Spatial mapping of disordered 2D systems: the conductance Sudoku

Motivated by recent advances on local conductance measurement techniques at the nanoscale, timely questions are being raised about what possible information can be extracted from a disordered graphene sheet by selectively interrogating its transport properties. We demonstrate how an inversion technique originally developed to identify the number of scatterers in a quantum device can be adapted to a multi-terminal set up in order to provide detailed information about the spatial distribution of impurities on the surface of graphene, as well as other 2D material systems. The methodology input is conductance readings (for instance, as a function of the chemical potential) between different electrode pairs, the output being the spatially resolved impurity density. We show that the obtained spatial resolution depends on the number of such readings. Furthermore, by separating the impurity locations into partitions arranged in a grid-like geometry, this inversion procedure resembles a Sudoku puzzle in which the compositions of different sectors of a device are found by imposing that they must add up to specific constrained values established for the grid rows and columns. We argue that this technique may be used with other quantities besides the conductance, paving the way to alternative new ways of extracting information from a disordered material through the selective probing of local quantities.