[Paper Review] Universal quantum transport and impurity band super metallicity in self-similar graphene carpets

G. Bouzerar, Didier Mayou|arXiv (Cornell University)|Mar 30, 2020

Graphene research and applications52 references4 citations

TL;DR

This paper investigates quantum transport in self-similar graphene Sierpinski carpets using tight-binding models and the Chebyshev Polynomial Green's Function method. It reveals a zero-energy flat band (ZEM) in a gapped spectrum, leading to a supermetallic phase at the neutrality point with universal conductivity σ(0) ≈ 1.315 e²/h, independent of inelastic scattering, driven by inter-band transitions despite vanishing state velocities.

ABSTRACT

Fractals, a fascinating mathematical concept made popular in the eighties, remained for decades a beautiful scientific curiosity mainly. With the tremendous advances in nanofabrication techniques, such as nanolithography, it has become possible to design self-similar materials with fine structures down to nanometer scale. Here, we investigate the effects of self similarity on quantum electronic transport in graphene Sierpinski carpets. We find that a gap opens up in the electron spectrum in the middle of which lies a flat band of zeros energy modes. Although these states have a zero velocity, a supermetallic phase is found at the neutrality point. For Fermi energy located in the valence/conduction band and in the presence of a small inelastic scattering the system stays metallic and the transport is found strongly anisotropic.

Motivation & Objective

• To explore the impact of self-similarity on quantum electronic transport in graphene-based fractal structures.
• To understand how fractal geometry influences the electronic spectrum and transport properties in graphene.
• To investigate the nature of conductivity at the neutrality point in graphene Sierpinski carpets (GSCs).
• To determine whether a supermetallic phase emerges due to zero-energy modes (ZEMs) in the presence of a gap.
• To analyze the anisotropy and scattering dependence of transport in the valence and conduction bands.

Proposed method

• Modeling electrons in GSCs using a nearest-neighbor tight-binding Hamiltonian with t = 2.7 eV.
• Applying iterative Sierpinski masks to create self-similar graphene structures with varying fractal levels (ic = 3 to 7).
• Using the Chebyshev Polynomial Green’s Function (CPGF) method for large-scale calculations with linear scaling in system size.
• Employing exact diagonalization for small systems (e.g., (4,4) GSC with 26,833 atoms) to validate ZEM states and charge density distributions.
• Calculating conductivity via the Kubo formula with the current operator defined through the commutator [H, x].
• Analyzing the diffusivity at E=0 using the relation Dx(E) ∝ η⟨Cx,β⟩, where ⟨Cx,β⟩ ≈ 4.1.

Experimental results

Research questions

• RQ1 Does self-similarity in graphene carpets lead to a gap in the electronic spectrum with a flat band of zero-energy modes?
• RQ2 Can a supermetallic phase emerge at the neutrality point despite the presence of a gap and vanishing group velocity of ZEM states?
• RQ3 What is the origin of finite conductivity σ(0) in the absence of intra-band contributions from ZEM states?
• RQ4 How does inelastic scattering rate η affect transport at the neutrality point and in the valence/conduction bands?
• RQ5 Is the transport in GSCs strongly anisotropic, and how does this depend on the Fermi energy position?

Key findings

• A finite energy gap opens in the electronic spectrum of graphene Sierpinski carpets, with a flat band of zero-energy modes (ZEMs) at the center.
• Despite zero velocity of ZEM states, a supermetallic phase emerges at the neutrality point with universal conductivity σ(0) ≈ 1.315 e²/h, matching the universal value 4e²/(πh) within 3%.
• Conductivity at E=0 arises exclusively from inter-band transitions between the ZEM band and the valence/conduction bands, not from intra-band processes.
• The diffusivity at E=0 is proportional to the inelastic scattering rate η, with a characteristic length scale Ld ≈ 4a, indicating extended ZEM states.
• For Fermi energy in the valence or conduction band and finite η, the system remains metallic with strongly anisotropic transport properties.
• The probability distribution of the matrix element Cx,β has a mean of 4.1 and width ~0.5, supporting the robustness of the conductivity mechanism.