Anderson transition in disordered graphene

Abstract

In a recent comment by Schleede and coworkers [1], they have correctly pointed out presence of small negative spectral weights of the order 10 −4 in RKPM expansion of the Dirac delta function. However, we found that these negative values are not responsible for the vanishing of the typical density of states (DOS) near the charge neutrality point.To clarify this point, in Fig. 1 we have plotted a Dirac delta function with Jackson attenuation factors (Solid line), and with RKPM (dashed line). The Gibbs oscillation seen in the dotted line are washed out both with Jackson as well as RKPM damping methods. As can be seen in the figure, the height of Dirac delta peak is underestimated by RKPM relative to the one obtained with Jackson g-factor. Such underestimation of the LDOSs, along with small σ broadening of kernel might be a possible reason for vanishing of typical DOS in regions with larger level spacing. Since largest level spacing in the tight-binding spectrum of graphene appears around the Dirac point, one expects to obtain vanishingly small LDOSs giving rise to the mobility edge reported in our letter [2]. We repeated our calculations with larger kernel width, and we obtain results similar to those obtained in Fig. 2 . Therefore we agree that the existence of mobility edge can be due to subtle details of the kernel used in the expansion.However, we do not agree with the main conclusion of the comment that smallest amount of uncorrelated on-site disorder in 2D honeycomb lattice is capable of localizing the entire spectrum. To discuss this claim, we present the following three arguments: (i) Let us repeat the same scaling analysis presented in right panel of Fig. 3 in Ref. [1]. In Fig. 2 we show the scaling analysis at fixed energies for W/t = 12 for a tightbinding model on a cubic lattice for which the critical value W c /t ∼ 16.5. In agreement with previous works of the above authors [3], for this value of disorder, one expects all states to -1000 -500 0 500 1000 1500 2000 2500 3000 3500 4000 0.49 0.495 0.5 0.505 0.51 jackson rkpm delta Fig. 1: Comparison of Dirac delta expansion corresponding to the same expansion order, N = 10 4 .. remain extended. As can be seen in Fig. 2, although the ratio R(E) = ρ typ (E)/ρ av (E) decreases by increasing system size, it finally saturates for larger sizes. In absence of a rigorous theory for the size dependence of R(E), the scaling result presented in right panel of Fig. 3 of the comment does not conclusively imply the localization of all states for the small value of disorder considered.(ii) The log-normal fitting to the distribution of LDOSs presented in left panel of Fig. 3 of the comment, shows that peak location of the distribution stays more or less fixed at ln(ρ/ρ av ) ≈ 0 by increasing the lattice size. This means that majority of LDOSs are equal to the mean value of DOS; a characteristic of extended states. On the other hand, the width of the distribution seems to saturate in the limit of large lattice sizes, which indicates states with energy 0.25t at W/t = 0.3 ar...

Direct observation of a dispersionless impurity band in hydrogenated graphene

“…However, since the position of H adsorbates is random (see supplementary information), the question of the localization of impurity band states arises. To answer this question, we use the KPM method to calculate the typical electron density of states, wherein the vanishing of this DOS indicates localization 16 . In Fig.…”

“…However, when the H impurity concentration is comparable to η c , in addition to the energy-dependent random potential (Anderson type), the hydrogenic wave functions will have substantial overlap to give rise to an impurity band. Moreover, for moderate values of the diagonal on-site disorder W ≈ γ, where W is the range of the on-site energies and γ is the C-C hopping energy, a mobility edge emerges in the conduction and valence bands 16 . The acceptor band survives the randomness regarding the position of the H atoms, and the states in the center of the acceptor band remain extended and a localization of these states is not expected.…”

“…Calculations of the typical and average density of states (DOS) using the kernel polynomial method (KPM) with a tight-binding (TB) band structure calculation 11,16 indicate that the new state does not localize easily despite the randomness in the H chemisorption sites.…”

Comment on “Anderson transition in disordered graphene” by Amini M. et al.

“…In a recent letter [1], Amini et al claim having found a mobility edge in graphene, a truly two-dimensional (2D) system. Their mobility edge ought to be induced by onsite uncorrelated (Anderson-type) disorder and-unlike to 3D systems-shall separate localized states in the band center, from the remaining extended states.…”

RKKY interaction in heavily vacant graphene

“…When the disorder is introduced within the Anderson model, both atomic scale oscillations and the Friedel oscillations due to Fermi surface survive [8]. Note that the Dirac picture remains quire robust against the weak disorder [9]. Therefore given the dimensional consistency the overall physics of the RKKY interaction is expected to survive in weak disorders considered in [8].…”