# Rippled Graphene as an Ideal Spin Inverter

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## Abstract

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## 1. Introduction

## 2. The Model Hamiltonian and the Eigenvalue Problem

## 3. Transmission through the Superlattice

- $\eta =1$ if $E\ge {E}_{\times}$;
- $\eta =-1$ if $E<{E}_{\times}$.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Avsar, A.; Ochoa, H.; Guinea, F.; Özyilmaz, B.; van Wees, B.J.; Vera-Marun, I.J. Spintronics in graphene and other two-dimensional materials. Rev. Mod. Phys.
**2020**, 92, 021003. [Google Scholar] [CrossRef] - Kane, C.L.; Mele, E.J. Quantum Spin Hall Effect in Graphene. Phys. Rev. Lett.
**2005**, 95, 226801. [Google Scholar] [CrossRef] [PubMed] - Huertas-Hernando, D.; Guinea, F.; Brataas, A. Spin-orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps. Phys. Rev. B
**2006**, 74, 155426. [Google Scholar] [CrossRef] - Yao, Y.; Ye, F.; Qi, X.-L.; Zhang, S.-C.; Fang, Z. Spin-orbit gap of graphene: First-principles calculations. Phys. Rev. B
**2007**, 75, 041401. [Google Scholar] [CrossRef] - Rashba, E.I. Graphene with structure-induced spin–orbit coupling: Spin-polarized states, spin zero modes, and quantum Hall effect. Phys. Rev. B
**2009**, 79, 161409(R). [Google Scholar] [CrossRef] - Katsnelson, M.; Geim, A. Electron scattering on microscopic corrugations in graphene. Philos. Trans. R. Soc. A
**2008**, 366, 195–204. [Google Scholar] [CrossRef] [PubMed] - Guinea, F.; Katsnelson, M.I.; Vozmediano, M.A.H. Midgap states and charge inhomogeneities in corrugated graphene. Phys. Rev. B
**2008**, 77, 075422. [Google Scholar] [CrossRef] - Pudlak, M.; Nazmitdinov, R.G. Klein collimation by rippled graphene superlattice. J. Phys. Condens. Matter
**2019**, 31, 495301. [Google Scholar] [CrossRef] - Ando, T. Spin-Orbit Interaction in Carbon Nanotubes. J. Phys. Soc. Jpn.
**2000**, 69, 1757–1763. [Google Scholar] [CrossRef] - Pichugin, K.N.; Pudlak, M.; Nazmitdinov, R.G. Spin-orbit effects in carbon nanotubes—Analytical results. Eur. Phys. J. B
**2014**, 87, 124. [Google Scholar] [CrossRef] - Smotlacha, J.; Pudlak, M.; Nazmitdinov, R.G. Spin transport in a rippled graphene periodic chain. J. Phys. Conf. Ser.
**2019**, 1416, 012035. [Google Scholar] [CrossRef] - Pudlak, M.; Smotlacha, J.; Nazmitdinov, R.G. On Symmetry Properties of The Corrugated Graphene System. Symmetry
**2020**, 12, 533. [Google Scholar] [CrossRef] - Feng, T.; Wang, Z.; Zhang, Z.; Xue, J.; Lu, H. Spin selectivity in chiral metal–halide semiconductors. Nanoscale
**2021**, 13, 18925–18940. [Google Scholar] [CrossRef] [PubMed] - Alyobi, M.M.; Barnett, C.J.; Rees, P.; Cobley, R.J. Modifying the electrical properties of graphene by reversible point-ripple formation. Carbon
**2019**, 143, 762–768. [Google Scholar] [CrossRef] - Vázquez de Parga, A.L.; Calleja, F.; Borca, B.; Passeggi, M.C.G.; Hinarejos, J.J.; Guinea, F.; Miranda, R. Periodically Rippled Graphene: Growth and Spatially Resolved Electronic Structure. Phys. Rev. Lett.
**2008**, 100, 056807. [Google Scholar] [CrossRef] [PubMed] - Ni, G.X.; Zheng, Y.; Bae, S.; Kim, H.R.; Pachoud, A.; Kim, Y.S.; Tan, C.L.; Im, D.; Ahn, J.H.; Hong, B.H. Quasi-Periodic Nanoripples in Graphene Grown by Chemical Vapor Deposition and Its Impact on Charge Transport. ACS Nano
**2012**, 6, 1158–1164. [Google Scholar] [CrossRef] [PubMed] - Vasić, B.; Zurutuza, A.; Gajić, R. Spatial variation of wear and electrical properties across wrinkles. in chemical vapour deposition graphene. Carbon
**2016**, 102, 304–310. [Google Scholar] [CrossRef] - Entin, M.V.; Magarill, L.I. Spin-orbit interaction of electrons on a curved surface. Phys. Rev. B
**2001**, 64, 085330. [Google Scholar] [CrossRef] - De Martino, A.; Egger, R.; Hallberg, K.; Balseiro, C.A. Spin-Orbit Coupling and Electron Spin Resonance Theory for Carbon Nanotubes. Phys. Rev. Lett.
**2002**, 88, 206402. [Google Scholar] [CrossRef] - Kuemmeth, F.; Ilani, S.; Ralph, D.C.; McEuen, P.L. Coupling of spin and orbital motion of electrons in carbon nanotubes. Nature
**2008**, 452, 448–452. [Google Scholar] [CrossRef] - Bulaev, D.V.; Trauzettel, B.; Loss, D. Spin-orbit interaction and anomalous spin relaxation. in carbon nanotube quantum dots. Phys. Rev. B
**2008**, 77, 235301. [Google Scholar] [CrossRef] - Chico, L.; López-Sancho, M.P.; Muñoz, M.C. Curvature-induced anisotropic spin–orbit splitting in carbon nanotubes. Phys. Rev. B
**2009**, 79, 235423. [Google Scholar] [CrossRef] - Jeong, V.; Lee, H.-W. Curvature-enhanced spin–orbit coupling in a carbon nanotube. Phys. Rev. B
**2009**, 80, 075409. [Google Scholar] [CrossRef] - Izumida, W.; Sato, K.; Saito, R. Spin–Orbit Interaction in Single Wall Carbon Nanotubes: Symmetry Adapted Tight-Binding Calculation and Effective Model Analysis. J. Phys. Soc. Jpn.
**2009**, 78, 074707. [Google Scholar] [CrossRef] - del Valle, M.; Margańska, M.; Grifoni, M. Signatures of spin–orbit interaction in transport properties. of finite carbon nanotubes in a parallel magnetic field. Phys. Rev. B
**2011**, 84, 165427. [Google Scholar] [CrossRef] - Klinovaja, J.; Schmidt, M.J.; Braunecker, B.; Loss, D. Carbon nanotubes in electric and magnetic fields. Phys. Rev. B
**2011**, 84, 085452. [Google Scholar] [CrossRef] - Wallace, P.R. The Band Theory of Graphite. Phys. Rev.
**1947**, 71, 622–634. [Google Scholar] [CrossRef] - Katsnelson, M.I. Graphene: Carbon in Two Dimensions; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Foa Torres, L.E.F.; Roche, S.; Charlier, J.-C. Introduction to Graphene-Based Nanomaterials: From Electronic Structure to Quantum Transport; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Pudlak, M.; Nazmitdinov, R.G. Spin-dependent electron transmission across the corrugated graphene. Physica E
**2020**, 118, 113846. [Google Scholar] [CrossRef] - Ghising, P.; Biswas, C.; Lee, Y.H. Graphene Spin Valves for Spin Logic Devices. Adv. Mater.
**2023**, 35, 2209137. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Sketch of the superlattice. The ballistic electron, coming from the left of the flat graphene sheet, incidents on the superlattice structure at an arbitrary angle $\phi $. (

**b**) Cross-section of the system that consists of two flat graphene sheets and superlattice. The two flat surfaces are the region L, defined in the intervals $-\infty <x<0$; and the region R, defined in the intervals $4NRcos{\theta}_{0}<x<\infty $. The region I (the concave arc) is a part of a nanotube of radius R, defined as $0<x<2Rcos{\theta}_{0}$. At ${\theta}_{0}=0$, the up surface is half that of the nanotube, while at ${\theta}_{0}=\pi /2$ the curvature does not exist. For the sake of analysis, we introduce the angle $\varphi =\pi -2{\theta}_{0}$. The region II (the convex arc with the radius R) is characterized by similar parameters to those of region I. Here, we have $-\infty <y<\infty $. To describe the scattering phenomenon, one has to define wave functions in different regions: flat (L,R) and curved (I, II) graphene surfaces.

**Figure 2.**The energy spectrum (14) as a function of the magnetic quantum number m. For a given energy E, the magnetic quantum number values ${m}_{-}$ and ${m}_{+}$ are determined from the crossing points of the dashed and solid lines by the horizontal line (E), presented as an example. The results are obtained at $R=12$ Å, $\varphi =\pi $, $\phi =\pi /6$. The values of spin–orbital strengths ${\lambda}_{x}\approx 0.267$ eV, ${\lambda}_{y}=0.00355$ eV (see Equation (10)) follow from the values of the parameters $\delta =0.01$, $p=0.1$, $\gamma =(4.5\times 1.42)$ eV·Å, ${\gamma}^{\prime}=\frac{8}{3}\gamma $.

**Figure 3.**The maximal spin-flip probabilities ${P}_{N,\phi}$ in the superlattice for various combinations $\{N,\phi \}$ (see discussion below) in the energy interval $E=0.02,\dots ,1$ eV for $R=12$ Å. For the sake of illustration, the points ${P}_{N,\phi}^{\{i\}}$ for corresponding energies ${E}_{i}=\mathsf{\Delta}E\times {N}_{i}$ ($\mathsf{\Delta}E=0.02$ eV, ${N}_{i}=1,\dots ,50$) are connected by a solid line.

**Figure 4.**The spin inversion probabilities ${P}_{N,\phi}\ge 0.99$ (yellow domain) as a function of number units $N=50,\dots ,450$ in the superlattice for angles $\phi ={5}^{\xb0},{5.01}^{\xb0},\dots ,{37}^{\xb0}$, for $R=12$ Å, at the incident energy beam $E=0.1$ eV (

**left**) and $E=1.0$ eV (

**right**).

**Figure 5.**Spin-flip points on the $N\times \phi $ mesh: (

**a**) $R=6$ Å, $E=0.1$ eV; (

**b**) $R=18$ Å, $E=0.6$ eV; (

**c**) $R=24$ Å, $E=0.8$ eV; (

**d**) $R=36$ Å, $E=1$ eV.

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**MDPI and ACS Style**

Buša, J.; Pudlák, M.; Nazmitdinov, R.
Rippled Graphene as an Ideal Spin Inverter. *Symmetry* **2023**, *15*, 1593.
https://doi.org/10.3390/sym15081593

**AMA Style**

Buša J, Pudlák M, Nazmitdinov R.
Rippled Graphene as an Ideal Spin Inverter. *Symmetry*. 2023; 15(8):1593.
https://doi.org/10.3390/sym15081593

**Chicago/Turabian Style**

Buša, Ján, Michal Pudlák, and Rashid Nazmitdinov.
2023. "Rippled Graphene as an Ideal Spin Inverter" *Symmetry* 15, no. 8: 1593.
https://doi.org/10.3390/sym15081593