1. Introduction

Photonic spin Hall effect (PSHE) is a photonic analogy of electronic spin Hall effect (SHE), in which the photon spin plays the role of the electron spin, and the refractive index gradient plays the role of the external field [1–5]. PSHE manifests itself as spin-dependent splitting arising from the spin orbit coupling of light [3–5]. In particular, when a beam passes through an inhomogeneous medium with index gradient, photons with opposite spin angular momentum i.e., the left and right circular polarization, are spatially separated from each other, resulting in a spin dependent lateral shift, i.e., the PSHE [3–7]. PSHE provides a new way to manipulate photons and has many applications in nano-optics, quantum information, and semiconductor physics [5,8,9]. In 2004, Onoda et al. theoretically explained PSHE by considering the geometric Berry phase and the angular momentum conservation [3]. Subsequently, in 2008, Hosten and Kwiat firstly detect and verify this phenomenon for photons passing through an air–glass interface by using the weak measurement technique [10,11]. Since then, PSHE has attracted great scientific research interests and technological upgrading on spinoptics [5].

At first, the research on PSHE focuses on the surface of three-dimensional (3D) materials, especially the metasurfaces [1,2,12–17]. With the rapid development of two-dimensional (2D) materials in the past decades, the research on PSHE has been extended to atomically thin 2D materials, such as the magnetically modulated PSHE in graphene [18,19], and the anisotropic optical response induced PSHE in black phosphorus [20,21]. Unfortunately, it is found that there is no in-plane spin Hall shift in nonmagnetic isotropic 2D materials due to the lack of cross conductivity arising from the restriction of time reversal symmetry [22,23]. Usually, for nonmagnetic 2D materials, we need to exert an external magnetic field to break the time reversal symmetry and obtain in-plane spin Hall shifts [18,19,21,24]. However, previous works [12–17,20] show that there are finite in-plane and transversal spin Hall shifts in anisotropic materials such as hyperbolic metamaterials, tilted linear-dichroic plate, and 2D black phosphorus in the absence of magnetic fields due to their anisotropic optical response.

Recently, a new kind of highly anisotropic electron system called semi-Dirac material which have both the properties of the Dirac electron in graphene and Schördinger electron in convectional semiconductors attracts intensive attention [25–30]. Semi-Dirac material hosts highly anisotropic two dimensional electron gas, of which the low energy dispersion is linear along one direction and parabolic along the perpendicular direction [31,32]. Semi-Dirac electron is firstly predicted in strained graphene [31] and recently observed in potassium-doped few-layer black phosphorus [33]. The unique dispersion results in highly anisotropic optical response to linearly polarized light [34]. The optical conductivity excited by linearly polarized light along the direction of linear dispersion is about two order larger than that of linearly polarized light along the direction of the parabolic dispersion [35]. From the previous work on the PSHE in monolayer black phosphorus [20], this highly anisotropic optical conductivity in semi-Dirac material surely indicates a strong PSHE with large in-plane spin Hall shift even in the absence of external magnetic fields. A large PSHE may allow us to observe the spin Hall shifts directly [1,36–38], and avoid to use the quantum weak measurement which desires complex experimental setup. Based on the measured spin Hall shifts, one can extract the optical parameters of the system such as the thickness [39] and the optical conductivity [40]. However, up to date, the PSHE in semi-Dirac materials still remains elusive.

In this work, we study the PSHE in 2D semi-Dirac materials. In our model, the beam is irradiated on a semi-Dirac material, then the reflected beam will undergo in-plane and transverse spin splitting, in which the centroids of left and right circular polarized light are spatially separated. By changing various incident conditions, we find that there is a large PSHE on the surface of the semi-Dirac material in the absence of external magnetic field resulting from its anisotropic optical response. The in-plane and transverse spin Hall shift are dozens times of the wavelength of the incident photon, which can be effectively tuned by the optical axis angle and the energy of incident photon. The maximum of the total in-plane (transverse) spin Hall shift is 83.91 (19.65) times of the wavelength of the incident photon, which is larger than that on the surface on conventional anisotropic 2D materials [20] and isotropic Dirac materials [18,19,24]. Our results provide further understanding on the spin-orbit coupling of light in non-magnetic anisotropic materials and useful information in designing spin-optical devices based on them.

2. Model and method

In our work, we establish a general model to describe the PSHE on the surface of the semi-Dirac materials as shown in Fig. 1(a). We consider a Gaussian wave packet with monochromatic frequency irradiate from the air to the surface of semi-Dirac material with incident angle $\theta _{i}$. We choose laboratory Cartesian frame $(x,y,z)$ to describe this question. The coordinates $({x_{i}},{y_{i}},{z_{i}})$ and $({x_{r}},{y_{r}},{z_{r}})$ represent the centers of the incident and reflected wave packets, respectively. The low-energy Hamiltonian of semi-Dirac material can be described by a two band model, which is given by [34]

 

Fig. 1. (a) Schematic illustration on the beam reflection on the surface of semi-Dirac materials in a Cartesian coordinate system with the incident angle $\theta _{i}$. The optical axis angle $\phi$ is defined as the angle between the $xoz$ plane and the incident plane which is determined by the incident light and the $z$-axis. The PSHE occurs on the reflecting surface and exhibits in-plane and transverse Hall shifts. (b) Schematic plot of the low energy dispersion of semi-Dirac electrons.

Download Full Size | PDF

Then, the conductivity tensor is given by

The off diagonal elements $\sigma _{xy}$ and $\sigma _{yx}$ are zero because of the time reversal symmetry [34,35]. The conductivity matrix is expressed as [20]

According to the electromagnetic boundary condition, the Fresnel reflection coefficients are obtained as [22]

Here $M_{\pm }^{1}=k_{tz}\pm k_{iz}+\omega \mu _{0}\sigma _{ss}$ and $M_{\pm }^{2}=\left ( k_{iz}\varepsilon \pm k_{tz}\varepsilon _{0}+k_{iz}k_{tz}\sigma _{pp}/\omega \right ) /\varepsilon _{0}$, $N=-\mu _{0}k_{iz}k_{tz}\sigma _{ps}^{2}/\varepsilon _{0}$, $R=2Z_{0}k_{iz}k_{tz}\sigma _{ps}$, $k_{iz}=k_{i}\cos \theta _{i}$ and $k_{tz}=k_{t}\cos \theta _{t}$, $k_{i}=k_{t}=\omega \left ( \varepsilon _{0}\mu _{0}\right ) ^{1/2}$, $Z_{0}= (\mu _{0}/\varepsilon _{0})^{1/2}$, and $\cos \theta _{t}=(1-\sin ^{2}\theta _{i}/4)^{1/2}$. While, $\theta _{i}\left ( \theta _{t}\right )$ is the incidence (refraction) angle, $\varepsilon _{0}\left ( \mu _{0}\right )$ the permittivity (permeability) in the vacuum, $\varepsilon _{1}$ the permittivity of refractive media, and $Z_{0}$ the impedance in the vacuum. The $r_{ps}$ and $r_{sp}$ sensitively depend on the crossed conductivity. A finite $r_{ps}$ and $r_{sp}$ require none zero crossed conductivity $\sigma _{sp}$.

Considering horizontal polarization $(H)$ and vertical polarization $(V)$ Gaussian incident beam reflected by the surface between the air and semi-Dirac materials, the angular spectrum of the incident Gaussian beam in momentum space is defined by

According to Eqs. (12)–(16) and taking into account the paraxial-approximation, we can evaluate the spin Hall shifts of the reflected beam. Generally, the spin Hall shifts are produced by horizontally and vertically polarized incidence [44]. The shift contributed by the vertically polarized incidence is usually small and can be neglected. Hence, we can only focus on the shifts of the horizontal polarization. The in-plane and transverse spin Hall shifts are

3. Results and discussions

In this section, we present the numerical results for the conductivity spectra, Fresnel coefficients and photonic spin Hall shifts as a function of photon energy or optical axis angle. Hereafter, unless explicitly specified, the conductivities are all in units of $\sigma _0$ = 2$e^2/h$, the temperature is set as T = 4 K, the level broadening factor is $\Gamma$=1 meV, and the Fermi energy is $E_F$ =0 eV.

First, we consider how the conductivities vary with the photon energy. Based on our calculation, we find the intraband transition is very weak and it can be ignored for certain Fermi energy, which is consistent with previous work [34,35]. Hence, we mainly present the conductivity contributed by interband transitions. Figure 2 plots the real and imaginary part of the conductivity as a function of photon energy with different optical axis angle $\phi$. According to Eqs. (6)–(7), the conductivities $\sigma _{pp}$ and $\sigma _{ss}$ equal to $\sigma _{yy}$ and $\sigma _{xx}$, respectively, when the optical axis angle $\phi$ is $0^{\circ }$. As shown in Figs. 2(a)-(b), arising from the highly anisotropic dispersion expressed by Eq. (2), the optical conductivity ($\sigma _{yy}$) excited by linearly polarized light along the linear dispersion direction is about two order larger than ($\sigma _{xx}$) excited by linearly polarized light along the parabolic dispersion direction in low frequency regime. By contrast, they tend to be the same in high frequency regime. For $\phi =0^{\circ }$, originating from the highly anisotropic behavior of $\sigma _{xx}$ and $\sigma _{yy}$, the longitudinal conductivity $\sigma _{pp}$ is also about two order larger than the transverse conductivity $\sigma _{ss}$ in low frequency regime. However, the cross conductivity $\sigma _{ps}$ is zero in this case according to Eq. (8) , which means there is no in-plane spin Hall shifts [22]. For $\phi =60^{\circ }$ [see Figs. 2(c)-(d)], the longitudinal conductivity $\sigma _{pp}$ is dominated by the optical conductivity $\sigma _{yy}$. It is larger than the transverse conductivity $\sigma _{ss}$ in this case. More importantly, there is a finite cross conductivity $\sigma _{ps}$ which induces the coupling between p- and s-polarised waves, due to the anisotropic optical conductivities ($\sigma _{xx}$ and $\sigma _{yy}$) resulting from the highly anisotropic dispersion.

 

Fig. 2. The real (a) and imaginary (b) part of the conductivity as a function of photon energy with optical axis angle $\phi =0^{\circ }$ and chemical potential $E_{F}$=0 eV. (c)/(d) is the counterpart to (a)/(b) with optical axis angle $\phi =60^{\circ }$.

Download Full Size | PDF

Figures 3(a)-(b) plot the real and imaginary part of the conductivity as a function of Fermi energy with photon energy $\hbar \omega$=0.3 eV and optical axis angle $\phi =60^{\circ }$. As shown in the figures, two different regimes in the conductivity spectra can be identified. There is a prominent inflection point at the half of the photon energy ($\hbar \omega$/2=0.15 eV). When the Fermi energy is lower (higher) than half of the photon energy [see Eq. (3)], the conductivity is dominated by the interband (intraband) transitions [34]. The result shows that the intraband conductivity is much weaker than the interband one. In our work, we focus on the PSHE on the surface between air and the pristine semi-Dirac material ($E_{F}$=0 eV). In this case, the electromagnetic response of the system is dominated by the interband transitions. For certain photon energy and Fermi level, the conductivity can be modulated by the optical axis angle as shown in Figs. 3(c)-(d). The conductivities versus optical axis angle is a trigonometric function. Importantly, there is a finite crossed conductivity $\sigma _{ps}$($\sigma _{sp}$) except for $\phi =0^{\circ }$ and $180^{\circ }$ arising from the intrinsic anisotropy of the semi-Dirac materials, which means it can be effectively tuned by the optical axis angle. The crossed conductivity will result in the in-plane and transverse spin Hall shifts and play an important role in spin-orbit interaction of light.

 

Fig. 3. The real (a) and imaginary (b) part of the conductivity as a function of Fermi energy with photon energy $\hbar \omega$=0.3 eV and optical axis angle $\phi =60^{\circ }$. (c)/(d) counterpart to (a)/(b) as a function of Fermi energy with photon energy $\hbar \omega$=0.3 eV and Fermi energy $E_{F}$=0 eV.

Download Full Size | PDF

Substituting various conductivities into Eqs. (9)–(11), we obtain the Fresnel coefficients which are directly related to the PSH shifts. Figs. 4(a)-(b) plot the Fresnel coefficients and the ratio between the transverse and longitudinal one $|r_{ss}/r_{pp}|$ as a function of the incident angle with photon energy $\hbar \omega$=0.3eV and optical axis angle $\phi =60^{\circ }$. From Fig. 4(a), we find the Brewster angle is $\theta _i=63.4^{\circ }$ where the longitudinal Fresnel coefficient ($|r_{pp}|$) takes the minimum [8]. However, the transverse Fresnel coefficient $|r_{ss}|$ increases with the increase of incident angle. It equals to 1 for glancing incidence ($\theta _i=90^{\circ }$) according to Eqs. (10). Importantly, there is a finite cross Fresnel coefficient ($|r_{sp}|$) arising from the anisotropic optical response of semi-Dirac material. Although, $|r_{sp}|$ is very small compared with $|r_{pp}|$ or $|r_{ss}|$, it directly determines whether there is an in-plane spin Hall shift and plays important role in the PSHE according to Eqs. (17)–(18). Further, the first derivative of cross Fresnel coefficient is closely related to the magnitude of the in-plane spin Hall shift. The first derivative takes the maximum at the Brewster angle, corresponding to the inflection point of $|r_{sp}|$. Another quantity closely related to the magnitude of the spin Hall shift is the ratio between the longitudinal Fresnel coefficient and the transverse one $|r_{ss}/r_{pp}|$ as depicted in Fig. 4(b). The ratio $|r_{ss}/r_{pp}|$ has a peak around the Brewster angle because the transverse Fresnel coefficient $|r_{pp}|$ takes the minimum at Brewster angle [see Fig. 4(a)]. Since both the first derivative of $|r_{sp}|$ and the ratio $|r_{ss}/r_{pp}|$ take the maximum at Brewster angle, hereafter we only focus on the spin Hall shifts around the Brewster angle. Figures 4(c)-(d) plot the Fresnel coefficients $r_{pp}$ and the ratio $|r_{ss}/r_{pp}|$ as functions of photon energy and optical axis angle by setting the incident angle near the Brewster angle. As shown in the figure, $|r_{pp}|$ and $|r_{ss}/r_{pp}|$ sensitively depend on the optical axis angle and photon energy because the reflection coefficients are closely related to the conductivity matrix. The $|r_{pp}|$ has a similar variation tendency with the $\sigma _{pp}$ as a function of optical axis angle.

 

Fig. 4. (a) the Fresnel coefficients and (b) the ratio between the transverse and longitudinal one $|r_{ss}/r_{pp}|$ as a function of the incident angle with photon energy $\hbar \omega$=0.3eV and optical axis angle $\phi =60^{\circ }$. The Fresnel’s coefficients (c) $r_{pp}$ and (d) the ratio $|r_{ss}/r_{pp}|$ as functions of photon energy and optical axis angle with incident angle $\theta _i=63.4^{\circ }$.

Download Full Size | PDF

To verify whether the PSH shifts take the maximum at the Brewster angle, we consider the total PSH shifts as a function of incident angle for certain photon energy and optical axis angle as presented in Fig. 5. From the figure, apart from the usually observed transverse shift, we find that there is a large in-plane shift due to the anisotropic optical response of the semi-Dirac material, which is similar to that in hyperbolic metamaterials [13,14,16,17] and tilted linear-dichroic plates [12,15], and monolayer black phosphorus [20]. Both the in-plane and transverse spin Hall shifts take the maximum near the Brewster’s angle because the first derivative of $r_{sp}$ and the ratio $|r_{ss}/r_{pp}|$ take the maximum at the Brewster angle. The maximum of the total in-plane and transverse spin Hall shift is 83.91$\lambda _i$ and 19.65$\lambda _i$, respectively, where $\lambda _i$ is the wavelength of the incident photon. In particular, for photon energy $\hbar \omega$=0.03 eV, the wavelength $\lambda _i$=41 $\mu$m. Then, the maximum of the total in-plane shift ($\langle \Delta X\rangle$) is 3.4 mm, which may be observable directly. However, the maximum of ($\langle \Delta Y\rangle$) is smaller than the beam diameter (2$w_0$=40$\lambda _i$) and it is difficult to observe them directly. Fortunately, the quantum weak measurement technique can amplify the spin Hall shifts and detect them precisely [10,11]. The first derivative of transverse Fresnel coefficients $\partial r_{pp} / \partial \theta _{i}$ also takes the maximum at the Brewster angle, which can be inferred from Fig. 4(a). Moreover, according to Eq. (18), the magnitude of transverse shift is closely related to the value of the ratio $r_{ss}/r_{pp}$.

 

Fig. 5. The total (a) in-plane and (b) transverse spin-dependent shifts in unit of incident wavelength $\lambda _i$ as a function of the incident angle with optical axis angle $\phi =5^{\circ }$, $0^{\circ }$ and photon energy $\hbar \omega$=0.03, 0.15 eV, respectively.

Download Full Size | PDF

Figure 6 shows the total (a) in-plane $\left \langle \Delta X\right \rangle$ and (b) transverse $\left \langle \Delta Y\right \rangle$ shifts on the surface of semi-Dirac materials as functions of optical axis angle and photon energy with incident angle setting near the Brewster angle and Fermi energy $E_F$=0 eV. As shown in the figure, the $\left \langle \Delta X\right \rangle$ is zero for $\phi =0$ or $\pi$ due to the absence of the cross conductivity, which can also be inferred from Eqs. (8), (11), (17) and (19). The $\left \langle \Delta X\right \rangle$ ($\left \langle \Delta Y\right \rangle$) is antisymmetrical (symmetrical) with respect to $\phi =\pi /2$ under certain photon energy. The total in-plane spin Hall shift is larger than the transverse one arising from the anisotropic optical response of semi-Dirac material. The spin Hall shifts depend more sensitively on the optical axis angle than the energy of the incident photon. The $\left \langle \Delta Y\right \rangle$ takes the maximum when the optical axis angle is set around $\phi ={0}^{\circ }$ or ${180}^{\circ }$ while the $\left \langle \Delta X\right \rangle$ takes the maximum when the optical axis angle is set around $\phi ={5}^{\circ }$ or ${175}^{\circ }$. The maximum of $\left \langle \Delta X\right \rangle$ and $\left \langle \Delta Y\right \rangle$ is 83.91$\lambda _i$ and 19.65$\lambda _i$, respectively, where $\lambda _i$ is the wavelength of the incident photon. The in-plane and transverse shift together contribute to a large PSH shift, when compared with those in traditional anisotropic two-dimensional materials [20] and isotropic Dirac materials [18,19,24]. This large photonic spin Hall shift is easier to be measured in real experiment. It is also useful in studying the spin-orbit coupling of light and designing spin-optical devices based on semi-Dirac materials.

 

Fig. 6. The total (a) in-plane and (b) transverse shifts on the surface between air and the semi-Dirac materials as functions of optical axis angle and photon energy with incident angle $\theta _i=63.4^{\circ }$ and Fermi energy $E_F$=0 eV.

Download Full Size | PDF

4. Conclusions

In conclusion, we established a general model to study the PSHE on the surface between air and semi-Dirac materials. We found there is a large PSHE originating from the intrinsic anisotropy of electronic states in semi-Dirac materials. Owing to the highly anisotropic optical response of semi-Dirac materials, there is a large in-plane spin Hall shift compared with that in the isotropic systems [22,23]. It was found that the in-plane and transverse spin Hall shift are dozens times of the incident wavelength, which can be effectively tuned by the optical axis angle and the energy of incident photon. The maximum of the total in-plane and transverse spin Hall shift is 83.91$\lambda _i$ and 19.65$\lambda _i$, respectively, resulting in a large PSH shift compared with those in conventional anisotropic two-dimensional materials [20] and isotropic Dirac materials [18,19,24]. This large PSH shift is more conveniently to be measured in real experiment. Our result is useful in understanding the spin-orbit coupling of light in semi-Dirac material and designing spin-optical devices based on it.