Independent degrees of freedom in two-dimensional materials

In hexagonal two-dimensional (2D) materials such as graphene, hexagonal boron nitride, or transition metal dichalcogenides, electrons possess intrinsic degrees of freedom (DOFs): sublattice pseudospin, spin, and valley. Independent polarization of the DOFs for carrying information can be harnessed in future semiconductor industry. Here we propose a general Hamiltonian based on the Haldane model with spin-orbit coupling (SOC) which realizes the independence of all the DOFs that can tune optical absorption spectra as a function of the SOC. The present result suggests a new 2D material or a new 2D electronic device which offers any combination of pseudospintronics, spintronics, and valleytronics applications.

Figure 1

(a) The 2D honeycomb lattice with nonequivalent $\mathrm{A}$ and $\mathrm{B}$ sublattices. The next-nearest-neighbor hopping with positive (negative) phase in the Haldane model is marked by the solid (dashed) arrows. (b) The first Brillouin zone (BZ) of the hexagonal lattice with the high-symmetry points $\mathrm{\Gamma },M$ and the two nonequivalent points $K,K^{\prime }$.

Figure 2

Diagram of the conditions for eight cases in the Haldane model with spin-orbit coupling in the $|t_2^b|-|\lambda |$ plane for (a) $\mathrm{\Delta }=0$ and (b) $\mathrm{\Delta }>0$. The numbers indicate each case in Table 1, with the missing cases underlined (4, 7, and 3). Cases 7 and 8 are the entire unannotated regions in (a) and (b), respectively.

Figure 3

Missing cases 4, 7, and 3 for any existing 2D materials and theories. (a), (c), and (e) are spin-dependent energy dispersion with PP in color and the Fermi energy $E_{\mathrm{F}}=0$ (horizontal dashed line), while in (b), (d), and (f) we plot $|M_{+}^{\uparrow }{\left(\mathbf{k}\right)|}^2$ (left) and $|M_{-}^{\uparrow }{\left(\mathbf{k}\right)|}^2$ (right) over the hexagonal BZ (white dashed line). $t_1=1$ eV, $t_2^a=0, t_2^b=0.3\times \sqrt{3}$ eV, and $\varphi =\pi /2$ for all cases. (a),(b) Case 4, $\lambda =0.3$ eV, $\mathrm{\Delta }=t_2^b-\lambda$; (c),(d) case 7, $\lambda =0.3$ eV, $\mathrm{\Delta }=0$; (e),(f) case 3, $\lambda =t_2^b, \mathrm{\Delta }=0$.