*5.1.1. Some remarks*

Interest in graphene-based superlattices (SL) has increased in recent years. Calculations of graphene-based SL with periodic rows of vacancies were performed using the molecular dynamics method [70]. Calculations of single-atom-thick SL formed by lines of pairs of adsorbed hydrogen atoms on graphene were carried out with the density functional theory [71].

Rippled graphene that can be treated as a SL with the one-dimensional periodic potential of ripples was investigated in [72–74]. SL obtained when a periodic electrostatic potential [75–78] or periodically located magnetic barriers [79–82] were applied to graphene were analytically examined.

However, the investigation of the graphene-based SL with a periodic electrostatic potential disregarded the fact that the application of the electrostatic potential to a gapless semiconductor (graphene) results in the production of electron-hole pairs and the redistribution of charges: electrons move from the region where the top of the valence band lies above the Fermi level to the region where the bottom of the conduction band lies

**Figure 17.** Graphene on the strip substrate consisting of alternating SiO2 and h-BN strips.

Finally, interface states can manifest themselves in the I-V curve of the planar heterostructure carrying a current parallel to the gapless graphene nanoribbon. An increase in applied electric field may cause charge carriers to "drop" into interface states (preferable energy-wise), giving rise to a region of negative differential conductivity in the I-V curve.

Before proceeding to the next section, let us recall some results obtained in this section. We

We have analyzed the characteristics of planar graphene nanostructures. On the one hand, they retain the unique properties of infinite graphene sheets. On the other hand, bandgap opening makes them important building blocks in carbon-based nanoelectronics, which can be used to control electron motion. Parameters of graphene QWs can easily be manipulated by varying the gapless nanoribbon width or the potential barriers in the adjoining gapped

We predict pseudospin splitting to occur in asymmetric graphene QWs and interface states to arise from the crossing of dispersion curves of gapless and gapped graphene materials. We have performed calculations of optical properties of planar graphene nanostructures and

Analysis of pseudospin (valley) characteristics in the heterostructure is simplified by using an effective Hamiltonian having a pseudospin-split energy spectrum. Note that an analogous spectrum was discussed in [63–65]. Therefore, the effective Hamiltonian must contain a Rashba-like spin-orbit coupling. We have developed the effective theory for describing

Interest in graphene-based superlattices (SL) has increased in recent years. Calculations of graphene-based SL with periodic rows of vacancies were performed using the molecular dynamics method [70]. Calculations of single-atom-thick SL formed by lines of pairs of adsorbed hydrogen atoms on graphene were carried out with the density functional theory

Rippled graphene that can be treated as a SL with the one-dimensional periodic potential of ripples was investigated in [72–74]. SL obtained when a periodic electrostatic potential [75–78] or periodically located magnetic barriers [79–82] were applied to graphene were

However, the investigation of the graphene-based SL with a periodic electrostatic potential disregarded the fact that the application of the electrostatic potential to a gapless semiconductor (graphene) results in the production of electron-hole pairs and the redistribution of charges: electrons move from the region where the top of the valence band lies above the Fermi level to the region where the bottom of the conduction band lies

**4.8. Some intermediate conclusions**

216 Graphene - New Trends and Developments

graphene sheets.

*5.1.1. Some remarks*

analytically examined.

[71].

think that it is important for understanding the next results.

suggested possible experiments to study the effects in question.

**5.1. Superlattice based on graphene on a strip substrate**

graphene-based systems with the pseudospin splitting.

**5. Planar graphene superlattices**

below the Fermi level. The SL becomes a structure consisting of positively charged regions, where the electrostatic potential displacing the Dirac points upward in energy is applied, alternating with negatively charged regions. The strong electrostatic potential of induced charges appears and strongly distorts the initial step electrostatic potential and, therefore, the electronic structure of SL calculated disregarding the electrostatic potential of induced charges.

To avoid the production of electron-hole pairs, SL appearing due to the periodic modulation of the band gap is considered.

SL in the form of the periodic planar heterostructure of graphene nanoribbons between which nanoribbons of h-BN are inserted was previously proposed in [83]. The band structure of such SL was numerically calculated. However, it is very difficult to implement this SL even using the advances of modern lithography, because problems inevitably arise with the control of periodicity in the process of the etching of nanoribbons in a graphene sheet and the insertion of h-BN nanoribbons. Moreover, h-BN is an insulator with a band gap of 5.97 eV, which significantly hinders the tunneling of carriers between graphene nanoribbons. Such a heterostructure is most probably a set of QWs where the wavefunctions of carriers from neighboring QWs almost do not overlap.

Here, SL formed by a graphene sheet deposited on a strip substrate is proposed. The strip substrate is made of periodically alternating strips of SiO2 (or any other material that does not affect the band structure of graphene) and h-BN, as shown in Fig. 17. The h-BN layers are located so that its hexagonal crystal lattice is under the hexagonal crystal lattice of graphene. Owing to this location, a band gap of 53 meV appears in the band structure of graphene in the graphene-sheet regions under the h-BN layers [37, 84].

It is assumed that all of the contacts between the regions of different band gaps are first-kind contacts (Dirac points of graphene are located in the band gaps of the gap modification of graphene). Such SL is a first-type SL (classification of SL can be found, e.g., in [85]).

The main advantage of the proposed SL is the simplicity of the manufacture and control of its periodicity. It is worth noting that some problems can arise in SL. The difference between the lattice constants of h-BN and graphene is about 2% [37]. If about 100 hexagonal graphene cells are packed into one period of the superlattice, the formation of the band gap in the gap modification of graphene in the graphene sheet regions above h-BN is violated owing to the inaccurate arrangement of carbon atoms above boron or nitrogen atoms. Since contacts between graphene and its gap modification are not heterocontacts (contacts between substances with different chemical compositions), the edges of quantum wells can be insufficiently sharp and QWs cannot be considered as square QWs. A transient

**Figure 18.** One-dimensional periodic Kronig-Penney potential of SL shown in Fig. 17: the periodically alternating gap modification of graphene on h-BN with a band gap of 2∆<sup>0</sup> = 53 meV and gapless graphene on SiO2.

layer with a spatially varying band gap can exist instead of the sharp edge. Finally, the substrate can be stressed. The appearing periodic stress field of the substrate can also affect the band structure of the proposed SL, but this effect is very small [86].

#### *5.1.2. The model for describing SL*

The *x* and *y* axes are perpendicular and parallel to the interfaces of h-BN and SiO2 strips, respectively (see Fig. 16). SL is described by the Dirac equation

$$\left(\left(\upsilon\_F \sigma \hat{\mathbf{p}} + \Delta \sigma\_z + V\right) \Psi(x \, y) = E \Psi(x, y)\_\prime \tag{81}$$

where *vF* <sup>≈</sup> <sup>10</sup><sup>8</sup> cm/s is the Fermi velocity, <sup>σ</sup> = (*σx*, *<sup>σ</sup>y*) and *<sup>σ</sup><sup>z</sup>* are the Pauli matrices, and **<sup>p</sup>** <sup>=</sup> <sup>−</sup>*i*<sup>∇</sup> is the momentum operator (the system of units with ¯*<sup>h</sup>* = 1 is used). The half-width of the band gap is periodically modulated:

$$\Delta = \begin{cases} 0, & d(n-1) < \infty < -d\_{II} + dn\_{\prime} \\ \Delta\_{0\prime} & -d\_{II} + dn < \infty < dn\_{\prime} \end{cases}$$

where *n* is an integer enumerating the supercells of the superlattice; *dI* and *dI I* are the widths of the SiO2 and h-BN strips, respectively; and *d* = *dI* + *dI I* are the period of the superlattice (the size of the supercell along the x axis). The periodic scalar potential *V* can appear due to the difference between the energy positions of the middle of the band gap of the gap modification of graphene and conic points of the Brillouin zone of gapless graphene (see Fig. 18):

$$V = \begin{cases} 0, & d(n-1) < \infty < -d\_{II} + dn\_{\star} \\ V\_{0\nu} & -d\_{II} + dn < \infty < dn. \end{cases}$$

In order for SL to be a first-type superlattice, the inequality |*V*0| ≤ 0 should be satisfied. The solution of Eq. (81) for the first supercell has the form

$$\Psi(\mathfrak{x}, y) = \psi\_1(\mathfrak{x}) e^{ik\_y y}, \quad 0 < \mathfrak{x} < d.$$

For the *n*th supercell, in view of the periodicity of the superlattice,

$$
\psi\_n(\mathbf{x}) = \psi\_1(\mathbf{x} + (n-1)d).
$$

In the region of QW (0 *< x < dI*), the solution of Eq. (81) is a plane wave

$$\psi\_n^{(1)}(\mathbf{x}) = \mathcal{N}\_{k\_1} \binom{a\_n^{(1)}}{b\_n^{(1)}} e^{i k\_1 \mathbf{x}} + \mathcal{N}\_{k\_1} \binom{c\_n^{(1)}}{d\_n^{(1)}} e^{-i k\_1 \mathbf{x}} \tag{82}$$

where *Nk*<sup>1</sup> is the normalization factor. The substitution of Eq. (82) into Eq. (81) provides the relation between the lower and upper spinor components

$$d\_n^{(1)} = \lambda\_+ a\_n^{(1)}, \quad d\_n^{(1)} = -\lambda\_- c\_n^{(1)}, \quad \lambda\_\pm = \frac{v\_F(k\_1 \pm ik\_y)}{E}.$$

The relation of *E* with *k*<sup>1</sup> and *ky* has the form

$$E = \pm \upsilon\_F \sqrt{k\_1^2 + k\_y^2}.$$

It is convenient to represent Eq. (82) in a more compact form [76]

$$\begin{aligned} \psi\_n^{(1)}(\mathbf{x}) &= \Omega\_{k\_1}(\mathbf{x}) \begin{pmatrix} a\_n^{(1)} \\ c\_n^{(1)} \end{pmatrix}, \\ \Omega\_{k\_1}(\mathbf{x}) &= \mathcal{N}\_{k\_1} \begin{pmatrix} 1 & 1 \\ \lambda\_+ & -\lambda\_- \end{pmatrix} e^{ik\_1 \mathbf{x} \sigma\_z}. \end{aligned} \tag{83}$$

When the inequality

**Figure 18.** One-dimensional periodic Kronig-Penney potential of SL shown in Fig. 17: the periodically alternating gap

layer with a spatially varying band gap can exist instead of the sharp edge. Finally, the substrate can be stressed. The appearing periodic stress field of the substrate can also affect

The *x* and *y* axes are perpendicular and parallel to the interfaces of h-BN and SiO2 strips,

where *vF* <sup>≈</sup> <sup>10</sup><sup>8</sup> cm/s is the Fermi velocity, <sup>σ</sup> = (*σx*, *<sup>σ</sup>y*) and *<sup>σ</sup><sup>z</sup>* are the Pauli matrices, and **<sup>p</sup>** <sup>=</sup> <sup>−</sup>*i*<sup>∇</sup> is the momentum operator (the system of units with ¯*<sup>h</sup>* = 1 is used). The half-width

where *n* is an integer enumerating the supercells of the superlattice; *dI* and *dI I* are the widths of the SiO2 and h-BN strips, respectively; and *d* = *dI* + *dI I* are the period of the superlattice (the size of the supercell along the x axis). The periodic scalar potential *V* can appear due to the difference between the energy positions of the middle of the band gap of the gap modification of graphene and conic points of the Brillouin zone of gapless graphene (see Fig.

0, *d*(*n* − 1) *< x <* −*dI I* + *dn*, ∆0, −*dI I* + *dn < x < dn*,

0, *d*(*n* − 1) *< x <* −*dI I* + *dn*, *V*0, −*dI I* + *dn < x < dn*.

In order for SL to be a first-type superlattice, the inequality |*V*0| ≤ 0 should be satisfied. The

(*vF*σ**<sup>p</sup>** <sup>+</sup> <sup>∆</sup>*σ<sup>z</sup>* <sup>+</sup> *<sup>V</sup>*) <sup>Ψ</sup>(*x y*) = *<sup>E</sup>*Ψ(*x*, *<sup>y</sup>*), (81)

modification of graphene on h-BN with a band gap of 2∆<sup>0</sup> = 53 meV and gapless graphene on SiO2.

the band structure of the proposed SL, but this effect is very small [86].

respectively (see Fig. 16). SL is described by the Dirac equation

∆ = 

*V* = 

solution of Eq. (81) for the first supercell has the form

*5.1.2. The model for describing SL*

218 Graphene - New Trends and Developments

18):

of the band gap is periodically modulated:

$$
\Delta\_0^2 + v\_F^2 k\_y^2 - (E - V\_0)^2 \ge 0 \tag{84}
$$

is satisfied, the solution of Eq. (81) in the barrier region (*dI < x < d*) has the form

$$\begin{aligned} \psi\_n^{(2)}(\mathbf{x}) &= \Omega\_{k\_2}(\mathbf{x}) \begin{pmatrix} a\_n^{(2)} \\ c\_n^{(2)} \end{pmatrix}, \\ \Omega\_{k\_2}(\mathbf{x}) &= \mathcal{N}\_{k\_2} \begin{pmatrix} 1 & 1 \\ -\widetilde{\lambda}\_- \ \widetilde{\lambda}\_+ \end{pmatrix} e^{k\_2 \mathbf{x} \sigma\_z} \end{aligned} \tag{85}$$

where

$$
\widetilde{\lambda}\_{\pm} = \frac{i v\_F (k\_2 \pm k\_y)}{E + \Delta\_0 - V\_0},
\\
k\_2 = \frac{1}{v\_F} \sqrt{\Delta\_0^2 + v\_F^2 k\_y^2 - (E - V\_0)^2}.
$$

The solution of Eq. (81) in the barrier region under the condition

$$
\Delta\_0^2 + v\_F^2 k\_y^2 - (E - V\_0)^2 < 0 \tag{86}
$$

is given by Eq. (85) with the change *k*<sup>2</sup> → *i*κ2, i.e., it is oscillating.

The possibility of existing *Tamm* minibands formed by localized states near the interface between graphene and its gap modification will be considered below. In this case, *k*<sup>1</sup> → *i*κ<sup>1</sup> *k*<sup>2</sup> is real. A necessary condition for existing Tamm states has the form

$$|k\_{\mathcal{Y}}| \ge |\varkappa\_1|;$$

under this condition, the energy *E* = ±*vF k*2 *<sup>y</sup>* <sup>−</sup> <sup>κ</sup><sup>2</sup> <sup>1</sup> is real.

#### *5.1.3. The derivation of the dispersion relation*

The dispersion relation is derived using the transfer matrix (*T* matrix) method. The *T* matrix relates the spinor components for the *n*th supercell to the spinor components of the solution of the same type for the (*n* + 1)th supercell. For example, for the solution in the quantum well region,

$$
\begin{pmatrix} a\_{n+1}^{(1)} \\ \boldsymbol{\varsigma}\_{n+1}^{(1)} \end{pmatrix} = \boldsymbol{T} \begin{pmatrix} a\_n^{(1)} \\ \boldsymbol{\varsigma}\_n^{(1)} \end{pmatrix}. \tag{87}
$$

To determine the *T* matrix, the following conditions of the continuity of the solution of the Dirac equation describing the considered superlattice are used:

$$
\begin{aligned}
\psi\_n^{(1)}(d\_I - 0) &= \psi\_n^{(2)}(d\_I + 0), \\
\psi\_n^{(2)}(d - 0) &= \psi\_{n+1}^{(1)}(+0).
\end{aligned}
$$

These conditions provide the equalities

$$
\begin{aligned}
\binom{a\_n^{(2)}}{c\_n^{(2)}} &= \Omega\_{k\_2}^{-1}(d\_I)\Omega\_{k\_1}(d\_I) \binom{a\_n^{(1)}}{c\_n^{(1)}}, \\
\binom{a\_{n+1}^{(1)}}{c\_{n+1}^{(1)}} &= \Omega\_{k\_1}^{-1}(0)\Omega\_{k\_2}(d) \binom{a\_n^{(2)}}{c\_n^{(2)}}.
\end{aligned}
$$

According to definition (87) of the *T* matrix and the last two equalities6,

$$T = \Omega\_{k\_1}^{-1}(0)\Omega\_{k\_2}(d)\Omega\_{k\_2}^{-1}(d\_I)\Omega\_{k\_1}(d\_I). \tag{88}$$

The substitution of Eqs. (84) and (85) with the corresponding arguments into Eq. (88) yields the expressions

$$T\_{11} = a e^{ik\_{\mathrm{I}}d\_{\mathrm{I}}} \left[ (\lambda\_{-} + \widetilde{\lambda}\_{+})(\lambda\_{+} + \widetilde{\lambda}\_{-})e^{-k\_{2}d\_{\mathrm{II}}} - \right.$$

$$-(\lambda\_{-} - \widetilde{\lambda}\_{-})(\lambda\_{+} - \widetilde{\lambda}\_{+})e^{k\_{2}d\_{\mathrm{II}}} \Big|\_{\prime} \, \prime \, \prime \, \tag{89}$$

$$T\_{12} = 2ae^{-ik\_{\mathrm{I}}d\_{\mathrm{I}}}(\lambda\_{-} + \widetilde{\lambda}\_{+})(\lambda\_{-} - \widetilde{\lambda}\_{-})sh(k\_{2}d\_{\mathrm{II}})\_{\prime} \, \prime \, \} $$

$$T\_{21} = T\_{12}^{\*} \quad T\_{22} = T\_{11}^{\*} \, \prime $$

where

where

well region,

220 Graphene - New Trends and Developments

under this condition, the energy *E* = ±*vF*

*5.1.3. The derivation of the dispersion relation*

These conditions provide the equalities

*<sup>λ</sup>*<sup>±</sup> <sup>=</sup> *ivF*(*k*<sup>2</sup> <sup>±</sup> *ky*) *E* + ∆<sup>0</sup> − *V*<sup>0</sup>

The solution of Eq. (81) in the barrier region under the condition

∆2 <sup>0</sup> <sup>+</sup> *<sup>v</sup>*<sup>2</sup> *Fk*2

is given by Eq. (85) with the change *k*<sup>2</sup> → *i*κ2, i.e., it is oscillating.

*k*<sup>2</sup> is real. A necessary condition for existing Tamm states has the form

*a* (1) *n*+1 *c* (1) *n*+1

Dirac equation describing the considered superlattice are used:

*a* (2) *n c* (2) *n*

*a* (1) *n*+1 *c* (1) *n*+1  = *T a* (1) *n c* (1) *n*

, *<sup>k</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> *vF* ∆2 <sup>0</sup> <sup>+</sup> *<sup>v</sup>*<sup>2</sup> *Fk*2

The possibility of existing *Tamm* minibands formed by localized states near the interface between graphene and its gap modification will be considered below. In this case, *k*<sup>1</sup> → *i*κ<sup>1</sup>


The dispersion relation is derived using the transfer matrix (*T* matrix) method. The *T* matrix relates the spinor components for the *n*th supercell to the spinor components of the solution of the same type for the (*n* + 1)th supercell. For example, for the solution in the quantum

To determine the *T* matrix, the following conditions of the continuity of the solution of the

*<sup>ψ</sup>*(1) *<sup>n</sup>* (*dI* <sup>−</sup> <sup>0</sup>) = *<sup>ψ</sup>*(2) *<sup>n</sup>* (*dI* <sup>+</sup> <sup>0</sup>),

*<sup>k</sup>*<sup>2</sup> (*dI*)Ω*k*<sup>1</sup> (*dI*)

*<sup>k</sup>*<sup>1</sup> (0)Ω*k*<sup>2</sup> (*d*)

*<sup>ψ</sup>*(2) *<sup>n</sup>* (*<sup>d</sup>* <sup>−</sup> <sup>0</sup>) = *<sup>ψ</sup>*(1)

= Ω−<sup>1</sup>

= Ω−<sup>1</sup>

<sup>1</sup> is real.

*<sup>n</sup>*+1(+0).

*a* (1) *n c* (1) *n*

*a* (2) *n c* (2) *n*

 ,

 .

 *k*2 *<sup>y</sup>* <sup>−</sup> <sup>κ</sup><sup>2</sup> *<sup>y</sup>* − (*<sup>E</sup>* − *<sup>V</sup>*0)2.

*<sup>y</sup>* <sup>−</sup> (*<sup>E</sup>* <sup>−</sup> *<sup>V</sup>*0)<sup>2</sup> *<sup>&</sup>lt;* <sup>0</sup> (86)

. (87)

$$\alpha = \frac{1}{(\lambda\_+ + \lambda\_-)(\tilde{\lambda}\_+ + \tilde{\lambda}\_-)}.$$

The last two relations in Eqs. (89) are the general properties of the *T* matrix.

The derivation of the dispersion relation with the use of the *T* matrix is briefly as follows.

Let *N* = *L*/*d* be the number of supercells in the entire SL, where *L* is the length of SL along the *x* axis, i.e., the direction of the application of the periodic potential. The Born-Karman cyclic boundary conditions for SL have the form

$$
\psi\_N^{(1,2)}(\mathfrak{x}) = \psi\_1^{(1,2)}(\mathfrak{x}).
$$

At the same time,

$$
\psi\_N^{(1,2)}(\mathfrak{x}) = T^N \psi\_1^{(1,2)}(\mathfrak{x})\_\prime
$$

from which, *<sup>T</sup><sup>N</sup>* <sup>=</sup> <sup>I</sup>, where <sup>I</sup> is the 2 <sup>×</sup> 2 unit matrix.

It is convenient to diagonalize the *T* matrix by means of the transition matrix *S*:

$$T\_d = \mathcal{S}TS^{-1} = \begin{pmatrix} \lambda\_1 & 0\\ 0 & \lambda\_2 \end{pmatrix}.$$

where *λ*1,2 are the eigenvalues of the *T* matrix and have the property *λ*<sup>2</sup> = *λ*<sup>∗</sup> <sup>1</sup>. According to *T<sup>N</sup> <sup>d</sup>* = I

$$
\lambda\_1 = e^{2\pi i n/N}, \quad -N/2 < n \le N/2.
$$

<sup>6</sup> Note that the cyclic permutations of the factors of Ω matrices are possible in the definition of the *T* matrix; these permutation do not change dispersion relation (90). This can be verified by comparing Eq. (88) with Eq. (23) in [76].

In view of the property *TrT* = *TrTd* and in terms of the notation *kx* = 2*πn*/*L* (−*π*/*d < kx* ≤ *π*/*d*), the dispersion relation is obtained in the form

$$TrT = 2\cos(k\_x d).\tag{90}$$

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

$$\operatorname{Re} T\_{11} = \cos(k\_x d).$$

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

$$\frac{v\_{\rm F}^2 k\_2^2 - v\_{\rm F}^2 k\_1^2 + V\_0^2 - \Delta\_0^2}{2v\_{\rm F}^2 k\_1 k\_2} \sinh(k\_2 d\_{II}) \sin(k\_1 d\_I) + \cosh(k\_2 d\_{II}) \cos(k\_1 d\_I) = \cos(k\_3 d). \tag{91}$$

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

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

$$\frac{-v\_{\text{F}}^{2}\varkappa\_{2}^{2} - v\_{\text{F}}^{2}k\_{1}^{2} + V\_{0}^{2} - \Delta\_{0}^{2}}{2v\_{\text{F}}^{2}k\_{1}\varkappa\_{2}}\sin(\varkappa\_{2}d\_{II})\sin(k\_{1}d\_{I}) + \cos(\varkappa\_{2}d\_{II})\cos(k\_{1}d\_{I}) = \cos(k\_{\text{x}}d).\tag{92}$$

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

$$\frac{v\_{\text{F}}^2 k\_2^2 + v\_{\text{F}}^2 \varkappa\_1^2 + V\_0^2 - \Delta\_0^2}{2v\_{\text{F}}^2 \varkappa\_1 k\_2} \sinh(k\_2 d\_{II}) \sinh(\varkappa\_1 d\_I) + \cosh(k\_2 d\_{II}) \cosh(\varkappa\_1 d\_I) = \cos(k\_2 d). \tag{93}$$

Equation (93) has the solution only under the condition

$$v\_F^2 k\_2^2 + v\_F^2 \varkappa\_1^2 + V\_0^2 - \Lambda\_0^2 < 0.$$

This condition can be rewritten as

$$
v\_F^2 k\_y^2 - E^2 < -E V\_0.\tag{94}$$

At the same time, for Tamm minibands *E*<sup>2</sup> = *v*<sup>2</sup> *Fk*2 *<sup>y</sup>* <sup>−</sup> *<sup>v</sup>*<sup>2</sup> *F*κ<sup>2</sup> <sup>1</sup> , i.e. the left-side of (94) is positive. The allowed values of the energy should be negative if *V*<sup>0</sup> *>* 0 and vice versa. It is not difficult to show that the inequality (94) has the solutions when [88]

$$
v\_F^2 k\_y^2 < \frac{\Delta\_0^2 (\Delta\_0^2 - V\_0^2)}{V\_0^2}.\tag{95}$$

Formally, this condition coincides with the condition of the intersection of the dispersion curves for graphene and its gap modification [44].
