Abstract
As the most prevalent element on our planet, carbon manifests a wide variety of allotropic phases, significantly contributing to its complex physical properties. Recently, several carbon allotropes have been reported to possess abundant topological phases in theory and experiment. This work focuses on a sp3 carbon allotrope, Z-ACA allotrope, which consists of 5-6-7-type Z-ACA carbon rings. This allotrope has been reported previously as a superhard material comparable to diamond. In this study, we report that it is a candidate for both an obstructed atomic insulator and a real Chern insulator. It is worth mentioning that Z-ACA exhibits an unconventional bulk-boundary correspondence due to its hinge boundary state manifestation. Our current research indicates that Z-ACA is a suitable carbon phase platform for studying the real topology and second-order bulk-boundary correspondence.
Introduction
Carbon [1–3] is widely regarded as the most active element inside the periodic table owing to its extensive capacity for sp, sp2, and sp3 hybridization. The flexibility of carbon’s bond hybridizations enables it to exhibit a wide range of allotropes, such as diamond, graphite, and amorphous carbon. There is considerable variety observed in the physicochemical features of these allotropes. Researchers are presently involved in the theoretical and experimental production of new carbon allotropes [4–10] with the aim of identifying materials that exhibit desirable electronic, magnetic, mechanical, thermal, and topological properties.
Discovering the topological properties and their physical origins for carbon materials is currently a very active area in condensed matter physics [11]. In recent years, there has been a significant surge in interest surrounding the phenomenon of magic-angle twisted bilayer graphene [12, 13]. The recently developed 2D material class, which is based on carbon, demonstrates a wide range of appealing electronic states, such as superconductivity and topological nontrivial electronic states. Several three-dimensional (3D) carbon allotropes [14–17] have been shown to display intriguing topological electronic states in addition to 2D twisted bilayer graphene. Note that 1D carbon systems, like certain graphene nanoribbons, may also host topological behaviors [18–20].
The field of topological quantum chemistry theory has made significant progress in reclassifying insulators that were previously regarded as trivial ones. These reclassifications of trivial insulators now include obstructed atomic insulators (OAIs) and atomic insulators [21–28]. This reclassification is based on the Wyckoff positions (WPs) of the orbitals that induce the band representations (BRs). Orbitals situated at atomic-occupied WPs induce the BRs of atomic insulators. On the other hand, the BRs of OAIs are generated by orbitals situated at both atom-occupied and atom-unoccupied WPs. The term “Obstructed Wannier charge centers (OWCCs)” [29, 30] pertains to atom-unoccupied WPs in OAIs.
In addition, it is crucial that the band eigenstates exhibit real topology [31–35] due to specific symmetry constraints, including spacetime inversion symmetry and the lack of spin-orbit coupling (SOC) phenomenon. The real Chern insulator (RCI) state has been discovered in numerous two-dimensional systems [36–41]. The notion is expanded to include 3D systems, which is of considerable importance. 3D RCIs are commonly linked to unconventional bulk-boundary correspondence, namely, hinge states [42–44], as they depict boundary states in two dimensions that are smaller than the 3D bulk.
In this work, using first-principle calculations, we will report that the sp3 carbon allotrope, Z-ACA allotrope, is a new candidate for OAI and RCI. We would like to point out that the stability, electronic, and mechanical properties of the Z-ACA allotrope have been investigated using a first-principles method by He et al. [45]. However, other researchers have not investigated the OAI and RCI states for Z-ACA due to materials with nontrivial real band topology having become a focus of current physics research within the last 2 years. Figure 1 shows the crystal structure of Z-ACA with Pmmn space group; the C atoms occupy an equivalent Wyckoff position of 2a (0.00000, 1.00000, 0.95706), 2b (0.00000, 0.50000, 0.93126), and 4e (0.00000, 0.76406, 0.82886). The optimized lattice constants a = 2.521 Å, b = 4.759 Å, and c = 7.930 Å in this work are comparable with others (a = 2.521 Å, b = 4.76 Å, c = 7.930 Å) [45].
FIGURE 1
Methods
The Vienna ab initio simulation package (VASP) [46] was employed to carry out the calculations within the framework of density functional theory. The Perdew-Burke-Ernzerhof [47] functional with generalized gradient approximations [48] was adopted to describe the exchange-correlation interactions. The computations were conducted using a plane-wave cutoff of 500 eV. The convergence criterion for the electronic self-consistence loop was established at 10−7 eV on the 13 × 7 × 4 Monkhorst-Pack k-point mesh. In terms of structural relaxation, the Hellmann-Feynman forces acting on each atom were assumed to be −0.01 eV/Å. The real Chern number νR can also be evaluated by using the Wilson-loop method, similar to the well-known Wilson-loop method for conventional topological insulators [49, 50]. The Wilson loop method traces the Wannier charge centre (Berry phase) evolution of valence wavefunctions between time-reversal-invariant momentum. The Wannier tight-binding method utilizes localized Wannier functions as basis orbitals to capture the compound’s physics. These basis Wannier functions are obtained from first-principles simulations. This procedure is implemented in the Wannier90 code [51].
Results and discussion
Firstly, we will assess the feasibility of identifying Z-ACA as an OAI. The OAI state of Z-ACA can be determined by applying the theory of elementary band representation (EBR). The EBRs correspond to the smallest sets of band structures that can be obtained using maximally localized atomic-like Wannier functions. The EBRs for the space group I4/mmm can be derived by combining the EBRs of the maximal WPs (2a, 2b, 4c, and 4d). Hence, we select the maximal WPs 2a, 2b, 4c, and 4d for the EBR decomposition of Z-ACA (see Figure 2).
FIGURE 2
The electronic bands of trivial insulators can be represented by nonnegative integer Linear Combinations of Elementary Band Representations (LCEBRs) according to the topological quantum chemistry (TQC) theory [52]. Supplementary Table S1 presents the LCEBRs derived from the populated electronic bands of Z-ACA. Curiously, in the instance of Z-ACA, all the LCEBRs display a nonzero integer combination for EBR (Ag@4d). Hence, it may be deduced that the EBR is inseparable and must be linked to the electron-containing WPs by 4d. The carbon atom is situated at the 2a, 2b, and 4e locations, while the 4d position remains unoccupied. Z-ACA fulfills the requirements to be categorized as an OAI, and the 4d WP is the OWCC. The locations of OWCC are illustrated in Figure 2 with green pentagram. Figure 3 depicts the band structure of Z-ACA along the selected high-symmetry paths. Figure 3 clearly displays an insulating gap. Hence, Z-ACA can be viewed as a promising candidate for 3D OAI.
FIGURE 3
In addition to its OAI nature, Z-ACA exhibits spacetime inversion symmetry and lacks the spin-orbit coupling (SOC) effect (see Supplementary Figure S1), making it a potential candidate for RCI. In order to verify the real Chern topologies of the 3D Z-ACA, it is necessary to calculate the vR values for the 2D kz-slices. In light of the system’s global band gap, it can be inferred that all 2D kz-slices are adiabatically connected, hence necessitating a shared νR.
That is, one can select a particular slice, such as kz = 0 or kz = π plane, to determine the νR. To calculate the vR for the planes mentioned above, one can use the parity eigenvalues at the time-reversal invariant momentum points on each plane, as expressed by the following formula [53, 54]:
The is the floor function, and is the number of occupied bands with negative spacetime inversion symmetry eigenvalue at time-reversal invariant momentum point Γi.
Table 1 presents the vR values of Z-ACA for the 2D kz = 0 and kz = π planes. A nonzero vR, which reflects the real topology, is discovered for Z-ACA. Actually, every 2D slice normal to kz has a nontrivial νR.
TABLE 1
| kz = 0 | kz = π | |||||||
|---|---|---|---|---|---|---|---|---|
| Γ | X | M | Y | Z | U | R | T | |
| n+ | 18 | 16 | 16 | 16 | 14 | 16 | 16 | 16 |
| n− | 14 | 16 | 16 | 16 | 18 | 16 | 16 | 16 |
| νR | 1 | 1 | ||||||
Parity information of the Z-ACA at the eight time-reversal invariant momentum points.
The nontrivial real topology of both planes can also be assessed using the Wilson-loop method. The computation will yield N curves in the θ-ky diagram, which represents the Wilson loop spectrum. As illustrated in Figure 4, the Wilson loop spectrum overlaps at one point and three points (an odd number of points) for θ = π in the kz = 0 and kz = π planes, showing nontrivial real topology in both planes.
FIGURE 4
Typically, 3D RCIs will demonstrate unconventional relationships between their bulk and boundary properties [44]. In other words, the 3D Z-ACA will have boundary states that exist in two dimensions less than the 3D bulk. Specifically, these boundary states will be confined to 1D hinges. In order to determine the behavior of possible hinges in Z-ACA, a tight-binding model (TB) was created for a one-dimensional tube with the geometry of Z-ACA. The sample used in the model retained its spacetime inversion symmetry. Later, the Wannier function [55–58] was used to calculate the band structure of the nanotube.
The band structure spectrum for the 1D tube geometry of Z-ACA is displayed in Figure 5. The presence of the hinge band is shown by the red line. Figure 6 depicts the spatial distributions of charges for the degenerate states (represented by red dots) at kz = 0. By examining the wave functions and seeing their localization at the two hinges of the nanotube (see Figure 6), we confirm that the degenerate states are indeed hinge states. As mentioned by Ref. [38], the hinge band for 3D RCI can be viewed as formed by stacking the corner modes of 2D RCI layers along z. Moreover, Hossain et al. [59] pointed out that the periodicity evinces quantum interference of electrons circumnavigating observed around the hinges, which can be applied to efficient topological electronic devices.
FIGURE 5
FIGURE 6
Summary
Based on first-principles calculations, we suggest that the carbon allotrope Z-ACA, with the 4d site as OWCC, is a potential candidate for OAI. Further analysis reveals that Z-ACA possesses a real topology characterized by a nontrivial vR, which is safeguarded by the symmetry of spacetime inversion. The presence of hinge states, which are confined to the two hinges of the Z-ACA tube, was exhibited. The Z-ACA shows excellent potential as a subject for studying the fascinating physics related to real topological phases and hinge states.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.
Author contributions
LW: Investigation, Software, Writing–original draft, Writing–review and editing. YL: Conceptualization, Investigation, Software, Writing–original draft, Writing–review and editing.
Funding
The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This work is supported by the Science and Technology Research Program of the Chongqing Municipal Education Commission (Grant No. KJZD-K202104101).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2024.1437146/full#supplementary-material
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Summary
Keywords
topological carbon material, obstructed Wannier charge center, real topology, hinge states, Z-ACA allotrope
Citation
Wu L and Li Y (2024) Z-ACA allotrope: a topological carbon material with obstructed Wannier charge center, real topology, and hinge states. Front. Phys. 12:1437146. doi: 10.3389/fphy.2024.1437146
Received
23 May 2024
Accepted
24 June 2024
Published
19 July 2024
Volume
12 - 2024
Edited by
Dipanjan Mazumdar, Southern Illinois University Carbondale, United States
Reviewed by
Saurabh Basu, Indian Institute of Technology Guwahati, India
Dongfei Wang, Beijing Institute of Technology, China
Updates
Copyright
© 2024 Wu and Li.
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*Correspondence: Lunsheng Wu, lswu@cqyu.edu.cn; Yang Li, liyang@cqwu.edu.cn
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.