Symmetry, distorted band structure, and spin-orbit coupling of group-III metal-monochalcogenide monolayers

The electronic structure of (group-III) metal-monochalcogenide monolayers exhibits many unusual features. Some, such as the unusually distorted upper valence-band dispersion we describe as a “caldera,” are primarily the result of purely orbital interactions. Others, including spin splitting and wave-function spin mixing, are directly driven by spin-orbit coupling. We employ elementary group theory to explain the origin of these properties, and use a tight-binding model to calculate the phenomena enabled by them, such as the band-edge carrier effective $g$ factors, optical absorption spectrum, conduction electron spin orientation, and a relaxation-induced upper-valence-band population inversion and spin polarization mechanism.

Figure 1

Unit cell of single-layer group-III metal monochalcogenide in (a) perspective- and (b) plan view. The red and green spheres correspond to chalcogen anions and metal cations, respectively. The dashed blue frame represents the unit cell boundaries. The three solid lines in (b) are axes for ${180}^{\circ }$ in-plane rotation operations. Panel (c) shows the reciprocal lattice points and reduced Brillouin zone, where different colored markers indicate symmetry-related points of different zone-center character, resulting in degeneracies of $\mathrm{\Gamma }$-point plane-wave eigenstates in the (d) nearly-free electron band structure along $K-\mathrm{\Gamma }-M$ axes.

Figure 2

Spin-independent tight-binding band structure around the zone center along the $K-\mathrm{\Gamma }-M$ directions, in the vicinity of the band gap. Here, the $y$ axis (energy) is contracted within the gap region for better illustration. IRs of zone center eigenstates are shown. The double-headed arrows indicate dominant $\mathbf{k}·\widehat{\mathbf{p}}$ interactions responsible for a caldera-shaped highest valence band depicted by the 3D illustration inset.

Figure 3

Evolution of valence bands before (left) and after (right) introducing spin-orbit interaction. Gray arrows indicate spin-orbit induced broken degeneracy of the ${\mathrm{\Gamma }}_3^{\pm }$ valence band. The symmetries of zone-center spin-dependent eigenstates are described by mixed basis functions given in Table 1. Here, $X=yz$ and $Y=xz$.

Figure 4

(a) Spin-orbit-induced $k$-cubic Dresselhaus splitting [Eq. (8)] of the spin-up and spin-down subbands is represented by the red manifold. The green hexagonal plane is similar to the reduced Brillouin zone with $\mathrm{\Gamma }-K$ and $\mathrm{\Gamma }-M$ axes shown. Blue vectors represent the Dresselhaus internal magnetic field. Note that the out-of-plane field direction is reversed for $K$ and $K^{\prime }$ and vanishes along $\mathrm{\Gamma }-M$. (b) Dominant perturbation paths (the two dashed loops) for the calculation of the coefficient ${\gamma }_1$ in Eqs. (8) and (9). Horizontal lines represent the spin-dependent $\mathrm{\Gamma }$-point states, with their single group origin listed on the right. $E_{1–4}$ are the energy values of the spin-split states with ${\mathrm{\Gamma }}_3^{+}$ single group origin, with respect to the energy of ${\mathrm{\Gamma }}_{1v}^{+}$, in ascending order. $P_2$ and $P_3$ are the same as they are in Fig. 2, and $Q$ is the momentum matrix element between conduction and valence bands with the same ${\mathrm{\Gamma }}_3^{+}$ symmetry.

Figure 5

$K-\mathrm{\Gamma }-M$ orbital magnetic moment of both spin states in the ${\mathrm{\Gamma }}_1^{+}$ valence band. The units chosen indicate the orbital contribution to the effective $g$ factor. (Inset) The orbital magnetic moment of the uppermost valence band within the full Brillouin zone, highlighting $K/K^{\prime }$ valley magnetism.

Figure 6

Spin-resolved optical absorption of right-handed circular polarized light resulting in excitation of electrons into the ${\mathrm{\Gamma }}_2^{-}$ conduction band, and the resulting spin polarization (black line, right axis). Critical band-edge energies are indicated by dashed or dotted lines.

Figure 7

(a) Schematic illustration showing optical orientation of spin-polarized conduction electrons via circularly-polarized electromagnetic excitations from the lower valence band and subsequent relaxation dynamics resulting in polarization of the upper valence band. (b) Spin-dependent carrier density in an equivalent three-level system modeled by Eqs. (12, 13, 14, 15, 16, 17, 18) and (c) the corresponding spin polarizations. In normalized units, the parameters are ${\tau }_c=80, {\tau }_v=1, {\tau }_{\ell }=1, R_z=10, {\tau }_h=1, D_c=5, D_{\ell }=5$, and $G^{\downarrow }=0$.