# Totally Spin-Polarized Currents in an Interferometer with Spin–Orbit Coupling and the Absence of Magnetic Field Effects

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## Abstract

**:**

## 1. Introduction

## 2. Interferometer Design and Model Description

## 3. Current and Conductance Calculation

**c**) shows the conductance for a small value of the distance between contacts, $R=10a$, and $\beta ={10}^{-2}t$. In this configuration, the value of $\alpha $ that satisfies the full spin polarization conditions is $\alpha =0.158t$. The Rashba SOC $\alpha $ manipulation ensures that when the conductance for one spin is zero, the conductance for the other could be very close to the maximum possible value, such that the device operates with a large and fully polarized current.

**a**), and the spin polarization p, panel (

**b**), as a function of the Rashba SOC intensity $\alpha $, for the parameters $R=100a$, ${t}^{\prime}=t$, and $\beta ={10}^{-2}t$. The Fermi level is assumed to be ${\u03f5}_{F}=9.422\times {10}^{-2}t$ and corresponds to the first positive value where the spin-down conductance is zero, as shown in Figure 3b. Analyzing the results, we see that by increasing the Rashba SOC intensity and keeping the other parameters fixed, ${\mathcal{G}}^{\downarrow}$ and ${\mathcal{G}}^{\uparrow}$ change their value until ${\mathcal{G}}^{\uparrow}$ becomes close to ${e}^{2}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}h$ and ${\mathcal{G}}^{\downarrow}=0$, when $\alpha =1.212\times {10}^{-2}t$. This $\alpha $ value satisfies the condition imposed by Equation (13), reproducing the maximum of ${\mathcal{G}}^{\uparrow}$ and the minimum of ${\mathcal{G}}^{\downarrow}$ as shown in Figure 3b. We notice that by increasing $\alpha $, the ${\mathcal{G}}^{{\sigma}_{r}}$ oscillates and other points of maximum polarization appear with conductance near ${e}^{2}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}h$, which corresponds to other solutions of Equation (13). In panel (

**b**), when $\alpha $ satisfies Equation (13), we observe that p alternatively takes the values of +1 or −1. This figure illustrates that, by changing the Rashba SOC intensity, it is also possible to tune the system so that for one spin the conductance is zero while for the other the results are very close to the maximum possible value.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SOC | Spin–Orbit Coupling |

QPC | Quantum Point Contact |

CISS | Chiral-Induced Spin Selective |

1DLSOC | One-Dimensional Lead with Spin–Orbit Coupling |

## References

- Wolf, S.A.; Awschalom, D.D.; Buhrman, R.A.; Daughton, J.M.; von Molnár, S.; Roukes, M.L.; Chtchelkanova, A.Y.; Treger, D.M. Spintronics: A Spin-Based Electronics Vision for the Future. Science
**2001**, 294, 1488–1495. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Žutić, I.; Fabian, J.; Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod. Phys.
**2004**, 76, 323–410. [Google Scholar] [CrossRef] [Green Version] - Chappert, C.; Fert, A.; Van Dau, F.N. The emergence of spin electronics in data storage. Nat. Mater.
**2007**, 6, 813–823. [Google Scholar] [CrossRef] - Hirohata, A.; Takanashi, K. Future perspectives for spintronic devices. J. Phys. D Appl. Phys.
**2014**, 47, 193001. [Google Scholar] [CrossRef] - Žutić, I.; Matos-Abiague, A.; Scharf, B.; Dery, H.; Belashchenko, K. Proximitized materials. Mater. Today
**2019**, 22, 85–107. [Google Scholar] [CrossRef] - Burkard, G.; Loss, D.; DiVincenzo, D.P. Coupled quantum dots as quantum gates. Phys. Rev. B
**1999**, 59, 2070–2078. [Google Scholar] [CrossRef] [Green Version] - Chiappe, G.; Anda, E.V.; Costa Ribeiro, L.; Louis, E. Kondo regimes in a three-dots quantum gate. Phys. Rev. B
**2010**, 81, 041310. [Google Scholar] [CrossRef] - Ladd, T.D.; Jelezko, F.; Laflamme, R.; Nakamura, Y.; Monroe, C.; O’Brien, J.L. Quantum computers. Nature
**2010**, 464, 45–53. [Google Scholar] [CrossRef] [Green Version] - Manchon, A.; Koo, H.C.; Nitta, J.; Frolov, S.M.; Duine, R.A. New perspectives for Rashba spin–orbit coupling. Nat. Mater.
**2015**, 14, 871–882. [Google Scholar] [CrossRef] [Green Version] - Datta, S.; Das, B. Electronic analog of the electro-optic modulator. Appl. Phys. Lett.
**1990**, 56, 665–667. [Google Scholar] [CrossRef] - Schliemann, J.; Egues, J.C.; Loss, D. Nonballistic Spin-Field-Effect Transistor. Phys. Rev. Lett.
**2003**, 90, 146801. [Google Scholar] [CrossRef] [Green Version] - Cartoixà, X.; Ting, D.Z.Y.; Chang, Y.C. A resonant spin lifetime transistor. Appl. Phys. Lett.
**2003**, 83, 1462–1464. [Google Scholar] [CrossRef] [Green Version] - Kunihashi, Y.; Kohda, M.; Sanada, H.; Gotoh, H.; Sogawa, T.; Nitta, J. Proposal of spin complementary field effect transistor. Appl. Phys. Lett.
**2012**, 100, 113502. [Google Scholar] [CrossRef] - Kohda, M.; Salis, G. Physics and application of persistent spin helix state in semiconductor heterostructures. Semicond. Sci. Technol.
**2017**, 32, 073002. [Google Scholar] [CrossRef] - Bernevig, B.A.; Orenstein, J.; Zhang, S.C. Exact SU(2) Symmetry and Persistent Spin Helix in a Spin-Orbit Coupled System. Phys. Rev. Lett.
**2006**, 97, 236601. [Google Scholar] [CrossRef] [Green Version] - Bardarson, J.H. A proof of the Kramers degeneracy of transmission eigenvalues from antisymmetry of the scattering matrix. J. Phys. A Math. Theor.
**2008**, 41, 405203. [Google Scholar] [CrossRef] [Green Version] - Aronov, A.G.; Lyanda-Geller, Y.B. Spin-orbit Berry phase in conducting rings. Phys. Rev. Lett.
**1993**, 70, 343–346. [Google Scholar] [CrossRef] - Larsen, M.H.; Lunde, A.M.; Flensberg, K. Conductance of Rashba spin-split systems with ferromagnetic contacts. Phys. Rev. B
**2002**, 66, 033304. [Google Scholar] [CrossRef] [Green Version] - Sugahara, S.; Tanaka, M. A spin metal–oxide–semiconductor field-effect transistor using half-metallic-ferromagnet contacts for the source and drain. Appl. Phys. Lett.
**2004**, 84, 2307–2309. [Google Scholar] [CrossRef] [Green Version] - Aharony, A.; Tokura, Y.; Cohen, G.Z.; Entin-Wohlman, O.; Katsumoto, S. Filtering and analyzing mobile qubit information via Rashba–Dresselhaus–Aharonov–Bohm interferometers. Phys. Rev. B
**2011**, 84, 035323. [Google Scholar] [CrossRef] - Shmakov, P.M.; Dmitriev, A.P.; Kachorovskii, V.Y. High-temperature Aharonov-Bohm-Casher interferometer. Phys. Rev. B
**2012**, 85, 075422. [Google Scholar] [CrossRef] [Green Version] - Shmakov, P.M.; Dmitriev, A.P.; Kachorovskii, V.Y. Aharonov-Bohm conductance of a disordered single-channel quantum ring. Phys. Rev. B
**2013**, 87, 235417. [Google Scholar] [CrossRef] [Green Version] - Nagasawa, F.; Takagi, J.; Kunihashi, Y.; Kohda, M.; Nitta, J. Experimental Demonstration of Spin Geometric Phase: Radius Dependence of Time-Reversal Aharonov-Casher Oscillations. Phys. Rev. Lett.
**2012**, 108, 086801. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Saarikoski, H.; Reynoso, A.A.; Baltanás, J.P.; Frustaglia, D.; Nitta, J. Spin interferometry in anisotropic spin-orbit fields. Phys. Rev. B
**2018**, 97, 125423. [Google Scholar] [CrossRef] [Green Version] - Nagasawa, F.; Reynoso, A.A.; Baltanás, J.P.; Frustaglia, D.; Saarikoski, H.; Nitta, J. Gate-controlled anisotropy in Aharonov-Casher spin interference: Signatures of Dresselhaus spin-orbit inversion and spin phases. Phys. Rev. B
**2018**, 98, 245301. [Google Scholar] [CrossRef] [Green Version] - Ringer, S.; Rosenauer, M.; Völkl, T.; Kadur, M.; Hopperdietzel, F.; Weiss, D.; Eroms, J. Spin field-effect transistor action via tunable polarization of the spin injection in a Co/MgO/graphene contact. Appl. Phys. Lett.
**2018**, 113, 132403. [Google Scholar] [CrossRef] [Green Version] - Lopes, V.; Anda, E.V. Spin polarized current in a quantum dot connected to a spin-orbit interacting Fermi sea. J. Phys. Chem. Solids
**2019**, 128, 188–195. [Google Scholar] [CrossRef] - Jonson, M.; Shekhter, R.I.; Entin-Wohlman, O.; Aharony, A.; Park, H.C.; Radić, D. DC spin generation by junctions with AC driven spin-orbit interaction. Phys. Rev. B
**2019**, 100, 115406. [Google Scholar] [CrossRef] [Green Version] - Debray, P.; Rahman, S.M.S.; Wan, J.; Newrock, R.S.; Cahay, M.; Ngo, A.T.; Ulloa, S.E.; Herbert, S.T.; Muhammad, M.; Johnson, M. All-electric quantum point contact spin-polarizer. Nat. Nanotechnol.
**2009**, 4, 759–764. [Google Scholar] [CrossRef] - Das, P.P.; Cahay, M.; Kalita, S.; Mal, S.S.; Jha, A.K. Width dependence of the 0.5×(2e
^{2}/h) conductance plateau in InAs quantum point contacts in presence of lateral spin-orbit coupling. Sci. Rep.**2019**, 9, 12172. [Google Scholar] [CrossRef] - Qin, Z.; Qin, G.; Shao, B.; Zuo, X. Rashba spin splitting and perpendicular magnetic anisotropy of Gd-adsorbed zigzag graphene nanoribbon modulated by edge states under external electric fields. Phys. Rev. B
**2020**, 101, 014451. [Google Scholar] [CrossRef] - Chico, L.; Latgé, A.; Brey, L. Symmetries of quantum transport with Rashba spin–orbit: Graphene spintronics. Phys. Chem. Chem. Phys.
**2015**, 17, 16469–16475. [Google Scholar] [CrossRef] [Green Version] - Farghadan, R.; Saffarzadeh, A. Generation of fully spin-polarized currents in three-terminal graphene-based transistors. RSC Adv.
**2015**, 5, 87411–87415. [Google Scholar] [CrossRef] [Green Version] - Santos, H.; Latgé, A.; Brey, L.; Chico, L. Spin-polarized currents in corrugated graphene nanoribbons. Carbon
**2020**, 168, 1–11. [Google Scholar] [CrossRef] - Sarkar, K.; Aharony, A.; Entin-Wohlman, O.; Jonson, M.; Shekhter, R.I. Effects of magnetic fields on the Datta-Das spin field-effect transistor. Phys. Rev. B
**2020**, 102, 115436. [Google Scholar] [CrossRef] - Ngo, A.T.; Debray, P.; Ulloa, S.E. Lateral spin-orbit interaction and spin polarization in quantum point contacts. Phys. Rev. B
**2010**, 81, 115328. [Google Scholar] [CrossRef] [Green Version] - Wan, J.; Cahay, M.; Debray, P.; Newrock, R. Possible origin of the 0.5 plateau in the ballistic conductance of quantum point contacts. Phys. Rev. B
**2009**, 80, 155440. [Google Scholar] [CrossRef] [Green Version] - Eto, M.; Hayashi, T.; Kurotani, Y. Spin Polarization at Semiconductor Point Contacts in Absence of Magnetic Field. J. Phys. Soc. Jpn.
**2005**, 74, 1934–1937. [Google Scholar] [CrossRef] [Green Version] - Liu, Y.; Xiao, J.; Koo, J.; Yan, B. Chirality-driven topological electronic structure of DNA-like materials. Nat. Mater.
**2021**, 20, 638–644. [Google Scholar] [CrossRef] - Reynoso, A.; Usaj, G.; Balseiro, C.A. Detection of spin polarized currents in quantum point contacts via transverse electron focusing. Phys. Rev. B
**2007**, 75, 085321. [Google Scholar] [CrossRef] - Aharonov, Y.; Bohm, D. Significance of Electromagnetic Potentials in the Quantum Theory. Phys. Rev.
**1959**, 115, 485–491. [Google Scholar] [CrossRef] [Green Version] - Nitta, J.; Meijer, F.E.; Takayanagi, H. Spin-interference device. Appl. Phys. Lett.
**1999**, 75, 695–697. [Google Scholar] [CrossRef] - Meijer, F.E.; Morpurgo, A.F.; Klapwijk, T.M. One-dimensional ring in the presence of Rashba spin-orbit interaction: Derivation of the correct Hamiltonian. Phys. Rev. B
**2002**, 66, 033107. [Google Scholar] [CrossRef] [Green Version] - Splettstoesser, J.; Governale, M.; Zülicke, U. Persistent current in ballistic mesoscopic rings with Rashba spin-orbit coupling. Phys. Rev. B
**2003**, 68, 165341. [Google Scholar] [CrossRef] [Green Version] - Sheng, J.S.; Chang, K. Spin states and persistent currents in mesoscopic rings: Spin-orbit interactions. Phys. Rev. B
**2006**, 74, 235315. [Google Scholar] [CrossRef] [Green Version] - Sheng, J.S.; Chang, K. Spin states and persistent currents in a quantum ring with an embedded magnetic impurity. J. Phys. Condens. Matter
**2007**, 20, 025222. [Google Scholar] [CrossRef] [Green Version] - Berche, B.; Chatelain, C.; Medina, E. Mesoscopic rings with spin-orbit interactions. Eur. J. Phys.
**2010**, 31, 1267–1286. [Google Scholar] [CrossRef] [Green Version] - Frustaglia, D.; Nitta, J. Geometric spin phases in Aharonov-Casher interference. Solid State Commun.
**2020**, 311, 113864. [Google Scholar] [CrossRef] - Zainagutdinov, A.; Telezhnikov, A.; Maksimova, G. Aharonov-Bohm nanoring with periodically modulated Rashba interaction: Energy spectrum and persistent currents. Phys. Lett. A
**2022**, 430, 127972. [Google Scholar] [CrossRef] - Kenmoe, M.B.; Kayanuma, Y. Transmission of a single electron through a Berry ring. Phys. Rev. B
**2022**, 105, 155117. [Google Scholar] [CrossRef] - Kozin, V.K.; Iorsh, I.V.; Kibis, O.V.; Shelykh, I.A. Quantum ring with the Rashba spin-orbit interaction in the regime of strong light-matter coupling. Phys. Rev. B
**2018**, 97, 155434. [Google Scholar] [CrossRef] [Green Version] - Tong, J.; Luo, F.; Ruan, L.; Qin, G.; Zhou, L.; Tian, F.; Zhang, X. High and reversible spin polarization in a collinear antiferromagnet. Appl. Phys. Rev.
**2020**, 7, 031405. [Google Scholar] [CrossRef] - Lopes, V.; Martins, G.B.; Manya, M.A.; Anda, E.V. Kondo effect under the influence of spin–orbit coupling in a quantum wire. J. Phys. Condens. Matter
**2020**, 32, 435604. [Google Scholar] [CrossRef] - Okuda, T.; Miyamoto, K.; Takeichi, Y.; Miyahara, H.; Ogawa, M.; Harasawa, A.; Kimura, A.; Matsuda, I.; Kakizaki, A.; Shishidou, T.; et al. Large out-of-plane spin polarization in a spin-splitting one-dimensional metallic surface state on Si(557)-Au. Phys. Rev. B
**2010**, 82, 161410. [Google Scholar] [CrossRef] [Green Version] - Park, J.; Jung, S.W.; Jung, M.C.; Yamane, H.; Kosugi, N.; Yeom, H.W. Self-Assembled Nanowires with Giant Rashba Split Bands. Phys. Rev. Lett.
**2013**, 110, 036801. [Google Scholar] [CrossRef] [Green Version] - Takayama, A.; Sato, T.; Souma, S.; Oguchi, T.; Takahashi, T. One-Dimensional Edge States with Giant Spin Splitting in a Bismuth Thin Film. Phys. Rev. Lett.
**2015**, 114, 066402. [Google Scholar] [CrossRef] [Green Version] - Brand, C.; Pfnür, H.; Landolt, G.; Muff, S.; Dil, J.H.; Das, T.; Tegenkamp, C. Observation of correlated spin–orbit order in a strongly anisotropic quantum wire system. Nat. Commun.
**2015**, 6, 8118. [Google Scholar] [CrossRef] [Green Version] - Tanaka, T.; Gohda, Y. First-principles prediction of one-dimensional giant Rashba splittings in Bi-adsorbed In atomic chains. Phys. Rev. B
**2018**, 98, 241409. [Google Scholar] [CrossRef] [Green Version] - Kopciuszynski, M.; Stepniak-Dybala, A.; Dachniewicz, M.; Zurawek, L.; Krawiec, M.; Zdyb, R. Hut-shaped lead nanowires with one-dimensional electronic properties. Phys. Rev. B
**2020**, 102, 125415. [Google Scholar] [CrossRef] - Mihalyuk, A.N.; Chou, J.P.; Eremeev, S.V.; Zotov, A.V.; Saranin, A.A. One-dimensional Rashba states in Pb atomic chains on a semiconductor surface. Phys. Rev. B
**2020**, 102, 035442. [Google Scholar] [CrossRef] - Han, J.; Zhang, A.; Chen, M.; Gao, W.; Jiang, Q. Giant Rashba splitting in one-dimensional atomic tellurium chains. Nanoscale
**2020**, 12, 10277–10283. [Google Scholar] [CrossRef] [PubMed] - Żurawek, L.; Kopciuszyński, M.; Dachniewicz, M.; Stróżak, M.; Krawiec, M.; Jałochowski, M.; Zdyb, R. Partially embedded Pb chains on a vicinal Si(113) surface. Phys. Rev. B
**2020**, 101, 195434. [Google Scholar] [CrossRef] - Gerstmann, U.; Vollmers, N.J.; Lücke, A.; Babilon, M.; Schmidt, W.G. Rashba splitting and relativistic energy shifts in In/Si(111) nanowires. Phys. Rev. B
**2014**, 89, 165431. [Google Scholar] [CrossRef] [Green Version] - Nakamura, T.; Ohtsubo, Y.; Tokumasu, N.; Le Fèvre, P.; Bertran, F.; Ideta, S.I.; Tanaka, K.; Kuroda, K.; Yaji, K.; Harasawa, A.; et al. Giant Rashba system on a semiconductor substrate with tunable Fermi level: Bi/GaSb(110)--(2×1). Phys. Rev. Mater.
**2019**, 3, 126001. [Google Scholar] [CrossRef] [Green Version] - Ohtsubo, Y.; Tokumasu, N.; Watanabe, H.; Nakamura, T.; Le Fevre, P.; Bertran, F.; Imamura, M.; Yamamoto, I.; Azuma, J.; Takahashi, K.; et al. One-dimensionality of the spin-polarized surface conduction and valence bands of quasi-one-dimensional Bi chains on GaSb(110)-(2×1). Phys. Rev. B
**2020**, 101, 235306. [Google Scholar] [CrossRef] - Nakamura, T.; Ohtsubo, Y.; Yamashita, Y.; Ideta, S.i.; Tanaka, K.; Yaji, K.; Harasawa, A.; Shin, S.; Komori, F.; Yukawa, R.; et al. Giant Rashba splitting of quasi-one-dimensional surface states on Bi/InAs(110)-(2×1). Phys. Rev. B
**2018**, 98, 075431. [Google Scholar] [CrossRef] [Green Version] - Quay, C.H.L.; Hughes, T.L.; Sulpizio, J.A.; Pfeiffer, L.N.; Baldwin, K.W.; West, K.W.; Goldhaber-Gordon, D.; de Picciotto, R. Observation of a one-dimensional spin–orbit gap in a quantum wire. Nat. Phys.
**2010**, 6, 336–339. [Google Scholar] [CrossRef] [Green Version] - Heedt, S.; Traverso Ziani, N.; Crépin, F.; Prost, W.; Trellenkamp, S.; Schubert, J.; Grützmacher, D.; Trauzettel, B.; Schäpers, T. Signatures of interaction-induced helical gaps in nanowire quantum point contacts. Nat. Phys.
**2017**, 13, 563–567. [Google Scholar] [CrossRef] - Zubarev, D.N. Double-time green functions in statistical physics. Sov. Phys. Uspekhi
**1960**, 3, 320–345. [Google Scholar] [CrossRef] - Keldysh, L.V. Diagram technique for nonequilibrium processes. Sov. Phys. JETP
**1965**, 20, 1018–1026. [Google Scholar]

**Figure 1.**Diagram of the interferometer composed of a one-dimensional lead on the SOC side coupled through two contacts to a metallic conductor.

**Figure 2.**Dispersion relation ${\u03f5}_{k{\sigma}_{r}}$ as a function of k and spin ${\sigma}_{r}$. The SOC intensities are taken to be $\alpha =\beta =1t$. The colored shadows illustrate, for each spin and ${\u03f5}_{F,R}<\u03f5<{\u03f5}_{F,L}$, the electronic states that participate in the current.

**Figure 3.**The conductance ${\mathcal{G}}^{{\sigma}_{r}}$ for each spin ${\sigma}_{r}$, panels (

**a**–

**c**), and the spin polarization of the current p, panel (

**d**), as a function of the Fermi energy ${\u03f5}_{F}$. Panels (

**a**,

**b**,

**d**) correspond to a system with $R=100a$, $\alpha =1.212\times {10}^{-2}t$ and $\beta ={10}^{-2}t$, while in panel (

**c**) the parameters are $R=10a$, $\alpha =0.158t$, and $\beta ={10}^{-2}t$. For both configurations, ${t}^{\prime}=t$.

**Figure 4.**The conductance ${\mathcal{G}}^{{\sigma}_{r}}$, panel (

**a**), and the spin polarization of the current p, panel (

**b**), as a function of the Rashba SOC intensity $\alpha $. The distance between contacts is $R=100a$ and the Dresselhaus SOC parameter is $\beta ={10}^{-2}t$. The Fermi level is given by Equation (12), ${\u03f5}_{F}=9.422\times {10}^{-2}t$, and when $\alpha $ satisfies Equation (13), ${\mathcal{G}}^{{\sigma}_{r}}$ is completely spin-polarized.

**Figure 5.**Electronic current ${J}^{{\sigma}_{r}}$ injected into the metallic conductor as a function of the applied potential V. As in Figure 4, the distance between contacts is $R=100a$ and the Rashba and Dresselhaus SOC parameters are $\alpha =1.212\times {10}^{-2}t$ and $\beta ={10}^{-2}t$, respectively. The Fermi level is taken to be ${\u03f5}_{F}=9.422\times {10}^{-2}t$, value for which the spin-down conductance is zero, and $t=1e$V. The black dotted line corresponds to the slope $e\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}h$.

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**MDPI and ACS Style**

Lopes, V.; Chiappe, G.; Ribeiro, L.C.; Anda, E.V.
Totally Spin-Polarized Currents in an Interferometer with Spin–Orbit Coupling and the Absence of Magnetic Field Effects. *Nanomaterials* **2022**, *12*, 4082.
https://doi.org/10.3390/nano12224082

**AMA Style**

Lopes V, Chiappe G, Ribeiro LC, Anda EV.
Totally Spin-Polarized Currents in an Interferometer with Spin–Orbit Coupling and the Absence of Magnetic Field Effects. *Nanomaterials*. 2022; 12(22):4082.
https://doi.org/10.3390/nano12224082

**Chicago/Turabian Style**

Lopes, Victor, Guillermo Chiappe, Laercio C. Ribeiro, and Enrique V. Anda.
2022. "Totally Spin-Polarized Currents in an Interferometer with Spin–Orbit Coupling and the Absence of Magnetic Field Effects" *Nanomaterials* 12, no. 22: 4082.
https://doi.org/10.3390/nano12224082