Dirac fermions on wires confined to the graphene Möbius strip

Since its discover, graphene has startled researchers due to its outstanding electronic, mechanical and thermal properties [1, 2]. This single layer sheet of carbon exhibits high carrier mobility (vanishing effective electron mass) [3] and thus, the electron is described as a massless chiral Dirac fermion on a flat surface [4]. As a result, graphene offers a bridge between condensed matter physics and quantum field theory in two dimensions [5]. Indeed, graphene produces well-known relativistic effects, such as the zitterbewegun [6] and the Klein paradox [7]. More recently, relativistic effects were found in Weyl semimetals [8], Majorana fermions [9], Bogoliubov particles [10] and Kagome crystals [11].

By bending the two dimensional sheet or considering the strain effects on the membrane, graphene also becomes a table-top laboratory for curved spaces phenomena [12, 13]. In fact, the curved geometry or the strain tensor modify the effective Hamiltonian leading to a position-dependent Fermi velocity [14]. Moreover, the coupling between the Dirac fermion and the curved geometry/strain provides pseudo gauge fields whose effects depend on the particular geometry/strain [15, 16, 17, 18]. Despite being two dimensional, graphene naturally displays ripples[19, 20] and corrugations [21]. These deformations of the surface significantly modify the electronic and thermal graphene properties [22]. The effects of the curved geometry on the electronic properties, the so called curvatronics has been explored in different geometries, such as the cone [23], helicoid strip [24, 25], the catenoid bridge [26, 27] and the torus [28].

Another noteworthy effect produced by the curved geometry is the so-called geometric phase [29]. The edge states [30] or the presence of defect, such as disclinations [31], produce geometric phases modifying the electronic properties. On conical graphene surfaces, a constant geometric gauge field yields to a geometric phase depending on the conical deficit angle [32]. Furthermore, the out-of-plane deformations of the graphene layer also lead to modifications on the optical conductivity due to Aharonov-Bohm type interference [33].

Besides the curved geometry, nontrivial topology also plays a central role on the electronic features of the low-dimensional system [35]. For instance, performing a $\pi$ twist on one end of a graphene ribbon and connecting one end to the other we obtain a graphene Möbius strip [36, 37]. This graphene membrane behaves as a topological insulator with stable edge states [38]. Moreover, the half twist modifies the rotational invariance of the fields and particles constrained to move along the Möbius band [39]. Several investigations on the formation [40], stability [41], charge transfer [42], magnetic properties [43], quantum spin-Hall effect [44, 45] of the graphene Möbius strip were performed. These properties are rather different from those obtained from usual cylindrical graphene rings [46]. Indeed, the curvature of the Möbius strip modifies the energy spectrum of non-relativistic electrons, whose isotropic and parity symmetries are broken [47]. For relativistic electrons, the influence of the a flat Möbius strip was investigate by assuming nontrivial boundary conditions [48].

In this work we study the effects of the graphene Möbius strip curved geometry on effective massless relativistic electrons. Unlike some previous works that used discrete tight-bind approach [38, 40, 41], we employed a continuum analysis wherein an effective massless Dirac fermion is constrained to a curved surface. Since Dirac equation on a curved surface does not couple with the curvature [49, 50], the effects of the curved geometry steam from the spinorial connection which acts as a pseudo-magnetic potential induced by the curved geometry [51]. As a result, a geometric phase determines the density of states for wires along the Möbius strip width. The breaking of the parity and isotropy symmetries yields to states localized on the inner or outer portion of the strip depending on the angle chosen for the wire. For wires along the strip length (angular direction), the geometric phase has an Aharonov-Bohm like effect, not modifying the density of states for a single electron. Nevertheless, the ground state wave function has a period of $4\pi$, whereas the first excited state is a $2\pi$ periodic function, due to the strip twist. Furthermore, the nontrivial topology also leads to an energy spectrum given as half-integer multiples of the ground state energy. These results agree with the modified boundary conditions analysis performed in Ref.([47, 48]).

This work is organized as the following. In the section (II) we briefly review the main definitions and properties of the Möbius strip geometry, such as the metric, curvatures and connection. In the section (III), we obtain the effective Hamiltonian for the massless Dirac particle on the surface and derive the expression for the geometric potential which depends on the geometric connection. In the section (IV) we obtain the exact solutions for the electron on wires along the width and the length of the strip. The energy levels and the effects of the nontrivial topology on the states and spectrum is discussed. Finally, additional comments and perspectives are outlined in section (V).

In this section we present the coordinate system to describe the graphene Möbius strip and study its main geometric features, such as the gaussian and the mean curvatures and the connection one-forms.

The graphene Möbius strip is a surface constructed by joining the two ends of a graphene ribbon after twist one end by a $\pi$ rotation, as shown in the fig.(1) [36]. This surface can be described by the following coordinate system [36, 47]

where $2L$ is the strip width , $-L\leq u\leq L$ is a coordinate along the width, and $0\leq\theta\leq 2\pi$ is the angular coordinate. The coordinate $u$ is the distance between the center of the strip and its endpoint, as shown in the Fig. 1. Furthermore, the parameter $a$ is the radius of the central circle and the radial unit vector is $\hat{\mathbf{r}}=\cos\theta\hat{\mathbf{i}}+\sin\theta\hat{\mathbf{j}}$.

A remarkable feature of the Möbius strip is its non-orientability. Indeed, performing a $2\pi$ rotation on the outer edge of the strip ($u=L$), the outer edge becomes the inner edge $(u=-L)$. Throughout the work, we use $L=1$.

The tangent vectors, defined as $e_{i}=\frac{\partial\mathbf{r}}{\partial x^{i}}$, have the form

From the tangent vectors, we can define the Möbius strip metric $g_{ij}$ as

where the angular metric factor $\beta$ is given by [47]

In the fig.(2) we plot the angular metric function $\beta(u,\theta)$ for some values of the inner radius $a$ with $L$ fixed to $L=1$. In fig.(2)(a) we choose $a$ such that $L/a=0.375$, i.e., $a=2.66$, for which nanorings were obtained [34]. As we increase the ratio $L/a$, as for $L/a=1$ in fig. (2)(b) and for $L/a=1.89$ in fig. (2)(c) the Möbius strip becomes more compact. The ratio $L/a=1.89$ is a critical value for a Möbius strip due to mechanical properties [40].

Thus, the $2+1$ infinitesimal line element has the form

It is worthwhile to mention that the metric tensor for a cylinder (nanotubes) has the form shown in Eq.(6) with $\beta=a$, whereas for the conical surface (graphitic cone), $\beta=\alpha u$, where $\alpha$ is the so-called angular deficit [23]. Other graphene based surface, such as the helicoid [24], catenoid [26, 27] and the torus [28] can also be described by the metric in Eq.(6).

The anisotropy of the Möbius strip breaks some surfaces symmetries, such as the parity symmetry, as we can see by $\beta(-u,-\theta)\neq\beta(u,\theta)$. However, a sort of modified Möbius parity symmetry holds, where

This modified parity symmetry steams from the twist performed on the strip. In the fig.(2), we plotted the angular metric component $\beta$ for $a=1$ and $-1\leq u\leq 1$, where the Möbius parity symmetry in Eq.(7) is shown.

The Möbius strip curvatures also exhibit the twisted parity symmetry in Eq.(7). Indeed, the expressions for the mean curvature $M$ and the gaussian curvature $K$ are given by [47]

and

The behaviour of the gaussian and the mean curvatures are shown in the fig.(3) and in the fig.(4), respectively. We adopt the same values for the ratio $L/a$ as done for the $\beta$ function, i.e., $L/a=0.375$ (a), $L/a=1$ (b) and $L/a=1.89$ (c). Note that the curvatures profiles depend on the ratio $L/a$. Unlike the cylindrical nanorings, where the gaussian curvature vanishes and the mean curvature is constant, the distribution of the curvatures on the Möbius strip depends on the position $u,\theta$ and the ration $L/a$.

As we will see in the next section, the massless relativistic electron couples with the surface connection instead of the curvature. Accordingly, in order to define the fermion dynamics on the surface, we have to adopt the so-called vielbein formalism [13]. Indeed, consider a set of matrices $e^{a}_{\mu}$ such that [23]

In $(2+1)-D$, the matrices $e^{a}_{\mu}$ are $SO(1,2)$ invariant and they are called dreinbeins. Using the dreinbeins we can define a local moving co-frame as $e^{a}=e^{a}_{\mu}dx^{\mu}$. For the Möbius metric in Eq.(6), the moving co-frame has the form

The presence of the trigonometric functions in eq.(11) reveals the local rotational invariance on the surface. Using the torsion-free condition, $de^{a}+\omega^{a}_{b}\wedge e^{b}=0$, the only non-vanishing connection 1-form $\omega^{a}_{b}=\Gamma^{a}_{\phantom{d}b\mu}dx^{\mu}$ is given by

Note that for a conical graphitic surface, $\beta=\alpha u$, and then, $\omega^{1}_{2}=-(\alpha-1)d\theta$ [23].

Once we reviewed the main geometric features of the graphene Möbius strip, in this section we describe how the effective massless electron couples to the graphene Möbius strip. We study the electron properties in the continuum limit, where the electron dynamics is governed by a Dirac equation defined on the curved graphene strip [13]. The curved geometry induces a pseudo-magnetic potential vector $\Gamma_{\mu}$, known as the spinorial connection. Then, we explore some features and effects of the spinorial connection, such as the geometric potential.

We employ the intrinsic coupling of the electron to the curved surface, wherein the massless Dirac equation on the surface is given by [13]

In the Eq.(13), the gamma matrices on the surface $\gamma^{\mu}$ as defined as

where $e^{a}_{\mu}$ are the dreinbeins. The spinorial covariant derivative $D_{\mu}$ is defined as

where the spinorial connection $\Gamma_{\mu}$ is defined as

Since the curved covariant derivative in Eq.(15) is similar to the minimal coupling between an electron and a magnetic vector $A_{\mu}$, the spinorial connection can be interpreted as a pseudo-magnetic potential vector induced by the curved geometry. In the spinorial connection in Eq.(16), $\omega^{ab}_{\mu}$ are the connection 1-form and $\gamma_{a}$ are the flat Dirac matrices. In this work we use the following representation of the Dirac matrices $\gamma^{0}=-i\sigma_{3}$, $\gamma^{1}=-\sigma_{2}$, and $\gamma^{2}=\sigma_{1}$ [27].

From Eq.(12), since $\omega^{1}_{2}=-(\partial_{u}\beta-1)d\theta$, the only non-vanishing component of the spinorial connection is given by

Thus, the pseudo-magnetic potential has only one component along the angular direction. Incidentally, the corresponding pseudo-magnetic field should point into the normal direction and it should be proportional to the Gaussian curvature [13].

The behaviour of the spinorial connection along the graphene Möbius strip is shown in the fig.(5). We adopt the same values for the ratio $L/a$ as done before, i.e. $L/a=0.375$ (a), $L/a=1$ (b) and $L/a=1.89$ (c). Note that the spinorial connection profile is rather different from the Gaussian and mean curvatures. Indeed, the connection is greater for inner points $(u<0)$ than for outer points $(u>0)$ of the strip. For the sake of comparison, the spinorial connection for a cone is constant $\Gamma_{\theta}=-\frac{i}{2}(\alpha-1)\sigma_{3}$ [23] and it vanishes for a single layer graphene, where $\alpha=1$.

In the last section we described how the electron dynamics is modified by the curved geometry of the graphene Möbius strip. In this section we explore the electronic properties of the Dirac fermion constrained to wires along the graphene Möbius strip. By doing so, the fermion dynamics is dependent not only on the wire curvature but on the geometric potential $U_{g}$ as well.

In this work we found exact solutions for an electron constrained to wires along a graphene Möbius strip. Such a surface can be formed by performing a twist on one graphene ribbon end and connecting it to the another end. By considering the electron effectively described as a massless Dirac fermion, the Möbius strip curved geometry provides a geometric potential depending on the geometric connection instead of the curvature. The geometry of the wires depends not only on the wire metric but on the Möbius geometry and symmetries, as well.

For wires along the strip width, i.e., for a given $\theta_{0}$, the effects of the curved geometry are encoded into a geometric phase which steams from the geometric potential. As a result, exact solutions for the Dirac equation were found which shows the formation of edge states. As we increase the energy, additional edge states forming rings appear. Moreover, the electronic probability density is highly dependent on the direction $\theta_{0}$ chosen for the wire. In fact, for some $\theta_{0}$ the electron is more concentrated near the inner edge, whereas for other values of $\theta_{0}$ the wave function is localized towards the outer edge of the band. This behaviour agrees with the lack of parity symmetry of the Möbius band, i.e., $\beta(-u,\theta_{0})\neq\beta(u,\theta_{0})$.

On the other hand, the wires along the strip length have a rather different geometry to those along the width. For $u=0$, the wire forms a ring of radius $a$, whereas for $u=\pm L$ (strip edges) the wire is open in the interval $0\leq\theta\leq 2\pi$. Due to the Möbius strip geometry, the wires for $u_{0}\neq 0$ are actually close if we consider the period $4\pi$. Accordingly, we expect that the wave function to be a periodic function of period $4\pi$. It turns out that, for the central ring at $u=0$, the ground state is a $4\pi$ periodic function whose probability density is localized around $\theta=\pi$. For $n$ odd, the wave function has a period that is a non-integer multiple of $2\pi$. Similarly, the energy levels are half-integer multiple of the quantum $\hbar v_{F}/a$. These non-integer features of both the electronic states and the spectrum are the result of the non-trivial geometry of the Möbius band. For a ring with the same inner radius as the cylindrical ring studied in the Ref.([34]), i.e., for $a=200nm$, the energy of the ground state is $E_{0}\approx 1,64\times 10^{-3}eV$. Since the ratio $L/a=0.375$ is smaller than the critical ratio, the non-trivial topology of the Möbius strip could be used to generate a geometric Aharonov-Bohm effect due to pseudo-magnetic field.

For the wires along the edges of the Möbius strip, the ground states $n=1$ exhibit a localized probability density around $\theta=2\pi$. Once again, the break of the parity symmetry yields to a different profile between the ground state at the inner edge $(u_{0}=-L)$ and at the outer edge $(u_{0}=L)$. The electronic states for the outer edge are more concentrate than those at the inner edge.

A remarkable result we found is the role played by the geometric phase on the electronic states. For the wires along the width, the Dirac equation can be simplified by considering a geometric phase depending on the geometric potential. It turns out that the energy levels and the period of the electronic states are determined as if the strip were flat. A similar result was found numerically by Ref.([47]), though the authors considered a non-relativistic electron. On the other hand, the geometric phase provides a damping of the wave function, what leads to the parity breaking profile discussed above. Along the angular direction, the geometric phase is given by the geometric (spinor) connection. Likewise the well-known Aharonov-Bohm phase, the geometric phase does not alter the density of states for a single electron. Hence, the period of the electronic states and the energy levels are determined by the angular metric function along the wire, i.e., $\beta(u_{0},\theta)$. This shows that the electronic properties on wires along the Möbius strip are different of those in usual circular wires, for the wires inherit the anisotropic Möbius geometry.

This work suggests as a perspective the investigation of the effects of the geometric phase on the interference pattern between electrons. Indeed, Aharanov-Bohm effects on graphene gated cylindrical nanorings of width $L=75nm$ and inner radius $a=200nm$ $(L/a=0.375)$ were investigated [34]. By considering a Möbius ring with the same size, similar effects could be observed due to the geometric phase. Moreover, the inclusion of external fields might lead to an additional parameter to control the density of states. In addition, the interaction of the confined electron to the external electric or magnetic fields should leave a peculiar signature which could be used to characterize the material. In fact, as proposed in Ref.[13] transmission electron microscopy (TEM) or scanning tunneling microscopy (STM) can be used to probe the morphology and relate it to the density of states. For the Möbius graphene strip, these microscopic techniques could be used to determine the ratio $L/a$ of the samples.

The authors thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), grants n^o 312356/2017-0 (JEGS), n^o 309553/2021-0 (CASA) for financial support.