Shape of the Landau subbands in disordered graphene

Abstract

The question of why different experiments obtain different shapes of Landau subbands of two-dimensional electron gases (2DEG's) is investigated. In particular, we consider disordered graphene within the tight-binding formulation in a high magnetic field and in the presence of impurity potentials of a finite interaction range. It is found that the shape of the total density of states (DOS) of Landau subbands is well described by a Gaussian function, while the shape of the local density of states (LDOS) can be fitted to either a simple Lorentzian function or a sum of several Lorentzian functions. This finding explains well why the experiments that measure the DOS of a Landau subband appear to be Gaussian, and those experiments that are sensitive to the LDOS can only be fitted well by Lorentzian functions. Thus, our results provide a natural explanation to this long-standing puzzle involving 2DEG's in the quantum Hall regime. Graphene has attracted a lot of attention due to its remarkable properties and its potential applications in nanoelectronics. 1 Its electronic properties are mainly due to its two-dimensional nature and the fact that the low-energy excitations are governed by the massless Dirac equation, 2 confirmed by the interesting, uneven Landau level (LL) distribution that varies with both the square root of the magnetic field B and the LL index n: E n = sgn(n) 2ehv 2 F |n|B, where v F , e, andh are the Fermi velocity, electron charge, and the Planck constant, respectively. This emblematic LL spectrum and its characteristic zero-energy state (n = 0) are directly related to the anomalous quantum Hall effect (QHE) in graphene. 3 Many efforts have been made to determine the electronic properties of the disorder-broadened Landau subbands (LS's). However, the shape of both the total density of states (DOS) and the local DOS (LDOS) of an LS of graphene is a controversial issue in the literature. Historically, it has generally been believed that the shape of the DOS of LS's in a conventional two-dimensional electron gas (2DEG) is Gaussian-like, 4,5 but experiments seem to present a different picture. Initially, this consensus was supported by careful measurements of specific heat, 6 magnetization, 7 and magnetocapacitance 8 on GaAs-Ga x Al 1−x As heterostructures. However, a recent scanning-tunneling-microscopy-scanningtunneling-spectroscopy (STM-STS) measurement on conventional 2DEG's of the n-InSb(110) ] surface, 9 as well as the cyclotron resonance (CR) absorption line shape on conventional 2DEG's, 10,11 observed a Lorentzian shape of an LS. The situation for disordered graphene is similar. All CR 13-15 and STM-STS 16-19 measurements suggested clearly a Lorentzian shape for an LS. However, recent magnetocapacitance experiments produced conflicting results: In one experiment, 20 an LS could be fitted well by the Lorentzian functions, while another experiment 21 observed a Gaussian shape for an n = 0 LS. On the theoretical side, the DOS of an LS of graphene is initially found to be Gaussian-like. [22][23][24] C...

Magnetic field independent shape of the zero-energy landau levels in a disordered T3 model

“…Equation (1) also reflects that the width of the LLs Γ depends on both magnetic field strength and disorder strength as Bg G~. In our previous studies, we numerically computed the shape of the zero-energy LL in disordered graphene, whose low-energy quasiparticle is governed by the Dirac-Weyl equation with pseudospin-1/2 [18,19]. Interestingly, our results are similar to the predictions of Wegner in the conventional two-dimensional electron systems.…”

“…The DOS is calculated with the Lanczos recursion method [18,19]. The averaged DOS can be evaluated from the Green's function…”

“…The shape of the Landau levels (LLs) in a disordered two-dimensional electronic system is important to many physical measurements, such as specific heat [1,2], magnetization [3], magnetocapacitance [4], and magnetoconductivity [5][6][7][8][9]. This subject has been theoretically and experimentally studied for many decades [10][11][12][13][14][15][16][17][18][19][20][21][22]. Based on the self-consistent Born approximation (SCBA), the shape of the lowest LL in the conventional two-dimensional electronic systems has been argued to be a semi-elliptical form, which is insensitive to the distribution of disorder [14].…”

“…It is shown that the shape of the LLs plays an important role in the transport properties, and the above scaling behavior leads to an interesting fact that the magnetoconductivity as a function of carrier density is independent of the disorder strength. In addition, the existence of the huge magnetic field independent macroscopic degeneracy of zero-energy LL results in blue shifts of the conduction and valence bands LLs, in contrast with what found in the disordered graphene [18,19]. This paper is structured as follows.…”

Effective low-energy description of the two-impurity Anderson model: RKKY interaction and quantum criticality

“…The transition between these two singlet phases in the TIAM is driven by the ratio between the Kondo temperatur T K and J RKKY [40,[44][45][46][47][48][49][50]. While a quantum critical point (QCP) separates both ground states in the presence of a special kind of particle-hole (P-H) symmetry [22,43,49], the quantum phase transition is replaced by a crossover if that symmetry is broken [51].…”