Electron localization in disordered graphene for nanoscale lattice sizes: multifractal properties of the wavefunctions

Abstract

An analysis of the electron localization properties in doped graphene is performed by doing a numerical multifractal analysis. By obtaining the singularity spectrum of a tight-binding model, it is found that the electron wave functions present a multifractal behavior. Such multifractality is preserved even for second neighbor interaction, which needs to be taken into account if a comparison is desired with experimental results. States close to the Dirac point have a wider multifractal character than those far from this point as the impurity concentration is increased. The analysis of the results allows to conclude that in the split-band limit, where impurities act as vacancies, the system can be well described by a chiral orthogonal symmetry class, with a singularity spectrum transition approaching freezing as disorder increases. This also suggests that in doped graphene, localization is in contrast with the conventional picture of Anderson localization in two dimensions, showing also that the common belief of the absence of quantum percolation in two dimensional systems needs to be revised.Ever since its discovery 1,2 , graphene has been considered as an ideal candidate to replace Silicon in electronics 3 , since this first truly two-dimensional crystal has the highest electrical and thermal conductivity known 4 . However, graphene per se is not a semiconductor. Several proposals have been made to solve this issue 5 . Experimentally, it has been found that doped graphene presents a metal to insulator transition 6 when doped with H, producing a kind of narrow band gap semiconductor. The increase in localization around the Dirac point was roughly predicted from an electron wavefunction frustration analysis in the graphene's underlying triangular lattice 7-9 . Such theoretical results were made under the supposition that Hydrogen bonds to the 2p z Carbon orbital, and thus impurities act as vacancies 10,11 . This case corresponds to the split-band limit. This approach has been useful to predict localization and the pseudogap size, i.e., the region in which the inverse participation ratio increases by one order of magnitude 7 , in very good agreement with experiments 6 , although vacancies and impurities are indeed different 12,13 . However, there is a theoretical nuance to the idea of having a metal-insulator transition in two dimensions (2D). According to the well known Abrahams's et. al. scaling analysis, in 1D and 2D all states are localized for any amount of disorder, excluding the possibility of a mobility edge 14 . The experimental and numerical analysis shows that not all states are localized, and there exists a kind of mobility edge associated with a pseudogap around the Fermi energy 9,15,16 . This means that there is a problem that has not been solved. There are two possibilities, either electron-electron interaction produces delocalization 17 , or somehow the Abrahams's et. al. analysis does not completely applies to this case. In graphene, electron-electron interaction is very weak 18 , so in principle, t...

Exotic multifractal conductance fluctuations in graphene

“…The most compelling explanation of the multifractal UCF we observe in our SLG samples is an incipient Anderson localization near the charge-neutrality point. Multifractality of the local amplitudes of critical eigenstates near Anderson localization has been studied, theoretically, in several quantum-condensed-matter systems [19,[22][23][24]. The multifractality of the eigenstates near the critical point directly affects the two-particle correlation function through the generalized diffusion coefficient [62,63], which, in turn, affects the local current fluctuations in the system via the Kubo formula.…”

“…Since the pioneering work of Mandelbrot [11], the detection and analysis of multifractal scaling in such systems have enhanced our understanding of several complex phenomena, e.g., the dynamics of the human heart-beat [12], the form of critical wave-functions at the Anderson-localization transition [1], the time series of the Sun's magnetic field [13], in medical-signal analysis (for instance, in pattern recognition, texture analysis and segmentation) [14], fully-developed turbulence and in a variety of chaotic systems [15,16]. In condensed-matter systems, signatures of multifractality are usually sought in the scaling of eigenfunctions at critical points [17][18][19][20][21][22][23][24]. Despite compelling theoretical predictions [25][26][27][28][29][30], there are no reports of the successful observation of multifractality in transport coefficients.…”

Multifractal Conductance Fluctuations in High-Mobility Graphene in the Integer Quantum Hall Regime

“…Multifractality, initially introduced to characterize the statistical properties of fluid turbulence [42,43], was later studied in a variety of fields ranging from heartbeats to cloud structures [44][45][46][47][48][49][50][51][52]. In condensed-matter science, most investigations of multifractality, which manifests itself at some phase transitions, employ a combination of theoretical and numerical techniques [36][37][38][53][54][55][56][57]. The experimental characterizations of multifractality in such condensed-matter settings require precision experiments in high-quality samples, often at very low temperatures.…”