**4.5. Electric field effect on excitons levels**

Interaction between an exciton and an external electrostatic field *E* is described by the operator

$$
\widehat{H}\_{\!i} = -\mathbf{d} \mathcal{E} = |\overrightarrow{e}| (\mathcal{E}\_{\!x} \mathbf{x} + \mathcal{E}\_{\!y} \mathbf{y}),
$$

where *x* = |*x*<sup>−</sup> − *x*+| and *y* = |*y*<sup>−</sup> − *y*+| are the electron-hole relative position vector components and **d** is dipole moment. The electric field is supposed to be weak enough to ensure that the energy level shift is not only smaller than the spacing between size-quantization levels but also smaller than the spacing between exciton levels. These conditions can be written as

$$d \ll a\_1 \ll a\_{\mathcal{E}'} $$

where *<sup>a</sup>*<sup>E</sup> = (*µ*∗|*<sup>e</sup>*|E)−1/3 is the electric length.

We consider two cases: (1) the electric field is applied parallel to the *x* axis and perpendicular to the nanoribbon edges in the graphene plane; (2) the electric field is applied along the *y* axis, parallel to the nanoribbon edges.

In the former case, the energy shift varies linearly with the difference between the average *x* components of the electron and hole position vectors:

$$E^{(1)}\_{\\\perp\lambda\lambda'} = |\overleftarrow{\varepsilon}| \mathcal{E} \left( \langle \mathbf{x}\_{-} \rangle\_{\lambda} - \langle \mathbf{x}\_{+} \rangle\_{\lambda'} \right),\tag{54}$$

where average *x* components are calculated by using electron and hole single-particle wave functions, generally depending on the electron and hole eigenvalues *λ* and *λ* of the operator *P*, respectively. Exciton energy shift (54) is independent of the principal quantum number *n*. It may vary with *λ* and *λ* , resulting in different exciton binding energies (more precisely, the binding energy of an electron–hole pair with *λ* = ±1 and *λ* = ±1 may have four different values).

In the latter case, the first-order electric field-induced correction is zero5,

<sup>5</sup> Note that *E*(1) <sup>⊥</sup>*<sup>n</sup>* <sup>≡</sup> 0 in the former case if the electron and hole spectra transform into each other under field inversion.

$$E\_{\parallel n}^{(1)} = |\vec{e}| \mathcal{E} \langle y \rangle\_n \equiv 0,\tag{55}$$

because the integral of *y*|*ψn*(*y*)| <sup>2</sup> with respect to *y* vanishes. To evaluate the second-order electric field-induced correction, we make use of the Dalgarno–Lewis perturbation theory [61]. Defining a Hermitian operator such that

$$[\![\overleftarrow{F}, \,\,\hat{H}\_0] | n\rangle = \hat{H}\_i | n\rangle,\tag{56}$$

where

**Figure 14.** Exciton ground-state energy calculated by formula (50) (curve *1*) and by formula (52) after Eq. (53) is solved

Interaction between an exciton and an external electrostatic field *E* is described by the

*<sup>H</sup><sup>i</sup>* <sup>=</sup> <sup>−</sup>**d***<sup>E</sup>* <sup>=</sup> <sup>|</sup>*<sup>e</sup>*|(E*<sup>x</sup> <sup>x</sup>* <sup>+</sup> <sup>E</sup>*yy*) where *x* = |*x*<sup>−</sup> − *x*+| and *y* = |*y*<sup>−</sup> − *y*+| are the electron-hole relative position vector components and **d** is dipole moment. The electric field is supposed to be weak enough to ensure that the energy level shift is not only smaller than the spacing between size-quantization levels but also smaller than the spacing between exciton levels. These

*<sup>d</sup> <sup>a</sup>*<sup>1</sup> *<sup>a</sup>*<sup>E</sup> ,

We consider two cases: (1) the electric field is applied parallel to the *x* axis and perpendicular to the nanoribbon edges in the graphene plane; (2) the electric field is applied along the *y*

In the former case, the energy shift varies linearly with the difference between the average *x*

where average *x* components are calculated by using electron and hole single-particle wave functions, generally depending on the electron and hole eigenvalues *λ* and *λ* of the operator *P*, respectively. Exciton energy shift (54) is independent of the principal quantum number *n*.

binding energy of an electron–hole pair with *λ* = ±1 and *λ* = ±1 may have four different

<sup>⊥</sup>*<sup>n</sup>* <sup>≡</sup> 0 in the former case if the electron and hole spectra transform into each other under field inversion.

<sup>⊥</sup>*λλ* <sup>=</sup> <sup>|</sup>*<sup>e</sup>*|E (*x*−*<sup>λ</sup>* − *x*+*λ*), (54)

, resulting in different exciton binding energies (more precisely, the

numerically for *a*<sup>0</sup> (curve *2*), *d*<sup>0</sup> = 0.22*d*.

206 Graphene - New Trends and Developments

conditions can be written as

It may vary with *λ* and *λ*

values).

<sup>5</sup> Note that *E*(1)

operator

**4.5. Electric field effect on excitons levels**

where *<sup>a</sup>*<sup>E</sup> = (*µ*∗|*<sup>e</sup>*|E)−1/3 is the electric length.

components of the electron and hole position vectors:

*E*(1)

In the latter case, the first-order electric field-induced correction is zero5,

axis, parallel to the nanoribbon edges.

$$\hat{H}\_0 = -\frac{1}{2\mu^\*} \frac{\partial^2}{\partial y^2} - \frac{\vec{e}^2}{|y|}$$

is the zeroth-order Hamiltonian, |*n* = *ψn*(*y*) is the zeroth-order wave function of the *n*th exciton level, and *<sup>H</sup><sup>i</sup>* <sup>=</sup> <sup>|</sup>*<sup>e</sup>*|E*y*, we obtain

$$E\_{\parallel n}^{(2)} = \langle n | \hat{H}\_i \hat{F} | n \rangle - \langle n | \hat{H}\_i | n \rangle \langle n | \hat{F} | n \rangle. \tag{57}$$

In the case in question, the second term in this formula vanishes by virtue of (55). Rewriting Eq. (56) as

$$
\psi\_n \frac{\partial^2 \widehat{F}}{\partial y^2} + 2 \frac{\partial \psi\_n}{\partial y} \frac{\partial \widehat{F}}{\partial y} = 2 \mu^\* \widehat{H}\_l \psi\_{n\prime} \tag{58}
$$

we find

$$
\hat{F}(y) = 2\mu^\* \int\_{-\infty}^{y} \frac{dy'}{|\psi\_n(y')|^2} \int\_{-\infty}^{y'} dy'' \psi\_n^\*(y'') \hat{H}\_l \psi\_n(y''). \tag{59}
$$

Combining (57) with (59), we have the exciton ground-state energy shift [42]

$$E\_{\parallel 0}^{(2)} = -\frac{5}{128} \frac{a\_1^3}{\ln^4 \frac{a\_1}{d}} \mathcal{E}^2 \,, \tag{60}$$

which is very small compared to *E*<sup>0</sup> given by (50) because of the fourth power of a logarithm in the denominator and a small numerical factor.

For comparison, we write out the energy correction to the first excited exciton state [42]

$$E\_{\parallel 1}^{(2)} = -\frac{3}{8} (31 - 6\gamma) a\_1^3 \mathcal{E}^2 \lambda$$

where *γ* = 0.577. . . is Euler's constant.

By analogy with layered heterostructures [62], the ionizing (exciton-breaking) field strength E*<sup>c</sup>* is estimated as

$$\mathcal{E}\_{\mathbb{C}} = \frac{|E\_0|}{8|\vec{\varepsilon}| \langle |y| \rangle\_0 \text{\textquotedblleft}},\tag{61}$$

where |*y*|<sup>0</sup> = *a*0/2 is the average electron-hole separation for the ground-state exciton. To logarithmic accuracy, it follows that [42]

$$\mathcal{E}\_{\mathfrak{c}} = \mu^{\*2} |\vec{e}|^{5} \ln^{3} \frac{a\_{1}}{d}. \tag{62}$$

To get the order of magnitude of E*c*, consider QW discussed abode. Setting *m*<sup>∗</sup> *<sup>e</sup>* = *m*<sup>∗</sup> *h* ≈ 0.0056*m*0, the SiO2 substrate dielectric constant *κeff* ≈ 5, *d* = 2.46 nm, and *a*<sup>1</sup> ≈ 81 nm, we use formula (62) to obtain E*<sup>c</sup>* = 9 kV/cm.

#### **4.6. The effective theory**

#### *4.6.1. The effective Hamiltonian*

According to the expressions (33) and (34) we have the approximations for dispersion curves of electrons and holes respectively

$$\begin{split} E\_{\lambda}^{\epsilon} &\approx E\_0^{\epsilon} + \frac{1}{2m\_{\epsilon}^\*} \left( k\_{\mathcal{Y}} - \lambda k\_{\mathcal{Y}\epsilon}^\* \right)^2 = \frac{k\_{\mathcal{Y}}^2}{2m\_{\epsilon}^\*} - \lambda a\_{\epsilon} k\_{\mathcal{Y}} + \Delta\_{\epsilon} \equiv \tilde{E}\_{\lambda'}^{\epsilon} \\ E\_{\lambda}^h &\approx E\_0^h - \frac{1}{2m\_{\tilde{h}}^\*} \left( k\_{\mathcal{Y}} + \lambda k\_{gh}^\* \right)^2 = -\frac{k\_{\mathcal{Y}}^2}{2m\_{\tilde{h}}^\*} - \lambda a\_h k\_{\mathcal{Y}} + \Delta\_h \equiv \tilde{E}\_{\lambda'}^h \end{split} \tag{63}$$

where the following notations are introduced

$$\alpha\_{\varepsilon,h} = \frac{k\_{ye,h}^\*}{m\_{\varepsilon,h}^\*}, \ \Delta\_{\varepsilon,h} = \pm \frac{k\_{ye,h}^{\*2}}{2m\_{\varepsilon,h}^\*} + E\_0^{\varepsilon,h}$$

(in the latter formula plus corresponds to the case of electrons, minus for the case of holes). We can write the effective Hamiltonians in the form including explicitly the parity *λ*

$$\begin{split} \hat{H}\_{eff}^{(\lambda)\varepsilon} &= \frac{\hat{p}\_y^2}{2m\_\varepsilon^\*} - \lambda \alpha\_\varepsilon \hat{p}\_y + \Delta\_{\varepsilon \prime} \\ \hat{H}\_{eff}^{(\lambda)h} &= \frac{\hat{p}\_y^2}{2m\_h^\*} - \lambda \alpha\_h \hat{p}\_y + \Delta\_h. \end{split} \tag{64}$$

We can also combine the Hamiltonians *<sup>H</sup>*(+1)*<sup>e</sup> eff* and *<sup>H</sup>*(−1)*<sup>e</sup> eff* into one 2×2 matrix Hamiltonian (analogously for *<sup>H</sup>*(+1)*<sup>h</sup> eff* and *<sup>H</sup>*(−1)*<sup>h</sup> eff* )

where *γ* = 0.577. . . is Euler's constant.

208 Graphene - New Trends and Developments

logarithmic accuracy, it follows that [42]

use formula (62) to obtain E*<sup>c</sup>* = 9 kV/cm.

**4.6. The effective theory**

*4.6.1. The effective Hamiltonian*

of electrons and holes respectively

*Ee <sup>λ</sup>* <sup>≈</sup> *<sup>E</sup><sup>e</sup>* <sup>0</sup> +

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

1 2*m*∗ *e* 

<sup>0</sup> <sup>−</sup> <sup>1</sup> 2*m*∗ *h* 

where the following notations are introduced

*ky* − *λk*<sup>∗</sup> *ye* 2 <sup>=</sup> *<sup>k</sup>*<sup>2</sup> *y* 2*m*∗ *e*

*ky* + *λk*∗ *yh* 2

> *ye*,*h m*∗ *e*,*h*

, ∆*e*,*<sup>h</sup>* = ±

(in the latter formula plus corresponds to the case of electrons, minus for the case of holes).

We can write the effective Hamiltonians in the form including explicitly the parity *λ*

*<sup>α</sup>e*,*<sup>h</sup>* <sup>=</sup> *<sup>k</sup>*<sup>∗</sup>

*<sup>H</sup>*(*λ*)*<sup>e</sup> eff* <sup>=</sup> *<sup>p</sup>*<sup>2</sup> *y* 2*m*∗ *e*

*<sup>H</sup>*(*λ*)*<sup>h</sup> eff* <sup>=</sup> *<sup>p</sup>*<sup>2</sup> *y* 2*m*∗ *h*

E*<sup>c</sup>* is estimated as

By analogy with layered heterostructures [62], the ionizing (exciton-breaking) field strength

<sup>E</sup>*<sup>c</sup>* <sup>=</sup> <sup>|</sup>*E*0<sup>|</sup>

<sup>E</sup>*<sup>c</sup>* <sup>=</sup> *<sup>µ</sup>*∗2<sup>|</sup>*<sup>e</sup>*<sup>|</sup>

To get the order of magnitude of E*c*, consider QW discussed abode. Setting *m*<sup>∗</sup>

where |*y*|<sup>0</sup> = *a*0/2 is the average electron-hole separation for the ground-state exciton. To

0.0056*m*0, the SiO2 substrate dielectric constant *κeff* ≈ 5, *d* = 2.46 nm, and *a*<sup>1</sup> ≈ 81 nm, we

According to the expressions (33) and (34) we have the approximations for dispersion curves

<sup>=</sup> <sup>−</sup> *<sup>k</sup>*<sup>2</sup> *y* 2*m*∗ *h*

> *k*∗<sup>2</sup> *ye*,*h* 2*m*∗ *e*,*h*

<sup>−</sup> *λα<sup>e</sup> <sup>p</sup><sup>y</sup>* <sup>+</sup> <sup>∆</sup>*e*,

<sup>−</sup> *λα<sup>h</sup> <sup>p</sup><sup>y</sup>* <sup>+</sup> <sup>∆</sup>*h*.

+ *Ee*,*<sup>h</sup>* 0

<sup>8</sup><sup>|</sup>*<sup>e</sup>*||*y*|<sup>0</sup>

<sup>5</sup> ln3 *<sup>a</sup>*<sup>1</sup>

, (61)

*<sup>d</sup>* . (62)

*λ*,

*λ*,

<sup>−</sup> *λαeky* <sup>+</sup> <sup>∆</sup>*<sup>e</sup>* <sup>≡</sup> *<sup>E</sup><sup>e</sup>*

<sup>−</sup> *λαhky* <sup>+</sup> <sup>∆</sup>*<sup>h</sup>* <sup>≡</sup> *<sup>E</sup><sup>h</sup>*

*<sup>e</sup>* = *m*<sup>∗</sup> *h* ≈

(63)

(64)

$$\begin{split} \hat{H}\_{eff}^{\varepsilon} &= \frac{\hat{p}\_{y}^{2}}{2m\_{\varepsilon}^{\*}} - \alpha\_{\varepsilon}\tau\_{z}\hat{p}\_{y} + \Delta\_{\varepsilon}, \\ \hat{H}\_{eff}^{h} &= \frac{\hat{p}\_{y}^{2}}{2m\_{h}^{\*}} - \alpha\_{h}\tau\_{z}\hat{p}\_{y} + \Delta\_{h}. \end{split} \tag{65}$$

Here we emphasize by using the matrix *τ<sup>z</sup>* that these Hamiltonians act in the valley space. The eigen wave functions of these Hamiltonians <sup>Ψ</sup>*<sup>e</sup> <sup>λ</sup>* and <sup>Ψ</sup>*<sup>h</sup> λ*,

$$
\hat{H}^{\varepsilon}\_{\varepsilon f} \tilde{\mathbf{Y}}^{\varepsilon}\_{\lambda} = \tilde{E}^{\varepsilon}\_{\lambda} \tilde{\mathbf{Y}}^{\varepsilon}\_{\lambda} \text{ and } \hat{H}^{h}\_{eff} \tilde{\mathbf{Y}}^{h}\_{\lambda} = \tilde{E}^{h}\_{\lambda} \tilde{\mathbf{Y}}^{h}\_{\lambda}.
$$

are spinors in a class of eigen functions of the operator <sup>P</sup><sup>ˆ</sup> <sup>=</sup> *<sup>τ</sup><sup>z</sup>* which can be considered as a "reduced" parity operator:

$$
\hat{\mathcal{P}}\tilde{\Psi}\_{\lambda}^{e\_{\mathcal{I}}h} = \lambda \tilde{\Psi}\_{\lambda}^{e\_{\mathcal{I}}h},
$$

$$
\tilde{\Psi}\_{+1}^{e\_{\mathcal{I}}h} = \begin{pmatrix} \tilde{\Psi}\_{+1}^{e\_{\mathcal{I}}h} \\ 0 \end{pmatrix}, \ \tilde{\Psi}\_{-1}^{e\_{\mathcal{I}}h} = \begin{pmatrix} 0 \\ \tilde{\Psi}\_{-1}^{e\_{\mathcal{I}}h} \end{pmatrix}.\tag{66}
$$

(Remember that the parity operator *P* = *τ<sup>z</sup>* ⊗ *σ*<sup>0</sup> where the matrix *σ*<sup>0</sup> acts in the sublattice space.)

We can also see that the second term in (65) is an analogue of spin-orbit (SO) coupling in the Rashba form:

$$
\hat{H}\_{SO}^{e,h} = \mathfrak{a}\_{e,h} \left[ \boldsymbol{\pi} \hat{\mathbf{p}} \right] \cdot \boldsymbol{\nu} \,, \tag{67}
$$

where τ = *τx*, *τy*, *τ<sup>z</sup>* is the matrix vector in the valley space (τ is the pseudospin operator), **<sup>p</sup>** <sup>=</sup> *<sup>p</sup><sup>x</sup>*, *<sup>p</sup><sup>y</sup>*, 0 is the vector operator of momentum in the *xy*-plane, and ν is the unit vector of the normal to the heterojunction interface (in the coordinate system used here, ν = **e***<sup>x</sup>* is the unit coordinate vector of the *x* axis). The constants *αe*,*<sup>h</sup>* can be named as the effective Rashba constants.

So, we have the effective theory with the non-relativistic Hamiltonian with the SO-like term describing the pseudospin splitting of the energy spectrum of charge carriers. An appearance of this term is due to an asymmetry of a QW potential profile. In an absence of the asymmetry in the case of the symmetrical QW, the effective Rashba constants *αe*,*<sup>h</sup>* tend to zero for both electrons and holes and there is no the pseudospin splitting in its energy spectra.

#### *4.6.2. The effective Hamiltonian in a presence of a magnetic field*

Let us set a problem about a quantization of a charge carriers energy in the planar heterostructure based on graphene in a magnetic field applied perpendicular to its plane. To begin with, we make some remarks.

Firstly, to separate the size quantization and the magnetic field quantization, we assume that a magnetic field *H* is enough weak one. Its condition can be expressed as the following inequality

$$a\_H \gg d,\tag{68}$$

where

$$a\_H = \sqrt{\frac{c}{|e|H}}$$

is the magnetic length, i.e. a restriction of the wave function along the *y* axis due to a magnetic field is significantly smaller than its restriction along the *x* axis (perpendicular to the potential barriers). A complexity of the problem consists in an absence of the usual Landau quantization whereas we have it in layered heterostructures at an application of a magnetic field perpendicular to layers (in this case problems about the size quantization and the magnetic field quantization are automatically separated).

Secondly, a vector potential **<sup>A</sup>** must be chosen so that decreasing wave function <sup>Ψ</sup>*<sup>e</sup>*,*<sup>h</sup> <sup>λ</sup>* (*x*, *y*) in the direction of the *y* axis is taken into account in an explicit form. The equation for <sup>Ψ</sup>*<sup>e</sup>*,*<sup>h</sup> <sup>λ</sup>* (*x*, *y*) in a presence the of a magnetic field must include a *y* coordinate. From general considerations, it is clear that <sup>Ψ</sup>*<sup>e</sup>*,*<sup>h</sup> <sup>λ</sup>* (*x*, *y*) must decrease along the *y* axis on the scale of *aH* (see the schematic picture on Fig. 15).

An appropriate choice of the vector potential is

$$\mathbf{A} = \begin{pmatrix} -Hy, \ 0, \ 0 \end{pmatrix}.$$

To have an opportunity to make the minimal substitution, the effective equation for <sup>Ψ</sup>*<sup>e</sup>*,*<sup>h</sup> <sup>λ</sup>* (*x*, *y*) must explicitly contain the momentum operator *<sup>p</sup><sup>x</sup>*. However, this operator was excluded from the effective Hamiltonian without a magnetic field (65). This circumstance makes us detach in the effective equation an artificial term corresponding to the "size-quantization energy" *Ee*,*<sup>h</sup>* <sup>0</sup> so that

$$\frac{\hat{p}\_\times^2}{2m\_{\varepsilon,h}^\*} \tilde{\Psi}\_\lambda^{\varepsilon,h}(\mathbf{x}, \mathbf{y}) = E\_0^{\varepsilon,h} \tilde{\Psi}\_\lambda^{\varepsilon,h}(\mathbf{x}, \mathbf{y}).$$

Moreover, we propose to use an approximate expression

$$E\_0^{\varepsilon,h} = \pm \frac{k\_{xe,h}^{\ast 2}}{2m\_{e,h}^{\ast}}.$$

**Figure 15.** A behavior of the envelope wave function <sup>Ψ</sup>*<sup>e</sup>*,*<sup>h</sup> <sup>λ</sup>* (*x*, *y*) of charge carriers in the planar heterostructure in a presence of a magnetic field applied perpendicular to its plane.

where *k*∗ *xe* and *k*<sup>∗</sup> *xh* are the size-quantized values of *kx* as eigen values of the operator *<sup>p</sup><sup>x</sup>* for electrons and holes near the extrema *ky* = *λk*∗ *ye* and *ky* = −*λk*<sup>∗</sup> *yh*, respectively.

So, we have in a presence of a magnetic field

$$\begin{split} \hat{H}\_{eff}^{\prime \epsilon} &= \frac{1}{2m\_{\epsilon}^{\ast}} \left( k\_{\text{xc}}^{\ast} + \frac{e}{c} H y \right)^{2} + \frac{1}{2m\_{\epsilon}^{\ast}} \hat{p}\_{\text{y}}^{2} + \hat{H}\_{\text{SO}}^{\varepsilon} + \tilde{\Delta}\_{\text{\epsilon}}, \\ \hat{H}\_{eff}^{\prime \hbar} &= \frac{1}{2m\_{\hbar}^{\ast}} \left( k\_{\text{x}\hbar}^{\ast} + \frac{e}{c} H y \right)^{2} + \frac{1}{2m\_{\hbar}^{\ast}} \hat{p}\_{\text{y}}^{2} + \hat{H}\_{\text{SO}}^{\hbar} + \tilde{\Delta}\_{\text{\hbar}}. \end{split} \tag{69}$$

with

*4.6.2. The effective Hamiltonian in a presence of a magnetic field*

the magnetic field quantization are automatically separated).

*p*2 *x* 2*m*∗ *e*,*h* <sup>Ψ</sup>*<sup>e</sup>*,*<sup>h</sup>*

Moreover, we propose to use an approximate expression

To begin with, we make some remarks.

210 Graphene - New Trends and Developments

considerations, it is clear that <sup>Ψ</sup>*<sup>e</sup>*,*<sup>h</sup>*

(see the schematic picture on Fig. 15).

An appropriate choice of the vector potential is

inequality

where

<sup>Ψ</sup>*<sup>e</sup>*,*<sup>h</sup>*

energy" *Ee*,*<sup>h</sup>*

<sup>0</sup> so that

Let us set a problem about a quantization of a charge carriers energy in the planar heterostructure based on graphene in a magnetic field applied perpendicular to its plane.

Firstly, to separate the size quantization and the magnetic field quantization, we assume that a magnetic field *H* is enough weak one. Its condition can be expressed as the following

> *c* |*e*|*H*

is the magnetic length, i.e. a restriction of the wave function along the *y* axis due to a magnetic field is significantly smaller than its restriction along the *x* axis (perpendicular to the potential barriers). A complexity of the problem consists in an absence of the usual Landau quantization whereas we have it in layered heterostructures at an application of a magnetic field perpendicular to layers (in this case problems about the size quantization and

*aH* =

Secondly, a vector potential **<sup>A</sup>** must be chosen so that decreasing wave function <sup>Ψ</sup>*<sup>e</sup>*,*<sup>h</sup>*

in the direction of the *y* axis is taken into account in an explicit form. The equation for

**A** = (−*Hy*, 0, 0).

must explicitly contain the momentum operator *<sup>p</sup><sup>x</sup>*. However, this operator was excluded from the effective Hamiltonian without a magnetic field (65). This circumstance makes us detach in the effective equation an artificial term corresponding to the "size-quantization

*<sup>λ</sup>* (*x*, *<sup>y</sup>*) = *<sup>E</sup>e*,*<sup>h</sup>*

*Ee*,*<sup>h</sup>* <sup>0</sup> = ± <sup>0</sup> <sup>Ψ</sup>*<sup>e</sup>*,*<sup>h</sup>*

*k*∗<sup>2</sup> *xe*,*h* 2*m*∗ *e*,*h* , *<sup>λ</sup>* (*x*, *y*).

To have an opportunity to make the minimal substitution, the effective equation for <sup>Ψ</sup>*<sup>e</sup>*,*<sup>h</sup>*

*<sup>λ</sup>* (*x*, *y*) in a presence the of a magnetic field must include a *y* coordinate. From general

*aH d*, (68)

*<sup>λ</sup>* (*x*, *y*) must decrease along the *y* axis on the scale of *aH*

*<sup>λ</sup>* (*x*, *y*)

*<sup>λ</sup>* (*x*, *y*)

$$
\tilde{\Delta}\_{e,h} = \pm \frac{k\_{ye,h}^{\*2}}{2m\_{e,h}^{\*}}.
$$

We can introduce operators of a generalized momentum in a magnetic field

$$\begin{aligned} \widehat{P}^{e,h}\_{\chi} &= k^\*\_{\ge e,h} - \frac{e}{c} \mathcal{A}\_{\chi} \\ \widehat{P}^{e,h}\_{\underline{y}} &= \widehat{p}\_{\underline{y}} - \frac{e}{c} \mathcal{A}\_{\underline{y}} \end{aligned} \tag{70}$$

The operators *<sup>P</sup><sup>e</sup>*,*<sup>h</sup> <sup>x</sup>* are c-numbers, whereas the operators *<sup>P</sup><sup>e</sup>*,*<sup>h</sup> <sup>y</sup>* coincide with the differential operator *<sup>p</sup><sup>y</sup>* <sup>=</sup> <sup>−</sup>*i∂<sup>y</sup>* in the determined above vector potential **<sup>A</sup>**. The commutation relation for these operators is

$$\mathbb{E}\left[\widehat{P}\_{\mathcal{X}}^{\varepsilon,\hbar}, \widehat{P}\_{\mathcal{Y}}^{\varepsilon,\hbar}\right] = \frac{ie}{c}H.\tag{71}$$

We can also consider the combinations of these operators

$$
\widehat{P}^{\varepsilon,h}\_{\pm} = \widehat{P}^{\varepsilon,h}\_{\mathfrak{x}} \pm i \widehat{P}^{\varepsilon,h}\_{\mathfrak{y}} \tag{72}
$$

with the commutation relation

$$\left[\hat{\mathcal{P}}\_{-}^{\mu}{}^{h}, \hat{\mathcal{P}}\_{+}^{\mu}\right] = -\frac{2e}{c}H.\tag{73}$$

Following the Rashba's paper [63] we can introduce the annihilation and creation operators: for electrons as

$$\mathfrak{a}\_{\mathfrak{c}} = \sqrt{\frac{c}{2|e|H}} \widehat{P}\_{-}^{\mathfrak{c}} , \mathfrak{a}\_{\mathfrak{c}}^{\dagger} = \sqrt{\frac{c}{2|e|H}} \widehat{P}\_{+}^{\mathfrak{c}} ,$$

for holes as

$$
\hat{a}\_h = \sqrt{\frac{c}{2|e|H}} \hat{P}\_{+\prime}^e \hat{a}\_h^\dagger = \sqrt{\frac{c}{2|e|H}} \hat{P}\_{-\prime}^h
$$

with the Bose commutation relation

$$\left[\mathfrak{a}\_{\mathfrak{e},h\prime} \,\,\mathfrak{a}\_{\mathfrak{e},h}^{\dagger}\right] = 1.$$

The effective Hamiltonians rewritten in the second-quantized representation are

$$\begin{split} \hat{H}\_{eff}^{\prime \mu\_{\varepsilon}} &= \omega\_{\varepsilon}^{\ast \varepsilon} \left( \mathfrak{d}\_{\varepsilon}^{\dagger} \hat{a}\_{\varepsilon} + \frac{1}{2} \right) - \frac{i}{\sqrt{2}} \frac{a\_{\varepsilon}}{a\_{H}} \mathfrak{r}\_{\varepsilon} \left( \hat{a}\_{\varepsilon} - \hat{a}\_{\varepsilon}^{\dagger} \right) + \widetilde{\Delta}\_{\varepsilon}, \\ \hat{H}\_{eff}^{\prime \hbar} &= \omega\_{\varepsilon}^{\ast \hbar} \left( \mathfrak{d}\_{\varepsilon}^{\dagger} \hat{a}\_{\varepsilon} + \frac{1}{2} \right) + \frac{i}{\sqrt{2}} \frac{a\_{\hbar}}{a\_{H}} \mathfrak{r}\_{\varepsilon} \left( \mathfrak{d}\_{\hbar} - \mathfrak{d}\_{\hbar}^{\dagger} \right) + \widetilde{\Delta}\_{\hbar \prime} \end{split} \tag{74}$$

where *<sup>ω</sup>*∗*e*,*<sup>h</sup> <sup>c</sup>* <sup>=</sup> <sup>|</sup>*e*|*<sup>H</sup> m*∗ *<sup>e</sup>*,*<sup>h</sup> <sup>c</sup>* are the cyclotron frequencies.

Let us solve the equation for <sup>Ψ</sup>*e*,*<sup>h</sup> <sup>λ</sup>* with the Hamiltonians (74) in the class of eigen functions of the operator <sup>P</sup>ˆ. The matrix *<sup>τ</sup><sup>z</sup>* should be replaced by corresponding eigen value *<sup>λ</sup>* in the equations for the components of <sup>Ψ</sup>*e*,*<sup>h</sup> <sup>λ</sup>* . After a nondimensionalization, these equations are written as

$$\begin{aligned} \left[\hat{\mathfrak{a}}\_{\varepsilon}^{\dagger}\hat{\mathfrak{a}}\_{\varepsilon} + \frac{1}{2} - i\lambda\delta\_{\text{SO}}^{\varepsilon}\left(\mathfrak{a}\_{\varepsilon} - \mathfrak{a}\_{\varepsilon}^{\dagger}\right)\right] \tilde{\psi}\_{\lambda}^{\varepsilon} &= \epsilon\_{\lambda}^{\varepsilon} \tilde{\psi}\_{\lambda'}^{\varepsilon} \\ \left[\hat{\mathfrak{a}}\_{h}^{\dagger}\hat{\mathfrak{a}}\_{h} + \frac{1}{2} + i\lambda\delta\_{\text{SO}}^{h}\left(\mathfrak{a}\_{h} - \mathfrak{a}\_{h}^{\dagger}\right)\right] \tilde{\psi}\_{\lambda}^{h} &= \epsilon\_{\lambda}^{h} \tilde{\psi}\_{\lambda'}^{h} \end{aligned} \tag{75}$$

where *δe*,*<sup>h</sup> SO* <sup>=</sup> <sup>√</sup> *<sup>α</sup>e*,*<sup>h</sup>* 2*aH ω*∗*e*,*<sup>h</sup> <sup>c</sup>* = √ 1 <sup>2</sup> *aHk*<sup>∗</sup> *ye*,*<sup>h</sup>* and <sup>±</sup>*<sup>e</sup>*,*<sup>h</sup> <sup>λ</sup>* <sup>=</sup> *<sup>E</sup><sup>e</sup>*,*<sup>h</sup> <sup>λ</sup>* −∆*<sup>e</sup>*,*<sup>h</sup> ω*∗*e*,*<sup>h</sup> <sup>c</sup>* , *<sup>e</sup>*,*<sup>h</sup> <sup>λ</sup> >* 0 and plus corresponds to electrons (its energy *<sup>E</sup><sup>e</sup>* <sup>=</sup> <sup>∆</sup>*<sup>e</sup>* <sup>+</sup> *<sup>ω</sup>*∗*<sup>e</sup> <sup>c</sup> <sup>e</sup> <sup>λ</sup>* has positive values), minus corresponds to holes (its energy *<sup>E</sup><sup>h</sup>* <sup>=</sup> <sup>∆</sup>*<sup>h</sup>* <sup>−</sup> *<sup>ω</sup>*∗*<sup>h</sup> <sup>c</sup> <sup>h</sup> <sup>λ</sup>* has negative values).

Now we make an expansion of *<sup>ψ</sup><sup>e</sup>*,*<sup>h</sup> <sup>λ</sup>* by the oscillator function basis *ψ<sup>n</sup>*

$$
\widetilde{\boldsymbol{\psi}}\_{\lambda}^{\varepsilon,\hbar} = \sum\_{n=0}^{\infty} a\_{\lambda n}^{\varepsilon,\hbar} \boldsymbol{\psi}\_n. \tag{76}
$$

We have also the normalization condition in the form of the integral

$$\int\_{-\infty}^{\infty} \left| \tilde{\psi}\_{\lambda}^{\varepsilon, \hbar} \right|^2 dy = 1$$

or in the form of the series

We can also consider the combinations of these operators

with the commutation relation

212 Graphene - New Trends and Developments

with the Bose commutation relation

*<sup>H</sup><sup>e</sup> eff* <sup>=</sup> *<sup>ω</sup>*∗*<sup>e</sup> c a*ˆ † *<sup>e</sup> a*ˆ*<sup>e</sup>* + 1 2 − *i* √2 *αe aH τz a*ˆ*<sup>e</sup>* − *a*ˆ † *e* + ∆*e*,

*<sup>H</sup><sup>h</sup> eff* <sup>=</sup> *<sup>ω</sup>*∗*<sup>h</sup> c a*ˆ † *<sup>e</sup> a*ˆ*<sup>e</sup>* + 1 2 + *i* √2 *αh aH τz a*ˆ*<sup>h</sup>* − *a*ˆ † *h* + ∆*h*,

for electrons as

for holes as

where *<sup>ω</sup>*∗*e*,*<sup>h</sup> <sup>c</sup>* <sup>=</sup> <sup>|</sup>*e*|*<sup>H</sup>*

written as

*m*∗

Let us solve the equation for <sup>Ψ</sup>*e*,*<sup>h</sup>*

equations for the components of <sup>Ψ</sup>*e*,*<sup>h</sup>*

 *a*ˆ † *<sup>e</sup> a*ˆ*<sup>e</sup>* + 1 <sup>2</sup> <sup>−</sup> *<sup>i</sup>λδ<sup>e</sup> SO a*ˆ*<sup>e</sup>* − *a*ˆ † *e ψe <sup>λ</sup>* <sup>=</sup> *<sup>e</sup> λψe λ*,

 *a*ˆ † *<sup>h</sup>a*ˆ*<sup>h</sup>* + 1 <sup>2</sup> <sup>+</sup> *<sup>i</sup>λδ<sup>h</sup> SO a*ˆ*<sup>h</sup>* − *a*ˆ † *h ψh <sup>λ</sup>* <sup>=</sup> *<sup>h</sup> λψh λ*,

*Pe*,*h* <sup>±</sup> <sup>=</sup> *<sup>P</sup><sup>e</sup>*,*<sup>h</sup>*

 *Pe*,*h* <sup>−</sup> , *<sup>P</sup><sup>e</sup>*,*<sup>h</sup>* + <sup>=</sup> <sup>−</sup>2*<sup>e</sup> c*

 *c* <sup>2</sup>|*e*|*<sup>H</sup> <sup>P</sup><sup>e</sup>*

 *c* <sup>2</sup>|*e*|*<sup>H</sup> <sup>P</sup><sup>e</sup>*

> *a*ˆ*e*,*h*, *a*ˆ † *e*,*h* = 1.

The effective Hamiltonians rewritten in the second-quantized representation are

*a*ˆ*<sup>e</sup>* =

*a*ˆ*<sup>h</sup>* =

*<sup>e</sup>*,*<sup>h</sup> <sup>c</sup>* are the cyclotron frequencies.

*<sup>x</sup>* <sup>±</sup> *iP<sup>e</sup>*,*<sup>h</sup>*

Following the Rashba's paper [63] we can introduce the annihilation and creation operators:

<sup>−</sup>, *<sup>a</sup>*<sup>ˆ</sup> † *<sup>e</sup>* =

<sup>+</sup>, *a*ˆ † *<sup>h</sup>* =

of the operator <sup>P</sup>ˆ. The matrix *<sup>τ</sup><sup>z</sup>* should be replaced by corresponding eigen value *<sup>λ</sup>* in the

 *c* <sup>2</sup>|*e*|*<sup>H</sup> <sup>P</sup><sup>e</sup>* +,

 *c* <sup>2</sup>|*e*|*<sup>H</sup> <sup>P</sup><sup>h</sup>* −

*<sup>λ</sup>* with the Hamiltonians (74) in the class of eigen functions

*<sup>λ</sup>* . After a nondimensionalization, these equations are

*<sup>y</sup>* (72)

*H*. (73)

(74)

(75)

$$\sum\_{n=0}^{\infty} \left| a\_{\lambda n}^{\varepsilon, h} \right|^2 = 1. \tag{77}$$

Taking into account the relations

$$\begin{aligned} \hat{a}\_{\varepsilon,h}\psi\_n &= \sqrt{n}\psi\_{n-1}, \\ \hat{a}\_{\varepsilon,h}^\dagger \psi\_n &= \sqrt{n+1}\psi\_{n+1}. \end{aligned}$$

we obtain two infinite systems of equations for coefficients *a<sup>e</sup> <sup>λ</sup><sup>n</sup>* and *<sup>a</sup><sup>h</sup> λn*

$$\begin{cases} \frac{1}{2}a\_{\lambda0}^{\varepsilon} - i\lambda \delta\_{SO}^{\varepsilon} a\_{\lambda1}^{\varepsilon} = \epsilon\_{\lambda}^{\varepsilon} a\_{\lambda0^{\prime}}^{\varepsilon} \\ \frac{3}{2}a\_{\lambda1}^{\varepsilon} - i\lambda\sqrt{2}\delta\_{SO}^{\varepsilon} a\_{\lambda2}^{\varepsilon} + i\lambda \delta\_{SO}^{\varepsilon} a\_{\lambda0}^{\varepsilon} = \epsilon\_{\lambda}^{\varepsilon} a\_{\lambda1^{\prime}}^{\varepsilon} \\ \frac{5}{2}a\_{\lambda2}^{\varepsilon} - i\lambda\sqrt{3}\delta\_{SO}^{\varepsilon} a\_{\lambda3}^{\varepsilon} + i\lambda\sqrt{2}\delta\_{SO}^{\varepsilon} a\_{\lambda1}^{\varepsilon} = \epsilon\_{\lambda}^{\varepsilon} a\_{\lambda2^{\prime}}^{\varepsilon} \\ \quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ \left(n + \frac{1}{2}\right)a\_{\lambda\mu}^{\varepsilon} - i\lambda\sqrt{n+1}\delta\_{SO}^{\varepsilon} a\_{\lambda\mu+1}^{\varepsilon} + i\lambda\sqrt{n}\delta\_{SO}^{\varepsilon} a\_{\lambda\mu-1}^{\varepsilon} = \epsilon\_{\lambda}^{\varepsilon} a\_{\lambda\mu}^{\varepsilon} \\ \quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \end{cases} \tag{78}$$

 1 2 *ah <sup>λ</sup>*<sup>0</sup> <sup>+</sup> *<sup>i</sup>λδ<sup>h</sup> SOa<sup>h</sup> <sup>λ</sup>*<sup>1</sup> <sup>=</sup> *<sup>h</sup> λah λ*0, 3 2 *ah <sup>λ</sup>*<sup>1</sup> + *iλ* √ 2*δ<sup>h</sup> SOa<sup>h</sup> <sup>λ</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>λδ<sup>h</sup> SOa<sup>h</sup> <sup>λ</sup>*<sup>0</sup> <sup>=</sup> *<sup>h</sup> λah λ*1, 5 2 *ae <sup>λ</sup>*<sup>2</sup> + *iλ* √ 3*δ<sup>h</sup> SOa<sup>h</sup> <sup>λ</sup>*<sup>3</sup> − *iλ* √ 2*δ<sup>h</sup> SOa<sup>h</sup> <sup>λ</sup>*<sup>1</sup> <sup>=</sup> *<sup>h</sup> λah λ*2, ............................................................................................. *n* + 1 2 *ah <sup>λ</sup><sup>n</sup>* + *iλ* <sup>√</sup>*<sup>n</sup>* <sup>+</sup> <sup>1</sup>*δ<sup>h</sup> SOa<sup>h</sup> <sup>λ</sup>n*+<sup>1</sup> − *iλ* <sup>√</sup>*nδ<sup>h</sup> SOa<sup>h</sup> <sup>λ</sup>n*−<sup>1</sup> <sup>=</sup> *<sup>h</sup> λah λn*, ............................................................................................. (79)

On the other hand, it is clear from the physical considerations that the energy spectrum of charge carriers should be independent on the quasimomentum shift. Here, this shift equals to the position of the dispersion curve extrema *ky* = ±*k*<sup>∗</sup> *ye* for electrons and *ky* = ±*k*<sup>∗</sup> *yh* for holes. The Landau levels spectrum in (quasi)one-dimensional system should also be the same as it in the case when we choose the dispersion curve extrema as the point of reference for quasimomentum:

$$
\epsilon\_{\lambda}^{e,h} = n + \frac{1}{2}.\tag{80}
$$

The systems of equations for coefficients *a<sup>e</sup> <sup>λ</sup><sup>n</sup>* (78) and *<sup>a</sup><sup>h</sup> <sup>λ</sup><sup>n</sup>* (79) are the recurrence relations for a calculation of these coefficients. So, the dependence on the parameters *δ<sup>e</sup> SO* and *<sup>δ</sup><sup>h</sup> SO* in the problem appears only in the wave function *<sup>ψ</sup>e*,*<sup>h</sup> λ* .

Lastly, our remark concerns a matter about a convergence of the series (77). Each term <sup>|</sup>*ae*,*<sup>h</sup> λn*| 2 contains a summand <sup>∼</sup> 1/(*δe*,*<sup>h</sup> SO*)2*n*, therefore it is necessary *<sup>δ</sup>e*,*<sup>h</sup> SO >* 1. This inequality should be valid due to the condition (68) because *k*∗ *ye*,*<sup>h</sup>* 1/*d* for QW with enough strong asymmetry.

### **4.7. Possible experiments on the heterostructure**

Pseudospin splitting can be observed by means of Raman spectroscopy. The *D* peak of interest for the present study (alternatively called 2*D* peak to emphasize that it is due to a two-phonon-assisted process) is located at 2700 cm−<sup>1</sup> [66]. It arises from intervalley scattering involving phonons with wavenumbers *<sup>q</sup> <sup>&</sup>gt; <sup>K</sup>*, where *<sup>K</sup>* <sup>=</sup> <sup>4</sup>*π*/3√3*<sup>a</sup>* <sup>≈</sup> 1.7 <sup>×</sup> <sup>10</sup><sup>8</sup> cm−<sup>1</sup> is the spacing between adjacent *K* and *K* points. One process of this kind is indicated as *A* → *B* → *C* → *D* → *A* in Fig. 16.

Pseudospin splitting enables intervalley scattering involving phonons with *q* ≈ *q* ∓ ∆*k* (with plus for electrons and minus for holes), where ∆*k* = 2*k*∗ *ye* and ∆*k* = 2*k*<sup>∗</sup> *yh* in electron and hole scattering, respectively. These processes contribute to a peak blueshifted from *<sup>D</sup>* by <sup>∆</sup>*ω*(+) *R* and a peak redshifted from *<sup>D</sup>* by <sup>∆</sup>*ω*(−) *<sup>R</sup>* , giving rise to a doublet structure of the *D* peak.

**Figure 16.** Possible double resonant Raman processes involving electron scattering between valleys. To simplify presentation, analogous processes involving hole scattering between valleys are not shown.

An estimate for ∆*ω<sup>R</sup>* can be obtained by using optical phonon dispersion *ωph*(*q*). The Raman shift is twice the optical phonon frequency:

$$
\delta\omega\_R(q) = 2\omega\_{ph}(q).
$$

The change in the Raman shift caused by pseudospin splitting is

214 Graphene - New Trends and Developments

1 2 *ah <sup>λ</sup>*<sup>0</sup> <sup>+</sup> *<sup>i</sup>λδ<sup>h</sup>*

3 2 *ah <sup>λ</sup>*<sup>1</sup> + *iλ*

5 2 *ae <sup>λ</sup>*<sup>2</sup> + *iλ*

 *n* + 1 2 *ah <sup>λ</sup><sup>n</sup>* + *iλ*

The systems of equations for coefficients *a<sup>e</sup>*

be valid due to the condition (68) because *k*∗

**4.7. Possible experiments on the heterostructure**

plus for electrons and minus for holes), where ∆*k* = 2*k*∗

contains a summand <sup>∼</sup> 1/(*δe*,*<sup>h</sup>*

as *A* → *B* → *C* → *D* → *A* in Fig. 16.

and a peak redshifted from *<sup>D</sup>* by <sup>∆</sup>*ω*(−)

problem appears only in the wave function *<sup>ψ</sup>e*,*<sup>h</sup>*

*SOa<sup>h</sup>*

√ 2*δ<sup>h</sup> SOa<sup>h</sup>*

√ 3*δ<sup>h</sup> SOa<sup>h</sup>*

to the position of the dispersion curve extrema *ky* = ±*k*<sup>∗</sup>

*<sup>λ</sup>*<sup>1</sup> <sup>=</sup> *<sup>h</sup> λah λ*0,

*<sup>λ</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>λδ<sup>h</sup>*

*<sup>λ</sup>*<sup>3</sup> − *iλ*

<sup>√</sup>*<sup>n</sup>* <sup>+</sup> <sup>1</sup>*δ<sup>h</sup>*

*SOa<sup>h</sup>*

√ 2*δ<sup>h</sup> SOa<sup>h</sup>*

.............................................................................................

.............................................................................................

*SOa<sup>h</sup>*

On the other hand, it is clear from the physical considerations that the energy spectrum of charge carriers should be independent on the quasimomentum shift. Here, this shift equals

holes. The Landau levels spectrum in (quasi)one-dimensional system should also be the same as it in the case when we choose the dispersion curve extrema as the point of reference

> *<sup>e</sup>*,*<sup>h</sup> <sup>λ</sup>* = *n* +

a calculation of these coefficients. So, the dependence on the parameters *δ<sup>e</sup>*

*<sup>λ</sup>*<sup>0</sup> <sup>=</sup> *<sup>h</sup> λah λ*1,

> *<sup>λ</sup>*<sup>1</sup> <sup>=</sup> *<sup>h</sup> λah λ*2,

*<sup>λ</sup>n*+<sup>1</sup> − *iλ*

1 2

*<sup>λ</sup><sup>n</sup>* (78) and *<sup>a</sup><sup>h</sup>*

*λ* .

Lastly, our remark concerns a matter about a convergence of the series (77). Each term <sup>|</sup>*ae*,*<sup>h</sup>*

*SO*)2*n*, therefore it is necessary *<sup>δ</sup>e*,*<sup>h</sup>*

Pseudospin splitting can be observed by means of Raman spectroscopy. The *D* peak of interest for the present study (alternatively called 2*D* peak to emphasize that it is due to a two-phonon-assisted process) is located at 2700 cm−<sup>1</sup> [66]. It arises from intervalley scattering involving phonons with wavenumbers *<sup>q</sup> <sup>&</sup>gt; <sup>K</sup>*, where *<sup>K</sup>* <sup>=</sup> <sup>4</sup>*π*/3√3*<sup>a</sup>* <sup>≈</sup> 1.7 <sup>×</sup> <sup>10</sup><sup>8</sup> cm−<sup>1</sup> is the spacing between adjacent *K* and *K* points. One process of this kind is indicated

Pseudospin splitting enables intervalley scattering involving phonons with *q* ≈ *q* ∓ ∆*k* (with

scattering, respectively. These processes contribute to a peak blueshifted from *<sup>D</sup>* by <sup>∆</sup>*ω*(+)

<sup>√</sup>*nδ<sup>h</sup> SOa<sup>h</sup>*

*<sup>λ</sup>n*−<sup>1</sup> <sup>=</sup> *<sup>h</sup>*

*λah λn*,

*ye* for electrons and *ky* = ±*k*<sup>∗</sup>

. (80)

*ye*,*<sup>h</sup>* 1/*d* for QW with enough strong asymmetry.

*ye* and ∆*k* = 2*k*<sup>∗</sup>

*<sup>R</sup>* , giving rise to a doublet structure of the *D* peak.

*<sup>λ</sup><sup>n</sup>* (79) are the recurrence relations for

*SO* and *<sup>δ</sup><sup>h</sup>*

*yh* in electron and hole

*SO >* 1. This inequality should

(79)

*yh* for

*SO* in the

*λn*| 2

*R*

for quasimomentum:

$$
\Delta\omega\_R^{(\pm)} \approx |\delta\omega\_R(\Delta K \mp \Delta k) - \delta\omega\_R(\Delta K)|.
$$

which amounts to <sup>∆</sup>*ω<sup>R</sup>* <sup>≈</sup> 24 cm−<sup>1</sup> for characteristic values of the heterostructure parameters. This value essentially exceeds the Raman spectral resolution of 1 cm−<sup>1</sup> [67] and compares to the *D* peak width for gapless graphene, Γ<sup>0</sup> = 30 cm−<sup>1</sup> [68, 69].

Note that a blue shift of the *D* peak has also been observed in the Raman spectrum of epitaxial graphene on a SiC substrate [67]. This effect is attributed to the strain induced by the substrate in quasi-free graphene since the SiC lattice constant exceeds substantially that of graphene.

Raman scattering contributions from gapped graphene sheets can be avoided either by using a laser beam whose width is smaller than that of the gapless graphene nanoribbon (*d* 10 nm) or by pumping at a frequency *ω* such that the beam cannot be absorbed by gapped graphene materials,

$$E\_{\mathcal{S}}^{eff} + 2\omega\_{ph} < \omega < \min\{2\Delta\_1, 2\Delta\_3\}$$

The positions of the luminescence lines corresponding to exciton levels can be determined from optical experiments and compared to theoretical predictions. The splitting of exciton lines in an electric field applied in the plane of the heterostructure along the normal to its boundaries is evaluated by using formula (54).

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.
