Mobility enhancement and temperature dependence in top-gated single-layer MoS${}_2$

The deposition of a high-$\kappa$ oxide overlayer is known to significantly enhance the room-temperature electron mobility in single-layer MoS${}_2$ (SLM) but not in single-layer graphene. We give a quantitative account of how this mobility enhancement is due to the nondegeneracy of the two-dimensional electron gas system in SLM at accessible temperatures. Using our charged impurity scattering model [Ong and Fischetti, Phys. Rev. B 86, 121409 (2012)] and temperature-dependent polarizability, we calculate the charged impurity-limited mobility (${\mu }_{\mathrm{imp}}$) in SLM with and without a high-$\kappa$ (HfO${}_2$) top-gate oxide at different electron densities and temperatures. We find that the mobility enhancement is larger at low electron densities and high temperatures because of finite-temperature screening, thus explaining the enhancement of the mobility observed at room temperature. ${\mu }_{\mathrm{imp}}$ is shown to decrease significantly with increasing temperature, suggesting that the strong temperature dependence of measured mobilities should not be interpreted as being solely due to inelastic scattering with phonons. We also reproduce the recently seen experimental trend in which the temperature scaling exponent ($\gamma$) of ${\mu }_{\mathrm{imp}}\propto T^{-\gamma }$ is smaller in top-gated SLM than in bare SLM. Finally, we show that $\sim 37\%$ mobility enhancement can be achieved by reducing the HfO${}_2$ thickness from 20 to 2 nm.

Figure 1

Basic model used in our calculation. The SLM is an infinitely thin layer at the interface ($z=0$) between a semi-infinite substrate and a top oxide layer of thickness $t_{\mathrm{ox}}$. The dielectric is capped with metal, which we assume to be a perfect conductor. The charged impurity at the interface has image charges under and above it in the substrate and top gate, respectively.

Figure 2

Plot of the normalized polarizability $\Pi (q,T,E_F)/\Pi (0,0,E_F)$ for (a) $n={10}^{12}$ cm${}^{-2}$ and (b) $n={10}^{13}$ cm${}^{-2}$ at $T=0$ K (solid), 50 K (dash-dot), 100 K (dotted), and 300 K (dashed). We also plot the corresponding normalized scattering potential ${\varphi }_q^{\mathrm{scr}}\left(T\right)/{\varphi }_{q=0}^{\mathrm{scr}}(T=0)$ for (c) $n={10}^{12}$ cm${}^{-2}$ and (d) $n={10}^{13}$ cm${}^{-2}$ in top-gated SLM.

Figure 3

Plot of ${\mu }_{\mathrm{imp}}^0$ (“Bare”) and ${\mu }_{\mathrm{imp}}^{\mathrm{TG}}$ (“HfO${}_2$”) at $T=10$ K (hollow symbols) and $T=300$ K (solid symbols) for $n={10}^{12}$ to $2\times {10}^{13}$ cm${}^{-2}$ for $n_{\mathrm{imp}}=4.0\times {10}^{12}$ cm${}^{-2}$. At 300 K, the mobility scales almost linearly with the electron density.

Figure 4

Plot of the mobility enhancement ${\mu }_{\mathrm{imp}}^{\mathrm{TG}}/{\mu }_{\mathrm{imp}}^0$ at 10 K (circle) and 300 K (triangle). The mobility enhancement increases at higher temperatures or lower electron densities.

Figure 5

Plot of (a) ${\mu }_{\mathrm{imp}+\mathrm{phon}}^0$ and (b) ${\mu }_{\mathrm{imp}+\mathrm{phon}}^{\mathrm{TG}}$ for $n={10}^{12}$ to $5\times {10}^{12}$ cm${}^{-2}$ in steps of $\Delta n={10}^{12}$ cm${}^{-2}$ (dashed lines) and $n={10}^{13}$ to $2\times {10}^{13}$ cm${}^{-2}$ in steps of $\Delta n=2\times {10}^{12}$ cm${}^{-2}$ (solid lines) from $T=10$ to 300 K. The arrows indicate the direction of increasing $n$. The thick dashed (solid) line corresponds to $n={10}^{12}$ cm${}^{-2}$ (${10}^{13}$ cm${}^{-2}$). As $n$ increases, $|d{\mu }_{\mathrm{imp}+\mathrm{phon}}/dT$| becomes smaller. (c) Plot of the exponent $\gamma$ for ${\mu }_{\mathrm{imp}+\mathrm{phon}}^0$ (solid circles), ${\mu }_{\mathrm{imp}}^0$ (open circles), ${\mu }_{\mathrm{imp}+\mathrm{phon}}^{\mathrm{TG}}$ (solid diamonds), and ${\mu }_{\mathrm{imp}}^{\mathrm{TG}}$ (open diamonds) from fitting to ${\mu }_e\propto T^{-\gamma }$ over the range $T=200$–300 K. The shaded region bounded by the dashed lines covers the range of $\gamma$ values (0.3–0.73) extracted for top-gated SLM in Ref. 8.

Figure 6

Plot of ${\mu }_{\mathrm{imp}}^{\mathrm{TG}}\left(t_{\mathrm{ox}}\right)/{\mu }_{\mathrm{imp}}^{\mathrm{TG}}\left(\infty \right)$ for $n={10}^{12}$ to $5\times {10}^{12}$ cm${}^{-2}$ and $t_{\mathrm{ox}}=2$ to 20 nm at 300 K. At $n={10}^{12}$ cm${}^{-2}$, a $\sim$37 percent mobility enhancement can be achieved by reducing the HfO${}_2$ thickness from 20 to 2 nm.