Spin-orbit coupling in elemental two-dimensional materials

The fundamental spin-orbit coupling and spin mixing in graphene and rippled honeycomb lattice materials silicene, germanene, stanene, blue phosphorene, arsenene, antimonene, and bismuthene is investigated from first principles. The intrinsic spin-orbit coupling in graphene is revisited using multiband $k·p$ theory, showing the presence of nonzero spin mixing in graphene despite the mirror symmetry. However, the spin mixing itself does not lead to the the Elliott-Yafet spin relaxation mechanism, unless the mirror symmetry is broken by external factors. For other aforementioned elemental materials we present the spin-orbit splittings at relevant symmetry points, as well as the spin admixture $b^2$ as a function of energy close to the band extrema or Fermi levels. We find that spin-orbit coupling scales as the square of the atomic number $Z$, as expected for valence electrons in atoms. For isolated bands, it is found that $b^2$ follows a scaling law close to $b^2\sim Z^4$. The spin-mixing parameter also exhibits giant anisotropy which, to a large extent, can be controlled by tuning the Fermi level. Our results for $b^2$ can be directly transferred to spin relaxation time due to the Elliott-Yafet mechanism, and therefore provide an estimate of the upper limit for spin lifetimes in materials with space inversion center.

Figure 1

(a) Nonrelativistic band structure of graphene with identified irreducible representations of the bands at the $K$ point. The labeling of the energy bands follows the irreducible representations of the $D_{3h}$ symmetry group of the $K$ point in graphene. Red and blue arrows visualize inter- and intraband couplings of the SOC Hamiltonian. The insets show a perspective and a top view of the crystalline structure of honeycomb 2D materials. Lattice vectors are labeled ${\vec{v}}_1$ and ${\vec{v}}_2$, ${\delta }_{\text{z}}$ is the out-of-plane lattice distortion (${\delta }_{\text{z}}=0$ for graphene), and the unit cell is the shaded gray. (b) A sketch of effects of intra- and interband SO coupling on the band structure and spin expectation values $\langle {\widehat{s}}_z\rangle$.

Figure 2

Relativistic band structures from first principles plotted along high symmetry points of the first Brillouin zone, shown as the inset in (b). The insets in (a), (c) and (e) visualize the splitting of degenerate orbital states at the $\mathrm{\Gamma }$ (${\mathrm{\Delta }}_{\text{so}}^{\mathrm{\Gamma }}$) and $K$ (${\mathrm{\Delta }}_{\text{so}}^K$) points upon the inclusion of spin–orbit coupling. The corresponding values of ${\mathrm{\Delta }}_{\text{so}}^{\mathrm{\Gamma }}$ and ${\mathrm{\Delta }}_{\text{so}}^K$ are collected in Table 2. The fundamental gap $\mathrm{\Delta }$ is depicted in (e). (f) Ordering of four bands close to the band gap at the $\mathrm{\Gamma }$-point ($D_{3d}$ symmetry group) with (blue) and without (red) SOC for blue phosphorene (d) (${\mathrm{\Gamma }}_3^{-}$ irrep for the conduction band without SOC), arsenene (e) and antimonene (g) (${\mathrm{\Gamma }}_2^{-}$ irrep for the conduction band without SOC). (i) Same as is (f) but for bismuthene (h). The strong SOC induces crossing of the two top-most bands and leads to band gap inversion.

Figure 3

Spin-orbital gap at the $K$ (${\mathrm{\Delta }}_{\text{so}}^K$) and $\mathrm{\Gamma }$ (${\mathrm{\Delta }}_{\text{so}}^{\mathrm{\Gamma }}$) points versus the atomic number $Z$ for materials of group 14 (${\mathrm{\Delta }}_{\text{so,14}}^K$, ${\mathrm{\Delta }}_{\text{so,14}}^{\mathrm{\Gamma }}$) and of group 15 (${\mathrm{\Delta }}_{\text{so,15}}^K$, ${\mathrm{\Delta }}_{\text{so,15}}^{\mathrm{\Gamma }}$). The names of elements are shown on the top $x$ axis. A quadratic function of $Z$ is plotted for the reference (dotted black line).

Figure 4

Calculated average spin-mixing parameter $b^2$ versus Fermi energy relative to the valence (conduction) band maximum (minimum) for materials of group 14. Materials are labeled by the element name: C graphene, Si silicene, Ge germanene, and Sn stanene. (a) Valence band and $\text{SQA}=Z$. (b) Conduction band and $\text{SQA}=Z$. The solid gray vertical line marks the values of $b^2$ plotted in Fig. 6. (c) Same as (a) but for $\text{SQA}=X/Y.$ (d) Same as (b) but for $\text{SQA}=X/Y$.

Figure 5

Calculated average spin-mixing parameter $b^2$ versus Fermi energy relative to the valence (conduction) band maximum (minimum) for materials of group 15. Materials are labeled by the element names: P blue phosphorene, As arsenene, Sb antimonene, and Bi bismuthene. (a) Valence band and $\text{SQA}=Z$. (b) Conduction band and $\text{SQA}=Z$. The solid gray vertical line marks the values of $b^2$ plotted in Fig. 6. (c) Same as (a) but for $\text{SQA}=X/Y$. (d) Same as (b) but for $\text{SQA}=X/Y$.

Figure 6

Averaged spin-mixing parameter $b^2$ in the conduction band and $\text{SQA}=Z$ versus the atomic number $Z$. Symbols represent the first-principles data, while lines are fits of a power function to the data, where $\alpha =4.6\times {10}^{-11}$ and $\beta =3.8\times {10}^{-11}$. The names of elements are shown on the top $x$ axis. The values of $b^2$ were taken from Figs. 4 and 5 at $E_F=60$ meV (marked by vertical lines in the corresponding figures).

Figure 7

Anisotropy of spin-mixing parameter $b_{\text{SQA}=X/Y}^2/b_{\text{SQA}=Z}^2$ versus Fermi energy for materials made of elements of group 14: (a) valence band and (b) conduction band. The Fermi energy is given with respect to the valence band maximum for the valence band and with respect to the conduction band minimum for the conduction band.

Figure 8

Anisotropy of spin-mixing parameter $b_{\text{SQA}=X/Y}^2/b_{\text{SQA}=Z}^2$ versus Fermi energy for materials made of elements of group 15: (a) valence band and (b) conduction band. The Fermi energy is given with respect to the valence band maximum for the valence band and with respect to the conduction band minimum for the conduction band.