**3. Graphene heterojunctions**

**Figure 3.** The energy surface for gapless graphene.

where charge carriers belong.

186 Graphene - New Trends and Developments

*<sup>λ</sup>* and *<sup>ψ</sup>*(+)

we rewrite the equality above as

heterostructure plane. It is necessary condition.

where *<sup>ψ</sup>*(−)

**2.3. The boundary conditions**

*<sup>ψ</sup>*(−)† *<sup>λ</sup>* (−*l*)*v*

and *x >* 0, respectively). Representing these functions as

 *v* (−) *F <sup>ψ</sup>*(−)† *<sup>λ</sup>* (−*l*)

(−) *<sup>F</sup> <sup>ψ</sup>*(−)

*<sup>ψ</sup>*(±) *<sup>λ</sup>* = *<sup>ψ</sup>*(±) *λ* exp *iϕ*(±) ,

Equations (8) and (9) are equivalently rewritten as the 2×2 matrix equation

−*ivFjσx∂<sup>x</sup>* + *λvFjkyσ<sup>y</sup>* + ∆*jσ<sup>z</sup>* + *Vj*

Hereinafter, valley indices *K* and *K* are omitted as unnecessary since *λ* specifies the valley

Now, let us discuss the boundary conditions at the interfaces between different graphene materials. At the outset, we note that they are easier to formulate than those used at the graphene–free-space interface in models of edge states [6, 35]. To derive boundary conditions in the present model, we must find a relation between *ψλ*(*l*) and *ψλ*(−*l*) as *l* → 0 in the neighbor-hood of x = 0, where *l* goes down to an atomic scale (condition at x = d is derived similarly). Multiplying Eq. (10) by *<sup>ψ</sup>*†(*x*) on the left, we integrate it over [−*l*, *<sup>l</sup>*]. Since a is small, we neglect all terms except those containing a derivative with respect to *x* to obtain<sup>2</sup>

*<sup>λ</sup>* (−*l*) = *<sup>ψ</sup>*(+)†

 = *v* (+) *F <sup>ψ</sup>*(+)† *<sup>λ</sup>* (*l*) .

<sup>2</sup> From the given equality, we have the continuity of the current component normal to the interface in the

*<sup>λ</sup>* (*l*)*v*

*<sup>λ</sup>* are defined on the left- and right-hand sides of the boundary (at *x <* 0

(+) *<sup>F</sup> <sup>ψ</sup>*(+) *<sup>λ</sup>* (*l*),

*ψλ*(*x*) = *Eλψλ*(*x*). (10)

We consider a planar heterojunction composed of graphene and a gap modification of graphene [36]. When we say a gap modification of graphene we imply a graphene with an energy gap in the Dirac spectrum of charge carriers. There are several gap modifications of graphene.

First, the energy gap can open because graphene sheets are located not on SiO2 substrate but on some other material, for example, h-BN, when two triangular sublattices of graphene become nonequivalent and a gap modification of graphene is formed with an energy gap of 53 meV [37]. Second, the energy gap opens in the epitaxially grown graphene on the SiC substrate [38], which is equal to 0.26 eV according to experimental results obtained by angular-resolved photoemission spectroscopy [39]. Third, recently another modification of graphene, i.e., graphane, was synthesized by hydrogenation [40], which has a direct energy gap of 5.4 eV at the Γ point according to the calculations [41]. In the first two cases, a graphene film deposited on inhomogeneous SiO2–h-BN or SiO2–SiC substrates can be used (Fig. 4a shows the case with h-BN). In the third case, an inhomogeneously hydrogenated graphene is used (a part of the graphene sample is left without hydrogenation, Fig. 4b).

We assume that the energy gap in the gap modifications of graphene opens at *K* and *K* points of the first Brillouin zone, which correspond to the Dirac points of gapless graphene.

**Figure 4.** Two variants of the system in question: **(a)** graphene layer on the substrate composed of h-BN and SiO2 and **(b)** nonuniformly hydrogenated graphene on the SiO2 substrate. Open circles are hydrogen atoms, which are located so that they are bound to carbon atoms of one sublattice on one side of graphene sheet and to carbon atoms of the other sublattice on the other side.

**Figure 5.** Graphene heterojunction under consideration.

Let us assume that the *x* axis is directed along the plane of the heterojunction perpendicular to the boundary between graphene and its gap modification and the *y* axis is directed along the boundary. The *z* axis is directed perpendicular to the plane of the heterojunction. The half-plane *x <* 0 is occupied by the gap-less graphene and the half-plane *x >* 0 belongs to the gap modification of graphene. So, the line *x* = 0 is the boundary under consideration.

In this case, the parameters in Eq. (5) with *j* = 1 are related to the gapless graphene and the parameters with *j* = 2 are related to the gap modification of the graphene: *vF*<sup>1</sup> and *vF*<sup>2</sup> are the Fermi velocities (in the general case, *vF*<sup>2</sup> <sup>=</sup> *vF*1, and *vF*<sup>1</sup> <sup>≈</sup> 108 cm/s); <sup>∆</sup><sup>1</sup> <sup>=</sup> 0 and <sup>∆</sup><sup>2</sup> <sup>=</sup> 0 are the half-widths of the energy gaps; *V*<sup>1</sup> and *V*<sup>2</sup> are the work functions (*V*<sup>2</sup> determines position of the middle of the energy gap for the gap modification of the graphene with respect to the Dirac points of the gapless graphene, and *V*<sup>1</sup> = 0 is chosen for the origin, see Fig. 5).

In order to avoid spontaneous generation of electron-hole pairs, we assume that the heterojunction in question is a junction of the first kind, i.e., the Dirac points of gapless graphene are located inside the energy gap of its gap modification. This limits value of the work function |*V*2| *<* ∆2.

Equation (5) is solved within the class of wave eigenfunctions Ψ*λ*(*x*) of the parity operator (6) for two areas on both sides from the boundary.

Equation (5) can be easily represented as two 2 × 2 matrix equations

$$\left(-i\nu\_{Fj}\sigma\_x \frac{d}{d\mathbf{x}} + \nu\_{Fj}k\_y\sigma\_y + \lambda\Delta\_j\sigma\_z + V\_j\right)\Psi\_{\lambda K}(\mathbf{x}) = E\_\lambda \Psi\_{\lambda K}(\mathbf{x})\_\prime \tag{12}$$

$$\left(-i v\_{Fj} \sigma\_x \frac{d}{d\mathbf{x}} - v\_{Fj} k\_y \sigma\_y - \lambda \Delta\_j \sigma\_z + V\_j\right) \Psi\_{\lambda K'}(\mathbf{x}) = E\_\lambda \Psi\_{\lambda K'}(\mathbf{x}).\tag{13}$$

In this case, we have *λ* = +1 in Eq. (12) and *λ* = −1 in Eq. (13).

The solution to Eq. (12) for boundary states has the form

$$\Psi\_{\lambda K}(\mathbf{x}) = \begin{cases} \mathbb{C}(\frac{1}{a}) \exp(\kappa\_1 \mathbf{x}), & \mathbf{x} < \mathbf{0}, \\ \mathbb{C}(\frac{b}{qb}) \exp(-\kappa\_2 \mathbf{x}), & \mathbf{x} > \mathbf{0}, \end{cases} \tag{14}$$

where

**Figure 5.** Graphene heterojunction under consideration.

188 Graphene - New Trends and Developments

work function |*V*2| *<* ∆2.

−*ivFjσ<sup>x</sup>*

−*ivFjσ<sup>x</sup>*

(6) for two areas on both sides from the boundary.

*d*

*d*

In this case, we have *λ* = +1 in Eq. (12) and *λ* = −1 in Eq. (13).

Equation (5) can be easily represented as two 2 × 2 matrix equations

*dx* <sup>+</sup> *vFjkyσ<sup>y</sup>* <sup>+</sup> *<sup>λ</sup>*∆*jσ<sup>z</sup>* <sup>+</sup> *Vj*

*dx* <sup>−</sup> *vFjkyσ<sup>y</sup>* <sup>−</sup> *<sup>λ</sup>*∆*jσ<sup>z</sup>* <sup>+</sup> *Vj*

Let us assume that the *x* axis is directed along the plane of the heterojunction perpendicular to the boundary between graphene and its gap modification and the *y* axis is directed along the boundary. The *z* axis is directed perpendicular to the plane of the heterojunction. The half-plane *x <* 0 is occupied by the gap-less graphene and the half-plane *x >* 0 belongs to the gap modification of graphene. So, the line *x* = 0 is the boundary under consideration. In this case, the parameters in Eq. (5) with *j* = 1 are related to the gapless graphene and the parameters with *j* = 2 are related to the gap modification of the graphene: *vF*<sup>1</sup> and *vF*<sup>2</sup> are the Fermi velocities (in the general case, *vF*<sup>2</sup> <sup>=</sup> *vF*1, and *vF*<sup>1</sup> <sup>≈</sup> 108 cm/s); <sup>∆</sup><sup>1</sup> <sup>=</sup> 0 and <sup>∆</sup><sup>2</sup> <sup>=</sup> 0 are the half-widths of the energy gaps; *V*<sup>1</sup> and *V*<sup>2</sup> are the work functions (*V*<sup>2</sup> determines position of the middle of the energy gap for the gap modification of the graphene with respect to the

Dirac points of the gapless graphene, and *V*<sup>1</sup> = 0 is chosen for the origin, see Fig. 5).

In order to avoid spontaneous generation of electron-hole pairs, we assume that the heterojunction in question is a junction of the first kind, i.e., the Dirac points of gapless graphene are located inside the energy gap of its gap modification. This limits value of the

Equation (5) is solved within the class of wave eigenfunctions Ψ*λ*(*x*) of the parity operator

Ψ*λK*(*x*) = *Eλ*Ψ*λK*(*x*), (12)

Ψ*λK*(*x*) = *Eλ*Ψ*λK*(*x*). (13)

$$a = i \frac{v\_{F1}(k\_y - \kappa\_1)}{E\_\lambda}, \quad q = i \frac{v\_{F2}(k\_y + \kappa\_2)}{E\_\lambda - V\_2 + \lambda \Delta\_2}.$$

*C* is the normalization factor, *b* = *vF*<sup>1</sup> *vF*<sup>2</sup> is the constant obtained when matching solutions for *x <* 0 and *x >* 0 at the line *x* = 0 under condition (11),

$$E\_{\lambda} = \pm v\_{F1} \sqrt{k\_{\mathcal{Y}}^2 - \kappa\_{1\prime}^2} \tag{15}$$

from which it follows that the necessary condition for the existence of the boundary states is given by inequality

$$|\kappa\_1 < |k\_Y|. \tag{16}$$

Equation (15) can be rewritten as

$$\kappa\_1 = \sqrt{k\_y^2 - E\_\lambda^2 / v\_{F1'}^2}$$

Therefore, the following inequality should also be valid

$$|E\_{\lambda}| < \upsilon\_{F1}|k\_{\mathcal{Y}}|.\tag{17}$$

Expression for *κ*<sup>2</sup> is represented in the form

$$\kappa\_2 = \frac{1}{v\_{F2}}\sqrt{\Delta\_2^2 - (E\_\lambda - V\_2)^2 + v\_{F2}^2 k\_\mathcal{Y}^2}$$

Moreover, the matching leads to the inequality

$$\frac{v\_{F1}(k\_y - \kappa\_1)}{E\_\lambda} = \frac{v\_{F2}(k\_y + \kappa\_2)}{E\_\lambda - V\_2 + \lambda \Delta\_2}. \tag{18}$$

The solution to Eq. (13) is produced from Eq. (14) by the following substitutions in factors *a* and *q*: *ky* → −*ky* and *λ* → −*λ*.

Let us discuss separately the case of zero mode *E<sup>λ</sup>* = 0. Components of the envelope wave function in *x <* 0 region (gapless graphene) Ψ*λ<sup>K</sup>* = ( *a*1 *a*2 ) exp(*κ*1*x*) satisfy equations:

$$\begin{aligned} (\kappa\_1 - k\_{\mathcal{Y}})a\_1 &= 0, \\ (\kappa\_1 + k\_{\mathcal{Y}})a\_2 &= 0, \end{aligned}$$

i.e., either *κ*<sup>1</sup> = *ky* (*ky >* 0) and *a*<sup>2</sup> = 0, or *κ*<sup>1</sup> = −*ky* (*ky <* 0) and *a*<sup>1</sup> = 0. Then it follows from the matching condition (11) that both components of the envelope wave function are zero in *x >* 0 region (*b* = 0); therefore, we have *a*<sup>1</sup> = 0 and *a*<sup>2</sup> = 0, i.e., Ψ*λK*(*x*) ≡ 0. Thus, there is no zero mode for the boundary states in question.

The following equations are easily obtained from Eq. (18):

$$\kappa\_1 \kappa\_2 = \frac{E\_\lambda (E\_\lambda - V\_2)}{v\_{F1} v\_{F2}} - k\_{y\prime}^2 \tag{19}$$

$$
\lambda \Delta\_2 E\_\lambda = \upsilon\_{F1} \upsilon\_{F2} k\_y(\kappa\_1 + \kappa\_2). \tag{20}
$$

The two latter equations are valid for either value of *λ* (for both valleys), because they are invariant in respect to simultaneous substitutions *ky* → −*ky* and *λ* → −*λ*.

Since *κ*<sup>1</sup> *>* 0 and *κ*<sup>2</sup> *>* 0, right-hand side of Eq. (19) should be positive. Let us denote by *ε*0(*ky*) such value of *E<sup>λ</sup>* that the right-hand side of Eq. (19) turns zero,

$$
\varepsilon\_0(k\_y) = \frac{V\_2}{2} \pm \sqrt{\frac{V\_2^2}{4} + v\_{F1}v\_{F2}k\_{y'}^2} \tag{21}
$$

where "+" corresponds to electrons and "−" to holes. Then, the condition *κ*1*κ*<sup>2</sup> *>* 0 is equivalent to the inequality

$$|E\_{\lambda}| > |\varepsilon\_{0}(k\_{\mathcal{Y}})|.\tag{22}$$

It follows from Eq. (20) that inequality *λky >* 0 holds for electron boundary states (*E<sup>λ</sup> >* 0), and *λky <* 0 holds for hole boundary states (*E<sup>λ</sup> <* 0). The boundary states are not degenerate in parity. That means that there is no Kramers degeneracy of energy spectrum for them. This is also true for boundary states in a planar quantum well based on graphene nanoribbon [42] and for boundary states localized on zigzag edges of gapless graphene [43]. Since parity determines charge carrier attribution to one of two valleys, the property mentioned above means also that there is a "valley polarization" of boundary states: electrons that move along the heterojunction boundary with *ky >* 0 are located near *K* point and electrons with *ky <* 0 are near *K* point and vise versa in case of holes. Because of that, current that flows along the heterojunction boundary would be "valley-polarized" [36].

By squaring Eq. (20) we get a quadratic equation, solution of which produces dependence of energy on *ky*:

$$E\_{\lambda}(k\_{\mathcal{Y}}) = \frac{v\_{F1}v\_{F-}k\_{\mathcal{Y}}^2V\_2 + \lambda v\_{F1}k\_{\mathcal{Y}}\Delta\_2\sqrt{\Delta\_2^2 + v\_{F-}^2k\_{\mathcal{Y}}^2 - V\_2^2}}{\Delta\_2^2 + v\_{F-}^2k\_{\mathcal{Y}}^2},\tag{23}$$

where *vF*<sup>−</sup> = *vF*<sup>1</sup> − *vF*2. Equation (23) takes into account that sign of *λky* determines type of charge carriers in the boundary states.

It is easy to verify that inequality (17) is always true if the energy is given by Eq. (23). Therefore, inequality (16) also holds.

Now, it is simple to analyze inequality (22). Let us introduce the following notation:

$$k\_{y1} = \frac{|V\_2|}{|v\_{F-}|},$$

$$k\_{y2,3} = \sqrt{\frac{v\_{F2}V\_2^2 + 2v\_{F-}\Delta\_2^2 \mp |V\_2|\sqrt{v\_{F2}^2V\_2^2 + 4v\_{F1}v\_{F-}\Delta\_2^2}}{2v\_{F2}v\_{F-}^2}}.$$

Under the condition

Let us discuss separately the case of zero mode *E<sup>λ</sup>* = 0. Components of the envelope wave

(*κ*<sup>1</sup> − *ky*)*a*<sup>1</sup> = 0, (*κ*<sup>1</sup> + *ky*)*a*<sup>2</sup> = 0,

i.e., either *κ*<sup>1</sup> = *ky* (*ky >* 0) and *a*<sup>2</sup> = 0, or *κ*<sup>1</sup> = −*ky* (*ky <* 0) and *a*<sup>1</sup> = 0. Then it follows from the matching condition (11) that both components of the envelope wave function are zero in *x >* 0 region (*b* = 0); therefore, we have *a*<sup>1</sup> = 0 and *a*<sup>2</sup> = 0, i.e., Ψ*λK*(*x*) ≡ 0. Thus, there is no

> *<sup>κ</sup>*1*κ*<sup>2</sup> <sup>=</sup> *<sup>E</sup>λ*(*E<sup>λ</sup>* <sup>−</sup> *<sup>V</sup>*2) *vF*1*vF*<sup>2</sup>

The two latter equations are valid for either value of *λ* (for both valleys), because they are

Since *κ*<sup>1</sup> *>* 0 and *κ*<sup>2</sup> *>* 0, right-hand side of Eq. (19) should be positive. Let us denote by

 *V*2 2

where "+" corresponds to electrons and "−" to holes. Then, the condition *κ*1*κ*<sup>2</sup> *>* 0 is

It follows from Eq. (20) that inequality *λky >* 0 holds for electron boundary states (*E<sup>λ</sup> >* 0), and *λky <* 0 holds for hole boundary states (*E<sup>λ</sup> <* 0). The boundary states are not degenerate in parity. That means that there is no Kramers degeneracy of energy spectrum for them. This is also true for boundary states in a planar quantum well based on graphene nanoribbon [42] and for boundary states localized on zigzag edges of gapless graphene [43]. Since parity determines charge carrier attribution to one of two valleys, the property mentioned above means also that there is a "valley polarization" of boundary states: electrons that move along the heterojunction boundary with *ky >* 0 are located near *K* point and electrons with *ky <* 0 are near *K* point and vise versa in case of holes. Because of that, current that flows along the

By squaring Eq. (20) we get a quadratic equation, solution of which produces dependence of

2 ±

invariant in respect to simultaneous substitutions *ky* → −*ky* and *λ* → −*λ*.

*ε*0(*ky*) such value of *E<sup>λ</sup>* that the right-hand side of Eq. (19) turns zero,

*<sup>ε</sup>*0(*ky*) = *<sup>V</sup>*<sup>2</sup>

heterojunction boundary would be "valley-polarized" [36].

*a*1 *a*2

<sup>−</sup> *<sup>k</sup>*<sup>2</sup>

<sup>4</sup> <sup>+</sup> *vF*1*vF*2*k*<sup>2</sup>

*λ*∆2*E<sup>λ</sup>* = *vF*1*vF*2*ky*(*κ*<sup>1</sup> + *κ*2). (20)


) exp(*κ*1*x*) satisfy equations:

*<sup>y</sup>*, (19)

*<sup>y</sup>*, (21)

function in *x <* 0 region (gapless graphene) Ψ*λ<sup>K</sup>* = (

190 Graphene - New Trends and Developments

zero mode for the boundary states in question.

equivalent to the inequality

energy on *ky*:

The following equations are easily obtained from Eq. (18):

$$v\_{\rm F1} < v\_{\rm F2} < 2v\_{\rm F1} \quad \frac{2}{v\_{\rm F2}} \sqrt{v\_{\rm F1}|v\_{\rm F-}|} \\ \Delta\_2 < |V\_2| < \Delta\_2 \tag{24}$$

the boundary states exist in the ranges3

$$0 < |k\_{\mathcal{Y}}| < k\_{y2\prime} \quad k\_{\mathcal{Y}3} < |k\_{\mathcal{Y}}| < k\_{y1\prime}$$

either for electrons, if *V*<sup>2</sup> *<* 0, or for holes, if *V*<sup>2</sup> *>* 0.

Under condition

$$
v\_{F1} < v\_{F2} < 2v\_{F1\prime} \ |V\_2| < \frac{2}{v\_{F2}} \sqrt{v\_{F1}|v\_{F-}|} \Delta\_2 \tag{25}$$

the boundary states exist in the range

$$0 < |k\_y| < k\_{y1}$$

either for electrons, if *V*<sup>2</sup> *<* 0, or for holes, if *V*<sup>2</sup> *>* 0. Under the condition

$$
v\_{F1} > v\_{F2}, \ 0 < V\_2 < \Delta\_2 \tag{26}$$

<sup>3</sup> Here and below, we exclude the point *ky* = 0, because it corresponds to *E<sup>λ</sup>* = 0.

the electron boundary states exist in the range

$$k\_{y3} < |k\_y| < k\_{y1\*}$$

and the hole boundary states exist in the range

$$0 < |k\_y| < k\_{y2}.$$

Under condition

$$v\_{F1} > v\_{F2'} \quad -\Delta\_2 < v\_2 < 0$$

the electron boundary states exist in the range

$$0 < |k\_{y}| < k\_{y2\prime}$$

and the hole boundary states exist in the range

$$k\_{y3} < |k\_y| < k\_{y1}.$$

Let us consider three special cases.

(1) Under condition *V*<sup>2</sup> = 0 and *vF*<sup>−</sup> = 0, the boundary states exist for both electrons and holes in the following range if *vF*<sup>1</sup> *> vF*<sup>2</sup>

$$0 < |k\_{\mathcal{Y}}| < \frac{\Delta\_2}{\sqrt{\overline{\upsilon\_{F1}\upsilon\_{F-}}}}.\tag{27}$$

(2) Under condition *vF*<sup>1</sup> = *vF*2, 0 *<* |*V*2| *<* ∆<sup>2</sup> the boundary states exist in the range

$$0 < |k\_{\mathcal{Y}}| < \frac{\Delta\_2 \sqrt{\Delta\_2^2 - V\_2^2}}{v\_{\mathcal{F}2} |V\_2|} \tag{28}$$

either for electrons, if *V*<sup>2</sup> *<* 0, or for holes, if *V*<sup>2</sup> *>* 0.

(3) Under condition *vF*<sup>1</sup> = *vF*2, *V*<sup>2</sup> = 0, the boundary states are absent both for electrons and holes, because |*Eλ*(*ky*)| = |*ε*0(*ky*)|, which is in contradiction with inequality (16).

the electron boundary states exist in the range

192 Graphene - New Trends and Developments

and the hole boundary states exist in the range

the electron boundary states exist in the range

and the hole boundary states exist in the range

Let us consider three special cases.

holes in the following range if *vF*<sup>1</sup> *> vF*<sup>2</sup>

either for electrons, if *V*<sup>2</sup> *<* 0, or for holes, if *V*<sup>2</sup> *>* 0.

Under condition

*ky*<sup>3</sup> *<* |*ky*| *< ky*1,

0 *<* |*ky*| *< ky*2.

*vF*<sup>1</sup> *> vF*2, −∆<sup>2</sup> *< V*<sup>2</sup> *<* 0

0 *<* |*ky*| *< ky*2,

*ky*<sup>3</sup> *<* |*ky*| *< ky*1.

(1) Under condition *V*<sup>2</sup> = 0 and *vF*<sup>−</sup> = 0, the boundary states exist for both electrons and

∆2 ∆2 <sup>2</sup> <sup>−</sup> *<sup>V</sup>*<sup>2</sup> 2

(3) Under condition *vF*<sup>1</sup> = *vF*2, *V*<sup>2</sup> = 0, the boundary states are absent both for electrons and

∆2 <sup>√</sup>*vF*1*vF*<sup>−</sup>

. (27)

*vF*2|*V*2<sup>|</sup> (28)

0 *<* |*ky*| *<*

(2) Under condition *vF*<sup>1</sup> = *vF*2, 0 *<* |*V*2| *<* ∆<sup>2</sup> the boundary states exist in the range

0 *<* |*ky*| *<*

holes, because |*Eλ*(*ky*)| = |*ε*0(*ky*)|, which is in contradiction with inequality (16).

**Figure 6.** Dispersion curves *Ee*,*<sup>h</sup> <sup>λ</sup>* (*ky* ) and *ε e*,*h* <sup>0</sup> (*ky* ): **(a)** there are no boundary states for electrons and holes at *V*<sup>2</sup> = 0, **(b)** there are only hole boundary states in the range 0 *<* |*ky* | *< ky*<sup>1</sup> at *V*<sup>2</sup> = 100 meV, and **(c)** there are only hole boundary states in the ranges 0 *<* |*ky* | *< ky*<sup>2</sup> and *ky*<sup>3</sup> *<* |*ky* | *< ky*<sup>1</sup> at *V*<sup>2</sup> = 250 meV.

Fig. 6 shows dispersion curves *Ee*,*<sup>h</sup> <sup>λ</sup>* (*ky*) and *ε e*,*h* <sup>0</sup> (*ky*) for the electron and hole boundary states for three values of *V*<sup>2</sup> in the model of graphene-based heterojunction with ∆<sup>2</sup> = 260 meV and *vF*<sup>2</sup> = 1.2 <sup>×</sup> 108 cm/s for gap modification of graphene.

Our results remain in essence the same if instead of a sharp heterojunction we consider a smooth heterojunction. Indeed, let *vF*(*x*) and ∆(*x*) vary smoothly from their values for gapless graphene to their values in gap modification of graphene over a strip with the width *d κ*−<sup>1</sup> 1, 2. Then change in energy of the boundary states is |*δEλ*(*ky*)| 1 meV. Such insignificant variation in energy of the boundary states produces no noticeable qualitative changes. A similar result has been obtained for boundary states in heterojunctions of narrow-gap semiconductors with intercrossing dispersion curves in [44].

To conclude, we would like to point out that the new type of boundary states in graphene heterojunctions can be studied in experiment by tunnel spectroscopy of angular-resolved photoemission spectroscopy similar to how it have been done for boundary states in gapless graphene [45–47].
