Orbital magnetic susceptibility of zigzag carbon nanobelts: a tight-binding study

Low-dimensional materials with honeycomb structures such as C₆₀ Elser1987 ; Coffey1992 , carbon nanotube Ajiki1993 ; Lu1995 ; Chen2005 , and graphene Sepioni2010 ; Ortega2013 , have attracted attention. Recently, a circular graphene nanoribbon, called a carbon nanobelt Povie2017 , has been added to allotropes of carbon. As defined by Itami et al. Segawa2016 ; Itami2023 , a carbon nanobelt is a ring-shaped segment of a carbon nanotube, and the cleavage of at least two C-C bonds is necessary to open it. The first carbon nanobelt is a segment of (6,6) carbon nanotubes and consists of 12 benzene rings. Longer and complex carbon nanobelts can be achieved using current synthesis techniques Li2021 ; Wang2023 ; Wang2024 . Although many physical properties of carbon nanobelts remain unknown, researchers have reported high electric conductance between the tip of a scanning tunneling microscope and carbon nanobelts; this excellent property is expected to benefit electric applications Li2022 .

A carbon nanotube can be regarded as a graphene sheet rolled into a tube Li2013 . Similarly, a carbon nanobelt can be regarded as a graphene nanoribbon that can be seamlessly bent into cylindrical shapes. Graphene exhibits unique magnetic properties not observed in other materials Goerbig2011 . For example, divergent orbital diamagnetism at the Dirac point has been theoretically predicted and was recently detected Bustamante2021 . The magnetic properties of graphene ribbons are primarily determined by two factors: orbital magnetization and spin angular momentum Sepioni2010 . This diamagnetic property originates from a previous contribution. Unlike the spin, the orbital magnetization of a graphene sheet is generated by a global electric current, which depends on the shape of the sheet. In this study, we focus on the magnetization of carbon nanobelts caused by the orbital current circulating around them.

Extensive theoretical studies on the orbital magnetization of graphene disks have revealed that it strongly depends on their size and type of edges, temperature, and defects Ezawa2007 ; Liu2008 ; Ominato2013 ; Deyo2021 . However, little is known about the magnetic properties of carbon nanobelts. Thus, we introduce a tight-binding model for carbon nanobelts and investigate the dependence of the magnetic susceptibility on their size and temperature. The advantage of using a tight-binding model over other theoretical methods, such as density functional theory Shi2020 ; Wu2013 ; Negrin2023 , is that the energy eigenvalue can often be expressed in analytical forms for arbitrary sizes. The magnetic susceptibility is obtained by calculating the second derivative of the free energy Ominato2013 . Thus, the dependence of the magnetic susceptibility on the size can be precisely discussed.

The remainder of this paper is organized as follows: In Section 2, we introduce a tight-binding model for a carbon nanobelt based on cyclacene, which is the first proposed carbon nanobelt Turker2004 . Although cyclacene has not yet been synthesized, its structure is the simplest among carbon nanobelts Gleiter2009 . Cyclacene can be considered as two interacting one-dimensional atomic arrays, hereafter referred to as rings. Therefore, we address the energy eigenvalue problem of a ring and explain its size dependence. In Section 3, the dependence of the magnetic susceptibility of the ring on its size and temperature is discussed. In Section 4, the magnetic properties of the carbon nanobelts, hereafter referred to as belts, are investigated. In particular, the difference in the magnetic susceptibility between the rings and the belts is discussed. In Section 5, the relations between the change in the energy level by applying a magnetic field and the magnetic properties is summarized.

The structure of the belt considered in this study consists of upper and lower rings, as shown in Fig. 1(a), where the solid circles indicate the atoms. A uniform magnetic field is applied parallel to the central axis ($z$-axis) of the belts. Figure 1(b) shows the position of atoms in the upper ring projected onto the $xy$-plane. We assume that atoms in a ring exist on a circle. The electrons can hop along the circumference and is characterized by the transfer integral $\gamma$. Additionally, electrons can hop between rings, whose transfer integral $\gamma^{\prime}$ may be different from $\gamma$. Because the magnetic response to the these parameter dependencies cannot be easily considered simultaneously, we prohibited hopping between rings in this section and considered the magnetic response of a single ring using a tight-binding model.

Let the number of sites (atoms) in the ring be $N$. The radius of a ring, whose nearest distance between sites in the $xy$-plane is $a$ is expressed as

Each site in the ring is labeled with an integer $j$ $\in$ $\{1,2,\ldots,N\}$ as shown in Fig. 1(b), and the position of the $j$-th site on the $xy$-plane $\vec{r}_{j}$ = $(x_{j},y_{j})$ is defined by

The Hamiltonian of the tight-binding model Ezawa2007 , in which the contribution of spin is neglected, is defined by

where the operator $c_{j}^{{\dagger}}$ creates an electron at the $j$-th site, and $c_{j}$ annihilates an electron at the $j$-th site. The periodic condition requires $c_{N+1}^{{\dagger}}$ = $c_{1}^{{\dagger}}$ and $c_{N+1}$ = $c_{1}$. The effect of the magnetic field in the Hamiltonian is expressed through the Peierls phase $\phi_{j}$ between the $j$-th site and the $(j+1)$-th site defined by

where $\vec{A}$ is the vector potential and is related to the applied magnetic flux density $\vec{B}$ in the form of $\vec{B}$ = $\nabla\times\vec{A}$. Using the Landau gauge, the vector potential can be expressed as

In the continuous limit of $a$ $\rightarrow$ 0, the Hamiltonian (58) can be approximately regarded as $(\hat{p}+e\vec{A})^{2}/2m$, where $\hat{p}$ and $m$ are the momentum operator and effective mass related to $\gamma$, respectively Boykin2001 ; Inui2023 . Combining Eq. (7) with Eq. (6) leads to

where $\alpha$ is a dimensionless parameter defined by

If the magnetic flux density through the ring is defined by $\phi$ $\equiv$ $\pi R_{N}^{2}B$, then the parameter $\alpha$ is expressed as $\phi/\phi_{0}$ where $\phi_{0}$ $\equiv$ $h$/(2$e$) is magnetic flux quantum.

The energy eigenvalue $\epsilon_{n,N}(\alpha)$ of the Hamiltonian $H_{\mbox{\tiny Ring}}$ with quantum number $n$ = 1, 2, $\ldots$ $N$ is given by

The $j$-th component of the eigenstate $\psi_{n}$ corresponding to $\epsilon_{n,N}(\alpha)$ is given by

where

The combination of the eigenvalue and eigenstate shown above satisfies the Schrödinger equation $H_{\mbox{\tiny Ring}}\psi_{n}$ = $\epsilon_{n,N}\psi_{n}$. Applying the Hamiltonian on the eigenstate, we have

where $[.]_{j}$ denotes the $j$-th component of $H_{\mbox{\tiny Ring}}\psi_{n}$. The eigenstates at the sites $j+1$ and $j-1$ can be expressed as

where

Combining Eqs. (14), (15), and (16), we obtain

We define the energy eigenvalue normalized by $\gamma$ as $\lambda_{n,N}(\alpha)$ ($\equiv$ $\epsilon_{n,N}(\alpha)/\gamma$). Figure 2 shows $\lambda_{n,N}(\alpha)$ for $N$ = 4, 5, and 6 as a function of $\alpha$. In the absence of the magnetic field, i.e., $\alpha$ = 0, each energy level has a degeneracy of two except for $n$ = 1 (grand state) and $n$ = $N$. Note that $\lambda_{2,4}(\alpha)$ and $\lambda_{3,4}(\alpha)$ are zero at $\alpha$ = 0. This holds for any $N$ if $N$ mod 4 = 0, i.e., if $N$ is a multiple of 4.

The number of six-membered rings in carbon nanobelts reported in Ref. Povie2017 is 12, which is greater than that in the site considered above. Although we showed the eigenvalues of the rings with a small $N$ for visibility, the exact eigenvalues can be obtained for an arbitrary $N$ and magnetic flux $B$ using Eq. (11).

Electron hopping between two rings is possible in a belt. We distinguish between the upper and lower rings with integers $k$ = 1 and 2, respectively. Each site in the belt is labeled by $j$ ($1$ $\leq$ $j$ $\leq$ $N$) and $k$. The total number of sites is 2$N$. Unlike the rings in Section 2, $N$ must be even for the belt. The Hamiltonian of a belt is expressed as

where $-\gamma^{\prime}$ is the transfer integral between the odd sites of the upper and lower rings. We assume that the transition between even sites is prohibited.

The energy eigenvalues of $H_{\mbox{\tiny Belt}}$ normalized by $\gamma$ are given by

where $t$ = $\gamma^{\prime}/\gamma$, and $\{p_{l},q_{l}\}$ = $\{-1,-1\}$, $\{-1,1\}$, $\{1,-1\}$, and $\{1,1\}$ for $l$ = 1, 2, 3, and 4, respectively where $t$ = $\gamma^{\prime}/\gamma$, and $p_{l}$ =$1$ and $-1$ for $l$ = 1, and 2, respectively. The components of eigenstates labeled with $(l,j,k)$ and $p_{l}$, $\psi_{n,l,j,k}$ are expressed as

where $C_{n,l}$ is a normalized constant determined from the condition that $\sum_{l,j}(|\psi_{n,l,j,1}|^{2}+|\psi_{n,l,j,2}|^{2})$ = 1, and $v_{n,l,j}$ is defined by

In contrast to the eigenstate presented in Section 2, these expressions of the eigenstates do not form an orthogonal basis but they satisfy the Schrödinger equation as follows. If $j$ is even, the state after applying $H_{\mbox{\tiny Belt}}$ to the eigenstate $\psi_{n}$ is written as

If $j$ is odd and $k$ = 1, we have

Here, to obtain the last equation, we used the following equation:

Similarly, the Schrödinger equation is satisfied for the case where $j$ is odd and $k$ = 2.

Figure 5 shows the normalized energy eigenvalues of the belts for $N$ = 4 and 6 at $t$ = 0.2. For $N$ = 4, the energy eigenvalues of $(n,l)$ = $(1,2)$ and $(1,4)$ are zero at $\alpha$ = 0, and this degeneracy remains even when the transfer of electrons between rings occurs.

The first and second derivatives of the normalized energy eigenvalues are, respectively, given by

Figure 6 shows the temperature dependence of the magnetic susceptibility of the belts with $t$ = 1 for $N$ = 4, 6, and 10. In contrast to Fig. 3, the susceptibility at absolute zero for $N$ = 4 is finite. We recall that the second term in the left-hand side of Eq. (23) causes the divergence for rings. In the case of the belt with $N$ = 4 and $t$ = 0, the derivative of the energy eigenvalues with $n$ = 2 and 3 near $\alpha$ = 0 is not zero (see Fig. 2), whereas, if $t$ is not zero, the first derivative of the energy eigenvalues of the belt with $N$ = 4 for $(n,l)$ = $(1,2)$ and $(1,4)$ converges to zero (see Fig. 5), as $\alpha$ approaches zero. Thus, the second term vanishes in the limit of $\alpha$ $\rightarrow$ 0, and the divergence does not occur. This finiteness of magnetic susceptibility for belts holds for any size, if $N$ is a multiple of four.

Interestingly, the sign of susceptibility for $N$ = 10 changes from negative to positive as the temperature increases. Figure 7(a) shows the phase diagram of the sign in a plane of the transfer integral $t$ and the normalized temperature for $N$ = 10. The sign is always negative for small $t$ values, independent of temperature. Above a critical value of $t$ near 0.76, the sign changes from negative to positive as the temperature increases from absolute zero. Conversely, the sign of the belt with $N$ = 4 changes from positive to negative as the temperature increases from absolute zero as shown in Fig. 7(b).

We showed that the magnetic properties of carbon nanobelts strongly depend on the divisibility of the number of sites by four. If the number of sites in the ring is a multiple of four, the magnetic susceptibility diverges as the temperature approaches absolute zero. This singularity occurs when the following conditions are satisfied. First, the energy eigenvalues converges to zero as the magnetic field decreases to zero. Second, their first derivative with respect to the magnetic field at $B$ = 0 is finite. If the chemical potential $\mu$ is not zero, the first condition should be written as the condition that the energy eigenvalues converge to the chemical potential as the magnetic field decreases. As we assume that $\mu$ = 0, the divergence occurs for $N$ = 4. The number of distinct energy eigenvalues increases as the number of sites increases. Thus, the divergence can occur for various values of the chemical potential as $N$ increases. For example, if $\mu$ = $\gamma$, the divergence occurs for $N$ = 6 (see Fig. 2(c)).

The energy eigenvalues of the belts change depending on the parameter $t$, which is a ratio of the transfer integral along the circumference and width directions. For a belt consisting of two carbon rings with $N$ = 4, the derivative of energy eigenvalues at zero becomes zero by the presence of the transfer integral along the width direction. It follows that the divergence at absolute zero does not occurs. However, this does not imply that no divergence occur for $t$ $\neq$ 0. The magnetic susceptibility of the belt with $N$ = 6 diverges if the parameter $t$ satisfying $\gamma\lambda_{n,N}(0,t)$ = $\mu$ exists.

This study focuses primarily on the interaction between electrons and an external magnetic field, and the many-body effect of electrons on the magnetic properties is considered indirectly via Pauli’s exclusion principle. To describe the electron systems in the carbon nanobelt more accurately, the interactions between electrons should be considered. The most important interaction is likely the Coulomb interaction between electrons, which may be considered using the Hubbard model, whose interaction is described by $c_{j\uparrow}^{{\dagger}}c_{j\uparrow}c_{j\downarrow}^{{\dagger}}c_{j\downarrow}$, where $\uparrow$ and $\downarrow$ denote spin up and spin down, respectively. For graphene nanoribbonsKumar2020; Lou2024, more realistic transport results have been obtained using the Hubbard model Hancock2010 .

Various carbon nanobelts and rings shapes have been proposed and synthesized. For example, Möbius carbon nanobelts, with a twisted structure are smoothly connected at the front and back Segawa2022 ; Heine2020 . Their magnetic properties may be significantly different from that the belts considered in this study.

This research was supported by the Ministry of Education, Culture, Sports, Science and Technology, through a Grant-in-Aid for Scientific Research(C), MEXT KAKENHI Grant Number 21K04895.