We report on the interlayer screening effect of graphene using Kelvin probe force microscopy (KPFM). By using a gate device configuration that enables the supply of electronic carriers in graphene sheets, the vertical screening properties were studied from measuring the surface potential gradient. The results show layer-dependence of graphene sheets, as the number of graphene layers increases, the surface potential decreases exponentially. In addition, we calculate the work function-related information of the graphene layers using KPFM.

Among two-dimensional (2D) transition metal dichalcogenides (TMDs), platinum diselenide (PtSe2) stands at a unique place in the sense that it undergoes a phase transition from type-II Dirac semimetal to indirect-gap semiconductor as thickness decreases. Defects in 2D TMDs are ubiquitous and play crucial roles in understanding and tuning electronic, optical, and magnetic properties. Here we investigate intrinsic point defects in ultrathin 1T-PtSe2 layers grown on mica through the chemical vapor transport (CVT) method, using scanning tunneling microscopy and spectroscopy (STM/STS) and first-principles calculations. We observed five types of distinct defects from STM topography images and obtained the local density of states of the defects. By combining the STM results with the first-principles calculations, we identified the types and characteristics of these defects, which are Pt vacancies at the topmost and next monolayers, Se vacancies in the topmost monolayer, and Se antisites at Pt sites within the topmost monolayer. Our study shows that the Se antisite defects are the most abundant with the lowest formation energy in a Se-rich growth condition, in contrast to cases of 2D molybdenum disulfide (MoS2) family. Our findings would provide critical insight into tuning of carrier mobility, charge carrier relaxation, and electron-hole recombination rates by defect engineering or varying growth condition in fewlayer 1T-PtSe2 and other related 2D materials.

We discuss a variety of codimension-one, non-invertible topological defects in general 3+1d QFTs with a discrete one-form global symmetry. These include condensation defects from higher gauging of the one-form symmetries on a codimension-one manifold, each labeled by a discrete torsion class, and duality and triality defects from gauging in half of spacetime. The universal fusion rules between these non-invertible topological defects and the one-form symmetry surface defects are determined. Interestingly, the fusion coefficients are generally not numbers, but 2+1d TQFTs, such as invertible SPT phases, Z N gauge theories, and U (1) N Chern-Simons theories. The associativity of these algebras over TQFT coefficients relies on nontrivial facts about 2+1d TQFTs. We further prove that some of these non-invertible symmetries are intrinsically incompatible with a trivially gapped phase, leading to nontrivial constraints on renormalization group flows. Duality and triality defects are realized in many familiar gauge theories, including free Maxwell theory, non-abelian gauge theories with orthogonal gauge groups, N = 1, and N = 4 super Yang-Mills theories.

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