Wave-particle duality of electrons with spin-momentum locking

Double-slit interference and single-slit diffraction effects of electrons on the surface of three-dimensional topological insulators


We investigate the effects of spin-momentum locking on the interference and diffraction patterns due to a double- or single-slit in an electronic Gedankenexperiment. We show that the inclusion of the spin-degree-of-freedom, when coupled to the motion direction of the carrier—a typical situation that occurs in systems with spin–orbit interaction—leads to a modification of the interference and diffraction patterns that depend on the geometrical parameters of the system.

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Data availability statement

This manuscript has associated data in a data repository. [Authors’ comment: All data generated or analysed during this study are included in this published article.]


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We acknowledge useful discussions with Alessandro De Martino, Erwann Bocquillon, M. Reyes Calvo, Geza Giedke, Andreas Inhofer, and Ingmar Swart. The work of TB and DB is supported by the Spanish Ministerio de Ciencia, Innovation y Universidades (MICINN) through the Project FIS2017-82804-P, and by the Transnational Common Laboratory Quantum-ChemPhys. DB thanks the University of Aix-Marseille for hosting him during the genesis of this work.

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Correspondence to Dario Bercioux.

Appendix A: On the diffraction formula

Appendix A: On the diffraction formula

In this appending, we present the derivation for the diffraction formula in the leading order in \(L^{-1}\) presented in Eq. (8) or (20). The main step is to expand all the \(\cos [(n-m)\varphi ]=\cos \ell \varphi \) into exponential functions so that

$$\begin{aligned} \mathcal {D}_\text {SE}&\longrightarrow \text {e}^{-\text {i} (N-1)\varphi } \left( \sum _{m=0}^N \text {e}^{\text {i} m \varphi }\right) ^2, \end{aligned}$$
$$\begin{aligned}&=\text {e}^{-\text {i} (N-1)\varphi } \left( \frac{1-\text {e}^{\text {i} N \varphi }}{1-\text {e}^{\text {i}\varphi }}\right) ^2, \end{aligned}$$
$$\begin{aligned}&= \text {e}^{\text {i} (N-1)\varphi }\left( \text {e}^{\text {i}\frac{(N-1)\varphi }{2}}\frac{\sin (N\varphi /2)}{\sin (\varphi /2)}\right) ^2, \end{aligned}$$
$$\begin{aligned}&= \left[ \frac{\sin \left( \frac{N\varphi }{2}\right) }{\sin \left( \frac{\varphi }{2}\right) }\right] ^2, \end{aligned}$$

where in (A.1a) we have used the properties of geometric series: \(\sum _{n=0}^N r^n=(1-r^{N+1})/(1-r)\). A similar expression is obtained starting by Eq. (19) with the introduction of the spin parameter \(\chi \).

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Bercioux, D., van den Berg, T.L., Ferraro, D. et al. Wave-particle duality of electrons with spin-momentum locking. Eur. Phys. J. Plus 135, 811 (2020). https://doi.org/10.1140/epjp/s13360-020-00837-3

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