Spin polarization and Fano–Rashba resonance in nonmagnetic graphene

We study the symmetry of spin transport in graphene with the Rashba spin–orbit coupling (SOC) and the staggered potential, which can be produced by depositing the graphene on a transition-metal dichalcogenides substrate. The results show that all three spin polarization components along the x, y and z directions are achieved with a measurable conductance in such a nonmagnetic graphene. The spin transport property near the two valleys is discussed in the light of the symmetry of the system. Both conductance and spin polarization present some certain symmetries with respect to the Rashba SOC (RSOC) and staggered potential. The system could work as a valley-spin polarization transverter which combines valleytronics and spintronics. Furthermore, the asymmetric Fano–Rashba resonance of the conductance and spin polarization could occur in a resonant structure due to interference of spin-polarized discrete and continuum states induced by the RSOC. The Fano–Rashba resonance can be effectively controlled by the gate voltage. The derived symmetry relations and numerical results could provide a guideline for the design of spin-valley-based devices.


Introduction
Graphene spintronics has received great attention in recent years due to its long spin relaxation time and promising applications [1,2]. The spin-orbit coupling (SOC) plays a crucial role in a variety of spintronics phenomena. However, in pristine graphene, the intrinsic SOC is predicted to be too weak for practical purposes [3]. The extrinsic Rashba SOC (RSOC) could compensate the weak SOC of graphene. The RSOC originates from the lack of inversion symmetry along the growth direction of the semiconductor heterostructure and it can be tuned by the external electric field [3,4]. A large RSOC λ with strength up to 13 meV has been experimentally reported for graphene grown on Ni(111) intercalated with a Au layer [5], or even larger (∼100 meV) [6]. Alternatively, graphene on transition-metal dichalcogenides (TMDs), such as MoS 2 or WSe 2 , could also achieve a RSOC λ on milli-electron volt scale that can be controlled by the transverse electric field [7,8].
The strong RSOC enables the spin manipulation in graphene. Recently, the RSOC in graphene has been utilized to research various topics, such as electronic transport [9][10][11][12][13][14][15], Andreev reflection [16][17][18][19], electron optics [20][21][22], quantum anomalous Hall effect [23][24][25], and band structure [26][27][28][29][30]. In the presence of RSOC, chiral tunneling in monolayer graphene behaves like in bilayer graphene and the spin separation is observed [10]. The RSOC could induce the spin double refraction and the spin polarization of transmission probability in graphene which strongly depend on the incident angle [9], in analogy to the case of two-dimensional electron gas [31]. In the ferromagnet-RSOC-superconductor contact, an anomalous equal-spin Andreev reflection is found due to the RSOC which yields a finite tunable spin-polarized subgap conductance [17,18]. It is demonstrated that the efficiency of optical spin injection in graphene with RSOC is of the order of the ratio of Zeeman and Rashba splittings [21]. The combination of RSOC and exchange field could break the time-reversal symmetry and open a nontrivial bulk gap in graphene, predicting a quantum anomalous Hall effect [23]. When the RSOC is stronger than the intrinsic SOC, the low-energy bands would undergo trigonal-warping deformation and anisotropic spin splitting [27]. The trigonal warping is essential for producing spin polarization of the transmitted current [29].
Hybrids of graphene and TMDs open new venues for spintronics applications [7,[32][33][34][35][36][37][38]. The proximity effects resulting from depositing a graphene layer on a TMDs substrate layer could change the dynamics of the electronic states in graphene, inducing SOC and staggered potential effects. The first-principles calculations of graphene on MoS 2 predict a giant and field-tunable proximity SOC for Dirac electrons which offer a platform for optospintronics [7]. The inverted bands and topologically protected helical edge states are formed due to SOC for graphene on WSe 2 , demonstrating the quantum spin Hall effect [34]. The valley-Hall conductivity would change the sign and the band gap is tunable in the presence of the RSOC, which leads to interband transitions of graphene on WSe 2 [38].
Motivated by the hybrids of graphene and TMDs, in this paper we study the spin transport through a graphene/RSOC/graphene (G/RSOC/G) heterostructure with a staggered potential applied on the RSOC region, as shown in figure 1(a). Here the G and RSOC in G/RSOC/G represent the pristine graphene region and the graphene region with the RSOC, respectively. The RSOC and staggered potential in graphene can be induced by monolayer TMDs [34][35][36][37][38]. Generally, the spin polarization is generated by the magnetic field which breaks the time-reversal symmetry [1,2]. It is found that the conductance is greatly spin-polarized due to the RSOC and staggered potential, even though the system still holds time-reversal symmetry. Symmetry is important to the experimental design and theoretical research. We discuss the spin transport based on the symmetry analysis by considering the spatial inversion operations, the spin rotation operations, the pseudospin rotation operations, and the time-reversal operations. In addition, we also find a Fano-Rashba resonance in the G/RSOC/G/RSOC/G structure and discuss the property of Fano-Rashba resonance in detail. The spin polarization and Fano-Rashba resonance could be controlled by the gate voltage.
The paper is organized as follows. In section 2 we introduce the theoretical model and analyze the symmetry of the model. The spin polarization in the G/RSOC/G and Fano-Rashba resonance in the G/RSOC/G/RSOC/G are discussed in section 3. Section 4 devotes to a conclusion.

Theoretical model
For the proposed G/RSOC/G model and G/RSOC/G/RSOC/G model, considering the RSOC and the staggered potential, the effective Hamiltonian describing low-energy electronic states near the valleys can be written in the form [2] where v F is the Fermi velocity, λ is the RSOC strength, V is the gate-induced potential, and η = ±1 represent the valleys K and K ′ . ∆ is the staggered potential describing the energy difference on A and B sublattices of graphene, which opens a band gap. σ x,y,z and s x,y are Pauli matrices for pseudospin (A and B sublattices) and spin, respectively. σ 0 and s 0 are unit matrices. The staggered potential ∆ and the potential V are constant in the RSOC region and vanish otherwise. The RSOC removes the spin degeneracy of the bands. Unlike in conventional semiconductor, the RSOC in graphene does not depend on the momentum [39]. In order to simplify the calculation, a dimensionless form is employed by replacing ℏvF , λ → λ d0 ℏvF , and V → V d0 ℏvF . d 0 is the length unit and ℏv F /d 0 is the energy unit. For graphene, ℏv F = 3at/2 ≈ 0.586 eV nm with the carbon-carbon distance a ≈ 0.142 nm and the nearest-neighbor hopping energy t ≈ 2.75 eV. If the length unit is d 0 = 100 nm, the energy unit is ℏv F /d 0 ≈ 5.86 meV.
The eigenvalue of equation (1) for both valleys reads (see appendix A for the details) with α = +(−) standing for the bands of spin up (spin down) and β = +(−) standing for the conduction (valence) band. Notice that equation (2) has a valley degeneracy. The wave function has the form Ψ(x, y) = ψ(x)e i kyy with ψ(x) = (ψ A↑ , ψ B↑ , ψ A↓ , ψ B↓ ) T and k y is the conserved y component of the momentum. The states ψ(x) at K and K ′ valleys in the RSOC region can be expressed as and , and the longitudinal wave vector k α satisfies the relation The superscripts ± in the wave functions represent the right-going and left-going propagating waves. In addition, the wave functions for spin-up and spin-down electrons in the G region (pristine graphene) are respectively, with k 2 x + k 2 y = E 2 . Considering an incident electron with spin s at valley η, the scattering states in the three regions of the G/RSOC/G model are given by withs = −s andᾱ = −α. r ss,ss η and a 3,4 are reflection coefficients, while t ss,ss η and a 1,2 are transmission coefficients. In the G region, the spin direction of incident electrons is out of plane and the z component of the spin is a good quantum number. The spin can be flipped in the RSOC region due to RSOC and so transmitted electrons may have opposite spin. We set t sLsR in the following discussion, which is the transmission coefficient for an incident electron at valley η with spin s L in the left lead scattered into spin s R in the right lead. s L(R) =↑ (+1) is for spin up and s L(R) =↓ (−1) is for spin down. By matching the wave functions at the interfaces x = 0 and d (see appendix B), the transmission coefficient t sLsR η can be obtained. Thus, the transmission probability is T sLsR η = |t sLsR η | 2 . Based on the Landauer-Büttiker formula, the spin dependent conductances at zero temperature are given by [40] G sLsR where θ is incident angle, and G 0 = e 2 L y k/π h is the conductance unit with the sample width L y in the y direction. We consider a short and wide junction that can be regarded as a one-dimensional system [41], Table 1. Symmetry operations on the Hamiltonian Hη(∆) and the transmission probability T sLsR η (ky, ∆). T sRsL η represents the transmission probability for electron with spin sR at valley η incident from the right lead scattered into the left lead with spin sL. Here, the spin direction of s ′ L(R) (s ′ ′ L(R) ) is the same as that of s R(L) (s R(L) ).s = −s andη = −η for spin and valley indices, respectively.

Operator
Hamiltonian Transmission probability and the value of L y does not affect the results. The total charge conductance for valley η is The spin polarizations for valley η in the right lead are defined as y, z and the conductances G ±j for the spin pointing to ±j direction. These spin polarizations can be calculated by [40] Below we will show that the RSOC would give rise to the spin flip and the finite value of spin polarizations P x,y,z η .

Symmetry of the model
Based on the symmetry of the system, we could analyze the effect of λ and ∆ on the spin transport.
Considering the spatial inversion operators R x,y , the spin rotation operators s x,y,z , the pseudospin rotation operators σ x,y,z , and the time-reversal operator T = −i s y C with the complex conjugation C, we could find all the symmetries of Hamiltonian (1) when λ is invariable (λ → λ), as shown in table 1. In particular, the system satisfies the time-reversal symmetry and the Hamiltonian between the two valleys is invariant with respect to this transformation, TH η T −1 = Hη. The transformed state in the G region is Thus, the transmission probabilities for electrons incident from left/right lead scattered into the right/left lead have T sLsR η (k y , ∆) = Ts RsL η (−k y , ∆). For a certain valley K or K ′ , the system is invariant with respect to the transformation T H η T −1 = H η at ∆ = 0.0 with the joint time-reversal operator The staggered potential σ z ∆ in H η breaks this symmetry, but reversing the sign of ∆ could lead to . The RSOC and the potential V are invariant under the reflection transformation R y with respect to the y axis, and so the system has a spatial inversion symmetry related with the operator The Hamiltonian has M 1 H η M −1 1 = H η at ∆ = 0.0 and the state has M 1 φ η (k y , s) = iφ η (−k y ,s). This indicates that M 1 not only changes k y to −k y but also flips the up spin (down spin) to the down spin (up spin). In the presence of staggered potential ∆, one may get Thus, it can be concluded that the transmission probability and the conductance for one valley satisfy Let us consider the transformation between the two valleys: and Hence, M 2 could change the state at K (K ′ ) valley to that at K ′ (K) valley with no spin-flip. This indicates that the spin transports between the two valleys are related by Obviously, the spin transport properties of the two valley are the same when ∆ = 0.0. The transformation could lead to the relation The relationship between two valleys can be also given by other joint operations, such as In addition, when λ changes the sign, i.e. λ → −λ, the Hamiltonian (1) remains unchanged under the operator s z ⊗ σ 0 or (s 0 ⊗ σ z )R x R y . The Hamiltonian would also be unchanged when (1) λ → −λ and ∆ → −∆ under the operator (s x ⊗ σ x )R y or (s y ⊗ σ y )R x ; (2) λ → −λ and η →η under the operator (s x ⊗ σ z )R y or (s y ⊗ σ 0 )R x ; (3) λ → −λ, η →η, and ∆ → −∆ under the operator (s 0 ⊗ σ x )R x R y or s z ⊗ σ y . The transformation and suggests that the sign change of λ has no effect on the conductance and the z-direction spin polarization. Besides the RSOC and the staggered potential, the intrinsic SOC λ i ηs z ⊗ σ z and the valley-Zeeman SOC λ v ηs z ⊗ σ 0 could also be induced by depositing graphene on a TMDs substrate [7,8]. The effect of the intrinsic SOC and the valley-Zeeman SOC on spin transport at the two valleys is also discussed based on the symmetry analysis in appendix C. The derived symmetry relations are expected to provide guidelines for the experimental measurements and the design of spin-valley-based devices.

Results and discussions
In the following calculation, we set the incident energy E = 20 meV, the width of RSOC region d = 100 nm, and the potential V = 0.0, unless otherwise noted, which do not affect the conclusion. We consider a ballistic transport in graphene where the device size is small compared to the mean free path. In experiments, the mean free paths of graphene have been estimated to be 600 nm [42]. The mean free path can be further enhanced by the lower temperature. We will study the spin polarization of the G/RSOC/G model in section 3.1 and then the Fano-Rashba resonance of the G/RSOC/G/RSOC/G model in section 3.2.

Spin polarization in G/RSOC/G model
First, we discuss the spin transport through the G/RSOC/G structure controlled by λ and ∆, taking the electron near K valley and incident from the left lead for instance. Figure When ϕ α = π/2, we can obtain the critical angle Thus, only electronic state |ψ − ⟩ exists in the RSOC region when |θ| > θ c α (see figure 1(b)). The four transmissions are the same and the peak value is 1/4, since only one channel is allowed in the RSOC region. . However, T ↑↑ K and T ↓↓ K are no longer equivalent for a given ∆ and k y , i.e. T ↑↑ K (k y , ∆) ̸ = T ↓↓ K (k y , ∆), although T ↑↓ K (k y , ∆) = T ↓↑ K (−k y , ∆) (see figure 2(b)), which is important to the spin polarizations P x,z η . Thus, the spin polarization p z as shown in figure 3(b). The spin polarizations gradually tend to saturation as λ increases and high spin polarizations can be achieved in the range |λ| > √ E 2 − ∆ 2 /2 where only state |ψ − ⟩ or |ψ + ⟩ exists in the RSOC region.
Next, the relationship between spin transports of the two valleys is discussed in figure 4. As shown in figure 4(a), for K or K ′ valley, the conductances are G ↑↑ . In consequence, the spin polarizations take on The effect of staggered potential ∆ is similar to that of the magnetic field on spin transport in two-terminal quantum waveguide structures [40]. The conductances between the two valleys have G sLsR . Thus, the spin polarizations for the two valleys are symmetric with respect to ∆ = 0.0, that is P x,y,z Because of M 3 H η (∆)M −1 3 = Hη(∆), the conductances for given ∆ have G sLsR K (∆) = Gs LsR K ′ (∆), and so as shown in figure 4(b). At the staggered potential ∆ = 0.0, all the conductances are equivalent and they are only polarized in the y direction. With the appearance and increase of ∆, the conductances become different  and present oscillation behavior (see figure 4(a)), thus they could be polarized in the x and z directions (see figure 4(b)). Although the spin polarizations in the x and z directions can cancel due to K and K ′ valley electrons, they are spin-valley related. For example, for ∆ > 0, the K and K ′ valley electrons are polarized in the +x and −x direction, respectively. By tuning the staggered potential ∆, the high spin polarizations may be observed.
Considering RSOC and staggered potential of graphene induced by different substrates, figure 5 discusses the effect of different values of RSOC λ and staggered potential ∆ on spin polarization. One can see that all the spin polarizations P x,y,z K increase with λ. As ∆ increases, P x,z K increases but P y K decreases. The variation of spin polarization is nonmonotonic. Interestingly, P x K presents a linear change with ∆ in the region [−10 meV, 10 meV] for large λ. The result indicates that one could gain a large spin polarization along certain direction by adjusting λ and ∆.
When a gate-induced potential V is applied in the RSOC region, figure 6(a) exhibits the dependence of spin polarization on the potential. We can see that the spin polarizations possess because the potential does not destroy the discussed symmetries of the system. The results manifest that the spin polarizations could be effectively enhanced and converted from positive value to negative value by adjusting the potential. As an application, the symmetry analysis and numerical results indicate that the G/RSOC/G model could work as a spin filter along the y direction and work as a valley-spin polarization transverter along the x and z directions which may transform a valley-polarized incident electron from the left lead to a spin-polarized outgoing electron in the right lead, as shown in figure 6(b). The valley-polarized current and spin-polarized current can be converted to each other by the gate voltage. Valleytronics and spintronics have caught much attention since the discovery of graphene [1,2,43,44]. We provide an approach to combine valleytronics and spintronics in the nonmagnetic graphene.

Fano-Rashba resonance in G/RSOC/G/RSOC/G model
Fano resonance effect is important phenomena in quantum physics, describing the quantum interference of discrete and continuum states, the main feature of which is the asymmetric line profile. The Fano resonances have been observed in various systems, such as atomic physics, Raman scattering, and electron transport [45]. It is predicted that a Fano resonance could be generated by RSOC in semiconductor quantum wire, named as Fano-Rashba resonance [46]. However, to our knowledge, the work on Fano-Rashba resonance in two-dimensional atomic crystals (e.g. graphene, silicene, TMDs, and boron nitride) is still blank. In this subsection, we study the Fano-Rashba resonance in graphene, which could occur in the G/RSOC/G/RSOC/G device. The widths of the two RSOC regions are the same and are set as d. The width of the middle G region is w. The transport near the K valley is taken as an example to discuss the Fano-Rashba resonance in the following.
From figure 7(a) we can clearly see that the transmission probability displays an asymmetric Fano-Rashba resonance, which mainly occurs in the range θ > θ c α . Such a phenomenon can be understood as follow. When θ > θ c α , |ψ + ⟩ is annihilated state and |ψ − ⟩ is propagating state in the RSOC region. Meanwhile, |φ ↑,↓ ⟩ is propagating state in the middle G region. As a result, the bound state related to |ψ + ⟩ and the continuous state related to |ψ − ⟩ are formed in the whole system, and their interference effect would lead to the Fano-Rashba resonance. Interestingly, the Fano-Rashba resonances of the four transmissions appear at the same location and the shapes are sharp. There also exist a smooth resonance between two Fano-Rashba resonances, which is induced by the resonant mode in the RSOC region. In addition, the total transmission probability T K and spin polarization p z K also show Fano-Rashba resonance (see figure 7(b)). The effect of structural parameters on Fano-Rashba resonance is discussed in figure 8. From figure 8(a) one may find that the position of the smooth resonance can be controlled by ∆, since it arises from the resonant mode in the RSOC region. As the width d of the RSOC region increases, more discrete energy levels of the resonant mode would appear, and so more smooth resonances are formed (see figure 8(b)). However, the Fano-Rashba resonance is independent to the change of ∆ and d. With the increase of the width w of the middle G region, the number and the position of the Fano-Rashba resonance could be regulated (see figure 8(c)), since it is related to the bound state in the middle G region.
In addition, figure 9(a) shows the transmission as a function of the potential V. Note that the potential V is only applied on the middle G region of the G/RSOC/G/RSOC/G model. The red dashed curves are Fano-Rashba resonances fitted by the normalized Fano curve. The formula for the shape of the Fano resonance profile can be expressed as with ϵ = (E − E 0 )/Γ [45]. E 0 and Γ determine the position and width of the Fano curve, respectively. T 0 is the maximum value of the peak and q is a phenomenological shape parameter. Distinctly, the Fano-Rashba resonance curves calculated from the Hamiltonian (1) and by equation (30) match each other perfectly.  The transmission exhibits a sharp dip is followed by a sharp peak, which is the typical characteristic of asymmetric Fano resonance. It can be seen that around a Fano-Rashba resonance, a small change of the potential would switch the transmission from the maximum value of unit to zero, which is the basis of a digital switch. Consequently, the Fano-Rashba resonance can be used to realize a digital switch or electron switch that can be turned on and off with a small gate voltage, as shown in figure 9(b). Although the Fano-Rashba resonance is angle dependent, it has been experimentally demonstrated that the angle-dependent transmission in graphene could be observed directly [47,48]. Therefore, the proposed digital binary/electron switch is feasible in experiments. Note that the Fano-Rashba resonance still exists for a positive potential. Finally, we discuss the Fano-Rashba resonance in the conductance and spin polarization, which are the measurable quantities in experiment. Figure 10 shows the conductance and spin polarization as a function of (a) the width w and (b) the potential V. We can clearly see that both conductance and spin polarization present Fano-Rashba resonance. Observably, the position of Fano-Rashba resonance in the conductance is the same as that in the spin polarization. Experimentally, a sudden change of conductance suggests the appearance of a Fano-Rashba resonance. Based on the dispersion relation E = ±ℏv F √ k 2 x + k 2 y + V and the resonant condition k x w = mπ, we can approximately obtain the potential V FR where the Fano-Rashba resonance takes place, and m is an integer. The result indicates that the Fano-Rashba resonance could be effectively controlled by the width and the potential.

Conclusion
In summary, we have predicted two intriguing features of transport in the nonmagnetic graphene, i.e. the spin polarization and Fano-Rashba resonance. The results indicate that not only the RSOC but also the staggered potential are important to generate the spin polarization. The properties of conductance and spin polarization are discussed detailedly based on the symmetry relations in the presence of staggered potential, RSOC, intrinsic SOC, and valley-Zeeman SOC. In addition, the RSOC could also give rise to the asymmetric Fano-Rashba resonance in the transport. As an application, the systems could work as a valley-spin polarization transverter and a digital/electron switch. The symmetry relations of spin polarization and Fano-Rashba resonance should be beneficial to the experimental measurements and the spin-valley-based device designs.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).
T sLsR η (k y , ∆, λ v ) = Ts RsL η (−k y , −∆, −λ v ). The Hamiltonian H η under the operators M 1 , M 2 , and M 3 has the relations: Similar to RSOC, the intrinsic SOC λ i is invariable under these transformations. These relations suggest that the conductance for the two spins at the two valleys satisfies Therefore, based on the symmetry analysis we could predict the effect of intrinsic SOC and valley-Zeeman SOC on the main property of spin transport at the two valleys.