Bias-dependent D’yakonov-Perel’ spin relaxation in bilayer graphene

We calculate the spin relaxation time of mobile electrons due to spin precession between random impurity scattering (D’yakonov-Perel’ mechanism) in electrically gated bilayer graphene analytically and numerically. Due to the trigonal warping of the band structure, the spin relaxation time exhibits an interesting nonmonotonic behavior as a function of both the Fermi energy and the interlayer bias potential. Our results are in good agreement with recent four-probe measurements of the spin relaxation time in bilayer graphene and indicate the possibility of an electrically switched spin device.

Figure 1

(a) Spin-orbit field ${\mathit{\Omega }}_{+}\left(\mathbf{k}\right)$ of electrons in bilayer graphene. (b) Spin splitting $\Delta E$ where circles/diamonds refer to the energy difference between the two electron-like low-energy bands $\Delta E=|E_{+,1}-E_{+,2}|$ obtained from a numerical diagonalization of the full Hamiltonian including ${\lambda }_4$-like SOI and lines to the splitting of the spin-orbit field $\Delta E=\hslash |{\mathit{\Omega }}_{+}\left(\mathbf{k}\right)|/2$ given by Eq. (5).

Figure 2

In-plane spin relaxation time ${\tau }_{S,\parallel }$ in bilayer graphene with ${\lambda }_4$-type SOI in the isotropic limit ($v_3=0$), (a) as a function of the Fermi energy $E_F$ for different bias voltages $U$, and (b) as a function of the bias voltage $U$ at a constant Fermi energy $E_F=0.06$eV. In both plots the momentum relaxation time is chosen to be ${\tau }_p={10}^{-13}$s. Dashed lines correspond to the large Fermi energy approximation given in Eq. (17). Note that we choose the Fermi energy to be larger than the $\mathbf{k}=0$ bias offset $E(\mathbf{k}=0)=U/2$, thus lines start at different Fermi energies for different values of the bias $U$.

Figure 3

In-plane spin relaxation in bilayer graphene with ${\lambda }_4$-type SOI. The plots show a comparison of the numerical data (diamonds) and the zeroth- (orange dashed) and first- (continuous purple lines) order estimates, ${\tau }_{S,\parallel }^0$ (15) and ${\tau }_{S,\parallel }^1$ (B6), for different values of the interlayer bias $U$ and the Fermi energy $E_F$. All curves are calculated in the limit of $a\ll R\ll 1/k_F$, using a mean scattering time ${\tau }_{\mathrm{sc}}={\tau }_p=0.1$ps. The insets show the trigonal warping of the Fermi surfaces. The corresponding spin relaxation times are listed in Table .