was not until 2015 that the concept of type-II WSMs was theoretically proposed by studying the topological properties of WTe_{2} and MoTe_{2} [13], [17]. Even if type-II WSM phase has been experimentally demonstrated in MoTe_{2}, WTe_{2}, and LaAlGe, among other materials [3], [11], [18], [19], [20], many controversial interpretations of the experimental results still exists. In theory, it allows the tilt of two Weyl nodes to be different [11], [21]. Nevertheless, most of the experimentally observed WSMs are oppositely tilted so far [11]. Attempt to search for more type-II WSMs with various tilts is ongoing from both the theoretical and experimental communities [17], [22].

In photoincs, when a polarized Gaussian wave packet is reflected from or transmitted through an interface, its left- and right-circular polarization components will split in directions perpendicular to the refractive index gradient [23], [24]. This phenomenon originates from the spin-orbital interaction (or coupling) and the angular momentum conservation, which is referred to as the photonic spin Hall effect (PSHE) [23], [24], [25], [26]. The PSHE, which is a photonic counterpart of the spin Hall effect in electronic systems, was predicted in theory by Onoda et al. in 2004 and was first confirmed in experiment by Hosten et al. in 2008 [27], [28]. Nowadays, the integration between topological Weyl systems and the PSHE is leading to a burgeoning research interest into hunting for more new and anomalous phenomena in spin-photonics. For instance, Ye et al. recently observed an enhanced PSHE in the vicinity of the Weyl point in a synthetic Weyl system and a helical Zitterbewegung effect when one wave packet traverses very close to the Weyl point [29]. Chen et al. found that the photonic spin Hall shifts on the surface of a WSM thin film was closely related to the distance of Weyl nodes, thus can be utilized to estimate the separation of Weyl nodes in momentum space [30]. Among the research on topological optics, optical conductivity is a key parameter to control and ameliorate the PSHE occurring at a topological interface [30], [31], [32], [33]. However, most of the previous studies do not involve the effects of tilt of Weyl nodes and chemical potential on the complex optical conductivity matrices of WSMs for all frequency ranges [12], [33], [34], [35]. Therefore, modifying and/or manipulating the PSHE via controlling the tilt and impurity in WSMs, to the best of our knowledge, have not yet been reported so far. The relationship between spin-dependent splitting and the tilt of Weyl nodes (or chemical potential) has remained ambiguous.

In 2019, Sonowal et al. analytically derived the complex optical conductivity matrices of type-I and type-II WSMs as functions of the tilt and chemical potential, which opens up the possibility to solely examine the impacts of tilt *α _{t}* of Weyl nodes and chemical potential

*μ*on the PSHE without using any small angle approximation [12]. Besides, the decisive criterion for a type-II WSM is the direct observation of its tilted band crossing in three directions in the momentum space [18]. However, traditional measuring methods including angle-resolved photoemission spectroscopy and scanning tunneling spectroscopy are sometimes invalid to directly observe the tilting Weyl cone and Fermi arcs in WSMs due to their limited resolution [16], [18]. To exploit more flexible and sensitive detection approaches is currently necessary. Motivated by the advances in topological optics, in this work, we theoretically investigated the PSHE on the surfaces of type-I and type-II WSM films by taking into the factors of

*α*and

_{t}*μ*. Evolutions of in-plane and transverse spin-Hall shifts with

*α*and/or

_{t}*μ*as well as the possible influencing mechanism are discussed in detail. This work may provide one new strategy to distinguish the topological phases in WSMs.

## 2 Theoretical model and optical conductivity

As illustrated in Figure 1A, one bulk WSM contains a pair of oppositely tilted Weyl nodes with broken time reversal symmetry [12], [36], [37]. These two Weyl nodes with chirality *ξ*=±1 are located in the Brillouin zone at ** k**={0, 0, ∓

*Q*}. The projection of nodes connects the ending points of Fermi arcs on the surfaces of Brillouin zone. The tilt degree of Weyl nodes is expressed by

*α*=

_{t}*v*, with |

*/v*_{t}_{F}*α*|<1 being a type-I Weyl node, and |

_{t}*α*|>1 being a type-II Weyl node. Here,

_{t}*v*is the tilt velocity of the

_{t}*ξ*=+1 Weyl node and

*v*is the Fermi velocity. According to the Kubo formula, the interband optical conductivity of bulk WSM can be written as [12]

_{F}where *m*, *n*∈{*x*, *y*}, the parameters of *γ*, *ω* and *ħω***_{k}** stand for the phenomenological damping factor, angular frequency of incident photons and transition energy, respectively.

On the basis of Eq. (1), one can deduce that *σ_{xx}*=

*σ*and

_{yy}*σ*=–

_{xy}*σ*. For simplicity, our model will work in the limit at zero temperature where the Fermi-Dirac distribution function has a Heaviside-step form. Then the real and imaginary parts of the longitudinal conductivity

_{yx}*σ*for a type-I WSM with |

_{xx}*α*|<1 are given by

_{t}with *σ _{ω}*=

*e*

^{2}

*ω*/(6

*hv*),

_{F}*ħω*=2

_{l}*μ*/(1+|

*α*|),

_{t}*ħω*=2

_{u}*μ*/(1–|

*α*|) and

_{t}In the above equations, *μ* is the chemical potential and *ω_{c}*≡

*v*is the ultraviolet cutoff frequency. For a type-II WSM with |

*k*_{F}_{c}*α*|>1, we have

_{t}with *ħω*′* _{u}*=2

*μ*/(|

*α*|–1) and

_{t}The Weyl nodes in a WSM can be viewed as a source or sink of the Berry curvature, which acts as a magnetic field in the momentum space. This finite Berry phase permits nontrivial band topology which can induce an anomalous Hall effect and contributes to the anomalous conductivity [7], [12], [38]. In the cases of type-I and II WSMs, the components of anomalous Hall conductivity are given by

respectively. Thus, the off-diagonal Hall conductivity *σ_{xy}*(

*ω*) including its real and imaginary parts for a type-I WSM with |

*α*|<1 can be calculated by

_{t}with *σ_{μ}*=

*e*

^{2}

*μ*/(

*h*

^{2}

*v*),

_{F}*σ*=

_{Q}*e*

^{2}

*Q*/(π

*h*), and

For a type-II WSM with |*α _{t}*|>1, we have

We considered an ultrathin WSM film whose thickness *d* is much smaller than the wavelength *λ*_{0} of incident photons while being larger than the atomic separation *a*, i.e. *a*<<*d*<<*λ*_{0}. In this limit, the ultrathin WSM can be treated as a 2D atomic layer when it interacts with incident photons [12], [30], [36]. And the surface conductivity of 2D WSM depositing upon a homogeneous substrate can be approximated as *σ_{mns}*=

*dσ*[12], [30], [36]. The nontrivial properties of 2D WSM enter only through the boundary condition for electromagnetic fields into two non-topological media on either side [36]. Here, the parameters of

_{mn}*λ*

_{0}=632.8 nm,

*d*=10 nm and

*v*=1×10

_{F}^{6}m/s are chosen. The Weyl node separation is set as

*Q*=3.2×10

^{8}m

^{-1}which is at the scale of

*Q*in both type-I and II WSMs and matching with that value in WTe

_{2}[18], [38].

Figure 2A, B describe the relative orientation of a pair of Weyl nodes for *α _{t}*>0 and for

*α*<0, respectively. The complex optical conductivities

_{t}*σ*and

_{xxs}*σ*as a function of tilt

_{xys}*α*are shown in Figure 2C and D. It is seen that both

_{t}*σ*and

_{xxs}*σ*of type-I WSM film (|

_{xys}*α*|<1) are less dependent upon the tilt

_{t}*α*and the ratio

_{t}*ħω*/

*μ*. This could imply that doping has little effect on the conductivity of type-I WSM film for a certain frequency. For type-II WSM film (|

*α*|>1), the absolute value of Re(

_{t}*σ*) or Im(

_{xxs}*σ*) gradually decreases with increasing |

_{xxs}*α*| while being nearly not affected by

_{t}*ħω*/

*μ*. As for the Hall conductivity

*σ*of type-II WSM film, it is sensitive to both

_{xys}*α*and

_{t}*ħω*/

*μ*. In order to more clearly reveal the evolution of optical conductivity with the ratio

*ħω*/

*μ*, Figure S1 in the Supplementary Material presents the complex optical conductivities of

*σ*and

_{xxs}*σ*as a function of

_{xys}*ħω*/

*μ*for WSM films with

*α*=±2.5. It further corroborates that the ratio

_{t}*ħω*/

*μ*has little effect on the longitudinal conductivity

*σ*(the difference induced by changing

_{xxs}*ħω*/

*μ*is smaller than 0.0025

*σ*

_{0}). Nevertheless, the function curves of Hall conductivity

*σ*for type-II WSM films with ±

_{xys}*α*are nearly symmetrically varied with increasing the ratio

_{t}*ħω*/

*μ*. In the following Section, we will show that the diversity of optical conductivity of WSM films gives rise to versatile behaviors of photonic spin Hall shifts.

In additon, to verify the accuracy of our calculations, Figure S2 in the Supplementary Material gives the variations of *σ _{xxs}* and

*σ*with different

_{xys}*α*and

_{t}*ħω*/

*μ*values for a certain chemical potential

*μ*=0.021 eV, in which the conductivity spectrum of

*σ*has ever been reported in [12]. One can see that the real part of Hall conductivity

_{xys}*σ*is less sensitive to the photonic energy. However, the

_{xys}*σ*as well as the imaginary part of

_{xxs}*σ*is sensitive to the changes of both

_{xys}*α*and

_{t}*ħω*/

*μ*if the frequency of incident photons is large enough. They are well in line with the results in [12]. In this work, we have mainly focused on the PSHE when the incident wavelength is 632.8 nm. Influences of different photonic energies on the PSHE for a certain chemical potential will be studied elsewhere. All the numerical calculation is performed in MATLAB R2019a.

## 3 Photonic spin Hall shifts

We posit that one Gaussian beam is illuminated on the surface of ultrathin WSM film without Fermi arc states, as depicted in Figure 1B. The incident plane of light lies in the *x*-*z* plane of laboratory Cartesian coordinate (*x*, *y*, *z*). Besides, we utilized the coordinate frames (*x_{i}*,

*y*,

_{i}*z*) and (

_{i}*x*,

_{r}*y*,

_{r}*z*) to signify the central wave vectors of incident and reflected wave packets, respectively. By solving the electromagnetic boundary conditions on either side of the thin film, the Fresnel reflection coefficients of WSM-substrate are expressed as functions of complex conductivity of WSM film [12]

_{r}where *σ*_{1}* _{s}*≡cos

*θ*/(

_{i}*c*

_{0}

*μ*

_{0})+

*n*cos

_{t}*θ*/(

_{t}*c*

_{0}

*μ*

_{0})+

*σ*and

_{xxs}*σ*

_{2}

*≡cos*

_{s}*θ*/(

_{t}*c*

_{0}

*μ*

_{0})+

*n*cos

_{t}*θ*/(

_{i}*c*

_{0}

*μ*

_{0})+

*σ*cos

_{yys}*θ*cos

_{i}*θ*. Here, the parameters of

_{t}*c*

_{0},

*μ*

_{0},

*θ*and

_{i}*θ*are the velocity of light in vacuum, vacuum permeability, incident and refraction angles, respectively. The refractive index

_{t}*n*of substrate are set as 1.46, coinciding with that value of amorphous SiO

_{t}_{2}substrate at visible light.

It is assumed that the angular spectrum of the incident Gaussian beam in momentum space is specified by *w*_{0} is the width of wave function. For horizontal |*H*(*k_{i}*)⟩ and vertical |

*V*(

*k*)⟩ polarization states of photons incident upon the surface of WSM thin film, after reflection, the polarization states of wave packets are determined by [30]

_{i}where *k*_{0} is the wave vector in vacuum, *k_{ix}* and

*k*(

_{iy}*k*and

_{rx}*k*) represent the wave-vector components of incident (reflected) beam along

_{ry}*x*and

_{i}*y*(

_{i}*x*and

_{r}*y*) axes, respectively. According to the boundary conditions,

_{r}*k*=−

_{rx}*k*and

_{ix}*k*=

_{ry}*k*. In the spin basis set, linearly polarized states of |

_{iy}*H*⟩ and |

*V*⟩ are decomposed into two orthogonal spin components

where |+⟩ and |−⟩ stand for the left- and right-circular polarization components, respectively.

Under the paraxial approximation, spin-dependent Hall shifts of these two orthogonal components in reflected beam can be mathematically derived according to Eqs. (17)–(19). For simplicity, incident photons are assumed to be only horizontally polarized in this work. The corresponding in-plane and transverse spin Hall shifts for left- and right-circular polarization components, as indicated in Figure 1B, are given by [30], [39]

Figure 3 shows the variations of in-plane and transverse spin Hall shifts (i.e. ⟨Δ*x*_{+}⟩ and ⟨Δ*y*_{+}⟩) for the left-circular component with respect to the tilt *α _{t}* and the incident angle when

*μ*=0.125 eV. The colorbars including those appearing in Figures 4 and 5 are all scaled in the unit of

*λ*

_{0}. Previous studies have demonstrated that the spin Hall shifts are sensitive to the minimum values of |

*r*| such that they generally give extreme values around the Brewster’s angles [39], [40]. This change tendency is reproduced through comparing Figure 3 with Figure S3 in the Supplementary Material. It is found that the Brewster’s angle for type-I WSM film (|

_{pp}*α*|<1) appears at about 57.5°, and the maximum values of ⟨Δ

_{t}*x*

_{+}⟩ and ⟨Δ

*y*

_{+}⟩ are 2.65

*λ*

_{0}and 1.40

*λ*

_{0}, respectively. As the tilt |

*α*| increases from 1 to 3 (type-II WSM film), the Brewster’s angle decreases from ~57.5° to 56.1° (see Figure S3 in the Supplementary Material), and the maximum absolute values of ⟨Δ

_{t}*x*

_{+}⟩ (36.31

*λ*

_{0}) and ⟨Δ

*y*

_{+}⟩ (−10.25

*λ*

_{0}) occur around

*α*=−2.82 with

_{t}*θ*=56.1°.

_{i}Figure 4 presents the in-plane and transverse spin Hall shifts as a function of the tilt *α _{t}* and

*ħω*/

*μ*at Brewster’s angles of type-I (A, B) and type-II (C, D) WSM films. We can see that the spin Hall shifts with |

*α*|<1 are relatively flat and have a similar variation tendency with the optical conductivity of type-I WSM film (see Figure 2). Thus both ⟨Δ

_{t}*x*

_{+}⟩ and ⟨Δ

*y*

_{+}⟩ are less sensitive to the variance of tilt

*α*and chemical potential

_{t}*μ*. And the in-plane and transverse spin Hall shifts for type-I WSM film in Figure 4A and B are about 2.66

*λ*

_{0}and −0.25

*λ*

_{0}, respectively. As for the thin film of type-II WSM with |

*α*|>1, the maximum spin Hall shifts in the in-plane and transverse directions are ⟨Δ

_{t}*x*

_{+}⟩=38.47

*λ*

_{0}and ⟨Δ

*y*

_{+}⟩=−9.79

*λ*

_{0}. Their corresponding tilts are

*α*=−2.80 and −2.31, respectively, and both the in-plane and transverse spin Hall shifts gradually decrease as the chemical potential (or

_{t}*ħω*/

*μ*) decreases (or increases), as shown in Figure 4C and D.

From Figures 3 and 4, we drew a conclusion that the maximum spin Hall shifts in both in-plane and transverse directions favor emerging at the tilts between −2 and −3. In view of this, Figure 5A and B exhibit the in-plane and transverse spin Hall shifts as a function of the incident angle and *ħω*/*μ* with *α _{t}*=−2.5. For comparison, corresponding spin Hall shifts with

*α*=2.5 are also shown in Figure 5C and D. It is seen that the largest in-plane spin Hall shift with

_{t}*α*=−2.5 is 30.95

_{t}*λ*

_{0}which appears at

*θ*=56.2° (Figure 5A). At incident angles being a little larger and smaller than 56.2°, the sign of transverse spin Hall shift changes from positive to negative (Figure 5B). And the spin Hall shifts in both in-plane and transverse directions gradually decrease with the reducing (or increasing) of chemical potential (or

_{i}*ħω*/

*μ*) for a fixed incident angle.

In marked contrast, the maximum spin Hall shifts in the in-plane and transverse directions for WSM film with *α _{t}*=2.5 are only several times of incident wavelength (see Figure 5C and D). Besides, the in-plane spin Hall shift at incident angles near 56.3° gradually increases as the chemical potential decreases, exhibiting an inverse variation trend comparing with that ⟨Δ

*x*

_{+}⟩ in Figure 5A. To further confirm these differences between type-II WSM films with different signs of

*α*, photonic spin Hall shifts ⟨Δ

_{t}*x*

_{+}⟩ and ⟨Δ

*y*

_{+}⟩ as functions of the incident angle and

*ħω*/

*μ*for type-II WSM films with

*α*=±1.5 (see Figure S4) and

_{t}*α*=±3.0 (see Figure S5) are also shown in the Supplementary Material. They demonstrate that the maximum ⟨Δ

_{t}*x*

_{+}⟩ (or ⟨Δ

*y*

_{+}⟩) value of negtive

*α*is larger than that of positive

_{t}*α*. For a fixed incident angle, photonic spin Hall shifts at negative

_{t}*α*mainly show a decreasing tendency with reducing the chemical potential while the in-plane spin Hall shift at positive

_{t}*α*may gradually increase as the chemical potential decreases.

_{t}These different behaviors in PSHE between type-II WSM films with ±*α _{t}* might be attributed to different dependences of optical Hall conductivity

*σ*on the ratio

_{xys}*ħω*/

*μ*induced by reversing the sign of

*α*. In general, a finite Hall conductivity of WSM stems from the breaking of the time-reversal symmetry. This manifests in the separation of two Weyl nodes with opposite chirality in the momentum space [11], [12]. As a result, the real part of

_{t}*σ*with negative

_{xys}*α*is monotonically decreased while that with positive

_{t}*α*is monotonically increased as the

_{t}*ħω*/

*μ*increases, as shown in Figure S1B in the Supplementary Material. Besides, the imaginary part of

*σ*reverses its sign upon changing the tilt orientation (

_{xys}*α*→−

_{t}*α*). As for the longitudinal conductivity

_{t}*σ*, as mentioned above, it is less sensitive to the sign of

_{xx}*α*and the ratio

_{t}*ħω*/

*μ*. More specifically, the Re(

*σ*) values are exactly the same, and the largest difference between the Im(

_{xxs}*σ*) values is only ~9×10

_{xxs}^{-4}

*σ*

_{0}for WSM films with

*α*=±2.5 (see Figure S1A in the Supplementary Material). Therefore, different behaviors of spin Hall shifts for opposite tilt orientations could be ascribed to the remarkable discrepancy in optical Hall conductivities induced by the broken time reversal symmetry in type-II WSM thin films.

_{t}## 4 Conclusions

In summary, the spin-dependent splitting induced by the spin-orbit interaction is investigated in detail, when one linearly polarized light is incident upon the WSM surface without Fermi arc states. We show that the photonic spin Hall shifts on the surface of type-I WSM film are little sensitive to the variance of tilt *α _{t}* and chemical potential

*μ*due to the weak susceptibility of optical conductivity to

*α*and

_{t}*μ*. By contrast, both the in-plane and transverse photonic spin Hall shifts on the surface of type-II WSM film present versatile dependent behaviors on the tilt orientation and the chemical potential. In particular, the maximum spin Hall shifts appear at the tilts between −2 and −3, and the largest values of available ⟨Δ

*x*

_{+}⟩ and ⟨Δ

*y*

_{+}⟩ are ~40 and ~10 times of incident wavelength, respectively. Nonetheless, the maximum spin Hall shifts for type-II WSM film with positive

*α*are only several times of incident wavelength. Moreover, the photonic spin Hall shifts at negative

_{t}*α*mainly show a decreasing tendency with reducing the chemical potential, however, the in-plane spin Hall shift at positive

_{t}*α*gradually increases as the chemical potential decreases at some special incident angles. This dependence of photonic spin Hall shifts on tilt orientation may arise from the striking differences in optical Hall conductivities induced by the broken time reversal symmetry in type-II WSM thin films. In consequence, measuring the photonic spin Hall shifts may provide an alternative way to determine the types of WSMs. These findings may also open up new opportunities to investigate and characterize the doping and the tilt orientation of Weyl nodes in WSMs.

_{t}The authors acknowledge the National Natural Science Foundation of China (No. 11804251) and the program of study abroad for young scholar sponsored by TJCU for financial support.

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## Footnotes

## Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2019-0468).