Ab Initio Studies of Exciton $g$ Factors: Monolayer Transition Metal Dichalcogenides in Magnetic Fields

The effect of a magnetic field on the optical absorption in semiconductors has been measured experimentally and modeled theoretically for various systems in previous decades. We present a new first-principles approach to systematically determine the response of excitons to magnetic fields, i.e., exciton $g$ factors. By utilizing the $GW$-Bethe-Salpeter equation methodology we show that $g$ factors extracted from the Zeeman shift of electronic bands are strongly renormalized by many-body effects which we trace back to the extent of the excitons in reciprocal space. We apply our approach to monolayers of transition metal dichalcogenides (${\mathrm{MoS}}_2$, ${\mathrm{MoSe}}_2$, ${\mathrm{MoTe}}_2$, ${\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2$) with strongly bound excitons for which $g$ factors are weakened by about 30%.

Figure 1

DFT band structure of ${\mathrm{MoSe}}_2$. In (a) the bands are colored according to the expectation values of the local orbital magnetic moments $m_{n\mathbf{k}}^{\mathrm{orb},\mathrm{loc}}$ [Eq. (3)]. In contrast to this, in (b) the orbital magnetic moment $m_{n\mathbf{k}}^{\mathrm{orb}}$ including the contribution from the Bloch states is shown [Eq. (5)]. The corresponding color scale is shown in units of ${\mu }_{\mathrm{B}}$. In (c) an enlargement along $\mathrm{K}\mathrm{\Gamma }-\mathrm{K}$ is shown in which the spin contributions have been added. The colors refer to the total magnetic moments $m_{n\mathbf{k}}=m_{n\mathbf{k}}^{\mathrm{orb}}+m_{n\mathbf{k}}^{\mathrm{spin}}$. On the left side we break down the different contributions of $m$ at the K point (see main text).

Figure 2

(a),(b) Difference of the magnetic moments of ${\mathrm{MoSe}}_2$’s first and second conduction band (CB, $\mathrm{CB}+1$, respectively) and the valence band (VB). The results are shown on a $(24\times 24)$ mesh with colors from red to blue denoting the differences. Note that the abrupt changes (e.g., red to blue) are related to band crossings. (c) $k$-dependent exciton wave function of the first bright (A) exciton on a logarithmic scale. Note that due to the magnetic field the $\pm \mathrm{K}$ degeneracy is lifted. The resulting exciton $g$ factor is calculated by multiplying the magnetic moments with the exciton wave functions [see Eq. (7)].