Spatial mapping of disordered 2D materials: the conductance Sudoku

Abstract

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 material by selectively interrogating its transport properties. Here 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 setup in order to provide detailed information about the spatial distribution of impurities on the surface of a 2D material. The methodology input are 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 not only on the number of conductance measurements but also on the electrode dimensions. Furthermore, when implemented with electrodes 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.