Acoustic phonon limited mobility in two-dimensional semiconductors: Deformation potential and piezoelectric scattering in monolayer MoS${}_2$ from first principles

We theoretically study the acoustic phonon limited mobility in $n$-doped two-dimensional MoS${}_2$ for temperatures $T<100$ K and high carrier densities using the Boltzmann equation and first-principles calculations of the acoustic electron-phonon (el-ph) interaction. In combination with a continuum elastic model, analytic expressions and the coupling strengths for the deformation potential and piezoelectric interactions are established. We furthermore show that the deformation potential interaction has contributions from both normal and umklapp processes and that the latter contribution is only weakly affected by carrier screening. Consequently, the calculated mobilities show a transition from a high-temperature $\mu \sim T^{-1}$ behavior to a stronger $\mu \sim T^{-4}$ behavior in the low-temperature Bloch-Grüneisen regime characteristic of unscreened deformation potential scattering. Intrinsic mobilities in excess of ${10}^5$ cm${}^2$ V${}^{-1}$ s${}^{-1}$ are predicted at $T<10$ K and high carrier densities ($n\gtrsim {10}^{11}$ cm${}^{-2}$). At 100 K, the mobility does not exceed $\sim 7\times {10}^3$ cm${}^2$ V${}^{-1}$ s${}^{-1}$. Our findings provide new and important understanding of the acoustic el-ph interaction and its screening by free carriers, and is of high relevance for the understanding of acoustic phonon-limited mobilities in general.

Figure 1

Top: Illustration of acoustic phonon scattering in the $K,K^{\prime }$ valleys of the Brillouin zone showing the phase space available for scattering below and above the Bloch-Grüneisen temperature $T_{\mathrm{BG}}$. The part of the hexagonal Brillouin zone marked by the gray-shaded box indicates the plotting range for the contour plots in Fig. 2. Bottom: Lattice and primitive unit cell of 2D hexagonal MoS${}_2$.

Figure 2

Calculated deformation potential and piezoelectric interactions in the $K$ valley of the conduction band in monolayer MoS${}_2$. The contour plots show the absolute value of the calculated coupling matrix elements $M_{\mathbf{q}\lambda }^{\mathrm{DP}/\mathrm{PE}}$ at $\mathbf{k}=\mathbf{K}$ for the TA (left) and LA (right) phonons as a function of the two-dimensional phonon wave vector $\mathbf{q}$. The absolute value of the relative phase $\varphi$ between the two interactions [see Eq. (8)] is shown in the bottom plots.

Figure 3

Normal and umklapp contributions to the deformation potential interaction for the TA (left) and LA (right) phonon. The plots show the angular average of the deformation potential interactions in Fig. 2 and their contributions from normal and umklapp processes given by Eq. (10) (full lines). The dashed lines show the analytic deformation potential interaction in Eq. (11).

Figure 4

Temperature and density dependence of the dimensionless screening parameter $q_s\left(T\right)$ in Eq. (24) for acoustic phonon (full lines) and charged impurity (dashed lines) scattering in 2D MoS${}_2$.

Figure 5

Scattering rate for deformation potential (dashed lines) and piezoelectric (full lines) interaction in 2D MoS${}_2$ at temperatures $T=10$ K and $T=50$ K and carrier densities $n={10}^{11}$ cm${}^{-2}$ (left) and $n={10}^{13}$ cm${}^{-2}$ (right). This corresponds to Fermi energies of $E_F\approx 0.25$ meV and $E_F\approx 25$ meV, respectively. In the right plot, the BG temperature is $T_{\mathrm{BG}}\approx 36$ $\left(57\right)$ K for the TA (LA) phonon. At $T=10$ K, the dip in the scattering rate that appears at the Fermi energy is a clear fingerprint of Bloch-Grüneisen physics.

Figure 6

Acoustic phonon limited mobility vs temperature. Top: Mobility vs temperature for different carrier densities. Bottom: Temperature dependence of the exponent $\gamma$ in $\mu \sim T^{-\gamma }$ for the same set of carrier densities. The dots mark the BG temperatures of the TA ($•$) and LA ($\circ$) phonons, respectively.

Figure 7

Acoustic phonon limited mobility vs carrier density for temperatures $T=4$, 20, 50 K. The dots mark the quantum-classical crossover from a nondegenerate to a degenerate carrier distribution at $T=T_F$.

Figure 8

Piezoelectric interaction in a 2D hexagonal lattice. The plots show the absolute value of the coupling matrix element $M_{\mathbf{q}\lambda }^{\mathrm{PE}}$ in Eq. (B19) for the TA (left) and LA (right) phonon as a function of the phonon wave vector $\mathbf{q}$. The parameters for MoS${}_2$ in Table  have been used.

Figure 9

Schematic illustration of the LCAO supercell matrix involved in the calculation of the el-ph coupling in Eq. (C2). The square lattice indicates the unit cells of the crystal lattice. The real-space cutoff $r_{\mathrm{cut}}$ measured from the position of the atomic site where the gradient of the potential is taken is used to separate out the short- and long-range part of the el-ph interaction. Matrix elements involving LCAO orbitals located beyond the cutoff as the one shown are defined as long range [see also Eq. (C3)].

Figure 10

Deformation potential (upper row) and piezoelectric (lower row) interactions in monolayer MoS${}_2$ obtained with the real-space partitioning scheme for different values of the real-space cutoff $r_{\mathrm{cut}}$. The plots show the absolute value of the matrix elements $M_{\mathbf{q}\lambda }^{\mathrm{DP}/\mathrm{PE}}$ averaged over the high-symmetry directions of the hexagonal lattice. The analytic couplings in Eqs. (11) and (13) are shown with dashed lines for the MoS${}_2$ parameters listed in Table .

Figure 11

Schematic illustration of an umklapp process involving a Fourier component of the scattering potential at $\mathbf{q}+\mathbf{G}$. The square lattice denotes the reciprocal lattice with the shaded cell indicating the first Brillouin zone. The dashed arrows show a pair of reciprocal lattice vectors ${\mathbf{G}}_{1/2}$ from the electronic Bloch functions that conserve crystal momentum by bringing $\mathbf{q}+\mathbf{G}$ back onto $\mathbf{q}$ in the first Brillouin zone.