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Quantum Transport of Dirac fermions in graphene with a spatially varying Rashba spin-orbit coupling

We theoretically study electronic transport through a region with inhomogeneous Rashba spin-orbit (RSO) coupling placed between two normal regions in a monolayer graphene. The inhomogeneous RSO region is characterized by linearly varying RSO strength within its borders and constant RSO strength in the central region. We calculate the transmission properties within the transfer matrix approach. It is shown that the amplitude of conductance oscillations reduces and at the same time the magnitude of conductance increases with increasing border thickness. We also investigate how the Fano factor can be modified by the border thickness of RSO region.


Introduction
Quantum transport of graphene-based nanostructures has attracted much of interest due to potential applications [1]. One of the interesting features of graphene originates from the possibility of engineering the structural and electronic properties of graphene [2]. In this regard, although in pristine graphene, the strength of spin-orbit coupling is weak, but the possibility of enhancing the strength of spin-orbit coupling in graphene layer has recently attracted a great interest from theoretical [3,4,5,6] and experimental [7,8] viewpoints. It has been shown that spin-orbit coupling in graphene can be increased, giving rise to the so-called spin Hall effect [9,10] at zero external magnetic fields at room temperature. This opens up new opportunity to manipulate charge-and spin-related phenomena efficiently same as manipulation of pseudospin [11] and valley [12,13,14,15] degrees of freedom for carbon-based electronics applications [16].
Basically, the spin-orbit coupling in graphene can be divided into intrinsic and extrinsic origins [10]. The intrinsic spin-orbit coupling comes from carbon intra-atomic spin-orbit coupling and it can be raised up to 17mev by proximity effect between monolayer graphene and few-layers semiconducting tungsten disulfide [17]. The extrinsic spin-orbit coupling, which is known as Rashba spinorbit (RSO), can be induced through breaking the inversion symmetry of the lattice structure. A giant Rashba splitting has been observed as large as 100 meV through Au intercalation at the graphene − Ni interface [18]. In realistic cases, when RSO coupling induces in a specific region of a sample through external sources such as gate configuration [19,20], substrate [21,22,23] or adatoms [24,25,26,27], the strength of RSO coupling may not change abruptly at the region's boundaries, and its strength modulates spatially from full value to the zero [28,29], due to the leakage of the source outside the region.
Transport properties under modulated potential [30,31,32] and energy gap [33,34,35,36] in graphene layer have been considered, but the role of spatially varying RSO coupling on the transport properties of graphene still lacks a clear understanding [37,38]. While most of the studies of transport properties from RSO region focus on abrupt change of strength of spin-orbit coupling at the borders [39,40,41,42,43,44,45], in this paper, we will address the effects of variation of RSO coupling with a constant strength gradient within its border regions on the transmission probability, the conductance and the Fano factor of graphene. We find that the magnitude and the angular dependence of the transmission probabilities depend strongly on the border thickness. In addition, it is shown that if the border thickness is finite, its effect on the conductance is not only to reduce the oscillations amplitude but also to enhance the conductance magnitude. Also, depending on the values of border thickness, the Fano factor of the system can be controlled fundamentally.

Model and Theory
We consider a monolayer graphene in the xy plane with an inhomogeneous RSO region between two normal regions. The interfaces of the regions are perpendicular to the x direction as shown schematically in Fig. 1(a). The inhomogeneous RSO region is comprised of border regions that are close to the normal regions, characterizing by linearly varying strength of RSO coupling of thickness b and the central region with thickness L over which RSO strength is constant (see Fig. 1(b)). So the spatial profile of RSO strength throughout the spin-orbit region, 0 ≤ x ≤ 2b + L, can be modeled as, where λ 0 is the height of RSO strength. The spatial variation of RSO coupling is assumed to be small on the scale of the graphene's lattice constant (a = 0.246 nm). In this semiclassical approach, the long-wavelength components of Hamiltonian will be valid, subsequently, states near the valley K+ and those near K− can be considered as two separate valleys in constructing the continuum approximation of Hamiltonian. Also, due to the absence of inter-valley scattering processes, we adopt single valley picture.
Within the continuum limit, the Hamiltonian describing the total system reads as [39,40,41,42,43,44,45] where v f ≈ 10 6 m/s is the Fermi velocity, the Pauli's matrices σ and s act on the pseudospin (sublattice) and spin spaces, respectively. Also, Θ(x) is the Heaviside step function and s 0 is the identity matrix.
Note that in conventional Rashba systems, Rashba Hamiltonian is momentumdependent [46], subsequently, in the case of non-uniform Rashba coupling, there will be terms in the Hamiltonian containing derivatives of the Rashba coupling in order to ensure hermiticity of the Rashba operator [46]. But in the case of graphene due to triangular symmetry, by expanding Rashba Hamiltonian around Dirac points (low-energy expansion), the lowest order of expansion is momentum-independent [10] which is in contrast to the conventional systems.
However, if we retain higher order terms in the expansion which are momentumdependent [47], the contribution of terms including derivatives of Rashba coupling is negligible, because in our analysis we have made the assumption that the strength of spin-orbit coupling changes linearly and smoothly.
For the numerical implementation, it is convenient to consider RSO region as a sequence of slices perpendicular to the direction of transport with nearly constant strength in each slice. By solving Eq. (2) in the ith slice of RSO region, one can determine the eigenvalues as where n = +(−) represents upper (lower) subband in the l = +(−) conduction (valance) band, k (q) is the x (y) component of wave vector which will be specified below and λ i is RSO strength in the ith slice. Also, the corresponding eigenvectors are given by, where m = 1 (-1) indicates the right-moving (left-moving) carrier. In the left and right normal regions, Eqs. (3) and (4) reduce, respectively, as where φ = tan −1 q/k.
In the calculation, an incident carrier is supposed to be injected from the left to the right upon incidence angle φ. Due to particle-hole symmetry in the system, without loss of generality, we focus on the conduction band states, l = +, and choose a positive value for the carrier energy (E > 0). Translational invariance in the y direction implies that the y component of the wave vector is a good quantum number, which it can be written in terms of the incidence But the x component of wave vector depends on the region under consideration.
Using Eqs. (3), (5) and (7), the x component of wave vector can be obtained as for the normal regions, and for the ith slice of the RSO region. From Eq. (9), it is easy to see that for n = − and E > 0, k in has pure real values, consequently, we have traveling modes. But for n = + and 0 < E < 2λ i , k in becomes pure imaginary and the wave is evanescent, whereas for n = + and E > 2λ i the wave is a traveling mode [45,48].
where the coefficient t n,m (r n,m ) represents the transmission (reflection) amplitude from incident subband n into subband m. Also, with the help of Eq.
(9), (7) and (4), the total wave function in the ith slice of RSO region can be expressed as, where the coefficients a i n,m and b i n,m are the scattering amplitudes from subband n into subband m.
Using the transfer-matrix method [49], transmission amplitude in the left normal region, t n,m , can be determined. Then transmission probability T n,m can be evaluated by T n,m = | t n,m | 2 . It should be noted that since the RSO interaction does not couple the states of spin-chiral carriers with opposite subband indices, so transmission probabilities through different subbands vanish, i.e., T +,− = 0 and T −,+ = 0 [44]. Having obtained transmission probabilities, the conductance of carriers at zero temperature in subband n can be obtained by [50], where G 0 = 2e 2 W E F /h 2 v f , W is the width of system, e is the electron charge, and E F is the Fermi energy.
Finally, we investigate the subband Fano factor of this system which can be given as follows [51],

Numerical results and discussions
Throughout this paper, the value of λ 0 is fixed to be 10 meV and L = 100 nm. In the numerical calculations, we have chosen the thickness of each slice 1nm. It should be noted that we have examined larger slice thickness and same qualitative results were found.

Conclusion
In conclusion, we studied transport properties through inhomogeneous RSO region in graphene. We investigated transmission probabilities in terms of border thickness, Fermi energy and incidence angle. Also, in the analysis of the conductance as a function of the Fermi energy, we identified that the main influence of border thickness of RSO region is the smearing out of the conductance oscillations and at the same time the making an enhancement of conductance magnitude. Also, the effect of border thickness is to reduce the magnitude and oscillations amplitude of both Fano factors of upper and lower subbands. Further, we analyzed the low energy behaviors of the Fano factors for different subbands in a wide range of border thicknesses ranging from small thickness at which transmission takes place via tunneling effect, to a large one at which transport is nearly ballistic.

Acknowledgements
The authors are grateful to A. Darudi for fruitful discussions.