Characterizing critical behavior and band tails on the metal-insulator transition in structurally disordered two-dimensional semiconductors: Autocorrelation and multifractal analysis

Our previous study observed the localization-delocalization transition and critical quantum fluctuations of the local density of states (LDOS) on the structurally disordered two-dimensional (2D) semiconductor MoS_{2}. This transition corresponds to the metal-insulator transition (MIT) reported in transport measurements. The structural disorder in MoS_{2} caused curvature-induced band gap fluctuations, leading to charge localization and unusual band edge flattening through doping. The critical behavior for the MIT was analyzed using autocorrelation and multifractality of LDOS mapping results. However, the effect of structural disorder on critical points has not been fully explored. Here, we systematically investigated the impact of structural disorder on band tail formation and critical doping concentration by examining the radial-averaged autocorrelation and multifractality of LDOS in 2D semiconductors. Our finding indicates that the radial-averaged autocorrelation and multifractality of LDOS characterize the band tail ranges and band edge flattening in disordered 2D semiconductors. Decaying regions in the radial-averaged autocorrelation profile and first-order derivative of singularity peak positions determine band tail ranges. Increased structural disorder led to larger band tail widths near valence and conduction band edges, while the doping-induced band edge flattening altered band tail widths for each valence and conduction band. As the band edge is flattened due to doping, the LDOS map near the critical energy becomes uniform, exhibiting a divergence in the localization length. The average value of conduction and valence band tail widths remained almost constant regardless of doping control, serving as a representative value for the degree of structural disorder. For the MIT, we found that the critical doping concentration depends on the degree of structural disorder in 2D semiconductors. Our findings provide valuable insights into the fundamental physics of structurally disordered 2D semiconductors in relevance to quantum phase transitions, which could have important implications for designing and optimizing electronic/optoelectronic devices based on 2D materials.