Design of quasiperiodic magnetic superlattices and domain walls supporting bound states

Abstract

We study the simplest Lamé magnetic superlattice in graphene, finding its allowed and forbidden energy bands and band-edge states explicitly. Then, we design quasiperiodic magnetic superlattices supporting bound states using Darboux transformations. This technique enables us to add any finite number of bound states, which we exemplify with the most straightforward cases of one and two bound states in the designed spectrum. The topics of magnetic superlattices and domain walls in gapped graphene turn out to be connected by a unitary transformation in the limit of significantly large oscillation periods. We show that the generated quasiperiodic magnetic superlattices are also linked to domain walls, with the bound states keeping their nature in such a limit.

1 Introduction

Graphene is a two-dimensional material composed of a single layer of carbon atoms placed in a hexagonal arrangement with remarkable mechanical and electrical properties. At low energy, its charge carriers are described as massless Dirac electrons with constant Fermi velocity [1, 2]. For this reason, this material can be used to perform bench-top experiments of analogs of quantum electrodynamics [3]. The quantum Hall effect [4], Klein tunneling [5,6,7], Zitterbewegung [8, 9], and Schwinger mechanism [10, 11] are examples of relativistic phenomena studied in graphene.

Let us notice that a superlattice appears when graphene is placed in periodic external fields or interactions. The study of periodic systems goes back to 1931 when Kronig and Penney introduced a model of electrons in crystal lattices in non-relativistic quantum mechanics [12]. In the context of graphene, we can find nowadays in the literature electrostatic [13,14,15,16] or magnetic [17,18,19,20] superlattices; there are also gapped graphene superlattices where there is a mass term involved [21]. Such fluctuations can be caused directly by periodic external electromagnetic fields or by the deposition of a monolayer graphene on specific substrates. When graphene is deposited on such substrates, the symmetry in the hexagonal lattice can be broken if electrons on a triangular sublattice have different hopping energy to the other sublattice. This behavior is observed for boron-nitride: This material has the same graphene lattice structure with two different types of atoms breaking down the symmetry [22]. Epitaxial graphene growth on a silicon-carbide substrate also allows obtaining the same behavior [23,24,25,26,27,28]. A mass term in the Dirac equation models such asymmetry and produces a spectral gap. An interesting limit occurs when the oscillation period of the mass term is large: If we get a mass-type barrier, it is said that we deal with domain walls [23].

On the other hand, supersymmetric quantum mechanics (SUSY) for physicists, or Darboux transformation (DT) for mathematicians, is an algorithm to expand the family of solvable potentials in non-relativistic quantum mechanics by using a known solvable Schrödinger equation as an input. In general, the initial Hamiltonian and its SUSY partners share some similarities, such as the asymptotic behavior of their potentials and their domain of definition. The corresponding spectra differ only in a finite number of energy levels. In particular, the Schrödinger equation for periodic potentials and its supersymmetric extensions, or SUSY partners, has been studied in [29,30,31,32,33,34,35,36,37]. Moreover, the SUSY technique has been successfully adapted to the Dirac equation in different settings; see for example [38,39,40,41,42,43,44,45,46]. The introduction of magnetic or pseudo-magnetic fields modifies the spectrum and may confine the electrons into a finite space region [47,48,49,50].

This work aims to address a Lamé magnetic superlattice in graphene, an exactly solvable system whose spectrum is composed of allowed and forbidden energy bands. Moreover, we will analyze how to insert bound states using Darboux transformations. The technique will introduce aperiodicities in the superlattice, which are responsible for the inserted bound states. Furthermore, by means of a unitary rotation, we will connect the problem of graphene in an external magnetic field with the problem of gapped graphene, which is modeled by adding a position-dependent mass term in the Dirac equation. An interesting limit is obtained for the period of the superlattice going to infinity, when the rotated system converts into a domain wall which keeps the added bound states.

2 Lamé magnetic superlattices

Low-energy charge carriers in graphene satisfy the Dirac equation $H \Psi = v_{\rm F} \vec {\sigma } \cdot \textbf{p} \Psi = E \Psi$, where $\textbf{p}$ is the momentum operator, $\vec {\sigma }=(\sigma _1, \sigma _2)$ is a vector array whose components are the Pauli matrices, and $v_{\rm F} \approx c/300$ is called Fermi velocity, and we placed the graphene layer in the $x-y$ plane. To introduce the effect of a magnetic field perpendicular to the graphene layer $\textbf{B}= B_z \hat{{\textbf {k}}} = \nabla \times \textbf{A}$, we use Peierls substitution (or minimal coupling) $\textbf{p} \rightarrow \textbf{p}+e \textbf{A}$; we restrict ourselves to problems with translational invariance along $y-$axis and use the Landau gauge, $\textbf{A}=(0,A_y(x),0)$; thus the wavefunction can be written as $\Psi (x,y)= e^{i k_y y} \psi (x)$. Then, the 2-dimension (2D) Dirac equation reduces to the 1-dimension (1D) one:

$$\begin{aligned} \left( -i \sigma _1 \partial _x + k_y \sigma _2 + \frac{e}{\hbar } A_y \sigma _2 \right) \psi = \frac{E}{\hbar v_{\rm F}} \psi . \end{aligned}$$

(1)

To obtain a dimensionless equation, we introduce the magnetic length $\ell _B = \sqrt{\hbar /e B_0}$, express $A_y= B_0 \ell _B (-\ell _B k_y+W_0(x))$ where $B_0$ is the magnetic field strength, and replace $x \rightarrow x/\ell _B, ~k_y \rightarrow \ell _B k_y,~ E \rightarrow \ell _B E/ \hbar v_{\rm F}$, resulting in

$$\begin{aligned} H \psi = \left[ -i \sigma _1 \partial _x + W_0 \sigma _2 \right] \psi = E \psi . \end{aligned}$$

(2)

With the previous replacements, the length x and momentum $k_y$ are measured in units of $\ell _B$ and $\ell _B^{-1}$, respectively, and the energy in units of $\ell _B/\hbar v_{\rm F}$. Typical values of magnetic field strengths $B_0$ in ferromagnetic materials range from 0.1T to 1T, which corresponds with $\ell _B\approx 80$–25 nm [20]. As a remark, if a Hamiltonian H as in (2) has an eigenvector $\psi _{E^+}$ with eigenvalue $E^+\ge 0$ in the positive part of the spectrum, then $\psi _{E^-} = \sigma _3 \psi _{E^+}$ is an eigenvector with energy $E^- = - E^+$.

Since $\psi (x)=(\psi ^+(x), \psi ^-(x))^T$ is a two-entry spinor, we can write (2) as a coupled system of equations:

$$\begin{aligned} - i \psi ^-_x - i W_0\psi ^-&= E \psi ^+, \nonumber \\ - i \psi ^+_x + i W_0\psi ^+&= E \psi ^-. \end{aligned}$$

(3)

Throughout this paper, we will use the notation $f_x = \partial _x f(x)$. Such a system can be decoupled by inserting one of these equations into the other and vice versa, resulting in

$$\begin{aligned} h^\pm \psi ^\pm = \varepsilon \psi ^\pm , \end{aligned}$$

(4)

where $h^\pm =- \partial _x^2 + V^\pm (x)$, $V^\pm = W_0^2 \pm W_{0 x}$, and $\varepsilon =E^2$.

Now, let us introduce the simplest Lamé magnetic superlattice and the corresponding vector potential amplitude

$$\begin{aligned} B_z(x)=B_0 \frac{m-1+ \text {dn}(x|m)^4}{\text {dn}(x|m)^2} ,\qquad W_0(x)=m \frac{\text {cn}(x|m) \text {sn}(x|m)}{\text {dn}(x|m)}, \end{aligned}$$

(5)

where m is the modulus of the Jacobi elliptic functions $\text {sn}(x|m),~ \text {cn}(x|m),~\text {dn}(x|m)$, such that $0 \le m \le 1$. The two Schrödinger potentials $V^- = -m+2\,m ~{\text {sn}^2(x|m)}$ and $V^+ = -m+2\,m ~{\text {sn}^2(x+K|m)}$ are called Lamé potentials, whose period is $\mathcal {T}=2K(m)$, where $K(m)=\int _0^{\pi /2} (1-m \sin ^2\theta )^{-1/2}\hbox{d}\theta$. By solving the Schrödinger equation $h^- \psi ^- = \varepsilon \psi ^-$ (see “Appendix 1” for more details) and using (3) we can obtain the eigenspinors and the allowed and forbidden bands of the Dirac problem. The allowed positive energies are then $E^+ = [0, \sqrt{1-m}] \cup [1, \infty )$. Note that the spectrum does not depend on the wavenumber $k_y$. The positive band-edge states are

$$\begin{aligned} \psi _{E_0} = \begin{pmatrix} -i A_1 ~\text {dn}(x+K|m) \\ A_2~ \text {dn}(x|m) \end{pmatrix}, \quad \psi _{E_{\sqrt{1-m}}} = \begin{pmatrix} -i ~\text {cn}(x+K|m) \\ \text {cn}(x|m) \end{pmatrix}, \quad \psi _{E_1} =\begin{pmatrix} -i~ \text {sn}(x+K|m) \\ \text {sn}(x|m) \end{pmatrix}, \end{aligned}$$

(6)

where $A_1,~A_2$ are arbitrary constants (one of them can be fixed from normalization); due to this, the zero-energy state is twofold degenerate. Figure 1 shows normalized probability densities of the band-edge states $\psi _{E_0},~\psi _{E_{\sqrt{1-m}}},~\psi _{E_1}$ (black, green, and yellow lines, respectively) in units of $\ell _B$. The three probability densities are periodic, with a period $T=K(m)$. The vertical lines represent multiples of such a period T. To plot, we used the modulus $m=1/2$ and the constants $A_1=A_2=1$.

Fig. 1

Lamé magnetic superlattices. Probability densities of the band edge-states, $|\psi _{E_0}|^2$ (black line), $|\psi _{E_{\sqrt{1-m}}}|^2$ (green line), $|\psi _{E_1}|^2$ (yellow line) for $m=1/2$ and $A_1=A_2=1$

3 Quasiperiodic magnetic superlattices supporting bound states

The model presented in the previous section is periodic and has a positive (and negative) spectrum composed of two allowed plus one forbidden band. We will derive two families of magnetic superlattices supporting bound states and keeping their band-structured spectrum. We use a Darboux transformation applied to a Dirac equation of the form (2) to insert the bound states. For the sake of completeness, we first introduce the Darboux transformation and then present two quasiperiodic magnetic superlattices supporting bound states.

3.1 Darboux transformation of a Dirac equation

To apply Darboux transformations, we take the dimensionless Dirac equation (2) and write it in the form (4). We choose one of the decoupled Hamiltonians $h^{\pm }$, let us say $h^-$. To add a bound state, we first define the operator $h_1^+= h^--\epsilon _1$, where $\epsilon _1$ is a real constant called factorization energy. Then, we assume that the intertwining relation $h_1^- L_1^+=L_1^+ h_1^+$ is fulfilled, where $L_1^\pm$ are known as intertwining operators ($L_1^-$ is the adjoint of $L_1^+$). The proposed expressions for the operators $L^\pm _1$ and $h_1^-$ are

$$\begin{aligned} L_1^\pm ={\mp }\partial _x+W_1(x,\epsilon _1), \quad h_1^-=-\partial _x^2+V_1^-(x,\epsilon _1), \end{aligned}$$

(7)

where $W_1(x, \epsilon _1)$ is called superpotential. From the definition of $h_1^+$ and the intertwining relation, the Schrödinger potentials take the form

$$\begin{aligned} V_1^+=V^--\epsilon _1, \quad V_1^-=V_1^+-2W_{1x}(x,\epsilon _1). \end{aligned}$$

(8)

Each potential is related to a Riccati equation, namely

$$\begin{aligned} V_1^\pm = W_1(x,\epsilon _1)^2\pm W_{1x}(x,\epsilon _1). \end{aligned}$$

(9)

We can map such Riccati equations into Schrödinger ones through the ansatz $W_1(x,\epsilon _1)=\partial _x \ln (u_{\epsilon _1})$, leading to:

$$\begin{aligned} (-\partial _x^2+V^+_1)u_{\epsilon _1}=0, \quad (-\partial _x^2+V^-_1)u_{\epsilon _1}^{-1}=0. \end{aligned}$$

(10)

The function $u_{\epsilon _1}$ is often referred to as seed solution. Since it fulfills a second-order differential equation, it can be written in terms of two linearly independent solutions as $u_{\epsilon _1}=u_1(x,\epsilon _1)+\eta \, u_2(x,\epsilon _1)$, where in principle, $\eta$ is an arbitrary constant. Once we fix the seed solution $u_{\epsilon _1}$, we can reconstruct a new Dirac Hamiltonian in the form

$$\begin{aligned} H_1=-i \sigma _1 \partial _x + W_1 \sigma _2. \end{aligned}$$

(11)

The zero-energy eigenspinor of $H_1$ is

$$\begin{aligned} \psi ^{(1)}_{\epsilon _1} \propto \left( \begin{matrix}0 \\ u_{\epsilon _1}^{-1}\end{matrix}\right) , \end{aligned}$$

(12)

The remaining solutions are expressed in terms of the intertwining operators and the solutions of the initial Dirac equation as follows:

$$\begin{aligned} \psi ^{(1)}=\left( \begin{array}{l} u_{\epsilon } \\ \frac{i}{\sqrt{\epsilon -\epsilon _1}} L_1^+ u_{\epsilon }\end{array}\right) , \end{aligned}$$

(13)

where $u_{\epsilon }$ fulfills the equation $-\partial _x^2u_{\epsilon }+V^-u_{\epsilon }=\epsilon u_{\epsilon }$ for any $\epsilon \ne \epsilon _1$. The corresponding vector potential is given by $\textbf{A}^{(1)} =(0,A^{(1)}_y,0)$, where $A^{(1)}_y= B_0 \ell _B(-\ell _B k_y+ W_1(x))$, and thus the new magnetic field is $B_1(x,\epsilon _1)=B_0 \partial _x W_1(x,\epsilon _1).$

Figure 2 summarizes the described Darboux transformation. First, we decouple the Dirac Hamiltonian H into two Schrodinger Hamiltonians $H\rightarrow (h^+,h^-)$. Second, we choose the Schrödinger Hamiltonian $h^-$ to build two Hamiltonians, $h_1^+=h^--\epsilon _1$ and $h_1^-$, where the last one arises from applying the Darboux transformation to $h^+_1$. Finally, we construct the new Dirac Hamiltonian, $(h_1^+,h_1^-)\rightarrow H_1$.

Fig. 2

Diagram of the Darboux transformation for a Dirac equation

3.2 Quasiperiodic superlattice with an added bound state

As a first example, we take the Lamé magnetic superlattice described in Sec. 2 and perform a Darboux transformation to add a single bound state. To this end, we select a seed solution $u_{\epsilon _1}=u_1(x,\epsilon _1)+\eta \,u_2(x,\epsilon _1)$, where $u_1$, $u_2$ are the Bloch functions

$$\begin{aligned} u_1=\frac{\sigma (x_0+\omega ') \sigma (x+\delta +\omega ')}{\sigma (x+\omega ') \sigma (x_0+\delta +\omega ')} e^{-\zeta (\delta )(x-x_0)}, \quad u_2=\frac{\sigma (x_0+\omega ') \sigma (x-\delta +\omega ') }{\sigma (x+\omega ') \sigma (x_0-\delta +\omega ')}e^{\zeta (\delta )(x-x_0)}, \end{aligned}$$

(14)

and $\sigma$, $\zeta$ are non-elliptic Weierstrass functions with $\omega '(m)=i K(1-m)$ being the imaginary half-period of $\wp (x)$. Throughout this paper we use $x_0=0$. We take $\eta > 0$ to avoid singularities in the magnetic field. The energy parameter $\epsilon _1$ and the displacement $\delta$ are related by $\epsilon _1=\frac{2}{3}(m+1)-\wp (\delta )-m,$ where $\delta$ is calculated by taking the inverse of $\wp (\delta )$ (see more details in “Appendix 1”). For this example $\epsilon _1<0$, thus, the spectrum of this system is given by the ground state energy $E^{(1)}_{\epsilon _1}=0$, which emerges as a consequence of the deformation of the external magnetic field, and the allowed energy bands $E^{+(1)}=[\sqrt{-\epsilon _1},\sqrt{1-m-\epsilon _1}]\cup [\sqrt{1-\epsilon _1},\infty )$ and $E^{-(1)}=-E^{+(1)}$. The expression of the bound-state spinor $\psi ^{(1)}_{\epsilon _1}$ can be obtained directly by the substitution of (14) into (12) while the positive-energy band-edge states become

$$\begin{aligned} \psi ^{(1)}_{E_0}= \begin{pmatrix} \text {dn}(x|m) \\ \frac{i}{\sqrt{-\epsilon _1}}L_1^+\text {dn}(x|m) \end{pmatrix}, \quad \psi ^{(1)}_{E_{\sqrt{1-m}}}= \begin{pmatrix} \text {cn}(x|m) \\ \frac{i}{\sqrt{1-m-\epsilon _1}}L_1^+\text {cn}(x|m) \end{pmatrix}, \quad \psi ^{(1)}_{E_1}= \begin{pmatrix} \text {sn}(x|m) \\ \frac{i}{\sqrt{1-\epsilon _1}}L_1^+\text {sn}(x|m) \end{pmatrix}. \end{aligned}$$

(15)

As a final step, the corresponding external magnetic field $B_1(x,\epsilon _1)= B_0 \partial _x W_1 = B_0 \partial _x^2 \ln (u_{\epsilon _1})$ (see “Appendix 2”) is recovered.

Figure 3 (left) shows the probability density $|\psi ^{(1)}_{\epsilon _1}|^2$ of the added bound state, which is confined to this region by the aperiodicity in the magnetic field. In Fig. 3 (center), we plot the probability density of the band-edge states, where the vertical lines indicate multiples of the initial Lamé system’s period $\mathcal {T}=2K(m)$. We see that the band-edge states recover their periodicity far from the origin. Figure 3 (right) presents a plot of the generated quasiperiodic magnetic superlattice with an added bound state (red curve) and the initial Lamé superlattice (gray shadow) for comparison [see (5)], in units of $B_0$. The magnetic field $B_1$ shows an aperiodicity around the origin introduced by the Darboux transformation, which is the cause for the confined state to appear. An impurity or defect in the superlattice could cause this modification in an experimental setup. As an important remark, the magnetic field $B_1(x,\epsilon _1)$ is actually a biparametric family of quasiperiodic magnetic fields, depending on the parameters $\epsilon _1 < 0$ and $\eta >0$. In the case $\eta =0$ or $\eta \rightarrow \infty$, it returns just a displacement of the Lamé magnetic superlattice and the bound state at $E_{\epsilon _1}^{(1)}=0$ disappears.

Fig. 3

Probability densities of the added bound state $|\psi ^{(1)}_{\epsilon _1}|^2$ (left) and of the band-edge states (center): $|\psi ^{(1)}_{E_0}|^2$ (black line), $|\psi ^{(1)}_{E_{\sqrt{1-m}}}|^2$ (green line) and $|\psi ^{(1)}_{E_1}|^2$ (yellow line). (Right) Quasiperiodic magnetic superlattice with an added bound state (red) and the Lamé magnetic superlattice (gray shadow). The used parameters are $m=1/2$, $\eta =1$, $\epsilon _1=-1/2$

At large values of x, the magnetic field $B_1$ recovers its periodicity with an added phase. To see this, note that the functions $u_{1,2}$ grow exponentially in our case of interest $\epsilon _{1} < 0$ and fulfill the relation $u_1(x)u_2(x+\delta )=c$, where c is some constant. We can define an $x_c>0$ such that $u_1(x_c)= c/u_2(x_c+\delta ) \sim 0$ and $u_2(-x_c + \delta )= c/u_1(-x_c) \sim 0$. Then, for $x>x_c$, the seed solution $u_{\epsilon _1} \rightarrow u_2$. In the same way, for a negative large value of x, ($x<-x_c$), the seed solution $u_{\epsilon _1} \rightarrow u_1$. Therefore the superpotential has the following asymptotic behaviors when $\pm x > x_c$:

$$\begin{aligned} W_1(x,\epsilon _1) \rightarrow \,m\,\,\text {sn}(x|m)\,\text {sn}(x\pm \delta |m)\,\text {sn}(\mp \delta |m)+\beta _\pm , \end{aligned}$$

(16)

where $\beta _\pm =\zeta (\omega '\pm \delta )-\zeta (\omega ')+\zeta (\mp \delta )$ are constants. The superpotential tends to be a periodic function in these regions, $W_1(x+\mathcal {T},\epsilon _1)= W_1(x,\epsilon _1)$, where $\mathcal {T}=2K(m)$. The above implies the periodicity of the intertwining operator $L^+_1$, the new band-edge states, and the magnetic field outside the origin:

$$\begin{aligned} B_1(x,\epsilon _1) \rightarrow B_0(\text {dn}(x\pm \delta |m)^2-\text {dn}(x|m)^2). \end{aligned}$$

(17)

3.3 Quasiperiodic superlattices with two bound states

We can iterate the Darboux transformation to add more than one bound state. In particular, the second-order transformation will add two of such states to the superlattice. The second iteration begins once we reconstruct $H_1$. Then we decouple the Dirac equation into two Schrödinger equations with Hamiltonians $h_1^\pm$. Now, we take $h^+_2 = h^-_1 - \epsilon _2$ and propose the new intertwining relation $h_2^- L_2^+=L_2^+ h_2^+$. The Hamiltonian $h_2^-$ and the intertwining operator $L_2^+$ are given by

$$\begin{aligned} L_2^+ = - \partial _x + \frac{\vartheta _x}{\vartheta }, \qquad h_2^-= - \partial _x^2 + V^-_2(x). \end{aligned}$$

(18)

The seed solution $\vartheta$ of this second transformation satisfies $h_1\vartheta =\epsilon _2\vartheta$, for $\epsilon _2\ne \epsilon _1$, then there is a $u_{\epsilon _2}$ such that $\vartheta =L_1^+ u_{\epsilon _2}$ fulfilling $-\partial _x^2 u_{\epsilon _2} + V^- u_{\epsilon _2} = (\epsilon _1+\epsilon _2) u_{\epsilon _2}$. Thus, the potential $V^-_2$ takes the form

$$\begin{aligned} V^-_2= V^+_2 - 2 \partial _x^2(\ln \vartheta ) = V^+_2 -2 \partial _x^2\ln \left( \frac{W(u_{\epsilon _1},u_{\epsilon _2})}{u_{\epsilon _1}}\right) , \end{aligned}$$

(19)

where $W(f,g)=f g_x-f_x g$ is the Wronskian of f and g. There are now two bound states, one for each first-order transformation. The second-order superpotential and magnetic field are given by

$$\begin{aligned} W_2(x,\epsilon _1,\epsilon _2)= \partial _x \ln \left( \frac{W(u_{\epsilon _1},u_{\epsilon _2})}{u_{\epsilon _1}}\right) , \quad B_2(x,\epsilon _1,\epsilon _2)=B_0 \partial _x \,W_2(x,\epsilon _1,\epsilon _2). \end{aligned}$$

(20)

This second-order transformation produces a quasiperiodic magnetic superlattice supporting two bound states. We start from the Lamé magnetic superlattice; the seed solutions that we consider $u_{\epsilon _{j}}=u_1(x,\epsilon _{j})+\eta _{j}u_2(x,\epsilon _{j})$, $j=1,2$, $\epsilon _2<0$, $\epsilon _1<0$ fulfill the Lamé equation (see (14)). The system under construction will support two bound states with energies $E^{\pm (2)}_{\epsilon _1}=\pm \sqrt{-\epsilon _2}, \quad E^{(2)}_{\epsilon _2}=0$. The bound states are given by

$$\begin{aligned} \Psi ^{(2)}_{\epsilon _1} \propto \left( \begin{array}{l} u_{\epsilon _1}^{-1} \\ L_2^+u_{\epsilon _1}^{-1} \end{array}\right) , \quad \Psi ^{(2)}_{\epsilon _2} \propto \left( \begin{array}{l} 0 \\ \frac{u_{\epsilon _1}}{W(u_{\epsilon _1},u_{\epsilon _2})} \end{array}\right) . \end{aligned}$$

(21)

The allowed energy bands in the positive part of the spectrum become

$$\begin{aligned} E^{+(2)}=[\sqrt{(-\epsilon _1-\epsilon _2)}, \sqrt{(1-m-\epsilon _1-\epsilon _2)}]\cup [\sqrt{(1-\epsilon _1-\epsilon _2)},\infty ), \end{aligned}$$

(22)

and $E^{-(2)}=-E^{+(2)}$. The positive-energy band-edge states are expressed as

$$\begin{aligned} \psi ^{(2)}_{E_0}&= \begin{pmatrix} \frac{i}{\sqrt{(-\epsilon _1)}}L_1^+\text {dn}(x|m) \\ -\frac{1}{\sqrt{(-\epsilon _1-\epsilon _2)}\sqrt{(-\epsilon _1)}}L_2^+L_1^+\text {dn}(x|m) \end{pmatrix}, \nonumber \\ \psi ^{(2)}_{E_{\sqrt{1-m}}}&= \begin{pmatrix} \frac{i}{\sqrt{(1-m-\epsilon _1)}}L_1^+\text {cn}(x|m) \\ -\frac{1}{\sqrt{(1-m-\epsilon _1-\epsilon _2)}\sqrt{(1-m-\epsilon _1)}}L_2^+L_1^+\text {cn}(x|m) \end{pmatrix}, \nonumber \\ \psi ^{(2)}_{E_1}&= \begin{pmatrix} \frac{i}{\sqrt{(1-\epsilon _1)}}L_1^+\text {sn}(x|m) \\ -\frac{1}{\sqrt{(1-\epsilon _1-\epsilon _2)}\sqrt{(1-\epsilon _1)}}L_2^+L_1^+\text {sn}(x|m) \end{pmatrix}. \end{aligned}$$

(23)

Figure 4 shows the probability densities of the described states: on the left for the bound states (21), on the center for the band-edge states (23). The corresponding magnetic field, calculated using Eq. (20), is plotted on the right (red curved) besides the Lamé magnetic field (gray shadow) for comparison, in units of $B_0$. Similar to the quasiperiodic magnetic superlattice with a single bound state, the period of the magnetic field far from the origin is $\mathcal {T}=2K(m)$. The deformations in the magnetic field now increase in magnitude with respect to the first transformation.

Fig. 4

Probability density of the two bound states in (21), $|\Psi _{\epsilon _2}|^2$ (purple line) and $|\Psi _{\epsilon _1}|^2$ (blue dashed line) (left) and of the band-edge states $|\psi ^{(2)}_{m}|^2$ (black line), $|\psi ^{(2)}_{1}|^2$ (green line) and $|\psi ^{(2)}_{1+m}|^2$ (yellow line), with period $\mathcal {T}=2K(m)$ (center). (Right) Quasiperiodic magnetic superlattice with two added bound states (red line) and Lamé magnetic superlattice (gray shadow). We use the parameters $\eta _1=1$, $\eta _2=-1$, $\epsilon _1=-1/2$, $\epsilon _2=-1$

4 Domain walls

When a graphene layer is deposited on a substrate, like boron nitrate, the behavior of the charge carriers can be modeled by the Dirac equation with an effective mass,

$$\begin{aligned} \mathcal{H} \Phi = E \Phi , \quad \mathcal{H} = \hbar v_{\rm F}(-i\sigma _1\partial _x+\sigma _2 k_y) +\mu (x) v_{\rm F}^2\sigma _3 . \end{aligned}$$

(24)

where the mass term represents a break of the symmetry between the two triangular sublattices of graphene. A domain wall occurs when there is a change in the symmetry breaking of the triangular sublattices, which is modeled by a position-dependent mass term $\mu =\mu (x)$ fulfilling [23]

$$\begin{aligned} \lim _{x\rightarrow -\infty } \mu (x)=-\mu < 0, \quad \lim _{x\rightarrow \infty } \mu (x)=\mu > 0. \end{aligned}$$

(25)

We can use the results of Sects. 2 and 3 to obtain different mass profiles $\mu (x)$ having domain walls and the corresponding solutions of the Dirac equation. Since the proposed mass profile depends only on one coordinate, we can use separation of variables to simplify the problem as follows, $\Phi (x,y)=e^{-ik_yy}X(x)$. Then we introduce the unitary transformation $\mathcal{R}=\exp (i\frac{\pi }{4}\sigma _1)$, such that $H=\mathcal{R} \mathcal{H}\mathcal{R}^{-1}$ and $\psi (x)=\mathcal{R} X(x)$. Therefore, H acquires a form similar to Eq. (2), and the Darboux transformation can be as well applied. To obtain a dimensionless Hamiltonian, the variables are renamed as $x\rightarrow x/l_{\mu }$, $k_y\rightarrow l_{\mu }k_y$ and $E\rightarrow l_{\mu }E/\hbar v_{\rm F}$, where $l_{\mu }= \hbar / v_{\rm F}\mu _0$ is the characteristic length, similar to the magnetic case for which $\mu _0$ represents the mass amplitude $\mu (x)=\mu _0 W_0(x)$ (see Sect. 2). This leads to the Dirac equation

$$\begin{aligned} (- i \sigma _1\partial _x+W_0(x)\sigma _2+k_y\sigma _3)\psi =E\psi . \end{aligned}$$

(26)

This is analog to equation (2), where the momentum $k_y$ in the rotated system can be interpreted as an extra mass term. We can also write (26) in terms of its components $\psi =(\psi ^+,\psi ^-)^T$, in order to obtain instead of (2) and (4),

$$\begin{aligned} \psi ^\pm =-\frac{i}{E\pm k_y}\left( \pm \partial _x+W_0(x)\right) \psi ^\pm , \quad -\psi ^{\mp }_{xx}+\left( W_0(x)^2\pm W_{0x}(x)\right) \psi ^{\mp }=\left( E^2-k_y^2\right) \psi ^{\mp }. \end{aligned}$$

(27)

We now use the Lamé magnetic field (5) and take the limit when the period of oscillation goes to infinity, $m\rightarrow 1$. Then:

$$\begin{aligned} \lim _{m\rightarrow 1} W_0(x)= \tanh (x). \end{aligned}$$

(28)

There is a single bound state $\psi _{E_0}=(0,\text {sech}(x))^T$, with associate energy $E=|k_y|$.

Moreover, we can use the results of Sect. 3 to generate new domain walls. For example, we can take the quasiperiodic magnetic superlattice with an added bound state, take the limit $m \rightarrow 1$ and apply at the end the unitary rotation $\mathcal R$. The mass amplitude is then $\mu (x) = \mu _0 W_1(x,\epsilon )$, where

$$\begin{aligned} W_1(x,\epsilon _1)=\tanh (x)+\left( \frac{\text {csch}(\delta ) \cosh (x) \left( \eta \,e^{2 x \coth (\delta )}-1\right) }{\cosh (\delta +x)+\eta \cosh (x-\delta ) e^{2 x \coth (\delta )}}-\frac{4\tanh (x/2)}{1+\tanh ^2(x/2)}\right) , \end{aligned}$$

(29)

with $\epsilon _1=1/3-\wp (\delta )$, $\eta > 0$. This mass profile fulfills the domain wall conditions, as it is shown in Fig. 5. Also, it is presented the probability density associated with the generated bound states $\psi ^{(1)}_{E_0}= \psi ^{(1)}_{E_{\sqrt{1-m}}}$ and $\psi ^{(1)}_{\epsilon _1}$. The first bound state arises from the collapse of the band-edge states, with energy $E^{(1)}=\sqrt{k_y^2-\epsilon _1}$, the second appears from the mass term modification, which is a consequence of the Darboux transformation with energy $E^{(1)}_{\epsilon _1}=0$. The corresponding eigenspinors have the explicit form

$$\begin{aligned} \psi ^{(1)}_{E_0,E_{\sqrt{1-m}}}&= \begin{pmatrix} \text {sech}(x) \\ \frac{i}{\sqrt{k_y^2-\epsilon _1}-k_y} L_1^+\text {sech}(x) \end{pmatrix}, \end{aligned}$$

(30)

$$\begin{aligned} \psi ^{(1)}_{\epsilon _1}&= \begin{pmatrix} 0 \\ \frac{\cosh (\delta ) \cosh (x) e^{x \coth (\delta )}}{\cosh (\delta +x)+\eta _1 \cosh (x-\delta ) e^{2 x \coth (\delta )}} \end{pmatrix}, \end{aligned}$$

(31)

where the intertwining operator is given by $L_1^+=-\partial _x+W_1(x,\epsilon _1)$. Different mass profiles with domain walls can be constructed taking the limit $T\rightarrow \infty$ of the quasiperiodic magnetic superlattices generated through the technique presented in this section.

Fig. 5

(Left) Bound state generated from the modification of the mass profile at zero energy $|\psi ^{(1)}_{\epsilon _1}|^2$. (Center) Bound state $|\psi ^{(1)}_1|^2$, which originates from the band collapse when $m\rightarrow 1$. (Right) Modified mass profile from the first-order Darboux transformation in the limit $m\rightarrow 1$ (red line) as compared with the initial mass profile (gray line). We have taken $\eta =2/3$, $\epsilon _1=-1/2$, $k_y=1$

5 Summary

In this work, we have presented a model that describes a monolayer graphene interacting with a Lamé magnetic superlattice which is periodic and smooth in the whole domain. The system has a band-structured spectrum with its eigenspinors having a closed explicit form.

Moreover, we have shown how to include magnetic field deformations by adding discrete energy levels to the spectrum without destroying the solvability of the system. The resulting magnetic fields turn out to be quasiperiodic, in the sense that far from the localized deformations, they become asymptotically periodic. The technique allows iterations; thus, adding any finite number of discrete energies to the spectrum turns out to be possible. We exemplified the technique through the simplest Lamé magnetic superlattice.

It was shown that the topics of magnetic superlattices and domain walls in gapped graphene can be connected if we first take the limit when the period of the superlattice goes to infinity and then a unitary rotation to the Dirac equation is applied. This connection also holds for the quasiperiodic magnetic superlattices constructed through Darboux transformations. The energy levels in the discrete spectrum keep their nature in the domain wall limit, but some extra levels could arise due to the collapse of energy bands. The deformations of the magnetic field translate into deformations of the mass term in gapped graphene.

Appendices

Appendix 1: Schrödinger equation with Lamé potentials

For the sake of completeness, we summarize the solution of Schrödinger equation with Lamé potentials in this appendix; the reader can find definitions of the involved functions in [51, 52], a review of the problem in [53], and a complete pedagogical approach in [54]. First, we start from a Schrödinger Hamiltonian $h=-\partial _x^2 + V(x)$ where the potential V(x) is periodic, nonsingular, with period $\mathcal {T}$. Thus, the solutions of the Schrodinger equation $h \psi = E \psi$ fulfill the Bloch theorem,

$$\begin{aligned} \psi (x)=e^{i\kappa x} u_\kappa (x), \end{aligned}$$

(32)

where $u_\kappa$ is a periodic function $u_\kappa (x+\mathcal {T})=u_\kappa (x)$ and $\kappa$ denotes the momentum of the crystal. Note also that $\psi (x+\mathcal {T})=e^{i\kappa \mathcal {T}}\psi (x)$. The spectrum consists of allowed energy bands with edges such that $\kappa \,\mathcal {T} =0,\pi$, then the band edge eigenfunctions fulfill $\psi (x + \mathcal {T}) = \pm \psi (x)$.

Now, the Lamé potentials are given by

$$\begin{aligned} V(x)=n(n+1)\,m\, \text {sn}(x|m)^2, \end{aligned}$$

(33)

where m is the modulus of the Jacobi elliptic function $\text {sn}(x|m)$ such that $0 \le m \le 1$, and n is a natural number. The period of the potential is $\mathcal {T}=2K(m)$, where

$$\begin{aligned} K(m)=\int _0^{\pi /2}\frac{\hbox{d}\theta }{\sqrt{1-m \sin ^2\theta }}. \end{aligned}$$

(34)

This system has $2n+2$ bands, $n+1$ allowed plus $n+1$ forbidden bands. An equivalent expression in terms of the Weierstrass elliptic function $\wp (x)$ is given by

$$\begin{aligned} V(x)=n(n+1)\left( \wp (x+iK(1-m))+\frac{1}{3}(m+1)\right) . \end{aligned}$$

(35)

We are using the short notation $\wp (x,g_2,g_3)\equiv \wp (x)$, and in general the Weierstrass functions $\sigma (x,g_2,g_3)\equiv \sigma (x)$ and $\zeta (x,g_2,g_3)\equiv \zeta (x)$ in the variable x for parameters, or elliptic invariants, $g_2$, and $g_3$. In this case

$$\begin{aligned} g_2(m)=\frac{4}{3} \left( m^2-m+1\right) , \quad g_3(m)=\frac{4}{27} (m-2) (2 m-1) (m+1). \end{aligned}$$

(36)

In this paper we will consider $n=1$, the simplest case of the Lamé potentials. The solution of the stationary Schrödinger equation $-\psi _{xx}+V(x)\psi =\epsilon \psi$ for any value of the energy parameter $\epsilon$ is given by

$$\begin{aligned} \psi =u_1(x,\epsilon )+\eta \,u_2(x,\epsilon ), \end{aligned}$$

(37)

where $\eta$ is an arbitrary constant and $u_1$, $u_2$ are the Bloch functions associated to $\epsilon$ given by

$$\begin{aligned} u_1(x,\epsilon )=\frac{\sigma (x_0+\omega ') \sigma (x+\delta +\omega ')}{\sigma (x+\omega ') \sigma (x_0+\delta +\omega ')} e^{-\zeta (\delta )(x-x_0)}, \quad u_2(x,\epsilon )=\frac{\sigma (x_0+\omega ') \sigma (x-\delta +\omega ') }{\sigma (x+\omega ') \sigma (x_0-\delta +\omega ')}e^{\zeta (\delta )(x-x_0)}, \end{aligned}$$

(38)

with $\omega '(m)=i K(1-m)$ being the imaginary half-period of $\wp (x)$ and $x_0$ a reference point in the x axis. The energy parameter $\epsilon$ and the displacement $\delta$ are related by $\epsilon =2(m+1)/3-\wp (\delta )$, i.e., the parameter $\delta$ is calculated through the inverse of $\wp (\delta )$. The functions $u_1$, $u_2$ fulfill the traditional Bloch relation

$$\begin{aligned} u_1(x+\mathcal {T})=e^{i\kappa \mathcal {T}} u_1(x), \quad u_2(x+\mathcal {T})=e^{-i\kappa \mathcal {T}} u_2(x), \end{aligned}$$

(39)

where $\kappa =2i(\omega \zeta (\delta )-\delta \zeta (\omega ))$. The band-edge states are given in terms of the Jacobi elliptic functions as follows

$$\begin{aligned} \psi _m = \text {dn}(x|m),\quad \psi _1 = \text {cn}(x|m), \quad \psi _{1+m}=\text {sn}(x|m). \end{aligned}$$

(40)

Their corresponding energies are $E=m,~1,~1+m.$ Therefore, the spectrum is composed by the allowed bands [m, 1], $[1+m,\infty )$, and the gaps $(-\infty ,m)$, $(1,1+m)$.

Appendix 2: Magnetic field expression for the quasiperiodic superlattice with an added bound state

The explicit expressions of the designed magnetic fields are, in general, lengthy. Here we present the corresponding expression for the Sect. 3.2:

$$\begin{aligned} B_1(x,\epsilon _1)=B_0\partial _x W_1(x,\epsilon _1), \end{aligned}$$

(41)

where $W_1=\frac{\partial _x(u_1+\eta \,u_2)}{u_1+\eta \, u_2}$ is the superpotential, and $u_1$, $u_2$ are the Bloch functions (see “Appendix 1”). Then, the derivative of the superpotential is

$$\begin{aligned} \partial _x W_1(x,\epsilon _1)=\frac{\partial _{xx}u_1+\eta \,\partial _{xx}u_2}{u_1+\eta \, u_2}-\left( \frac{\partial _xu_1+\eta \,\partial _xu_2}{u_1+\eta \, u_2}\right) ^2. \end{aligned}$$

The explicit form of the derivatives are

$$\begin{aligned}&\partial _xu_1= -\frac{e^{-x \zeta (\delta )}\sigma (\omega ') \sigma (x+\delta +\omega ')}{\sigma (\delta +\omega ') \sigma (x+\omega ')} \left( \zeta (\delta )-\zeta (x+\delta +\omega ')+\zeta (x+\omega ')\right) ,\\&\partial _xu_2=-\frac{e^{x \zeta (\delta )}\sigma (\omega ') \sigma (x-\delta +\omega ')}{\sigma (\delta -\omega ') \sigma (x+\omega ')} \left( \zeta (\delta )+\zeta (x-\delta +\omega ')-\zeta (x+\omega ')\right) ,\\&\partial _{xx}u_1=\frac{e^{-x \zeta (\delta )}\sigma (\omega ') \sigma (x+\delta +\omega ') \left( \wp (x+\omega ')-\wp (x+\delta +\omega ')+(\zeta (\delta )-\zeta (x+\delta +\omega ')+\zeta (x+\omega '))^2\right) }{\sigma (\delta +\omega ') \sigma (x+\omega ')},\\&\partial _{xx}u_2=-\frac{e^{x \zeta (\delta )} \sigma (\omega ') \sigma (x-\delta +\omega ') \left( \wp (x+\omega ')-\wp (x-\delta +\omega ')+(\zeta (\delta )+\zeta (x-\delta +\omega ')-\zeta (x+\omega '))^2\right) }{\sigma (x+\omega ') \sigma (\delta -\omega ')}. \end{aligned}$$

Therefore, the magnetic field is given by

$$\begin{aligned} B_1(x,\epsilon _1)=B_0 \left( \wp \left( x+\omega '\right) -\frac{P_1+P_2+P_3}{\left( \eta e^{2 x \zeta (\delta )} \sigma \left( \delta +\omega '\right) \sigma \left( x-\delta +\omega '\right) -\sigma \left( \delta -\omega '\right) \sigma \left( x+\delta +\omega '\right) \right) ^2}\right) , \end{aligned}$$

(42)

where

$$\begin{aligned} P_1&=\eta ^2 \wp \left( x-\delta +\omega '\right) e^{4 x \zeta (\delta )} \sigma \left( \delta +\omega '\right) ^2 \sigma \left( x-\delta +\omega '\right) ^2,\\ P_2&= \eta e^{2 x \zeta (\delta )} \sigma \left( \delta -\omega '\right) \sigma \left( \delta +\omega '\right) \sigma \left( x-\delta +\omega '\right) \sigma \left( x+\delta +\omega '\right) \\& \quad \left( \left( 2 \zeta (\delta )+\zeta \left( x-\delta +\omega '\right) -\zeta \left( x+\delta +\omega '\right) \right) ^2-\wp \left( x-\delta +\omega '\right) -\wp \left( x+\delta +\omega '\right) \right) ,\\ P_3&=\wp \left( x+\delta +\omega '\right) \sigma \left( \delta -\omega '\right) ^2 \sigma \left( x+\delta +\omega '\right) ^2. \end{aligned}$$