Exciton $g$ factors of van der Waals heterostructures from first-principles calculations

External fields are a powerful tool to probe optical excitations in a material. The linear energy shift of an excitation in a magnetic field is quantified by its effective $g$ factor. Here we show how exciton $g$ factors and their sign can be determined by converged first-principles calculations. We apply the method to monolayer excitons in semiconducting transition metal dichalcogenides and to interlayer excitons in ${\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$ heterobilayers and obtain good agreement with recent experimental data. The precision of our method allows us to assign measured $g$ factors of optical peaks to specific transitions in the band structure and also to specific regions of the samples. This revealed the nature of various, previously measured interlayer exciton peaks. We further show that, due to specific optical selection rules, $g$ factors in van der Waals heterostructures are strongly spin- and stacking-dependent. The calculation of orbital angular momenta requires the summation over hundreds of bands, indicating that for the considered two-dimensional materials the basis set size is a critical numerical issue. The presented approach can potentially be applied to a wide variety of semiconductors.

Figure 1

Properties of transition metal dichalcogenide monolayers ${\mathrm{MX}}_2$. (a) Top view of the atomic structure, large and small balls represent M (metal) and X (chalcogen) atoms, respectively. (b) The Brillouin zone with the points $\mathrm{\Gamma }$ at the center and K at the corners. The sign of the K points (valley index) alternates. (c) Schematic band structure at the $+\mathrm{K}$ point. Small arrows next to the colored bands indicate the spin orientation of the conduction (c, $\mathrm{c}+1$) and valence (v-1, v) bands. Double arrows indicate dipole-allowed optical transitions, where the polarization $\sigma +$ is shown in red, $z$ in black and the dashed line represents a forbidden transition. In summary: the spin-conserving transitions at $+\mathrm{K}$ couple to $\sigma +$ polarized light, one of the spin-flip transitions is optically dark and the other one couples to $z$-polarized light.

Figure 2

Properties of transition metal dichalcogenide heterobilayers for interlayer twist angles $\theta$ close to $0^{\circ }$ (top line) and ${60}^{\circ }$ (bottom line), as exemplified by ${\mathrm{MoSe}}_2/{\mathrm{WSe}}_2$. (a) Scheme of the periodic atomic structure reconstruction, indicating strong deviations from ideal moiré patterns. The area of low-energy, high-symmetry stacking configurations (blue) is significantly enlarged and $0^{\circ }$ and ${60}^{\circ }$ have different reconstructions. (b) The geometry of high-symmetry stacking configurations, where purple corresponds to ${\mathrm{WSe}}_2$ and orange to ${\mathrm{MoSe}}_2$ layers. Metal atoms are depicted by bigger circles and chalcogenes by smaller ones. (c) Schematic band structures of the stacking configurations at the $+\mathrm{K}$ point of the heterobilayer Brillouin zone. The color code indicates that ${\mathrm{MoSe}}_2$ is the electron layer and ${\mathrm{WSe}}_2$ is the hole layer. Small arrows to the left of the bands indicate the spin-orientation. Double arrows indicate dipole-allowed optical transitions (selection rules), where $\sigma +, \sigma -$ and $z$ are the corresponding polarizations. The selection rules are strongly spin- and stacking-dependent where, contrary to monolayers (see Fig. 1), spin-flip transitions can couple to $\sigma +$ or $\sigma -$ polarized light.

Figure 3

Impact of basis set size $N$, band gap correction $\mathrm{\Delta }$, exchange-correlation functional (PBE, LDA), and electronic-structure method (PAW, LAPW) on orbital angular momenta and exciton $g$ factors in ${\mathrm{WS}}_2$ monolayer. (a) Convergence of the (dimensionless) orbital angular momenta $L_{n,+\mathrm{K}}$ of the two highest valence band states ($n=v,v-1$) and two lowest conduction band states ($n=c,c+1$) at the $+\mathrm{K}$ point with respect to the number of bands $N$ included in the calculation [Eq. (8)]. (b) Convergence of the intervalley $g$ factors of A and B excitons $g_{\mathrm{A}}^{1\mathrm{L}}=2(L_{c+1,+K}-L_{v,+K})$ and $g_{\mathrm{B}}^{1\mathrm{L}}=2(L_{c,+K}-L_{v-1,+K})$. $N=1$ is the lowest-energy state of the valence shell, the valence band maximum is indicated by a dashed vertical line. A large number of bands ($N\ge 300$) is required to converge the $g$ factors to a precision of 0.1. (c) Impact of the band gap correction $\mathrm{\Delta }$ on the orbital momenta $L_{n,+\mathrm{K}}$ and (d) the $g$ factor. The dashed vertical line indicates the ${\mathrm{G}}_0{\mathrm{W}}_0$ quasiparticle band gap. While the $L_{n,+\mathrm{K}}$ depend on $\mathrm{\Delta }$, the exciton $g$ factors are almost insensitive to it.

Figure 4

Convergence of the orbital angular momenta $L_{n,+\mathrm{K}}$ of the two highest valence band states ($n=v,v-1$) and two lowest conduction band states ($n=c,c+1$) at the $+\mathrm{K}$ point with respect to the number of states $N$ included in the calculation and the convergence of the intervalley exciton $g$ factor $g^{1\mathrm{L}}=2(L_{c(+1),+K}-L_{v(-1),+K})$ for A and B excitons in transition metal dichalcogenide monolayers. A large number of states $N$ is required converge the $g$ factors. For a precision of 0.1/0.01 $N=322/695$ states are necessary in ${\mathrm{MoS}}_2$, 321/771 in ${\mathrm{MoSe}}_2$, 547/881 in ${\mathrm{MoTe}}_2$, 376/604 in ${\mathrm{WS}}_2$ and 303/746 in ${\mathrm{WSe}}_2$. The $g$ factors of different TMD monolayers are similar and the value of the B exciton is always lower than that of the A exciton. In ${\mathrm{WSe}}_2{g}_{\mathrm{A}}^{1\mathrm{L}}$ and $g_{\mathrm{B}}^{1\mathrm{L}}$ differ the most (see Table 1). $N=1$ is the lowest-energy state of the valence shell, the valence band maximum is indicated by a dashed vertical line and all values are PBE-PAW results.