*5.1.4. The results of the numerical calculation*

In view of the property *TrT* = *TrTd* and in terms of the notation *kx* = 2*πn*/*L* (−*π*/*d < kx* ≤

Taking into account the last relation in Eqs. (89), Eq. (90) can also be written in the form

Dispersion relation (90) under condition (84) gives the equation [86]

*ReT*<sup>11</sup> = cos(*kxd*).

According to this equation, the passage to the single-band limit is performed by two methods: first, *V*<sup>0</sup> = ∆<sup>0</sup> (QW only for electrons) and, second, *V*<sup>0</sup> = −∆<sup>0</sup> (QW only for holes). The result of the passage coincides with the known nonrelativistic dispersion relation (see, e.g., [87]),

If inequality (86) is satisfied, the change *k*<sup>2</sup> → *i*κ<sup>2</sup> should be made in Eq. (91)

For Tamm minibands, the change *k*<sup>1</sup> → *i*κ<sup>1</sup> should be made in Eq. (91):

*TrT* = 2 cos(*kxd*). (90)

sinh(*k*2*dI I*) sin(*k*1*dI*) + cosh(*k*2*dI I*) cos(*k*1*dI*) = cos(*kxd*). (91)

sin(κ2*dI I*) sin(*k*1*dI*) + cos(κ2*dI I*) cos(*k*1*dI*) = cos(*kxd*). (92)

sinh(*k*2*dI I*) sinh(κ1*dI*) + cosh(*k*2*dI I*) cosh(κ1*dI*) = cos(*kxd*). (93)

<sup>0</sup> <sup>−</sup> <sup>∆</sup><sup>2</sup>

<sup>0</sup> *<* 0.

*<sup>y</sup>* <sup>−</sup> *<sup>E</sup>*<sup>2</sup> *<sup>&</sup>lt;* <sup>−</sup>*EV*0. (94)

*π*/*d*), the dispersion relation is obtained in the form

<sup>0</sup> <sup>−</sup> <sup>∆</sup><sup>2</sup> 0

although the expressions for *k*1, *k*2, and *E* are different.

<sup>0</sup> <sup>−</sup> <sup>∆</sup><sup>2</sup> 0

<sup>0</sup> <sup>−</sup> <sup>∆</sup><sup>2</sup> 0

Equation (93) has the solution only under the condition

*v*2 *Fk*2 <sup>2</sup> <sup>+</sup> *<sup>v</sup>*<sup>2</sup> *F*κ<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>V</sup>*<sup>2</sup>

> *v*2 *Fk*2

*v*2 *Fk*2 <sup>2</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *Fk*2 <sup>1</sup> <sup>+</sup> *<sup>V</sup>*<sup>2</sup>

<sup>−</sup>*v*<sup>2</sup> *F*κ<sup>2</sup> <sup>2</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *Fk*2 <sup>1</sup> <sup>+</sup> *<sup>V</sup>*<sup>2</sup>

*v*2 *Fk*2 <sup>2</sup> <sup>+</sup> *<sup>v</sup>*<sup>2</sup> *F*κ<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>V</sup>*<sup>2</sup>

2*v*<sup>2</sup> *<sup>F</sup>k*1*k*<sup>2</sup>

222 Graphene - New Trends and Developments

2*v*<sup>2</sup> *<sup>F</sup>k*1κ<sup>2</sup>

2*v*<sup>2</sup> *<sup>F</sup>*κ1*k*<sup>2</sup>

This condition can be rewritten as

The numerical calculations of the dependence of the energy on *kx* were performed for two values *ky* = 0 and 0.1 nm−<sup>1</sup> at *V*<sup>0</sup> = 0 (see Fig. 19). The energy of carriers is assumed to be low, |*E*| 1 eV, because the Dirac dispersion relation for carriers and, correspondingly, Dirac equation (81) are invalid for high energies.

The electron minibands are separated from hole minibands by a band gap, which increases with |*ky*|. For *dI* = *dI I* at *ky* = 0, it is *Eg* 10–30 meV when d=10–100 nm. In this case, the solution of Eq. (91) is transformed to the solution of Eq. (92). The band gap can increase strongly when *dI I* increases with respect to *dI*: *Eg* 100 meV, i.e., is several times larger than 2∆0.

The width of the minibands decreases with an increase in the period of the superlattice *d*. The dependence of the width of the minibands on *V*<sup>0</sup> was also examined. The widths of the electron and hole minibands increase and decrease, respectively, at *V*<sup>0</sup> *>* 0 and vice versa at *V*<sup>0</sup> *<* 0.

## *5.1.5. The possible applications of SL*

The described superlattice can be used as FET where the substrate serves as a gate. The ratio of the current through the superlattice to the current through the gate at a substrate thickness of about 10 nm can reach <sup>∼</sup> 106 as for FET based on graphene nanoribbons [11]. The main advantage of the considered superlattice is the absence of the effect of the scattering of carriers on the edges of a nanoribbon on their mobility. The mobility of the carriers in gapless graphene reaches *<sup>µ</sup>*<sup>0</sup> <sup>=</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup><sup>5</sup> cm2/(V s) [4, 5]. However, the mobility of carriers in FET based on the graphene nanoribbon with a width of *w* ∼ 3 nm is three orders of magnitude smaller than *µ*0. The cause of such a strong decrease is possibly the scattering of carriers at the edges of graphene nanoribbons. The mean free path between two acts of the scattering of carriers at the edge of the graphene nanoribbon *λedge* ∝ *w*/*P*, where *P* is the probability of backscattering [11]. For sufficiently good edges, P1. The problem of scattering on edges is absent for the proposed superlattice; for this reason, the mobility of the carriers in the superlattice is expected to be ∼ *µ*<sup>0</sup> in the absence of the problems with the periodicity of the potential. At the same time, a sufficiently large *Eg* value, which provides the operation of FET at room temperature, can be reached.

If an Au film is deposited on the lower side of the substrate and graphene is optically pumped, the superlattice can be used as a terahertz laser similar to a terahertz laser based

**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 the SiO2 substrate.
