Charged impurity scattering in two-dimensional materials with ring-shaped valence bands: GaS, GaSe, InS, and InSe

The singular density of states and the two Fermi wave vectors resulting from a ring-shaped or “Mexican hat” valence band give rise to unique trends in the charged impurity scattering rates and charged impurity limited mobilities. Ring-shaped valence bands are common features of many monolayer and few-layer two-dimensional (2D) materials including the III-VI materials GaS, GaSe, InS, and InSe. The wave-vector dependence of the screening, calculated within the random phase approximation, is so strong that it is the dominant factor determining the overall trends of the scattering rates and mobilities with respect to temperature and hole density. Charged impurities placed on the substrate and in the 2D channel are considered. The different wave-vector dependencies of the bare Coulomb potentials alter both the magnitudes and the temperature dependencies of the mobilities. Moving the charged impurities 5 Å from the center of the channel to the substrate increases the mobility by an order of magnitude by suppressing the large wave-vector backscattering across the outer Fermi ring.

Figure 1

Monolayer GaS between ${\mathrm{SiO}}_2$ substrate and BN capping layer. The black atom depicts an ionized impurity at the substrate interface 5.5 Å from the center of the channel. The blue atom depicts an ionized impurity at the center of the monolayer GaS layer.

Figure 2

(a) Comparison of a Mexican hat dispersion (red) and a parabolic dispersion (blue). The height of the Mexican hat band at $k=0$ is ${\epsilon }_h=0.11\mathrm{eV}$. The two concentric rings show the two Fermi circles that exist for energies below ${\epsilon }_h$. (b) Density of states of the Mexican hat dispersion (red) and the parabolic dispersion (blue). The parabolic and Mexican hat dispersion both have an effective mass of $0.409{\mathrm{m}}_0$.

Figure 3

Carrier concentration of GaS, GaSe, InS, and InSe as a function of Fermi level $E_F$ for (a) $5\mathrm{K}$, (b) $77\mathrm{K}$, and (c) $300\mathrm{K}$. Parameters used for the materials are tabulated in Table 1.

Figure 4

(a)–(c) $q_{\lambda }\left(q\right)\equiv -\frac{e^2}{2\epsilon }\mathrm{\Pi }\left(q\right)$ at three different temperatures: 5 K, 77 K, and 300 K. (a) $q_{\lambda }\left(q\right)$ for a parabolic dispersion with $E_F=40\mathrm{meV}$. (b) $q_{\lambda }\left(q\right)$ for a Mexican hat dispersion with $E_F=+5\mathrm{meV}$. (c) $q_{\lambda }\left(q\right)$ for a Mexican hat dispersion with $E_F=-3\mathrm{meV}$. The inset shows the two iso-energy rings in momentum space of the Mexican hat dispersion. The momentum transfer $q$ between two rings is shown. (d) $q_{\lambda }\left(q\right)$ for a Mexican hat dispersion for a fixed hole density of ${10}^{13}{\mathrm{cm}}^{-2}$. The corresponding Fermi energies at each temperature are shown in the legend. For both the parabolic and Mexican hat dispersions, the band structure parameters are the same as those used in Fig. 2, and the dielectric constant is $\epsilon =3.1{\epsilon }_0$.

Figure 5

Polar plots of the matrix elements of the RPA screened Coulomb potential as a function of scattering angle at (a) $T=5\mathrm{K}$ and $E=2.5\mathrm{meV}$ and (b) $T=300\mathrm{K}$ and $E=25\mathrm{meV}$. The polar angle $\theta$ is the angle between $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$. The legend refers to the three curves in each polar plot. “Inner” denotes matrix elements with $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$ both on the inner ring, “Outer” denotes matrix elements with $\mathbf{k}$ and ${\mathbf{k}}^{\prime }$ both on the outer ring, and “Inter” denotes matrix elements with $\mathbf{k}$ on the inner ring and ${\mathbf{k}}^{\prime }$ on the outer ring. (c) $q, q_{\lambda }$, and $q+q_{\lambda }$ as a function of $q$ corresponding to (a). (d) $q, q_{\lambda }$, and $q+q_{\lambda }$ as a function of $q$ corresponding to (b). The hole density is fixed at ${10}^{13}{\mathrm{cm}}^{-2}$ for all figures.

Figure 6

Polar plots of $1/\tau \left(\theta \right)$ given by the integrand of Eq. (14) for (a) $k$ on the inner ring and (b) $k$ on the outer ring. Inter-ring and intra-ring rates are indicated by the legend. Inter-ring contributions can be either positive or negative. (c) Four components of the total scattering rate as a function of energy. The components are indicated by the legends where, for example, “Inner-outer” denotes the initial state on the inner ring and the final state on the outer ring.

Figure 7

Momentum scattering rates for a charge density of ${10}^{13}{\mathrm{cm}}^{-2}$ at (a) $5\mathrm{K}$, (b) $77\mathrm{K}$, and (c) $300\mathrm{K}$. The band parameters correspond to the ones used in Fig. 2, and the polarization functions are shown in Fig. 4. (d) Maximum values of $q$ for scattering from the outer ring ($2k_2$) or within the inner ring ($2k_1$) as a function of energy.

Figure 8

Energy dependence of the total momentum relaxation rates for (a) GaS and (b) InSe for three different temperatures. The charge density is fixed at ${10}^{13}{\mathrm{cm}}^{-2}$. Parameters used for GaS and InSe are tabulated in Table 1.

Figure 9

Charged impurity limited hole mobility of GaS (a) as a function of temperature for a carrier density of ${10}^{12}{\mathrm{cm}}^{-2}$ (blue) and ${10}^{13}{\mathrm{cm}}^{-2}$ (red) and (b) as a function of carrier density at $5\mathrm{K}$ (blue), $77\mathrm{K}$ (red), and $300\mathrm{K}$ (green). The impurity density $n_I$ is fixed at ${10}^{12}{\mathrm{cm}}^{-2}$.

Figure 10

(a) Charged impurity limited monolayer hole mobility as a function of temperature of GaS, GaSe, InS, and InSe with the charged impurities in the middle of the channel ($d=0$, dashed line) and on the substrate ($d=5.5\AA$, solid line). (b) $q{e}^{qd}, q_{\lambda }, q+q_{\lambda }$, and $q{e}^{qd}+q_{\lambda }$ for GaS at $T=77\mathrm{K}$ where $d=5.5\AA$. The hole density $n_s={10}^{13}{\mathrm{cm}}^{-2}$, and the charged impurity density $n_I$ is fixed at ${10}^{12}{\mathrm{cm}}^{-2}$.

Figure 11

Charged impurity limited monolayer hole mobilities as a function of carrier density at (a) $T=77\mathrm{K}$ and (b) $T=300\mathrm{K}$ for the four III-VI materials as indicated by the legends. Solid lines result from charged impurities on the substrate ($z_0=5.5\AA$), and the dashed lines result from charged impurities in the middle of the channel ($z_0=0$). The impurity density $n_I$ is fixed at ${10}^{12}{\mathrm{cm}}^{-2}$.