Persistent charge and spin currents in a quantum ring using Green׳s function technique: Interplay between magnetic flux and spin–orbit interactions

doi:10.1016/j.physe.2014.07.024

Physica E: Low-dimensional Systems and Nanostructures

Volume 64, November 2014, Pages 169-177

https://doi.org/10.1016/j.physe.2014.07.024 Get rights and content

Highlights

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A new approach based on Green׳s function formalism is given to determine persistent currents.
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Like conventional methods, evaluations of eigenvalues and eigenfunctions are not required.
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Magnetic properties of large molecular rings encountered in biopolymers can be investigated.
•
Persistent currents in large size rings can be determined with a very high degree of accuracy.

Abstract

We put forward a new approach based on Green׳s function formalism to evaluate precisely persistent charge and spin currents in an Aharonov–Bohm ring subjected to Rashba and Dresselhaus spin–orbit interactions. Unlike conventional methods our present scheme circumvents direct evaluation of eigenvalues and eigenstates of the system Hamiltonian to determine persistent currents which essentially reduces possible numerical errors, especially for larger rings. The interplay of Aharonov–Bohm flux and spin–orbit interactions in persistent charge and spin currents of quantum rings is analyzed in detail and our results lead to a possibility of estimating the strength of any one of the spin–orbit fields provided the other one is known. All these features are exactly invariant even in the presence of impurities, and therefore, can be substantiated experimentally.

Graphical abstract

Introduction

The promise of new technological breakthroughs has been a major driving force for studying transport in meso- and nano-structures whose dimensions are comparable to and even smaller than the mean free paths or wavelengths of electrons [1], [2], [3]. Simultaneously, inspection of electronic transport in low-dimensional systems comprising simple and complex structures has brought up several new underlying questions. The progress in experimental techniques has allowed for systematic investigations of artificially made nanostructures whose transport properties are affected or even governed by quantum effects and this makes it possible to perform experiments that directly probe quantum properties of phase coherent many-body systems.

The appearance of circular currents, induced by external magnetic fields in isolated (no source and drain electrodes) quantum rings, commonly known as persistent currents, is an astonishing quantum effect which reveals the significance of phase coherence of electronic wave functions in low-dimensional quantum systems. The phenomenon of persistent current in normal metal rings in the presence of Aharonov–Bohm (AB) flux ϕ has been first exposed [4] in the early 60s, and then, in 1983 Büttiker et al. [5] have successfully revived it and they have established that an isolated normal metal mesoscopic ring threaded by an AB flux ϕ carries an equilibrium current which does not decay over time and circulates within the sample. Following this pioneering work, interest in this subject has rapidly picked up with substantial theoretical [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] and experimental [20], [21], [22], [23] works. Still, many open questions persist in this particular issue. For instance, persistent current examined in disordered rings is considerably larger than the corresponding theoretical predictions [23], [24], [25]. In 2009, Bluhm et al. [26] have made in situ measurements using scanning SQUID microscope for studying magnetic properties of 33 discrete mesoscopic gold rings, taking one ring at a time. Their experimental results fit reasonably well with the theoretically predicted value [6] only in an ensemble of 16 nearly ballistic rings [24] and in a single ballistic ring [25]. But, the current amplitudes in single isolated diffusive gold rings [23] are still order of magnitude larger than the theoretical predictions. In the presence of electron–electron (e–e) interaction and disorder an explanation has been proposed [7], [27], [28], [29], based on a perturbative calculation, which reveals persistent currents with greater amplitude compared to the non-interacting case, but still off by an order of magnitude. Moreover, the origin of the e–e interaction parameters taken into account in the theory is not suitably unraveled. Thus, it demands further studies to resolve these controversial issues.

Another challenging topic is the possible existence of a spin current [30] in mesoscopic rings with spin–orbit (SO) interaction, even in the absence of a magnetic field. This phenomenon may be observed through the recently developed Doppler and spin modulation relaxation techniques [31], [32], [33]. The SO interaction is a rudimentary mechanism that is manifested in several fascinating properties that pertain to the anticipations of semiconducting structures as potential quantum devices. In conventional semiconducting materials two typical SO interactions are encountered. One is called as Rashba spin–orbit interaction (RSOI) [34] and the other is named as Dresselhaus spin–orbit interaction (DSOI) [35]. The previous one is associated with electric field that is generated from the structural inversion asymmetry at interfaces, while the later results from the bulk inversion asymmetry [36]. An additional contribution can also arise from surface anisotropies [37] together with simple Rashba SO interaction, which is associated with the interfacial electric field normal to the surface that results from the band offset at the interface of two different semiconductors. Quantum rings – ring-shaped quantum wells fabricated at such heterojunctions – comprise such anisotropies are exemplary candidates for examining SO coupling effects in persistent currents. Note that if one of the components that make the interface is characterized by bulk asymmetry, a corresponding contribution of the DSOI will also exist at the interface [38]. In such quantum rings electronic transport will exhibit the interplay between these different contributions to the SO coupling at the interface. A sizable amount of related theoretical work has already revealed the distinctive features of persistent charge and spin currents in mesoscopic rings subjected to Rashba and Dresselhaus SO fields [30], [39], [40], [41], however a well defined methodology for the prediction of persistent current in large samples is still missing and the magneto-transport properties of such structures are fascinating and remain controversial. The accurate determination of persistent current in such systems, in the presence of an AB flux, is a route for analyzing its magnetic properties.

A clear understanding of the role played by SO interactions in the phenomena of persistent charge and spin currents necessitates proper estimation of the strength of these interactions. The Rashba SO interaction which is controlled by an external gate voltage placed in the vicinity of the sample [36], [37] can be determined by the structure of the interface. This yields, in principle, a wide range of possible values of RSOI and its determination in any given material is crucial [42]. The feasible routes of measuring the strength of DSOI are mainly based on the photo-galvanic methods [43], measurement of electrical conductance of nano-wires [44], and an optical monitoring of the spin precession of the electrons [45]. A unitary transformation has been explored [46], [47] which brings out a hidden symmetry, when applied to the SO Hamiltonian, that has been used to establish that by making the strengths of the two SO interactions equal one achieves a zero spin current in the material [48], and this vanishing spin current is a robust effect which is observed even in the presence of disorder [48], and thus, can be established experimentally. Observing the persistent charge current [49] one can estimate the strength of DSOI, and, by monitoring the vanishing of persistent spin current one can determine both the RSOI and DSOI in a single mesoscopic ring [48], [50].

The established approach to the determination of persistent charge [6], [40], [49], [51], [52], [53], [54], [55], [56] and spin [30], [40], [50], [57] currents in isolated conducting rings is based on the evaluation of eigenvalues and eigenvectors of the system Hamiltonian. For large size rings such an approach becomes highly numerically unreliable, and most importantly – hard to speculate in the presence of interaction with external baths. Here we propose a new approach, based on Green׳s function formalism, that circumvents the need to evaluate system eigenvalues and eigenfunctions. In particular, this Green׳s function methodology for determining persistent currents should give us access to evaluation of the magnetic properties of large conducting rings as well as molecular rings encountered in biopolymers. We firmly believe that the Green׳s function technique will yield persistent charge and spin currents a very high degree of accuracy, and this will definitely make it possible to consider the interplay between molecular structure and geometry and the resulting persistent currents obtained in the presence of an AB flux ϕ and SO interactions.

The rest of the paper is organized as follows. In Section 2, the model quantum system and the calculation method are described. In Section 3, the numerical results are presented which describe the (i) behavior of persistent charge current, (ii) characteristic features of persistent spin current, and (iii) possible route of estimating the strength of RSOI and DSOI in a single mesoscopic ring. Finally, in Section 4, we summarize our essential results.

Section snippets

Model and hamiltonian

We consider a mesoscopic ring which is subjected to both Rashba and Dresselhaus SO fields. The ring is threaded by an AB flux ϕ which is measured in unit of $ϕ_{0} = ch / e$ , the elementary flux-quantum. A schematic view of this ring is illustrated in Fig. 1. The TB Hamiltonian of such a N-site ring in the site representation reads as [48], [49], [50] $H_{R} = \sum_{n} c_{n}^{†} ϵ_{n} c_{n} + \sum_{n} (c_{n + 1}^{†} t e^{i θ} c_{n} + h . c .) - \sum_{n} (c_{n + 1}^{†} (i σ_{x}) α \cos φ_{n, n + 1} e^{i θ} c_{n} + h . c .) - \sum_{n} (c_{n + 1}^{†} (i σ_{y}) α \sin φ_{n, n + 1} e^{i θ} c_{n} + h . c .) + \sum_{n} (c_{n + 1}^{†} (i σ_{y}) β \cos φ_{n, n + 1} e^{i θ} c_{n} + h . c .) + \sum_{n} (c_{n + 1}^{†} (i σ_{x}) β \sin$

Numerical results and discussion

In what follows, we will present numerical results computed for circulating charge and spin currents in mesoscopic rings based on Green׳s function formalism. In all calculations we measure the energy scale in unit of the hopping integral t which is set equal to 1. The Rashba and Dresselhaus SO coupling strengths are also scaled in unit of this hopping parameter t. Throughout the numerical analysis we restrict ourselves to absolute zero temperature and fix $c = h = e = 1$ .

First, we focus on the

Summary and outlook

In the present work, we have proposed a new approach based on Green׳s function formalism within a tight-binding framework to evaluate precisely the persistent charge and spin currents in spin–orbit interaction induced AB rings. The essential results are summarized as follows.

As already pointed out that the standard methodology to the determination of circulating charge and spin currents in isolated conducting loops is based on the evaluation of eigenvalues and eigenvectors of the system

Acknowledgments

The first author is thankful to Prof. A. Nitzan for useful conversations, and S. Saha and P. Dutta for helpful discussions.

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Persistent charge and spin currents in a quantum ring using Green׳s function technique: Interplay between magnetic flux and spin–orbit interactions

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Model and hamiltonian

Numerical results and discussion

Summary and outlook

Acknowledgments

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Solid State Commun.

Synth. Met.

Phys. Lett. A

Solid State Commun.

Electronic Transport in Mesoscopic Systems

Introduction to Mesoscopic Physics

Quantum Transport: Atom to Transistor

Phys. Rev. Lett.

Phys. Rev. B

Phys. Rev. Lett.

Phys. Rev. Lett.

Europhys. Lett.

Phys. Rev. B

J. Phys.: Condens. Matter

Phys. Rev. B

Phys. Status Solidi B

Int. J. Mod. Phys. B

Quantum Matter

Phys. Rev. Lett.

Phys. Rev. Lett.

Science

Phys. Rev. Lett.

Phys. Rev. Lett.

Phys. Rev. Lett.

Phys. Rev. Lett.

Europhys. Lett.

J. Phys. (Paris)