We report the direct measurement of the persistent current carried by a single electron by means of magnetization experiments on self-assembled InAs=GaAs quantum rings. We measured the first Aharonov-Bohm oscillation at a field of 14 T, in perfect agreement with our model based on the structural properties determined by cross-sectional scanning tunneling microscopy measurements. The observed oscillation magnitude of the magnetic moment per electron is remarkably large for the topology of our nanostructures, which are singly connected and exhibit a pronounced shape asymmetry. DOI: 10.1103/PhysRevLett.99.146808 PACS numbers: 73.21.La, 73.23.Ra, 78.67.Hc In quantum mechanics, particular attention is paid to phenomena occurring due to the phase coherence of charge carriers in doubly connected (ring) topologies. Electrons confined to a submicron ring manifest a topologically determined quantum-interference phenomenon, known as the Aharonov-Bohm (AB) effect [1], as a result of the oscillatory behavior of their energy levels as a function of an applied magnetic field. This behavior is usually associated with the occurrence of oscillatory persistent currents in the ring [2 -4]. Experimental evidence for AB oscillations has been detected in the mesoscopic regime in metallic [5,6] and semiconducting [7,8] rings, containing many electrons. We address the occurrence of the AB effect in defect-free self-assembled semiconductor nanostructures [9][10][11][12][13]. The ability to fill nanostructures with only a few (1-2) electrons offers the unique possibility to detect magnetic field induced oscillations in the persistent current carried by single electron states. We report the first direct measurement by means of ultrasensitive magnetization experiments of the oscillatory persistent current carried by a single electron in self-assembled InAs/GaAs ''volcanolike'' nanostructures. Remarkably, this single electron current occurs even in the absence of an opening [14] in our nanostructures, which is required for the AB effect in the standard treatment [1]. The magnetic field at which the first oscillation in the magnetic moment arises is much higher than expected from the diameter of the quantum rings as determined by atomic force microscopy [13]. However, the experiments are in good agreement with a model based on the structural parameters as determined with cross-sectional scanning tunneling microscopy (XSTM) measurements.The persistent current was determined via the magnetic moment of electrons in a highly homogeneous ensemble of InAs self-assembled nanostructures. The sample was grown by molecular beam epitaxy and contains 29 mutually decoupled periods [ Fig. 1(a)] [15]. Each period consists of a nanostructured InAs layer, between two 24 nm GaAs layers, and a 2 nm doped (7 10 16 cm ÿ3 Si) GaAs layer that provides electrons to the InAs nanostructures. We used a one-dimensional Poisson solver [16] to estimate the average number of electrons per nanostructure to be about 1.5. Considering the two possible spin orientations we ...

Offermans, P.; Koenraad, P.M.; Wolter, J.H.; Granados, D.; Garcia, J.M.; Fomin, V.; Gladilin, V.N.; Devreese, J.T.

We derive a nonsymmetrized 8-band effective-mass Hamiltonian for quantumdot heterostructures (QDHs) in Burt's envelope-function representation. The 8×8 radial Hamiltonian and the boundary conditions for the Schrödinger equation are obtained for spherical QDHs. Boundary conditions for symmetrized and nonsymmetrized radial Hamiltonians are compared with each other and with connection rules that are commonly used to match the wave functions found from the bulk k ·p Hamiltonians of two adjacent materials. Electron and hole energy spectra in three spherical QDHs: HgS/CdS, InAs/GaAs, and GaAs/AlAs are calculated as a function of the quantum dot radius within the approximate symmetrized and exact nonsymmetrized 8×8 models. The parameters of dissymmetry are shown to influence the energy levels and the wave functions of an electron and a hole and, consequently, the energies of both intraband and interband transitions.PACS numbers: 73.20. Dx, 73.40.Kp, 73.40.Lq This model explicitly includes eight bands around the Γ point of the Brillouin zone, namely, electron, heavy-, light-, and spin-orbit split-off hole bands (each of them is twice-degenerate due to the spin), and treats all other bands as remote. Along with more simple models, the 8×8 k ·p Hamiltonian has been used to investigate different QDs (see, e. g. .Recently, one has begun to apply multiband effective-mass Hamiltonians to investigate elastic, electronic, and optical properties of multilayer nanostructures such as quantumdot heterostructures (QDHs): CdS/HgS 11 , InAs/GaAs 12,13 , GaAs/Al x Ga 1−x As 14,15 , and CdS/HgS/CdS/H 2 O 16,17 . However, it should be emphasized, that multiband k ·p Hamiltonians are derived for homogeneous bulk materials, i.e. under the assumption that all effective-mass parameters are constant. This is important, because at a certain step of the derivation, wavenumbers k are declared as operatorsp/h that do not commute with the functions of coordinates. But, at the heterointerfaces of the multilayer nanostructures, there occurs an abrupt change of effective-mass parameters from their values in one material to those in the adjacent material. Inside a thin transitional layer that contains the heterointerface, the ordering of the differential operators and coordinate-dependent effective-mass parameters in the multiband Hamiltonian becomes crucial. In QDs with an infinitely high confining potential for electrons and holes, all components of the wave function vanish at the heterointerface, and there remains a possibility of applying the bulk multiband k ·p Hamiltonian straightforwardly. [3][4][5][6][8][9][10][11] There are two ways to proceed from QDs to QDHs.(i) The first way is to use an appropriate bulk multiband Hamiltonian for each constituent material separately, and then to match the obtained homogeneous solutions at the abrupt heterojunctions applying the connection rules (CRs) that are usually obtained by imposing the continuity of the wave function envelopes and of the normal to the heterointerface component of the velocity. 11,16...

Citation for published version (APA):Fomin, V. M., Gladilin, V. N., Klimin, S. N., Devreese, J. T., Kleemans, N. A. J. M., & Koenraad, P. M. (2007). Theory of electron energy spectrum and Aharonov-Bohm effect in self-assembled Inx Ga1-x As quantum rings in GaAs. Physical Review B, 76(23) Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. Theory of electron energy spectrum and Aharonov-Bohm effect in self-assembled In x Ga 1−x As quantum rings in GaAs We analyze theoretically the electron energy spectrum and the magnetization of an electron in a strained In x Ga 1−x As/ GaAs self-assembled quantum ring ͑SAQR͒ with realistic parameters, determined from the crosssectional scanning-tunneling microscopy characterization of that nanostructure. The SAQRs have an asymmetric indium-rich craterlike shape with a depression rather than an opening at the center. Although the real SAQR shape differs strongly from an idealized circular-symmetric open ring structure, the Aharonov-Bohm oscillations of the magnetization survive.