Misorientation-angle-dependent electrical transport across molybdenum disulfide grain boundaries

Grain boundaries in monolayer transition metal dichalcogenides have unique atomic defect structures and band dispersion relations that depend on the inter-domain misorientation angle. Here, we explore misorientation angle-dependent electrical transport at grain boundaries in monolayer MoS2 by correlating the atomic defect structures of measured devices analysed with transmission electron microscopy and first-principles calculations. Transmission electron microscopy indicates that grain boundaries are primarily composed of 5–7 dislocation cores with periodicity and additional complex defects formed at high angles, obeying the classical low-angle theory for angles <22°. The inter-domain mobility is minimized for angles <9° and increases nonlinearly by two orders of magnitude before saturating at ∼16 cm2 V−1 s−1 around misorientation angle≈20°. This trend is explained via grain-boundary electrostatic barriers estimated from density functional calculations and experimental tunnelling barrier heights, which are ≈0.5 eV at low angles and ≈0.15 eV at high angles (≥20°).

Therefore, the way to define the low-angle tilt GB is even easier, just drat two lines directly which divide the two angles. However, the above method assumes that the edges of the grains grow at the same speed (near equilibrium conditions). The kinetic factors like the transportation of sources take effect during real growth made the straight line bended. Here, the analysis is based on kinetics, thermodynamic points aren't included. c, TEM image of transferred MoS 2 on TEM grid, red and yellow triangles marked the grown flake in (a). d, DF-TEM image acquired in sample. Due to the atomic steps or kinks, the edges of these triangles deviated from the zigzag (Moterminated) edges. Firstly, we can skip the kinks on the edges of (a), and simply select the section right before or after the kink. Secondly, when the triangles are not exactly equatorial triangles (some of the angles are not quite 60), we use the three sets of edge pairs from the two triangles (c) to measure the misorientation angles and then average the results to minimize error. Scale bar is 10 µm.  I   II   I1   II1   II3  I3   I2   II2   III   IV   III1   IV1   III3   IV3   IV2 III2 are not equilateral triangles. The misorientation angle from these flakes was defined following Supplementary Fig. 2; the angle is 19. After electrical measurement, the device was transferred to a TEM grid to confirm the misorientation angle between the two flakes. b, High-resolution STEM image of the GB between two flakes; each domain is marked by red and blue triangles, which correspond to three Mo atoms location. The angle difference between the red and blue triangles indicates the misorientation angle between these two flakes is ~ 19, which was consistent with the SAED pattern (inset image). this distance is similar to that between the two dislocation core in (e). The distance between two dislocation cores is also consistent with low angle GB theory, as shown by the blue function in the inset of (f).

Supplementary Note 1 ‫|‬ Band-tail state description
Using Fermi-Dirac statistics and setting the conduction band edge as a reference (E C = 0), the delocalized charge density associated with transport at the conduction band edge in n-type MoS 2 is described by the following exponential relationship: where band N is the band-edge charge density. However, the band edge mobility in MoS 2 is far less than that expected for a perfect lattice with phonon-limited scattering 2 (μ band ≈400 cm 2 V -1 s -1 ).
Due to the temperature-independent nature of the SS (~230 mV/dec) ( Supplementary Fig. 4), a large concentration of localized charges (n loc ) associated with a band-tail states exists in MoS 2 and thus, 1 band loc  n n for applicable experimental conditions. The percentage of this localized charge that participates in transport is negligibly small and limits carrier mobility. In this case, the effective field-effect mobility is 3 : By taking the natural log and substituting 1 band loc  n n , Eq (2) simplifies to: Localized band edge states are also modeled by an exponentially decaying density of states: where defect N is the local defect density, E defect is the defect level with respect to the conduction band edge, and kT β  1 describes the exponential decay of band tail states for a specific device.
For a single device, the above relationship holds true. However, the defect density is likely not constant both within a flake and when comparing different flakes in spite of special care taken to reduce sample variation. The defect density depends on the growth, transfer, and fabrication procedures. Similarly, substrate disorder should also have a small spatial dependence on the same substrate and for devices on different substrates. The impact of the spatial inhomogeneity is accounted for by assuming a normal distribution of the quantities E defect and β 1 .
To introduce this statistical distribution of spatially inhomogeneous defects (which are further related to statistical mobility distribution), can be replaced with an arbitrary normal distribution function. As a consequence, Eq. (4) is replaced by: Eq. (5) represents a statistical distribution of the localized defect density, where is a normal distribution function of defects. Upon substituting Eq. (5) into Eq.
(3), we obtain the log-normal distribution of the mobility: , as all other parameters are constant or vary trivially due to the natural logarithm. While was defined as normal distribution, this function should at least be symmetric about an expected/mean value. This must be true in order for the experimental data to be statistically meaningful due to acceptable sample reliability, consistent fabrication, and processing conditions. A Gaussian function is a good example for simplicity, but the specific type of symmetric distribution is not critical here.
If band-tail states are the limiting factor for transport, a large set of devices should then have a measured FE  that follows the FE  dispersion in Eq. (6), e.g. a log-normal distribution. This is precisely what was observed in Fig. 1D in the main text. With an expected mobility of 44 FE   cm 2 V -1 s -1 , 9 band loc n n , which is consistent with one of the major initial assumptions when deriving the log-normal relationship. It should be noted that the inter-domain results do not fit within this log-normal distribution.

Supplementary Note 2 ‫|‬ Relating V th to inter-domain charging
The presence of charge within the MoS 2 is due to intrinsic self-capacitance of the Here, V gs at maximum SS is designated at V th for purpose of analysis here, leading to the following expression: As evident in Fig. 3A Thus, a difference in the median V th is a direct result of Fermi level variations in the two systems.
The GB region is the primary source of mobility degradation in inter-domain transport and thus, the Fermi level difference results from the boundary region specifically. As a result, the GB is the source of an electrostatic potential boundary.

Supplementary Note 3 ‫|‬ MoS 2 -Ti/Au Contacts
The Ti contacts utilized in the field-effect measurements were analyzed by comparing the results of 4-terminal with 2-terminal measurements. Contact resistance was roughly estimated by R C = R 2T -2αR S with α = L/W as a geometric normalization parameter. The obtained results are shown in Fig. S6. The data are symmetrically distributed with an average of 52kΩ and no dependence on boundary position, as is expected. These resistances are orders of magnitude too large for practical applications, they do not influence the 4-terminal conclusions in this work. We note that recent progress in the field of local contact phase changes and the use of graphene or Al as contacts is promising in reducing these values 4,5 .

Supplementary Methods
Dark field optical microscopy (DF-OM). Optical microscopy (ZEISS, Axio Imager 2) was used to obtain images of the GBs surface morphologies of the transferred TMD samples.