Marc Josien
email: marc.josien@enpc.fr

Mathematical properties of the Weertman equation

Keywords: Reaction-advection-diffusion equation, traveling waves, integrodifferential equation, the Weertman equation, fractional Laplacian

come

Introduction

Motivation We derive here some mathematical properties of the Weertman equation and show it is the limit of an evolution equation. Our motivation comes from our interest in materials science. The problem we consider, however classical, enjoys the following specificity that it involves the dissipative integrodifferential operator -|∂ x | (also denoted as -(-∆) 1/2 ), which has -|k| as Fourier symbol. In addition to this purely theoretical interest, the results proven here give a solid ground to a numerical approach that we have implemented in [START_REF] Josien | Fourier-based numerical approximation of the Weertman equation for moving dislocations[END_REF].

Our starting point is the so-called Weertman equation (see [START_REF] Rosakis | Supersonic dislocation kinetics from an augmented Peierls model[END_REF]):

-|∂ x | η(x) + cη (x) = F (η(x)) for x ∈ R, (1)
with boundary conditions

lim x→-∞ η(x) = η l and lim x→+∞ η(x) = η r , (2)
where both the scalar c ∈ R (called velocity) and the function η ∈ C 2 (R) are the unknowns, and where η l < η r . The function F ∈ C 3 (R) is a bistable potential; namely, it satisfies F (η l ) = F (η r ) = 0, F (η l ) > 0, and F (η r ) > 0.

(

) 3
From a physical point of view, Equation ( 1) is a nondimensionalized form of the Weertman equation for steadily-moving dislocations in materials science (see [START_REF] Rosakis | Supersonic dislocation kinetics from an augmented Peierls model[END_REF]). Dislocations are linear defects in crystals, the motion of which is responsible for the plasticity of metals. From a physical standpoint, the function η represents a discontinuity between the local relative material displacement u(x, y) in the upper and in the lower half-spaces surrounding the glide plane on which moves the dislocation line (see Figure 1); see, e.g., [START_REF] Hirth | Theory of dislocations[END_REF] for details. In [START_REF] Achleitner | Traveling waves for a bistable equation with nonlocal diffusion[END_REF], the term |∂ x | η accounts for the long-range elastic self-interactions that tend to spread the core. This repulsive interaction is counterbalanced by the nonlinear pull-back force F (η), which binds together the upper and lower half-spaces. Moreover, the moving dislocation is subjected to various drag mechanisms encoded into the term cη . From a broader perspective, the function η can be understood as a moving phasetransformation front between the states η l and η r (see Figure 1), which are local minimizers of the potential F .

Traveling wave of reaction-diffusion equation Equation ( 1) is a special case of the general equation

A[η](x) + cη (x) = 0 for x ∈ R, η(-∞) = η l and η(+∞) = η r , (4)
where A[η] = Lη -F (η) is a nonlinear operator, in which L is a diffusive operator and F is a bistable potential. As is easily seen, Equation (4) describes the traveling waves of the following reaction-diffusion equation:

∂ t u(t, x) = A[u(t, •)](x) for x ∈ R, u(0, x) = u 0 (x) for x ∈ R, (5)
in the sense that, if u(t, x) = η(xct) is a traveling wave satisfying [START_REF] Cabré | Nonlinear equations for fractional Laplacians II[END_REF], then (η, c) solves [START_REF] Cabré | Nonlinear equations for fractional Laplacians I[END_REF]. Ipso facto, finding a solution to (4) amounts to finding traveling waves solutions to [START_REF] Cabré | Nonlinear equations for fractional Laplacians II[END_REF]. Natural questions thus arises:

(i) Does Equation ( 4) have one and only one solution (η, c)?

(ii) Which properties does the solution to Equation (4) enjoy?

(iii) Is the traveling wave η(xct), for (η, c) solution to (4), an attractor of the dynamical system (5)?

These questions have been addressed by many authors for various operators L and for bistable potential F satisfying -most of the time-the extra condition that F does not admit any local minimum between η l and η r . Other types of nonlinearities, not considered here, have attracted much attention. See [START_REF] Volpert | Traveling wave solutions of parabolic systems[END_REF] for the classification of traveling waves and an overview of reaction-diffusion equations.

In the seminal article [START_REF] Sattinger | On the stability of waves of nonlinear parabolic systems[END_REF], Sattinger remarked that if (η, c) is a solution to (4), then (η(•+ξ), c) is also a solution to (4), for arbitrary ξ ∈ R. Therefore, solutions to (4) can at most be unique up to a translation. In this regard, he introduced the notion of asymptotic stability of traveling wave and proved that the solution (η, c) to (4), if it exists, is asymptotically stable under general assumptions about the spectrum of A.

In the celebrated article [START_REF] Fife | The approach of solutions of nonlinear diffusion equations to travelling front solutions[END_REF], Fife and McLeod answered the Questions (i) and (iii) in the case where L is the Laplacian. They proved indeed that if F satisfies (3) and has no local minimum between η l and η r , then there exists a solution (η, c) to (4), which is unique up to a translation, and that this solution is globally asymptotically stable. Namely, for all initial conditions u 0 taking values in [η l , η r ] such that u 0 (-∞) = η l and u 0 (+∞) = η r , there exist ξ ∈ R, K > 0 and κ > 0 such that the solution u of (5) satisfies

u(t, •) -η(• -ξ -ct) L ∞ (R) ≤ Ke -κt , (6)
for all t ∈ R + . Among other important concepts, all amenable to a wide class of dissipative operators, it is observed in [START_REF] Fife | The approach of solutions of nonlinear diffusion equations to travelling front solutions[END_REF] that A satisfies a comparison principle. Thus, any solution u(t, x) of ( 5) can be squeezed between a supersolution w +1 (t, x) and a sub-solution w -1 (t, x), both at a controlled distance from η(xξct).

In a more recent article [START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF], Chen combined this squeezing approach with an iterative technique. Under technical assumptions about the operator A, he proved the global asymptotic stability of the traveling waves of (5), provided that there exists a monotonic solution η to (4). In this context, a positive answer to Question (i) and technical assumptions imply, using Chen's squeezing technique, a positive answer to Question (iii). We use Chen's approach in the present article.

The article [START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF] also provides tools for establishing the existence and the uniqueness of a solution to (4). They have been used in [START_REF] Chmaj | Existence of traveling waves in the fractional bistable equation[END_REF] to positively answer to Question (i) in the case where L is the fractional Laplacian -|∂ x | α , for α ∈ (0, 2) (the latter operator has -|k| α as Fourier symbol). Also, in [START_REF] Achleitner | Traveling waves for a bistable equation with nonlocal diffusion[END_REF], the authors have adapted Chen's squeezing technique to prove that the solution (η, c) to ( 4) is globally asymptotically stable in the sense of [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF], in a general framework including the case L = -|∂ x | α , for α ∈ (1, 2). However, they underlined the fact that the case α ≤ 1 (and in particular α = 1), is still an open question. This motivates our study.

With an approach different from [START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF], the existence and the uniqueness of a solution to (4), for L = -|∂ x | α and α ∈ (0, 2), has been proved in [START_REF] Cabré | Nonlinear equations for fractional Laplacians I[END_REF][START_REF] Cabré | Nonlinear equations for fractional Laplacians II[END_REF] in the special case where c = 0 (the so-called balanced case). These results have been generalized by [START_REF] Gui | Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian[END_REF]. Assuming that F ∈ C 3 (R) satisfy (3) and the following extra condition:

F (u) > F (η l ), ∀u ∈ (η l , η r ), F (u) > 0 or F (u) > F (η r ), ∀u ∈ (η l , η r ), (7)
it is showed in [10, Th. 1.1] that there exist a unique c ∈ R and an increasing function η ∈ C 2 (R), which is unique up to a translation, that solve (1). Conditions ( 3) and [START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF] mean that the potential F (u) has two major wells in u = η l and u = η r (the states η l and η r are therefore stable), and that its behavior is controlled between these wells ; for example, the potential can have minor wells (see Figure 2). The proof of [START_REF] Gui | Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian[END_REF] relies on special solutions to (4) built in [START_REF] Cabré | Nonlinear equations for fractional Laplacians I[END_REF][START_REF] Cabré | Nonlinear equations for fractional Laplacians II[END_REF], which, by homotopy techniques, are used to find the solutions to the general case. The result of [START_REF] Gui | Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian[END_REF] will be our starting point for proving the global asymptotic stability of the traveling waves of ( 5)

for L = -|∂ x |.
Additionally, the authors of [START_REF] Cabré | Nonlinear equations for fractional Laplacians I[END_REF][START_REF] Cabré | Nonlinear equations for fractional Laplacians II[END_REF][START_REF] Gui | Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian[END_REF] have also studied some properties of the solutions to (4). In particular, when L = -|∂ x |, they have shown that η > 0 and that there exist constants B > A > 0 such that, for all |x| ≥ 1,

A|x| -2 ≤ η (x) ≤ B|x| -2 . ( 8
)
See [START_REF] Cabré | Nonlinear equations for fractional Laplacians II[END_REF]Th. 2.7] for the special case where c = 0 and [10, Prop. 3.2] for the general case. Moreover, the following identity is proved (see [START_REF] Gui | Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian[END_REF]Prop. 4.1]):

c = [F (η r ) -F (η l )] ˆR |η | 2 -1 . (9)
Formula ( 9) is useful because it provides the sign of c just by considering the values F (η l ) and F (η r ).

As remarked in [START_REF] Cabré | Nonlinear equations for fractional Laplacians I[END_REF], if c = 0, then (1) can be interpreted as the restriction to the boundary of an elliptic problem with Neumann boundary condition. If indeed u solves the following problem:

∆u(x, y) + c∂ y u(x, y) = 0 in R × R + , ∂ y u(x, 0) = F (u(x, 0)) on R × {0}, (10)
for c = 0, then η(x) := u(x, 0) is a solution to [START_REF] Achleitner | Traveling waves for a bistable equation with nonlocal diffusion[END_REF]. However, we stress that, when c = 0, Equation (10) describes a diffusive traveling wave in the

F (η) η • η l • η r Figure 2: A double-well "camel-hump" potential F , η l = -1, η r = 1.
half-space. In this case, (1) is not the restriction to the boundary of the problem [START_REF] Gui | Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian[END_REF], which instead reads

(-∆ -c∂ x ) 1/2 η(x) = -F (η(x)). (11)
We refer to [START_REF] Cabré | Traveling wave solutions in a halfspace for boundary reactions[END_REF] for a mathematical study of [START_REF] Hirth | Theory of dislocations[END_REF]. But, we mention for completeness that (1) is in fact the restriction to the boundary of the following elliptic equation

∆u(x, y) = 0 in R × R + , ∂ y u(x, 0) + c∂ x u(x, 0) = F (u(x, 0)) on R × {0}. (12)
In a physical context, the latter is envisioned as an elastic equation in the half-plane with a nonlinear boundary condition. We briefly justify it. If indeed we take the Fourier transform with respect to x, denoted as F x , of the first equation of [START_REF] Josien | [END_REF], and if we restrict on bounded solutions, we obtain

F x {u(•, y)} (k) = e -|k|y F x {u(•, 0)} (k).
Injecting the above information in the second equation of (12) then yields (1) if we denote η(x) = u(x, 0) (recall that |∂ x | is an operator which has |k| as Fourier symbol).

Main results

Our first result concerns the asymptotic expansion of the solution η to (1). The following proposition is a refinement of results of [START_REF] Cabré | Nonlinear equations for fractional Laplacians I[END_REF][START_REF] Cabré | Nonlinear equations for fractional Laplacians II[END_REF][START_REF] Gui | Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian[END_REF]: [START_REF] Chmaj | Existence of traveling waves in the fractional bistable equation[END_REF]. Then η has the following asymptotes:

Proposition 1.1. Let η l < η r and F ∈ C 3 (R) satisfy (3). Assume that (η, c) is a solution to (1) and (2) such that η ∈ C 2 (R) is an increasing function satisfying η > 0 and (
η(x) -η r ∼ x→+∞ η l -η r πF (η r ) x -1 , and η(x) -η l ∼ x→-∞ η l -η r πF (η l ) x -1 . ( 13
)
In addition to their theoretical interest, these asymptotes also allow for getting more accurate numerical approximations of η, as shown in [START_REF] Josien | Fourier-based numerical approximation of the Weertman equation for moving dislocations[END_REF].

Our second result is:

Proposition 1.2.
Under the hypotheses of Proposition 1.1, (c, η) satisfies the following identity:

c = 1 η r -η l lim R→+∞ ˆR -R F (η). (14)
The above identity is formally obtained by integrating Equation ( 1) over R; we rigorously prove it. Notice that, by Proposition 1.1 and using a Taylor expansion, F (η) / ∈ L 1 (R). As mentioned above in the concise form (5), we consider the following dynamical system:

∂ t u(t, x) + |∂ x | u(t, x) = -F (u(t, x)) for x ∈ R, u(0, x) = u 0 (x) for x ∈ R, (15)
for an initial condition

u 0 ∈ L ∞ (R). We say that u ∈ L ∞ loc (R + , L ∞ (R)) is a weak solution to (15) if, for all T > 0, for all φ ∈ C 1 c ([0, T ), C ∞ c (R)), there holds ˆT 0 ˆR u(t, x) (-∂ t + |∂ x |) φ(t, x)dx dt = ˆR u 0 (x)φ(0, x)dx - ˆT 0 ˆR F (u(t, x))φ(t, x)dx dt. ( 16
)
Our third and final result is that (1) is the long-time limit of (15), for general initial conditions u 0 with suitable behavior at infinity (see Figure 3 for an example). We prove the following:

Theorem 1.3. Let η l < η r , F ∈ C 3 (R) satisfy (3) and ∆ 0 > 0 be such that F > 0 on [η l -∆ 0 , η l + ∆ 0 ] ∪ [η r -∆ 0 , η r + ∆ 0 ]. ( 17
)
Assume that u 0 ∈ L ∞ (R) takes values in [η l -∆ 0 , η r + ∆ 0 ] and satisfies lim sup x→-∞ u 0 (x) < η l + ∆ 0 and lim inf x→+∞ u 0 (x) > η r -∆ 0 . (18)
Then:

(i) Equation (15) has a unique weak solution u ∈ L ∞ loc (R + , L ∞ (R)). More- over, for all T 0 > 0, u ∈ C (T 0 , +∞), C 2 (R) ∩ C 1 ((T 0 , +∞), C(R)) . ( 19
) (ii) In addition, for all t > 0 and x ∈ R, u(t, x) ∈ [η l -∆ 0 , η r + ∆ 0 ]. (iii) Assume that (η, c) is a solution to (1) and (2) such that η ∈ C 2 (R) is an increasing function satisfying η > 0 and lim |x|→+∞ η (x) = 0. ( 20
)
Then, there exist constants κ > 0, K > 0 and ξ such that

u(t, •) -η(• -ct + ξ) L ∞ (R) ≤ Ke -κt , (21)
for all t ∈ R + . In the above estimate, κ only depends on F , η and ∆ 0 , whereas K and ξ also depend on u 0 .

Theorem 1.3 suggests that simulating ( 15) is sufficient to obtain in the long time a numerical approximation of the solution (η, c) to [START_REF] Achleitner | Traveling waves for a bistable equation with nonlocal diffusion[END_REF]. In this regard, it is significant that c, which is an unknown of (1), does not appear in [START_REF] Pellegrini | [END_REF]. This in particular allows for constructing an approximation of the traveling wave velocity, which is unknown before the end of the simulation. We refer the reader to our study [START_REF] Josien | Fourier-based numerical approximation of the Weertman equation for moving dislocations[END_REF], where we explain the details of the numerical strategy, and to a forthcoming article [START_REF] Pellegrini | [END_REF] for the multi-dimensional case. Outline Our contribution is organized as follows. In Section 2, we introduce notations and give essential properties of the operator |∂ x |. In Section 3, we prove Propositions 1.1 and 1.2. In Section 4, we justify the existence and the uniqueness of a weak solution to Equation [START_REF] Pellegrini | [END_REF], which satisfies (19), establishing thus (i) of Theorem 1.3. In Section 5, we use Chen's approach for proving (ii) and (iii) of Theorem 1.3. The key ingredients are a comparison principle and specific sub-solutions and super-solutions. Although we could check the technical assumptions and apply Chen's theorem, we prefer to restrict Chen's proof to our special case for self-consistency and simplicity.

2∆ 0 2∆ 0 η l + η r+ x u 0 (x)

Notations and definitions

Notations We denote by C ∞ c (R) the space of smooth functions with compact supports in R and by D the space of distributions over R. For u ∈ C ∞ c (R), we denote the Fourier transform by F {u} (k) := ´R e -ikx u(x)dx. For two functions u and v, we denote by * the convolution. Henceforth, the Fourier transform and the convolution are only taken with respect to the space variable x (and never with respect to the time variable t). We make use of the principal value of 1

x , denoted by p.v. 1

x , which is the distribution defined by

p.v. 1 x , u = lim →0 + ˆ+∞ u(y) y dy + ˆ- -∞ u(y) y dy , for u ∈ C ∞ c (R).
Definition and properties of the operator |∂ x | For convenience, we recall some elementary properties of the operator |∂ x |. The Hilbert trans-form H of u ∈ L 2 (R) is defined by

H {u} := F -1 {-i sign(k)F {u} (k)} . ( 22
) It is immediate that, if u ∈ L 2 (R), then H 2 {u} = -u. Next, the operator |∂ x | is defined as |∂ x | u(x) := H {u } (x) = F -1 {|k|F {u} (k)} (x), ( 23
) for u ∈ C ∞ c (R). As F {p.v. (1/x)} (k) = -iπsign(k), the operator |∂ x | can be rewritten as |∂ x | u(x) = - 1 π ˆ+∞ 0 u (x + y) -u (x -y) y dy (24) = - 1 π ˆ+∞ 0 u(x + y) -2u(x) + u(x -y) y 2 dy, (25)
the last expression being obtained from the previous one by integrating by parts. We see from (23) that the operator |∂ x | is symmetric and positive, like the Laplacian. But, unlike the Laplacian, it is clear from (25) that |∂ x | u(x) does not only depend on u in the neighborhood of x but also on each value u(y), for y ∈ R; put differently,

|∂ x | is non-local. A straightforward computation yields that |∂ x | φ ∈ L 1 (R) whenever φ ∈ C ∞ c (R).
Hence, one can extend |∂ x | over L ∞ (R) by duality, defining |∂ x | u as the following distribution:

|∂ x | u : φ ∈ C ∞ c (R) → ˆR u(y) |∂ x | φ(y)dy. ( 26
)
When u is sufficiently regular, explicit expressions of

|∂ x | u are available. Namely, if u ∈ L ∞ (R) ∩ C 2 (R), then Expression (25) is valid. In particu- lar, |∂ x | u ∈ C(R) ∩ L ∞ (R). The proof can be done by density of C ∞ c (R) in C 2 loc (R)
, using the fact that (25) is true for smooth functions. If we assume furthermore that u ∈ L 1 (R), then Expression (24) is also valid; this is deduced from (25) by integration by parts.

Asymptotes and an identity about velocity

The proof of Proposition 1.1 relies on the asymptotic behavior of Cauchy integrals (see [14, p. 267]) and involves the following technical lemma: Lemma 3.1. Under the hypotheses of Proposition 1.1, there holds

η ∈ L ∞ (R). ( 27
)
Remark 3.2. Remark that it is also possible to establish by technical arguments that there exists a constant C > 0 such that, for all |x| > 1,

|η (x)| ≤ C|x| -2 (1 + ln(|x|)) . (28)
(We refer the reader to [START_REF] Josien | [END_REF] for the proof of (28)). However, ( 27) is sufficient to prove Proposition 1.1. Yet, if F is sinusoidal, one can derive analytical solutions η to (1), as is shown in [START_REF] Rosakis | Supersonic dislocation kinetics from an augmented Peierls model[END_REF], which are of the form

η(x) = η l + η r -η l π π 2 + arctan (ax) , for a > 0. Whence η (x) = - η r -η l π 2a 3 x (a 2 x 2 + 1) 2 ∼ x→+∞ - 2(η r -η l ) πax 3 .
Thus (28) is probably not optimal.

We postpone the proof of Lemma 3.1 until the end of the proof of Proposition 1.1 and temporarily admit Lemma 3.1.

Proof of Proposition 1.1. We focus on the case x → +∞. Provided that

x |∂ x | η(x) -→ x→+∞ 1 π ˆ+∞ -∞ η (y)dy = 1 π (η r -η l ) , (29)
then, using (1), ( 2), ( 3) and ( 8), we get by Taylor expansion

F (η r ) (η(x) -η r ) ∼ x→+∞ - 1 π x -1 (η r -η l ) ,
which is [START_REF] Josien | Fourier-based numerical approximation of the Weertman equation for moving dislocations[END_REF].

Let us now prove (29). By assumption, η ∈ C 1 (R), and by (8), we have η ∈ L 1 (R). As a consequence, there holds

x |∂ x | η(x) = x π lim →0 + ˆx- -∞ + ˆ+∞ x+ η (y)
xy dy.

Let R > 0 and x > 2R. We split the integral into three parts

πx |∂ x | η(x) = ˆR -∞ η (y) 1 -y/x dy + x ˆx-R R + ˆ+∞ x+R η (y) x -y dy + x lim →0 + ˆx- x-R + ˆx+R x+ η (y) x -y dy. (30)
The first right-hand term in (30) is dealt with the dominated convergence theorem, the second one avoids the singularity of |x -y| -1 and is bounded thanks to [START_REF] Chmaj | Existence of traveling waves in the fractional bistable equation[END_REF], and the third one is on the singularity of |x -y| -1 and is controlled thanks to ( 27) and ( 8).

As η ∈ L 1 (R) and since (recall that x > 2R)

η (y) 1 -y/x ≤ 2 |η (y)| if y < R, and η (y) 1 -y/x -→ x→+∞ η (y),
then, by the dominated convergence theorem

ˆR -∞ η (y) 1 -y/x dy -→ x→+∞ ˆR -∞ η (y)dy. (31)
Next, we split the second integral of (30) into three parts. Invoking (8), we deduce that, as

|x| > 2R, ˆx-R R + ˆ+∞ x+R η (y) x -y dy ≤ Cx -1 ˆx/2 R |η (y)| dy + CR -1 ˆx-R x/2 + ˆ+∞ x+R |η (y)| dy ≤ Cx -1 ˆx/2 R dy y 2 + CR -1 ˆx-R x/2 + ˆ+∞ x+R dy y 2 ≤ CR -1 x -1 . ( 32
)
Last, we split the last integral of (30) into two parts, namely:

ˆx- x-R + ˆx+R x+ η (y) x -y dy = ˆx-2 + ˆR x -2 η (x -z) -η (x + z) z dz .
The first part of the right-hand side of the above equation is dealt with by using (27), and the second one by using [START_REF] Chmaj | Existence of traveling waves in the fractional bistable equation[END_REF]. Whence, as x > 2R and for x -2 ≥ , ˆx-

x-R + ˆx+R x+ η (y) x -y dy ≤ Cx -2 ˆR x -2 z -1 dz + C ˆx-2 dz ≤ Cx -2 ln(Rx 2 ) + 1 .

Therefore

x lim

→0 + ˆx- x-R + ˆx+R x+ η (y) x -y dy ≤ Cx -1 ln(Rx 2 ) + 1 . (33)
Convergence (31) and Estimates (32) and (33) finally yield

lim sup x→+∞ πx |∂ x | η(x) - ˆ+∞ -∞ η (y)dy ≤ ˆ+∞ R η (y)dy + CR -1 ,
which, thanks to (8), implies (29) upon letting R → +∞.

We then proceed with the:

Proof of Lemma 3.1. We first remark that if g ∈ L 2 (R) and if

-|∂ x | h(x) + ch (x) = g(x), (34)
then h ∈ L 2 (R). Indeed, the Fourier transform turns (34) into

(-|k| + cik)F{h}(k) = F{g}(k).
Therefore, kF{h}(k) ∈ L 2 (R), whence h ∈ L 2 (R). We use this result to prove that η ∈ L ∞ (R).

Upon differentiating (1), we obtain

-|∂ x | η (x) + cη (x) = F (η(x))η (x). ( 35
) As η ∈ L ∞ (R), F ∈ C 3 (R) and η ∈ L 2 (R) (thanks to (8)), then the right- hand side of (35) is in L 2 (R). Therefore η ∈ L 2 (R). Differentiating (35) yields -|∂ x | η (x) + cη (x) = F (η(x))η (x) + F (η(x)) (η (x)) 2 . ( 36
) As η ∈ L ∞ (R), F ∈ C 3 (R), η ∈ L 2 (R)
, and, thanks to (8), (η ) 2 ∈ L 2 (R), then the right-hand side of (36) is in L 2 (R). Therefore η ∈ L 2 (R). As a consequence, since η ∈ L 2 (R), we deduce by Sobolev injection that η ∈ L ∞ (R), whence (27).

We now focus on Proposition 1.2. Both Identities ( 9) and ( 14) are formally obtained by testing (1) against a certain function g, namely g = η for (9), and g = 1 for [START_REF] Muskhelishvili | Some basic problems of the mathematical theory of elasticity[END_REF]. We justify below this formal integration.

Proof of Proposition 1.2. Let R > 2. We integrate (1) over [-R, R]:

- ˆR -R |∂ x | η(x)dx + c (η(R) -η(-R)) = ˆR -R F (η(x))dx. (37)
Thus, Identity ( 14) stems from (37), provided that

lim R→+∞ ˆR -R |∂ x | η(x)dx = 0. (38)
We prove (38) using ( 8) and ( 27). As η ∈ L 1 (R) ∩ C 1 (R), there holds

|∂ x | η(x) = 1 π ˆ+∞ 0 η (x -y) -η (x + y) y dy = 1 π ˆ|y|<R (η (x -y) -η (x)) y dy + 1 π ˆ|y|>R η (x -y) y dy. ( 39
) Remark that ˆ|y|>R η (x -y) y dy ≤ 2R -1 η L 1 (R) ,
and that, using (27

), ˆ|y|<R (η (x -y) -η (x)) y dy ≤2R η L ∞ (R) ≤ CR.
Therefore, integrating (39) thanks to Fubini's theorem yields

ˆR -R |∂ x | η(x)dx = 1 π ˆ|y|<R (η(R -y) -η(R)) -(η(-R -y) -η(-R)) y dy + 1 π ˆ|y|>R η(R -y) -η(-R -y) y dy = : T 1 + T 2 . (40)
First, we bound T 1 . If |y| < R/2, thanks to (8), we obtain

|η(R -y) -η(R)| ≤ C|y|R -2 and |η(-R -y) -η(-R)| ≤ C|y|R -2 .
As a consequence, ˆ|y|<R/2

(η(R -y) -η(R)) -(η(-R -y) -η(-R)) y dy ≤ CR -1 . (41)
Note that, as underlined in [START_REF] Gui | Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian[END_REF], a consequence of ( 8) is that

A|x| -1 ≤ η r -η(x) ≤ B|x| -1 , if x > 1, A|x| -1 ≤ η(x) -η l ≤ B|x| -1 , if x < -1. (42) Therefore, if |y| < R |η(R -y) -η(R)| ≤ C R -|y| + 1 and |η(-R -y) -η(-R)| ≤ C R -|y| + 1 . Whence ˆR/2<|y|<R (η(R -y) -η(R)) -(η(-R -y) -η(-R)) y dy ≤ C R ˆR R/2 dy R + 1 -y ≤ CR -1 ln(R). (43)
We deduce from (41) and (43) that

|T 1 (R)| ≤ CR -1 (1 + ln(R)). (44)
Thanks to [START_REF] Chmaj | Existence of traveling waves in the fractional bistable equation[END_REF]

and since η ∈ L ∞ (R), if |y| > R, we have |η(R -y) -η(-R -y)| ≤ C min R (|y| -R) -2 , 1 .
Whence, splitting T 2 into two parts,

|T 2 (R)| ≤C ˆR+ √ R R dy y + ˆ+∞ R+ √ R R y(y -R) 2 dy ≤C √ R R + C ˆ+∞ √ R dz z 2 ≤ CR -1/2 . ( 45
)
Bearing (40) in mind, we observe that (44) and ( 45) imply (38).

4 Existence, uniqueness and regularity of the solution to the evolution equation [START_REF] Pellegrini | [END_REF] We now justify the existence, the uniqueness and the regularity of a weak solution u to [START_REF] Pellegrini | [END_REF]. We proceed in the classical way; as the methods as well as the type of results are well-known, we only give a few hints of proofs.

We refer the interested reader to [START_REF] Josien | [END_REF] for some technical details and extra materials about the proofs, and to [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF] for a reference on evolution equations involving m-dissipative operators.

Using the Fourier transform, the solution to the homogeneous linear equation

∂ t u(t, x) + |∂ x | u(t, x) = 0 for x ∈ R, and u(0, x) = u 0 (x) (46)
is given by u(t, x) = {K t * u 0 } (x), where the kernel K t is defined by

K t (x) = t π(t 2 + x 2 ) if t > 0 and K 0 = δ 0 , (47)
the Fourier transform of which is e -|k|t . Before getting to the inhomogeneous linear equation, we underline some interesting properties of the kernel K t .

First, for all t ≥ 0, K t is a probability measure. Moreover, for all t > 0, K t is a smooth function. In particular, the space derivative of K t satisfies

d dx K t L 1 (R) ≤ Ct -1 , ( 48
)
where C is a universal constant. In all these aspects, K t is similar to the Gaussian kernel K t (x) = e -x 2 2t 2 /(t √ 2π). The semi-group generated by K t allows for solving the inhomogeneous equation

∂ t u(t, x) + |∂ x | u(t, x) = g(t, x) for x ∈ R, u(0, x) = u 0 (x) for x ∈ R. ( 49
) Indeed, let T > 0, u 0 ∈ L ∞ (R) and g ∈ L ∞ ([0, T ] × R).
Then there exists a unique weak solution u ∈ L ∞ ([0, T ] × R) to (49) in the sense that, for

all φ ∈ C 1 c ([0, T ), C ∞ c (R))
, the following identity holds:

ˆT 0 ˆR u(t, x) (-∂ t + |∂ x |) φ(t, x)dxdt -ˆR u 0 (x)φ(0, x)dx = ˆT 0 ˆR g(t, x)φ(t, x)dxdt. ( 50
)
This solution can be written thanks to the Duhamel formula as

u(t, x) = K t * u 0 (x) + ˆt 0 K t-s * g(s, •)ds (x), (51)
with the convention that K 0 * u 0 = u 0 , even if u 0 is not regular. The existence of a solution u to (49) is a consequence of the fact that ( 51) is well-defined; its uniqueness is showed using the adjoint problem of (49). We now turn to the semi-linear equation [START_REF] Pellegrini | [END_REF]. Let F ∈ W 2,∞ (R) and u 0 ∈ L ∞ (R). Then there exists a unique weak solution u ∈ L ∞ loc (R + , L ∞ (R)) to [START_REF] Pellegrini | [END_REF]. Moreover, u can be expressed as

u(t, x) = {K t * u 0 } (x) - ˆt 0 K t-s * F (u(s, •))ds (x). ( 52
)
The proof is done by a classical fixed-point argument on (52) (see for example [6, Sec. 4.3 p. 56]). Finally, we justify that the evolution equation ( 15) has a regularizing effect; in other words, the weak solution to [START_REF] Pellegrini | [END_REF] becomes instantly a classical solution. Assume indeed that F ∈ C 3 (R) ∩ W 3,∞ (R) and u 0 ∈ L ∞ (R), and let u ∈ L ∞ loc (R + , L ∞ (R)) be the weak solution to [START_REF] Pellegrini | [END_REF]. Then, for all T 0 > 0, we have (19). Therefore, for all t > 0, there holds

∂ t u(t, x) + |∂ x | u(t, x) = -F (u(t, x)), (53)
in the strong sense. Finally

u ∈ C ([0, +∞[ , weak- * -L ∞ (R)) . ( 54
)
The proof of (19) relies on an iterative argument based on (52), using the fact that, for all t > 0, K t is a smooth probability measure that satisfies (48). Last, a straightforward adaptation of the proof of [2, Ex. 4.24 p. 126] yields (54). Note that, similarly to [1, Th. 2.9.], t → u(t, •) may not be continuous at

0 in L ∞ (R). If indeed u 0 is discontinuous, then u(t, •) cannot tend to u 0 in L ∞ (R) when t → 0 because, invoking (19), u(t, •) is a continuous function for all t > 0.
5 Convergence of the evolution equation [START_REF] Pellegrini | [END_REF] to the Weertman equation (1

)
In this section, we prove (ii) and (iii) of Theorem 1.3. The proof can be summarized in two steps: first, we show that (15) satisfies a comparison principle, then we use Chen's method of squeezing, establishing respectively (ii) and (iii) of Theorem 1.3. For the sake of self-consistency, simplicity and conciseness (Chen's theory being quite general), we prefer to restrict the whole proof of [START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF]Th. 3.1] to our specific case rather than to check that the hypotheses of Chen's theory are satisfied (precisely Hypotheses (A1), (A2), (A3), (B1), (B2) and (B3) of [START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF], which are indeed satisfied in our case). We henceforth assume that 3) and ( 17). We introduce the non-linear operator A[u] := -|∂ x | u -F (u), of which we now discuss some immediate properties. By the results of Section 4, A generates a semi-group on the Banach space L ∞ (R). A is translation invariant; namely, for all h ∈ R, and for any function u(x), there holds

F ∈ C 3 (R) ∩ W 3,∞ (R) satisfies (
A[u(h + •)](x) = A[u](x + h), ∀x ∈ R. (55)
Moreover, A maps constant functions to constant functions; namely

A[α • 1] = -F (α) • 1,
for all α ∈ R, where 1 above denotes the function identically equal to 1.

The operator A satisfies the following comparison principle:

Proposition 5.1. Let F ∈ W 2,∞ (R). Let u and u ∈ L ∞ loc (R + , L ∞ (R)) be such that ∂ t u(t, x) -A[u(t, •)](x) = g(t, x) ≤ 0, ∂ t u(t, x) -A[u(t, •)](x) = g(t, x) ≥ 0, (56)
where g and g ∈ L ∞ loc (R + , L ∞ (R)), and u(0, •) ≤ u(0, •) with u(0, •) = u(0, •) on a non-negligible set. Then, for almost every t > 0, x ∈ R, there holds u(t, x) < u(t, x).

(57)

Remark 5.2. Proposition 5.1 has an immediate corollary: Assume that

u 0 ∈ L ∞ (R) takes values in [η l -∆ 0 , η r + ∆ 0 ] and let u ∈ L ∞ loc (R + , L ∞ (R)
) be the unique solution to [START_REF] Pellegrini | [END_REF] with the initial condition u 0 . By (3), u := x → η r +∆ 0 and u : x → η l -∆ 0 are respectively supersolutions and subsolutions to [START_REF] Pellegrini | [END_REF]. Therefore, Proposition 5.1 implies that u(t, x) ∈ [η l -∆ 0 , η r + ∆ 0 ], for almost every t ∈ R + and x ∈ R, establishing thus (ii) of Theorem 1.3.

Proposition 5.1 is shown thanks to the Duhamel Formula (51) and Grönwall's Lemma.

Proof. Let M > F L ∞ (R) . We set v(t, x) := e M t (u(t, x) -u(t, x)) , (58)
and prove that v ≥ 0. In view of (56), we have

∂ t v(t, x) + |∂ x | v(t, x) = M v(t, x) -e M t (F (u(t, x)) -F (u(t, x)) + g(t, x) ,
where g(t, x) := e M t g(t, x)g(t, x) ≥ 0. Since the right-hand side of the latter equation is in

L ∞ loc (R + , L ∞ (R)), then, using (51), one can express v(t, x) as v(t, x) = {K t * v(0, •)} (x) + ˆt 0 K t-s * M v(s, •) -e M s (F (u(s, •)) -F (u(s, •)) + g(s, •)) ds (x). (59)
We introduce v -(t) := -ess inf {min(0, v(t, x)), x ∈ R}. By Taylor expansion, for almost every (s, y) ∈ [0, t] × R, there holds

M v(s, y) -e M s (F (u) -F (u)) (s, y) ≥M v(s, y) -F L ∞ (R) e M s |u(s, y) -u(s, y)| ≥M v(s, y) -M |v(s, y)| ≥ -2M v -(s). (60)
Therefore, using (59), since K t , g and v(0, •) are nonnegative, and since K t is a probability measure for all t ≥ 0, we obtain, for almost every t ∈ R + and x ∈ R,

-v(t, x)) ≤ 2M ˆt 0 v -(s)ds, whence v -(t) ≤ 2M ˆt 0 v -(s)ds. ( 61
) Since u, u ∈ L ∞ loc (R + , L ∞ (R)), then, v -∈ L ∞ loc (R + ).
Hence, by Grönwall's Lemma, we deduce from (61) that v -(t) = 0, for almost every t > 0. Injecting this information in (60), and next in (59), yields v(t, x) ≥ {K t * v(0, •)} (x). As a consequence, as K t is positive if t > 0 and as v(0, •) = u(0, •)u(0, •) is nonnegative and positive on a non-negligible set, we deduce that v(t, x) > 0, for almost every t > 0 and x ∈ R. This implies (57).

Then, we establish a stronger version of the comparison principle, the proof of which mimicks that of Proposition 5.1: Corollary 5.3. Under the assumptions of Proposition 5.1, there exists a positive decreasing function ρ such that, for all R > 1,

ess inf x∈[-R,R] (u(1, x) -u(1, x)) ≥ ρ(R) ˆ1 0 (u(0, y) -u(0, y)) dy. (62)
Proof. Introducing v defined by (58), and using ( 59) and (60), we obtain

v(t, x) ≥ {K t * v(0, •)} (x),
since v is nonnegative. By definition of K t and of v, it implies (62).

The proof of (iii) of Theorem 1.3 is done after [START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF]Th. 3.1], the proof of which we restrict here to our particular case. By the previous steps, we already know that there exists a unique weak solution u(t, x) to (15) (see Remark 5.2), and we aim at establishing (21).

Namely, we build special sub-solutions and super-solutions to (15) that are based on the existing solution to (1) (see Lemma 5.4 below). Then, we prove that both the vertical and the horizontal distances between these solutions are controlled (respectively 2δ j and 2l j on Figure 5, see also Lemma 5.5 below). Using the fact that ( 15) is an autonomous system, we use the established control to iteratively build successive sub-solutions w j -1 and supersolutions w j +1 surrounding the actual solution u(t j , x) to [START_REF] Pellegrini | [END_REF]. At each step j, the distance between these sub-solutions and super-solutions is lowered. Thus, the solution u is squeezed between these sub-solutions and supersolutions. As a consequence, when t goes to infinity, the solution u(t, •) tends, up to a translation, to the solution to [START_REF] Achleitner | Traveling waves for a bistable equation with nonlocal diffusion[END_REF]. Because of the iterative nature of the squeezing, this convergence is achieved with exponential speed.

2δ j 2l j + + w j +1 (t j , •) w j -1 (t j , •) u(t j , •) 2δ j+1 2l j+1 + + w j+1 +1 (t j+1 , •) w j+1 -1 (t j+1 , •) u(t j+1 , •)
Figure 4: Squeezing of u(t, x) solution to (5).

Lemma 5.4 (Lemma 2.2 of [START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF]). Under the hypotheses of Theorem 1.3, let ∆ 1 < ∆ 0 . Then, there exist positive constants σ and β such that, for all δ ∈ (0, ∆ 1 ) and l ∈ R, the functions w -1 (t, x) and w +1 (t, x) defined by

w i (t, x) = η (ζ i (t, x)) + iδe -βt for i ∈ {-1, +1}, (63)
where

ζ i (t, x) = x -ct + il + iσδ 1 -e -βt for i ∈ {-1, +1}, (64)
are respectively a sub-solution and a super-solution to [START_REF] Pellegrini | [END_REF].

Proof of Lemma 5.4. As A is invariant by translation, the variable l in the definition of ζ plays no role. Hence, we take l = 0 in the proof below. We also impose for the moment σ ≤ 1.

A straightforward computation yields

(∂ t + |∂ x |) w i (t, x) = iσβδe -βt/2 -c η (ζ i (t, x)) -iδβe -βt + |∂ x | η (ζ i (t, x)) ,
and, as η satisfies (1),

(∂ t + |∂ x |) w i (t, x) = iσβδe -βt η (ζ i (t, x)) -iβδe -βt -F (η (ζ i (t, x))) .
Thus

(∂ t -A) w i (t, x) =iβδe -βt [ση (ζ i (t, x)) -1] + F (w i (t, x)) -F (η (ζ i (t, x))) .
By Taylor-Lagrange expansion, there exists a convex combination θ i (t, x) of η (ζ i (t, x)) and w i (t, x) such that

(∂ t -A) w i (t, x) =iδe -βt βση (ζ i (t, x)) -β + F θ i (t, x) . (65)
Recall that ∆ 0 > ∆ 1 . Then, by (2), there exists R 0 > 0 such that

|η(y) -η l | < (∆ 0 -∆ 1 )/2 if y < -R 0 , |η(y) -η r | < (∆ 0 -∆ 1 )/2 if y > R 0 .
Therefore, as σδ < ∆ 1 , we also have

|w i (t, x) -η l | ≤ (∆ 0 + ∆ 1 )/2 < ∆ 0 if ζ i (t, x) < -R 0 , |w i (t, x) -η r | ≤ (∆ 0 + ∆ 1 )/2 < ∆ 0 if ζ i (t, x) > R 0 . Let β := inf F (v), |v -η r | ≤ ∆ 0 or |v -η l | ≤ ∆ 0 (66) Thus, if |ζ i (t, x)| > R 0 , by definition of β and of θ i (t, x), F (θ i (t, x)) -β ≥ 0. Moreover, η > 0. Therefore βση (ζ i (t, x)) + F θ i (t, x) -β ≥ 0. (67)
Now, we set

σ := min β -1 inf |y|<R 0 η (y) -1 F L ∞ (R) + β , 1 . (68)
Therefore, if |ζ i (t, x)| ≤ R 0 , we also have (67). As a conclusion, in any case, (65) and (67) yield

i (∂ t -A) w i (t, x) ≥ 0,
which implies that w +1 and w -1 are respectively a super-solution and a subsolution to [START_REF] Pellegrini | [END_REF].

Using Lemma 5.4, it is eventually possible to squeeze a solution u(t, x) of ( 15) between a sub-solution and a super-solution. The following Lemma explains how this squeezing is tightened: Lemma 5.5 (Lemma 3.3 of [START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF]). Under the hypotheses of Theorem 1.3, let ∆ 1 < ∆ 0 . Assume that there exist ξ ∈ R, δ ∈ (0, ∆ 1 ) and l ∈ [0, L] for fixed L such that, for all x ∈ R,

η(x -l) -δ ≤ u(0, x) ≤ η(x + l) + δ. (69)
Then, taking β and σ as in Lemma 5.4, there exist a positive constant * depending only on η, F , and L, and parameters ξ, δ, l satisfying ξ ≤ l, δ = δ + * min(1, l)e β , and

0 ≤ l ≤ l + σδ - σ * min(1, l) 2 ,
such that, for all t ≥ 1 and x ∈ R,

η(x -ξ -l -ct) -δe -βt ≤ u(t, x) ≤ η(x -ξ + l -ct) + δe -βt . (70)
Proof. Thanks to Lemma 5.4, the functions w +1 and w -1 defined by (63), for ζ i defined by (64), are respectively a super-solution and a sub-solution to [START_REF] Pellegrini | [END_REF]. Using (69), it follows from Proposition 5.1 that, for all t ≥ 0 and x ∈ R,

w -1 (t, x) ≤ u(t, x) ≤ w +1 (t, x). (71)
Let l := min(1, l) and 1 := inf x∈[-1,2] η (x). Since η is increasing, a Taylor expansion yields ˆ1 0

(η(x + l) -η(x -l)) dx ≥ ˆ1 0 η x + l -η x -l dx ≥ 2 1 l.
Therefore, at least one of the following estimates is true

ˆ1 0 (u(0, x) -η(x -l)) dx ≥ 1 l or ˆ1 0 (η(x + l) -u(0, x)) dx ≥ 1 l.
Hereafter, we only consider the first case, as the second one is similar. First, using (20), there exists R 1 such that

2ση (x) ≤ 1 if |x| > R 1 (72) Let R 2 := R 1 + L + |c| + 1 + σ∆ 0 .
On the one hand, invoking Proposition 5.1, we compare u and w

-1 on [-R 2 , R 2 ] inf x∈[-R 2 ,R 2 ] u(1, x) -η (ζ -1 (1, x)) + δe -β ≥ ρ(R 2 ) ˆ1 0 [u(0, y) -η(y -l) + δ] dy ≥ ρ(R 2 ) 1 l. (73)
and bearing in mind that η is increasing, we deduce from (77) and (78) that u(t, x) ≥η xct -ξlδe -βt .

Moreover, recalling (71), we have u(t, x) ≤η (xct + l + σδ) + δe -βt ≤ η xct + lξ + δe -βt .

(80)

As a consequence, we obtain the desired result (70) from ( 79) and (80).

We are now in position to finish the proof of Theorem 1.3. The proof is done while iterating Lemma 5.5, which gradually tightens the squeezing around u(t, x).

Proof of (iii) of Theorem 1.3 (restriction of the proof of Theorem 3.1 of [START_REF] Chen | Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations[END_REF]). We proceed in four steps, lowering iteratively in time the values δ and l such that, for all x ∈ R, there holds η (xctξl)δ ≤ u(t, x) ≤ η (xctξ + l) + δ.

(81)

Step 1 By assumption [START_REF] Volpert | Traveling wave solutions of parabolic systems[END_REF] and since η is increasing from η l to η r , there exist ∆ 1 < ∆ 0 , and L > 1 sufficiently large such that (81) holds with t = t 1 := 0, δ = δ 1 := ∆ 1 , ξ = ξ 1 := 0, and l = l 1 := L -σ∆ 0 .

Step 2 We define

Using Lemma 5.4, we deduce from the previous step that there exists t 2 sufficiently high such that (81) holds with t = t 2 , δ = δ 2 = δ * and for a certain ξ ∈ R and l = l 2 ≤ L (as * implicitly depends on L, we further ensures that l j ≤ L, for all j). If l 2 ≤ 1, one directly goes to Step 3. Otherwise, as long as l j > 1, one applies Lemma 5.5 at time t j = t 2 + (j -2)t * (recall that ( 15) is an autonomous evolution equation), δ j = δ * and get, by (82) and (83), that (81) holds for t = t j+1 with l ≤ l jκ * and δ ≤ (1κ * )δ * . Therefore, one can take δ j+1 := δ * , l j+1 := l jκ * and iterate until l j < 1.

Step 3 By Step 2, we have an index j 0 such that (81) holds for t = t j 0 , δ = δ j 0 = δ * , ξ j 0 ∈ R and l = l j 0 = 1. Using Lemma 5.5 and Definitions (82) and (83), a straightforward computation inductively shows that, for all j ≥ 0, Inequalities (81) hold for t = t j+j 0 , δ = δ j+j 0 , and l = l j+j 0 being defined by t j+j 0 := t j 0 + jt * , δ j 0 +j := (1κ * ) j δ * and l j+j 0 := (1κ * ) j , (

and for ξ = ξ j such that |ξ j 0 +j+1ξ j 0 +j | ≤ (1κ * ) j .

Step 4 We have shown that (81) holds for (t, δ, l) = (t j , δ j , l j ), for all j ≥ 0.

For t > 0, we associate j implicitly defined by t ∈ [t j , t j+1 ). Thus, we deduce from Lemma 5.4 that (81) also holds for t, δ = δ j , ξ = ξ j , and l = l j + σδ j . Taking Step 3 into account yields, for all j > j 0 , δ ≤ δ * (1κ * ) j-j 0 and l ≤ (1 + σδ * ) (1κ * ) j-j 0 .

Moreover, ξ j converges to ξ ∞ and, for all j > j 0 ,

|ξ j -ξ ∞ | ≤ κ -1 * (1 -κ * ) j-j 0 .
Yet, a simple calculation shows

(1κ * ) j-j 0 = (1κ * ) -j 0 -t j 0 /t * exp (t ln(1κ * )/t * ) .

Setting κ :=ln(1κ * )/t * > 0 and

K := δ * + 1 + σδ * + κ -1 *
η L ∞ (R) (1κ * ) -j 0 -t j 0 /t * , we obtain, for all t ≥ t j 0 , t ∈ [t j , t j+1 ),

sup x∈R |η(x -ct -ξ ∞ ) -u(t, x)| ≤δ j + (|l j | + |ξ j -ξ ∞ |) η L ∞ (R) ≤ Ke -κt .
This concludes the proof of Theorem 1.3.

ηFigure 1 :

 1 Figure 1: Typical shape of η(x), solution to (1); here, u(x, y) is the material displacement.

Figure 3 :

 3 Figure 3: A possible initial condition u 0 in Theorem 1.3.

δ

  * := min (∆ 1 , * /4) and κ * := σ * /2σδ * ≥ σ * /4 > 0. (82) Also, we set t * ≥ 2 such that e -βt * (1 + * /δ * ) e β ≤ 1κ * .

Acknowledgements We would like to thank Claude Le Bris for his advice, and Yves-Patrick Pellegrini for his kindness and for providing a physical insight into the Weertman equation.

We define * := min ∆ 1 1e -β ,

As a consequence, if |x| < R 2 , (73) yields

On the other hand, if

Inequality (71) and Definition (72) then imply that

Therefore, from (75) and from ( 76), it appears that, for all x ∈ R, there holds

We set δ := δe -β + * l e β , which, thanks to (74), satisfies δe -β ≤ ∆ 1 . Applying once more Lemma 5.4 yields, for all t ≥ 1,

By definition of δ and ζ -1 , the argument of η in the above estimate is


V A Saroka

K G Batrakov

V A Demin

L A Chernozatonskii

Band Gaps in Jagged and Straight Graphene Nanoribbons Tunable by an External Electric Field

Band gap control by an external field is useful in various optical, infrared and THz applications. However, widely tunable band gaps are still not practical due to variety of reasons. Using the orthogonal tight-binding method for π-electrons, we have investigated the effect of the external electric field on a subclass of monolayer chevron-type graphene nanoribbons that can be referred to as jagged graphene nanoribbons. A classification of such ribbons was proposed and band gaps for applied fields up to the SiO2 breakdown strength (1 V/nm) were calculated. According to the tight-binding model, band gap opening (or closing) takes place for some type of jagged graphene nanoribbons in the external electric field that lays in the plane of the structure and perpendicular to its longitudinal axis. Tunability of the band gap up to 0.6 eV is attainable for narrow ribbons. In the case of jagged ribbons with armchair edges larger jags forming a chevron pattern of the ribbon enhance the controllability of the band gap. For jagged ribbons with zigzag and armchair edges regions of linear and quadratic dependence of the band gap on the external electric field can be found that are useful in devices with controllable modulation of the band gap.

I. INTRODUCTION

Since it was obtained in a freestanding form graphene 1 has been attracting the attention of the scientific community both as an interesting object for fundamental study (due to its massless Dirac fermions 2 ) and as a base for future technological advances, due to its chemical stability, mechanical strength 3 , high electrical and thermal 4,5 conductivities. Numerous graphene applications 6 include field effect transistors 7 and their interconnects 8,9 , sensors 10,11 , hydrogen storage [START_REF] Tozzini | Physical chemistry chemical physics[END_REF] , terahertz emitters 13,14 , transparent electrodes etc. Many of these applications would benefit from a full control over the band gap. For instance, great efforts have been undertaken to develop novel tunable sources and detectors of THz radiation 15 . Speaking of electronic applications, one must confess that there is a problem of contact resistance 16 that can be easily overcome in all carbon electronic devices 17 , but this again requires a complete control over the band gaps of the nanostructures. Additionally, all carbon electronics can be easily recycled and used in a closed-loop production cycle. Therefore it is desirable to control the band gap both by structural modification and external fields.

A number of techniques have been proposed for band gap engineering in graphene: patterning of graphene 18,19 , straining of graphene [20][21][22][23][24] , lateral confinement of charge carries in one dimension in graphene nanoribbon (GNR), vertical inversion symmetry breaking in bilayer graphene 25,26 or trilayer graphene 27 . All of them have some advantages and disadvantage, for instance, patterning allows higher current compared to GNR, while symmetry breaking by an external electric field provides tunability of the band gap, giving rise to tunable devices which is not the case for patterning or carrier confinement.

The universal approach to band gap control was also attempted in a combination of strategies. Recently, after extensive theoretical study 28,29 , simultaneous lateral carrier confinement and vertical inversion symmetry breaking in bilayer GNRs to tune their band gaps and improve the on/off ratio in logic devices has been experimentally verified and reported 30 .

However, as was noticed earlier in the literature [31][32][33] , owing to the presence of edges for GNR electric field can have not only an out-of-plane direction that is normal to the ribbon plane, but also an in-plane one that is transverse to the ribbon longitudinal axis. A crucial difference between these two geometries of applied field is that the latter may affect properties of one-layer systems. Even though width dependence was reported and band gap increase for zigzag GNR width decrease was shown 31,33 , quite wide ribbons were chosen for the sample study and the limit of this increase for narrow (sub-5nm) GNR was not demonstrated. Probably, it could be explained by the natural desire to be closer to experimentalists who were mostly limited at that time by electron beam lithography and etching techniques of GNR production. Taking into account the latest advances in GNR synthesis, when GNRs with atomically precise edges were synthesised and a bottom-up approach to synthesis of chevron-type GNRs was demonstated 34 , one must feel free to consider narrow GNRs. It is worth noticing that considerable steps forward have been made in the technique scalability and processability. At the moment it is possible to synthesize over 1 gram of ribbons in a single synthesis 35 . Quite long (> 200 nm) ribbons can be synthesized and modified in solution 36 that allows their deposition on any conductive or isolative substrate. Recently heterojunctions in chevron-type nanoribbons were also produced and investigated 37 . All of these experimental efforts have resulted in the introduction of a new class of carbon nanostructures of a chevron type and their derivatives which are referred to by some authors as graphene nanowiggles. These structures have been attracting much attention in last years 5,[38][39][40][41][42][43] , although their history can be traced back to the pioneering theoretical works 44,45 .

Here we investigate an effect of the in-plane homogeneous electric field for a subclass of chevron-type GNRs and compare it with the effect for ordinary GNRs having straight edges, e.g. GNRs of zigzag and armchair types. In Section II we introduce the structures under consideration and discuss their similarities and differences from those in other recent works [38][39][40][41][42][43] . We also discuss the connection of their description with that of straight GNRs. Section III is devoted to the method used. In Section IV we report on our obtained results. Finally, in Section V we summarize our findings and present our conclusions.

II. STRUCTURES

In what follows we will refer to structures being studied as jagged graphene nanoribbons (JGNRs). The main novelty of JGNR is so-called jags introduced in Ref. 46 . However, in our case jags are placed asymmetrically on both side of the ribbon so that the mirror reflection symmetry with respect to the ribbon longitudinal axis is broken. Breaking of the reflection symmetry admits the existence of very narrow ribbons of such class without a host ribbon. They can be referred to as assembled graphene nanoribbons, if one wants to highlight the fact that they can be represented as a sequence of ordinary GNRs with straight edges connected at a specific angle that can be referred to as an apex one. In this regard JGNRs are like GNWs in papers [38][39][40][41][42][43] . However, GNWs can have a truncated jag form, while jag arms are always of equal lengths so that they are reflection symmetric with respect to the axis that is orthogonal to the longitudinal one. Conversely, JGNR edge form is always zigzag, whereas jag arms can be of different lengths. Zigzag just mentioned must not be confused with zigzag type of edge.

To describe a jag one needs two vectors L 1 and L 2 . Their magnitudes correspond to the jag arm lengths, while the angle between them is the apex angle. In one respect, JGNR is even more simple than GNR with symmetric positioning of jags 46 because one edge of JGNR can be obtained from another by mere translation on a certain vector, which is reasonable to determine as width vector W (See Fig. 1). These three vectors describing the macrostructure of the superlattice are the chief vectors of JGNR, but they are still not connected with the microstructure of JGNR, e.g. there is no reference to the graphene hexagon lattice. In the most general case they could be any lattice vectors having the form C = na 1 + ma 2 ;

(1)

where a 1 and a 2 are the primitive translations of the graphene lattice. Introducing the greatest common divisor (d) of n and m, one can rewrite Eq.( 1) in the form

C = d (v 1 a 1 + v 2 a 2 ) = dv; (2)
Z60 Z120 A60 A120 1 (1, 0) (1, -1) (2, -1) (2, -1) 2 (0, 1) (0, 1) (1, 1) (-1, 2) w (1, 1) (1, 0) (1, 0) (1, 1)
where v can be referred to as an elementary vector for the specified direction as v 1 and v 2 do not have common divisor except for unity. For two particular types of directions in the graphene lattice that are referred to as zigzag (Z) and armchair (A), the elementary vectors are the smallest possible having magnitudes of √ 3a 0 and 3a 0 , respectively, where a 0 is the distance between the nearest carbon atoms. Therefore, in this work we restrict the chief vectors L 1 , L 2 and W only to these directions depicted by 2-2 and 1-1 lines in the Fig. 1.

As a result, one can label any JGNR by a set of three integers corresponding to the chief vectors L 1 , L 2 and W . These integer indexes uniquely define JGNR if the set of elementary vectors is known (see Table I). For one set of elementary vectors there is a manifold of index sets, therefore the set of elementary vectors can be considered as characteristic of a JGNR type. But the set of vectors are not so convenient for structure type labelling as an index set for ribbon one. Therefore, taking into account that structures corresponding to various vector sets differ from one another only by their edge types and apex angles, we propose the following notation "t"ϕ, where "t" stands for "type" and ϕ -for the apex angle. If one demands the width vector to be a bisector of the apex angle, then both jag arms have the same type of edges, e.g. zigzag or armchair, and can be denoted as Z or A. Thus, all variety of JGNR is reduced to four types Z60, Z120, A60, A120 and within each type any ribbon is specified by index set 1 , 2 ; w (see Fig. 2).

By means of the introduced vectors and indexes such quantities as JGNR superlattice translation vector T and the number of atoms in the unit cell N can be expressed:

T = L 1 -L 2 ;
(3)

N = λw( 1 + 2 ); ( 4
)
where λ = 2 in case of JGNR Z60, Z120, A60, and λ = 6 in case of JGNR A120. The difference in values of λ arises due to our intention to preserve inversion symmetry (There are two centres of such symmetry for each JGNR, see Fig. 1) of the structures. A smaller step in the A-direction to increment width or smaller elementary vector for width can be determined, but then inversion symmetry on microstructure level for JGNR will be lost, as shown in Fig. 3. As one can see, there are three possible subtypes of JGNR A120 symmetric with respect to inversion operation, but only one of them, see Fig. 3 a), can be rolled into a nanotube like JGNRs of other types and only such ribbons will be considered further.

a 1 a 2 L 1 L 2 W T 1 2 x y I 1 I 2 1 2 1 2 1 2 1 2 1 2 1 2 a a 1 a 2 L 1 L 2 W T 1 2 x y I 1 I 2 1 2 1 2 1 2 1 2 1 2 1 2 b FIG. 1.
JGNRs characteristic vectors for zigzag a) and armchair b) ribbon orientations on a graphene sheet represented by ribbons Z60 3, 3; 2 and A60 2, 2; 3 , respectively. Lines 2-2 and 1-1 (insets) show all possible zigzag (Z) and armchair (A) directions, respectively. L1, L2 and W are chief vectors, 1, 2 and w are elementary vectors, a1 and a2 are primitive translations of the graphene lattice, T is the JGNR translation vector and I1,I2 are the centres of JGNR inversion symmetry. So far, only two points of view on the structure of JGNR were mentioned. On one hand, it can be represented as a sequence of ordinary straight GNR pieces concatenated at specific angles, but on the other hand it can be considered as an ordinary straight GNR with some triangle areas cut to produce jags, but there are some more of them, see Fig. 4. Therefore the width introduced above is not so a complete characteristic. However, on its basis one can determine others. For instance, an effec-tive width, e.g. the width of the GNR that one cut some parts from to produce jagged edges, can be expressed as follows:

W ef f = T × P 1(2) × T T 2 , (5)
where erties of JGNR. Other characteristics closely related to the width are widths of the JGNR arm ribbons which have the form of

P 1(2) = L 1(2) + W .
W 1(2) = L 1(2) × W × L 1(2) L 2 1(2) . (6)
For the present paper, as the width vector was chosen to be the bisector of the apex angle, these two quantities defined by Eq.( 6) are equivalent. Finally, one can think about some JGNRs as a host ribbon with attached triangle fragments on both sides (see Fig. 4), therefore for quite wide JGNR one can introduce a host ribbon width:

W h = 2W -W ef f . (7)
Equating the right hand side of Eq.( 7) to zero, one can easily obtain the criterion for the host ribbon presence in the structure for each type of JGNRs: Z60 ≥ 2w, Z120 ≥ 2w, A60 > 2w/3, A120 ≥ 2w. Such simple mapping of JGNR onto ordinary straight GNRs is only possible in symmetric case when 1 = 2 . As for the asymmetric case, it is more complicated, because the effective, the host and the straight ribbons become specific JGNRs with low values of indexes. Additionally, this mapping would be of no practical use for ribbons from the same class that can be compared straightforwardly.

III. METHOD

The electronic structure of the presented JGNRs was investigated within the orthogonal tight-binding model for π-electrons. Electronic bands were obtained as eigenvalues for the matrix Hamiltonian H with elements of the following form:

H nn = i,j,q t ni exp (ikr nij ) δ r n ij ,r n 1 -r n q (8)
where r nij is a vector pointing the j-th position of the i-th order nearest neighbour from the n-th atom position in the unit cell of the structure, r nq is a radius-vector pointing the position of the n-th atom in the q-th unit cell. Within the tight-binding model, the so-called hopping integral t ni for n-th atom in the unit cell of the structure is expressed as

t ni = φ n | Ĥ|φ n+i , (9)
where Ĥ is a system Hamiltonian, φ n is an atomic orbital of the n-th atom and i can be referred to as the neighbour order. In the presence of an external field the Hamiltonian takes form

Ĥ = Ĥ0 + Û , ( 10
)
where Ĥ0 is the Hamiltonian of the system without external field, Û = -eEr is the potential energy operator for a homogeneous electric field. Taking this into account one has

t ni = t ni,0 + δt ni , with δt ni = φ n | Û |φ n+i .
Note that the external electric field is much less than the atomic one, i.e.,

eE t n1,0 a 0 , (11)
where t n1,0 ≈ 3 eV, a 0 = 1.42 Å. This allows one to neglect any change in the atomic orbitals φ n , φ n+i due to the field. It is also worth mentioning that we do not expect large field enhancement at the sharp ends of jags as their sizes in the structures considered are about 5nm, which is quite small, and they are arranged periodically in an infinite line so that their influence on each other averages and reduces the resulting field.

As we are interested in the pure effect of the jagged edges and its influence on electronic properties of JGNRs, we eliminate possible differences of hopping integrals at various sites within unit cell so that t ni = t i ; t ni,0 = t i,0 ; δt ni = δt i . In fact, the model of ideal geometry was implemented, which is close to the real geometry of the structure when dangling sp 2 bonds at JGNR edges are passivated with H atoms to form a strong C-H σbonds. Nearest-neighbor C-H matrix elements associated with these H-atoms do not appear explicitly in H 0 because these σ orbitals are symmetric with respect to the nodal plane of the π-orbitals and hence decouple from the π-conduction bands described by H 0 . Also, because C forms a strong covalent bond with H, the σ and σ * bands associated with these bonds lie far from E F and hence need not be considered further.

It is worth noting that on one hand hopping integral values at the edge can differ from those in the GNR interior so that it could cause band gap opening in armchair GNR 47 . But on the other hand, the same band gap opening can be explained by accounting for higher order hopping integrals 48 . A more accurate model taking into account both effects was reported latter 49 . Interestingly, a more precise model accounting for edge distortions showed that this effect leads to a small correction only for armchair ribbons of 3n series. However, it is very unlikely that this small correction to the band gap does affect its tunability. Therefore, admitting an error for the band gap of no more than 10%, the nearest neighbours up to the third order were taken into account and model parameters were chosen as proposed in the paper 48 : t 0,0 = 0 eV, t 1,0 = -3.2 eV, t 2,0 = 0 eV t 3,0 = -0.3 eV.

In the present paper we neglect screening effect and assume that applied field influences directly each lattice site. It seems reasonable for strong electric fields. However, for weak fields one should keep in mind that the values presented should be interpreted as a value of an effective field. The applied electric field contributes mainly to the site energy t 0 . In fact, there are additional terms for higher order hopping integrals δt 1...∞ . To assess them one can approximate carbon π-orbitals by p z wave functions of the hydrogen like atom 50 :

φ n (|r -r n |, θ) = 1 4 √ 2π Z a B 5/2 exp - Z|r -r n | 2a B |r -r n | cos θ, (12)
where Z = 6 (for carbon), a B is the Borh radius and θ is the polar angle measured from the z-direction (in spherical coordinate system), and calculate the ratios δt i /t i,0 . Obviously, the value obtained for i = 0 is of greater significance than any other. That is why we adopted δt 1...∞ ≈ 0, δt 0 = -Er eV.

The band gap dependence on applied electric field was investigated for electric strength magnitudes up to the 0.1 V/ Å = 1 V/nm. This value is 10 times less than typical strength used in field emission calculations 51,52 , but corresponds to the breakdown strength of electric field in SiO 2 53 . Even though it seems to be a quite high magnitude of field, it must be noticed that it still meets the requirement of Eq. (11). It is often chosen as a natural upper limit by others 26,29,33 for the electric field applied normally to the plane of the structure. Thus, our choice makes for an easier comparison of the two field arrangements. It is worth noting that break down values are very sensitive to the material used. For instance, in experimental work 30 it was possible to increase the value more than twice by using a combination of SiO 2 with HfO 2 , so that the resulting value was ∼ 0.033 V/ Å, that is only 3 times lower than mentioned above.

IV. RESULTS

Let us now consider typical JGNR band structures and their changes in an external electric field. Throughout this paper all ribbons are assumed to be laying in the xy-plane and the y-axis is collinear with the ribbon translation vector so that transverse electric field strength has only one non-zero component ε x . In Fig. 5, one can see that the most profound changes take place for JGNR Z60 and A120. In the first case, one deals with a band gap opening while in the second case -with a band gap closing. Surprisingly, there are no observable changes for band gaps of JGNR Z120 and A60. However, some alterations in Z120 and A60 band structures can be noticed. For the sake of the clarity of the discussion, let us take the following convention: all band are numbered by indexes J so that the first conduction band corresponds to J = 1 and first valence to J = -1 and so on. Then it can be pointed out that a profound splitting takes place at the edge of the Z120 Brillouin zone for bands J = ±1 and J = ±2, respectively. Similar splitting but of less magnitude and shifted towards the center of Brillouin zone is observed for A60 bands J = ±2 and J = ±3, correspondingly. With respect to the ribbon A60 one must confess that the least variations occur for bands J = ±1 compared to equivalent bands of other ribbons presented. It means that the effective mass of charge carriers changes only slightly due to the presence of a transverse electric field. In the case of Z120 ribbons the change of the effective mass for low doping concentrations should also be negligible. However, more significant changes take place for ribbons Z60 and A120. In both cases profound band bending can be observed for the magnitude of electric field ε x = 0.05 V/ Å. In general it can be seen that the influence of the field is stronger at the edge of Brillouin zone, as has already been shown in nanotube and nanohelix superlattice properties analysis, which is very relevant to a JGNR in the external electric field as it creates for JGNR's electron a periodic electrostatic potential similar to that discussed in papers [54][55][56] .

Next we consider the band gap dependence on the magnitude of the transverse electric field in more detail. Let us proceed with the JGNR Z60. For Fig. 6 we took a bit wider ribbon Z60 3, 3; 9 to show that for a high value of the electric field strength of about 0.06 V/ Å the band gap closing can be observed. The next feature that can be noticed in Fig. 6 is dispersionless bands, whose positions clearly correlate with the magnitude of the external field. These bands arise from zigzag edge states reported in the paper 57 . However, as JGNR is a superlattice with a longer translation period, its band structure if one does not take into account interactions leading to band anticrossing is a folded structure of straight GNR with zigzag edges (ZGNR). As one can see, due to such folding these bands are absolutely dispersionless through out the whole Brillouin zone compared to ZGNR where edge bands are dispersionless only though the 1/3 of Brillouin zone. Changing the superlattice period of translation T by means of jag arms L 1 , L 2 (see Eq.3) one can control a number of foldings or put in other words a number of dispersionless bands. Without the external electric field almost all of them are degenerate, but with a switched on field their splitting is observed. These bands must result in sharp peaks in the density of states, whose positions in turn must be field dependent as well. However, as one can see from Fig. 6 for the high magnitude of electric strength of about 0.06 V/ Å they drown among multitude of other bands. But for narrower ribbons Z60 the density of states arising from all other bands except edge ones around the Fermi level is low in a quite wide region from -1 to 1 eV, see Fig. 7. It means that the peaks should act in this region as energy levels and we showed it for JGNR Z60 with w = 6 in Fig. 7. Taking into account all the just mentioned results, we predict that for JGNR Z60 with w = 4, .., 8 in an external transverse electric field, a series of electromagnetic emission and absorption lines will be observed and the number of lines will be consistent with T . We tested that this result remains true for asymmetric JGNRs Z60 characterized by indexes 1 = 2 and for JGNRs Z60 of a slightly different structure, see the Appendix .

Considering in Fig. 8 the band gap closing for JGNR A120, one must notice that this closing takes place owing to the splitting of bands J = ±1, ±2 at the edge of the Brillouin zone. This splitting leads to the band gap shift in k-space from k = 0 for ε x = 0 V/ Å to the region k > π/2 for ε x = 0.06 V/ Å. Although there are some bands with low dispersion near energies ±1 eV their positions are not affected by the applied electric field. On the contrary, the almost flatness of the bands J = ±1 can be achieved for ε x = 0.06 V/ Å in k-space from k = π/2 to k = π. It is very useful because it must increase the probability of interband transitions with frequency exactly corresponding to the band gap due to higher density of states on its both sides. It is very interesting to see how the dependence of the band gap on transverse electric field is affected by symmetric JGNR parameters w and 1 = 2 . These results are presented for JGNRs Z60 and A120 in Fig. 9. As it can be seen from Fig. 9 a) the maximum value of the band gap opening (∼ 0.6 eV) is higher for the narrower ribbon with w = 7 but it is attainable for stronger transverse electric field ( 0.04 V/ Å) compared to Z60 3, 3; 8 and 3, 3; 9 , besides for the specified ribbon the band gap cannot be closed within the restricted range < 0.1 V/ Å as for ribbon Z60 3, 3; 9 . For a low strength field < 0.02 V/ Å the band gap opening for narrower Z60 is a bit less than for wider ones, but it results in more sig- 5. The effect of a transverse electric field εx on the band structure of four types of JGNRs, where EF -the Fermi energy.

0 Π 2 Π 2 1 0 1 2 k E E F , eV Z60 3,3;6 0 Π 2 Π k Z120 3,3;6 0 Π 2 Π k A60 3,3;6 0 Π 2 Π k A120 3,3;6 x 0. V x 0.06 V FIG.
k is expressed in the units of 1/T everywhere. nificant divergence for greater difference of width indexes. The influence of jag arm index on the band gap opening of the ribbon Z60 shown in Fig. 9 b) is opposite to what was just mentioned. For higher indexes the maximum value increases and drifts to greater values of ε x while band gap closing at ε x = 0.1 V/ Å is not affected by them.

0 Π 2 Π 2 1 0 1 2 k E E F , eV x 0 V 0 Π 2 Π k x 0.02 V 0 Π 2 Π k x 0.04 V 0 Π 2 Π k x 0.06 V
In low strength fields the difference in band gap opening is almost negligible. According to Fig. 9 c) the band gap closing does not take place immediately, and JGNRs A120 3, 3; 6 are quite resistant to low electric fields with ε x < 0.02 V/ Å. However this resistance decreases if the width index increases that leads to lower values of electric strength required to close the band gap completely: ε x = 0.055 V/ Å for A120 3, 3; 8 and ε x = 0.08 V/ Å for A120 3, 3; 6 . Another fascinating feature of the ribbons is the peak in the band gap dependence on electric field 7. The splitting of dispersionless bands in the transverse electric field for JGNRs Z60 3, 3; 6 , 5, 5; 6 , 7, 7; 6 . DOSnormalized density of states. strength observed for A120 3, 3; 7 at ε x = 0.85 V/ Å. This peak position shifts to lower magnitudes of ε x as the width index w increases while its height seems to remain unaffected. This is quite strange taking into account that there is an obvious decrease of initial values of the band gap, e.g. for ε x = 0 V/ Å, for greater indexes w.

0 Π 2 Π 2 1 0 1 2 k E E F , eV Z60 3,3;6 0 1 D E DOS 0 Π 2 Π k Z60 5,5;6 0 1 D E DOS 0 Π 2 Π k Z60 7,7;6 0 1 D E DOS x 0. V x 0.05 V FIG.
0 Π 2 Π 2 1 0 1 2 k E E F , eV x 0 V 0 Π 2 Π k x 0.02 V 0 Π 2 Π k x 0.04 V 0 Π 2 Π k x 0.06 V
Compared to this case the data presented in Fig. 9 d) are very different. While the decrease of initial values of the band gap for incremented jag arm indexes 1 and 2 is less than in Fig. 9 c) for increasing w the resistance to the band gap closing is of the same measure, e.g. achieved at ε x = 0.053 V/ Å for A120 7, 7; 6 . This value is compared to the value for A120 3, 3; 8 , however in the case of A120 7, 7; 6 one closes a wider band gap. In Fig. 9 d) a peak similar to that one in Fig. 9 c) in dependence of E g on ε x is manifested. Its position shifts to lower ε x for greater 1 , 2 and it resembles behaviour of the peak in Fig. 9 c) but its height decreases if jag arms indexes 1 , 2 both increase.

It is obvious that the dependences E g (ε x ) for Z60 and A120 JGNRs presented in Fig. 9 cannot be described by simple functions, however they can be approximated by polynomials that are truncated Taylor series. Quite easily one can specify the regions where two or three terms of Taylor expansion are necessary for a reasonable approximation. These regions correspond to liner and quadratic dependences E g (ε x ) and a good knowledge about them is of great importance for possible application in linear and non-linear devices, therefore we calculated the parameters characterizing the rate of the band gap opening/closing and presented them in Table II. Moreover, Table II enables comparison between two types of ribbons, which is difficult to do by means of plotting. Although it might seem strange to compare different patterns we want to notice that ribbon A120 placed in sufficiently strong electric field so that the point of complete band gap closing, for instance ε x = 0.08 V/ Å for A120 3, 3; 6 , is achieved can work in the same regime as Z60 ribbon -metal-dielectric transition for decreasing field. As can be clearly seen from the Table II the absolute value of β is larger for all Z60 than for A120 ribbons presented in the Table II, that means their band gaps can be controlled more efficiently.

Having studied the dependence of E g on ε x for JGNRs Z60 and A120, we proceed with a comparison of found patterns with similar ones for simple GNR with II. Fitting of linear and quadratic regions of Eg(εx) curves for JGNRs presented in Fig. 9. As can be seen from Fig. 10 a) the dependence of E g on ε x for Z60 3, 3; 9 lays between curves for the straight and the effective ribbons. There is no likeliness with the arm ribbon that allows one to obtain high value for E g of about 0.8 eV but requires close to SiO 2 breakdown electric strength magnitude. Comparing curves for the host and Z60 ribbons, one see that the former lays below the latter for low values of ε x ∼ 0.025 V/ Å and above for greater values of ε x . In general, the curve for Z60 is to a great extend similar to curves for effective, straight and host ribbons especially in region of low values of ε x < 0.02 V/ Å. However, Fig. 10 b) provides quite different results. The curve describing E g vs. ε x dependence for Z60 12, 12; 6 almost coincides with the curve for the arm ribbon ZGNR(6) for ε x < 0.02 V/ Å and considerably deviate from it at higher ε x . It does not approach curves for the effective ZGNR (24) or straight ZGNR (12). The case of JGNR A120 3, 3; 6 shown in Fig. 10 c) significantly differs from that one of JGNRs Z60, because A120 ribbons have armchair edges while they are mapped onto ZGNRs. However, in spite of the fact that the arm ribbon for A120 ribbons has the same type of edges, one notice a crucial difference between them. Firstly, it is impossible to control band gap of AGNR (18), which is the arm one for the A120 3, 3; 6 , by means of a transverse electric field. And secondly, the value of the band gap is greater for a jagged ribbon. In the Fig. 10 d) one can see the effect of deeper jags which leads to the absence of the host ribbon in A120 8, 8; 4 . This JGNR is a bit narrower than that presented in the Fig. 10 c), but its band gap is closed nearly at the same magnitude of ε x ∼ 0.08 V/ Å as for A120 3, 3; 6 . It is a straightforward result of jag arm elongation that shifts this point of closing back to lower values of ε x after its shift to higher ones due to decrementing of width index w. It is also useful to compare curves for A120 ribbons in Fig. 10 c) and d) in the region close to zero magnitude of electric field. One sees that while the initial value of the band gap is greater for A120 8, 8; 4 , E g ∼ 0.8 eV compared to E g ∼ 0.65 eV, the closing point is almost the same. Moreover, the drop in the band gap of A120 8, 8; 4 in the region of ε x < 0.02 V/ Å is greater than for A120 3, 3; 6 that means the band gap of the former can be more readily controlled. In all other respects Fig. 10 c), d) demonstrate very similar curves behaviour, where observable distinctions can be attributed only to quantitative differences of jagged ribbon parameters and, consequently, the straight, effective and arm ribbons ones. For instance, curve for the arm ribbon in Fig. 10 d) is also field independent while corresponds to a higher value of the band gap of about 0.6 eV compared to 0.4 eV in Fig. 10 c). Finally, we report on the effect of asymmetry 1 = 2 on both JGNRs Z60 and A120. Having analysed results for JGNRs and ordinary straight GNRs, it is easy to comprehend data shown in Fig. 11 a) and b). The increment of just one index 2 results in E g vs. ε x curve evolution to the form inherent for the arm ZGNR and AGNR, respectively. To be more persuasive, we suggest the reader to look again at the curves for ZGNR (14) and AGNR (18) in Fig. 10 a) and c), correspondingly. After that results in Fig. 11 are easy to understand. As in the case of Fig. 9 for ribbons in Fig. 11 we specified linear and quadratic regions of E g (ε x ) and calculated parameters (see Table III). In contrast to the data in Table II

βx + α γx 2 + βx + α JGNR α, eV β, eV• Å/V εmin, V/ Å εmax, V/ Å α, eV β, eV• Å/V γ, eV• Å2 /V 2 εmin, V/ Å εmax, V/ Å Z60

V. CONCLUSIONS

To summarize, in this paper we have studied a subclass of chevron-type GNRs -jagged graphene nanoribbons (JGNRs). It was shown that it is possible to control the band gap of two types of JGNRs, namely Z60 TABLE III. Fitting of linear and quadratic regions of Eg(εx) curves for JGNRs presented in Fig. 11. Fitting for ribbons Z60 3, 3; 7 , A120 3, 3; 6 is presented in Table II. and A120, by an external transverse electric field, e.g. applied in the plane of the ribbon normally to its longitudinal axis. The band gap opening is possible for Z60 ribbons, while the band gap closing is characteristic for A120 ribbons. In both cases the value of the band gap opening/closing is greater for narrower ribbons and requires stronger electric fields. As there is a natural limit for electric strength values due to the breakdown phenomenon, there must be an optimal value of ribbon width providing the highest possible value of the band gap. We argue that for ε x,max = 0.1 V/ Å, e.g. SiO 2 as a substrate, the optimal width is about w ∼ 6. For these values of width Z60 ribbons in a transverse electric field behave like a quantum dot system and series of emission and absorption lines must be observed if they are not forbidden by optical selection rules. Some enhancement in the controllability of the band gap in the transverse electric field is achievable for A120 ribbons as the longer jag arms the lower field is required to close the band gap, while it is not the case for Z60 ones. Also we must mentioned the second peak for A120 JGNR that is seemed to be unique for this type of structures as it was not mentioned for ordinary straight GNRs in the paper 31 . Finally, we notice that the value of band gap opening for Z60 is about 0.6 eV and so high that even taking into account possible error in band gap evaluation (≈ 10%) and assuming the same field screening (∼ 25% for ε x = 0.05 V/ Å) as for bilayer graphene one can assess band gap opening as no less than ∼ 0.4 eV that is about twice larger than for bilayer graphene in a normal electric field 26 . The value is also large compared to that one for carbon nanotubes and can be implemented for fabrication of all metallic transistors proposed firstly for nanotubes in the paper 58 . Following the just mentioned work, it seems reasonable even to combine normal and transverse geometry to control conductivity of the channel and carrier concentration separately. Actually, separate control of the band gap and the Fermi energy paves the way to a new type of devices that could build a bridge between optics and electronics. Speaking again of A120, one must stress also the quite wide range of band gap variation of about 0.6 eV. This property of A120 ribbons could make them suitable candidates for infrared or THz laser media with widely tunable irradiation frequency.

βx + α γx 2 + βx + α JGNR α, eV β, eV• Å/V εmin, V/ Å εmax, V/ Å α, eV β, eV• Å/V γ, eV• Å2 /V 2 εmin, V/ Å εmax, V/ Å Z60
Although only a homogeneous transverse electric field was considered we deem that the results obtained give a hint on the effect of charge impurities on some types of GNRs, however to prove the last statement another detailed study is necessary. It is quite important to notice that in spite of the fact that only graphene nanoribbons were considered in this study, its result can be relevant to other structures with similar geometry produced by pat-terned evaporation 59 of H, O, F, atoms from the surface of graphane, graphene oxide or graphene fluoride, respectively, or substitution of BN atoms on C atoms in h-BN layer 60 . As was shown in papers 19,24,61 , such structures exhibit similar electronic properties to GNRs of the same shape. After having written this paper we became aware of recent results confirming spin ordering on the zigzag edges at room temperature 62 . This effect can reduce the range of tunability for Z60 ribbons and needs a separate detailed investigation. Appendix: Realistic Z60 structure Ribbons of Z60 type are good as a theoretical model but unfortunately it is very likely that such configurations are difficult to realize experimentally. Therefore we decided to provide additional results to prove that the found effect is not due to the peculiar structural configuration associated with one carbon atom missing in hexagons on the border between two jags but rather due the ribbon chevron-type pattern and zigzag edge. A structure with one carbon atom removed from the jag apex and placed at the opposite edge to complete hexagons seems to be more stable. Such structure can be referred to as Z60r, where "r" -stand for "realistic". As one can see in Fig. 12, band gap opening and dispersionless band splitting in the external transverse electric field can be found as well in structures of more energetically favourable configuration.

FIG. 2 .

 2 FIG.2. JGNRs types represented by ribbons Z60 3, 3; 5 , Z120 3, 3; 5 , A60 3, 3; 5 and A120 3, 3; 5 , respectively.

FIG. 3 .

 3 FIG. 3. JGNRs A120 subtypes. Ribbons shown on a), b), c) are symmetric, because their opposite edges highlighted by red (top) and blue (bottom) can be superimposed by simple translation, e.g. they have inversion centres like I1,I2 in Fig.1. Ribbons depicted on d), e), f) are asymmetric due to different sorts of their opposite edges. Only a) and f) ribbons can be smoothly rolled into a nanotube. The framed ribbon is of the same A120 type as the one shown in Fig.2 d).

FIG. 6 .

 6 FIG.6. The band structure evolution for increasing magnitude of transverse electric field for JGNRs Z60 3, 3; 9 (left) and atomic structure of the ribbon (right).

FIG. 8 .

 8 FIG.8. The band structure evolution for increasing magnitude of transverse electric field for JGNRs A120 3, 3; 6 (left) and atomic structure of the ribbon (right).

FIG. 9 .

 9 FIG. 9. The band gap Eg dependence on the magnitude of transverse electric field εx for symmetric JGNR Z60 a), b) and A120 c), d) with various width and jag arm indexes w, 1 = 2, correspondingly.

12 FIG. 10 .

 1210 FIG. 10. The comparison of JGNRs Z60 a), b) and A120 c), d) with various straight GNRs, where (n) denotes the number of atom pairs in the unit cell of the

FIG. 11 .

 11 FIG. 11. The influence of asymmetry 1 = 2 on the electric field effect in JGNRs Z60 a) and A120 b).

ACKNOWLEDGMENTS

  This work was supported by EU FP7 ITN NOTEDEV (through Grant No. FP7-607521); IRSES projects CA-COMEL (Grant No. FP7-247007), FAEMCAR (Grant No. FP7-318617) and CANTOR (Grant No. FP7-612285); Graphene Flagship (Grant No. 604391) and the Ministry of Education of the Republic of Belarus (Grant No. 20140773). The authors are very grateful to Prof. Philippe Lambin and Prof. Mikhail Portnoi for useful advice and Charles Downing for careful reading of the manuscript .

FIG. 12 .

 12 FIG.12. The splitting of dispersionless bands in the transverse electric field for JGNRs Z60r.

TABLE I .

 I The coordinates of JGNR elementary vectors 1, 2 and w in the basis of primitive translations of graphene lattice a1 and a2.

  atom pairs in the unit cell of the structure. The mapping mentioned at the end of Sec.II is very useful for doing this. Using the mapping based on Eqs.(5-7) results presented in Fig.10 a)-d) were obtained. For the sake of completeness ribbons Z60 and A120 with and without host ribbons were considered. The presence of the host ribbon in JGNR is our criterion for large values of jag arm indexes 1 , 2 with respect to width index w.

	3, 3; 7	0	23.3	0	0.010	0	29.8	-361.4	0.010	0.040
	Z60 3, 3; 8	0	25.9	0	0.010	0	33.7	-526.3	0.010	0.035
	Z60 3, 3; 9	0	29.6	0	0.007	0	35.7	-680.7	0.007	0.030
	Z60 5, 5; 9	0	28.4	0	0.007	0	29.2	-425.5	0.007	0.045
	Z60 7, 7; 9	0	27.0	0	0.007	0	27.7	-365.0	0.007	0.055
	A120 3, 3; 6	-	-	-	-	0.7	0.6	-108.4	0.010	0.080
	A120 3, 3; 7	0.8	-12.5	0.035	0.060	0.6	1.4	-177.8	0.010	0.035
	A120 3, 3; 8	0.7	-12.5	0.025	0.050	0.5	2.1	-264.4	0	0.025
	A120 5, 5; 6	1.0	-15.7	0.035	0.060	0.6	8.0	-349.5	0.010	0.035
	A120 7, 7; 6	0.8	-14.8	0.020	0.050	0.5	9.5	-616.2	0.007	0.020
	zigzag and armchair edges, which will be referred to as
	ZGNR(n) and AGNR(n), respectively, where n is a num-
	ber of carbon

  , absolute values of β for different ribbons in Table III are very close and imply approximately equal efficiency in the band gap control.


ZIGZAG-SHAPED SUPERLATTICES ON THE BASIS OF GRAPHENE NANORIBBONS: STRUCTURE AND ELECTRONIC PROPERTIES

Keywords: graphene nanoribbons, superlattices, electronic properties, band gap engineering

The paper focuses on superlattices consisting of two coplanar fragments of one-layer graphene nanoribbons that have different width and are connected at an angle. Classification of such superlattices was carried out; their electronic properties were studied using the tight-binding method. It was demonstrated that in superlattices consisting of two fragments of graphene nanoribbons with armchair edges connected at an angle of 60º, the band gap can be regulated by the number of dimeric carbon atom chains of one of the fragments. In that case one can observe a periodic dependence of the band gap on the number of chains with a characteristic period equal to three dimeric chains. The number of dimeric chains of the second superlattice fragment regulates the average band gap value near which the periodic oscillations occur, as well as the amplitude of those oscillations. Therefore, one can accomplish a sufficiently precise band gap tuning for such structures. Such tuning can find its wide application in the booming carbon nanoelectronics industry when creating generators, amplifiers and sensors in the nanochains.

INTRODUCTION

Over the past decade, the international research community has paid a lot of attention to graphene and its nanostructured derivatives, such as graphene nanoribbons [1][2][3], regarding them as the foundation for future carbon electronics [4], that is expected to replace today's silicon electronics. Electronic applications of graphene are likely to be associated with its use as a flexible substrate in VLSI schemes (VLSI stands for Very Large Scale Integration), on which the optoelectronic chains will be created by methods of nanolithography [5,6]. Therefore, the basic components of carbon electronics will consist of quasi-one-dimensional graphene structures that graphene nanoribbons essentially are. Today, researchers are working on the theoretical and experimental studies of their application as passive elements, namely interconnectors [7,8], waveguides [9,10], and dividers [11]. They also pay attention to the active elements of electronic circuits, such as amplifiers and switches [12][13][14]. The band gap of the structures used plays an important role in both cases: in the case of passive elements it is important for resistance matching, while in the case of active elements it is crucial for creation of heterojunctions. Band gap can be engineered by two methods: by doping individual ribbon sections [15,16] and by engineering its shape [17][18][19][20]. Taking into account significant progress in the synthesis of graphene nanoribbons and one-dimensional structures with more complex shapes [21], this paper offers a theoretical study of the functional capabilities presented by band gap engineering in superlattices on the basis of graphene nanoribbons by means of engineering their structure.

CRYSTALLINE STRUCTURE

The crystalline structure of the examined zigzag-shaped superlattices on the basis of graphene nanoribbons is presented in Fig. 1. Three vectors can describe the unit cell of such a structure: two vectors 1 L and 2 L that describe the shape of the superlattice edge and one vector W that characterizes its width. We shall call 1 L , 2 L the shoulder vectors and W the width vector. Each of the vectors introduced can be presented by some translation vector of the two-dimensional hexagonal graphene lattice and written as follows: a are elementary translation vectors of the hexagonal graphene lattice, , n m are whole numbers. In the general case n and m take independent values. Therefore, in order to describe the suggested structures, one needs to set six numbers. However, it is difficult to carry out classification in this case, so let us consider the following approach. For crystallographic directions, for which n and m have the greatest common divisor, vector C can be presented as follows:

12

nm cc cc       C a a v ,
where nc, mc are whole numbers, v is some vector, let us call it "elementary vector", c is the greatest common divisor for n and m . Therefore, by means of elementary vector introduction one can fixate the crystallographic direction. This can be conveniently done for vectors L in such a way that they would be fixated along the zigzag or the armchair direction. Such fixation could also be performed for the width vector, as it was done in [22,23], but in this case it is better not to do that for the following reason. As one can see from Fig. 1, the unit cell can be presented as two fragments of ordinary ribbons 1

F , 2
F lying in the same plane and connected at an angle, and the crystallographic direction of the vector W determines the width of fragments w to characterize the width of the corresponding fragments in length units of elementary vectors 1  , 2  . In the present paper let us limit ourselves to the symmetrical case, when both vectors ) for each superlattice type. As it is shown in Fig. 1, the origin of coordinates coincides with one of the atoms of the A -sublattice.

Z60 Z120 A60 A120 1    1, 0   1, 1    2, 1    2, 1  2    0,1   0,1   1,1   1, 2  i S 1; i    1 3, 1 3  1; i    1 3, 1 3  2; i    1 6, 1 3  3; i    1 6,1 6
Within each type, a specific superlattice is determined by the set of four   w can be fractional numbers divisible by13 for structures А60 and А120. Increase by 1 in one of the indices 1 w , 2 w for structures Z60 and Z120 corresponds to adding one zigzag-type carbon chain to the corresponding fragment, and increase by 1/3 in one of the indices for structures A60 and A120 -to adding one dimeric chain. As it is demonstrated further on, the electronic properties of superlattices can vary considerably depending on the number and type of the above-mentioned elements.

ELECTRONIC PROPERTIES

Using the tight-binding method, we have studied the dependence of the band gap of the described zigzagshaped superlattices on their geometric parameters. The calculations took into account only the π-orbits, the overlap integrals of which were set to zero. The nearest neighbors were taken into account up to the third order. Hopping integrals were selected as in [24] Fig. 2 presents the typical band spectra obtained by calculations. As one can see from the figure, some of the structures are metallic or almost metallic. For Z60 structures this metallicity is ensured by flat multiple-degeneracy bands that correspond to the states, the wave function of which is localized on the structure edges and on the atom closest to the apex point. Similar edge states are known for zigzag-type graphene nanoribbons [25,26]. The same kinds of states occur for Z120 structures, but for them in many cases degeneracy is lifted and a small band gap ~30 meV is formed between the edge states. For all semiconductor structures, both wide-and narrow-gap ones, one should note the presence of the direct band gap. One should particularly note the periodic regularities for index 1 w in the band gap of some structures. In the case of small indices i w and large i  such superlattices are equivalent to the quantum dot chains and their spectra, as a rule, have many flat bands. However, this does not affect the periodic dependence. Let us now examine the indicated periodic dependences in greater detail, excluding Z60 metallic structures from consideration. for A60 and A120. One can see from the general picture that oscillations can be regular and irregular. In the case of regular oscillations, the series are well divided by value; the lines that can connect points within the same series are practically not intertwisted; one can identify at least a maximal or a minimal series. In the case of irregular oscillations, the lines that can connect the points within the same series are intertwisted and it is impossible to identify a minimal or a maximal series. The regular oscillations are better visualized by sequential connection of points As one can see from Fig. 3, there are qualitative differences for small (

 , 3 , 2 ,

   

) and large (

   

) shoulder index values for Z120 structures. In the first case, one can observe the regular oscillations that slowly abate with the increase in 1 w . One should note that it is slower than the abating 1/w known for graphene nanoribbons, where w is ribbon width [24,27]. The maximal series is clearly distinguished and manifests the following dependence on index 2 w :

   

, the amplitude of oscillations practically does not change. One should also emphasize that its absolute value is about 10 times lower and is characterized by milli-electronvolts. In all the cases, the results are given for symmetrical superlattices ( 2 1    ). However, one should note that the results do not change qualitatively for A60 and A120 structures and all the conclusions remain in force for the asymmetrical structures, for which 2 1    . In the case of Z120 structures, when (see Fig. 3).

CONCLUSION

This paper examines the new class of carbon nanostructures -zigzag-shaped superlattices on the basis of two graphene nanoribbon fragments. Classification of such superlattices was performed. The structure of the unit cells was described by means of elementary translations of the crystalline graphene lattice. The π-electron tight-binding method was used to study the dependence of the band gap on the width of ribbon fragments that make up the unit cell. It follows from the data obtained, that the simplest regularities in the band gap behavior occur for superlattices with armchair edges, where fragments connect at an angle of 60 . In particular, one observes the characteristic periodic dependence of the band gap on the width of one of the superlattice fragments with a period equal to three dimeric chains of carbon atoms. It, in turn, manifests the same dependence on the second fragment width. This simple combination of properties of ordinary graphene nanoribbons with armchair edges allows greater flexibility in regulating the band gap. After technology of synthesis of such superlattices reaches a certain level, this phenomenon should find its application when creating integrated circuits on the basis of carbon nanostructures. Let us note that optical applications in the infrared frequency range can also be possible, provided that dipole transitions between the valence and conduction bands are allowed by the corresponding optical selection rules, but this issue should become the subject of future research.

This research was supported by projects FP7 ITN NOTEDEV(FP7-607521), CACOMEL(FP7-247007), FAEMCAR (FP7-318617) and CANTOR (FP7-612285); EU H2020 RISE project CoExAN (H2020-644076); European Graphene Flagship project (604391), as well as by the Belarusian Ministry of Education (grant 20140773), Belarusian State University (grant 11, 2014), and the international grant AFOSR "Nanosized Cherenkov-Type Terahertz Light Emitter Based on Double-Walled Carbon Nanotubes and Bi-graphene Nanoribbons".

Fig. 1 .

 1 Fig. 1. Structure of the zigzag-shaped superlattice A60(2,2; 1,5/3) and the main vectors characterizing it: A and B are atoms from different graphene sublattices. The apex points are located on different positions ( 1 S , 2 S and 3 S) for different superlattice types as indicated in Table1.

1 L , 2 L 1  , 2  corresponding to shoulder vectors 1 L , 2

 121212 , in order to set the superlattice type. Two crystallographic direction are identified in the hexagonal graphene lattice; they are called zigzag and armchair. In Fig.1, they are indicated with a dotted line 2-2 and a dashed line 1-1 respectively. Elementary vector v has the lowest values along those directions. Let us introduce vectors

1

 1

F and 2 F, 2 w 1  , 2 

 2212 . That is why, in order to study the dependence of electronic properties on the width of each fragment, it is necessary to save its setting by means of two indices 1 w . Furthermore, it is convenient to set the vector W via elementary vectors , in order for indices 1 w , 2

  :

Fig. 2 .

 2 Fig. 2. Typical band spectra of zigzag-shaped graphene superlattices in the first Brillouin zone Г-Х.

Fig. 3 иE ) on parameters 1 w , 2 w

 312 4 present the results of the study of band gap dependence ( g for superlattices Z120, A60 and A120. In all the presented cases, characteristic oscillations are observed in dependences Three series were identified in each structure type for their analysis:

  line. The irregular oscillations are better visualized by connecting the points of the same series with a line.

Fig. 3 ., 2 w

 32 Fig. 3. Dependence of the band gap on the width indices

   . In the second case, one can observe the irregular oscillations for 120 w  , that are replaced by regular ones at 1 20 w  . In the same way as in the

Fig. 4 ., 2 w 1 w

 421 Fig. 4. Dependence of the band gap on width indices

1 

 1  , one deals with the case close to Z60, i.e. the structures are metallic, and when 2

Table

  are characterized by directions of the same type: either zigzag or armchair. This imposes limitations on the possible angles between them. As a result, one can distinguish four structure types: Z60, Z120, A60 and A120. The table presents the elementary vectors and a coordinate of the apex point

	S (	i 	1, 2, 3
	i

1  , 2 

TABLE 1 .

 1 Coordinates of elementary vectors and apex point

	S in the basis	a ,	a for different superlattice types
	i	1	2