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Wave-Particle Duality of Electrons with Spin-Momentum Locking Dario Bercioux, Tineke Van Den Berg, Dario Ferraro, Jerome Rech, Thibaut Jonckheere, Thierry Martin

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Wave-Particle Duality of Electrons with Spin-Momentum Locking Dario Bercioux, Tineke Van Den Berg, Dario Ferraro, Jerome Rech, Thibaut Jonckheere, Thierry Martin Wave-particle duality of electrons with spin-momentum locking Dario Bercioux, Tineke van den Berg, Dario Ferraro, Jerome Rech, Thibaut Jonckheere, Thierry Martin To cite this version: Dario Bercioux, Tineke van den Berg, Dario Ferraro, Jerome Rech, Thibaut Jonckheere, et al.. Wave- particle duality of electrons with spin-momentum locking. The European Physical Journal Plus, Springer, 2020, 135 (10), pp.811. 10.1140/epjp/s13360-020-00837-3. hal-03131659 HAL Id: hal-03131659 https://hal.archives-ouvertes.fr/hal-03131659 Submitted on 5 Feb 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Eur. Phys. Special Topics manuscript No. (will be inserted by the editor) 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 Dario Berciouxa,1,2, Tineke L. van den Bergb,1, Dario Ferraro3,4,5, Jérôme Rech3, Thibaut Jonckheere3, Thierry Martin3 1Donostia International Physics Center (DIPC), Manuel de Lardizbal 4, E-20018 San Sebastián, Spain 2IKERBASQUE, Basque Foundation of Science, 48011 Bilbao, Basque Country, Spain 3Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France 4Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy 5SPIN-CNR, Via Dodecaneso 33, 16146 Genova, Italy October 15, 2020 Abstract We investigate the effects of spin-momentum of Physics selected these experiments to be the most locking on the interference and diffraction patterns due beautiful ones in physics of the past century [7]. to a double- or single-slit in an electronic Gedankenex- Although this type of research has now been in large periment. We show that the inclusion of the spin-degree- part delegated to the educational framework [8,9], sev- of-freedom, when coupled to the motion direction of the eral groups in recent years tried to push the limits of the carrier — a typical situation that occurs in systems with understanding of the validity of the wave-particle dual- spin-orbit interaction — leads to a modification of the ity towards large quantum objects and molecules. One interference and diffraction patterns that depend on the of the most complex attempts was realized by consid- geometrical parameters of the system. ering interference and diffraction of large C60 molecu- les [10,11,12]. This experiment was a breakthrough in 1 Introduction the understanding of the limits of quantum theory be- cause the C60 molecule is close to being a classical ob- The wave-particle duality is one of the fundamental ject when considering its many excited internal degrees- paradigms introduced by quantum mechanics, which of-freedom and also the large possibility of coupling tells us that every particle or quantum entity may be to the environment that can lead to decoherence ef- described as either a particle or a wave [1]. In one of fects [10,13,14]. More recent experiments verified in- his “Lectures on physics” books, Feynman et al. [2] pro- terference effects in more complex molecular aggrega- posed to verify the wave nature of electrons by per- tes [15]. The problem of addressing the “wave-particle forming a thought experiment analogous to the one con- duality” rationally in the framework of complex quan- ducted by Thomas Young, performed in the first decade tum systems has been recently investigated by Carnio of the 1800s to show the wave nature of light. The first et al. [16]. experiment implementing the Young experiment with In the standard setup for the double-slit experi- electrons was realized by Jönsson [3,4] contemporane- ment, the spin degree-of-freedom of the electrons does ously with the preparation of Feynman’s lecture notes; not play any role. Some recent experiments focusing on these results were confirmed a few years later by a team electron beams carrying orbital angular momentum [17] of researchers at the University of Bologna [5]. In this showed the appearance of dislocations in the interfer- arXiv:2007.01021v3 [cond-mat.mes-hall] 14 Oct 2020 experiment, the two slits of the setup by Jönsson were ence pattern. Here we note a strong analogy with pho- substituted by a biprism. Further refinement came af- tons carrying an orbital angular momentum; in fact ter more than a decade with an experiment performed these can be generated by a diffraction grating contain- at the Hitachi lab by Tonomura et al. [6]. The read- ing a dislocation [18]. Recently, the role of Rashba spin- ers of the magazine “Physics World” of the Institute orbit interaction (RSOI) [19] was investigated in the ae-mail: [email protected] interference pattern in two-dimensional electron gases be-mail: [email protected] (2DEGs) [20] in a time-dependent fashion. The authors 2 conclude that the RSOI does not affect the interference a single slit for the case of spinless electrons. In Sec.3, pattern.However, from other investigations it is known we perform the same analysis by considering the surface spin-orbit interaction is of importance in various quan- state electrons of 3DTI. We first present the low-energy tum optics phenomena, such as in spin-dependent dou- physics of general 3DTIs, before moving on to the high- ble refraction and pumping [21,22]. energy physics of a more specific case valid for certain In this work, we focus on an extreme case of the 3DTIs. In Sec.4, we present a comparison between the results presented in Ref. [20] by considering the sur- different cases and show the presence of a finite correc- face states of three-dimensional topological insulators tion due to SML. We end the work with conclusions (3DTI). These surface states are described by an ef- and outlook in Sec.5. fective Hamiltonian similar to the one of a 2DEG with RSOI but with an infinite effective electron mass so that 2 Case of spinless electrons the parabolic kinetic term is zero [23]. In 2DEG with RSOI, the surface electron states are characterized by We start evaluating the interference and diffraction pat- the so-called spin-momentum locking (SML), i.e., the terns for SEs in a 2DEG. They are described by the spin orientation is strongly connected to the motion di- quadratic Hamiltonian: rection [24]. We study two fundamental mechanisms of interference: one arising from a wave passing through a p2 SE = + V (r); (1) double-slit opening and one by a single slit opening. In H 2m∗ the former case, we assume that the width of each slit where m∗ is the effective electron mass of the 2DEG is of the same order as the electron Fermi wavelength; under investigation. In this Hamiltonian, V (r) is a two- in the latter case, we relax this restriction. Convention- dimensional potential describing a wall with one or more ally, interference refers to the case of interaction of a few slits, given by the shape of the red lines in Fig.1. Even waves, whereas diffraction considers the case of a large though we do not explicitly need this potential in our number of interacting waves. Nevertheless, they repre- theory, as we work with point-like sources at the po- sent the same physical wave mechanisms. Contrary to sitions of the slits, some important physical considera- the results of Ref. [20], we work only with stationary tions must be kept in mind. The wall must have a width states, because the time-dependent part of the wave- w that is larger than the decay length ξ of the wave- function would lead only to an overall intensity pref- function inside the wall w > ξ, so that no tunneling actor [25]. Throughout this work, we compare the case will take place. Further, the edges have to be smooth, of spinless electrons (SEs), i.e., electrons that are spin- at least on the order of the long wavelength approxima- degenerate, with electrons with SML. We find that SML tion [21]. leads to a small but finite correction to the interference and diffraction pattern in comparison to a SE system. These corrections can be of the order of a few per cents 2.1 Interference from double-slit set-up and tend to disappear in the so-called far-field limit whereas they are more significant in the opposite near- We consider first a double-slit setup: these are separated field limit. Our results fall in the framework of the so- by a distance d and there is a distance L between the called electron quantum optics [26] for ballistic chiral slits and an observation plane. Here, we consider the conductors. Extensions to the case of the edge states case in which the opening of each single slit h λF, ∼ of two-dimensional topological insulators have already where λF = 2π=kF is the Fermi wavelength associated 2 2 ∗ been proposed [27,28,29,30,31]. The present work rep- with the electron Fermi energy F = ~ kF=2m . We resents an extension of electron quantum optics to the will relax this condition when weE consider the diffrac- realm of topological surface states. tion pattern in the next section. The two slits opening The mechanism of modification of the interference S1=2 are the source of two circular plane waves that patterns we study in this work plays an important role propagate to the observation point P on the screen at a when studying quasi-particle interference via scanning distance L — see Fig.1(a).
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