Phonon-limited transport of two-dimensional semiconductors: Quadrupole scattering and free carrier screening

Two-dimensional (2D) semiconductors have demonstrated great potential for next-generation electronics and optoelectronics. An important property for these applications is the phonon-limited charge carrier mobility. The common approach to calculate the mobility from first principles relies on the interpolation of the electron-phonon coupling (EPC) matrix. However, it neglects the scattering by the quadrupoles generated by phonons, limiting its accuracy. Here we present a first-principles method to incorporate the quadrupole scattering, which results in a much better interpolation quality and thus a more accurate mobility as exemplified by monolayer $\mathrm{Mo}{\mathrm{S}}_2$ and InSe. This method also allows for a natural incorporation of the effects of the free carriers, enabling us to efficiently compute the screened EPC and thus the mobility for electrostatically doped semiconductors. Particularly, we find that the electron mobility of InSe is more sensitive to the carrier concentration than that of $\mathrm{Mo}{\mathrm{S}}_2$ due to the stronger long-range scattering in intrinsic InSe. With increasing electron concentration, the InSe mobility can reach $\sim 4$ times the intrinsic value, then decrease owing to the involvement of heavier electronic states. Our work provides accurate and efficient methods to calculate the phonon-limited mobility in the intrinsic and electrostatically doped 2D materials, and improves the fundamental understanding of their transport mechanism.

Figure 1

(a) Intrinsic electron mobility of $\mathrm{Mo}{\mathrm{S}}_2$ and InSe, calculated with or without 2D quadrupoles. (b)–(f) Comparison of the DFPT-calculated EPC strengths (dots) and the interpolated ones (dashed lines: without quadrupoles; solid lines: with quadrupoles), for selected phonon modes of intrinsic $\mathrm{Mo}{\mathrm{S}}_2$ and InSe. For both materials, the initial states are selected to be 30 meV above the CBM along the $\mathrm{\Gamma }-M$ direction, and the final states are located at the isoenergy circle of the same valley. The vibration patterns are shown in the insets. Note that the discontinuity in (f) is due to the phonon band crossing and the resulting mixed eigenvectors near the crossing point.

Figure 2

EPC strengths of InSe (a)–(c) and $\mathrm{Mo}{\mathrm{S}}_2$ (d),(e) for selected phonon modes over large carrier concentration range, from intrinsic to $3\times {10}^{13}\mathrm{c}{\mathrm{m}}^{–2}$. The line thickness indicates the carrier concentration, with thicker lines for larger carrier concentrations. The initial and final states are the same as those in Fig. 1. (f) Electron mobility of InSe and $\mathrm{Mo}{\mathrm{S}}_2$ as a function of carrier concentration.

Figure 3

(a) Electron mobility calculated using the total EPC matrix (tot), the long-range EPC only (LR), and the short-range EPC only (SR) (see main text for definitions). (b),(c) Scattering rates for intrinsic $\mathrm{Mo}{\mathrm{S}}_2$ and InSe. (d) The total EPC matrix element for LO mode of intrinsic $\mathrm{Mo}{\mathrm{S}}_2$ and LO/TO mode of InSe. The initial and final states are the same as those in Fig. 1, and are illustrated in the inset as well. (e),(f) Same as (b) and (c) but for a high carrier concentration $\left(n_e={10}^{13}\mathrm{c}{\mathrm{m}}^{–2}\right)$. The inset in (f) shows the zoomed-in view.

Figure 4

(a) Logarithm change of the mobility $\left[{log}_{10}\left(\mu /{\mu }_0\right)\right]$, “Drude inverse effective mass” $\left[{log}_{10}\left(\langle M^{–1}\rangle /{\langle M^{–1}\rangle }_0\right)\right]$ and “Drude relaxation time” (${log}_{10}(\overline{\tau }/{\overline{\tau }}_0)$) with carrier concentration for InSe. (c) Local inverse effective mass $m_{n\mathbf{k}}^{-1}$ (see main text for definition) for InSe. The dashed lines indicate the energies where the electronic states are the most important to the mobility under different carrier concentrations.