*5.2.1. Preliminary remarks*

Now, we suggest to consider SL based on gapless graphene with alternating regions characterized by different values of the Fermi velocity [90]. In our case, the *Fermi velocity engineering* is based on the usage of the surrounding graphene materials, which have different values of permittivity [91]. It should be pointed out that the idea to control the Coulomb interaction between charge carriers in graphene by the choice of substrate materials with the necessary values of dc permittivity was first put forward in [92].

In such heterostructures, it is possible to achieve the energy quantization for charge carriers even in the absence of potential barriers (regions with wider band gaps) and QWs (regions with narrower band gaps), and even without any variations in the work function [25]. Note that the Tamm minibands are absent here since the straight dispersion lines do not intersect anywhere except for the Dirac point.

Such structure can be produced by the deposition of graphene on striped substrates where either the composition parameter *x* in an alloy of SiO2−*x*, or the density of some

**Figure 20.** Three variants of SL under study: **(a)** graphene sheet placed on a striped substrate consisting of alternating layers of materials with substantially different values of the permittivity, e.g., SiO2 with *ε* = 3.9 (I) and HfO2 with *ε* = 25 (II); **(b)** graphene sheet placed on the HfO2 substrate with periodically arranged grooves; and **(c)** graphene sheet deposited on a periodic array of parallel metallic strips. A plate of heavily doped silicon *n*-Si is used as a gate.

(nonmagnetic) impurities, or dc permittivity *ε* exhibit periodic variations. Here, we treat in detail the latter possibility.

According to the results of the theoretical [93–97] and experimental [91, 98–102] studies, the Fermi velocity becomes substantially renormalized. To estimate the renormalized Fermi velocity, we can use the relation [95]

**Figure 19.** Numerically calculated dependence of the energy on *kx* for two *ky* values and two superlattice periods d. The dispersion curves for the superlattices with *dI* = (solid lines) *dI I*, (dashed lines) *dI I*/2, and (dotted lines) 2*dI I*.

on gapless graphene [89]. In this case, terahertz radiation will be emitted from the regions of

**5.2. Superlattice based on gapless graphene with the alternating Fermi velocity**

Now, we suggest to consider SL based on gapless graphene with alternating regions characterized by different values of the Fermi velocity [90]. In our case, the *Fermi velocity engineering* is based on the usage of the surrounding graphene materials, which have different values of permittivity [91]. It should be pointed out that the idea to control the Coulomb interaction between charge carriers in graphene by the choice of substrate materials with the

In such heterostructures, it is possible to achieve the energy quantization for charge carriers even in the absence of potential barriers (regions with wider band gaps) and QWs (regions with narrower band gaps), and even without any variations in the work function [25]. Note that the Tamm minibands are absent here since the straight dispersion lines do not intersect

Such structure can be produced by the deposition of graphene on striped substrates where either the composition parameter *x* in an alloy of SiO2−*x*, or the density of some

necessary values of dc permittivity was first put forward in [92].

the SiO2 substrate.

*5.2.1. Preliminary remarks*

224 Graphene - New Trends and Developments

anywhere except for the Dirac point.

$$\frac{v\_F}{v\_{F0}} = 1 - 3.28\alpha^\* \left[ 1 + \frac{1}{4} \ln \left( 1 + \frac{1}{4\alpha^\*} - 1.45 \right) \right] \text{.}$$

where *<sup>α</sup>*<sup>∗</sup> <sup>=</sup> *e*2/¯*hvF*<sup>0</sup> is the analog of the fine structure constant, *vF*<sup>0</sup> is the initial unrenormalized Fermi velocity (*vF*<sup>0</sup> <sup>=</sup> 0.85 <sup>×</sup> <sup>10</sup><sup>8</sup> cm/s) [91, 101], *e*<sup>2</sup> <sup>=</sup> *<sup>e</sup>*2/*εeff* , and *<sup>ε</sup>eff* <sup>=</sup> (*ε*<sup>1</sup> + *ε*2)/2 is the effective dc permittivity for the charge carriers in graphene depending on the values *ε*<sup>1</sup> and *ε*<sup>2</sup> of dc permittivity characterizing the materials surrounding graphene. Note that here the band gap is not open; this is confirmed in experiment with an accuracy of 0.1 meV [101].

Within the graphene region located over the strip with the lower value of *ε*, we have larger *α*∗. Hence, the corresponding renormalized Fermi velocity should be higher than that over the strip with the higher value of *ε*. This suggests the possibility of modulating *vF* by varying the substrate permittivity. Note that such a system is a one-dimensional photonic crystal.

The first version of the suggested system is a graphene sheet placed on a striped substrate consisting of alternating layers of materials with substantially different values of the permittivity. A schematic image of such a system is shown in Fig. 20a.

It is also possible to use a substrate with periodically arranged grooves prepared by etching. The graphene sheet placed on such substrate should have the periodically alternating regions

**Figure 21.** Fermi velocity profile in SL under study (*vF*<sup>1</sup> *> vF*<sup>2</sup> case). The enumeration of supercells in SL and the sizes of its regions are indicated in the lower part of the figure: *dI* is the width of the graphene strip with the Fermi velocity *vF*1, *dI I* is the width of the graphene strip with the Fermi velocity *vF*2, and *d* = *dI* + *dI I* is the SL period.

suspended over the grooves and those being in contact with the substrate material (see Fig. 20b). The renormalization of the Fermi velocity should be the most clearly pronounced just in the suspended graphene regions since here we have *εeff* = 1. According to the experimental data, the renormalized Fermi velocity in suspended graphene increases to 3 <sup>×</sup> 108 cm/s [101].

In the regions with graphene in contact with the narrow gap semiconducting material, where *εeff* 1, the renormalized Fermi velocity differs only slightly from the unrenormalized one. In addition, the substrate itself is a diffraction grating. Therefore, the system should exhibit rather interesting optical characteristics, demanding a separate study.

There is another version of the system under study. It is possible to deposit graphene on a periodic array of parallel metallic strips (Fig. 20c). This is the limiting case: in the suspended graphene regions, we have *εeff* = 1 (the strongest renormalization of the Fermi velocity), whereas in the regions with graphene in contact with metallic strips, we have *εeff* = ∞ (vanishing renormalization of the Fermi velocity [91]).

We see that a whole class of such type of systems, which were not studied earlier, is possible. Without doubt, the studies of such systems should lead to important advances in the implementation of the technologies based on the controlled Fermi velocity.
