Exciton band structure of monolayer ${\mathrm{MoS}}_2$

We address the properties of excitons in monolayer ${\mathrm{MoS}}_2$ from a theoretical point of view, showing that low-energy excitonic states occur both at the Brillouin-zone center and at the Brillouin-zone corners, that binding energies at the Brillouin-zone center deviate strongly from the ${(n-1/2)}^{-2}$ pattern of the two-dimensional hydrogenic model, and that the valley-degenerate exciton doublet at the Brillouin-zone center splits at finite momentum into an upper mode with nonanalytic linear dispersion and a lower mode with quadratic dispersion. Although monolayer ${\mathrm{MoS}}_2$ is a direct-gap semiconductor when classified by its quasiparticle band structure, it may well be an indirect gap material when classified by its excitation spectra.

Figure 1

Quasiparticle band structure of monolayer ${\mathrm{MoS}}_2$. The solid curves were obtained using the quantum espresso package [28] with fully relativistic pseudopotentials under the Perdew-Burke-Ernzerhof generalized-gradient approximation, and a $16\times 16\times 1\vec{k}$ grid. The dashed curves were calculated from the tight-binding model, with cyan (red) representing states that are even (odd) under mirror operation with respect to the Mo plane. $v_{1,2}$ and $c_{1,2}$ label the bands close to the valence- and conduction-band edges near the $K$ and $K^{\prime }$ points. The inset shows the hexagonal Brillouin zone (pink) associated with the triangular Bravais lattice of ${\mathrm{MoS}}_2$ and an alternate rhombohedral primitive zone (black), and labels the principle high-symmetry points in reciprocal space. Note that the valence-band maxima at $\Gamma$ is only slightly lower in energy than the valence-band maxima at $K,K^{\prime }$.

Figure 2

(a) Energies of type-III excitons as a function of center-of-mass momentum $\vec{Q}$. This figure is based on a calculation performed using a $45\times 45\vec{k}$ grid. The lines were added as a guide to the eye. Solid (dashed) lines represent states that are doubly (singly) degenerate. The labels of the excitons with $\vec{Q}=0$ are explained in the main text. Excitons with $\vec{Q}=K$ are labeled by ${\chi }_K^1$, ${\chi }_K^2$, and so on in ascending order of energy. The left inset is a $\vec{k}$-space map plot of $P_{\vec{Q}}\left(\vec{k}\right)$ [see Eq. (4)] for the $\vec{Q}=\frac{2}{45}M$ exciton in the lower-energy branch evolving from $A$. The right inset schematically illustrates the dominant electron-hole transitions which contribute to the ${\chi }_K^1$, ${\chi }_K^2$, and ${\chi }_K^3$ exciton states. (b) Binding energy $E_b$ for $A$, $B$, and $A_{2s}$ excitons at $\vec{Q}=0$ as a function of $N^{1/2}$, where $N$ is number of $\vec{k}$ points.

Figure 3

(a) Real part of the optical conductivity with (solid red curve) and without (dashed green curve) electron-hole interactions. For these calculations the BS equation was solved on a $51\times 51\vec{k}$ grid and transitions were broadened by 20 meV. The solid green arrow indicates the quasiparticle band gap. The two dashed gray arrows mark the energies of the $A_{2p}$ excitons. Note that we have rigidly shifted the excitation energy spectrum by a constant, so that the $A$ exciton energy is at 1.93 eV as measured by photoluminescence experiments [6, 9, 10, 11]. (b)–(e) $\vec{k}$-space maps of $P_{\vec{Q}=0}\left(\vec{k}\right)$ for low-energy excitons. (b)The $\mathcal{L}=1$ exciton $|A_{+}\rangle$. (c),(d) Two nondegenerate $A_{2p}$ excitons in valley $K$. (e) The $\mathcal{L}=1A_{2s}$ exciton.

Figure 4

The energy difference between the large-momentum exciton ${\chi }_K^3$ and zero-momentum exciton $A$ as a function of dielectric constant $\epsilon$. The calculation was performed on a $45\times 45\vec{k}$ grid. ${\chi }_K^3$ is the triplet exciton state with a hole in $\Gamma$ valley and an electron in $K$ valley.

Figure 5

Binding energy $E_b$ in massive Dirac model for $1s$, $2s$, and $2p$ excitons in valley $K(\tau =1)$ as a function of dielectric constant $\epsilon$. The parameter values are $\hslash v_F=1.105\text{eV}\times 3.193\text{Å}$, $\Delta =0.7925\text{eV}$ and $r_0=33.875\text{Å}/\epsilon$. The inset table compares $E_b$ obtained respectively from massive Dirac model (MD) and lattice model (LM) for $\epsilon =2.5$.

Figure 6

Binding energy $E_b$ in massive Dirac model with standard Coulomb $\mathrm{interaction}(r_0=0,F\left(q\right)=1)$ for $1s$, $2s$, and $2p$ excitons in valley $K(\tau =1)$ as a function of fine-structure constant $\alpha$.