Optical excitation of valley and spin currents of chiral edge states in graphene with Rashba spin-orbital coupling

Graphene on a substrate with a topological line defect possesses chiral edge states that exhibit linear dispersion and have opposite Fermi velocities for two valleys. The chiral edge states are localized at the line defect. With the presence of Rashba spin-orbital coupling, the dispersion of the chiral edge states splits into two. The optical excitation is modeled by the generalized semiconductor Bloch equation based on tight-binding theory. Charge, valley, and spin currents generated by normally incident plane waves through the photogalvanic effect as well as those generated by oblique light through the surface-plasmon drag effect are studied. Conditions for optical generation of purely localized valley or spin currents, which are solely originated from the chiral edge states, are discussed.

Figure 1

(a) Top view of the graphene nanoribbon on the h-BN substrate. The lattice sites on top of the boron and nitride atom are represented by the blue (dark) and red (gray) circles, respectively. The dashed line indicates the location of the topological line defect along the zigzag edge. The rectangle is the supercell for the tight-binding calculation. (b) Side view of the system sustaining the photogalvanic effect. (c) Side view of the system sustaining the surface-plasmon drag effect.

Figure 2

(a) Band structure of the chiral edge states in the K valley ($\tau =1$), which is calculated by the Dirac fermion model. The red (filled) and blue (empty) dots indicate the upper and lower branches of the chiral edge states. The solid (dotted) lines are the band edge of the first (second) conduction and valence bands, i.e., ${\epsilon }_{-}(0,k_y)$ $\left[{\epsilon }_{+}(0,k_y)\right]$. The dashed lines indicate the band gap of the bulk region. (b) The expectation values of spin $x$ of the corresponding chiral edge states. The red (filled) and blue (empty) dots are calculated with the continuous Dirac fermion model. The red (dash-dotted) and blue (dashed) lines are calculated with the discrete tight-binding model. (c) The band structure of the nanoribbon with the topological line defect in the middle, which is calculated with the tight-binding theory.

Figure 3

The wave function and spin distribution of a specific chiral edge state indicated by the horizontal arrow in Fig. 2. (a) The wave function of each spinor component and the modulus of the wave function. (b) The expectation value of spin $x$, $y$, and $z$ vs position $x$.

Figure 4

The valley currents (a), (b) and spin currents (c), (d), which are generated by the photogalvanic effect with the incident field polarization being ${\widehat{\mathbf{e}}}_x$ and the amplitude being $E_0={10}^{-3}\text{V}/\mu \text{m}$, vs frequency of the incident field. (a), (c) Intrinsic graphene with $E_F=0$. (b), (d) Graphene with $E_F=28\text{meV}$. The legend in figure (b) defines the corresponding line types for the localized chiral currents $J_{y,c}$, bulk currents $J_{y,b}$, and total currents, which applies to all four subfigures.

Figure 5

The interband optical transition channels among the bulks bands as well as between the bulk bands and the chiral bands, plotted as double arrows. The conduction- and valence-band edges are denoted by the thick black lines, and the chiral bands are denoted by the thin blue and red lines. The minimal frequency of the incident field for transition between the conduction and valence bulk bands in the photogalvanic effect (a) is the band gap of the bulk bands, $2{\mathrm{\Delta }}_0$; that in the plasmon drag effect (b) is 69 $\mathrm{meV}>2{\mathrm{\Delta }}_0$, because of the momentum transfer from the SPs to the electrons.

Figure 6

The charge currents (a), (b), valley currents (c), (d), and spin currents (e), (f), which are generated by the plasmon drag effect with the incident field wave number being $\left|\mathbf{q}\right|=0.085{\text{nm}}^{-1}$ and the amplitude being $E_0={10}^{-3}\text{V}/\mu \text{m}$, vs frequency of the incident field. (a), (c), (e) Intrinsic graphene with $E_F=0$. (b), (d), (f) Graphene with $E_F=28\text{meV}$. The legend in figure (b) applies to all six subfigures.