Spin-orbit coupling assisted by flexural phonons in graphene

We analyze the couplings between spins and phonons in graphene. We present a complete analysis of the possible couplings between spins and flexural, out-of-plane, vibrations. From tight-binding models, we obtain analytical and numerical estimates of their strength. We show that dynamical effects, induced by quantum and thermal fluctuations, significantly enhance the spin-orbit gap.

Figure 1

(a) Real-space lattice and Brillouin zone of flat graphene with 2 atoms per unit cell. (b) The same with 6 atoms per unit cell. Note that ${\mathbf{K}}_{\pm }$ are now equivalent to $\Gamma$.

Figure 2

Flat graphene and bent graphene. The arrows represent the $p_x$ (black), $p_y$ (red), and $p_z$ (blue) orbitals. The local basis of atomic $p$ orbitals in the bent graphene is uniquely defined by the isomorphism between them.

Figure 3

(a) Definition of the angle $\varphi$. (b) Sketch for the calculation of the new hoppings between $p_z$ and $p_i$ orbitals.

Figure 4

Effective Kane-Mele mass induced by the coupling with flexural phonons. In red (lower curve) the estimation neglecting the acoustic branch and the dispersion of the optical one. In blue (upper curve) the calculation within the model described in Appendix .

Figure 5

Dispersion of flexural phonons computed within the nearest-neighbor force model described in the text with $\alpha =8.5$ eV. In red (upper curve) are the dispersion for the optical branch, in blue (lower curve) are the acoustic branch.