Ab initio simulation of single- and few-layer ${\mathrm{MoS}}_2$ transistors: Effect of electron-phonon scattering

In this paper, we present full-band atomistic quantum transport simulations of single- and few-layer ${\mathrm{MoS}}_2$ field-effect transistors (FETs) including electron-phonon scattering. The Hamiltonian and the electron-phonon coupling constants are determined from ab initio density-functional-theory calculations. It is observed that the phonon-limited electron mobility is enhanced with increasing layer thicknesses and decreases at high charge concentrations. The electrostatic control is found to be crucial even for a single-layer ${\mathrm{MoS}}_2$ device. With a single-gate configuration, the double-layer ${\mathrm{MoS}}_2$ FET shows the best intrinsic performance with an ON current, $I_{\text{ON}}=685\mu \mathrm{A}/\mu \mathrm{m}$, but with a double-gate contact the transistor with a triple-layer channel delivers the highest current with $I_{\text{ON}}=1850\mu \mathrm{A}/\mu \mathrm{m}$. The charge in the channel is almost independent of the number of ${\mathrm{MoS}}_2$ layers, but the injection velocity increases significantly with the channel thickness in the double-gate devices due to the reduced electron-phonon scattering rates in multilayer structures. We demonstrate further that the ballistic limit of transport is not suitable for the simulation of ${MX}_2$ FETs because of the artificial negative differential resistance it predicts.

Figure 1

(a) Comparison of the band-structure features of single-layer ${\mathrm{MoS}}_2$ as obtained with the MLWF Hamiltonian and the DFT one. The plot shows the relative deviations of the band gap ($E_G=1.79$ eV), energy separation of the $\mathrm{\Sigma }$ and $K$ valleys in the conduction band ($E_{K\mathrm{\Sigma }}=0.048$ eV), and electron effective mass at the $K$ point ($m_K^{*}=0.52m_0$) and at the $\mathrm{\Sigma }$ point ($m_{\mathrm{\Sigma }}^{*}=0.62m_0$) along the $K-\mathrm{\Gamma }$ line as a function of the number of neighbor interactions included in the MLWF Hamiltonian. (b) Single-layer, (c) double-layer, and (d) triple-layer ${\mathrm{MoS}}_2$ band structures obtained with DFT and with the MLWF Hamiltonian including up to sixth-nearest-neighbor interactions. The zero energy level is set to the midgap energy.

Figure 2

(a) Side view of a double-layer ${\mathrm{MoS}}_2$ structure where $x$ denotes the transport direction and $y$ is the direction of confinement. (b) Top view of the same system as in (a). The $z$ direction is assumed to be periodic and modeled via 21 $k_z$ points. The device Hamiltonian $H\left(k_z\right)$ is computed from the $H_0$ Hamiltonian of an ${\mathrm{MoS}}_2$ strip with width $\mathrm{\Delta }$ and the $H_{-}$ and $H_{+}$ couplings as $H\left(k_z\right)=H_0+H_{-}{e}^{-i{k}_z\mathrm{\Delta }}+H_{+}{e}^{i{k}_z\mathrm{\Delta }}$. The shaded areas mark the primitive hexagonal unit cell of ${\mathrm{MoS}}_2$ used in the electronic-structure calculations and the generation of MLWFs, while the rectangular unit cell is utilized in the transport calculations.

Figure 3

(a) Single-layer and (b) double-layer ${\mathrm{MoS}}_2$ phonon band structure calculated with vasp.

Figure 4

Difference between the MLWF Hamiltonian matrix elements with and without strain. The couplings between one Wannier function situated on a Mo atom and three on one of its S neighbors are shown for a (a) double-layer ${\mathrm{MoS}}_2$ with the selected Wannier function on the molybdenum being the $d_{z^2}$ orbital-like one and (b) a triple-layer ${\mathrm{MoS}}_2$ with the $d_{xz}$-like orbital on the Mo atom. The green down-pointing triangles, red up-pointing triangles, and blue squares correspond to the $p_x$-, $p_y$-, and $p_z$-like MLWFs on the sulfur atom.

Figure 5

(a) Channel resistance in ${\mathrm{MoS}}_2$ as a function of its length at $n_{2\text{D}}={10}^{11}{\mathrm{cm}}^{-2}$ electron concentration. (b) Low-field phonon-limited mobility in single-, double-, and triple-layer ${\mathrm{MoS}}_2$ as a function of the carrier concentration.

Figure 6

Closeup of the (a) single-, (c) double-, and (d) triple-layer ${\mathrm{MoS}}_2$ band structures showing the energy separations between the $K$ and $\mathrm{\Sigma }$ valleys. At both the $K$ and $\mathrm{\Sigma }$ points, the highest transport effective masses of the degenerate valleys are shown. (b) Energy-resolved electron density in single-layer ${\mathrm{MoS}}_2$ at different charge concentrations. The height of the first peak at $E=0.91$ eV ($K$ valley) is normalized to 1. As the electron population increases, the satellite $\mathrm{\Sigma }$ valley situated at $E=0.95$ eV along the $K-\mathrm{\Gamma }$ direction becomes more and more filled.

Figure 7

(a) Schematic view of a single-gate monolayer ${\mathrm{MoS}}_2$ field-effect transistor (FET). (b) Schematics of a double-gate FET with triple-layer ${\mathrm{MoS}}_2$ as the channel material.

Figure 8

(a) Ballistic transfer characteristics $I_d-V_{gs}$ of the single-gate monolayer ${\mathrm{MoS}}_2$ FET at low (solid red) and high (dashed blue) $V_{ds}$. (b) Conduction-band structure of single-layer ${\mathrm{MoS}}_2$ in the primitive unit cell and (c) folded band structure along the transport direction $x$ at $k_z=0$ highlighting the four lowest subbands that contribute to electron transport. (d) Conduction-band edge of the simulated device at $V_{gs}=0.35$ V and $V_{ds}=0.68$ V, including the channels that the bands in (b) and (c) form. A reflectionless transmission is only possible if a given channel is available in the source, channel, and drain region [45]. Due to their narrow energy width, electrons in bands 1, 2, and 3 cannot propagate from source to drain at high $V_{ds}$, contrary to band 4. At low $V_{ds}$, states belonging to any of the four subbands can propagate, which explains the negative differential resistance observed in (a).

Figure 9

Spectral current $I_d(E,x)$ of a single-gate monolayer ${\mathrm{MoS}}_2$ FET at $V_{ds}=0.68$ V and $V_{gs}=0$ V in the case of (a) ballistic simulation and (b) when electron-phonon scattering is turned on. Red indicates high current concentrations, green no current, and the dashed line denotes the conduction-band edge.

Figure 10

(a) Transfer characteristics $I_d-V_{gs}$ at $V_{ds}=0.05$ and 0.68 V with electron-phonon scattering for the single-gate monolayer ${\mathrm{MoS}}_2$ FET. Transfer characteristics of single-, double-, and triple-layer ${\mathrm{MoS}}_2$ FETs at $V_{ds}=0.68$ V with (b) a single-gate and (c) a double-gate geometry.

Figure 11

Maximum of the electrostatic potential energy in the channel (ToB) as a function of the gate voltage in the (a) single-gate and (b) double-gate FETs at $V_{ds}=0.68$ V.