Electron transport properties of mirror twin grain boundaries in molybdenum disulfide: Impact of disorder

Grain boundaries in two-dimensional transition-metal dichalcogenides can strongly affect the transport properties by reducing the electron mobility or allowing gap conduction through extended grain boundary states. Here, by combining advanced modeling tools–density-functional-theory-calibrated tight-binding Hamiltonians and Green's function techniques–we investigate transport along and across mirror twin grain boundaries in ${\mathrm{MoS}}_2$. Our results show that the grain boundary conductive channels are strongly affected by sulfur vacancies, while short-range Anderson disorder has a moderate impact, which we quantitatively analyze, and long-range disorder has a very weak effect. As for transport across the grain boundaries, the system conductance turns out to be less than half that for the pristine system, and the spin-orbit coupling and intervalley scattering are found to play an important role. Our findings are beneficial to the understanding and the prediction of the impact of mirror twin grain boundaries in the transport phenomena and could be of help in designing electronic devices based on transition metal dichalcogenides.

Figure 1

(a) Side and top views of the relaxed atomic structure of a periodic MTB, highlighted by a dashed red line, in ${\mathrm{MoS}}_2$. Mo atoms are in purple color and S atoms are in yellow color. The dashed rectangle indicates a unit cell for the ribbon, which is periodic along the $y$ direction and whose lattice parameter is $a=0.316$ nm. (b) Band structure obtained with the calibrated TB model of a zigzag ribbon with width $W=10$ nm and a periodic MTB along its axis. $①$ and $②$ indicate the two spin-degenerate MTB bands within the bulk gap, which are highlighted by blue and green lines. The size of the red dots corresponds to the weight of the states on the molybdenum atoms in the MTB region. The indicated K, K', and $\mathrm{\Gamma }$ points correspond to the projection of the corners and the center of the hexagonal Brillouin zone of 2D ${\mathrm{MoS}}_2$ onto the 1D Brillouin zone of the ribbon. Inset: Band structures with (red line) and without (black line) SOC in the region indicated by a square. The energy gap induced by SOC is $\sim 80$ meV, which is close to that found by DFT calculations, see Fig. 9 for more details. (c) Probability density of the states at $E=2$ eV on bands $①$ and $②$ on the atoms close to MTB.

Figure 2

(a) Band structures of a zigzag ribbon with $W=10$ nm and the MTB along its axis (same as in Fig. 1) and for a pristine zigzag ribbon with half-width $W=5$ nm. Only the valence band region is shown. (b) Local density of states of the ribbon with an MTB as a function of the $x$ coordinate along the ribbon transverse section and the electron energy $E$. Edge states and MTB states within the bulk gap are indicated in white arrows. A white dashed circle displays the region with reduced density of states on the MTB.

Figure 3

(a) Three sulfur vacancy positions, indicated by circles, at different distances $d$ from the MTB. (b) Transmission coefficient as a function of energy for the pristine system and in the presence of a single sulfur vacancy at the three positions specified in (a). (c) Band structure of the pristine ribbon in the bulk gap region. The size of the dots indicates the weight of the sulfur component of the corresponding states, which is only present for band $①$. The dashed lines indicate the energies of the localized sulfur vacancy states.

Figure 4

(a) Example of Anderson disorder realization with random on-site energies for the atoms over the section of length $L$, and with strength $\mathrm{\Delta }=100$ meV. The MTB is highlighted by a dashed red line. (b) Average transmission coefficient as a function of the electron energy $E$ for the MTB system in the pristine case ($L=0$ nm) and in the presence of Anderson disorder with $\mathrm{\Delta }=100$ meV and $L$ varying from 50 nm to 200 nm. The continuous lines and shaded regions show the average and the 25th to the 75th percentile of the transmission over 100 disorder realizations, respectively. (c) Mean free path as a function of $E$ for different Anderson disorder strengths $\mathrm{\Delta }$. Inset: Mean free path as a function of $\mathrm{\Delta }$ for two representative energies $E=1.2$ eV and $E=2.1$ eV. The continuous lines correspond to the fit $\propto {\mathrm{\Delta }}^{-2}$. (d) Average transmission coefficient as a function of $E$ for the MTB system in the presence of Anderson disorder with $\mathrm{\Delta }=100$ meV and $L=50$ nm, and sum of the average transmission of the two MTB bands. $\langle T_1\rangle$ is the average transmission of band $①$ contribution and $\langle T_2\rangle$ is that of band $②$.

Figure 5

Main panel: Average transmission coefficient as a function of the energy and for different $L$ in the case of a ${\mathrm{SiO}}_2$ substrate. The averaging is performed over 100 disorder realizations. Inset: Example of long-range potential profile realization for a ${\mathrm{SiO}}_2$ substrate.

Figure 6

(a) Main panel: Zero-temperature conductance per unit of width as a function of the electron energy for pristine 2D ${\mathrm{MoS}}_2$ and in the presence of a transverse MTB. Inset: Conductance at the top of the valence band in logarithmic scale. (b) Zero-temperature conductance per unit of width at the top of the valence band for 50-nm-wide ${\mathrm{MoS}}_2$ ribbons with a transverse MTB in the absence (black line) and in the presence of short-range disorders in the region of the MTB, namely sulfur vacancies over an 80-nm-long section and Anderson disorder over a 2-nm-wide stripe surrounding the MTB with different strengths. To suppress edge contribution to the transmission, we introduced edge roughness. (c) Transmission coefficient as a function of the wave number and the electron energy for 2D ${\mathrm{MoS}}_2$ with a transverse MTB. The white lines correspond to the band profile of 2D ${\mathrm{MoS}}_2$, while the dashed circles indicate the strong suppression of transmission in the valence band.

Figure 7

Transmission coefficient as a function of the wave number and the electron energy in one of the two regions indicated by dashed circles in Fig. 6. The band structure of for a 20-nm-wide zigzag ribbon with a periodic MTB without (green lines) and with (white lines) SOC is superimposed.

Figure 8

Band structure obtained with the noncalibrated TB model and without SOC for the ribbon of Fig. 1. MTB bands within the bulk gap are indicated in blue, green, and purple lines. Orange lines display the MTB band above the bulk states at the $\mathrm{\Gamma }$ point. The MTB bands in blue and green lines correspond to bands $①$ and $②$ in Fig. 1. Red circles denote the weight of molybdenum atoms in the MTB region. (a) Contribution of all $d$ orbitals, (b) $d_{xy}$, (c) $d_{yz}$, (d) $d_{zx}$, (e) $d_{x^2-y^2}$, and (f) $d_{z^2}$. Contribution of S atoms to the MTB bands within the bulk gap is insignificant compared to that of Mo atoms and thus not displayed.

Figure 9

(a) Band structure obtained with the calibrated TB model for the ribbon of Fig. 1. The SOC-induced gap is indicated by an arrow. (b) Band structure obtained by DFT calculations for a ribbon with width $W=2.2$ nm. The SOC-induced gap, indicated by an arrow, is $\sim 60$ meV, i.e., slightly narrower than in the TB case. A narrow ribbon width is considered to reduce the computational burden. The shape of the MTB bands is not significantly affected by the ribbon width, provided the ribbon is large enough to avoid the coupling between edge and MTB states. Black and red lines display the bands without and with SOC, respectively.

Figure 10

Scaling analysis of the localized (a), (b) and diffusive (c), (d) transport regimes, for Anderson disorder with $\mathrm{\Delta }=100$ meV. (a) Average logarithm of the transmission coefficient as a function of $L$ at $E=1$ eV, which corresponds to the energy region A in Fig. 4. The continuous line shows the linear fit. The estimated localization length is $\xi \approx 58$ nm. (b) Frequency distribution of $\ln T$ for $L=300$ nm at the same energy. (c) Inverse of the average the transmission coefficient as a function of $L$ at $E=1.8$ eV chosen within the energy region C in Fig. 4. The continuous line shows the linear fit. The estimated mean free path is $\ell \approx 750$ nm. (d) Frequency distribution of $T$ for $L=300$ nm at the same energy.

Figure 11

Separation of the MTB bands, $①$ (green lines) and $②$ (blue lines) by modification of the Hamiltonian. (a) Band structure without any modification for the ribbon of Fig. 1. (b), (c) Band structures obtained by modification of the Hamiltonian for isolating bands $①$ and $②$, respectively.

Figure 12

Mean free path as a function of the electron energy in the presence of Anderson disorder with strength $\mathrm{\Delta }$ varying from 50 meV to 200 meV. The individual mean free paths of (a) band $①$ and (b) band $②$, calculated starting from the modified Hamiltonian corresponding to the band structures in Figs. 11 and 11, respectively.