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Electronic Structure of Quantum Dots REVIEWS OF MODERN PHYSICS, VOLUME 74, OCTOBER 2002 Electronic structure of quantum dots Stephanie M. Reimann Mathematical Physics, Lund Institute of Technology, 22100 Lund, Sweden Matti Manninen Department of Physics, University of Jyva¨skyla¨, 40351 Jyva¨skyla¨, Finland (Published 26 November 2002) The properties of quasi-two-dimensional semiconductor quantum dots are reviewed. Experimental techniques for measuring the electronic shell structure and the effect of magnetic fields are briefly described. The electronic structure is analyzed in terms of simple single-particle models, density-functional theory, and ‘‘exact’’ diagonalization methods. The spontaneous magnetization due to Hund’s rule, spin-density wave states, and electron localization are addressed. As a function of the magnetic field, the electronic structure goes through several phases with qualitatively different properties. The formation of the so-called maximum-density droplet and its edge reconstruction is discussed, and the regime of strong magnetic fields in finite dot is examined. In addition, quasi-one-dimensional rings, deformed dots, and dot molecules are considered. CONTENTS V. Magnetic Fields: Addition Energy Spectra and a Single-Particle Approach 1311 A. Harmonic oscillator in a magnetic field 1312 I. Introduction 1284 1. Fock-Darwin spectrum and Landau bands 1312 A. Shell structure 1284 2. Constant-interaction model 1312 B. ‘‘Magic numbers’’ in finite fermion systems 1286 B. Measurements of addition energy spectra 1313 II. Quantum Dot Artificial Atoms 1287 1. Tunneling spectroscopy of vertical dots 1313 A. Fabrication 1287 2. Gated transport spectroscopy in magnetic B. Coulomb blockade 1288 fields 1314 C. Probes of single-electron charging 1291 3. B-N phase diagram obtained by single- 1. Gated transport spectroscopy 1292 electron capacitance spectroscopy 1315 2. Single-electron capacitance spectroscopy 1292 4. B-N phase diagram of large vertical 3. Transport through a vertical quantum dot 1293 quantum dots 1315 III. Addition Energy Spectra 1293 VI. Role of Electron-Electron Interactions in Magnetic A. Many-body effects in quantum dots 1294 Fields 1316 B. Density-functional method 1295 A. Exact diagonalization for parabolic dots in C. Parabolic confinement 1296 magnetic fields 1316 D. Addition energy spectra described by mean-field B. Electron localization in strong magnetic fields 1319 theory 1296 1. Laughlin wave function 1319 E. Reproducibility of the experimental addition 2. Close to the Laughlin state 1319 energy spectra 1298 3. Relation to rotating Bose condensates 1320 F. Oscillator potential with flattened bottom 1298 4. Localization of electrons and the Laughlin G. Three-dimensionality of the confinement 1299 state 1321 H. Triangular quantum dots 1300 5. Localization and the many-body spectrum 1322 I. Elliptic deformation 1300 6. Correlation functions and localization 1323 J. Self-assembled pyramidal quantum dots in the VII. Density-Functional Approach for Quantum Dots in local spin-density approximation 1302 Magnetic Fields 1324 IV. Internal Electronic Structure 1302 A. Current spin-density-functional theory 1324 A. Classical electron configurations 1303 B. Parametrization of the exchange-correlation B. Spin polarization in the local spin-density energy in a magnetic field 1325 approximation 1304 C. Ground states and addition energy spectra 1. Circular and elliptic quantum dots 1304 2. Quantum wires and rings 1305 within the symmetry-restricted current-spin- a. Quantum wires 1305 density-functional theory 1325 b. Quantum rings 1305 1. Angular momentum transitions 1325 3. Artifacts of mean-field theory? 1306 2. Addition energy spectra in magnetic fields 1326 C. Mean-field versus exact diagonalization 1306 D. Reconstruction of quantum Hall edges in large D. Quasi-one-dimensional systems 1308 systems 1326 1. One-dimensional square well 1308 1. Edge reconstruction in current-spin-density- 2. Quasi-one-dimensional quantum rings 1309 functional theory 1327 E. Quantum Monte Carlo studies 1310 2. Phase diagram 1327 1. Hund’s rule 1310 E. Edge reconstruction and localization in 2. Wigner crystallization 1311 unrestricted Hartree-Fock theory 1328 0034-6861/2002/74(4)/1283(60)/$35.00 1283 ©2002 The American Physical Society 1284 S. M. Reimann and M. Manninen: Electronic structure of quantum dots F. Ensemble density-functional theory and charging energy spectra of small, vertical quantum dots noncollinear spins 1328 (Tarucha et al., 1996): the borderline between the phys- VIII. Quantum Rings in a Magnetic Field 1329 ics of bulk condensed matter and few-body quantum A. Electronic structure of quantum rings 1329 systems was crossed. Much of the many-body physics B. Persistent current 1331 that was developed for the understanding of atoms or IX. Quantum Dot Molecules 1331 nuclei could be applied. In turn, measurements on arti- A. Lateral-dot molecules in the local spin-density ficial atoms yielded a wealth of data from which a fun- approximation 1333 damental insight into the many-body physics of low- B. Vertical double dots in the local spin-density dimensional finite fermion systems was obtained approximation 1334 C. Exact results for vertical-dot molecules in a (Kouwenhoven, Austing, and Tarucha, 2001). With fur- magnetic field 1334 ther progress in experimental techniques, artificial atoms D. Lateral-dot molecules in a magnetic field 1335 will continue to be a rich source of information on X. Summary 1335 many-body physics and undoubtedly will hold a few sur- Acknowledgments 1336 prises. References 1336 The field of nanostructure physics has been growing rapidly in recent years, and much theoretical insight has been gained hand in hand with progress in experimental I. INTRODUCTION techniques and more device-oriented applications. Re- viewing the whole, very broad and still expanding field Low-dimensional nanometer-sized systems have de- would be an almost impossible task, given the wealth of fined a new research area in condensed-matter physics literature that has been published in the last decade. We within the last 20 years. Modern semiconductor process- thus restrict this review to a report on the discovery of ing techniques allowed the artificial creation of quantum shell structure in artificial atoms (with a focus on well- confinement of only a few electrons. Such finite fermion controlled dots in single-electron transistors) and sum- systems have much in common with atoms, yet they are marize aspects of theoretical research concerning the electronic ground-state structure and many-body physics man-made structures, designed and fabricated in the of artificial atoms. (A review of the statistical theory of laboratory. Usually they are called ‘‘quantum dots,’’ re- quantum dots, with a focus on chaotic or diffusive elec- ferring to their quantum confinement in all three spatial tron dynamics, was recently provided by Alhassid, dimensions. A common way to fabricate quantum dots is 2000.) In our analysis of shell structure, we shall be to restrict the two-dimensional electron gas in a semi- guided by several analogies to other finite quantal sys- conductor heterostructure laterally by electrostatic tems that have had a major impact on both theoretical gates, or vertically by etching techniques. This creates a and experimental research on quantum dots: the chemi- bowl-like potential in which the conduction electrons cal inertness of the noble gases, the pronounced stability are trapped. In addition to the many possibile techno- of ‘‘magic’’ nuclei, and enhanced abundances in the mass logical applications, what makes the study of these ‘‘ar- spectra of metal clusters. tificial atoms’’ or ‘‘designer atoms’’ (Maksym and Chakraborty, 1990; Chakraborty, 1992, 1999; Kastner, A. Shell structure 1992, 1993; Reed, 1993; Alivisatos, 1996; Ashoori, 1996; McEuen, 1997; Kouwenhoven and Marcus, 1998; Gam- The simplest approach to a description of finite quan- mon, 2000) interesting are the far-reaching analogies to tal systems of interacting particles is based on the idea systems that exist in Nature and have defined paradigms that the interactions, possibly together with an external of many-body physics: atoms, nuclei, and, more recently, confinement, create an average ‘‘mean field,’’ which, on metallic clusters (see, for example, the reviews by Brack, an empirical basis, can be approximated by an effective 1993 and de Heer, 1993) or trapped atomic gases [see potential in which the particles are assumed to move the Nobel lectures by Cornell, 2001, Ketterle, 2001, and independently. This a priori rather simple idea forms the Wieman, 2001, and, for example, the reviews by Dalfovo basis of Hartree, Hartree-Fock, and density-functional et al., 1999, and Leggett, 2001 and the recent book by theories. The last, with its many extensions, provides Pethick and Smith (2002)]. Quantum dots added an- powerful techniques for electronic structure calculations other such paradigm. Their properties can be changed in and is nowadays applied extensively in many different a controlled way by electrostatic gates, changes in the areas of both physics and chemistry (see the Nobel lec- dot geometry, or applied magnetic fields. Their techno- tures by Kohn, 1999, and Pople, 1999). logical realization gave access to quantum effects in fi- The distribution of single-particle energy levels of the nite low-dimensional systems that were largely unex- mean-field potential can be nonuniform, and bunches of plored. degenerate or nearly degenerate levels, being separated After the initial success in the fabrication and control from
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