Temperature-dependent excitonic effects in the optical properties of single-layer ${\mathrm{MoS}}_2$

Temperature influences the performance of two-dimensional (2D) materials in optoelectronic devices. Indeed, the optical characterization of these materials is usually realized at room temperature. Nevertheless, most ab initio studies are still performed without including any temperature effect. As a consequence, important features are thus overlooked, such as the relative height of the excitonic peaks and their broadening, directly related to the temperature and to the nonradiative exciton relaxation time. We present ab initio calculations of the optical response of single-layer ${\mathrm{MoS}}_2$, a prototype 2D material, as a function of temperature using density functional theory and many-body perturbation theory. We compute the electron-phonon interaction using the full spinorial wave functions, i.e., fully taking into account the effects of spin-orbit interaction. We find that bound excitons ($A$ and $B$ peaks) and resonant excitons ($C$ peak) exhibit different behavior with temperature, displaying different nonradiative linewidths. We conclude that the inhomogeneous broadening of the absorption spectra is mainly due to electron-phonon scattering mechanisms. Our calculations explain the shortcomings of previous (zero-temperature) theoretical spectra and match well with the experimental spectra acquired at room temperature. Moreover, we disentangle the contributions of acoustic and optical phonon modes to the quasiparticles and exciton linewidths. Our model also allows us to identify which phonon modes couple to each exciton state, which is useful for the interpretation of resonant Raman-scattering experiments.

Figure 1

Spectral functions of single-layer ${\mathrm{MoS}}_2$ for temperatures 0 K (left panel) and 300 K (right panel). The colored squares denote the points $T_c$ (red), $T_{c^{\prime }}$ (blue), $K_c$ (purple), and $K_v$ (green). Spectral function is normalized. The color scale bar indicates the maximum value in black and the minimum value in white.

Figure 2

Spectral functions for selected band states in Fig. 1: $T_c$ (red), $T_c^{\prime }$ (blue), $K_c$ (magenta), and $K_v$ (green), and for $T=0$ K (dotted line), 300 K (dashed line), and 1000 K (solid line).

Figure 3

Phonon band structure and density of states of single-layer ${\mathrm{MoS}}_2$. Eliashberg functions of the band states $T_c, T_c^{\prime }, K_c$, and $K_v$ (see definitions in the text).

Figure 4

Optical spectra of single-layer ${\mathrm{MoS}}_2$ as a function of temperature. The dashed line represents optical spectra without temperature effects, and the solid lines represent spectra at increasing temperature: 0, 100, 200, and 300 K (red, green, magenta, cyan). Dots are experimental data from Ref. [26].

Figure 5

Exciton energy as a function of temperature (solid lines). The shadow region shows the width of each exciton. The dashed line marks the exciton energy without electron-phonon interaction. The photoluminescence data of Ref. [44] are represented by black dots and the FWHM by the gray area.