Visualizing the multifractal wave functions of a disordered two-dimensional electron gas

Berthold Jäck,Fabian Zinser,Elio J König,Klaus Kern,Sune N P Wissing,Christian R Ast,Anke B Schmidt,Markus Donath

doi:10.1103/physrevresearch.3.013022

Abstract

The wavefunctions of a disordered two-dimensional electron gas at the quantum-critical Anderson transition are predicted to exhibit multifractal scaling in their real space amplitude. We experimentally investigate the appearance of these characteristics in the spatially resolved local density of states of a two-dimensional mixed surface alloy Bi_xPb_{1-x}/Ag(111), by combining high-resolution scanning tunneling microscopy with spin and angle-resolved inverse-photoemission experiments. Our detailed knowledge of the surface alloy electronic band structure, the exact lattice structure and the atomically resolved local density of states enables us to construct a realistic Anderson tight binding model of the mixed surface alloy, and to directly compare the measured local density of states characteristics with those from our model calculations. The statistical analyses of these two-dimensional local density of states maps reveal their log-normal distributions and multifractal scaling characteristics of the underlying wavefunctions with a finite anomalous scaling exponent. Finally, our experimental results confirm theoretical predictions of an exact scaling symmetry for Anderson quantum phase transitions in the Wigner-Dyson classes.

Highlights

Multifractality is an ubiquitous phenomenon in nature that characterizes a variety of physical phenomena from turbulence [1] to fluid flow and surface growth [2]
The band structure is visibly affected by strong spin-orbit coupling (SOC) [22], and it is confined within the first atomic layer at the sample surface [22], evidencing the 2D character of the surface alloy’s electronic properties
Spectroscopic measurements of the local electronic density of states (LDOS) with the Scanning tunneling microscopy (STM) reveal a strong influence of a locally varying disorder potential on the electronic states that is induced by the different valency of the Bi and Pb atoms

Summary

Introduction

Multifractality is an ubiquitous phenomenon in nature that characterizes a variety of physical phenomena from turbulence [1] to fluid flow and surface growth [2] It describes the power-law scaling of fractal quantities with not one but a continuous (infinite) set of anomalous scaling exponents. At Anderson criticality, multifractality characterizes the real space amplitude fluctuations of the underlying quantum mechanical wave functions (r) [6] and thereby the real space distribution of the corresponding local electronic density of states (LDOS) [7,8]

Results

Discussion

Conclusion