Spin-split electronic states in graphene: Effects due to lattice deformation, Rashba effect, and adatoms by first principles

The spin-dependent electronic structure of graphene is investigated by first-principles calculations, using relativistic full-potential linearized augmented plane wave and Korringa-Kohn-Rostoker methods. Our systematic study addresses various effects on the electronic states at the Dirac points: in-plane and out-of-plane deformation of graphene’s honeycomb lattice, external electric fields, doping and band filling due to heavy and magnetic adatoms (Au and Ni). Having revealed the underlying mechanisms, our findings open a route to manufacture graphene with sizable spin splittings.

Figure 1

(Color online) Schematic of the electric field implementation in FLAPW. The graphene layer (center) is sandwiched by a capacitor with charges $\pm q$ which imposes a transverse electrostatic field.

Figure 2

(Color online) (a) Transition from the honeycomb to the square lattice. The graphene layer is snapshot for $U=1.0$, 0.7, 0.3, and 0.0 [cf. Eqs. (1, 2)]. Yellow (light gray) and red (dark gray) spheres distinguish the A and B sublattices. The angle $\alpha$ between the basis vectors ${\vec{a}}_1$ and ${\vec{a}}_2$ is given in addition. (b) Width $\Delta$ of the fundamental band gap at $K$ $\left(K^{\prime }\right)$ of the two-dimensional Brillouin zone versus $U$. $\Delta$ is shown for calculations without spin-orbit coupling (“Without SO,” large, black), with spin-orbit coupling (“SO,” medium-sized, red), as well as with spin-orbit coupling and an external electric field of $0.1\text{V}/\text{Å}$ (“$\text{SO}+\text{E}$,” small, green), respectively. The top abscissa indicates the nearest-neighbor distance (in Bohr radii).

Figure 3

(Color online) Band structure of graphene near the Dirac point $K$ $\left(K^{\prime }\right)$ (a) without SO coupling and without external electric field, (b) including SO coupling and without electric field, and (c) including SO coupling and an external electric field of $0.1\text{V}/\text{Å}$. In (c), lines depict the model band structure, Eq. (5), whereas red small and blue big symbols represent the $\mu =1$ and $\mu =-1$ branches obtained from the self-consistent calculations, respectively.

Figure 4

Strength ${\lambda }_{\text{R}}$ of the Rashba spin-orbit coupling in graphene versus strength of an external electric field (dots). The line serves as guide to the eyes.

Figure 5

(Color online) Spin-resolved band structure of graphene near $K$ $\left(K^{\prime }\right)$ with an external electric field of about $0.8\text{V}/\text{Å}$. The upper (a) left and (b) right panels represent the band structure near $K$ for the spin quantization lying in-plane and out-of-plane (parallel to the surface normal), respectively. The lower panels, (c) and (d), are as (a) and (b) but near $K^{\prime }$. The color (grayscale) distinguishes between the two spin projections while the thickness indicates the degree of the spin polarization.

Figure 6

(Color online) Perspective view of the spin topology of two Dirac cones at the $K$ (schematic or artist’s view). At large $k$ offsets from $K$ the spins (arrows) are in-plane [Fig. 5a] and rotate counterclockwise along the circular cuts of the Dirac cones. The spin are rotated increasingly out of the plane [Fig. 5b] when approaching the tip of a cone, leading to spin vortices.

Figure 7

(Color online) Spin-orbit coupling and fundamental SO band gap in mini-ripple graphene. (a) Intrinsic SO splitting ${\lambda }_{\text{i}}$ versus buckling $d$. The latter is the $z$ distance between A and B carbon atoms (sublattices), in units of the lattice constant $a$. (b) Fundamental SO band gap versus external electric field for $d/a=1\mathrm{\%},\dots ,5\mathrm{\%}$ (as indicated). Lines are guides to the eyes.

Figure 8

Band structure of $\pi$ states in graphene with $\frac{1}{2}$ of a monolayer of Au adsorbed ($k$ relative to $K$), (a) without SO coupling and (b) including SO coupling.

Figure 9

(Color online) Electronic properties of graphene covered by $\frac{1}{2}$ of a monolayer of Au. The shift $\Delta E_{\text{F}}$ of the Dirac point with respect to the Fermi level $E_{\text{F}}$ (red small circles) and the SO splitting ${\Delta }_{\text{SO}}$ at the Dirac point (blue big circles) are displayed versus Au-graphene distance (equilibrium distance $d_{\text{Au}}=6.41a_0$). Lines are guides to the eyes.

Figure 10

(Color online) Electronic properties of graphene covered by $\frac{1}{2}$ of a monolayer of Ni. Band structure of $\pi$ states close to the Fermi level (a) with $k$ relative to $K$ and (b) that of deep-laying $\pi$ states at $\Gamma$. Colors differentiate between majority and minority spin bands, black thin and red thick bands represent the results without SO coupling, cyan thin and orange thick bands in panel (b) represent those including SO coupling.