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1998 | Buch

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Quantum Dots

verfasst von: Professor Lucjan Jacak, Dr. Arkadiusz Wójs, Dr. Paweł Hawrylak

Verlag: Springer Berlin Heidelberg

Buchreihe : NanoScience and Technology

Enthalten in: Professional Book Archive

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Über dieses Buch

We present an overview of the theoretical background and experimental re sults in the rapidly developing field of semiconductor quantum dots - systems 8 6 of dimensions as small as 10- -10- m (quasi-zero-dimensional) that contain a small and controllable number (1-1000) of electrons. The electronic structure of quantum dots, including the energy quan tization of the single-particle states (due to spatial confinement) and the evolution of these (Fock-Darwin) states in an increasing external magnetic field, is described. The properties of many-electron systems confined in a dot are also studied. This includes the separation of the center-of-mass mo tion for the parabolic confining potential (and hence the insensitivity of the transitions under far infrared radiation to the Coulomb interactions and the number of particles - the generalized Kohn theorem) and the effects due to Coulomb interactions (formation of the incompressible magic states at high magnetic fields and their relation to composite jermions), and finally the spin-orbit interactions. In addition, the excitonic properties of quantum dots are discussed, including the energy levels and the spectral function of a single exciton, the relaxation of confined carriers, the metastable states and their effect on the photoluminescence spectrum, the interaction of an exciton with carriers, and exciton condensation. The theoretical part of this work, which is based largely on original re sults obtained by the authors, has been supplemented with descriptions of various methods of creating quantum-dot structures.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

Scientific research into electronic systems was limited for a long period of time to naturally occurring isolated atoms or particles, metallic or semiconductor crystals, or beams of beta radiation. Most of these are three-dimensional systems, while an effective reduction of geometry to two or fewer dimensions — by a strong spatial localization to a plane, line, or point (i.e., confinement of an electron in at least one direction at the de Broglie wavelength) — occurs only in the case of atoms and electrons localized on crystal imperfections (e.g., on impurities).

Lucjan Jacak, Arkadiusz Wöjs, Paweł Hawrylak

2. Creation and Structure of Quantum Dots

Abstract

Unlike quantum wells, where the motion of carriers is restricted to a plane through the crystallization of thin epitaxial layers [64], the creation of quantum wires or dots, which confine the carriers to a space with at least two of three dimensions limited to the range of the de Broglie wavelength, requires far more advanced technology.

Lucjan Jacak, Arkadiusz Wöjs, Paweł Hawrylak

3. Single-Particle States of Quantum Dots

Abstract

The energy of an electron confined in an area as small as a quantum dot is strongly quantized, i.e., the energy spectrum is discrete. In typical structures, with characteristic dimensions in the range of 10–100 nm, the distance between neighboring energy levels is on the order of a few meV. As shown in Fig. 3.1, the quantization of energy, or alternatively, the reduction of the dimensionality of the system, is directly reflected in the dependence of the density of states on energy.

Lucjan Jacak, Arkadiusz Wöjs, Paweł Hawrylak

4. Properties of an Interacting System

Abstract

The advantage of the explicit separation of the center-of-mass motion from the relative dynamics in the description of quantum dots was first demonstrated by Laughlin for the case of three electrons [93]. The theory of a many-electron dot in the generalized Jacobi coordinates, utilizing both single-particle and many-particle, symmetries was formulated by Hawrylak [55]. In this chapter we present a derivation based on the standard Jacobi coordinates.

Lucjan Jacak, Arkadiusz Wöjs, Paweł Hawrylak

5. Intraband Optical Transitions

Abstract

The characteristic single-particle excitation energy of typical quantum dots lies in the far infrared (FIR) range. Depending on the dot dimensions, this energy varies between a fraction to tens of meV. As described previously, in Sect. 4.1, due to the fact that the wavelength of light from this range (≈ 1 mm) considerably exceeds the dot diameter (< 1µm), the FIR radiation cannot couple directly to the internal (relative) motion of the system of confined electrons, but only to their center-of-mass motion. Indirect interaction of the FIR light with the relative motion of electrons, which would allow for the observation of the effects of electron-electron interactions in the FIR experiment, requires a coupling between the relative (rel) and the center-of-mass (cm) motions.

Lucjan Jacak, Arkadiusz Wöjs, Paweł Hawrylak

6. Interband Optical Transitions

Abstract

The photoluminescence measurements are the basic tool that allow for the investigation of discrete energy levels of quantum dots. For this reason this is usually the first step in the studies of quantum dots obtained by various methods (described previously in Chap. 2). The idea of the photoluminescence experiment is presented in Fig. 6.1. A laser beam of appropriate wavelength excites the electrons from the valence band (VB) to the conduction band (CB) and creates electron-hole pairs (γ_in → e + h). These pairs can be excited directly into the discrete levels in the quantum dot, or above the discrete levels; to the two-dimensional quantum-well continuum or (even higher excitation energy) to the bulk semiconductor continuum. A fraction of the generated particles relax nonradiatively and fall to the ground state or to weakly excited states in quantum dots. The electron-hole pairs confined in the dots recombine, emitting photons, which are then registered (e + h →γ_Out)

Lucjan Jacak, Arkadiusz Wöjs, Paweł Hawrylak

7. Capacitance Spectroscopy

Abstract

The techniques of capacitance spectroscopy are based on the measurement of the electrical capacity of the system C = dQ/dV _G (which is related directly to the density of states dN/dµ) as a function of the gate voltage V _G applied perpendicularly to the quantum well. The quantum well, in which quantum dots are created, is placed above the doped layer (electrode) and serves as a reservoir of free carriers. The change of voltage V _G leads to a change in the number of electrons transferred from the well to the dot. Due to the discrete spectrum of energy levels in the quantum dot, the addition of a single electron to a dot occurs only at discrete values of V _G. The transfer of the Nth electron from the electrode to the quantum dot occurs when the two chemical potentials are equal: µ_E = µN. The chemical potential of the dot is defined as the energy required to add a single electron to the system,

$$\mu N = E_N - E_{N - 1,}$$

(7.1)

while the chemical potential of the electrode is equal (up to an additive constant) to its electrical potential:

$$\mu _E (V_G ) = \mu _E (0) + V_G .$$

(7.2)

Lucjan Jacak, Arkadiusz Wöjs, Paweł Hawrylak

8. Description of the Properties of Self-Assembled Quantum Dots Within the Band-Structure Model

Abstract

A method for creating self-assembled dots (SAD’s) is presented in Sect. 2.6. In this chapter we shall concentrate on self-assembled dots in the shape of spherical lenses shown in Fig. 8.1 (for the description of properties of pyramidal SAD’s see [47]). Such a dot is formed on a narrow wetting layer of thickness t _w, and modeled as a part of a sphere of height h and radius at the base _s. The conduction-band edge in the material of the wetting layer and the dot lies below that in the surrounding material [114, 115]. Thus, the electrons are confined in the narrow wetting-layer quantum well due to the step in the conduction-band edge at the interface, and they are further localized in the area of the dot due to the locally increased thickness of the layer. The effective lateral potential V(r,z) that acts on the electrons confined in the wetting layer is shown in the corner of Fig. 8.1.

Lucjan Jacak, Arkadiusz Wöjs, Paweł Hawrylak

9. Description of a Many-Electron Quantum Dot with the Inclusion of the Spin—Orbit Interaction

Abstract

In this chapter we shall present an analytical theory given by Jacak et al. [70], which is based on the Hartree-Fock approximation and describes a quantum dot that contains many electrons. The limitation of a large number of electrons (a few tens) results from the perturbational character of the calculation. However, it allows for the inclusion of the Hartree-Fock interaction with a controlled accuracy. In addition, the spin-orbit interaction is taken into account, which seems to explain many subtle characteristics that are measured for various dots. It should be added that the effects due to the spin-orbit interaction become more significant for larger numbers of electrons in a quantum dot. Its role in this artificial atom, which is squeezed in the plane, is even more important than in the case of ordinary, three-dimensional atoms.

Lucjan Jacak, Arkadiusz Wöjs, Paweł Hawrylak

10. Description of an Exciton in a Quantum Dot Within the Effective-Mass Approximation

Abstract

In the following description a quantum dot is modeled as a local perturbation of the periodic crystal field of a semiconducting structure that surrounds the dot. In the description of such a system, the effective-mass approximation can be applied if the perturbation potential varies slowly over the interatomic distance. In the case of quantum dots with sizes a few tens of nanometers, this assumption is expected to be satisfied. This also seems doubtless the case for the dots created by means of interlayer diffusion or by the application of a modulated electric field. It is probably a good approach for lens-shaped self-assembled dots. However, the cases of dots in the form of compact bubbles embedded in another semiconductor are disputable, since their sharply defined geometrical shapes may lead to a more rapid jump of the potential at the interface.

Lucjan Jacak, Arkadiusz Wöjs, Paweł Hawrylak

Backmatter

Titel: Quantum Dots
verfasst von: Professor Lucjan Jacak
Dr. Arkadiusz Wójs
Dr. Paweł Hawrylak
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-72002-4
Print ISBN: 978-3-642-72004-8
DOI: https://doi.org/10.1007/978-3-642-72002-4

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Inhaltsverzeichnis

Frontmatter

1. Introduction

2. Creation and Structure of Quantum Dots

3. Single-Particle States of Quantum Dots

4. Properties of an Interacting System

5. Intraband Optical Transitions

6. Interband Optical Transitions

7. Capacitance Spectroscopy

8. Description of the Properties of Self-Assembled Quantum Dots Within the Band-Structure Model

9. Description of a Many-Electron Quantum Dot with the Inclusion of the Spin—Orbit Interaction

10. Description of an Exciton in a Quantum Dot Within the Effective-Mass Approximation

Backmatter