Mobility of two-dimensional materials from first principles in an accurate and automated framework

We present a first-principles approach to compute the transport properties of 2D materials in an accurate and automated framework. We use density-functional perturbation theory in the appropriate bidimensional setup with open-boundary conditions in the third direction. The materials are charged by field effect via planar countercharges. In this approach, we obtain electron-phonon matrix elements in which dimensionality and doping effects are inherently accounted for, without the need for post-processing corrections. This treatment highlights some unexpected consequences, such as an increase of electron-phonon coupling with doping in transition-metal dichalcogenides. We use symmetries extensively and identify pockets of relevant electronic states to minimize the number of electron-phonon interactions to compute; the integrodifferential Boltzmann transport equation is then linearized and solved beyond the relaxation-time approximation. We apply the entire protocol to a set of much studied materials with diverse electronic and vibrational band structures: electron-doped ${\mathrm{MoS}}_2,{\mathrm{WS}}_2,{\mathrm{WSe}}_2$, phosphorene, arsenene, and hole-doped phosphorene. Among these, hole-doped phosphorene is found to have the highest mobility, with a room temperature value around $600{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$. Last, we identify the factors that affect most phonon-limited mobilities, such as the number and the anisotropy of electron and hole pockets, to provide a broader understanding of the driving forces behind high mobilities in two-dimensional materials.

Figure 1

(a) Schematic description of the first-principles workflow for electronic transport. (b) Interpolated electron-phonon couplings $g_{\mathbf{k},{\mathbf{k}}^{\prime }}$ for electron-doped ${\mathrm{WS}}_2$. The initial state $\mathbf{k}$ considered here is indicated by a white star; the other points are the possible final states in the finely sampled pockets, where the color of the point indicates the strength of the electron-phonon coupling matrix element. The index of the phonon mode indicated at the top of each subplot refers to a purely energetic ordering of the phonon modes associated with each transition. This implies that crossings in phonon dispersions may lead to discontinuities in the plots. This explains the set of seemingly out-of-place EPC matrix elements in the ${\mathrm{K}}^{\prime }$ valley for modes 5 and 6, and in lesser measure in the Q valleys for modes 2 and 3. (c) Scattering times interpolated on the fine grid of electronic states $\mathcal{F}$, shown using a color scale for each electronic state in the valleys of ${\mathrm{WS}}_2$. (d) Temperature-dependent resistivity and mobility of electron-doped ${\mathrm{WS}}_2\left(n=5\times {10}^{13}{\mathrm{cm}}^{-2}\right)$.

Figure 2

Representation of the different sampling grids. $\mathcal{F}$ is the fine grid used for the final integration of the conductivity and the sum over final states in the BTE. $\mathcal{I}$ is a coarser grid of irreducible states on which we solve the BTE and compute the scattering times. The grid $\mathcal{Q}$ of phonon momenta on which the linear response is computed is obtained by finding the irreducible set of momenta linking the electronic states from an even coarser grid, represented here in lime green.

Figure 3

Irreducible phonon momenta $\mathbf{q}\in \mathcal{Q}$ (in red, those actually computed via DFPT) and all relevant phonons (in light blue) that can be obtained from the latter by symmetry transformations. To help visualizing the corresponding pairs of initial and final electronic states, the dashed lines correspond to a BZ centered either the high-symmetry K point (green) or the bottom of the Q valley (orange) at the origin, the Q valley being approximately halfway between $\mathrm{\Gamma }$ and Q.

Figure 4

Interpolated $g_{\mathbf{k}{\mathbf{k}}^{\prime }}$ for electron-doped arsenene. The initial state considered is indicated by a white star. The rest of the points are the possible final states in the finely sampled pockets and the color of the point indicates the strength of the electron-phonon coupling matrix element. The index of the phonon mode indicated at the top of each subplot refers to a purely energetic ordering of the phonon modes associated with each transition.

Figure 5

Interpolated $g_{\mathbf{k}{\mathbf{k}}^{\prime }}$ for hole-doped phosphorene. The initial state considered is indicated by a white star. The rest of the points are the possible final states in the finely sampled pockets and the color of the point indicates the strength of the electron-phonon coupling matrix element. The index of the phonon mode indicated at the top of each subplot refers to a purely energetic ordering of the phonon modes associated with each transition.

Figure 6

Mobilities as a function of temperature for electron doped of ${\mathrm{MoS}}_2,{\mathrm{WS}}_2,{\mathrm{WSe}}_2$, arsenene, and phosphorene, as well as for the hole side of phosphorene. The carrier density is $5\times {10}^{13}{\mathrm{cm}}^{-2}$ for all systems except electron-doped phosphorene where it is $5/3\times {10}^{13}{\mathrm{cm}}^{-2}$.

Figure 7

Scattering times in ${\mathrm{MoS}}_2,{\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2$ (from left to right). The color scale is the same for all subplots. The scattering rate $\left(\hslash /\tau \right)$ clearly increases for states with energy high enough for the Q valley to be accessible. The Fermi levels are: $-2.83,-2.46$, and $-2.13$ eV, respectively.

Figure 8

Angle-dependent mobilities at room temperature for hole-doped and electron-doped phosphorene and electron-doped arsenene and ${\mathrm{WS}}_2$. The angle corresponding to the transport direction refers to the direction of the in-plane electric field driving the current with respect to the $\mathit{x}$ direction indicated in the plots of EPC. For phosphorene, $0\left(\pi /2\right)$ corresponds to the zigzag (armchair) direction. While phosphorene shows highly anisotropic transport, only very small variations $(\approx {10}^{-2}{\mathrm{cm}}^2/$(Vs), below numerical accuracy) can be observed in ${\mathrm{WS}}_2$ and arsenene.

Figure 9

Doping dependency of intravalley scattering in both electron valleys of ${\mathrm{WS}}_2$. The EPC is measured by the quantity $\langle g_{\alpha }^2\rangle$, described in the text.

Figure 10

From left to right, top to bottom: band structure along high symmetry path for electron-doped ${\mathrm{MoS}}_2,{\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2$, As, ${\mathrm{P}}_4$, and hole-doped ${\mathrm{P}}_4$ (black solid lines) compared with the undoped case (red dashed lines). In each case the two band structures are aligned with respect to the top of the valence band. The zero is at the Fermi level of the doped case.

Figure 11

From left to right, top to bottom: phonon dispersion along high symmetry path for electron-doped ${\mathrm{MoS}}_2,{\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2$, As, ${\mathrm{P}}_4$, and hole-doped ${\mathrm{P}}_4$ (black solid lines) compared with the undoped case (red dashed lines). Phonon softenings can be observed for phonons corresponding to possible electronic transitions with strong electron-phonon coupling. For example, in arsenene, the softening at K corresponds to the strong intervalley coupling $(g\approx 140$ meV) with mode number 4 observed in Fig. 4.

Figure 12

Interpolated $g_{\mathbf{k}{\mathbf{k}}^{\prime }}$ for electron-doped ${\mathrm{MoS}}_2$. The initial state considered is indicated by a white star. The rest of the points are the possible final states in the finely sampled pockets and the color of the point indicates the strength of the electron-phonon coupling matrix element. The index of the phonon mode indicated at the top of each subplot refers to a purely energetic ordering of the phonon modes associated with each transition.

Figure 13

Interpolated $g_{\mathbf{k}{\mathbf{k}}^{\prime }}$ for electron-doped ${\mathrm{WS}}_2$, with the initial state at the bottom of the Q valley.

Figure 14

Interpolated $g_{\mathbf{k}{\mathbf{k}}^{\prime }}$ for electron-doped ${\mathrm{WSe}}_2$, with the initial state at the bottom of the K valley.

Figure 15

Interpolated $g_{\mathbf{k}{\mathbf{k}}^{\prime }}$ for electron-doped phosphorene, with the initial state at the bottom of the $\mathrm{\Gamma }$ valley.