Soft Coulomb gap and asymmetric scaling towards metal-insulator quantum criticality in multilayer MoS2

Quantum localization–delocalization of carriers are well described by either carrier–carrier interaction or disorder. When both effects come into play, however, a comprehensive understanding is not well established mainly due to complexity and sparse experimental data. Recently developed two-dimensional layered materials are ideal in describing such mesoscopic critical phenomena as they have both strong interactions and disorder. The transport in the insulating phase is well described by the soft Coulomb gap picture, which demonstrates the contribution of both interactions and disorder. Using this picture, we demonstrate the critical power law behavior of the localization length, supporting quantum criticality. We observe asymmetric critical exponents around the metal-insulator transition through temperature scaling analysis, which originates from poor screening in insulating regime and conversely strong screening in metallic regime due to free carriers. The effect of asymmetric scaling behavior is weakened in monolayer MoS2 due to a dominating disorder.

0 to 8 T by 0.5 T step, we did backgate bias sweeps and measured Hall voltages. From the slope in V Hall vs. B plot for each V BG , we show an example for chosen backgate bias in Supplementary   Fig. 1b and the carrier density n 2D for each V BG was calculated as shown in Supplementary Fig.   1c. n 2D values are rather scattered but their V BG dependences show the decent linear behavior for both V ds = 0.5 and 1 V. The solid line in this figure is from the approximation 2D = ox ( BG − th )⁄ using V th = 11 V and C ox = 11.5×10 -8 F/cm 2 for 300 nm SiO 2 . The effective oxide capacitance from the Hall measurement is nearly identical to the geometric value while the carrier density obtained from geometry undereatimates the value of n 2D . However, we note that the determination of the critical exponent values is insensitive to the constant offset of n 2D due to δ ≡ ( 2D − c ) c ⁄ .  shows the total resistance R T (squares) and the contact resistance R c (circles) at T = 10K, where R T = R c +R ch , R ch is the channel resistance. At low V ds , R c is dominant due to the Schottky barrier.

Supplementary
As V ds increases, R c becomes smaller and negligible above 1 V because the Schottky barrier becomes thinner. Consequently, 2FE is determined mainly by R c for low V ds and R ch for high V ds . Since the maximal transconductance is found in the metallic phase, and σ for mobility calculation decreases as V ds increases in this metallic phase as shown in Fig. 1b of the manuscript, 2FE shows the non-monotonic behavior for V ds . In contrast, since 4FE involves only the channel, it monotonically decreases as V ds increases. In Supplementary Fig. 1a, the largest 4FE at 10K is ~720 cm 2 V -1 s -1 at V ds = 0.1 V. This voltage itself is certainly not small to give the zero voltage limit of conductivity. However, according to Supplementary Fig. 2b, most voltage drop for 0.1 V occurs in the contact, and only ~2% in the channel, which is ~2 mV. Therefore, we believe that the zero voltage limit of 4FE is not far from ~720 cm 2 V -1 s -1 , and also the temperature scaling analysis of conductivity obtained at 0.1 V is reasonable. We show the properties of 3.5 nm thick MoS 2 sample in this section. Supplementary Fig. 3 shows the optical and AFM images, and thickness profile of this sample.

Supplementary Note 3. Scaling analysis for 3.5 nm-thick
Supplementary Fig. 4 shows conductivities for temperature (4a) and electric field (4d), and scaling analysis. The temperature 4K, where E-scaling is performed, is low enough to be in the diffusive regime, i.e., it is lower than the Dingle temperature D = ℏ (2 * ) ⁄~10K for ~1000 cm 2 V -1 s -1 .
11.1 μm 4.6 μm 6.5 μm In the following, we show the simulation result for Joule heating of this device. It is performed using a COMSOL multiphysics modeling software. For the simulation, we used following parameters: • sample thickness: 3.5 nm, Au electrode thickness: 60 nm.
We simulate for three regions at 4K, i.e., insulating, near transition, and metallic phase. For simplicity, they are distinguished by the current level. Since the current level at transition point is ~ 1×10 -5 A, we choose 8×10 -6 A for insulating and 7×10 -5 A for metallic region. According to the results in Supplementary Fig. 6, Joule heating for all three regions in our measurement configuration is insignificant. To demonstrate that the excitation we used for the conductivity measurement in this device is close to the zero voltage limit so that the heating effect for temperature scaling is insignificant, we show two values in Supplementary Fig. 7 below, one in the zero voltage limit and the other at for insulating phase at the lowest temperature, but other than that, two values are quite similar.
Since the temperature scaling was performed for the broad temperature range, we do not think around 10% error at the lowest temperature does cause the significant errors in the temperature scaling.

Supplementary Note 4. Intermediate glass phase
Electron glass features were experimentally observed in strongly disordered Si-MOSFET 3 . In the ref. S3, the critical carrier density c ≈ 5.2 × 10 11 cm -2 . g ≈ 7.5 × 10 11 cm -2 is identified as a carrier density below which the 2D electron system freezes into an electron glass. s * is determined such that ⁄ = 0 at this carrier density. The data close to n c are well described c < s < g < s * (ref. 9 in the manuscript). In our data, we identified V BG = 10 V as a critical field corresponding to c ≈ 3.37 × 10 12 cm -2 . The real critical field could be in between 10 and 15 V as shown in Supplementary Fig. 8 below. If it really is, the temperature dependence of σ would be weaker than the one at V BG = 10 V . The power x for the trace at V BG = 10 V is 0.91 far from 1.5, and the intermediate region (colored), if there is, very narrow, i.e., δ g ≡ � g − c � c ⁄ ≈ 0.11, which is contrast to the Si case in ref. S3, δ g ≈ 0.44. In this sense, an apparent metallic glass feature is not visible or exists in a very narrow range. shown in the inset of Supplementary Fig. 9a. Supplementary Fig. 9a presents the backgate bias dependent conductivity in the unit of e 2 /h for selected temperatures. This conductivity was taken at V ds = 0.5 V which is not small but we believe that this does not change the scaling behavior significantly since our temperature range for scaling is rather high, T > 60K so that the field effect is relatively weak compared to the thermal effect. Conductivity crossover for temperature occurs around V BG ~ 23 V signifying the metal-insulator transition. The critical carrier density n c at this bias is estimated to be ~1.9 × 10 12 cm -2 using 2D = ox ( BG − th )/ at room temperature as in the manuscript. This n c yields r s~7 .8.

Supplementary
Temperature dependent 4-probe mobility is calculated at V BG = 50 V and shown in Supplementary Fig. 9b. The mobility at ~10K is approximately 170 cm 2 V -1 s -1 . Compared with multilayer, monolayer MoS 2 is less interacting and more disordered system. Supplementary Figs. 9c and 9d display the conductance and renormalized conductivities by the critical conductivity as a function of temperature, respectively. Supplementary Fig. 9e shows the collapse of renormalized conductivities after rescaling the temperatures.
Finally, the critical exponent ~2.87 is obtained from the linear fit of temperature scaling parameter T 0 for | | as shown in Supplementary Fig. 9f. In addition to this large value, more symmetric scaling parameter T 0 for the metal and insulating phase suggest that MIT in this monolayer MoS 2 is likely disorder driven. The deviation at higher | | in metallic regime is not clearly understood. The intermediate state may exist.
In another CVD-grown monolayer MoS 2 on h-BN, we measured voltage-dependent conductivity for several temperatures to see how much variation of conductivity for the voltage changes as the temperature increases. Supplementary Fig. 10a shows the optical image of monolayer MoS 2 on h-BN. Supplementary Fig. 10b displays the backgate bias-dependent conductivity for several different temperatures taken at V ds = 0.5 V. The metal-insulator crossover is not visible for BG ≤ 80 V but the larger curvature for smaller temperature indicates MIT to occur at higher V BG . Supplementary Figs. 10c and 10d show drain-source voltage-dependent conductivity for different temperatures at V BG = 35 V and 70 V, respectively.
The conductivity changes with V ds . The changing rate is stronger at lower temperature, while it becomes weaker as temperature increases. For T > 100K and V ds <1 V, it seems that the voltage dependence of conductivity is rather weak. Since the scaling for monolayer was performed for T > 60K as noted earlier and the collapse ( Supplementary Fig. 10e) is quite good up to 280K, this suggests that the scaling analysis with this voltage V ds = 0.5 V is still reliable for insulating phase. Accordingly, for metallic phase, we expect the scaling analysis for monolayer to be still valid, although we could not explicitly proved it due to the inaccessibility of metallic phase in