Band-structure topologies of graphene: Spin-orbit coupling effects from first principles

The electronic band structure of graphene in the presence of spin-orbit coupling and transverse electric field is investigated from first principles using the linearized augmented plane-wave method. The spin-orbit coupling opens a gap of $24\mu \text{eV}$ (0.28 K) at the $K\left(K^{\prime }\right)$ point. It is shown that the previously accepted value of $1\mu \text{eV}$, coming from the $\sigma \text{−}\pi$ mixing, is incorrect due to the neglect of $d$ and higher orbitals whose contribution is dominant due to symmetry reasons. The transverse electric field induces an additional (extrinsic) Bychkov-Rashba-type splitting of $10\mu \text{eV}$ (0.11 K) per V/nm, coming from the $\sigma \text{−}\pi$ mixing. A “miniripple” configuration with every other atom shifted out of the sheet by less than 1% differs little from the intrinsic case.

Figure 1

(Color online) Graphene’s essentials. Bottom-up: carbon atoms form a honeycomb lattice with two atoms in the unit cell. The first Brillouin zone of the reciprocal lattice contains two nonequivalent Dirac points, $K$ and $K^{\prime }$. The relevant states at the Fermi level form two touching cones with the tips at $K\left(K^{\prime }\right)$.

Figure 2

(Color online) Band-structure topologies of graphene. Transverse electric field drastically changes the topology of the bands near $K\left(K^{\prime }\right)$. (a) Zero electric field. (b) Electric field of magnitude $E=1.0\text{V}/\text{nm}$. (c) $E=2.44\text{V}/\text{nm}$. (d) $E=4.0\text{V}/\text{nm}$. The first principles results are represented by circles (the Fermi level is at zero). The curves are fits to the analytical model. The spin branch $\mu =1$ is shown in solid (red), $\mu =-1$ in dashed (blue). The calculated Fermi velocity is $v_{\text{F}}=0.833\times {10}^6\text{m}/\text{s}$.

Figure 3

(Color online) Bychkov-Rashba-type splitting in graphene and orbital-resolved density of states. (a) Transverse electric field is modeled with a zigzag potential. (b) Bychkov-Rashba spin-orbit induced splitting at $K\left(K^{\prime }\right)$ as a function of the electric field. The slope is $9.9\mu \text{eV}$ per V/nm. (c) Projected density of states to particular atomic orbitals. The Fermi level is at zero. The $d$-character is enhanced by a factor of ten.

Figure 4

(Color online) Calculated spectral gap at $K\left(K^{\prime }\right)$ in the miniripple configuration (inset), as a function of the relative corrugation strain with respect to the lattice constant $a$. Neglecting $d$ and higher orbitals results in an almost constant shift, proving that the main effects come from $\sigma \text{−}\pi$ hybridization.