1. Introduction

The photonic spin Hall effect (PSHE) refers to spin-dependent splitting perpendicular to the field played by the refractive index gradient, which can be regarded as a direct photonic analogy of electronic spin Hall effect [1]. The PSHE is generally at the subwavelength scale being too weak to be measured directly, and it is usually detected in the experiment based on the weak measurement technology [2–6]. The PSHE originates from an effective spin-orbital interaction, which describes the mutual influence of the spin (polarization) and trajectory of the beam [7]. In the past ten years, the PSHE has been intensively investigated in different physical systems, such as optical physics [8], high-energy physics [9], Metasurfaces [10], and Weyl semimetals [11]. With the emergence and in-depth study of graphene, its extraordinary electronic and photonic properties have received great attention [12,13]. Currently, there are two main models in use for understanding the light-matter interaction in the graphene structures, the first is a thin film model that can be either isotropic or anisotropic along the vertical direction and the second is a zero-thickness model with both null out-of-plane surface susceptibility and conductivity [14,15]. The experimental results show that the zero-thickness model can perfectly explain the light-matter interaction on the surface of monolayer or bilayer graphene, while the case of more than two layers, which can no longer be regarded as two dimensional thickness, should be described by the slab model or the thin film model [16,17]. Therefore, the zero-thickness model is often applied to the study of the PSHE of single-layer or double-layer graphene. In this case, monolayer graphene is isotropic when external fields are absent. Recently, the electrically tunable properties of monolayer graphene were used for active manipulation asymmetric PSHE [18]. The PSHE in the moiré superlattices of bilayer graphene was also studied and exhibited a strong dependence on moiré angle [19]. Remarkably, even if the out-of-plane surface conductivity is neglected, in the presence of electric and magnetic fields, monolayer or bilayer graphene is anisotropic and its surface conductivities are tensor $\boldsymbol {\sigma }$ [20,21]. The study of magneto-optical modulation of anisotropic monolayer graphene and its beam shifts has quickly become a hot issue chased. Tang et al. proposed a magneto-optical modulation method for PSHE based on transmitted beam in an anisotropic monolayer graphene-substrate system in the terahertz region [22]. On this basis, Yin et al. used a multilayer structure containing an anisotropic monolayer graphene to achieve electromagnetic tunable enhancement of PSHE at terahertz frequencies [23]. Interestingly, under the action of the external magnetic field, the quantized PSHE was also predicted to occur in anisotropic monolayer graphene-substrate system due to the spin-orbit interaction and the special properties of single-layer graphene [24]. However, the study of magneto-optical modulation and the quantized PSHE in anisotropic monolayer graphene faces a fundamental challenge: even at terahertz frequencies, the external magnetic field requires several Tesla, or even tens of Tesla, and the equipment that generates this magnetic field is bulky, which has seriously hindered the application development of electronic devices based on anisotropic monolayer graphene.

Strained graphene provides a new platform for the study of new physics, including but not limited to the singular topological quantum field and the quantum electrodynamics field. Unlike conventional monolayer graphene, even if the external fields are absent, single-layer strained graphene is intrinsically anisotropic, which greatly reduces the dependence on external fields. In recent years, there has been increasing interest in harnessing the strain properties of monolayer graphene to control its physical properties [25–27]. Suitable strain induces uniform pseudo-magnetic fields $B_{ps}$ in monolayer graphene, which can range from a few Tesla to hundreds of Tesla [28,29]. If the deformation of single-layer strained graphene is on the atomic spacing scale, its Brillouin region will be distorted, so that the electrons in the two valleys (K and K’ valley) move in opposite directions [30,31], which is similar to the movement of electrons under the action of a magnetic field. Typically, this motion is theoretically studied using the Green’s function method and through the numerical implementation of the Kubo formula [32], while Ferreira et al. proposed using equations of motion (EOM) to study such problems and generalize them to include magnetic fields, and the results showed that this method is not only more flexible, but also appropriately regularized EOM solutions are fully equivalent to Kubo’s formula [33]. Although both strained and external magnetic fields in the two-dimensional material system can be attributed to similar pseudopotential effects [34]. However, it is worth noting that the properties exhibited by these two fields acting on a single-layer graphene are significantly different [35,36]. The $B_{ps}$ induced by the deformation mode of the strained graphene lattice has opposite signs for the two valleys, i.e. the pseudo-magnetic field is $+ B_{ps}$ for the K valley while $- B_{ps}$ for K’, which means that the strained magnetic field, unlike external magnetic field, does not break the time-reversal symmetry of single-layer graphene as a whole, and it also can lead to the formation of pseudo-Landau levels (PLLs) [35]. Because of this property of strained graphene, it is often necessary to provide a small additional external magnetic field $B_{e}$ for breaking the time-reversal symmetry when studying its Faraday effect, and the results showed that large Faraday rotations in monolayer strained graphene can be achieved even at sub-Tesla magnetic field and infrared frequency [36]. This is not only very beneficial for the miniaturization of equipment based on strained graphene, but also can greatly save costs. Strained graphene has been extensively discussed in longitudinal transport [37], spin relaxation phenomena [38], and singular electron states [39]. Its special properties are expected to bring new opportunities for the study of PSHE. However, to our knowledge, the quantized PSHE in monolayer strained graphene has not been reported, and its large $B_{ps}$ may greatly reduce the dependence of the quantized PSHE on the $B_{e}$. Therefore, it is of great necessary to understand the in-depth magneto-optical transport characteristics behind strained graphene and the relationship between its intrinsic physical mechanism and the quantized PSHE.

In this work, we investigate the quantized PSHE in the strained graphene-substrate system, and find that its dependence on the external magnetic field $B_{e}$ is greatly reduced, where the $B_{e}$ is two orders of magnitude smaller than that previously reported in conventional graphene-substrate system. First, we discuss the PLLs splitting of monolayer strained graphene varing with $B_{e}$ and $B_{ps}$, and the quantized conductivities of it changing with $B_{e}$ and $E_{f}$. Then, we focus on the influences of the $B_{e}$ and the $E_{f}$ on the PSHE of the system. Finally, we analyze the response of pseudo-Brewster angles of the system to $E_{f}$ and $B_{e}$, and study the giant quantized PSHE near these angles. The giant quantized PSHE enables the direct optical measurements of quantized conductivities and PLLs in monolayer strained graphene.

2. Theoretical analysis

To investigate the quantized PSHE in a strained graphene-substrate system, a general model is established to describe the PSHE of the reflected beam in the system. Here, we only study the quantized PSHE in a single-layer strained graphene, and use a zero-thickness model to explain the light-matter interaction on its surface, which will not be emphasized later. Figure 1(a) is schematic representation the wave reflection at a strained graphene-substrate surface in a Cartesian coordinate frame. As shown in the figure, a monolayer strained graphene is placed on the top of a homogeneous and isotropic substrate. The z-axis of the laboratory Cartesian coordinate frame $\left ( {x ,y ,z} \right )$ is perpendicular to the surface, and an imposed static external magnetic field $B_{e}$ is applied along the positive z axis in the system. Considering a Gaussian beam with monochromatic frequency $\omega$ hits the surface of the system from air at an incident angle of $\theta _i$, and the in-plane $\Delta _{x \pm }^{H,V}$ and transverse $\Delta _{y \pm }^{H,V}$ spin-dependent splittings in the PSHE of the left-circular $\left | + \right \rangle$ or right-circular $\left | - \right \rangle$ polarized component for $\left | H \right \rangle$ or $\left | V \right \rangle$ polarization state occur on the reflecting surface. In addition, we use coordinate frames $\left ( {x_i, y_i, z_i } \right )$ and $\left ( {x_r, y_r, z_r } \right )$ to denote the incident beam and the reflected one, respectively. Figure 1(b) shows the total magnetic field distribution of a monolayer strained graphene under the $B_{e}$, where $\pm B_{ps}$ are the pseudo-magnetic field induced by strain (assuming that $\pm B_{ps}$ are evenly distributed on the surface). Figure 1(c) is diagram for the K and K’ valley resolved PLLs, where $\Delta _n$ represents the energy spacing of the two valleys for the $n_{th}$ PLL. In the presence of both the $\pm B_{ps}$ and the $B_{e}$, the total effective magnetic field is $B_{e}+ B_{ps}$ for the K valley while $B_{e}- B_{ps}$ for K’ valley.

 

Fig. 1. Beam propagation model of strained graphene-substrate system under different external magnetic fields $B_{e}$. (a) Schematic representation the wave reflection at the system surface in a Cartesian coordinate frame. A monolayer strained graphene is placed on the top of a homogeneous and isotropic substrate. An imposed $B_{e}$ is applied along the positive z axis in the system. $\theta _i$ is the incident angle. The in-plane $\Delta _{x \pm }^{H,V}$ and transverse $\Delta _{y \pm }^{H,V}$ spin-dependent splittings in the PSHE of the left-circular $\left | + \right \rangle$ or right-circular $\left | - \right \rangle$ polarized component for $\left | H \right \rangle$ or $\left | V \right \rangle$ polarization state occur on the reflecting surface. (b) Schematic illustrating of a strained graphene with induced pseudo-magnetic field $B_{ps}$, under the presence of the $B_{e}$. (c) Diagram for the K and K’ valley resolved PLLs. The total effective magnetic field is $B_e - B_{ps}$ for K valley and $B_e + B_{ps}$ for K’ valley; $\Delta _n$ represents the energy spacing of the two valleys for the $n_{th}$ PLL.

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Strained graphene induces the generation of $B_{ps}$, which has the opposite signs for K and K’ valleys. Under the action of this $B_{ps}$, the PLLs will split. When a $B_{e}$ (needed to break the time-reversal symmetry) is present, the valley degeneracy of the $n \ne 0$ PLLs is expected to be lifted and the energy spacing of the two valleys for the $n_{th}$ PLL can be expressed as [40]:

Here, $\sigma _{xx}$ and $\sigma _{yy}$ denote the longitudinal conductivities ($\sigma _{xx}$ = $\sigma _{yy}$); $\sigma _{xy}$ and $\sigma _{yx}$ represent the transverse (Hall) conductivities ($\sigma _{xy}$ = $\sigma _{yx}$). The conductivities related to K and K’ valley charge carriers become different due to the fact that the pseudo-magnetic field has opposite signs for two valleys: $+ B_{ps}$ for the K valley and $- B_{ps}$ for the K’ valley. Thus, the conductivities can be expressed as [33]

To study the interaction of light and matter on the surface of strained graphene, we need to obtain the Fresnel coefficients from the boundary conditions. The reflected and transmitted amplitudes meet the following equations:

Here, $E_i^{p,s}$ represents the amplitude of the incident $p$ wave or $s$ wave. $E_r^{p,s}$ indicates the amplitude of the reflected $p$ wave or $s$ wave. $E_t^{p,s}$ denotes the amplitude of the transmitted $p$ wave or $s$ wave. $\theta _i$ and $\theta _t$ represent the angle of incidence and refraction, respectively. $Z_0$ and $Z$ are the impedance in the air and the medium, respectively. The Fresnel reflection coefficients are determined by the incident and reflected amplitude: $r_{pp}=E_r^p/E_i^p$, $r_{ss}=E_r^s/E_i^s$, $r_{ps}=E_r^p/E_i^s$, $r_{sp}=E_r^s/E_i^p$; by solving the boundary conditions, the Fresnel reflection coefficients of the the strained graphene-substrate can be expressed as [24,41]:

Here, $\theta _{i,r}$ denote the incident and reflected angles. After reflection, $\left | {H\left ( {k_{i} } \right )} \right \rangle$ and $\left | {V\left ( {k_{i} } \right )} \right \rangle$ evolve as $\left [ {\left | {H\left ( {k_r } \right )} \right \rangle \left | {V\left ( {k_r } \right )} \right \rangle } \right ]^T = m_R \left [ {\left | {H\left ( {k_i } \right )} \right \rangle \left | {V\left ( {k_i } \right )} \right \rangle } \right ]^T$, where

Here, $k_{rx}$ and $k_{ry}$ are reflected wave vector in the $x_r$ and $y_r$ direction, respectively. Based on boundary conditions: $k_{rx}=-k_{ix}$, $k_{ry}=k_{iy}$. The polarizations associated with each angular spectrum components experience different rotations in order to satisfy the boundary condition after reflection:

It is well known that the PSHE manifests itself as spin-dependent splitting which appears in both position and momentum spaces. To reveal the PSHE, we now determine the in-plane and transverse shifts of the wave packet. In the spin basis set, the polarization state of $\left | H \right \rangle$ or $\left | V \right \rangle$ can be decomposed into two orthogonal spin components:

Here, $w_0 = 1$ mm is the width of the wave function [24]. The total wave function is made up of the packet spatial extent and the polarization description:

In addition, the spin-independent terms have been neglected, and only the spin-dependent ones are retained. The in-plane spin Hall shifts and the transverse spin Hall shifts can be written as [42]

Note that the spin Hall shifts are complex and can be written as $\delta _{x,y}^{H,V} = {\mathop {\rm Re}\nolimits } \left [ {\delta _{x,y}^{H,V} } \right ] + i{\mathop {\rm Im}\nolimits } \left [ {\delta _{x,y}^{H,V} } \right ]$. Here, the real part is related to the spin Hall shift in the position space, while the imaginary part is associated with the spin Hall shifts in the momentum space [42,43]. The in-plane spin Hall shifts of the wave packet at initial position ($z_{r} = 0$) are given by

Substituting Eqs. (22) and (23) into Eq. (28), the in-plane spin-dependent splitting of the two spin components in the PSHE can be written as:

In the following simulations, the influences of $E_{f}$ and $B_{e}$ on PSHE will be further investigated based on the theory.

3. Results and discussions

In this section, the quantized PSHE in the strained graphene-substrate system shown in Fig. 1 will be discussed in detail. The incident beam frequency is fixed at $\omega = 1$ THz; the refractive index of the substrate for Si in the terahertz range is $n_{si} = 3.415$; the electron relaxation time is $\tau = 0.1$ ps; and temperature is chosen as $T = 4$ K. From the previous theoretical analysis, it can be seen that the PSHE of the reflected beam on the surface of the system is affected by the reflection coefficients closely related to the PLLs splitting and conductivities of strained graphene, while the conductivities are mainly affected by the $E_{f}$ and $B_{e}$, and further affect the PSHE in this system. Therefore, we first discuss the variations of PLLs splitting of monolayer strained graphene with $B_{e}$ and $B_{ps}$, and the quantized conductivities of it changing with $B_{e}$ and $E_{f}$. Then, we analyze the PSHE in the system according to these variations. On this basis, we study the response of pseudo-Brewster angles of the system and the giant quantized PSHE near these angles to the $E_{f}$ and $B_{e}$.

Firstly, we discuss the PLLs splitting and quantized conductivities of monolayer strained graphene shown in Fig. 1. In order to clarify the influences of the $B_{e}$ and $B_{ps}$ on the splitting of PLLs, the energy spacing $\Delta _n$ of the two valleys for the $n_{th}$ PLL is plotted as functions of the $B_{e}$ and the $B_{ps}$ according to Eq. (1), as illustrated in Figs. 2(a) and 2(b). It can be found that the $\Delta _n$ gradually increases with increasing of $B_{e}$ and gradually decreases with increasing of $B_{ps}$, which indicates that the number of PLLs gradually increases, and the valley degeneracy of the $n \ne 0$ PLLs is lifted. It is noticeable that the PLLs splitting of monolayer strained graphene will further affect its conductivities. Although the Hall conductivity $\sigma _{xy}$ is affected by both $B_{ps}$ and $B_{e}$, $B_{ps}$ has opposite signs for K and K’ valleys, the overall response of the $\sigma _{xy}$ to magnetic field only shows a strong dependence on $B_{e}$ according to Eq. (5). For this reason, the longitudinal conductivity $\sigma _{xx}$ and Hall conductivity $\sigma _{xy}$ are respectively plotted as functions of the $B_{e}$ at different $E_{f} = 20,30,40,50$ meV, as shown in Figs. 2(c) and 2(d). From these two figures, it is found that the $\sigma _{xx}$ and the $\sigma _{xy}$ present obvious quantized behaviors with the change of $E_{f}$ and $B_{e}$. The quantization step widths of the $\sigma _{xx}$ and the $\sigma _{xy}$ gradually increase with increasing of $B_{e}$, and decrease as increasing of $E_{f}$. The values of them gradually decrease with increasing of $B_{e}$, and gradually increase with increasing of $E_{f}$. In fact, the quantized behaviors of the conductivities are inseparable from the PLLs splitting of strained graphene, and it is precisely because of the PLLs splitting on joint action of the $B_{ps}$ and the $B_{e}$ (needed to break time-reversal symmetry) that the conductivities will have these behaviors described above. It is worth noting that some recent experimental works have once again confirmed the unique valley-polarized Landau quantization [31,39,44]. Researchers have demonstrated experimentally that it is possible to realize valley polarization and valley inversion in strained graphene by using both strain-induced $B_{ps}$ and real $B_{e}$ [31]. The distinct nature of the PLLs enables one to realize novel electronic states beyond what is feasible with real LLs [39].

 

Fig. 2. The PLLs splitting and quantized conductivities of monolayer strained graphene. (a) and (b) represent the energy spacing $\Delta _n$ of the two valleys for the $n_{th}$ PLL varing with $B_{e}$ and $B_{ps}$ for $\left | n \right | \le 5$, respectively. (c) and (d) represent longitudinal $\sigma _{xx}$ and Hall $\sigma _{xy}$ conductivity as functions of $B_{e}$ at different $E_{f} = 20,30,40,50$ meV, respectively.

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Next, emphasis is put on the influences of $B_{e}$ and $E_{f}$ upon the behaviors of PSHE in a strained graphene-substrate system at a fixed incident angle based on the relationships between the conductivities and the two. To simplify the discussion, only the incidence of $\left | H \right \rangle$ polarization light is considered and the incident angle is selected as the air-substrate Brewster angle $\theta _i = 74^ \circ$. From the previous theoretical analysis, it can be seen that the Fresnel reflection coefficients of the system can be obtained by solving Maxwell’s equations with using the boundary conditions, and they are closely related to the PSHE in the system, so the modulus of reflection coefficients $\left | {r_{pp} } \right |$ and $\left | {r_{ps} } \right |$ are plotted as functions of the $B_{e}$ and $E_{f}$, as shown in Figs. 3(a) and 3(b). From these two figures, it is found that with the change of $B_{e}$ and $E_{f}$, the $\left | {r_{pp} } \right |$ and $\left | {r_{ps} } \right |$ both have obvious quantized behaviors, and their quantization step widths are consistent with that of the conductivities. Differently, with the variations of $B_{e}$ and $E_{f}$, the size of the two shows different behaviors. With increasing of $B_{e}$, the $\left | {r_{pp} } \right |$ gradually increases in the low Fermi energy region, and decreases slowly in the high Fermi energy region; while the $\left | {r_{ps} } \right |$ gradually decreases, and it has a relatively larger change amplitude in the high Fermi energy region. As increasing of $E_{f}$, the $\left | {r_{pp} } \right |$ first decreases and then gradually increases in the weak magnetic field region, and its change amplitude is very small in the strong magnetic field region; while the $\left | {r_{ps} } \right |$ gradually increases and its change amplitude in a region of weak magnetic field is greater than that in the strong magnetic field region. In fact, based on Eqs. (10)–(12), the quantized behaviors of the reflection coefficients with variations of $B_{e}$ and $E_{f}$ are very similar to that of the conductivities, owing to the close relationships between the two. According to Eqs. (29) and (31), the reflection coefficients further affect the PSHE of a strained graphene-substrate system. To describe the influences of the above parameters on PSHE in the system more clearly, the in-plane $\Delta _{x + }^H$ and transverse $\Delta _{y + }^H$ spin-dependent splittings in the PSHE of the left-circular $\left | + \right \rangle$ polarized component are plotted as functions of the $B_{e}$ and $E_{f}$, as illustrated in Figs. 3(c) and 3(d). From these two figures, we can find that the $\Delta _{x + }^H$ and the $\Delta _{y + }^H$ have obvious quantized behaviors with the change of $B_{e}$ and $E_{f}$. Intuitively, the quantized behaviors of $\Delta _{x + }^H$ are very similar to that of $\left | {r_{pp} } \right |$ in Fig. 3(a), and the quantized behaviors of $\Delta _{y + }^H$ have many resemblances with $\left | {r_{ps} } \right |$ in Fig. 3(b). At the same time, the $\Delta _{x + }^H$ is significantly greater than the $\Delta _{y + }^H$. This can be attributed to that the gradient of $\left | {r_ {pp}} \right |$ with the change of $B_ {e}$ and $E_ {f}$ is greater than that of $\left | {r_ {ps}} \right |$. However, it is worth noting that the size of $\Delta _{x + }^H$ and $\Delta _{y + }^H$ varing with $B_{e}$ and $E_{f}$ are opposite to that of the reflection coefficients changing with them. For example, the region where the size of $\Delta _{x + }^H$ and $\Delta _{y + }^H$ takes the maximum is instead the region where the reflection coefficients are zero, which can be explained mathematically by combining Eqs. (29) and (31).Interestingly, the quantization step widths of $\Delta _{x + }^H$ and $\Delta _{y + }^H$ are good agreement with that of the $\sigma _{xx}$ and the $\sigma _{xy}$ [see Figs. 2(c) and 2(d)]. In particular, the external magnetic field required to generate quantized PSHE in conventional graphene is on the order of Tesla, causing the splitting of real LLs, which is the essential reason for the formation of quantized PSHE in it [24]. The difference is that the quantized PSHE in the strained graphene-substrate system has a greatly reduced dependence on the $B_{e}$, which is two orders of magnitude smaller than that reported in Ref. [24], due to the presence of a large $B_{ps}$ in the strained graphene, which induces the splitting of PLLs. In this case, a sub-Tesla $B_{e}$ only needed is used to break the time-reversal symmetry of it ($B_{e}$ here does not cause the LLs splitting). Under the joint action of these two magnetic fields, the valley degeneracy of the $n \ne 0$ PLLs of K and K’ valley is lifted. This is the underlying physical mechanism of forming the quantized PSHE in strained graphene under a sub-Tesla $B_{e}$. In addition, the twisted bilayer graphene has been introduced into PSHE due to its extraordinary optical and electronic properties [45], and it has also been proved that there is a huge $B_{ps}$ in twisted bilayer graphene causing the PLLs splitting [27]. Therefore, we have reason to believe that, similar to the case of single-layer strained graphene, PSHE in twisted bilayer graphene will also have enhanced or even quantized behaviors and other novel phenomena under small external magnetic field.

 

Fig. 3. The reflection coefficients and the PSHE for the $\left | H \right \rangle$ polarization impinging on the strained graphene-substrate system are plotted as functions of $B_{e}$ and $E_{f}$. (a) and (b) represent modulus of reflection coefficients $\left | {r_{pp} } \right |$ and $\left | {r_{ps} } \right |$ as functions of $B_{e}$ and $E_{f}$, respectively. (c) and (d) represent the in-plane $\Delta _{x + }^H$ and transverse $\Delta _{y + }^H$ spin-dependent splittings in the PSHE of the left-circular $\left | + \right \rangle$ polarized component as functions of $B_{e}$ and $E_{f}$, respectively.

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Then, for exploring the regulation of PSHE in the strained graphene-substrate system, the relationships between the reflection coefficients and the above parameters at different incident angles $\theta _{i}$ are discussed. Further research finds that the PSHE in the system is not only affected by the $B_{e}$ and $E_{f}$, but also influenced by the incident angle $\theta _{i}$, especially at special angles, such as near the Brewster angles, where there will be a giant PSHE. Since the addition of strained graphene will modify the Brewster angle of air-substrate, this angle is referred to as the pseudo-Brewster angle by using the definition in Ref. [46], and the incident angles $\theta _{i}$ are guaranteed to be near the pseudo-Brewster angles in the subsequent simulation analysis. To illustrate the influences of the $\theta _{i}$, the reflection coefficients $\left | {r_{pp} } \right |$ and $\left | {r_{ps} } \right |$ are plotted as functions of the $B_{e}$, $E_{f}$ and $\theta _{i}$, as shown in Figs. 4(a)-(c). Figures 4(a) and 4(b) are the pseudo-color images of $\left | {r_{pp} } \right |$ and $\left | {r_{ps} } \right |$ with the changes of the $B_{e}$ and $\theta _{i}$ at fixed $E_{f} = 0.5$ emV. As can be seen from the figures, the white dashed line ($\left | {r_{pp} } \right | = 0$) presents a ladder distribution with the variations of $B_{e}$. Combined with Eq. (10), it can be seen that the angles corresponding to $\left | {r_{pp} } \right | = 0$ are the pseudo-Brewster angles which have a close relationship with the conductivities. Figure 4(c) is the pseudo-color image of the $\left | {r_{pp} } \right |$ with the variations of the $E_{f}$ and $\theta _{i}$ at fixed $B_{e} = 0.1$ T (white dashed line indicates $\left | {r_{pp} } \right | = 0$). Similarly, the angles corresponding to $\left | {r_{pp} } \right | = 0$ are the pseudo-Brewster angles. More importantly, compared with $\left | {r_{pp} } \right |$ and $\left | {r_{ps} } \right |$, it can be found that with the change of $\theta _{i}$, $\left | {r_{pp} } \right |$ has a greater angular gradient than $\left | {r_{ps} } \right |$. In other words, $\left | {r_{pp} } \right |$ is more sensitive to the change of $\theta _{i}$. Interestingly, the $\Delta _{x + }^H$ is proved to be related to the angular gradient of the reflection coefficient, while the $\Delta _{y + }^H$ originates from the spin-orbit interaction. Therefore, the $\Delta _{x + }^H$ may be more closely related to the $\left | {r_{pp} } \right |$. In order to more clearly explain the influences of $E_{f}$ and $B_{e}$ on the pseudo-Brewster angles of the system, a line graph is drawn as presented in Fig. 4(d). One can see from the figure, the pseudo-Brewster angles of the system have obvious quantized behaviors with the changes of $E_{f}$ and $B_{e}$. In the region of weak magnetic field and high Fermi energy, the pseudo-Brewster angles will change significantly due to the narrow quantization steps of the conductivities in Figs. 2(c) and 2(d), which greatly modifies the Brewster angles of the air-substrate. While in the region of strong magnetic field and low Fermi energy, the pseudo-Brewster angles are not sensitive to the changing magnetic field and Fermi energy because the quantization steps become wider. This phenomenon is mainly attributed to the splitting of PLLs. In fact, the pseudo-Brewster angles are proportional to the square of $E_{f}$ and inversely proportional to the $B_{e}$. This is consistent with the changes of PLLs.

Finally, giant quantized PSHE near the pseudo-Brewster angles of the strained graphene-substrate system is obtained. Figure 5 shows the in-plane $\Delta _{x + }^H$ and transverse $\Delta _{y + }^H$ spin-dependent splittings in the PSHE of the left-circular $\left | + \right \rangle$ polarized component for the $\left | H \right \rangle$ polarization varing with $\theta _{i}$, $E_{f}$, and $B_e$. As can be seen from the figure, the peak values of the PSHE occur at a specific angle (that is, the pseudo-Brewster angle), and the quantized behaviors are more pronounced for higher $B_e$ and lower $E_{f}$. As the $B_e$ increases, the peak values of the $\Delta _{x + }^H$ and the $\Delta _{y + }^H$ move at small angles [Figs. 5(a) and 5(b)], while they move at larger angles as the $E_{f}$ increases [Figs. 5(c) and 5(d)]. Combined with Fig. 4, it can be seen that the movement directions of the peak values are consistent with that of the pseudo-Brewster angles, and the quantization step widths of the PSHE correspond to that of the reflection coefficients. Judging from this, the giant quantized PSHE near the pseudo-Brewster angle in the strained graphene-substrate system can be attributed to the drastic variation and quantized of the reflection coefficients. Interestingly, unlike the PSHE in conventional graphene or at isotropic interfaces, the $\Delta _{x + }^H$ in the strained graphene-substrate system compared with the $\Delta _{y + }^H$ reveals a more giant shift value in Fig. 5. In an isotropic system, the $\Delta _{x + }^H$ is proved to be related to the angular gradient of the reflection coefficient, while the $\Delta _{y + }^H$ originates from the spin-orbit interactioin [47]. Based on the physical mechanism of spin-dependent splitting in isotropic systems, by analyzing formula (29) and (31), we have reason to believe that in the strained graphene-substrate system, the $\Delta _{x + }^H$ is mainly related to the angular gradients of Fresnel coefficients $\left | {r_{pp} } \right |$ [Figs. 4(a) and 4(c)], while the $\Delta _{y + }^H$ results from the spin-orbit interaction enhanced by the cross-reflection coefficient $\left | {r_{ps} } \right |$ [Fig. 4(b)]. Not only that, the angular gradients of $\left | {r_{pp} } \right |$ is greater than $\left | {r_{ps} } \right |$, and it causes that the enhancement of $\Delta _{x + }^H$ in the strained graphene-substrate system compared with the $\Delta _{y + }^H$ is more significant. It is worth mentioning that in an isotropic system, the limitation of spin-dependent splitting in PSHE is usually half of the beam waist [47]. While in the strained graphene-substrate system, $\Delta _{y + }^H$ is smaller than the half of beam waist and the $\Delta _{x + }^H$ breaks this value. More than that, some other studies have shown that the magnitude of the splitting can also break this value, especially in some anisotropic two-dimensional material systems [4,23]. So, the limitation of PSHE in strained two-dimensional material system is still worth exploring. In fact, the quantized PSHE is proportional to the $B_e$, and is inversely proportional to the square of the $E_{f}$. This rule of change seems to be a source of enlightenment for us, since the PSHE can be regulated flexibly and dynamically if we appropriately control the external conditions. Conversely, this giant quantized PSHE can be used to directly measure the optical parameters such as quantized conductivities and PLLs of strained graphene. Furthermore, weak measurement technology is an important and convenient approach for detecting the beam shifts, which can magnify the original shifts hundreds of times [2–6]. Nevertheless, a quantitative relation between the strain of graphene and the pseudo-magnetic field could be still lacking. Thus, combined with weak measurement technology, it is possible to detect the strain of graphene through measuring the PSHE occurring on the surfaces of strained graphene. This may also open up new ideas for strain measurement of other two-dimensional materials.

 

Fig. 4. The reflection coefficients and pseudo-Brewster angles in the strained graphene-substrate system. (a) and (b) represent modulus of reflection coefficients $\left | {r_{pp} } \right |$ and $\left | {r_{ps} } \right |$ as functions of $B_{e}$ and $\theta _i$ at fixed $E_{f}=50$ meV, respectively. (c) represents modulus of reflection coefficient $\left | {r_{pp} } \right |$ as functions of $E_{f}$ and $\theta _i$ at fixed $B_{e}=0.1$ T. (d) shows the magnitudes of the pseudo-Brewster angles changing with $B_{e}$ at different $E_{f} = 20,30,40,50$ meV.

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Fig. 5. The in-plane $\Delta _{x + }^H$ and transverse $\Delta _{y + }^H$ spin-dependent splittings in the PSHE of the left-circular $\left | + \right \rangle$ polarized component for the $\left | H \right \rangle$ polarization vary with $B_{e}$, $E_{f}$ and $\theta _i$. (a) $\Delta _{x + }^H$ and (b) $\Delta _{y + }^H$ are plotted as functions of $B_{e}$ and $\theta _i$ at fixed $E_{f}=50$ meV, respectively. (c) $\Delta _{x + }^H$ and (d) $\Delta _{y + }^H$ are plotted as functions of $E_{f}$ and $\theta _i$ at fixed $B_{e}=0.1$ T, respectively.

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4. Conclusion

To summarize, we have investigated the quantized PSHE in a strained graphene-substrate system under a sub-Tesla external magnetic field $B_e$, and found the $B_e$ required to produce this effect is two orders of magnitude smaller than that previously reported in conventional graphene-substrate system. It is revealed that the $\Delta _x^H$ and the $\Delta _y^H$ exhibit different quantized behaviors, and they correspond to the quantized reflection coefficients $\left | {r_{pp} } \right |$ and $\left | {r_{ps} } \right |$, respectively. Notably, the $B_e$ in conventional graphene cause real LLs split, thus forming the quantized PSHE. Differently, there is a large pseudo-magnetic field $B_{ps}$ in strained graphene inducing the splitting of PLLs, so the quantized PSHE in the system has a greatly reduced dependence on the $B_{e}$. In the presence of both the $B_{ps}$ and the $B_{e}$ (needed to break time-reversal symmetry of strained graphene), the total effective magnetic field is $B_{e}+ B_{ps}$ for the K valley while $B_{e}- B_{ps}$ for K’ valley, and the valley degeneracy of the $n \ne 0$ PLLs is lifted. The splitting of PLLs and the lifted of valley degeneracy are the essential reason for the formation of quantized PSHE in the strained graphene-substrate system under a sub-Tesla $B_e$. Further research has revealed that the pseudo-Brewster angles of the system are quantized with the change of Fermi energy $E_f$ and external magnetic field $B_e$, and the quantized peak values of PSHE appear near these angles. The giant quantized PSHE is expected to be used for direct optical measurements of quantized conductivities and PLLs in strained graphene. In addition, it may also open up new ideas for strain measurement of other two-dimensional materials.