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## Über dieses Buch

This book, now in its second edition, introduces readers to quantum rings as a special class of modern high-tech material structures at the nanoscale. It deals, in particular, with their formation by means of molecular beam epitaxy and droplet epitaxy of semiconductors, and their topology-driven electronic, optical and magnetic properties. A highly complex theoretical model is developed to adequately represent the specific features of quantum rings. The results presented here are intended to facilitate the development of low-cost high-performance electronic, spintronic, optoelectronic and information processing devices based on quantum rings.

This second edition includes both new and significantly revised chapters. It provides extensive information on recent advances in the physics of quantum rings related to the spin-orbit interaction and spin dynamics (spin interference in Rashba rings, tunable exciton topology on type II InAs/GaAsSb quantum nanostructures), the electron-phonon interaction in ring-like structures, quantum interference manifestations in novel materials (graphene nanoribbons, MoS2), and the effects of electrical field and THz radiation on the optical properties of quantum rings. The new edition also shares insights into the properties of various novel architectures, including coupled quantum ring-quantum dot chains and concentric quantum rings, topologic states of light in self-assembled ring-like cavities, and optical and plasmon m.odes in Möbius-shaped resonators.

## Inhaltsverzeichnis

### Chapter 1. Quantum Ring: A Unique Playground for the Quantum-Mechanical Paradigm

Abstract
The physics of quantum rings is reviewed from basic concepts rooted in the quantum-mechanical paradigm—via unprecedented challenges brilliantly overcome by both theory and experiment—to promising application perspectives.

### Chapter 2. Optical Berry Phase in Micro/Nano-rings

Abstract
Theoretical and experimental results are presented, which introduce topology into the field of optical and plasmonic resonances in ring resonators. Due to occurrence of the Berry phase in non-trivial evolution, plasmon/photon modes with non-integer numbers of wavelengths along the circumference are revealed in metallic/dielectric Möbius rings, which do not exist in conventional ring resonators. In cone-shaped anisotropic microtube resonators, the optical spin-orbit coupling is enabled for generation of the Berry phase acquired in a non-cyclic and non-Abelian evolution. These topology-induced effects imply promising applications related to manipulating photons in on-chip integrable quantum devices.
Libo Ma, Vladimir M. Fomin, Oliver G. Schmidt

### Chapter 3. From Dot to Ring: Tunable Exciton Topology in Type-II InAs/GaAsSb Quantum Dots

Abstract
We present an experimental and theoretical study about the carrier confinement geometry and topology in InAs/GaAsSb quantum dots. The investigated sample consists of a field-effect device embedding a single layer of dot-in-a-well InAs/GaAsSb nanostructures. These nanostructures exhibit large electron-hole dipole moments and radiative lifetimes under externally applied electric fields. Both phenomena are related to the type-II band alignment existing between the two materials which, in principle, could also result in a change of the hole orbital confinement topology from simply to doubly connected. The latter aspect will be confirmed by ensemble magnetophotoluminescence experiments at 4.2 K. The oscillations observed in the photoluminescence intensity and degree of circular polarization will be described by an axially symmetric $$\mathbf {k}\cdot \mathbf {p}$$ model combining vertical electric and magnetic fields. Due to the large spin-orbit coupling of III-Sb nanostructures, the modulation of the orbital confinement geometry and topology reported here shall open a venue to control the spin dynamics by external voltages. This exciting idea will be theoretically discussed through band-effective models including spin-orbit coupling and anisotropic confinement effects.
José M. Llorens, Vivaldo Lopes-Oliveira, Victor López-Richard, José M. Ulloa, Benito Alén

### Chapter 4. Self-organized Quantum Rings: Physical Characterization and Theoretical Modeling

Abstract
An adequate modeling of self-organized quantum rings is possible only on the basis of the modern characterization of those nanostructures. We discuss an atomic-scale analysis of the indium distribution in self-organized InGaAs quantum rings (QRs). The analysis of the shape, size and composition of self-organized InGaAs QRs at the atomic scale reveals that AFM only shows the material coming out of the QDs during the QR formation. The remaining QD material, as observed by Cross-Sectional Scanning Tunneling Microscopy (X-STM), shows an asymmetric indium-rich crater-like shape with a depression rather than an opening at the center and determines the observed ring-like electronic properties of QR structures. A theoretical model of the geometry and materials properties of the self-organized QRs is developed on that basis and the magnetization is calculated as a function of the applied magnetic field. Although the real QR shape differs strongly from an idealized circular-symmetric open-ring structure, Aharonov-Bohm-type oscillations in the magnetization have been predicted to survive. They have been observed using the torsion magnetometry on InGaAs QRs. Large magnetic moments of QRs are shown to originate from dissipationless circulating currents in the ground state of an electron or hole in the QR. Examples of prospective applications of QRs are presented that do and do not utilize the topological properties of QRs.
V. M. Fomin, V. N. Gladilin, J. van Bree, M. E. Flatté, J. T. Devreese, P. M. Koenraad

### Chapter 5. Scanning-Probe Electronic Imaging of Lithographically Patterned Quantum Rings

Abstract
Quantum rings patterned from two-dimensional semiconductor heterostructures exhibit a wealth of quantum transport phenomena at low temperature and in a magnetic field that can be mapped in real space thanks to dedicated scanning probe techniques. Here, we summarize our studies of GaInAs- and graphene-based quantum rings by means of scanning-gate microscopy both at low magnetic field, where Aharonov-Bohm interferences and the electronic local density-of-states are imaged, and at high magnetic field and very low temperatures, where the scanning probe can image Coulomb islands in the quantum Hall regime. This allows decrypting the apparent complexity of the magneto-resistance of a mesoscopic system in this regime. Beyond imaging and beyond a strict annular shape of the nanostructure, we show that this scanning-probe technique can also be used to unravel a new counter-intuitive behavior of branched-out rectangular quantum rings, which turns out to be a mesoscopic analog of the Braess paradox, previously known for road or other classical networks only.
F. Martins, D. Cabosart, H. Sellier, M. G. Pala, B. Hackens, V. Bayot, S. Huant

### Chapter 6. Functionalization of Droplet Etching for Quantum Rings

Abstract
We give an overview on various types of strain-free semiconductor quantum ring (QR) structures created in a self-assembled fashion with the local droplet etching (LDE) method. LDE is fully compatible with conventional molecular beam epitaxy (MBE) and utilizes liquid Ga or Al droplets which drill nanoholes into semiconductor surfaces. The nanohole openings are surrounded by walls composed of Arsenides of the droplet material. Here the nanohole and wall formation mechanisms and the tunability of their structural properties are discussed. Three different concepts for QR generation by LDE are addressed. In the first concept, GaAs recrystallized during LDE with Ga droplets on AlGaAs substrates forms directly GaAs quantum rings. The second concept is based on the wave-function tuning of V-shaped GaAs QDs by an applied gate-voltage. Here, either the electron or the hole wave function can be transformed into a ring-shape, whereas the respective other charge carrier type remains in a zero-dimensional QD state. The third concept considers the partial depletion of a near surface GaAs quantum well (QW) due to tunneling. The LDE-related wall increases locally the distance to the surface which reduces tunneling and generates a ring-shaped charge-carrier concentration in the QW below the wall. The fabrication and structural properties of these three types of QRs, simulations of the quantized electronic levels and wave functions, and first optical data are discussed.
Christian Heyn, Michael Zocher, Wolfgang Hansen

### Chapter 7. Fabrication of Ordered Quantum Rings by Molecular Beam Epitaxy

Abstract
Quantum rings have attracted a lot of attention due to their unique properties and have been under extensive theoretical and experimental investigations. For example, Aharonov-Bohm effect has been observed in quantum rings which shows potential to realize quantum computational devices. In addition, quantum rings have found application in optoelectronics. Due to the ring-shaped morphology altered from dots, the vertical confinement in nanorings is stronger than in quantum dots. Laser and infrared photodetectors have recently been demonstrated by using quantum rings. To meet the urgent demands for quantum rings, various effects have been devoted to quantum ring fabrication techniques. There are two of the most used bottom-up fabrication methods of self-assembled rings using molecular beam epitaxy (MBE). Semiconductor quantum rings can be created by conventional molecular beam epitaxy and Droplet Epitaxy technique. Despite great efforts devoted to quantum ring fabrication using these techniques, alignment of quantum rings is not well documented. Fabrication of ordered quantum rings is of high priority for theoretical as well as practical investigations, such as persistent current and photodetectors. Recently, both vertically and laterally ordered quantum rings have been demonstrated. In this chapter, the growth mechanisms and fabrication techniques for aligned quantum rings grown are reviewed.
Jiang Wu, Zhiming M. Wang

### Chapter 8. Self-assembled Semiconductor Quantum Ring Complexes by Droplet Epitaxy: Growth and Physical Properties

Abstract
Extremely complex semiconductor quantum ring based structures, as single ring, multiple concentric quantum rings and coupled ring-disk, dot-ring and dot-disk structures, can be easily designed and grown by Droplet Epitaxy. In this paper, the fabrication and the characterization of such complex quantum nanostructures are reviewed. Electronic structure, single photon emission, carrier dynamics and magnetic properties in ring structures will be discussed.
Stefano Sanguinetti, Takaaki Mano, Takashi Kuroda

### Chapter 9. Optical Aharonov-Bohm Oscillations with Disorder Effects and Wigner Molecule in a Single GaAs/AlGaAs Quantum Ring

Abstract
The optical Aharonov-Bohm effect in a single quantum ring is associated with disorder effects. In the presence of structure anisotropy, localisation, internal electric field, and impurity scattering, optical Aharonov-Bohm oscillations of an electron-hole pair become modulated. Additionally, provided that a strongly correlated exciton pair is formed in a single quantum ring similar to the Wigner molecule, novel oscillations can be observed for increasing magnetic field. In this case, the biexciton emission energy changes abruptly at transition magnetic fields with a fractional oscillation period compared to that of the exciton, the so-called fractional optical Aharonov-Bohm oscillations.
K. Kyhm, H. D. Kim, R. Okuyama, M. Eto, K. C. Je, R. A. Taylor, G. Nogues, L. S. Dang, A. A. L. Nicholet, M. Potemski, J. S. Kim, J. D. Song

### Chapter 10. Aharonov-Bohm Effect for Neutral Excitons in Quantum Rings

Abstract
Quantum interference patterns predicted by theory due to the finite structure of neutral excitons in a single layer of InAs quantum rings are corroborated experimentally in the magneto-photoluminescence spectra of these nanostructures. The effects associated to built in electric fields and to the temperature on these Aharonov-Bohm-like oscillations are described and confirmed by complementary experimental procedures. Also a similar behavior was observed in a hybrid structure composed by a topmaster single layer of InAs quantum rings grown on a vertically stacked and laterally aligned InGaAs quantum dot superlattice.
M. D. Teodoro, V. L. Campo Jr., V. López-Richard, E. Marega Jr., G. E. Marques, G. J. Salamo

### Chapter 11. Electronic, Magnetic and Optical Properties of Quantum Rings in Novel Systems

Abstract
In the past few decades, major advances in nanofabrication techniques have resulted in the creation of tunable few-electron nanoscale quantum rings with unique topology and the energy spectrum. These rings display many remarkable effects in magnetotransport and optical spectroscopy that were predicted earlier in theoretical studies. Having external control over the size and the number of electrons in such a ring offers intriguing possibilities to study the interplay between the confinement and the electron-electron interaction. Here we review the physics of quantum rings in a few novel situations and unravel some new physical phenomena involving these rings that go beyond our current understandings of physics derived from conventional nanoscale quantum rings.
Tapash Chakraborty, Aram Kh. Manaselyan, Manuk G. Barseghyan

### Chapter 12. Spin Interference Effects in Rashba Quantum Rings

Abstract
Quantum interference effects in rings provide suitable means to control spins at the mesoscopic scale. In this chapter we present the theory underlying spin-induced modulations of unpolarized currents in quantum rings subject to the Rashba spin-orbit interaction. We discuss explicitly the connection between the conductance modulations and the geometric phase acquired by the spin during transport, as well as pathways to directly control them.
Carmine Ortix

### Chapter 13. Quantum Rings in Electromagnetic Fields

Abstract
This chapter is devoted to optical properties of so-called Aharonov-Bohm quantum rings (quantum rings pierced by a magnetic flux resulting in Aharonov-Bohm oscillations of their electronic spectra) in external electromagnetic fields. It studies two problems. The first problem deals with a single-electron Aharonov-Bohm quantum ring pierced by a magnetic flux and subjected to an in-plane (lateral) electric field. We predict magneto-oscillations of the ring electric dipole moment. These oscillations are accompanied by periodic changes in the selection rules for inter-level optical transitions in the ring allowing control of polarization properties of the associated Terahertz radiation. The second problem treats a single-mode microcavity with an embedded Aharonov-Bohm quantum ring which is pierced by a magnetic flux and subjected to a lateral electric field. We show that external electric and magnetic fields provide additional means of control of the emission spectrum of the system. In particular, when the magnetic flux through the quantum ring is equal to a half-integer number of the magnetic flux quanta, a small change in the lateral electric field allows for tuning of the energy levels of the quantum ring into resonance with the microcavity mode, thus providing an efficient way to control the quantum ring-microcavity coupling strength. Emission spectra of the system are discussed for several combinations of the applied magnetic and electric fields.
A. M. Alexeev, M. E. Portnoi

### Chapter 14. Intense Terahertz Radiation Effect on Electronic and Intraband Optical Properties of Semiconductor Quantum Rings

Abstract
The current chapter aims to theoretically demonstrate that intense Terahertz (THz) laser field can be a powerful method for the controlling of electro-optical properties of quantum rings (QRs). We explore the electronic and impurity states, charge localization and intraband optical phenomena in $$\text {GaAs/Ga}\text {Al}\text {As}$$ QRs irradiated by the intense THz laser field. Single and concentric double QRs, as well as artificial molecules formed by the laterally aligned QRs are explored. It is demonstrated how the laser field modifies the energy spectrum and wave functions by the strong distortion of the original cylindrical geometry of quantum confinement. Moreover, our findings give an insight on the laser field-affected inter-ring coupling of concentric double QRs and dissociation of QR molecules. Additionally, the new way of control of quantum-confining Stark effect with intense THz laser field is introduced.
H. M. Baghramyan, M. G. Barseghyan, A. A. Kirakosyan, D. Laroze

### Chapter 15. Electron-Phonon Interaction in Ring-Like Nanostructures

Abstract
General expressions of the electron-phonon Hamiltonians in ring-like nanostructures are settled. A unified macroscopic continuum approach for the treatment of acoustical and optical phonon modes in semiconductor core-shell nanowires is established. A basis for the space of solutions is derived, and by applying the appropriate boundary conditions, the dispersion relation curves, as well as the displacement fields and the electrostatic potential for non-polar and polar optical modes are reported. Employing the methods of quantum field theory, the electron- and hole-zone-center optical phonon deformation potential and optical long-range as well as electron-acoustical phonon Hamiltonians are deduced in a systematic way. The results are valid for the study of polar and non-polar semiconductor based core-shell nanowires and to analyze the role of intrinsic strain and the geometric factors on the electron-phonon coupling strengths. Special emphasis is placed on the importance of the cylindrical symmetry in the interaction Hamiltonians.
C. Trallero-Giner, Darío G. Santiago-Pérez, Leonor Chico, R. Pérez-Álvarez

### Chapter 16. Differential Geometry Applied to Rings and Möbius Nanostructures

Abstract
Nanostructure shape effects have become a topic of increasing interest due to advancements in fabrication technology. In order to pursue novel physics and better devices by tailoring the shape and size of nanostructures, effective analytical and computational tools are indispensable. In this chapter, we present analytical and computational differential geometry methods to examine particle quantum eigenstates and eigenenergies in curved and strained nanostructures. Example studies are carried out for a set of ring structures with different radii and it is shown that eigenstate and eigenenergy changes due to curvature are most significant for the groundstate eventually leading to qualitative and quantitative changes in physical properties. In particular, the groundstate in-plane symmetry characteristics are broken by curvature effects, however, curvature contributions can be discarded at bending radii above 50 nm. A more complicated topological structure, the Möbius nanostructure, is analyzed and geometry effects for eigenstate properties are discussed including dependencies on the Möbius nanostructure width, length, thickness, and strain. In the final part of the chapter, we derive the phonon equations-of-motion of thin shells applied to 2D graphene using a differential geometry formulation.
Benny Lassen, Morten Willatzen, Jens Gravesen

### Chapter 17. Band Mixing Effects in InAs/GaAs Quantum Rings and in MoS Quantum Dots Ring-Like Behaving

Abstract
The physics of semiconductor quantum rings near the band edge is often well described considering decoupled bands. There are however instances where band coupling leads to relevant changes in the electronic structure and derived properties. In this chapter we analyze two such cases. First, we focus on the heavy hole-light hole band mixing in self-assembled InAs/GaAs quantum rings, which is important for current endeavour to develop quantum information science using the spin of holes. In InAs/GaAs quantum dots, the hole ground state is known to be mainly formed by the heavy hole subband. However, there is a finite spin-orbit coupling with the light-hole subband which is critical in determining the hole spin properties. Based on k$$\cdot$$p theory, in this chapter we study the influence of hole subband mixing in quantum rings. It is shown that the inner cavity of the ring enhances the light hole component of the ground state. As the quasi-1D limit is approached, the light-hole character becomes comparable to that of the heavy hole. Strain reduces the coupling, but it is still larger than in quantum dots. Second, we study the electronic structure of monolayer MoS$$_2$$ quantum dots subject to a magnetic field. Here, the coupling between conduction and valence band gives rise to mid-gap topological states which localize near the dot edge. These edge states are analogous to those of 1D quantum rings. We show they present a large, Zeeman-like, linear splitting with the magnetic field, anticross with the delocalized Fock-Darwin-like states of the dot, give rise to Aharonov-Bohm-like oscillations of the conduction (valence) band low-lying states in the K (K$$^{\prime }$$) valley, and modify the strong-field Landau levels limit form of the energy spectrum.
Carlos Segarra, Josep Planelles, Juan I. Climente

### Chapter 18. Circular n-p Junctions in Graphene Nanoribbons

Abstract
A characteristic feature of graphene as the Dirac conductor is that one can introduce doping by external voltages, so that the n-p junction can be defined and controlled by gating. The electrostatic n-p junctions in graphene act as waveguides that confine currents. The fact can be classically understood by the opposite orientation of the Lorentz force at both sides of the n-p junction, so that the carriers in both the conduction and valence band are shifted towards the junction by the external magnetic field. We describe our proposal for an Aharonov-Bohm interferometer at the n-p junction induced by the potential of the tip of an atomic force microscope. The conductance of the system exhibits Aharonov-Bohm oscillations provided that the persistent currents localized at the junction are coupled to the quantum Hall edge currents. The coupling is controlled by the Fermi energy and the tip potential. We discuss the Lorentz force effects in the system as compared to etched quantum rings in graphene and III-V semiconductors.
Alina Mreńca-Kolasińska, Bartłomiej Szafran

### Backmatter

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