Available online at www.sciencedirect.com Physica E 21 (2004) 1–35 www.elsevier.com/locate/physe Review Quantum rings for beginners: energy spectra and persistent currents S. Viefersa;∗, P. Koskinenb, P. Singha Deoc, M. Manninenb aDepartment of Physics, University of Oslo, P.O. Box 1048 Blindern, N-0316 Oslo, Norway bDepartment of Physics, University of Jyvaskyl$ a,$ FIN-40014 Jyvaskyl$ a,$ Finland cS.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake City, Kolkata 98, India Received 30 April 2003; accepted 12 August 2003 Abstract Theoretical approaches to one-dimensional and quasi-one-dimensional quantum rings with a few electrons are reviewed. Discrete Hubbard-type models and continuum models are shown to give similar results governed by the special features of the one-dimensionality. The energy spectrum of the many-body states can be described by a rotation–vibration spectrum of a “Wigner molecule” of “localized” electrons, combined with the spin-state determined from an e7ective antiferromagnetic Heisenberg Hamiltonian. The persistent current as a function of the magnetic 8ux through the ring shows periodic oscillations arising from the “rigid rotation” of the electron ring. For polarized electrons the periodicity of the oscillations is always the 8ux quantum 0. For nonpolarized electrons the periodicity depends on the strength of the e7ective Heisenberg coupling and changes from 0 ÿrst to 0=2 and eventually to 0=N when the ring gets narrower. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.23.Ra; 73.21.Hb; 75.10.Pq; 71.10.Pm Keywords: Quantum ring; Persistent current; Electron localization Contents 9.3. Bethe ansatz ................................. 14 9.3.1. Ground state of the t-model .............. 14 1. Introduction ...................................... 2 9.3.2. Higher bands .......................... 15 2. Experimental situation .............................. 3 9.4. Vibrational bands ............................. 16 2 3. Strictly 1D-ring with noninteracting spinless electrons ... 4 9.5. 1=dij interactions .............................. 17 4. Localization of noninteracting spinless fermions in 1D .. 5 9.6. Finite U ..................................... 18 5. Classical interacting electrons in a strictly 1D ring...... 7 10. Quasi-1D-continuum rings: Exact CI method and 6. E7ect of a magnetic 8ux—persistent currents .......... 8 e7ective Hamiltonian ............................... 20 7. Noninteracting particles with spin .................... 10 11. Exact diagonalization: ÿnite magnetic ÿeld ............ 21 8. Noninteracting electrons in a lattice .................. 11 11.1. Flux inside the ring ........................... 21 9. Interacting electrons on a lattice: the Hubbard model ... 11 11.2. Homogeneous magnetic ÿeld, no Zeeman splitting . 23 9.1. Model and exact diagonalization ................ 11 12. Periodicity of persistent current ...................... 24 9.2. The t-model .................................. 12 12.1. Strictly 1D rings: Spectrum of rigid rotation ...... 24 12.2. Periodicity change in quasi-1D rings ............. 25 13. Other many-body approaches for quasi-1D continuum rings 27 ∗ Corresponding author. 13.1. Quantum Monte Carlo ......................... 27 E-mail address: [email protected] (S. Viefers). 13.2. Local density approximation .................... 28 1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.08.076 2 S. Viefers et al. / Physica E 21 (2004) 1–35 14. Relationto Luttingerliquid ......................... 28 possibility is to assume a ring-shaped smooth external 15. Pair correlation .................................... 29 conÿnement where interacting electrons move. In this 16. Interactionofthe spinwith the magneticÿeld: the continuum model the electron–electron interaction is Zeemane7ect ..................................... 29 usually the normal Coulomb interaction (e2=4 r). 17. E7ect of animpurity ............................... 30 0 18. Summary ......................................... 32 In the case of a small number of electrons (typically N¡10) the many-particle problem is well deÿned in Acknowledgements ................................... 33 both models and can be solved with numerical diago- References .......................................... 33 nalization techniques for a desired number of lowest energy states. Other many-particle methods have also been ap- plied for studying the electronic structure of quantum 1. Introduction rings. The discrete rings can be solved by using the Bethe ansatz [10,11], which becomes powerful espe- The observationof the Aharonov–Bohmoscilla- cially in the case of an inÿnitely strong contact inter- tions [1] and persistent current [2] in small conducting action(the so-called t-model). In the case of contin- rings, on one hand, and the recent experimental devel- uum rings, quantum Monte Carlo methods and density opments in manufacturing quantum dots [3] and rings functional methods have been used (see Section 13). [4] with only a few electrons, on the other, have made The purpose of this paper is to give anintroductory quantum rings an ever increasing topic of experimen- review to the many-particle properties of rings with a tal research and a new playground for many-particle few electrons. Our aim is not to give a comprehensive theory in quasi-one-dimensional systems. review of all the vast literature published. The main Many properties of the quantum rings can be ex- emphasis is to clarify the relations between di7erent plained with single-electron theory, which in a strictly methods and to point out general features of the elec- one-dimensional (1D) system is naturally very sim- tronic structures of the rings and their origins. Most ple. On the contrary, the many-particle fermion prob- of the results we show inÿgures are our owncom- lem in1D systems is surprisinglycomplicated due putations made for this review. However, we want to to enhanced importance of the Pauli exclusion princi- stress that most of the phenomena shown have been ple. In general, correlations are always strong, leading published before (inmanycase by several authors) to non-fermionic quasiparticles as low energy excita- and we will give reference to earlier work. tions. It is then customary to say that strictly 1D sys- We take anapproach where we analysethe tems are not “Fermi liquids” but “Luttinger liquids” many-body excitation spectrum and its relation to with speciÿc collective excitations (for reviews see the single particle spectrum and electron localization Refs. [5–8]). along the ring. We will show that, irrespective of The many-particle approach normally used in study- the model, the excitationspectrum innarrowrings ing the properties of Luttinger liquids starts by assum- can be understood as a rotation–vibration spectrum ing an inÿnitely long strictly one-dimensional system. of localized electrons. The e7ect of the magnetic 8ux In small ÿnite rings a direct diagonalization of the penetrating the ring is also studied as the change of many-body Hamiltonian, using numerical techniques the spectrum as a function of the 8ux. This is used to and a suitable basis, is possible and can provide more analyze the periodicity of the persistent current as a direct informationonthemany-particlestates. Two function of the 8ux. Again, it is shown that similar di7erent theoretical models have been used for the ÿ- results are obtained with the discrete and continuum nite rings. In a discrete model the ring is assumed models. to consist of L discrete lattice sites (or atoms) with Throughout this paper we use the term “spinless N electrons,which canhop from site to site. Inthis electrons” to describe a system of completely polar- case the electron–electron interaction is usually as- ized electronsystem, i.e. the many-particlestate hav- sumed to be e7ective only when the electrons are at ing maximum total spin and its z-component. We use the same site. The many-particle Hamiltonian is then lower case letters to describe single particle proper- a Hubbard Hamiltonian [9] or its extension. Another ties and capital letters for many-particle properties S. Viefers et al. / Physica E 21 (2004) 1–35 3 (e.g. m and M for angular momenta). We use terms of nanoscopic rings containing only a few electrons like “rotational” and “vibrational” states quite loosely [4,13,14]. At the same time, many of the experi- for describing excitations, which in certain limiting ments still study mesoscopic rings with hundreds of cases have exactly those meanings. When we talk electrons. Methods of forming such rings include about the rotational state we will use the terminology lithographic methods for forming individual rings of the nuclear physics and call the lowest energy state ona semiconductorsurface. The spectroscopic tech- of a given angular momentum an yrast state. niques are based on the tunneling current through the Experimentally, the study of the spectra of quantum ring or capacitance spectroscopy. Another possibility ringis still inanearly state of animpressive devel- is to create a large number of self-organized rings opment. It is not yet the time to make detailed com- on a substrate. The large number of rings allows for parisonbetweenthe theory andexperiments.Never- observationof direct optical absorption. theless, we will give inSection 2 a short overview of One of the hallmarks in this ÿeld of research has the experimental situation. beenthe experimentalobservationof the Aharonov– We thenattempt to review