# Topological Atomic Chains on 2D Hybrid Structure

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model and Theoretical Description

_{1}), i.e., with topological mid-gap states at both chain ends. When one applies $V>W$ it is obtained a chain in the trivial topological phase without end states (SSH

_{0}). The second part of Equation (2) describes the chain-substrate coupling where electron transitions between the i-th chain state and the surface states are established by ${V}_{i,\overrightarrow{k}\alpha}$ matrix elements (hybridization terms).

## 3. Results and Discussions

#### 3.1. Straight SSH Atomic Chain

_{1}chain on the surface with rectangular DOS (panels a and d) the electronic structure of the chain is characterized by two sideband peaks (bulk bands) with the energy gap between them for the interior sites (e.g., for $i=2$) and the mid-gap state localized at $E={\epsilon}_{0}$ for the edge site ($i=1$ and $i=N$). In this case, the surface DOS is flat (energy independent) and can be effectively described within the wide-band limit approximation. Thus, the surface described within this approximation does not influence the chain topological states which is in agreement with the literature results [60]. The situation changes for realistic 2D square lattice substrate with the van Hove singularity in the middle of the band (panels b and e). This van Hove peak can be considered as a kind of sharp atomic state which is coupled with the mid-gap edge state of the SSH chain. It is the reason that the topological SSH state splits for $E=0$ (panel b) and it is not robust against the surface states. In addition, for the asymmetrical case (panel e, ${\epsilon}_{0}=4$) the energy of the van Hove peak corresponds to the energy of the lower (left) sideband in the chain and there is a small local minimum in the structure of this band at $E=0$. For other energies, i.e., beyond the van Hove singularity the substrate spectral density is relatively flat and thus the chain DOS is almost the same as for the rectangular case (see the upper panels). The most interesting case one can observe for the honeycomb lattice underneath the chain (panels c and f). Now, the substrate DOS is characterized by two van Hove peaks at $E=\pm 5$ and a local minimum in the middle of the band. As before these peaks are responsible for the local minima in the chain DOS, i.e., in both sidebands (panel c) or in the mid-gap state (panel f). However, for very low values of the substrate DOS (near $E=0$) the chain-surface effective coupling $\Gamma \left(E\right)$ tends to zero which leads to the atomic limit for this energy span. Thus, for the symmetrical case the mid-gap topological state is very narrow (and almost disperssionless) than for non-zero surface DOS (shown in panels a and b). One can see that 2D substrates essentially modify topological states and change effectively their shape. For the asymmetrical case (panel f) low values of the substrate DOS correspond to the left sideband of the chain DOS. The sidebands consist of many bulk states (due to the surface coupling they form relatively smooth function of the energy) but in this case the lower one reveals an atomic structure around the zero energy and has many sharp peaks. At the same time the second (upper) sideband of the chain DOS (around $E=9$) remains in the form of the bulk shape. As a consequence, very regular structure of the chain DOS can drastically change in the presence of the van Hove singularity or energy dips in the substrate DOS. Note that for ${\epsilon}_{0}=0$ (left panels) all local DOS functions are symmetrical in the energy scale which results from symmetrical structures of the surface DOS with respect to the Fermi energy and the particle-hole symmetry is not broken in this case. For different value of ${\epsilon}_{0}$ this symmetry still exists for a plane surface DOS (right upper panel) but it is broken for real 2D lattice DOS (panels e and f). Thus, strong asymmetry in the local DOS of atomic system can come from the peaked structure of the surface DOS which was not reported before and it leads to breaking of the system particle-hole symmetry.

#### 3.2. Boundary Effects in Chain DOS

_{0}(upper panel) and SSH

_{1}(bottom panel) in the form of heatmaps of the local DOS along the whole chain, i.e., at each chain site, $i=1,...,12$. The position of ${\epsilon}_{0}$ is very close to the surface DOS boundary, ${\epsilon}_{0}=14$, such that the bottom sideband of the SSH chain lies inside the surface DOS and the second one is outside this band. For the trivial SSH chain (upper panel) the boundary of the surface DOS ($E=15$) corresponds to the chain energy gap and there are still two sidebands of the chain DOS (inside and outside the surface band). The lower sideband is quite smooth (like the sidebands in Figure 2, upper panels) as its energy lies inside the surface band. On the other hand, the outside sideband is characterized by very sharp and high DOS peaks as in this case there are no corresponding energy states in the substrate and the effective coupling $\Gamma \left(E\right)=0$ for $E>15$. Thus, outside the substrate band the chain states tend to atomic limit and strong asymmetry of the local DOS appears (as a function of the energy). For nontrivial SSH

_{1}chain (bottom panel) both chain sidebands are also nonsymmetrical and they behave in the same way as for the SSH

_{0}chain. However, at both edge sites, $i=1$ and $i=N$, there exist topological states and as one can see for $E>15$ (outside the substrate band) these states suddenly disappear and no extra states are observed above this energy. In this case, the structure of topological state becomes asymmetrical and this interesting effect is going to be studied in more details.

_{0}chain the local DOS at the edge sites has only two sidebands structure and for the SSH

_{1}chain this structure still exists together with the mid-gap states.

#### 3.3. Zig-Zag and Armchair-Edge Chains between 2D Systems

_{1}chain is characterized by small but nonzero DOS in the energy gap region. This effect is known in the SSH chains where topological states appear mainly at both chain ends but they also symmetrically leak inside the chain with decreasing intensities [29,46]. Here one can observe asymmetrical values of DOS along the chain. It is a consequence of much higher topological state at the last site $i=20$ in comparison with the first site DOS, which allows for deeper penetration of this state into the chain. As an unexpected result it was found that nonzero local DOS at the Fermi level in the chain is observed for these sites which are directly coupled with the right electrode (which has no states at the Fermi energy).

## 4. Conclusions

- Surface with singularities in DOS essentially influences the spectral density function (local DOS) along the chain and is responsible for strong asymmetry in the topological chain energetic structure. It leads to the particle-hole symmetry breaking in the system.
- The surface van Hove singularities can split the SSH topological state of the chain. On the other hand dips in the surface DOS lead to dispersionless strongly localized states (topological or normal) in the chain.
- There was also discovered that topological mid-gap states can exist outside the surface band boundaries. It is important that when the chain on-site energies lie near the surface DOS edges topological state reveals partially localized behaviour with both wide dispersion due to continuous band states and sharp localized peak which comes from the surface band boundaries.
- Different geometries of the SSH atomic chain between two 2D electrodes systems show spatial and energetic asymmetry in the structure of chain DOS which leads to different energies of both topological edge states at the chain ends.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

**Figure A1.**Real and imaginary parts of the $\Sigma \left(E\right)$ function,

`Re`$\Sigma \left(E\right)=\Lambda \left(E\right)$ and

`Im`$\Sigma \left(E\right)=-\Gamma \left(E\right)/2$, respectively, for a rectangular lead DOS of the width $w=30$. The cross point of $E-{\epsilon}_{0}$ and $\Lambda \left(E\right)$ indicates a localized state in the system, here it is set ${\epsilon}_{0}=15$.

`Im`$\Sigma \left(E\right)=\Lambda \left(E\right)$ (broken blue curve), is nonzero even for vanishing $\Gamma \left(E\right)$, and the localized state appears for the crossing point of $E-{\epsilon}_{0}$ and $\Lambda \left(E\right)$ functions.

## References

- Yu, B. Graphene nanoelectronics: Overview from post-silicon perspective. In Proceedings of the 2012 IEEE 11th International Conference on Solid-State and Integrated Circuit Technology, Xi’an, China, 29 October–1 November 2012; pp. 1–2. [Google Scholar] [CrossRef]
- Shore, K.A. Introduction to Graphene-Based Nanomaterials: From Electronic Structure to Quantum Transport, by Luis E.F. Foa Torres, Stephan Roche and Jean-Christophe Charlier. Contemp. Phys.
**2014**, 55, 344–345. [Google Scholar] [CrossRef] - Li, D.; Liu, T.; Yu, X.; Wu, D.; Su, Z. Fabrication of graphene–biomacromolecule hybrid materials for tissue engineering application. Polym. Chem.
**2017**, 8, 4309–4321. [Google Scholar] [CrossRef] - Lieber, C.M. One-dimensional nanostructures: Chemistry, physics and applications. Solid State Commun.
**1998**, 107, 607–616. [Google Scholar] [CrossRef] - Saxena, S.K.; Nyodu, R.; Kumar, S.; Maurya, V.K. Current Advances in Nanotechnology and Medicine. In NanoBioMedicine; Saxena, S.K., Khurana, S.M.P., Eds.; Springer: Singapore, 2020; pp. 3–16. [Google Scholar] [CrossRef]
- Auslaender, O.M.; Steinberg, H.; Yacoby, A.; Tserkovnyak, Y.; Halperin, B.I.; Baldwin, K.W.; Pfeiffer, L.N.; West, K.W. Spin-Charge Separation and Localization in One Dimension. Science
**2005**, 308, 88–92. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kwapiński, T.; Taranko, R. Spin and charge pumping in a quantum wire: The role of spin-flip scattering and Zeeman splitting. J. Phys. Condens. Matter
**2011**, 23, 405301. [Google Scholar] [CrossRef] - Nadj-Perge, S.; Drozdov, I.K.; Li, J.; Chen, H.; Jeon, S.; Seo, J.; MacDonald, A.H.; Bernevig, B.A.; Yazdani, A. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science
**2014**, 346, 602–607. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pawlak, R.; Kisiel, M.; Klinovaja, J.; Meier, T.; Kawai, S.; Glatzel, T.; Loss, D.; Meyer, E. Probing atomic structure and Majorana wavefunctions in mono-atomic Fe chains on superconducting Pb surface. NPJ Quantum Inf.
**2016**, 2, 16035. [Google Scholar] [CrossRef] - Shin, J.S.; Ryang, K.D.; Yeom, H.W. Finite-length charge-density waves on terminated atomic wires. Phys. Rev. B
**2012**, 85, 073401. [Google Scholar] [CrossRef][Green Version] - Kurzyna, M.; Kwapiński, T. Non-local electron transport through normal and topological ladder-like atomic systems. J. Appl. Phys.
**2018**, 123, 194301. [Google Scholar] [CrossRef] - Kwapiński, T. Conductance oscillations and charge waves in zigzag shaped quantum wires. J. Phys. Condens. Matter
**2010**, 22, 295303. [Google Scholar] [CrossRef] - van der Wiel, W.G.; De Franceschi, S.; Elzerman, J.M.; Fujisawa, T.; Tarucha, S.; Kouwenhoven, L.P. Electron transport through double quantum dots. Rev. Mod. Phys.
**2002**, 75, 1–22. [Google Scholar] [CrossRef][Green Version] - Fujisawa, T.; Tokura, Y.; Hirayama, Y. Transient current spectroscopy of a quantum dot in the Coulomb blockade regime. Phys. Rev. B
**2001**, 63, 081304. [Google Scholar] [CrossRef][Green Version] - Hayashi, T.; Fujisawa, T.; Cheong, H.D.; Jeong, Y.H.; Hirayama, Y. Coherent Manipulation of Electronic States in a Double Quantum Dot. Phys. Rev. Lett.
**2003**, 91, 226804. [Google Scholar] [CrossRef] [PubMed][Green Version] - Arkinstall, J.; Teimourpour, M.H.; Feng, L.; El-Ganainy, R.; Schomerus, H. Topological tight-binding models from nontrivial square roots. Phys. Rev. B
**2017**, 95. [Google Scholar] [CrossRef][Green Version] - Jürß, C.; Bauer, D. High-harmonic generation in Su-Schrieffer-Heeger chains. Phys. Rev. B
**2019**, 99. [Google Scholar] [CrossRef][Green Version] - Huneke, J.; Platero, G.; Kohler, S. Steady-State Coherent Transfer by Adiabatic Passage. Phys. Rev. Lett.
**2013**, 110. [Google Scholar] [CrossRef][Green Version] - Kohler, S.; Lehmann, J.; Hänggi, P. Driven quantum transport on the nanoscale. Phys. Rep.
**2005**, 406, 379–443. [Google Scholar] [CrossRef][Green Version] - Kurzyna, M.; Kwapiński, T. Electron Pumping and Spectral Density Dynamics in Energy-Gapped Topological Chains. Appl. Sci.
**2021**, 11, 772. [Google Scholar] [CrossRef] - Wilczek, F. Quantum Time Crystals. Phys. Rev. Lett.
**2012**, 109, 160401. [Google Scholar] [CrossRef][Green Version] - Sacha, K. Modeling spontaneous breaking of time-translation symmetry. Phys. Rev. A
**2015**, 91, 033617. [Google Scholar] [CrossRef][Green Version] - Sacha, K.; Zakrzewski, J. Time crystals: A review. Rep. Prog. Phys.
**2017**, 81, 016401. [Google Scholar] [CrossRef] - Kurzyna, M.; Kwapiński, T. Nontrivial dynamics of a two-site system: Transient crystals. Phys. Rev. B
**2020**, 102, 245414. [Google Scholar] [CrossRef] - Lindner, N.H.; Refael, G.; Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys.
**2011**, 7, 490–495. [Google Scholar] [CrossRef][Green Version] - Perez-Piskunow, P.M.; Usaj, G.; Balseiro, C.A.; Torres, L.E.F.F. Floquet chiral edge states in graphene. Phys. Rev. B
**2014**, 89. [Google Scholar] [CrossRef][Green Version] - Kraus, C.V.; Dalmonte, M.; Baranov, M.A.; Läuchli, A.M.; Zoller, P. Majorana Edge States in Atomic Wires Coupled by Pair Hopping. Phys. Rev. Lett.
**2013**, 111, 173004. [Google Scholar] [CrossRef] [PubMed] - Kurzyna, M.; Kwapiński, T. Edge-state dynamics in coupled topological chains. Phys. Rev. B
**2020**, 102, 195429. [Google Scholar] [CrossRef] - Pérez-González, B.; Bello, M.; Gómez-León, Á.; Platero, G. SSH model with long-range hoppings: Topology, driving and disorder. arXiv
**2018**, arXiv:1802.03973. [Google Scholar] - Su, W.P.; Schrieffer, J.R.; Heeger, A.J. Solitons in Polyacetylene. Phys. Rev. Lett.
**1979**, 42, 1698–1701. [Google Scholar] [CrossRef] - Jałochowski, M.; Krawiec, M. Antimonene on Pb quantum wells. 2D Mater.
**2019**, 6, 045028. [Google Scholar] [CrossRef] - Stpniak-Dybala, A.; Dyniec, P.; Kopciuszyski, M.; Zdyb, R.; Jałochowski, M.; Krawiec, M. Planar Silicene: A New Silicon Allotrope Epitaxially Grown by Segregation. Adv. Funct. Mater.
**2019**, 29, 1906053. [Google Scholar] [CrossRef] - Drost, R.; Ojanen, T.; Harju, A.; Liljeroth, P. Topological states in engineered atomic lattices. Nat. Phys.
**2017**, 13, 668–671. [Google Scholar] [CrossRef][Green Version] - Le, N.H.; Fisher, A.J.; Curson, N.J.; Ginossar, E. Topological phases of a dimerized Fermi–Hubbard model for semiconductor nano-lattices. NPJ Quantum Inf.
**2020**, 6, 24. [Google Scholar] [CrossRef][Green Version] - Crain, J.N.; McChesney, J.L.; Zheng, F.; Gallagher, M.C.; Snijders, P.C.; Bissen, M.; Gundelach, C.; Erwin, S.C.; Himpsel, F.J. Chains of gold atoms with tailored electronic states. Phys. Rev. B
**2004**, 69, 125401. [Google Scholar] [CrossRef][Green Version] - Kopciuszyński, M.; Dyniec, P.; Krawiec, M.; Łukasik, P.; Jałochowski, M.; Zdyb, R. Pb nanoribbons on the Si(553) surface. Phys. Rev. B
**2013**, 88, 155431. [Google Scholar] [CrossRef] - Baski, A.; Saoud, K.; Jones, K. 1-D nanostructures grown on the Si(5 5 12) surface. Appl. Surf. Sci.
**2001**, 182, 216–222. [Google Scholar] [CrossRef] - Hensgens, T.; Fujita, T.; Janssen, L.; Li, X.; Van Diepen, C.J.; Reichl, C.; Wegscheider, W.; Das Sarma, S.; Vandersypen, L.M.K. Quantum simulation of a Fermi–Hubbard model using a semiconductor quantum dot array. Nature
**2017**, 548, 70–73. [Google Scholar] [CrossRef] [PubMed] - Zwanenburg, F.A.; Dzurak, A.S.; Morello, A.; Simmons, M.Y.; Hollenberg, L.C.L.; Klimeck, G.; Rogge, S.; Coppersmith, S.N.; Eriksson, M.A. Silicon quantum electronics. Rev. Mod. Phys.
**2013**, 85, 961–1019. [Google Scholar] [CrossRef] - Huda, M.N.; Kezilebieke, S.; Ojanen, T.; Drost, R.; Liljeroth, P. Tuneable topological domain wall states in engineered atomic chains. NPJ Quantum Mater.
**2020**, 5, 17. [Google Scholar] [CrossRef] - Ji, J.; Song, X.; Liu, J.; Yan, Z.; Huo, C.; Zhang, S.; Su, M.; Liao, L.; Wang, W.; Ni, Z.; et al. Two-dimensional antimonene single crystals grown by van der Waals epitaxy. Nat. Commun.
**2016**, 7, 13352. [Google Scholar] [CrossRef][Green Version] - Sun, X.; Song, Z.; Liu, S.; Wang, Y.; Li, Y.; Wang, W.; Lu, J. Sub-5 nm Monolayer Arsenene and Antimonene Transistors. ACS Appl. Mater. Interfaces
**2018**, 10, 22363–22371. [Google Scholar] [CrossRef] [PubMed] - Jałochowski, M.; Kwapinski, T.; Łukasik, P.; Nita, P.; Kopciuszyński, M. Correlation between morphology, electron band structure, and resistivity of Pb atomic chains on the Si(553)-Au surface. J. Phys. Condens. Matter
**2016**, 28, 284003. [Google Scholar] [CrossRef] - Krawiec, M.; Kwapiński, T.; Jałochowski, M. Double nonequivalent chain structure on a vicinal Si(557)-Au surface. Phys. Rev. B
**2006**, 73, 075415. [Google Scholar] [CrossRef][Green Version] - Kwapiński, T.; Kohler, S.; Hänggi, P. Electron transport across a quantum wire in the presence of electron leakage to a substrate. Eur. Phys. J. B
**2010**, 78, 75–81. [Google Scholar] [CrossRef][Green Version] - Asboth, J.K.; Oroszlany, L.; Palyi, A. A Short Course on Topological Insulators; Springer: Cham, Switzerland, 2016. [Google Scholar]
- Li, L.; Yang, C.; Chen, S. Winding numbers of phase transition points for one-dimensional topological systems. EPL
**2015**, 112, 10004. [Google Scholar] [CrossRef][Green Version] - Li, L.; Xu, Z.; Chen, S. Topological phases of generalized Su-Schrieffer-Heeger models. Phys. Rev. B
**2014**, 89, 085111. [Google Scholar] [CrossRef][Green Version] - Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge Studies in Semiconductor Physics and Microelectronic Engineering; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar] [CrossRef]
- Podloucky, R.; Desjonquères, M.C.; Spanjaard, D. Concepts in Surface Physics, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 1996; ISBN 978-3-540-58622-7. [Google Scholar]
- Newns, D.; Read, N. Mean-field theory of intermediate valence/heavy fermion systems. Adv. Phys.
**1987**, 36, 799–849. [Google Scholar] [CrossRef] - Kwapiński, T. Conductance oscillations of a quantum wire disturbed by an adatom. J. Phys. Condens. Matter
**2007**, 19, 176218. [Google Scholar] [CrossRef] [PubMed] - Choy, T.C. Density of states for a two-dimensional Penrose lattice: Evidence of a strong Van-Hove singularity. Phys. Rev. Lett.
**1985**, 55, 2915–2918. [Google Scholar] [CrossRef] [PubMed] - Horiguchi, T. Lattice Green’s Functions for the Triangular and Honeycomb Lattices. J. Math. Phys.
**1972**, 13, 1411–1419. [Google Scholar] [CrossRef] - Katsnelson, M.I. Graphene: Carbon in Two Dimensions; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar] [CrossRef]
- Kogan, E.; Gumbs, G. Green’s Functions and DOS for Some 2D Lattices. Graphene
**2021**, 10, 1–12. [Google Scholar] [CrossRef] - Yuhara, J.; Shichida, Y. Epitaxial growth of two-dimensional Pb and Sn films on Al(111). Thin Solid Film.
**2016**, 616, 618–623. [Google Scholar] [CrossRef] - Feng, H.; Liu, C.; Zhou, S.; Gao, N.; Gao, Q.; Zhuang, J.; Xu, X.; Hu, Z.; Wang, J.; Chen, L.; et al. Experimental Realization of Two-Dimensional Buckled Lieb Lattice. Nano Lett.
**2020**, 20, 2537–2543. [Google Scholar] [CrossRef] - Yuhara, J.; Schmid, M.; Varga, P. Two-dimensional alloy of immiscible metals: Single and binary monolayer films of Pb and Sn on Rh(111). Phys. Rev. B
**2003**, 67, 195407. [Google Scholar] [CrossRef] - Kurzyna, M.; Kwapiński, T. Electronic properties of atomic ribbons with spin-orbit couplings on different substrates. J. Appl. Phys.
**2019**, 125, 144301. [Google Scholar] [CrossRef]

**Figure 1.**Schematic view of the SSH atomic chain for different geometries: (

**a**) straight chain on 2D electrode, (

**b**,

**c**) zig-zag and armchair edge chains between two 2D electrodes called here Hybrid1 and Hybrid2, respectively. The broken and solid red lines represent different couplings between the nearest-neighbour atomic sites in the SSH chain.

**Figure 2.**Local DOS at two sites of the straight SSH chain (see panel a in Figure 1), $i=1$ (blue broken curves) and $i=2$ (red solid curves) as a function of energy, and for different surface DOS shown by the dashed black curves: rectangular DOS (panels

**a**and

**d**), 2D-DOS with one van Hove singularity (panels

**b**and

**e**) and for 2D-DOS with two van Hove peaks (panels

**c**and

**f**). The left (right) panels correspond to ${\epsilon}_{0}=0$ (${\epsilon}_{0}=4$). The chain length is $N=12$ sites, the couplings are $V=1$, $W=4$, and the surface DOS width is $w=30$ (all energies are expressed in $\Gamma $ units).

**Figure 3.**Heatmap plots of the energy and site dependent local DOS along the whole chain, $N=12$, for the same system as in Figure 2 for the rectangular surface DOS and for ${\epsilon}_{0}=14$ (in the vicinity of the edge surface DOS). The upper (bottom) panel corresponds to the SSH

_{0}chain with $V=4$, $W=1$ (SSH

_{1}chain with $V=1$, $W=4$).

**Figure 4.**Local DOS at the first site of the SSH

_{1}chain on a surface described by a rectangular DOS (the same as in Figure 2, upper panels) for different values of the chain on-site energy, ${\epsilon}_{0}=12,14,15,16$ and 18, respectively. The SSH

_{1}couplings are $V=1$, $W=4$.

**Figure 5.**Heatmap plots of the energy and site dependent local DOS along the whole zig-zag chain for the Hybrid1 system (see panel b in Figure 1) for the on-site energies ${\epsilon}_{0}=0$ (upper panel) and for ${\epsilon}_{0}=4$ (bottom panel). The left (right) 2D electrode is described by DOS with one van Hove (two van Hove) singularity and the chain length is $N=20$ sites, $V=1$, $W=4$.

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Kwapiński, T.; Kurzyna, M.
Topological Atomic Chains on 2D Hybrid Structure. *Materials* **2021**, *14*, 3289.
https://doi.org/10.3390/ma14123289

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Kwapiński T, Kurzyna M.
Topological Atomic Chains on 2D Hybrid Structure. *Materials*. 2021; 14(12):3289.
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**Chicago/Turabian Style**

Kwapiński, Tomasz, and Marcin Kurzyna.
2021. "Topological Atomic Chains on 2D Hybrid Structure" *Materials* 14, no. 12: 3289.
https://doi.org/10.3390/ma14123289