*5.2.2. The model*

The model for the description of the suggested SL is similar to that used earlier to study SL on the striped substrate with the periodic variation in the band gap [86].

In our case, we assume that the band gap remains unchanged and is equal to zero (gapless graphene) and the work function is the same over all regions of SL (its value is chosen as the energy reference point). We have only a modulation of the Fermi velocity. In gapless graphene, a change in the work function leads to the electrical breakdown and to the creation of electron-hole pairs. We also assume that the near-border region corresponding to the gradual change in the Fermi velocity is much narrower than the SL period. Therefore, the *vF* profile can be considered to be sharp enough (see Fig. 21).

We consider the charge carriers located close to the *K* point of the Brillouin zone (the results should be the same for the charge carriers located in the vicinity of the *K* point). Let the *x* axis be perpendicular to the strips as is shown in Fig. 21. The envelope of the wave-function Ψ(*x*, *y*) for the charge carriers obeys the Dirac-Weyl equation with variable Fermi velocity<sup>7</sup>

$$
v\_F \sigma \diamond \Psi(\mathbf{x}, y) = E \Psi(\mathbf{x}, y),
\tag{96}$$

,

$$v\_F = \begin{cases} v\_{F1\prime} & d(n-1) < x < -d\_{II} + dn \\ v\_{F2\prime} & -d\_{II} + dn < x < dn. \end{cases} \tag{97}$$

Here, **<sup>p</sup>** <sup>=</sup> <sup>−</sup>*i*<sup>∇</sup> is the momentum operator (here and further on, ¯*<sup>h</sup>* <sup>=</sup> 1). Integers *<sup>n</sup>* enumerate supercells (see Fig. 21). The Pauli matrices σ = (*σx*, *σy*) act in the space of two sublattices. The motion of charge carriers in SL along the *y* axis is free; hence, a solution to Eq. (96) has the form Ψ(*x*, *y*) = *ψ*(*x*)*eiky <sup>y</sup>*.

Similarly to [86, 90], we find a solution of Eq. (96) with respect to *ψ*(*x*) for the *n*th supercell (i) at 0 *< x < dI*

$$\begin{aligned} \psi\_n^{(1)}(\mathbf{x}) &= \Omega\_{k\_1}(\mathbf{x}) \begin{pmatrix} a\_n^{(1)} \\ c\_n^{(1)} \end{pmatrix}, \\\\ \Omega\_{k\_1}(\mathbf{x}) &= N\_{k\_1} \begin{pmatrix} 1 & 1 \\ \lambda\_+^{(1)} & -\lambda\_-^{(1)} \end{pmatrix} e^{ik\_1 \mathbf{x} \sigma\_\varepsilon}, \\\\ \lambda\_\pm^{(1)} &= \frac{\upsilon\_{F1}(k\_1 \pm ik\_y)}{E}, \ k\_1 = \frac{\sqrt{E^2 - \upsilon\_{F1}^2 k\_y^2}}{\upsilon\_{F1}} \end{aligned}$$

(ii) at *dI < x < d*

**Figure 21.** Fermi velocity profile in SL under study (*vF*<sup>1</sup> *> vF*<sup>2</sup> case). The enumeration of supercells in SL and the sizes of its regions are indicated in the lower part of the figure: *dI* is the width of the graphene strip with the Fermi velocity

suspended over the grooves and those being in contact with the substrate material (see Fig. 20b). The renormalization of the Fermi velocity should be the most clearly pronounced just in the suspended graphene regions since here we have *εeff* = 1. According to the experimental data, the renormalized Fermi velocity in suspended graphene increases to 3 <sup>×</sup> 108 cm/s [101]. In the regions with graphene in contact with the narrow gap semiconducting material, where *εeff* 1, the renormalized Fermi velocity differs only slightly from the unrenormalized one. In addition, the substrate itself is a diffraction grating. Therefore, the system should exhibit

There is another version of the system under study. It is possible to deposit graphene on a periodic array of parallel metallic strips (Fig. 20c). This is the limiting case: in the suspended graphene regions, we have *εeff* = 1 (the strongest renormalization of the Fermi velocity), whereas in the regions with graphene in contact with metallic strips, we have *εeff* = ∞

We see that a whole class of such type of systems, which were not studied earlier, is possible. Without doubt, the studies of such systems should lead to important advances

The model for the description of the suggested SL is similar to that used earlier to study SL

In our case, we assume that the band gap remains unchanged and is equal to zero (gapless graphene) and the work function is the same over all regions of SL (its value is chosen as the energy reference point). We have only a modulation of the Fermi velocity. In gapless graphene, a change in the work function leads to the electrical breakdown and to the creation of electron-hole pairs. We also assume that the near-border region corresponding to the gradual change in the Fermi velocity is much narrower than the SL period. Therefore, the *vF*

We consider the charge carriers located close to the *K* point of the Brillouin zone (the results should be the same for the charge carriers located in the vicinity of the *K* point). Let the *x*

in the implementation of the technologies based on the controlled Fermi velocity.

on the striped substrate with the periodic variation in the band gap [86].

*vF*1, *dI I* is the width of the graphene strip with the Fermi velocity *vF*2, and *d* = *dI* + *dI I* is the SL period.

rather interesting optical characteristics, demanding a separate study.

(vanishing renormalization of the Fermi velocity [91]).

profile can be considered to be sharp enough (see Fig. 21).

*5.2.2. The model*

226 Graphene - New Trends and Developments

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

$$\frac{1}{2} \left\{ v\_{\mathbb{F}}(\boldsymbol{x}), \,\,\sigma \Phi \right\} \,\, \Psi(\boldsymbol{x}, \, \boldsymbol{y}) = E \Psi(\boldsymbol{x}, \, \boldsymbol{y}) .$$

<sup>7</sup> In the general case, one should write the anticommutator of the Fermi velocity *vF* (*x*) with the term containing the momentum operator *p*ˆ*<sup>x</sup>*

Such symmetrization of the Hamiltonian is necessary for retaining its Hermitian form. Similar problems were considered in [103, 104]. In the case of the stepwise profile (97) of the Fermi velocity, we obtain the equation for *Psi*(*x*, *y*) in form (96). This limitation is not significant since allowance for a smooth dependence *vF* (*x*) will complicate the calculations, but will insignificantly change the final results.

**Figure 22.** Schematic image illustrating the behavior of the envelope of the wavefunction of charge carriers in SL under study: **(a)** the oscillatory solution in all regions and **(b)** the solution being oscillatory in one region and exhibiting exponential decay deep into another region (*vF*<sup>1</sup> *> vF*<sup>2</sup> case).

$$
\lambda\_{\pm}^{(2)} = \frac{v\_{\rm F2}(k\_2 \pm ik\_y)}{E}, \ k\_2 = \frac{\sqrt{E^2 - v\_{\rm F2}^2 k\_y^2}}{v\_{\rm F2}}.
$$

Here, *Nk*<sup>1</sup> and *Nk*<sup>2</sup> are the normalization factors.

For the case *vF*<sup>1</sup> *> vF*2, the condition for the existence of the solution of Eq. (96), which oscillates in all regions of the SL (it is schematically illustrated in Fig. 22a), is reduced to the inequality

$$k\_2^2 > \left(\frac{v\_{F1}^2}{v\_{F2}^2} - 1\right) k\_y^2. \tag{98}$$

The existence of a *solution of the mixed type* is also possible (see Fig. 22b). In this case, we have an oscillatory solution in some regions (effective QWs), whereas in the other regions, it exhibits exponential decay (effective potential barriers) deep into these regions. The condition for the existence of the mixed type solution is determined by the inequality inverse to (98) and it is met only for finite *ky* values.

The effective quantum barrier of the new type is the region with the higher Fermi velocity because the energy of the charge carriers with the same momentum **k** in it is higher than that in the effective QW with the lower Fermi velocity [25]. In contrast to the usual QW, which is formed owing to the change in the width of the band gap, the height of the barrier in SL under study grows with *ky*. At *ky*=0, the barrier vanishes and our problem is reduced to the *empty lattice model* [105]. In the latter model, the potential is absent, but the periodicity is retained. As a result, energy bands corresponding to the symmetry of the problem arise, but we have zero band gaps.

#### *5.2.3. The dispersion relation*

To derive the dispersion relation, we use the transfer matrix (*T*-matrix) method in the way similar to that employed in [86, 90].

The transfer matrix determines the relation between the coefficients appearing in the expressions for the envelopes of the wavefunctions for the neighboring supercells

$$
\begin{pmatrix} a\_{n+1}^{(1)} \\ c\_{n+1}^{(1)} \end{pmatrix} = T \begin{pmatrix} a\_n^{(1)} \\ c\_n^{(1)} \end{pmatrix}, \\
\begin{pmatrix} a\_{n+1}^{(2)} \\ c\_{n+1}^{(2)} \end{pmatrix} = T \begin{pmatrix} a\_n^{(2)} \\ c\_n^{(2)} \end{pmatrix}.
$$

We use the following boundary conditions for matching of the envelopes of the wavefunctions [27, 42]

$$\sqrt{\upsilon\_{F1}}\psi\_n^{(1)} = \sqrt{\upsilon\_{F2}}\psi\_n^{(2)}$$

and also the Bloch conditions in the form

$$
\psi\_n^{(1)}(\mathfrak{x} + d) = \psi\_n^{(1)}(\mathfrak{x})e^{i\mathbf{k}\_\chi d}
$$

and

**Figure 22.** Schematic image illustrating the behavior of the envelope of the wavefunction of charge carriers in SL under study: **(a)** the oscillatory solution in all regions and **(b)** the solution being oscillatory in one region and exhibiting

*<sup>E</sup>*<sup>2</sup> <sup>−</sup> *<sup>v</sup>*<sup>2</sup>

*vF*<sup>2</sup>

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

.

*<sup>y</sup>*. (98)

*<sup>E</sup>* , *<sup>k</sup>*<sup>2</sup> <sup>=</sup>

For the case *vF*<sup>1</sup> *> vF*2, the condition for the existence of the solution of Eq. (96), which oscillates in all regions of the SL (it is schematically illustrated in Fig. 22a), is reduced to the

The existence of a *solution of the mixed type* is also possible (see Fig. 22b). In this case, we have an oscillatory solution in some regions (effective QWs), whereas in the other regions, it exhibits exponential decay (effective potential barriers) deep into these regions. The condition for the existence of the mixed type solution is determined by the inequality inverse to (98)

The effective quantum barrier of the new type is the region with the higher Fermi velocity because the energy of the charge carriers with the same momentum **k** in it is higher than that in the effective QW with the lower Fermi velocity [25]. In contrast to the usual QW, which is formed owing to the change in the width of the band gap, the height of the barrier in SL under study grows with *ky*. At *ky*=0, the barrier vanishes and our problem is reduced to the *empty lattice model* [105]. In the latter model, the potential is absent, but the periodicity is retained. As a result, energy bands corresponding to the symmetry of the problem arise, but

To derive the dispersion relation, we use the transfer matrix (*T*-matrix) method in the way

The transfer matrix determines the relation between the coefficients appearing in the

expressions for the envelopes of the wavefunctions for the neighboring supercells

exponential decay deep into another region (*vF*<sup>1</sup> *> vF*<sup>2</sup> case).

228 Graphene - New Trends and Developments

Here, *Nk*<sup>1</sup> and *Nk*<sup>2</sup> are the normalization factors.

and it is met only for finite *ky* values.

we have zero band gaps.

*5.2.3. The dispersion relation*

similar to that employed in [86, 90].

inequality

*<sup>λ</sup>*(2)

<sup>±</sup> <sup>=</sup> *vF*2(*k*<sup>2</sup> <sup>±</sup> *iky*)

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

$$
\psi\_n^{(2)}(x+d) = \psi\_n^{(2)}(x)e^{ik\_xd}.
$$

Then, the expression for the *T*-matrix has the form [86, 90]

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

The dispersion relation is determined from Eq. (90), which for the oscillatory type solution, can be written as

$$\frac{v\_{\rm F1} v\_{\rm F2} k\_y^2 - E^2}{v\_{\rm F1} v\_{\rm F2} k\_1 k\_2} \sin(k\_1 d\_I) \sin(k\_2 d\_{II}) + \cos(k\_1 d\_I) \cos(k\_2 d\_{II}) = \cos(k\_x d). \tag{99}$$

For the solution of the mixed type, the dispersion relation is found from (99) through the use of the formal substitution *<sup>k</sup>*<sup>1</sup> <sup>→</sup> *<sup>i</sup>κ*1, where *<sup>κ</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> *vF*<sup>1</sup> *v*2 *F*1*k*<sup>2</sup> *<sup>y</sup>* − *<sup>E</sup>*2.

At *ky* = 0, transcendental equation (99) has the form

$$\cos\left(k\_1 d\_I + k\_2 d\_{II}\right) = \cos(k\_x d) \tag{100}$$

for which the exact solution can be found

$$E\_l(k\_\chi) = \pm v\_F^\* \left( k\_\chi + \frac{2\pi l}{d} \right), \ l = 0, 1, 2, \dots$$

Here, the effective Fermi velocity is introduced as

$$v\_F^\* = \frac{v\_{F1}v\_{F2}d}{v\_{F1}d\_{II} + v\_{F2}d\_I}.\tag{101}$$

For the *l*th miniband, the energy at the *K* point is equal to

$$E\_l^0 = \pm \frac{2\pi l v\_F^\*}{d}, \ l = 0, 1, 2, \dots$$

We can see that the lower electron miniband (*l* = 0) touches the upper hole miniband at the *K* point and graphene remains gapless.

From Eq. (100), we find that, at the edge of the *l*th miniband, the energy at *kx* = ±*π*/*d* is equal to

$$E\_l\left(\pm\frac{\pi}{d}\right) = \pm \frac{\pi(2l+1)v\_F^\*}{d}, \ l = 0, 1, 2, \dots$$

The minibands are separated by the direct band gaps

$$E\_G = E\_{l+1} \left( \pm \frac{\pi}{d} \right) - E\_l \left( \pm \frac{\pi}{d} \right) = \frac{2\pi v\_F^\*}{d}.$$

In the case of *ky* = 0, indirect gaps are absent

$$E\_l\left(\frac{\pi}{d}\right) = E\_{l+1}\left(-\frac{\pi}{d}\right),$$

which corresponds to the empty lattice model [105].

#### *5.2.4. The numerical calculation of the energy spectrum*

Let us calculate the lower electron miniband for SL shown in Fig. 19c. According to [101], for it we have *vF*<sup>1</sup> <sup>=</sup> <sup>3</sup> <sup>×</sup> 106 cm/s (suspended graphene) and *vF*<sup>2</sup> <sup>=</sup> 0.85 <sup>×</sup> <sup>10</sup><sup>6</sup> cm/s (in the region with the contact of graphene with the metal, the Fermi velocity coincides with *vF*0).

In the weak coupling model, the problem concerning the edge type at the interface turns out to be unimportant. Let us assume that we have a zigzag-type boundary at the interface (see Fig. 20) and, in each of two regions of the supercell, integer numbers *NI* and *NI I* of graphene unit cells are packed up. Then, we have *dI* = 3*NI a* and *dI I* = 3*NI I a*, where *a* = 1.42 Å is the lattice constant of graphene. For calculations, we assume that *NI* = *NI I* = 50, i.e., *dI* = *dI I* = 21.3 nm.

In the framework of the suggested model, it is necessary to introduce the upper limit on the wave vector component characterizing the free motion of charge carriers, |*ky*| *kc*. Momentum *kc* corresponds to the energy of the ultraviolet cutoff, Λ ≈ 3 eV [101]. As a result, we find *kc* <sup>≈</sup> 4.3 nm−1. This, in turn, imposes the limitation on the SL period, *<sup>d</sup> <sup>a</sup>*.

The results of numerical calculations are represented in the form of two *E*(*kx*, *ky*) plots for the lower electron miniband: (*i*) *E*(*kx*) at fixed values of *ky* (Fig. 23a) and (*ii*) *E*(*ky*) at fixed values of *kx* (Fig. 23b). In Fig. 23a, we can see, in particular, that *ky* = 0 corresponds to the

**Figure 23.** Numerical calculation of the dispersion curves for the lower electron miniband **(a)** versus *kx* at fixed *ky* values and **(b)** versus *ky* at fixed *kx* values.

linear dispersion law and the effective Fermi velocity is *v*∗ *<sup>F</sup>* <sup>≈</sup> 1.325 <sup>×</sup> 108 cm/s. The lower curve in Fig. 23b exhibits a nearly linear growth. This means that the *E*(*kx*, *ky*) surface has the conical shape near the Dirac point.

Thus, we confirm by numerical calculations that at *ky* = 0, the Fermi velocity of electrons (holes) has a constant value, does not vanish up to the boundaries of minibands, and is determined by Eq. (101) (this is true for all minibands). In this sense, the particles do not feel the boundaries of minibands. Note that, for *ky* = 0, the velocity of particles always vanishes at the miniband boundaries.

#### *5.2.5. The qualitative analysis of the current-voltage characteristics*

For the *l*th miniband, the energy at the *K* point is equal to

*K* point and graphene remains gapless.

230 Graphene - New Trends and Developments

*El* ±*π d* = ±

The minibands are separated by the direct band gaps

In the case of *ky* = 0, indirect gaps are absent

which corresponds to the empty lattice model [105].

*5.2.4. The numerical calculation of the energy spectrum*

*dI* = *dI I* = 21.3 nm.

*EG* = *El*<sup>+</sup><sup>1</sup>

 ±*π d* − *El* ±*π d* 

*El π d* 

equal to

*E*0 *<sup>l</sup>* = ± 2*πlv*∗ *F*

We can see that the lower electron miniband (*l* = 0) touches the upper hole miniband at the

From Eq. (100), we find that, at the edge of the *l*th miniband, the energy at *kx* = ±*π*/*d* is

*π*(2*l* + 1)*v*∗

= *El*<sup>+</sup><sup>1</sup>

Let us calculate the lower electron miniband for SL shown in Fig. 19c. According to [101], for it we have *vF*<sup>1</sup> <sup>=</sup> <sup>3</sup> <sup>×</sup> 106 cm/s (suspended graphene) and *vF*<sup>2</sup> <sup>=</sup> 0.85 <sup>×</sup> <sup>10</sup><sup>6</sup> cm/s (in the region with the contact of graphene with the metal, the Fermi velocity coincides with *vF*0). In the weak coupling model, the problem concerning the edge type at the interface turns out to be unimportant. Let us assume that we have a zigzag-type boundary at the interface (see Fig. 20) and, in each of two regions of the supercell, integer numbers *NI* and *NI I* of graphene unit cells are packed up. Then, we have *dI* = 3*NI a* and *dI I* = 3*NI I a*, where *a* = 1.42 Å is the lattice constant of graphene. For calculations, we assume that *NI* = *NI I* = 50, i.e.,

In the framework of the suggested model, it is necessary to introduce the upper limit on the wave vector component characterizing the free motion of charge carriers, |*ky*| *kc*. Momentum *kc* corresponds to the energy of the ultraviolet cutoff, Λ ≈ 3 eV [101]. As a result, we find *kc* <sup>≈</sup> 4.3 nm−1. This, in turn, imposes the limitation on the SL period, *<sup>d</sup> <sup>a</sup>*. The results of numerical calculations are represented in the form of two *E*(*kx*, *ky*) plots for the lower electron miniband: (*i*) *E*(*kx*) at fixed values of *ky* (Fig. 23a) and (*ii*) *E*(*ky*) at fixed values of *kx* (Fig. 23b). In Fig. 23a, we can see, in particular, that *ky* = 0 corresponds to the

 −*π d* ,

*F*

*<sup>d</sup>* , *<sup>l</sup>* <sup>=</sup> 0, 1, 2, . . .

<sup>=</sup> <sup>2</sup>*πv*<sup>∗</sup> *F <sup>d</sup>* .

*<sup>d</sup>* , *<sup>l</sup>* <sup>=</sup> 0, 1, 2, . . .

Let us briefly discuss at the qualitative level the effect of the SL potential on the transport phenomena.

Having in mind the aforementioned qualitative difference between the *ky* = 0 and *ky* = 0 cases, we should expect that the current-voltage characteristics (*I*–*V* curves) of SL under study should be significantly different for these two cases.

At *ky* = 0, the transport characteristics of SL under study should be the same as for effective gapless graphene with the average Fermi velocity *v*∗ *<sup>F</sup>* given by Eq. (101). In particular, at any arbitrarily low charge carrier density, we should observe nonzero minimum conductivity *σmin*. According to the experimental data, we have *σmin* = 4*e*2/*h* [2], which coincides with the ballistic conductivity of graphene. The *I*–*V* curve should exhibit a linear growth similar to that characteristic of graphene samples with high enough mobility of charge carriers, *µ* 104 cm2/(V s) [106].

In the case of *ky* = 0, the situation is more complicated. At a nonzero transverse field *Vy* and at a sufficiently small longitudinal field *Vx*, the *I*–*V* curve should be a growing one and the differential conductivity at small values of *Vx* is about or higher than the minimum conductivity

$$
\sigma\_{\mathrm{dif}}(V\_{\mathrm{x}} \approx 0) \gtrsim \sigma\_{\mathrm{min}}.
$$

Now, we calculate the velocity of electrons for the case of fixed longitudinal (E*x*) and nonzero transverse (E*y*) electric fields. For the corresponding implementation of such situation in experiment, it is possible to use the standard Hall device.

For simplicity, we assume that transport is ballistic; i.e., the mean free path *λ* is so large that an electron accelerated by the applied electric field can reach the miniband boundary without any scattering. To distinguish the spectrum related to the potential of the superlattice, the mean free path should be much larger than the period of SL [85]

$$
\lambda\_f \gg d.\tag{102}
$$

For the sufficiently pure graphene samples, we have *λ<sup>f</sup>* 1*µ*m.

The direction of the electron motion is characterized by the polar angle *φ* = arctan(*ky*/*kx*). Its value remains unchanged in the whole −*π*/*d* ≤ *kx* ≤ *π*/*d* range. The contribution to the conductivity related to the intraminiband transitions is determined by the electron velocity, which we seek:

$$v\_{\phi} = \left. \frac{\partial E}{\partial k} \right|\_{k\_y = k\_x \tan \phi}.$$

In Fig. 24, we illustrate the calculated dependence of the electron velocity on *kx* for the same SL parameters as above for the polar angles *φ* = 5*o*, 10*o*, and 15*o*. We can see that the velocity indeed vanishes at the miniband boundary and its abrupt decrease takes place within a quite narrow range near the miniband boundary. For low momenta, we have *v<sup>φ</sup>* ≈ *v*<sup>∗</sup> *F*.

An application of the superlattice at nonzero temperatures requires the existence of a quite clearly pronounced Fermi velocity profile; i.e., we should use rather large *φ* and *δvF* = |*vF*<sup>1</sup> − *vF*2| values:

$$
\pi \frac{\delta v\_F}{d} \sin \phi \gg T.
$$

However, at large *φ* values close to *π*/2, the condition according to which charge carriers pass a large number of supercells at the mean free path can be violated. Then, condition (102) turns out to be unimportant: condition *λ<sup>f</sup>* cos *φ d* should be met.

*σmin*. According to the experimental data, we have *σmin* = 4*e*2/*h* [2], which coincides with the ballistic conductivity of graphene. The *I*–*V* curve should exhibit a linear growth similar to that characteristic of graphene samples with high enough mobility of charge carriers,

In the case of *ky* = 0, the situation is more complicated. At a nonzero transverse field *Vy* and at a sufficiently small longitudinal field *Vx*, the *I*–*V* curve should be a growing one and the differential conductivity at small values of *Vx* is about or higher than the minimum

*σdi f*(*Vx* ≈ 0) *σmin*.

Now, we calculate the velocity of electrons for the case of fixed longitudinal (E*x*) and nonzero transverse (E*y*) electric fields. For the corresponding implementation of such situation in

For simplicity, we assume that transport is ballistic; i.e., the mean free path *λ* is so large that an electron accelerated by the applied electric field can reach the miniband boundary without any scattering. To distinguish the spectrum related to the potential of the superlattice, the

The direction of the electron motion is characterized by the polar angle *φ* = arctan(*ky*/*kx*). Its value remains unchanged in the whole −*π*/*d* ≤ *kx* ≤ *π*/*d* range. The contribution to the conductivity related to the intraminiband transitions is determined by the electron velocity,

In Fig. 24, we illustrate the calculated dependence of the electron velocity on *kx* for the same SL parameters as above for the polar angles *φ* = 5*o*, 10*o*, and 15*o*. We can see that the velocity indeed vanishes at the miniband boundary and its abrupt decrease takes place within a quite

An application of the superlattice at nonzero temperatures requires the existence of a quite clearly pronounced Fermi velocity profile; i.e., we should use rather large *φ* and *δvF* =

*<sup>d</sup>* sin *<sup>φ</sup> <sup>T</sup>*.

However, at large *φ* values close to *π*/2, the condition according to which charge carriers pass a large number of supercells at the mean free path can be violated. Then, condition

*ky*=*kx* tan *φ*

.

*<sup>v</sup><sup>φ</sup>* <sup>=</sup> *<sup>∂</sup><sup>E</sup> ∂k* 

narrow range near the miniband boundary. For low momenta, we have *v<sup>φ</sup>* ≈ *v*<sup>∗</sup>

*π δvF*

(102) turns out to be unimportant: condition *λ<sup>f</sup>* cos *φ d* should be met.

*λ<sup>f</sup> d*. (102)

*F*.

experiment, it is possible to use the standard Hall device.

mean free path should be much larger than the period of SL [85]

For the sufficiently pure graphene samples, we have *λ<sup>f</sup>* 1*µ*m.

*µ* 104 cm2/(V s) [106].

232 Graphene - New Trends and Developments

conductivity

which we seek:


**Figure 24.** Numerical calculation of the electron velocity in the lower miniband along the direction specified by the fixed polar angle *φ*.

Similarly to the situation occurring in semiconductor SLs, the motion of charge carriers at sufficiently strong electric field E*<sup>x</sup>* is finite. They oscillate with the Stark frequency [85]

$$
\Omega = e\mathcal{E}\_x d.
$$

This stems from the nonlinearity of the *I*–*V* curve manifesting itself in the negative differential conductivity at a certain section of it. Charge carriers in the nonlinear regime undergo a large number of the Bloch oscillations during the mean free time *τ*:

$$
\Omega \mathfrak{r} \gg 1.\tag{103}
$$

We estimate the mean free time as *τ* ≈ *λ<sup>f</sup>* /*v*<sup>∗</sup> *<sup>F</sup>* (the velocity of charge carriers is *v<sup>φ</sup>* ≈ *v*<sup>∗</sup> *F* everywhere except for a narrow range near the miniband boundaries). Then, condition (103) can be rewritten as

$$\mathcal{E}\_{\mathbf{x}} \gg \frac{v\_F^\*}{ed\lambda\_f}.\tag{104}$$

Condition (104) automatically gives an estimate for the minimum longitudinal voltage above which negative differential conductivity becomes possible

$$V\_{\chi\text{ min}} \simeq \frac{v\_F^\*}{ed} \frac{L\_{\chi}}{\lambda\_f}\_f$$

where *Lx* is the size of the system along the *x* axis. Assuming that *Lx λ<sup>f</sup>* , we arrive at the estimate *Vx min* 0.02 V for SL with the same parameters as above.

In Fig. 25, we represent the qualitative behavior of the *I*–*V* curve for SL under study. At *ky* = 0 (zero applied voltage in the transverse direction, *Vy* = 0), we observe its linear growth. At *ky* = 0 (nonzero transverse voltage, *Vy* = 0), a section with negative differential conductivity arises in the curve. In this case, for higher *Vy* values, this section is more pronounced and more shifted toward lower *Vx* values. However, as is mentioned above, this section can arise only at a sufficiently high longitudinal voltage

**Figure 25.** Qualitative behavior of the *I*–*V* curve for SL under study. Three *I*(*Vx* ) plots under the linear *I*–*V* curve correspond to the growth of the transverse voltage *Vy* (from top to bottom).
