**Abstract**

Anomalous charge carrier transport in graphene is studied within a topologically nontrivial quasi-relativistic graphene model. The model predicts additional topological contributions, such as the Majorana-like mass-term correction to the ordinary ohmic component of the current, the spin-orbital-coupling, "Zitterbewegung"-effect corrections to conductivity in space, and time dispersion regime. The corrections appear due to non-Abelian quantum statistics for the charge carriers in graphene. The chiral anomaly of electrophysical and optical properties may emerge due to a deconfinement of the pseudo-Majorana quasiparticles. It has been shown that phenomena of negative differential conductivity, loss of universal far-infrared optical conductivity, and nonzero "minimal" direct-current conductivity in graphene occur due to flattening and vorticity of the pseudo-Majorana model graphene energy bands.

**Keywords:** graphene, anomalous charge transport, conductivity, chiral anomaly, quasi-relativistic model, Dirac-Hartree-Fock self-consistent field approximation, pseudo-Majorana fermion graphene model

## **1. Introduction**

Graphene discovery is the driving force for progress in the current developments of quantum devices due to fascinating electrical and optical graphene properties. Graphene is an atomically thin carbon layer, in which valent and conduction bands touch each other in six points of hexagonal Brillouin zone, called Dirac points or valleys *K*,*K*<sup>0</sup> [1]. Graphene belongs to strongly correlated systems, in which many-body interactions occur. Thanks to the strong electron-hole correlations, the electrophysical and optical properties of the graphene are very unusual. According to the linear response theory, the temperature dependence of the Hall conductivity for Fermi liquid, to which most metals belong, does not depend upon temperature [2]. But, the strongly correlated systems exhibit an anomalous charge magnetotransport

[3]. A similar situation prevails for universal nonzero direct-current (DC) graphene minimum conductivity (scale range of <sup>4</sup>*e*<sup>2</sup> *<sup>h</sup>* [4] – <sup>6</sup>*e*<sup>2</sup> *<sup>h</sup>* [5, 6] depending on support type) when the linear response theory predicts vanishing electrical conductivity for direct current (DC) or <sup>4</sup>*e*<sup>2</sup> *<sup>π</sup><sup>h</sup>* for the low-frequency conductivity in pure graphene [7, 8]. Here, *h* is the Planck constant, and *e* is the electron charge.

The situation with optical conductivity in the far-infrared spectral range of 0*:*05 � 0*:*5 eV (25 *μ*m–350 *μ*m) turned out to be even more unpredictable. Theoretical calculations of the far-infrared optical conductivity within a pseudo-Dirac massless fermion model for graphene predict its universal value, *<sup>e</sup>*<sup>2</sup> <sup>4</sup>*h*, [9], which contradicts entirely experimental data on the existence of extrema in this spectral range, namely, an asymmetric peak at 0.15 eV (� <sup>150</sup> � 200 cm�<sup>1</sup> ) [10] and a shallow minimum in the spectral range of � 0*:*2 � 0*:*3 eV [11, 12] (see **Figure 1a** and **b**). Here, ℏ is the Planck constant divided by 2*π*. Slowly increasing the optical conductivity approaches the universal value in the mid-to-near-infrared spectral range of 0*:*5 � 1*:*2 eV [13] where optical transitions occur far enough from the valleys *K*,*K*<sup>0</sup> of the graphene Brillouin zone. Thus, the applicability of the pseudo-Dirac graphene model is doubtful in both spectral ranges.

The electron is a complex fermion; thus, if one decomposes its wavefunction into its real and imaginary parts, which would be Majorana fermions, they are rapidly remixed by electromagnetic interactions. However, such a decomposition could be reasonable for graphene because of the effective electrostatic screening. Experimental signatures of graphene Majorana states in graphene-superconductor junctions without the need for spin-orbital coupling (SOC) have been established in [14]. An analysis of Majorana-like graphene models also become relevant in connection with the discovery of unconventional superconductivity for twisted bilayer graphene at *θ* ≈1*:*05 angle of rotation of one monoatomic layer (monolayer) relative to another one [15]. A quantum statistics of graphene model with Majorana-like states should be a non-Abelian one, and the absence of that is the main obstacle to shed light on topologically nontrivial mechanisms of graphene conductivity. The non-Abelian pseudo-Majorana statistics of graphene charge carriers, which could clarify the anomalous effects, are behind several major theoretical approaches to graphene. The approaches are based on the massless pseudo-Dirac fermion model. The pseudo-Dirac graphene model Hamiltonian gives predictions that are very different from experimental data for a wide range of transport phenomena in graphene physics, such as the existence of topological currents in graphene superlattices [16], a sharp rise of Fermi velocity value *vF* in touching valent and conduction bands [17], and a lack of excitonic instability [18].

So, up to now, there is no consistent theory of topologically nontrivial graphene conductivity.

In this chapter, we show that charge transport coefficients in linear response to electromagnetic fields contain anomalous contributions arising from strong correlations between graphene charge carriers. The strong temperature dependencies that are observed in Hall, and optical and electrical conductivities are explained by non-Abelian statistics of topologically nontrivial graphene charge carriers of pseudo-Majorana nature within an earlier developed quasi-relativistic self-consistent Dirac-Hartree-Fock approach in a tight-binding approximation [19–25]. This quasirelativistic approach allowed to achieve success in calculations of electronic properties for quasi-circular graphene *p* � *n n*ð Þ � *p* junctions also [26].

Our goal is to study quantum transport of charge carriers with vortex dynamics within a quasi-relativistic graphene model using a high-energy *k* ! � *p* !-Hamiltonian and *Anomalous Charge Transport Properties and Band Flattening in Graphene: A Quasi… DOI: http://dx.doi.org/10.5772/intechopen.106144*

#### **Figure 1.**

*Graphene conductivity and transmission spectra [10, 12] in the far-infrared (a) and mid-to-near-infrared (b) spectral ranges. Graphene conductivity is obtained from transmittance contrast. The normalized change* Δ*T=TCNP in transmittance TCNP of the non-dopped graphene is induced by the presence of the graphene sample; TCNP is the transmittance at the applied voltage for charge-neutral Dirac point. The gate-induced change of transmission in graphene is obtained for different values of gate voltage. The Drude conductivity fitting is shown. (c) Electron (left) and hole bands (right) of a quasi-relativistic NF* ¼ 3*-flavors model graphene that were calculated with a pseudo-Majorana mass term. (d) The vortex texture in contour plots of electron (left) and hole (right) bands was* ! ! *.*

*calculated with the Majorana-like mass term on momentum scales q=KA* ¼ 0*:*002*; q* ! <sup>¼</sup> *<sup>k</sup>* � *K*

to perform simulations of the complex conductivity of the system in a number of experimentally interesting cases. The Wilson non-closed loop method will be used to prove dichroism of the band structure, which leads to valley Hall conductivity.
