**3. Electronic structure**

The band structure of graphene within the quasi-relativistic *NF* ¼ 3-flavors model has been calculated (see [25] and references therein) with the Majorana-like mass term and is presented in **Figure 1c**. The graphene bands are conical near the Dirac point at *q q*<sup>0</sup> ð Þ! 0, *q* ¼ *p* ! � *K* ! (*q*<sup>0</sup> <sup>¼</sup> *<sup>p</sup>* !0 � *<sup>K</sup>* !0 ) where *p* ! *p* !0 is a momentum of electron (hole). But, they flatten at large *q q*<sup>0</sup> ð Þ. The band structure of graphene within the quasi-relativistic model with pseudo-Majorana charge carriers hosts vortex and antivortex whose cores are in the graphene valleys *K* ! and *K* !0 of the Brillouin zone, respectively (**Figure 1c** and **d**). Touching the Dirac point *K K*<sup>0</sup> ð Þ, the cone-shaped valence and conduction bands of graphene are flattened at large momenta *p* of the graphene charge carriers [23]. It signifies that the Fermi velocity *vF* diminishes drastically to very small values at large momenta *p*. Since eight sub-replicas of the graphene band near the Dirac point degenerate into the eightfold conic band

**Figure 2.** *Non-Abelian phases* <sup>Φ</sup>1, … , <sup>Φ</sup><sup>4</sup> *of the Wilson loop eigenvalues in the units of <sup>π</sup> at nonzero gauge fields; q*! <sup>¼</sup> *<sup>k</sup>* ! � *K* !

*.*

(**Figure 1c** and **d**) the pseudo-Majorana fermions forming the eightfold degenerate vortex are confined by the hexagonal symmetry. In the state of confinement, the pseudo-Majorana fermions are linked with the formation of electron-hole pairs under the action of the hexagonal symmetry.

A global characterization of all Dirac touching with non-Abelian Zak phases Φ1, Φ2, Φ3, Φ<sup>4</sup> as arguments of the Wilson loop operator for our model is presented in **Figure 2** [23]. Accordingly, in the case of the nonzero gauge fields, the simulations predict the homotopy group *Z*<sup>12</sup> with the generator �*π=*6 in the Dirac points *K K*<sup>0</sup> ð Þ [23]. **Figure 1d** shows that the proposed model has the nontrivial topological properties revealing in the dichroism of its band structure. The vorticity of the band is originated from Majorana-like excitations. It is natural to assume that such peculiarities would lead also to observable consequences in charge transport in such a system.

## **4. Non-Abelian currents in quasi-relativistic graphene model**

Conductivity can be considered as a coefficient linking the current density with an applied electric field in a linear regime of response. To reach the goal, several steps should be performed. First, one has to subject the system to an electromagnetic field, and this can be implemented by a standard change to canonical momentum

*p* ! ! *<sup>p</sup>* ! � *<sup>e</sup> c A* ! in the Hamiltonian in the following way:

$$
\left[c\overleftarrow{\sigma\_{2D}} \cdot \left(\overrightarrow{p}\_{AB} - \frac{e}{c}\overrightarrow{A}\right) - \Sigma\_{AB}\overleftarrow{\Sigma\_{BA}}\left(\overrightarrow{p}\_{AB} - e\overrightarrow{A}/c\right)\right] \overleftarrow{\hat{\chi}^{+}\_{+\sigma\eta}}\left(\overrightarrow{r}\right) \left|0-\sigma\right\rangle = cE\_{qu}(p)\overleftarrow{\hat{\chi}^{+}\_{+\sigma\eta}}\left(\overrightarrow{r}\right) \left|0, -\sigma\right\rangle,\tag{5}
$$

$$
\Sigma\_{AB}\overleftarrow{\Sigma\_{BA}}\left(\overrightarrow{p}\_{AB} - e\overrightarrow{A}/c\right) = \Sigma\_{AB}\overleftarrow{\Sigma\_{BA}}(\mathbf{0}) + \sum\_{i} \frac{d\Sigma\_{AB}\overleftarrow{\Sigma\_{BA}}}{dp\_{i}^{\dagger}}\big|\_{p\_{i}^{\prime}=0}\left(p\_{i}^{AB} - \frac{e}{c}A\_{i}\right)\tag{6}
$$

$$+\frac{1}{2} \sum\_{i,j} \frac{d^2 \Sigma\_{AB} \overline{\Sigma}\_{BA}}{dp\_i' dp\_j'} \Bigg|\_{p\_i', \ p\_j'=0} \left( p\_i^{AB} - \frac{e}{c} A\_i \right) \left( p\_j^{AB} - \frac{e}{c} A\_j \right) + \dots \tag{6}$$

Here, *A* ! is a vector potential of the field. In what follows, we omit the cumbersome designation "AB" if this does not lead to a lack of sense.

Second, to find the quasi-particle current we use the perturbation theory [29, 30]. Taking into account the expression (6), a potential energy operator *V* for interaction between the secondary quantized fermionic field *χ*þ*σ<sup>B</sup>* ð Þ *x* with the electromagnetic field reads

$$\begin{split} \chi^{\dagger}\_{+\sigma\_{8}} \left[ -c\vec{\sigma}\_{BA} \cdot \frac{\boldsymbol{\varepsilon}}{c} \vec{A} - \boldsymbol{M}\_{BA}(\mathbf{0}) - \sum\_{i} \frac{d\mathbf{M}\_{BA}}{dp\_{i}^{\prime}} \vert \right. \quad \times \left( p\_{i}^{AB} - \frac{\boldsymbol{\varepsilon}}{c} \boldsymbol{A}\_{i} \right) \\ V = \\ -12 \sum\_{i,j} \frac{d^{2}\mathbf{M}\_{BA}}{dp\_{i}^{\prime} dp\_{j}^{\prime}} \vert \Bigg. \quad \times \left( p\_{i}^{AB} - \frac{\boldsymbol{\varepsilon}}{c} \boldsymbol{A}\_{i} \right) \left( p\_{j}^{AB} - \frac{\boldsymbol{\varepsilon}}{c} \boldsymbol{A}\_{j} \right) + \dots \right] \chi\_{+\sigma\_{8}} . \end{split} \tag{7}$$

*Anomalous Charge Transport Properties and Band Flattening in Graphene: A Quasi… DOI: http://dx.doi.org/10.5772/intechopen.106144*

Then, one can find a quasi-relativistic current [31] of charge carriers in graphene as:

$$\begin{split} &j\_i^{\text{SM}} = c^{-1} j\_i \equiv j\_i^{\text{O}} + j\_i^{\text{2b}} + j\_i^{\text{so}}, \\ &j\_i = c \chi\_{+\sigma\_{\text{B}}}^{\dagger}(\mathbf{x}^+) \boldsymbol{\nu}\_{\mathbf{x}^+ \cdot \mathbf{x}^-}^i \chi\_{+\sigma\_{\text{B}}}(\mathbf{x}^-) \\ &- \frac{c^2 A\_i}{c M\_{AB}} \chi\_{+\sigma\_{\text{B}}}^{\dagger}(\mathbf{x}^+) \chi\_{+\sigma\_{\text{B}}}(\mathbf{x}^-) \\ &+ \frac{e\hbar}{2 \mathcal{M}\_{AB}} \left[ \vec{\nabla} \times \chi\_{+\sigma\_{\text{B}}}^{\dagger}(\mathbf{x}^+) \vec{\sigma} \chi\_{+\sigma\_{\text{B}}}(\mathbf{x}^-) \right]\_i, \; i = \mathbf{1}, 2. \end{split} \tag{8}$$

Here,

$$\mathfrak{x}^{\pm} = \mathfrak{x} \pm \varepsilon,\ \mathfrak{x} = \left\{ \vec{r},\ t\_{0} \right\},\ \vec{r} = \{\mathfrak{x},\mathfrak{y}\},\ \ t\_{0} = \mathbf{0},\ \ \varepsilon \to \mathbf{0};\tag{9}$$

*vi <sup>x</sup>*þ*x*� is the velocity operator determined by a derivative of the Hamiltonian (1), *χ*þ*σ<sup>B</sup> x*<sup>þ</sup> ð Þ is the secondary quantized fermion field, the terms *j O <sup>i</sup>* , *j Zb <sup>i</sup>* , *j so <sup>i</sup>* , *i* ¼ *x*, *y* describe an ohmic contribution that satisfies the Ohm's law and contributions of the polarization and magneto-electric effects, respectively.

To perform quantum-statistical averaging for the case of nonzero temperature, we use a quantum-field method developed in references [32, 33]. After tedious but simple algebra one can find the conductivity in our model:

$$\begin{split} &\sigma\_{ii}^{O}(w,\ k) \\ &= \frac{ie^{2}\overline{\rho}^{2}}{(2\pi c)^{2}}\operatorname{Tr}\left[\left(1-M\_{\mathrm{BA}}\left(\overline{p}\right)\frac{\partial^{2}M\_{\mathrm{BA}}}{\partial p\_{i}^{2}}\right)\right. \\ &\quad\times\left(M\overline{v}^{i}(p),N\overline{v}^{i}(p)\right)d\overline{p}, \\ \sigma\_{II}^{\mathrm{2b}}(w,k) &= \frac{ie^{2}}{\overline{\rho}^{2}(2\pi c)^{2}} \\ &\times\operatorname{Tr}\left[\frac{M\_{\mathrm{BA}}\left(\overline{p}\right)^{2}}{2}\sum\_{i=1}^{2}\frac{\partial^{2}M\_{\mathrm{BA}}}{\partial p\_{i}^{2}}\left(M\overline{v}^{i}(p),N\overline{v}^{i}(p)\right)d\overline{p}, \\ \sigma\_{12(21)}^{\sigma}(w,k) &= (-1)^{1(2)}\frac{i}{2}\frac{ie^{2}\overline{\rho}^{2}}{(2\pi c)^{2}} \\ &\times\operatorname{Tr}\left[M\_{\mathrm{BA}}\left(\overline{p}\right)\frac{\partial^{2}M\_{\mathrm{BA}}}{\partial p\_{1}\partial p\_{2}}\left(M\overline{v}^{i-1}(p),N\overline{v}^{i-1}(p)\right)\sigma\_{z}d\overline{p} \end{split} \tag{12}$$

for the current *j O <sup>i</sup>* , *j Zb <sup>i</sup>* , *j so <sup>i</sup>* , respectively. Here, matrices *M*, *N* are given by the following expressions:

$$\begin{aligned} M &= \frac{f\left[\overline{\beta}((H(p^{+}) - \mu)/\hbar)\right] - f\left[\overline{\beta}(H^{\dagger}(-p^{-}) - \mu/\hbar)\right]}{\overline{\beta}z - \overline{\beta}(H(p^{+})/\hbar) + \overline{\beta}(H^{\dagger}(-p^{-})/\hbar)}, \\ N &= \frac{\delta(\hbar\omega + \mu)}{(\hbar z + H(p^{+}) - H^{\dagger}(-p^{-}))\overline{\beta}}. \end{aligned}$$

Here, *f* is the Fermi–Dirac distribution, *z* ¼ *ω* þ *iε*, *p* !� <sup>¼</sup> *<sup>p</sup>* ! � *<sup>k</sup>* ! , *ω* is a frequency, *μ* is a chemical potential, and *β* is an inverse temperature divided by *c*.

## **5. Method of conductivity calculation**

#### **5.1 Diagonalized representation of the Hamiltonian**

We perform the conductivity calculation in a representation where the Hamiltonian of the graphene model is diagonalized to find the velocity determined in the following way:

$$
\overrightarrow{v}\_{AB} \approx \frac{\partial H\_{AB}}{\partial \overrightarrow{p}}.\tag{13}
$$

The velocity operator *v* ! *AB BA* ð Þ should be transformed by the transformation of the form *S*�<sup>1</sup> *v* ! *ABS* with a matrix *S* constructed on the eigenvectors *χ* of the Hamiltonian. The corresponding matrix *S* should be constructed on the eigenvectors of the operator adjoined to the Hamiltonian. In every *p*-point the particle (hole) Hamiltonian is represented by 2 � 2 matrix, we denote matrix elements of the exchange operator *i*Σ*<sup>x</sup> rel* � � *AB BA* ð Þ formally as <sup>Σ</sup>*AB BA* ð Þ *ij* . Then, the eigenvectors *χi*, *i* ¼ 1, 2 of the Hamiltonian (1) being the rows of the appropriate matrix *S* can be expressed in an explicit way:

$$\chi\_1 = \left\{ \frac{ip\sin\left(\phi\right)\left(\Sigma\_{11}^{AB}\Sigma\_{21}^{AB} + \Sigma\_{12}^{AB}\Sigma\_{22}^{AB}\right) + p\cos\left(\phi\right)\left(\Sigma\_{12}^{AB}\Sigma\_{22}^{AB} - \Sigma\_{11}^{AB}\Sigma\_{21}^{AB}\right) - p\left(\Sigma\_{12}^{AB}\Sigma\_{21}^{AB} - \Sigma\_{11}^{AB}\Sigma\_{22}^{AB}\right)}{p\left(\left(\left(\Sigma\_{11}^{AB}\right)^2 - \left(\Sigma\_{12}^{AB}\right)^2\right)\cos\left(\phi\right) - i\left(\left(\Sigma\_{11}^{AB}\right)^2 + \left(\Sigma\_{12}^{AB}\right)^2\right)\sin\left(\phi\right)\right)}, 1\right) \tag{14}$$

$$\chi\_2 = \left\{ \frac{ip\sin\left(\phi\right)\left(\Sigma\_{11}^{AB}\Sigma\_{21}^{AB} + \Sigma\_{12}^{AB}\Sigma\_{22}^{AB}\right) + p\cos\left(\phi\right)\left(\Sigma\_{12}^{AB}\Sigma\_{22}^{AB} - \Sigma\_{11}^{AB}\Sigma\_{21}^{AB}\right) + p\left(\Sigma\_{12}^{AB}\Sigma\_{21}^{AB} - \Sigma\_{11}^{AB}\Sigma\_{22}^{AB}\right)}{p\left(\left(\left(\Sigma\_{11}^{AB}\right)^2 - \left(\Sigma\_{12}^{AB}\right)^2\right)\cos\left(\phi\right) - i\left(\left(\Sigma\_{11}^{AB}\right)^2 + \left(\Sigma\_{12}^{AB}\right)^2\right)\sin\left(\phi\right)\right)}, 1\right) \tag{15}$$

In this way, we can calculate numerically the velocity operator in every *p*-point with subsequent its substitution to the conductivity integral.

#### **5.2 Integral calculations**

Contributions to conductivity include 2D integrals over the Brillouin zone (BZ). For example, the integrals in the ohmic contribution given by formulae (10) have the following form:

$$\sigma\_{\vec{y}}^{\text{inru}}\left(\boldsymbol{\mu},\vec{k}\right) = \sum\_{a=1,2} \frac{i\boldsymbol{\pi}^{2}\nu\_{0}^{2}}{\pi^{2}} \int\_{\mathbb{R}\mathbb{Z}} d^{2}\vec{\boldsymbol{p}} \frac{\left(\left[\nu\_{\text{ad}}^{i}\left(\vec{\boldsymbol{p}}\right)\boldsymbol{\nu}\_{\text{ad}}^{i}\left(\vec{\boldsymbol{p}}\right)\right] \boldsymbol{f}\left[\boldsymbol{\varepsilon}\_{1}\left(\vec{\boldsymbol{p}} - \vec{k}\,/2\right)\right] - \boldsymbol{f}\left[\boldsymbol{\varepsilon}\_{1}\left(\vec{\boldsymbol{p}} + \vec{k}\,/2\right)\right]\right)}{\left(\boldsymbol{\varepsilon}\_{1}\left(\vec{\boldsymbol{p}} + \vec{k}\,/2\right) - \boldsymbol{\varepsilon}\_{1}\left(\vec{\boldsymbol{p}} - \vec{k}\,/2\right)\right)\left(\boldsymbol{\alpha} - \boldsymbol{\varepsilon}\_{1}\left(\vec{\boldsymbol{p}} + \vec{k}\,/2\right) + \boldsymbol{\varepsilon}\_{1}\left(\vec{\boldsymbol{p}} - \vec{k}\,/2\right)\right)}. \tag{16}$$

*Anomalous Charge Transport Properties and Band Flattening in Graphene: A Quasi… DOI: http://dx.doi.org/10.5772/intechopen.106144*

$$
\sigma\_{\vec{\eta}}^{\text{inter}}\left(\boldsymbol{\alpha},\vec{k}\right) = \frac{2i\alpha\kappa^{2}\nu\_{0}^{2}}{\pi^{2}} \int\_{\text{BZ}} d^{2}\vec{p} \, \frac{\nu\_{12}^{\prime}\left(\vec{p}\right)\nu\_{21}^{\prime}\left(\vec{p}\right)\left(f\left[\epsilon\_{1}\left(\vec{p} - \vec{k}\,/2\right)\right] - f\left[\epsilon\_{2}\left(\vec{p} + \vec{k}\,/2\right)\right]\right)}{\left(\epsilon\_{2}\left(\vec{p} + \vec{k}\,/2\right) - \epsilon\_{1}\left(\vec{p} - \vec{k}\,/2\right)\right)\left(\alpha^{2} - \left(\epsilon\_{2}\left(\vec{p} + \vec{k}\,/2\right) - \epsilon\_{1}\left(\vec{p} - \vec{k}\,/2\right)\right)^{2}\right)} \tag{17}
$$

Here, the first integral is for the intraband transitions, and the second one is for the interband ones. Now, we highlight the pole structure for the integrands for small but finite *k* ! , accounting for that *ε*<sup>1</sup> *p* !� � ¼ �*ε*<sup>2</sup> *<sup>p</sup>* !� �. The first integral can be rewritten as:

$$\sigma\_{\vec{\eta}}^{\text{intra}}\left(\boldsymbol{\alpha},\vec{k}\right) = \sum\_{a=1,2} \frac{i\epsilon^{2}v\_{0}^{2}}{\pi^{2}} \int\_{\text{BZ}} d^{2}\vec{p} \left[v\_{aa}^{i}\left(\vec{p}\right)v\_{aa}^{i}\left(\vec{p}\right)\right] \frac{d\vec{f}\left(\boldsymbol{\varepsilon}\right)}{d\boldsymbol{\varepsilon}}\Big|\_{\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}\_{1}\left(\vec{p}\right)} \frac{1}{\left(\boldsymbol{\alpha} - \vec{k} \cdot \nabla\boldsymbol{\varepsilon}\_{1}\left(\vec{p}\right)\right)},\tag{18}$$

where we have performed the Taylor series expansion on ∣*k* ! ∣ up to linear terms. In the second integral, only the second term in the denominator can produce poles, so we expand it into a power series on ∣*k* ! ∣, making a change to polar coordinates *px*, *py* � � ! ð Þ *p*, *ϕ* that results:

$$
\sigma\_{\vec{\eta}}^{\text{inter}}\left(\boldsymbol{\omicron},\overrightarrow{\boldsymbol{k}}\right) = \frac{2i\boldsymbol{\alpha}^{2}\nu\_{0}^{2}}{\pi^{2}} \int\_{\text{BZ}} \boldsymbol{p}d\boldsymbol{p}d\boldsymbol{\phi} \frac{\nu\_{12}^{i}\left(\overrightarrow{\boldsymbol{p}}\right)\nu\_{21}^{i}\left(\overrightarrow{\boldsymbol{p}}\right)\left(\boldsymbol{f}\left[\boldsymbol{e}\_{1}\left(\overrightarrow{\boldsymbol{p}} - \overrightarrow{\boldsymbol{k}}/2\right)\right] - \boldsymbol{f}\left[\boldsymbol{e}\_{2}\left(\overrightarrow{\boldsymbol{p}} + \overrightarrow{\boldsymbol{k}}/2\right)\right]\right)}{\left(\boldsymbol{e}\_{2}\left(\overrightarrow{\boldsymbol{p}} + \overrightarrow{\boldsymbol{k}}/2\right) - \boldsymbol{e}\_{1}\left(\overrightarrow{\boldsymbol{p}} - \overrightarrow{\boldsymbol{k}}/2\right)\right)\left(\boldsymbol{\alpha}^{2} - 4\boldsymbol{p}^{2} - \boldsymbol{k}^{2}\sin^{2}\phi\right)}.\tag{19}
$$

Pole structure of (18) and (19) is presented in **Figure 3**. In accordance with Eq. (18) and **Figure 3** at *k* ! ¼ 0 this integral is a regular one, whereas at finite *k* there is a line of poles (dashed lines in the **Figure 3**). At a finite *k* the pole structure of *σinter ij ω*, *k* � �! (19) is an elliptic one that results in the necessity to account for an infinite sum of poles as a contribution to conductivity: *σinter ij ω*, *k* � �! ∝Ð *dϕ* Re sð Þ *ϕ* , where *Re s*ð Þ *ϕ* is a residue in the pole located at angle *ϕ* on the poles line. The integral (19) in the case *k* ¼ 0 holds poles laying at a circumference that can be effectively reduced to a single one as *σinter ij ω*, *k* � �! ∝2*π* Re s1, where Re s1 is a residue in arbitrary point of the circumference. For every oblate ellipse at large wave numbers, *k*, the main contribution to the integral (19) gives the only points touching the circumference *σinter ij ω*, *k* � �! � 2 Re s1. Thus, the value of the optical conductivity decreases with the growth of *k*.

We define the upper integration limit for the model with linear dispersion in a way based on reasoning from the energy limit of tight-binding-approximation applicability (*ω max* <1 eV ≈ 10<sup>4</sup> K). Then, the upper integration limit *q max* on momentum is *q max* � *ω max =vF*, *ω max* <1 eV, and correspondingly for the massless Dirac fermions model, the integration should use the range from 0 to ∣*q* !∣<0*:*14 ∣*K* ! *A*∣. As the simulation results presented in **Figure 4** show the integration within this limit leads to the conductivity fall in the range *ω max* � 4000 K (0.3 eV) at the temperature of 3 K.

Thus, the range of momenta to predict conductivity in near-infrared and visible spectral ranges is outside the limits of applicability of the massless pseudo-Dirac fermion model.

#### **Figure 3.**

*Sketch of the poles structure of integrands for intra-(dashed lines) and inter-(solid lines) bands contribution to conductivity. Solid lines parameters at ω* ¼ 1 *(in units of vF*∣*K* ! *<sup>A</sup>*∣*) are k=*∣*K* ! *<sup>A</sup>*∣ ¼ 0*:*1 *(red),* 0*:*7 *(green),* 0*:*9 *(blue). Dashed line parameters at ω* ¼ 0*:*2 *(in units of vF*∣*K* ! *<sup>A</sup>*∣*) are k=*∣*K* ! *<sup>A</sup>*∣ ¼ 0*:*21 *(red),* 0*:*4 *(green),* 0*:*9 *(blue).*

#### **Figure 4.**

*Frequency dependencies of the real and imaginary parts of the massless ohmic term of conductivity at very small wave number q* <sup>¼</sup> <sup>10</sup>�<sup>8</sup>∣*<sup>K</sup>* ! *<sup>A</sup>*<sup>∣</sup> *in relative units of e*<sup>2</sup>*=*ℏ*; a cutting parameter q max* <sup>¼</sup> <sup>0</sup>*:*<sup>14</sup> <sup>∣</sup>*<sup>K</sup>* ! *<sup>A</sup>*∣*. The model [9, 32] is simulated at T = 3 K, chemical potential μ* ¼ 135*K (red-dashed lines in figures (a–b)) and at T = 200 K, μ* ¼ 33 *K (red-dashed line in figure (c)). Numerical results for our model are green-solid and black-dashed lines for the approximation with zero gauge phases at T = 3 K, μ* ¼ 135*K and T = 200 K, μ* ¼ 33*K, respectively, and blue line for the approximation with nonzero gauge phases at T = 3 K, μ* ¼ 135*K in figures (a–b). (c) Dependence of the real and imaginary parts of the massless ohmic term of conductivity on the damping γ for the approximation of the nonzero gauge fields with γ* ¼ 0*:*1 *(green curve) and γ* ¼ 1 *(blue curve) at T = 200 K, μ* ¼ 33*K.*

*Anomalous Charge Transport Properties and Band Flattening in Graphene: A Quasi… DOI: http://dx.doi.org/10.5772/intechopen.106144*

According to our calculation presented in **Figure 1c**, the energy bandstructure is flattened for the pseudo-Majorana fermion graphene model. It signifies that the Fermi velocity *vM* for our model pseudo-Majorana states tends to zero for high wave numbers *q*. Since *vM* ! 0 for large *q* the applicability condition *vMq* <*ω max* holds always. We choose ∣*q* !∣ ≤0*:*38∣*K* ! *<sup>A</sup>*∣ as the limit of integration over momentum ∣*q* !∣ because it is in the range of ∣*q* !∣ from 0*:*28∣*K* ! *<sup>A</sup>*∣ to 0*:*56∣*K* ! *<sup>A</sup>*∣ appropriate for our model. Such choice corresponds to *ω max* >7000 K.

In the approximation with zero-phases of the gauge fields, the analytical formulas for the integrands in the conductivity contribution terms have been used. The integrals have been calculated with adaptive integration steps in both directions (∣*k*∣, *ϕ*) providing high calculation accuracy (not less than 0.01%).

In the approximation with nonzero gauge fields, we have to calculate numerically by introducing into consideration the small-positive damping constant *γ* for the states as a small imaginary contribution to the energies. The values of *γ* define the extent of smoothness of the singular behavior of the integrand and does not influence the general form of the dependency curve in accordance with **Figure 4c**.

All quantities necessary for the calculation of the complex conductivity have been calculated on a grid in the space of wave vectors with 200-point discretization in the angle *ϕ* for every given wave number *q* and variable step (a denser grid at small wave

numbers and larger at large ones) to the maximum wave number *q max* ¼ 0*:*38∣*K* ! *A*∣. The 2D interpolation on this grid has been used for integrands evaluation in the intermediate points that are necessary for conductivity simulations. An error stipulated by the interpolation from the grid in wave vectors space has been roughly estimated by interpolation of the conic spectrum of the Dirac pseudo-fermion model on the same lattice with subsequent usage of the interpolation data for evaluation of the conductivity. Its value turns out to be less than 10�3%.

Total estimation of the conductivity error has been performed by variation of the number of points used for interpolation of the energy band spectrum (by diminishing this number at factor two and subsequent comparison of the simulation results in both cases). It turns out to be not exceeding 10% in the considered frequency region. It should be noted that the error bars for values of the Fermi velocity that were measured by different techniques, including transport experiments (Shubnikov–de Haas oscillations) [34], infrared measurements of the Pauli blocking in graphene [35], magneto-optics [36], were also of the order of 10%.

## **6. Far-infrared optical spectroscopy of graphene**

In this section, we study optical transitions in the Majorana-like fermion graphene model and compare the theoretical predictions with experimental data for the farinfrared spectral range. The frequency dependencies of the real and imaginary part of the ohmic optical conductivity ( *k* ! <sup>¼</sup> 0) for temperatures *<sup>T</sup>* <sup>¼</sup> 3, 200 K are shown in **Figure 5**. Chemical potentials, *μ*, are used to be 135 and 33 K for *T* ¼ 3 and 200 K, respectively. For comparison, far-infrared conductivity is shown in the pseudo-Dirac fermion graphene model [9].

The conductivity in the far-infrared spectral range calculated within the massless pseudo-Dirac fermion graphene model gains a constant value that contradicts the experimental data.

**Figure 5.**

*Frequency dependencies of real (green and blue solid lines) and imaginary (yellow and magenta solid lines) parts of ohmic contribution σ<sup>O</sup> xx yy* ð Þ *to the far-infrared conductivity in relative units of e*<sup>2</sup>*=*<sup>ℏ</sup> *for our model (1) (green and magenta lines) and for the Dirac massless fermion model [9, 32] (blue and yellow lines). The calculations were carried out at temperatures T* ¼ 3 *(a), 200(b) K and chemical potentials μ* ¼ 135 *(a), 33(b) K for two case: Taking into account (left) and neglecting (right) the pseudo-Majorana mass term.*

The real part of the ohmic contribution in the optical conductivity for the pseudo-Majorana fermion graphene model undergoes a steep increase finalizing a much slower fall in the spectral range of 1000 � 2000 K (see **Figure 5**, left). The maximum conductivity is at the frequency of� 2000 K (0.13 eV). The predicted presence of the asymmetric spectral band in the optical conductivity for graphene is confirmed experimentally that the asymmetric peak is really observed on a frequency of � 0.15 eV (� <sup>150</sup> � 200 cm�<sup>1</sup> )[10].

The slow decrease of the peak intensity compared to the rise finalizes with steep achieving the plateau, which in turn ends with a shallow minimum in the remaining higher-frequency part of the far-infrared spectrum shown in **Figure 5**, left. Verifying the prediction of low (antipeak) being on the frequency of � 0*:*27 eV (3000 K) the experiment [12] confirms that the experimental antipeak is really indicated in the spectral range of � 0*:*2 � 0*:*3 eV.

*Anomalous Charge Transport Properties and Band Flattening in Graphene: A Quasi… DOI: http://dx.doi.org/10.5772/intechopen.106144*

The appearance of the peak is accompanied by the tendency of the imaginary part of the ohmic contribution in the optical conductivity toward negative values. It testifies that the position of the peak in the real part correlates with the emergence of plasmon oscillations. The exclusion of the pseudo-Majorana mass term results in the loss of the asymmetry of the far-infrared conductivity peak and a shift to the highfrequency region (see **Figure 5**, right). It testifies that chiral symmetry is restored, and correspondingly the anomalous transport disappears. According to results presented in **Figure 5**, the far-infrared peak and antipeak exist at both low and high temperatures. The influence of temperature consists in their smoothing, which impedes their experimental detection.

## **7. Electronic transport and field effect**

In this section, we study essential features of the electric charge transport by Majorana-like carriers in graphene and compare theoretical predictions with experimental data.

Total current *J* ! in graphene is determined by electron and hole currents of valleys *K*, *K*<sup>0</sup> as *J* ! ¼ *J* ! *<sup>K</sup>* � *J* ! *<sup>K</sup>*<sup>0</sup> . The two currents flow on non-coinciding paths in the topologically nontrivial pseudo-Magorana graphene model because the jump of an electron (hole) from a site of sublattice *A* (*B*) to the nearest site of sublattice *B* (*A*) is equivalent to bypassing the lattice site with the acquisition of the carrier wave function of the phase, being a multiple of the group generator *π*6 of the homotopy group **Z**<sup>12</sup> (**Figure 2**) in addition to 60-degree rotation by virtue of the homotopy group of graphene Brillouin zone and the hexagonal symmetry. It signifies that at transition from one trigonal sublattice to another one, the direction of motion of the charge ! !

carrier is rotated at an angle of *<sup>π</sup>* <sup>2</sup>. As a result, the *J <sup>K</sup>*, *J <sup>K</sup>*<sup>0</sup> are orthogonal.

Let us denote the first and the second terms in the conductivity *σ<sup>O</sup> ii* (see Eq. 10) through *σ<sup>o</sup> ii* and *σadd ii* , respectively: *σ<sup>O</sup> ii* <sup>¼</sup> *<sup>σ</sup><sup>o</sup> ii* <sup>þ</sup> *<sup>σ</sup>add ii* . *σadd ii* and *σ<sup>o</sup> ii* depend and do not depend on the Majorana-like mass term *MAB*, respectively. Then, taking into account of the polarization effects (Eq. 11) one can determine the pseudo-Majorana corrections, *σ tp ii* , *i* ¼ *x*, *y* to the conductivity in the following way: *σ tp ii* <sup>¼</sup> *<sup>σ</sup>add ii* <sup>þ</sup> *<sup>σ</sup>Zb ii* , *σ tp xx* ¼ �*σ tp yy*.

Then, in the absence of the Majorana conductivity corrections, we get the Ohm's law because of the direction of the sum, *J* !*o* ¼ *J* !*o <sup>K</sup>* � *J* !*o <sup>K</sup>*<sup>0</sup> of the currents, *J* !*o <sup>K</sup>*, *J* !*o <sup>K</sup>*0, equal to ℜ*e σ<sup>o</sup> iiE* ! *<sup>i</sup>*, *i* ¼ *x*, *y* coincides with the direction of an applied electric field *E* ! , *E* ! ¼ *E* ! *<sup>x</sup>* þ *E* ! *y* as **Figure 6** shows. Meanwhile, the direction which is orthogonal to *E* ! the current is absent. Taking into account the pseudo-Majorana conductivity corrections, the total current changes in the following way: *J* ! *skew* � *J* ! *<sup>K</sup>* � *J* ! *<sup>K</sup>*<sup>0</sup> ¼ *J* !*o* þ *σ tp xxE* ! *<sup>x</sup>* � *<sup>σ</sup>tp yyE* ! *<sup>y</sup>* � P<sup>2</sup> *<sup>i</sup>*¼<sup>1</sup>ð Þ �<sup>1</sup> *<sup>i</sup>*�<sup>1</sup> *<sup>σ</sup><sup>O</sup> ii* <sup>þ</sup> *<sup>σ</sup>Zb ii* � �*<sup>E</sup>* ! *<sup>i</sup>*. As **Figure 6** shows the *J* ! *skew* is rotated in respect to *E* ! , and correspondingly a nonzero component of the current appears in the direction orthogonal to *E* ! . It proves that a topological current can exist in the pseudo-Majorana graphene model and the abnormal transport appears due to the presence of the Majorana mass term.

**Figure 7** demonstrates negative differential conductivity for the topological current *J tp* <sup>¼</sup> <sup>ℜ</sup>*e<sup>σ</sup> tp ii* ð Þ *<sup>ω</sup> <sup>U</sup>* assuming the increase of the system energy in a form <sup>ℏ</sup>*<sup>ω</sup>* � *<sup>U</sup>*<sup>2</sup> , where *U* is a bias voltage.

A total spin-orbital valley current *J* ! *VHE* <sup>¼</sup> *<sup>σ</sup>so xyB* ! *<sup>y</sup>* � *<sup>σ</sup>so yxB* ! *<sup>x</sup>* is produced under an action of a magnetic field *B* ! <sup>∥</sup> parallel to *E* ! . One can note that *J* ! *VHE* is always directed tangentially to the bias *E* ! and added to the current *J* !*tp* . Then, the total current in the

#### **Figure 6.**

*Sketch of currents in the topologically nontrivial semimetal: J<sup>o</sup> x y*ð Þ *is a massless ohmic current along the axis X Y*ð Þ*, Jtp is a sum,* P<sup>2</sup> *<sup>i</sup>*¼<sup>1</sup> *<sup>σ</sup>add ii* <sup>þ</sup> *<sup>σ</sup>Zb ii* � �*<sup>E</sup>* ! *<sup>i</sup> of the polarization and dynamical-mass corrections to J<sup>o</sup> x y*ð Þ*. E* ! *is an applied electric field.*

#### **Figure 7.**

*Dependencies of topological current J on bias voltage U: Simulation results (red solid curve) and their fitting (blue solid curve) for negative differential conductivity (NDC) in our quasi-relativistic graphene model at temperature T = 3 K, chemical potential μ* ¼ 135 *K; black solid and green dashed curves present experimental data and theoretical calculation for NDC in two graphene flakes twisted approximately at 90° to each other at 1.8° misalignment angle [37]. The bias voltage U is given in volts "V."*

*Anomalous Charge Transport Properties and Band Flattening in Graphene: A Quasi… DOI: http://dx.doi.org/10.5772/intechopen.106144*

direction of the vector *E* ! increases. It signifies that a negative magnetoresistance (NMR) appears at weak magnetic fields parallel to electric ones. NMR is a specific feature of topological materials and presents a phenomenon of chiral anomaly [38, 39].

Let us investigate longitudinal conductivity for low frequencies *ω* ! 0 and nonvanishing wave vectors *k* ! ¼ *p* !*=*ℏ � *K* ! *A B*ð Þ. The longitudinal conductivity *σ<sup>L</sup> ω*, *k* � �! is determined through the conductivity tensor splitting into longitudinal and transversal terms as [40]

$$
\sigma\_{\vec{\eta}}(\boldsymbol{\omega}, \vec{k}) = \left(\delta\_{\vec{\eta}} - \frac{k\_i k\_j}{\overline{k}^2}\right) \sigma\_T\left(\boldsymbol{\omega}, \vec{k}\right) + \frac{k\_i k\_j}{\overline{k}^2} \sigma\_L\left(\boldsymbol{\omega}, \vec{k}\right); i, j = \infty, \mathbf{y}. \tag{20}
$$

when choosing *k* ! ¼ ð Þ *kx*, 0 , *k* ! ¼ 0, *ky* � �, or *kx* <sup>¼</sup> *ky* <sup>¼</sup> *<sup>k</sup>* one always has:

$$
\sigma\_{\infty}\left(\boldsymbol{\alpha}, \stackrel{\dashv}{k}\right) = \sigma\_L(\boldsymbol{\alpha}, k), \text{ or } \\
\sigma\_{\mathcal{\mathcal{V}}}\left(\boldsymbol{\alpha}, \stackrel{\dashv}{k}\right) = \sigma\_L(\boldsymbol{\alpha}, k).
$$

Now, let us calculate the low-frequency conductivity *σ ω*, *r* ! � �. To do it, one has to perform the inverse Fourier transformation.

$$\sigma\left(\boldsymbol{\alpha},\,\overrightarrow{\boldsymbol{r}}\right) = \frac{1}{\left(2\pi\right)^{2}} \left[\sigma\_{L}^{\boldsymbol{\rho}}(\boldsymbol{\alpha},\boldsymbol{k})\epsilon^{i\overrightarrow{\boldsymbol{k}}\cdot\overrightarrow{\boldsymbol{r}}}\,\mathrm{d}^{2}\boldsymbol{k}.\tag{21}$$

Here, *ω* is a cyclic frequency. We consider the effects of spatial dispersion on the real part ℜ*e σ<sup>O</sup> <sup>L</sup>* ð Þ *ω*, *k* of the longitudinal complex conductivity at the low frequencies: *<sup>ω</sup>* <sup>¼</sup> <sup>10</sup>�10, 0*:*004, 13.3 K (kelvin) for the massless pseudo-Dirac graphene fermion model with the number of flavors *NF* ¼ 2 (pseudospin and spirality) and our graphene model with the *NF* ¼ 3 flavors. Conductivity for frequencies in the Hertz range, for example, 2.08 Hz (*<sup>ω</sup>* <sup>¼</sup> <sup>10</sup>�<sup>10</sup> K), can be considered as a minimum conductance for direct current. The numerical results are presented in **Figure 8**.

The function ℜ*e σ<sup>o</sup> <sup>L</sup>*ð Þ *ω*, *k* for the massless pseudo-Dirac fermion graphene model is a positive constant function. The ℜ*eσ<sup>o</sup> <sup>L</sup>*ð Þ*<sup>k</sup>* � � *<sup>k</sup>*!<sup>∞</sup> � <sup>ℜ</sup>*<sup>e</sup> <sup>σ</sup><sup>k</sup>*<sup>∞</sup> is the positive constant (ℜ*<sup>e</sup> <sup>σ</sup><sup>k</sup>*<sup>∞</sup> <sup>&</sup>gt;0) for the all frequencies (it is equal to � <sup>0</sup>*:*004 for *<sup>ω</sup>* <sup>¼</sup> <sup>0</sup>*:*004, 10�<sup>10</sup> K and � 0*:*017 for *ω* ¼ 13*:*3 K. The real part of the longitudinal complex conductivity in the *NF* <sup>¼</sup> 2-model becomes the very large positive constant, <sup>ℜ</sup>*<sup>e</sup> <sup>σ</sup><sup>o</sup> <sup>L</sup>*ð Þ*<sup>k</sup>* � � *<sup>k</sup>*!<sup>0</sup> � <sup>ℜ</sup>*<sup>e</sup> <sup>σ</sup><sup>k</sup>*<sup>0</sup> � <sup>44</sup>*:*443, at small values of *<sup>k</sup>*, *<sup>k</sup>* <sup>≪</sup> 1 for the frequencies *<sup>ω</sup>* <sup>¼</sup> <sup>10</sup>�10, 0.004 K. As **Figure 8** shows the function ℜ*e σ<sup>o</sup> <sup>L</sup>*ð Þ *ω*, *k* in the *NF* ¼ 2-model sharply increases in a very narrow range of *k=KA*. Since ℜ*e σ<sup>o</sup> <sup>L</sup>*ð Þ *ω*, *k* is constant almost everywhere for the *NF* ¼ 2-model of graphene does not oscillate and *eik* ! �*r* ! enters the integrand of the expression (21), the ℜ*e σ ω*, *r* ! � � is equal to zero, and correspondingly the minimum conductivity is equal to zero in this graphene model. But this prediction for the DC case contradicts the experimental facts that the value of the low-frequency conductivity is in the range 4 *<sup>e</sup>*<sup>2</sup> *<sup>h</sup>* –<sup>6</sup> *<sup>e</sup>*<sup>2</sup> *h* .

The function ℜ*e σ<sup>o</sup> <sup>L</sup>*ð Þ *ω*, *k* for the pseudo-Majorana graphene *NF* ¼ 3-models both with and without the pseudo-Majorana mass term is a sign-alternating one at *k* ! ∞. It signifies that the plasmon oscillations can emerge in the graphene *NF* ¼ 3-models.

**Figure 8.**

*Real part σ<sup>o</sup> <sup>L</sup>*ð Þ *<sup>ω</sup>*, *<sup>k</sup> of the longitudinal ohmic contribution to conductivity vs. wave number k, k*! ¼ *p* ! � *K* ! *A B*ð Þ *for our pseudo-Majorana fermion NF* ¼ 3*-model (1) (solid curves) and for the massless Dirac fermion model [32] (dashed curves), at the frequencies ω:* 13*:*3 *K (*0*:*27 *THz, blue color),* 0*:*004 *K (*83 *MHz, green color),* 10�<sup>10</sup> *K (*2*:*08 *Hz, red color). The calculations were carried out at temperatures T* ¼ 100 *K and chemical potentials μ* ¼ 1 *K for two case: Taking into account (left) and neglecting (right) the pseudo-Majorana mass term. σ<sup>o</sup> <sup>L</sup>*ð Þ *ω*, *k is measured in the relative units of e*<sup>2</sup>*=h and labeled as* ℜ*e σ*13*:*3, ℜ*e σ*0*:*<sup>004</sup> *and* ℜ*e σdc for the frequencies 13.3,* 0*:*004*, and* 10�<sup>10</sup> *K, respectively.*

Neglecting the pseudo-Majorana mass term the function ℜ*e σ<sup>o</sup> <sup>L</sup>*ð Þ *ω*, *k* for the pseudo-Majorana fermion graphene *NF* ¼ 3-model is weakly oscillating for all frequencies (see **Figure 8**). Since ℜ*e σ<sup>o</sup> <sup>L</sup>*ð Þ *ω*, *k* for this case is practically constant except for a very narrow interval, then as well as for the *NF* ¼ 2 model, the ℜ*e σ ω*, *r* ! � � is equal to zero, and correspondingly the graphene model with the chiral pseudo-Majorana charge carriers is not semimetal.

Now let us examine the *NF* ¼ 3 model with the nonzero pseudo-Majorana mass term. In this case, the ℜ*e σ<sup>o</sup> <sup>L</sup>*ð Þ *ω*, *k* strongly oscillates for all frequencies (**Figure 8**). In this case for *<sup>ω</sup>* <sup>¼</sup> <sup>0</sup>*:*004, 10�<sup>10</sup> K, since the <sup>ℜ</sup>*<sup>e</sup> <sup>σ</sup><sup>o</sup> <sup>L</sup>*ð Þ *ω*, *k* has the large maximum (� <sup>44</sup>*:*<sup>55</sup> *<sup>e</sup>*<sup>2</sup> *<sup>h</sup>* ), and oscillating tends to zero it behaves like the function ℜ*e σ<sup>k</sup> max* sin ð Þ *k*�*k max* ð Þ *<sup>k</sup>*�*<sup>k</sup> max* . Such sort of functions can be considered as a finite approximation of the Dirac *<sup>δ</sup>*�function in the form of the sinc function lim *<sup>ε</sup>*!0<sup>þ</sup> sin *<sup>x</sup> <sup>ε</sup><sup>x</sup>* , and the coefficient ℜ*e σ<sup>k</sup> max* is called the intensity or spectral power of the *δ*�function. In the DC limit (*<sup>ω</sup>* <sup>¼</sup> <sup>10</sup>�<sup>10</sup> K), the <sup>ℜ</sup>*<sup>e</sup> <sup>σ</sup><sup>o</sup> <sup>L</sup>*ð Þ *ω*, *k* possesses one maximum at *k* ! 0. Correspondingly, the DC minimum conductivity *σdc r* !� � for the *NF* <sup>¼</sup> 3 model with the chiral anomaly may be approximated by the integral with only one Dirac *δ*-function entering the integrand:

$$\sigma^{dc}\left(\boldsymbol{\alpha},\,\overrightarrow{\boldsymbol{r}}\right) = \frac{1}{\left(2\pi\right)^{2}} \int \mathfrak{Re}\sigma\_{L}^{\rho}(\boldsymbol{\alpha},\boldsymbol{k}) e^{i\vec{k}\cdot\overrightarrow{\boldsymbol{r}}} \,\mathrm{d}^{2}\boldsymbol{k} \approx \frac{1}{2\pi} \int \mathfrak{Re}\sigma\_{L}^{\rho}(\boldsymbol{\alpha},\boldsymbol{0}) \delta(\boldsymbol{k}) e^{i\vec{k}\cdot\overrightarrow{\boldsymbol{r}}} \,\mathrm{d}\boldsymbol{k} = \frac{7\epsilon^{2}}{\hbar}, \ \boldsymbol{\alpha} = 10^{-10} \,\mathrm{K}. \tag{22}$$

The minimum DC-conductivity of graphene in devices with a large area of a graphene monolayer on SiO2 turns out to be of 4*e*<sup>2</sup>*=<sup>h</sup>* [4] at low temperatures (� <sup>1</sup>*:*<sup>5</sup> K). The minimum DC-conductivity of suspended graphene [5, 6] and of graphene on boron nitride substrate [41] is of � <sup>6</sup>*e*<sup>2</sup>*=<sup>h</sup>* at *<sup>T</sup>* � 300 K. Since for *<sup>ω</sup>* <sup>¼</sup> <sup>0</sup>*:*004 K one obtains the same value of the conductivity the low-frequency conductivity behaves in *Anomalous Charge Transport Properties and Band Flattening in Graphene: A Quasi… DOI: http://dx.doi.org/10.5772/intechopen.106144*

a universal manner. Thus, our estimate (22) is in perfect agreement with the experimental data.
