Dynamical scaling analysis of the optical Hall conductivity in the graphene quantum Hall system with various types of disorder

Dynamical scaling of the optical Hall conductivity σxy(εF , ω) at the n = 0 Landau level in graphene is analyzed for the 2D effective Dirac fermion and honeycomb lattice models with various types of disorder. In the Dirac fermion model with potential disorder, σxy(εF , ω) obeys a well-defined dynamical scaling, characterized by the localization exponent ν and the dynamical critical exponent z. In sharp distinction, scaling behavior of σxy(εF , ω) in the honeycomb lattice model with bond disorder (preserving chiral symmetry), becomes anomalous. 1. Motivation Study of the quantum Hall effect (QHE) has been primarily focused on static properties. Yet, dynamical aspects are expected to display a wealth of interesting phenomena. For the QHE in graphene, investigation of dynamical response is rather scarce. Two of the present authors (in collaboration with Hatsugai), studied the ac Hall effect in the THz frequency regime. The Hall conductance (although not quantized in ac), is shown to display plateau structure resulting from the localization of states. [1] While the localization dominates the QHE already in the static case, here we are talking about the ac Hall conductivity, so that the most reliable way to examine the dynamics of localization is to apply the dynamical scaling argument. [2] The QHE in graphene is peculiar due to the occurrence of n = 0 Landau level (LL) around the Dirac point where electron and hole branches intersect. It has no counterpart in the ordinary QHE[3, 4], and hence, static and dynamic scaling are of special interest in the n = 0 LL. Analysis of localization delocalization transition in QHE has been mostly focused on the scaling behavior of the dc longitudinal conductivity σxx(εF ), where εF is the Fermi energy. [5] On the other hand, dynamical scaling properties of the longitudinal ac conductivity σxx(εF , ω) and ac Hall conductivity σxy(εF , ω) have not yet been thoroughly addressed, less so in the graphene QHE. Experimentally, scaling properties of σxx(εF , ω) was investigated in frequency range ω = 0 [6] up to the GHz regime [7], while the relevant (i.e., ∼ cyclotron energy) frequency range is THz. Recent advances in THz measurements (e.g., Faraday rotation in magnetic fields) enable us to study the dynamical response functions [8, 9]. Study of optical properties of graphene develops rather intensely, including experimental transmission spectra[10], or theoretical examination of the cyclotron emission[11]. Thus, the physics of dynamical scaling in graphene QHE is experimentally accessible in the THz regime. HMF-19 IOP Publishing Journal of Physics: Conference Series 334 (2011) 012045 doi:10.1088/1742-6596/334/1/012045 Published under licence by IOP Publishing Ltd 1 Theoretically, an intriguing question is how the static Hall conductivity, which may be regarded as a topological quantity[12, 13], evolves into the optical Hall conductivity, especially in the THz regime where the relevant energy scale is the cyclotron energy. [14, 1, 15] It has recently been shown that the plateau structure of σxy(ω) is retained in the ac (∼ THz) regime in both the ordinary two-dimensional electron gas and the massless Dirac model, although the plateau height in the ac conductivity deviates from the quantized values.[1] The plateaus remain remarkably robust against disorder, which can be attributed to localization. To gain a deeper understanding of the robust step structures in the ac regime, we need to elucidate the static and dynamical scaling behavior of the response functions near the plateau to plateau transition. For finite frequencies we have a new length scale, Lω ∼ ω− 1 z , where z is the dynamical critical exponent. A proper quantity for carrying scaling analysis is the width W of the transition region, which generically depends on ω through Lω and on the sample size L. An interest in graphene QHE also comes from the relationship between the chiral symmetry or sublattice symmetry in graphene and the anomalous n = 0 LL. If the disorder respects the chiral symmetry which is the case with bond disorder or random magnetic fields, n = 0 LL remains anomalously sharp [16]. Hence, for the first time in this paper, we also discuss what happens to the scaling behavior in the honeycomb lattice (tight-binding) model, with bond disorder that preserves chiral symmetry. 2. Formalism To study graphene QHE we first employ the two-dimensional effective 2D Dirac fermion model in a magnetic field with random potential, H = vFσ · π + V (r), (1) where σ = (σx, σy) is a two-component vector of Pauli matrices, and π = p+eA is the covariant momentum. Disorder is introduced by a random potential V (r) composed of Gaussian scattering centers of range d and amplitude ±u with a density nimp placed at randomly chosen positions Rj . A measure of disorder (the Landau level broadening) is Γ = 2u[nimp/2π(l + d2)]1/2, where l is the magnetic length and we chose d = 0.7l. Diagonalization of the Hamiltonian is carried out within subspace spanned by the five LLs around n = 0 LL. The system’s geometry is a square of size L × L with L/l = 25, 30, 35, 40. In terms of the eigenfunctions and the energy eigenvalues εa, the optical Hall conductivity is evaluated in terms of the Kubo formula, σxy(εF , ω) = ih̄e2 L2 ∑