**4. Graphene quantum well**

## **4.1. General consideration**

In this study, we examine a planar quantum well (QW) made of a graphene nanoribbon whose edges are in contact with gapped graphene sheets.

A bandgap opening in graphene can be induced by several methods. First, graphene can be deposited on h-BN substrate instead of a silicon-oxide one. This makes its two triangular sublattices nonequivalent, inducing in a bandgap of 53 meV [37]. Second, epitaxial graphene grown on a silicon-carbide substrate also has a nonzero bandgap [38]. According to angle-resolved photoemission data, a bandgap of 0.26 eV is produced by this method [39]. Third, a hydrogenated derivative of graphene synthesized recently, graphane [40], has been predicted to have a direct bandgap of 5.4 eV at the Γ point [41]. Fourth, *ab initio* calculations have shown that CrO3 adsorption on graphene induces a gap of 0.12 eV [48]. In the first two methods, a heterogeneous substrate can be used, such as an h-BN — SiO2 nanoribbon — h-BN or SiC — SiO2 nanoribbon — SiC one (Fig. 7a depicts a substrate with h-BN). The last two methods produce a graphene sheet containing a nanoribbon without hydrogenation (as the nonhydrogenated one in Fig. 7b) or a graphene strip without adsorbed CrO3 molecules, respectively. Furthermore, the bandgap can be varied by using partially hydrogenated graphene (where some carbon atoms are not bonded to hydrogen atoms). Combinations of these methods can also be employed. Extensive experimental studies of graphene on substrates made of various materials, including rare-earth metals, have been reported recently [49–51]. It may be possible to open a bandgap via adsorption of other molecules on graphene or by using other materials as substrates. The use of gapped graphene to create potential barriers opens up additional possibilities for bandgap engineering in carbon-based materials [52].

In the case of QW, in Eq. (10) ∆*<sup>j</sup>* = *Egj*/2 (*j* = 1, 2, 3) denotes half-width of bandgap (∆<sup>1</sup> = 0 and ∆<sup>3</sup> = 0 in regions *1* and *3*, whereas ∆<sup>2</sup> = 0 region *2*); the respective work functions *V*<sup>1</sup> and *V*<sup>3</sup> of regions *1* and *3* depend on the mid-gap energies relative to the Dirac points for the corresponding materials (we set *V*<sup>2</sup> = 0 to be specific, see Fig. 8).

The solution to Eq. (10) is expressed as follows.

**Figure 7.** Two configurations of the system under study: **(a)** graphene sheet on a substrate consisting of a SiO2 nanoribbon of width d inserted between h–BN nanoribbons; **(b)** a graphene sheet on a SiO2 substrate containing a nonhydrogenated nanoribbon of width *d*, where open and closed circles are hydrogen atoms bonded to carbon atoms in different sublattices on opposite sides of the sheet, respectively.

**Figure 8.** An energy scheme of QW under analysis.

**1.** At *x <* 0,

$$
\psi\_{\lambda}(\mathbf{x}) = \mathbb{C} \begin{pmatrix} 1 \\ q\_1 \end{pmatrix} e^{k\_1 \mathbf{x}},\tag{29}
$$

where

**4. Graphene quantum well**

whose edges are in contact with gapped graphene sheets.

In this study, we examine a planar quantum well (QW) made of a graphene nanoribbon

A bandgap opening in graphene can be induced by several methods. First, graphene can be deposited on h-BN substrate instead of a silicon-oxide one. This makes its two triangular sublattices nonequivalent, inducing in a bandgap of 53 meV [37]. Second, epitaxial graphene grown on a silicon-carbide substrate also has a nonzero bandgap [38]. According to angle-resolved photoemission data, a bandgap of 0.26 eV is produced by this method [39]. Third, a hydrogenated derivative of graphene synthesized recently, graphane [40], has been predicted to have a direct bandgap of 5.4 eV at the Γ point [41]. Fourth, *ab initio* calculations have shown that CrO3 adsorption on graphene induces a gap of 0.12 eV [48]. In the first two methods, a heterogeneous substrate can be used, such as an h-BN — SiO2 nanoribbon — h-BN or SiC — SiO2 nanoribbon — SiC one (Fig. 7a depicts a substrate with h-BN). The last two methods produce a graphene sheet containing a nanoribbon without hydrogenation (as the nonhydrogenated one in Fig. 7b) or a graphene strip without adsorbed CrO3 molecules, respectively. Furthermore, the bandgap can be varied by using partially hydrogenated graphene (where some carbon atoms are not bonded to hydrogen atoms). Combinations of these methods can also be employed. Extensive experimental studies of graphene on substrates made of various materials, including rare-earth metals, have been reported recently [49–51]. It may be possible to open a bandgap via adsorption of other molecules on graphene or by using other materials as substrates. The use of gapped graphene to create potential barriers opens up additional possibilities for bandgap engineering in

In the case of QW, in Eq. (10) ∆*<sup>j</sup>* = *Egj*/2 (*j* = 1, 2, 3) denotes half-width of bandgap (∆<sup>1</sup> = 0 and ∆<sup>3</sup> = 0 in regions *1* and *3*, whereas ∆<sup>2</sup> = 0 region *2*); the respective work functions *V*<sup>1</sup> and *V*<sup>3</sup> of regions *1* and *3* depend on the mid-gap energies relative to the Dirac points for the

**Figure 7.** Two configurations of the system under study: **(a)** graphene sheet on a substrate consisting of a SiO2 nanoribbon of width d inserted between h–BN nanoribbons; **(b)** a graphene sheet on a SiO2 substrate containing a nonhydrogenated nanoribbon of width *d*, where open and closed circles are hydrogen atoms bonded to carbon atoms

corresponding materials (we set *V*<sup>2</sup> = 0 to be specific, see Fig. 8).

The solution to Eq. (10) is expressed as follows.

in different sublattices on opposite sides of the sheet, respectively.

**4.1. General consideration**

194 Graphene - New Trends and Developments

carbon-based materials [52].

$$\begin{aligned} q\_1 &= -i \frac{v\_{F1}(k\_1 - \lambda k\_{\mathcal{Y}})}{E\_{\lambda} - V\_1 + \Delta\_1}, \\\\ v\_{F1} k\_1 &= \sqrt{\Delta\_1^2 - (E\_{\lambda} - V\_1)^2 + v\_{F1}^2 k\_{\mathcal{Y}}^2}. \end{aligned}$$

**2.** At 0 *< x < d*,

$$\psi\_{\lambda}(\mathbf{x}) = \mathbb{C}\begin{pmatrix} \kappa^\* \\ q\_{2}\kappa^\* \end{pmatrix} e^{i\mathbf{k}\_2\mathbf{x}} + \mathbb{C}\begin{pmatrix} \kappa \\ -q\_{2}\kappa \end{pmatrix} e^{-i\mathbf{k}\_2\mathbf{x}},\tag{30}$$

where

$$\begin{aligned} \kappa &= \frac{1}{2} \sqrt{\frac{v\_{F1}}{v\_{F2}}} \left[ 1 + i \left( \frac{\lambda k\_y}{k\_2} + \frac{v\_{F1} (k\_1 - \lambda k\_y) E\_\lambda}{v\_{F2} k\_2 (E\_\lambda - V\_1 + \Delta\_1)} \right) \right], \\\\ q\_2 &= \frac{v\_{F2} (k\_2 + i\lambda k\_y)}{E\_\lambda}, \qquad E\_\lambda = \pm v\_{F2} \sqrt{k\_2^2 + k\_{y'}^2} \end{aligned}$$

with plus and minus corresponding to electrons and holes, respectively. **3.** At *x > d*,

$$\psi\_{\lambda}(\mathbf{x}) = \mathbb{C}\begin{pmatrix} \zeta \\ q\_{\overline{\lambda}}\zeta \end{pmatrix} e^{-k\_{\overline{\lambda}}(\mathbf{x}-\overline{d})} ,\tag{31}$$

where

$$\begin{aligned} \mathcal{L} &= \sqrt{\frac{\upsilon\_{F1}}{\upsilon\_{F3}}} \left[ \cos(k\_2 d) + \left( \frac{\lambda k\_y}{k\_2} + \frac{\upsilon\_{F1} (k\_1 - \lambda k\_y) E\_\lambda}{\upsilon\_{F2} k\_2 (E\_\lambda - V\_1 + \Delta\_1)} \right) \sin(k\_2 d) \right] \\\\ q\_3 &= i \frac{\upsilon\_{F3} (k\_3 + \lambda k\_y)}{E\_\lambda - V\_3 + \Delta\_3}, \\\\ \upsilon\_{F3} k\_3 &= \sqrt{\Delta\_3^2 - (E\_\lambda - V\_3)^2 + \upsilon\_{F2}^2 k\_y^2}. \end{aligned}$$

The constant *C* is found by using the normalization condition for wavefunctions (29)—(31),

$$\int\_{-\infty}^{\infty} \Psi\_{\lambda}^{\dagger}(\mathbf{x}) \Psi\_{\lambda}(\mathbf{x}) d\mathbf{x} = 1.$$

The carrier energy spectrum is determined by the dispersion relation

$$
\tan(k\_2 d) = \upsilon\_{F2} k\_2 f(\lambda k\_{\circ}; k\_1, k\_{\circ}, E\_{\lambda}),
\tag{32}
$$

,

where

$$\begin{split} f(\lambda k\_{\mathcal{Y}} \boldsymbol{k}\_{1}, k\_{3}, E\_{\lambda}) &= \left[ v\_{\rm F1} (k\_{1} - \lambda k\_{\mathcal{Y}}) (E\_{\lambda} - V\_{3} + \Delta\_{3}) + v\_{\rm F3} (k\_{3} + \lambda k\_{\mathcal{Y}}) (E\_{\lambda} - V\_{1} + \Delta\_{1}) \right] \\ &\times \left[ \mathbb{E}\_{\lambda} (E\_{\lambda} - V\_{1} + \Delta\_{1}) (E\_{\lambda} - V\_{3} + \Delta\_{3}) - v\_{\rm F2} v\_{\rm F3} \lambda k\_{\mathcal{Y}} (k\_{3} + \lambda k\_{\mathcal{Y}}) (E\_{\lambda} - V\_{1} + \Delta\_{1}) \right] \\ &+ v\_{\rm F1} v\_{\rm F2} \lambda k\_{\mathcal{Y}} (k\_{1} - \lambda k\_{\mathcal{Y}}) (E\_{\lambda} - V\_{3} + \Delta\_{3}) - v\_{\rm F1} v\_{\rm F3} (k\_{1} - \lambda k\_{\mathcal{Y}}) (k\_{3} + \lambda k\_{\mathcal{Y}}) E\_{\lambda} \right]^{-1} \end{split}$$

is a function of *k*<sup>2</sup> as well. Equation (32) must be solved for *k*2, and then the energy *E<sup>λ</sup>* is found.

In the case of an asymmetric QW, the dependence of (32) on *λ* gives rise to pseudospin splitting as the extrema of the dispersion curves shift away from Brillouin-zone corners. The dispersion relation predicts that *<sup>E</sup>λ*(*ky*) = *<sup>E</sup>*−*λ*(*ky*), and an energy splitting appears near the conduction-band bottom at *ky* = *k*∗ *ye*:

$$
\delta E\_s^\varepsilon = |E\_{-1}^\varepsilon(k\_{ye}^\*) - E\_{+1}^\varepsilon(k\_{ye}^\*)|.
$$

A similar energy splitting appears near the valence-band top at *ky* = *k*∗ *ye*:

$$\delta E\_s^h = |E\_{-1}^h(k\_{yh}^\*) - E\_{+1}^h(k\_{yh}^\*)|$$

Thus, a graphene nanoribbon becomes an indirect band-gap semiconductor analogous to silicon and germanium, where an electron-hole plasma can exist [53]. In the case of a symmetric QW (∆<sup>1</sup> = ∆3, *V*<sup>1</sup> = *V*3, *vF*<sup>1</sup> = *vF*3) band structure is invariant under parity and there is no pseudospin splitting [42].

**Figure 9.** Energy spectra: **(a)** symmetric QW (no pseudospin splitting), with matching branches for *λ* = +1 and *λ* = −1 (*Eλ*(*ky* ) = *E*−*<sup>λ</sup>*(*ky* ) = *Eλ*(−*ky* )); **(b)** asymmetric QW, with pseudospin splitting manifested by the "spread-out" in quasimomentum between the extrema at *k*∗ *ye* for electrons, shown for *b*<sup>−</sup> = 1, and at *k*<sup>∗</sup> *yh* for holes, shown for *b*<sup>+</sup> = 1 (*Eλ*(*ky* ) = *E*−*<sup>λ</sup>*(*ky* )).

#### **4.2. Size quantization**

where

where

found.

*ζ* =

196 Graphene - New Trends and Developments

*<sup>f</sup>*(*λky*;*k*1, *<sup>k</sup>*3, *<sup>E</sup>λ*) =

conduction-band bottom at *ky* = *k*∗

and there is no pseudospin splitting [42].

×

*vF*<sup>1</sup> *vF*<sup>3</sup> cos(*k*2*d*) +

*vF*3*k*<sup>3</sup> =

 *λky k*2 +

*q*<sup>3</sup> = *i*

 ∆2

∞

Ψ†

−∞

The carrier energy spectrum is determined by the dispersion relation

*ye*:

A similar energy splitting appears near the valence-band top at *ky* = *k*∗

*δE<sup>e</sup> <sup>s</sup>* <sup>=</sup> <sup>|</sup>*E<sup>e</sup>*

*δE<sup>h</sup> <sup>s</sup>* <sup>=</sup> <sup>|</sup>*E<sup>h</sup>*

*vF*1(*k*<sup>1</sup> − *λky*)*E<sup>λ</sup> vF*2*k*2(*E<sup>λ</sup>* − *V*<sup>1</sup> + ∆1)

,

*F*2*k*<sup>2</sup> *y*.

tan(*k*2*d*) = *vF*2*k*<sup>2</sup> *f*(*λky*; *k*1, *k*3, *Eλ*), (32)

*vF*1(*k*<sup>1</sup> − *λky*)(*E<sup>λ</sup>* − *V*<sup>3</sup> + ∆3) + *vF*3(*k*<sup>3</sup> + *λky*)(*E<sup>λ</sup>* − *V*<sup>1</sup> + ∆1)

*Eλ*(*E<sup>λ</sup>* − *V*<sup>1</sup> + ∆1)(*E<sup>λ</sup>* − *V*<sup>3</sup> + ∆3) − *vF*2*vF*3*λky*(*k*<sup>3</sup> + *λky*)(*E<sup>λ</sup>* − *V*<sup>1</sup> + ∆1)

+*vF*1*vF*2*λky*(*k*<sup>1</sup> − *λky*)(*E<sup>λ</sup>* − *V*<sup>3</sup> + ∆3) − *vF*1*vF*3(*k*<sup>1</sup> − *λky*)(*k*<sup>3</sup> + *λky*)*E<sup>λ</sup>*

is a function of *k*<sup>2</sup> as well. Equation (32) must be solved for *k*2, and then the energy *E<sup>λ</sup>* is

In the case of an asymmetric QW, the dependence of (32) on *λ* gives rise to pseudospin splitting as the extrema of the dispersion curves shift away from Brillouin-zone corners. The dispersion relation predicts that *<sup>E</sup>λ*(*ky*) = *<sup>E</sup>*−*λ*(*ky*), and an energy splitting appears near the

*ye*) <sup>−</sup> *<sup>E</sup><sup>e</sup>*

*yh*) <sup>−</sup> *<sup>E</sup><sup>h</sup>*

+1(*k*<sup>∗</sup> *ye*)|.

+1(*k*<sup>∗</sup> *yh*)| *ye*:

−1(*k*<sup>∗</sup>

−1(*k*<sup>∗</sup>

Thus, a graphene nanoribbon becomes an indirect band-gap semiconductor analogous to silicon and germanium, where an electron-hole plasma can exist [53]. In the case of a symmetric QW (∆<sup>1</sup> = ∆3, *V*<sup>1</sup> = *V*3, *vF*<sup>1</sup> = *vF*3) band structure is invariant under parity

*vF*3(*k*<sup>3</sup> + *λky*) *E<sup>λ</sup>* − *V*<sup>3</sup> + ∆<sup>3</sup>

<sup>3</sup> <sup>−</sup> (*E<sup>λ</sup>* <sup>−</sup> *<sup>V</sup>*3)<sup>2</sup> <sup>+</sup> *<sup>v</sup>*<sup>2</sup>

*<sup>λ</sup>*(*x*)Ψ*λ*(*x*)*dx* = 1.

The constant *C* is found by using the normalization condition for wavefunctions (29)—(31),

sin(*k*2*d*)

 ,

−<sup>1</sup>

Solving Eq. (32), we determine the size-quantized energies

$$E\_{\lambda b\_{\mp}}(k\_y) = \pm \upsilon\_{F2} \sqrt{k\_{2b\_{\mp}}^2 (\lambda k\_y) + k\_{y'}^2}$$

where *b*<sup>∓</sup> = 1, 2, . . . labels electron (−) and hole (+) branches, respectively. The size-quantized energy spectra for symmetric and asymmetric QW are shown schematically in Fig. 9.

We now determine the carrier effective masses arising because of size quantization in the graphene nanoribbon in a planar heterostructure. Note that the effective masses are invariant under parity regardless of pseudospin splitting. Hereinafter, we omit indices *b*∓, restricting ourselves to a particular branch of the electron spectrum and a particular branch of the hole spectrum.

We write the dispersion law for electrons near an extremum at *λk*∗ *ye* as

$$\begin{split} E\_{\lambda}^{\varepsilon} &\approx E\_{0}^{\varepsilon} + \frac{1}{2m\_{\varepsilon}^{\*}} \left( k\_{\mathcal{Y}} - \lambda k\_{\mathcal{Y}\varepsilon}^{\*} \right)^{2}, \\ m\_{\varepsilon}^{\*} &= \frac{1}{v\_{F2}} \frac{\sqrt{k\_{20\varepsilon}^{2} + k\_{\mathcal{Y}\varepsilon}^{\*2}}}{1 + k\_{20\varepsilon}^{2} + k\_{20\varepsilon}k\_{20\varepsilon}^{\prime}}, \end{split} \tag{33}$$

where the respective values *k*20*e*, *k* <sup>20</sup>*e*, *k* <sup>20</sup>*<sup>e</sup>* of *k*2*e*(*ky*) and its first and second derivatives at *ky* = *λk*∗ *ye* are independent of *λ*; *E<sup>e</sup>* <sup>0</sup> = *vF*<sup>2</sup> *k*2 <sup>20</sup>*<sup>e</sup>* <sup>+</sup> *<sup>k</sup>*∗<sup>2</sup> *ye* is the energy at the extremum.

**Figure 10.** Electron (curves *1*) and hole (curves *2*) effective masses in the graphene nanoribbon (in units of free-electron mass *m*0) as functions of *V*<sup>1</sup> for *V*<sup>3</sup> = 0 **(a)** and as functions of *V*<sup>3</sup> for *V*<sup>1</sup> = 0 **(b)**.

Analogous expressions are obtained for hole energies:

$$\begin{split} E\_{\lambda}^{h} &\approx E\_{0}^{h} + \frac{1}{2m\_{h}^{\*}} \left( k\_{\mathcal{Y}} - \lambda k\_{yh}^{\*} \right)^{2}, \\ m\_{h}^{\*} &= \frac{1}{v\_{F2}} \frac{\sqrt{k\_{20h}^{2} + k\_{ye}^{\*2}}}{1 + k\_{20h}^{f2} + k\_{20e}k\_{20h}^{\prime}} \end{split} \tag{34}$$

where the respective values *k*20*h*, *k* <sup>20</sup>*h*, *k* <sup>20</sup>*<sup>h</sup>* of *k*2*h*(*ky*) and its first and second derivatives at *ky* = −*λk*<sup>∗</sup> *yh*; *<sup>E</sup><sup>h</sup>* <sup>0</sup> = −*vF*<sup>2</sup> *k*2 <sup>20</sup>*<sup>h</sup>* <sup>+</sup> *<sup>k</sup>*∗<sup>2</sup> *yh*.

To estimate characteristic values, we consider the planar heterostructure combining a gapless nanoribbon with gapped graphene sheets with ∆<sup>1</sup> = 0.75 eV, *vF*<sup>1</sup> = 1.1*vF*2, ∆<sup>3</sup> = 1 eV, and *vF*<sup>3</sup> = 1.2*vF*2. The nanoribbon width is *d* = 2.46 nm (ten hexagonal cells). Since the unknown values of *V*<sup>1</sup> and *V*<sup>3</sup> can be found by comparing our results with experimental data, we seek the dependence of energy spectrum parameters on *V*<sup>1</sup> and *V*3. Note that |*V*1| ≤ ∆<sup>1</sup> and |*V*3| ≤ ∆<sup>3</sup> to ensure that the heterostructure is type I.

Figures 10–13 show the results of numerical calculations of electron and hole effective masses in the graphe-ne nanoribbon, extremum energies, *k*∗ *xe* and *k*<sup>∗</sup> *xh* values, and pseudospin splitting *δEe*,*<sup>h</sup> <sup>s</sup>* plotted versus work function for one of the gapped graphene sheets given that the work function for the other is zero.

It is clear from Fig. 13 that the pseudospin splitting energy may amount to approximately 10 meV. To obtain a larger pseudospin splitting, QW must be more asymmetric. Both *V*<sup>1</sup> and *V*<sup>3</sup> can be varied by shifting the valley energies in gapped graphene under applied stress, with potential barriers playing the role of bandgaps in the gapped graphene sheets. An analogous effect is achieved by applying an electric field on the order of 10<sup>6</sup> V/cm perpendicular to the interfaces in the graphene plane [33].

As expected, the energy spectrum is symmetric under the change *E* → −*E* when *V*<sup>1</sup> and *V*<sup>3</sup> = 0; i.e., the electron and hole spectra have equal effective masses, extremum energies, extremum positions, and pseudospin splitting energies. The electron and hole effective masses in graphene are smaller than those in the gapped graphene sheets adjoining the gapless graphene nanoribbon (*m*∗ <sup>1</sup> <sup>=</sup> <sup>∆</sup>1/*v*<sup>2</sup> *<sup>F</sup>*<sup>1</sup> ≈ 0.11*m*<sup>0</sup> and *m*<sup>∗</sup> <sup>3</sup> <sup>=</sup> <sup>∆</sup>3/*v*<sup>2</sup> *<sup>F</sup>*<sup>3</sup> ≈ 0.15*m*0).

**Figure 11.** Electron (curves *1*) and hole (curves *2*) extremum energies and in the size-quantization spectra as functions of *V*<sup>1</sup> for *V*<sup>3</sup> = 0 **(a)** and as functions of *V*<sup>3</sup> for *V*<sup>1</sup> = 0 **(b)**. The effective bandgap *Eeff <sup>g</sup>* = *E<sup>e</sup>* <sup>0</sup> <sup>+</sup> *Eh* 0 ≈ 629 meV varies insignificantly.

**Figure 12.** Extremum points of size-quantization branches for electrons (*k*∗ *<sup>y</sup>* = *k*<sup>∗</sup> *ye*, curves *1*) and holes (*k*<sup>∗</sup> *<sup>y</sup>* = *k*<sup>∗</sup> *yh*, curves *2*) as functions *V*<sup>1</sup> for *V*<sup>3</sup> = 0 **(a)** and as functions of *V*<sup>3</sup> for *V*<sup>1</sup> = 0 **(b)**. Inserts show the relative positions of dispersion curves for *V*<sup>1</sup> = *V*<sup>3</sup> = 0 **(a)** and at *k*<sup>∗</sup> *ye* = *k*<sup>∗</sup> *yh* **(b)**; *K* and *K* points are set at the same position for simplicity.

**Figure 13.** Pseudospin splitting in electron (curves *1*) and hole (curves *2*) spectra, *δE<sup>e</sup> <sup>s</sup>* and *δE<sup>h</sup> <sup>s</sup>* , as functions *V*<sup>1</sup> for *V*<sup>3</sup> = 0 **(a)** and as functions of *V*<sup>3</sup> for *V*<sup>1</sup> = 0 **(b)**. Vanishing *δEe*,*<sup>h</sup> <sup>s</sup>* corresponds to vanishing *k*<sup>∗</sup> *ye*,*<sup>h</sup>* in Fig. 8, as shown in inserts to **(a)**. Inserts to **(b)** show positions of dispersion curves when *k*∗ *ye* and *k*<sup>∗</sup> *yh* coincide.

#### **4.3. Interface states**

**Figure 10.** Electron (curves *1*) and hole (curves *2*) effective masses in the graphene nanoribbon (in units of free-electron

1 2*m*∗ *h* 

> *k*2 <sup>20</sup>*<sup>h</sup>* <sup>+</sup> *<sup>k</sup>*∗<sup>2</sup> *ye*

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

To estimate characteristic values, we consider the planar heterostructure combining a gapless nanoribbon with gapped graphene sheets with ∆<sup>1</sup> = 0.75 eV, *vF*<sup>1</sup> = 1.1*vF*2, ∆<sup>3</sup> = 1 eV, and *vF*<sup>3</sup> = 1.2*vF*2. The nanoribbon width is *d* = 2.46 nm (ten hexagonal cells). Since the unknown values of *V*<sup>1</sup> and *V*<sup>3</sup> can be found by comparing our results with experimental data, we seek the dependence of energy spectrum parameters on *V*<sup>1</sup> and *V*3. Note that |*V*1| ≤ ∆<sup>1</sup> and

Figures 10–13 show the results of numerical calculations of electron and hole effective

It is clear from Fig. 13 that the pseudospin splitting energy may amount to approximately 10 meV. To obtain a larger pseudospin splitting, QW must be more asymmetric. Both *V*<sup>1</sup> and *V*<sup>3</sup> can be varied by shifting the valley energies in gapped graphene under applied stress, with potential barriers playing the role of bandgaps in the gapped graphene sheets. An analogous effect is achieved by applying an electric field on the order of 10<sup>6</sup> V/cm perpendicular to the

As expected, the energy spectrum is symmetric under the change *E* → −*E* when *V*<sup>1</sup> and *V*<sup>3</sup> = 0; i.e., the electron and hole spectra have equal effective masses, extremum energies, extremum positions, and pseudospin splitting energies. The electron and hole effective masses in graphene are smaller than those in the gapped graphene sheets adjoining the

<sup>1</sup> <sup>=</sup> <sup>∆</sup>1/*v*<sup>2</sup>

*<sup>s</sup>* plotted versus work function for one of the gapped graphene sheets given that

*<sup>F</sup>*<sup>1</sup> ≈ 0.11*m*<sup>0</sup> and *m*<sup>∗</sup>

*ky* − *λk*<sup>∗</sup> *yh* 2 ,

<sup>20</sup>*<sup>h</sup>* + *k*20*ek*

20*h* ,

<sup>20</sup>*<sup>h</sup>* of *k*2*h*(*ky*) and its first and second derivatives at

*xe* and *k*<sup>∗</sup>

<sup>3</sup> <sup>=</sup> <sup>∆</sup>3/*v*<sup>2</sup>

*<sup>F</sup>*<sup>3</sup> ≈ 0.15*m*0).

*xh* values, and pseudospin

(34)

mass *m*0) as functions of *V*<sup>1</sup> for *V*<sup>3</sup> = 0 **(a)** and as functions of *V*<sup>3</sup> for *V*<sup>1</sup> = 0 **(b)**.

*Eh <sup>λ</sup>* <sup>≈</sup> *<sup>E</sup><sup>h</sup>* <sup>0</sup> +

*m*∗ *<sup>h</sup>* <sup>=</sup> <sup>1</sup> *vF*<sup>2</sup>

<sup>20</sup>*h*, *k*

Analogous expressions are obtained for hole energies:

where the respective values *k*20*h*, *k*

<sup>0</sup> = −*vF*<sup>2</sup>

the work function for the other is zero.

interfaces in the graphene plane [33].

gapless graphene nanoribbon (*m*∗

 *k*2 <sup>20</sup>*<sup>h</sup>* <sup>+</sup> *<sup>k</sup>*∗<sup>2</sup> *yh*.


masses in the graphe-ne nanoribbon, extremum energies, *k*∗

*yh*; *<sup>E</sup><sup>h</sup>*

198 Graphene - New Trends and Developments

*ky* = −*λk*<sup>∗</sup>

splitting *δEe*,*<sup>h</sup>*

We consider interface states of a new type that arise in a narrow quasimomentum interval from the crossing of dispersion curves and are analogous to those in narrow-gap semiconductor heterostructures [44]. In the planar graphene-based heterostructure examined here, these states are localized near the heterojunction interfaces between the nanoribbon and the gapped graphene sheets. Interface states can exist not only in QWs but also in quantum barriers [10]. Note that interface states arise as well from the crossing of dispersion curves in a single heterojunction between different graphene materials [42].

The wave function describing an interface electronic state is expressed as follows.

$$\textbf{1. At } x < 0,$$

$$
\tilde{\psi}\_{\lambda}(\mathbf{x}) = \tilde{\mathcal{C}} \begin{pmatrix} 1 \\ \tilde{q}\_1 \end{pmatrix} e^{\kappa\_1 \mathbf{x}} \,\tag{35}
$$

where

$$
\tilde{q}\_1 = -i \frac{u\_1(\kappa\_1 - \lambda k\_y)}{E\_\lambda - V\_1 + \Delta\_1},
$$

$$
v\_{F1}\kappa\_1 = \sqrt{\Delta\_1^2 - (E\_\lambda - V\_1)^2 + v\_{F1}^2 k\_y^2}.
$$

**2.** At 0 *< x < d*,

$$
\tilde{\psi}\_{\lambda}(\mathbf{x}) = \tilde{\mathcal{C}} \begin{pmatrix} \tilde{\mathcal{K}}\_{-} \\ \tilde{q}\_{2}\tilde{\mathcal{K}}\_{-} \end{pmatrix} e^{-\mathsf{K}\_{2}\mathbf{x}} + \tilde{\mathcal{C}} \begin{pmatrix} \tilde{\mathcal{K}}\_{+} \\ \tilde{q}\_{2}^{\prime}\tilde{\mathcal{K}}\_{+} \end{pmatrix} e^{\mathsf{K}\_{2}\mathbf{x}},\tag{36}
$$

where

$$\begin{split} \widetilde{\varkappa}\_{\pm} &= \frac{1}{2} \sqrt{\frac{\upsilon\_{F1}}{\upsilon\_{F2}}} \left[ 1 \pm \frac{\lambda k\_{\mathcal{Y}}}{\kappa\_{2}} \pm \frac{\upsilon\_{F1} (\kappa\_{1} - \lambda k\_{\mathcal{Y}}) E\_{\lambda}}{\upsilon\_{F2} \kappa\_{2} (E\_{\lambda} - V\_{1} + \Delta\_{1})} \right], \\ \widetilde{q}\_{2} &= i \frac{\upsilon\_{F2} (\kappa\_{2} + \lambda k\_{\mathcal{Y}})}{E\_{\lambda}}, \widetilde{q}\_{2}' = -i \frac{\upsilon\_{F2} (\kappa\_{2} - \lambda k\_{\mathcal{Y}})}{E\_{\lambda}}. \end{split}$$

**3.** At *x > d*

$$
\tilde{\psi}\_{\lambda}(\mathbf{x}) = \tilde{\mathcal{C}} \begin{pmatrix} \tilde{\mathcal{G}} \\ \tilde{q}\_{3}\tilde{\zeta} \end{pmatrix} e^{-\kappa\_{3}(\mathbf{x}-\mathbf{d})} \, \tag{37}
$$

,

where

$$
\widetilde{\zeta} = \sqrt{\frac{\upsilon\_{F1}}{\upsilon\_{F3}}} \left[ ch(\kappa\_2 d) + \left( \frac{\lambda k\_y}{\kappa\_2} + \frac{\upsilon\_{F1} (\kappa\_1 - \lambda k\_y) E\_\lambda}{\upsilon\_{F2} \kappa\_2 (E\_\lambda - V\_1 + \Delta\_1)} \right) sh(\kappa\_2 d) \right],
$$

$$
\widetilde{q}\_3 = i \frac{\upsilon\_{F3} (\kappa\_3 + \lambda k\_y)}{E\_\lambda - V\_3 + \Delta\_3},
$$

$$
\upsilon\_{F3} k\_3 = \sqrt{\Delta\_3^2 - (E\_\lambda - V\_3)^2 + \upsilon\_{F3}^2 k\_y^2}.
$$

The relation for energy of interface states is

$$E\_{\Lambda} = \pm v\_{\rm F2} \sqrt{k\_{\rm y}^2 - \kappa\_{\rm 2}^2} \tag{38}$$

with plus and minus corresponding to electrons and holes, respectively.

The expression for energy in (38) implies that an interface state exists only if<sup>4</sup>

$$|\kappa\_2| < |k\_y|.$$

We obtain the dispersion relation

$$
\tanh(\kappa\_2 d) = \upsilon\_{F2} \kappa\_2 f(\lambda k\_{\mathcal{Y}}; \kappa\_1, \kappa\_{\mathcal{Y}}, E\_{\lambda}). \tag{39}
$$

which is similar to (32) up to the substitutions *k*<sup>1</sup> → *κ*1, *k*<sup>2</sup> → *iκ*2, and *k*<sup>3</sup> → *κ*3.

When *V*<sup>1</sup> = 0, the allowed quasimomenta for hole interface states (*λ* = −1) are similar to those for electron states, but the hole and electron energies have opposite signs; i.e., the spectrum is symmetric under the change *E<sup>λ</sup>* → −*Eλ*. When *V*<sup>1</sup> = 100 meV, the symmetry is broken and hole interface states exist only at negative quasimomenta.

#### **4.4. Excitons**

barriers [10]. Note that interface states arise as well from the crossing of dispersion curves in

 1 *q*1 *e*

*u*1(*κ*<sup>1</sup> − *λky*) *E<sup>λ</sup>* − *V*<sup>1</sup> + ∆<sup>1</sup>

<sup>1</sup> <sup>−</sup> (*E<sup>λ</sup>* <sup>−</sup> *<sup>V</sup>*1)<sup>2</sup> <sup>+</sup> *<sup>v</sup>*<sup>2</sup>

,

 <sup>κ</sup><sup>+</sup> *q* <sup>2</sup><sup>κ</sup><sup>+</sup>

*vF*1(*κ*<sup>1</sup> − *λky*)*E<sup>λ</sup> vF*2*κ*2(*E<sup>λ</sup>* − *V*<sup>1</sup> + ∆1)

> *vF*2(*κ*<sup>2</sup> − *λky*) *Eλ*

*vF*1(*κ*<sup>1</sup> − *λky*)*E<sup>λ</sup> vF*2*κ*2(*E<sup>λ</sup>* − *V*<sup>1</sup> + ∆1)

,

*F*3*k*<sup>2</sup> *y*.

*F*1*k*<sup>2</sup> *y*.

> *e*

> > ,

.

*sh*(*κ*2*d*)

<sup>2</sup>, (38)

*<sup>κ</sup>*<sup>1</sup> *<sup>x</sup>*, (35)

*<sup>κ</sup>*<sup>2</sup> *<sup>x</sup>*, (36)

, (37)

The wave function describing an interface electronic state is expressed as follows.

*ψ<sup>λ</sup>*(*x*) = *C*

*<sup>q</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>*<sup>i</sup>*

 ∆2

 <sup>κ</sup><sup>−</sup> *<sup>q</sup>*<sup>2</sup><sup>κ</sup><sup>−</sup>  *e* <sup>−</sup>*κ*<sup>2</sup> *<sup>x</sup>* <sup>+</sup> *<sup>C</sup>*

, *q* <sup>2</sup> = −*i*

 *ζ q*3*ζ e* −*κ*3(*x*−*d*)

*vF*3(*κ*<sup>3</sup> + *λky*) *E<sup>λ</sup>* − *V*<sup>3</sup> + ∆<sup>3</sup>

<sup>3</sup> <sup>−</sup> (*E<sup>λ</sup>* <sup>−</sup> *<sup>V</sup>*3)<sup>2</sup> <sup>+</sup> *<sup>v</sup>*<sup>2</sup>

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

a single heterojunction between different graphene materials [42].

*vF*1*κ*<sup>1</sup> =

*ψ<sup>λ</sup>*(*x*) = *C*

*vF*<sup>1</sup> *vF*<sup>2</sup>  1 ± *λky κ*2 ±

*vF*2(*κ*<sup>2</sup> + *λky*) *Eλ*

*ψ<sup>λ</sup>*(*x*) = *C*

 *λky κ*2 +

*<sup>q</sup>*<sup>3</sup> <sup>=</sup> *<sup>i</sup>*

 ∆2

*E<sup>λ</sup>* = ±*vF*<sup>2</sup>

<sup>κ</sup><sup>±</sup> <sup>=</sup> <sup>1</sup> 2

*<sup>q</sup>*<sup>2</sup> <sup>=</sup> *<sup>i</sup>*

**1.** At *x <* 0,

200 Graphene - New Trends and Developments

**2.** At 0 *< x < d*,

where

where

**3.** At *x > d*

where

*ζ* = *vF*<sup>1</sup> *vF*<sup>3</sup>

The relation for energy of interface states is

*ch*(*κ*2*d*) +

*vF*3*k*<sup>3</sup> =

In gapless graphene, the carrier effective mass is zero and excitons do not exist. The existence of excitons in gapless graphene would lead to excitonic instability and excitonic insulator transition to a gapped state [54, 55].

The energy gap arising in a graphene nanoribbon due to the size quantization makes it possible to generate excitons by optical excitation or electron-hole injection. Excitons in QW strongly affect optical properties of the system considered here.

Excitons in similar quasi-one-dimensional carbon-based systems (semiconducting single- and multi-walled nanotubes) have been studied theoretically in [56]. The exciton spectrum is calculated here for a planar graphene quantum well by using the model applied to quantum wires in [57]. This model yields simple analytical expressions for exciton binding energy.

Since formulas (33) and (34) are obtained in the nonrelativistic limit, the two-particle exciton wave function depending on the electron and hole coordinates *y*<sup>−</sup> and *y*<sup>+</sup> in a sufficiently narrow nanoribbon must obey the 1D Schrödinger equation with Coulomb potential:

$$\left(-\frac{1}{2m\_{\epsilon}^{\*}}\frac{\partial^{2}}{\partial y\_{-}^{2}} - \frac{1}{2m\_{h}^{\*}}\frac{\partial^{2}}{\partial y\_{+}^{2}} - \frac{\vec{\varepsilon}^{2}}{|y\_{-}-y\_{+}|}\right)\phi(y\_{-},y\_{+}) = E^{\prime}\phi(y\_{-},y\_{+}),\tag{40}$$

where *<sup>E</sup>* <sup>=</sup> *<sup>E</sup>* <sup>−</sup> *<sup>E</sup>eff <sup>g</sup>* and *<sup>e</sup>*<sup>2</sup> <sup>≡</sup> *<sup>e</sup>*2/*κeff* . The effective dielectric constant of graphene, *<sup>κ</sup>eff* <sup>=</sup> (*ε* + *ε* )/2, may vary widely with the dielectric constants *ε* and *ε* of the media in contact with graphene, such as free-space permittivity and substrate dielectric constant [58, 59].

<sup>4</sup> The zero mode corresponding to <sup>|</sup>*κ*2<sup>|</sup> <sup>=</sup> <sup>|</sup>*ky* <sup>|</sup> (with *<sup>E</sup><sup>λ</sup>* = 0) is irrelevant here because *<sup>ψ</sup><sup>λ</sup>*(*x*) <sup>≡</sup> 0.

The electron-hole Coulomb interaction in a 1D graphene nanoribbon is three-dimensional, but the problem can be reduced to one dimension (electron and hole *y* positions) for sufficiently narrow nanoribbons.

Rewriting Eq. (40) in terms of electron-hole separation *y* = *y*<sup>−</sup> − *y*<sup>+</sup> and center-of-mass coordinate

$$Y = \frac{m\_\mathcal{e}^\* y\_- + m\_h^\* y\_+}{m\_\mathcal{e}^\* + m\_h^\*}$$

and introducing the function

$$
\phi(y\_{-"\prime}y\_{+}) = \psi\_n(y)e^{iKY}\,\mu
$$

where *K* is the total exciton momentum, we obtain

$$\left(-\frac{1}{2\mu^\*} \frac{\partial^2}{\partial y^2} - \frac{\vec{e}^2}{|y|}\right) \psi\_n(y) = E\_n \psi\_n(y),\tag{41}$$

where *µ*∗ = *m*∗ *<sup>e</sup> m*<sup>∗</sup> *h*/(*m*<sup>∗</sup> *<sup>e</sup>* + *m*<sup>∗</sup> *<sup>h</sup>*) is the reduced mass and *En* is the energy of the *n*th exciton level (*n* = 0, 1, 2, . . . is the principal quantum number). The total exciton energy *E* is obtained by adding the total kinetic energy of the electron-hole pair to *En*:

$$E' = E\_n + \frac{K^2}{2(m\_\varepsilon^\* + m\_h^\*)}.$$

To find the solution at *y >* 0, we substitute *ψn*(*y*) represented as

$$
\psi\_n(y) = B\_n \exp\left(-y/a\_n\right) F\_n\left(\frac{2y}{a\_n}\right).
$$

into Eq. (41) and obtain the confluent hypergeometric differential equation

$$
\xi F\_n'' - \xi F\_n' + \eta F\_n = 0,\tag{42}
$$

where *ξ* = <sup>2</sup>*<sup>y</sup> an* and *<sup>η</sup>* <sup>=</sup> *<sup>µ</sup>*<sup>∗</sup>*<sup>e</sup>*2*an*. We also have

$$E\_n = -\frac{1}{2\mu^\* a\_n^2}.\tag{43}$$

Equation (42) with *η* = *n* is solved by the associated Laguerre polynomial

$$F\_n(\mathfrak{f}) = \frac{1}{n!} \mathfrak{f} e^{\mathfrak{f}} \frac{d^n}{d\mathfrak{f}^n} \left( \mathfrak{f}^{n-1} e^{-\mathfrak{f}} \right) \equiv L\_n^{-1}(\mathfrak{f}).$$

and the wavefunction is expressed as

The electron-hole Coulomb interaction in a 1D graphene nanoribbon is three-dimensional, but the problem can be reduced to one dimension (electron and hole *y* positions) for

Rewriting Eq. (40) in terms of electron-hole separation *y* = *y*<sup>−</sup> − *y*<sup>+</sup> and center-of-mass

*m*∗ *<sup>e</sup>* + *m*<sup>∗</sup> *h*

*φ*(*y*−, *y*+) = *ψn*(*y*)*e*

level (*n* = 0, 1, 2, . . . is the principal quantum number). The total exciton energy *E* is obtained

*K*2 2(*m*∗

*<sup>e</sup>* + *m*<sup>∗</sup> *h*) .

*<sup>e</sup> y*<sup>−</sup> + *m*<sup>∗</sup>

*<sup>h</sup>y*<sup>+</sup>

*iKY*,

*<sup>h</sup>*) is the reduced mass and *En* is the energy of the *n*th exciton

2*y an* .

*ψn*(*y*) = *Enψn*(*y*), (41)

*<sup>n</sup>* + *ηFn* = 0, (42)

. (43)

*<sup>Y</sup>* <sup>=</sup> *<sup>m</sup>*<sup>∗</sup>

*∂*2 *<sup>∂</sup>y*<sup>2</sup> <sup>−</sup> *<sup>e</sup>*<sup>2</sup> |*y*| 

*E* = *En* +

*ψn*(*y*) = *Bn* exp (−*y*/*an*) *Fn*

*En* <sup>=</sup> <sup>−</sup> <sup>1</sup>

2*µ*∗*a*<sup>2</sup> *n*

into Eq. (41) and obtain the confluent hypergeometric differential equation

*ξF <sup>n</sup>* − *ξF*

Equation (42) with *η* = *n* is solved by the associated Laguerre polynomial

*n*! *ξe <sup>ξ</sup> <sup>d</sup><sup>n</sup> dξ<sup>n</sup> ξn*−1*e* −*ξ* <sup>≡</sup> *<sup>L</sup>*−<sup>1</sup> *<sup>n</sup>* (*ξ*).

*Fn*(*ξ*) = <sup>1</sup>

sufficiently narrow nanoribbons.

202 Graphene - New Trends and Developments

and introducing the function

*<sup>e</sup> m*<sup>∗</sup> *h*/(*m*<sup>∗</sup>

where *K* is the total exciton momentum, we obtain

*<sup>e</sup>* + *m*<sup>∗</sup>

 − 1 2*µ*∗

by adding the total kinetic energy of the electron-hole pair to *En*:

To find the solution at *y >* 0, we substitute *ψn*(*y*) represented as

*an* and *<sup>η</sup>* <sup>=</sup> *<sup>µ</sup>*<sup>∗</sup>*<sup>e</sup>*2*an*. We also have

coordinate

where *µ*∗ = *m*∗

where *ξ* = <sup>2</sup>*<sup>y</sup>*

$$
\psi\_n(y) = B\_n \exp\left(-y/a\_n\right) L\_n^{-1}\left(\frac{2y}{a\_n}\right) \dots
$$

Analogously, we find the solution to Eq. (41) at *y <* 0:

$$\psi\_{\boldsymbol{n}}(\boldsymbol{y}) = \pm B\_{\boldsymbol{n}} \exp\left(\boldsymbol{y}/a\_{\boldsymbol{n}}\right) L\_{\boldsymbol{n}}^{-1} \left(-\frac{2\boldsymbol{y}}{a\_{\boldsymbol{n}}}\right) \boldsymbol{\lambda}$$

where "+" and "−" are taken for *n* = 0 and *n* = 0, respectively, and the continuity of *ψn*(*y*) and its first derivative *ψ <sup>n</sup>*(*y*) are used as boundary conditions. Since *ψn*(0) = 0 and *ψ* (0) = 0 for *n* = 0, the excited-state wavefunction *ψn*(*y*) is odd (otherwise, it would be discontinuous at the origin), whereas the ground-state wavefunction is even.

The normalization condition

$$\int\_{-\infty}^{\infty} |\psi\_n(y)|^2 dy = 1$$

is used to determine the coefficient *Bn* in the expression for *ψn*(*y*):

$$B\_n = \left[ a\_n \int\_0^\infty (L\_n^{-1}(\xi))^2 e^{-\xi} d\xi \right]^{-1/2}$$

and *Bn* <sup>=</sup> 1/√2*an* for *<sup>n</sup>* <sup>=</sup> 1, 2, . . . and *<sup>B</sup>*<sup>0</sup> <sup>=</sup> 1/√*a*<sup>0</sup> for *<sup>n</sup>* = 0. Here,

$$a\_{ll} = \frac{n}{\mu^\* \tilde{c}^2} \tag{44}$$

(*n* = 1, 2 . . .) is the Bohr radius of an exciton in the *n*th excited state. Combining (43) with (44), we find the exciton energy spectrum:

$$E\_n = -\frac{\mu^\* \overleftarrow{e}^4}{2n^2}.\tag{45}$$

The 1D ground-state (*n* = 0) Coulomb energy exhibits a logarithmic divergence at short distances [60]. Therefore, the lateral spread of the exciton wave function (along the *x* axis) due to the three-dimensional nature of Coulomb interaction should be taken into account by introducing a cutoff parameter *d*<sup>0</sup> *d*. Averaging the kinetic energy operator

$$
\widehat{T} = -\frac{1}{2\mu^\*} \frac{\partial^2}{\partial y^2}
$$

and the potential

$$V(y) = -\frac{\vec{e}^2}{|y|}\theta\left(|y| - d\_0\right)$$

over ground-state wave functions

$$
\psi\_0(y) = \frac{1}{\sqrt{a\_0}} e^{-|y|/a\_0},
\tag{46}
$$

where the ground-state Bohr radius *a*<sup>0</sup> plays the role of a variational parameter, we express the ground-state exciton energy as [42]

$$E\_0 = \frac{1}{2\mu^\* a\_0^2} - \frac{2\tilde{e}^2}{a\_0} \ln \frac{a\_0}{d}. \tag{47}$$

Minimizing (47) with respect to *a*0, we obtain an equation for *a*0:

$$a\_0 = \frac{a\_1}{2\left(\ln\frac{a\_0}{d} - 1\right)}.\tag{48}$$

To logarithmic accuracy, when

$$
\ln \frac{a\_1}{d} \gg 1,\tag{49}
$$

we find the relations

$$E\_0 = 4E\_1 \ln^2 \frac{a\_1}{d} \,\tag{50}$$

$$a\_0 = \frac{a\_1}{2\ln\frac{a\_1}{d}}.\tag{51}$$

Using (48), we easily obtain the next-order correction to *E*0:

$$
\delta E\_0^{(1)} = -8E\_1 \ln \frac{a\_1}{d} \ln \left( 2 \ln \frac{a\_1}{d} \right) .
$$

We now examine the applicability of the formulas derived here. The semiconducting state induced in a graphene nanoribbon is stable with respect to spontaneous electron-hole pair creation (excitonic insulator transition) only if the exciton binding energy |*E*0| is smaller than the effective bandgap in the graphene nanoribbon,

$$|E\_0| < E\_{\mathfrak{g}}^{eff}.$$

Furthermore, the quantum well width *d* must be much smaller than the exciton Bohr radius *a*1,

$$d \ll a\_1.$$

Logarithmically accurate formula (50) is correct only if condition (49) holds. However, the asymmetric QW analyzed here to examine pseudospin effects may not admit even a single size-quantization level if the graphene nanoribbon width *d* is too narrow. As *d* decreases, the effective bandgap increases, approaching ∆<sup>+</sup> + ∆−, where ∆<sup>±</sup> = min{∆<sup>1</sup> ± *V*1, ∆<sup>3</sup> ± *V*3} (with plus and minus corresponding to electrons and holes, respectively). When a certain *dc* is reached, the size-quantization levels are pushed into the continuum. This imposes a lower limit on *d*:

$$d > d\_{\infty}$$

where *dc* can be estimated as [42]

*<sup>T</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup> 2*µ*∗


<sup>√</sup>*a*<sup>0</sup> *e*

where the ground-state Bohr radius *a*<sup>0</sup> plays the role of a variational parameter, we express

<sup>−</sup> <sup>2</sup>*<sup>e</sup>*<sup>2</sup> *a*0

ln *<sup>a</sup>*<sup>0</sup>

*<sup>V</sup>*(*y*) = <sup>−</sup> *<sup>e</sup>*<sup>2</sup>

*<sup>ψ</sup>*0(*y*) = <sup>1</sup>

*<sup>E</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> 2*µ*∗*a*<sup>2</sup> 0

> *<sup>a</sup>*<sup>0</sup> <sup>=</sup> *<sup>a</sup>*<sup>1</sup> 2 ln *<sup>a</sup>*<sup>0</sup> *<sup>d</sup>* − 1

> > ln *<sup>a</sup>*<sup>1</sup>

*<sup>E</sup>*<sup>0</sup> <sup>=</sup> <sup>4</sup>*E*<sup>1</sup> ln<sup>2</sup> *<sup>a</sup>*<sup>1</sup>

*<sup>a</sup>*<sup>0</sup> <sup>=</sup> *<sup>a</sup>*<sup>1</sup> 2 ln *<sup>a</sup>*<sup>1</sup> *d*

> *<sup>d</sup>* ln 2 ln *<sup>a</sup>*<sup>1</sup> *d* .

We now examine the applicability of the formulas derived here. The semiconducting state induced in a graphene nanoribbon is stable with respect to spontaneous electron-hole pair creation (excitonic insulator transition) only if the exciton binding energy |*E*0| is smaller than

<sup>0</sup> <sup>=</sup> <sup>−</sup>8*E*<sup>1</sup> ln *<sup>a</sup>*<sup>1</sup>

Minimizing (47) with respect to *a*0, we obtain an equation for *a*0:

Using (48), we easily obtain the next-order correction to *E*0:

the effective bandgap in the graphene nanoribbon,

*<sup>δ</sup>E*(1)

and the potential

over ground-state wave functions

204 Graphene - New Trends and Developments

the ground-state exciton energy as [42]

To logarithmic accuracy, when

we find the relations

*∂*2 *∂y*<sup>2</sup>

*θ* (|*y*| − *d*0)

−|*y*|/*a*<sup>0</sup> , (46)

*<sup>d</sup>* . (47)

. (48)

*<sup>d</sup>* 1, (49)

*<sup>d</sup>* , (50)

. (51)

$$d\_{\mathcal{C}} \simeq \frac{\pi v\_{F2}}{\Delta\_{+} + \Delta\_{-}}.$$

As *d* increases, condition (49) is violated. In this case, a more accurate variational calculation should be performed using the modified three-dimensional Coulomb potential

$$
\tilde{V}(y) = -\frac{\vec{c}^2}{\sqrt{y^2 + d\_0^2}},
$$

where *d*<sup>0</sup> is a cutoff parameter. We average the Hamiltonian with potential *V*(*y*) over trial functions (46) to obtain

$$E\_0 = \frac{1}{2\mu^\* a\_0^2} - \frac{2\tilde{e}^2}{a\_0} I(\rho),\tag{52}$$

where *I*(*ρ*) = *<sup>π</sup>* <sup>2</sup> [*H*0(*ρ*) − *Y*0(*ρ*)], *Hν*(*ρ*) is a Struve function, *Yν*(*ρ*) is a Bessel function of the second kind, and *ρ* = 2*d*0/*a*<sup>0</sup> (*ν* is 0 here and 1 below).

Minimizing (52) with respect to *a*0, we obtain an equation for *a*0:

$$\frac{2a\_0}{a\_1}I(\rho) + \frac{4d\_0}{a\_1}I(\rho) = 1,\tag{53}$$

where *<sup>J</sup>*(*ρ*) = <sup>1</sup> <sup>−</sup> *<sup>π</sup>* <sup>2</sup> [*H*1(*ρ*) − *Y*1(*ρ*)].

Figure 14 shows the numerical results obtained by using both methods to calculate *E*0(*d*) for the heterostructure, with *d*<sup>0</sup> ∝ *d* adjusted to match the curves at small *d*. Discrepancy at large *d* increases as ln(*a*1/*d*) approaches unity.

**Figure 14.** Exciton ground-state energy calculated by formula (50) (curve *1*) and by formula (52) after Eq. (53) is solved numerically for *a*<sup>0</sup> (curve *2*), *d*<sup>0</sup> = 0.22*d*.
