## Introduction

The real confined systems, also called low-dimensional systems or nanostructures, are any three-dimensional quantum systems in which the carriers are free to move in only two, one, or even zero dimensions.

In nanostructured semiconductors the concept of quasi-particles is related to electrons, holes, and excitons; the characteristic length that defines the confinement degree is the effective Bohr radius of excitons, a*B. This magnitude may be considerably larger than lattice constant, aL. Therefore, it is possible to create a mesoscopic structure which is, in one, two, or three dimensions, comparable to or even lesser than a*B but still larger than aL. In these structures the elementary excitations will experience quantum confinement, resulting in a finite motion along the confinement axis and an infinite motion in other directions. This way, one deals with the so-called nanostructured systems or, in a broad accepted classification, the quasi-two-dimensional systems (single het-erostructures, quantum wells, multiple quantum wells, and superlattices), quasi-one-dimensional systems (quantum well wires), and quasi-zero-dimensional systems (quantum dots, crystallites, and quantum boxes).

The confinement degree of quasi-particles is related to the magnitudes a*B and the quantum confinement size d (or r0); two regimes can be readily distinguished: the weak confinement regime and the strong confinement regime. The weak confinement regime corresponds to the case when the nanostructure confinement size [d (or r0)] is smaller but still a few times larger than a*B. The mathematical condition is a*B ^ d (or r0). In this case, the electron and hole are correlated; the exciton can be envisioned as a quasi-particle moving around inside the nanostructure with only little energy increment due to confinement. In this case, the infinite potential well model (within the single band effective mass approximation) gives a reasonable description of the experimentally observed shift in the exciton ground state energy.

The strong confinement regime corresponds to the condition a*B » d (or r0). In this case the confinement effect dominates over the Coulomb potential, and the electron and hole should be viewed as individual particles predominantly in their respective single particle ground states with only little spatial correlation between them. In this regime the exciton in the nanostructure "feels" the boundary effects strongly, and the inclusion of a finite height for the confining potential barrier has become an important requirement in order to account for recent experiments on the optical properties of small nanostructures.

In general the nanostructures are fabricated using two different materials A and B, considering that A and B are semiconductors with bandgaps eA and eB, respectively. In the nanostructure growth process, one of the materials is deposited over the other one and a potential barrier is formed in the interface due to the difference of bandgaps. According to the gap positions and values, three types of

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Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 4: Pages (43-111)

nanostructures can be obtained which we will identify as type I, II, and III nanostructures (see Fig. 1).

In the type I nanostructures, B material is a barrier for both the valence and conduction electrons, which are localized within the same A material. Examples of type I nanostructures are GaAs-Ga(Al)As, Ga0 47Al0 53As-InP, GaSb-AlSb, etc.

In the type II nanostructures (also called staggered nano-structures) one material acts as a well for conduction electrons but as a barrier for valence electrons. Examples of the type II nanostructures are InP-Al0 53In048As where the electrons are mostly in the InP material and the holes in Al053In047As, and InAs-GaSb where the electrons are mostly in the InAs semiconductor and the holes are mostly in the GaSb semiconductor. In type II nanostructures, we deal with interface excitons where the interacting particles are spatially separated, a situation which is reminiscent of that of the bound impurity states created by impurities placed in the barriers of the quantum structures.

The general importance of the confined systems and nanostructured materials has been widely suggested to be a key process in the future of nanotechnology and of interest in diverse fields including magnetics, pharmaceutics, aerospace, nanoelectronics, optoelectronics, etc.

Several methods for obtaining the confined systems exist, namely, the reduction of some spatial directions to quantum scale, by the application of any kind of fields or by limiting the borders in the synthesis of the materials.

Several forms of classifying the confined systems exist; the most universal considers the number of directions where the particle could move freely. For example, quasi-two-dimensional systems (Q2D) have two directions for the free movement of the carriers and one confined spatial direction (see Fig. 2). The Q1D system has only one direction for free movement and two directions of confined movement where the carriers are compelled to move in a reduced space of quantum scale (see Fig. 3). The Q0D systems have no directions for free movement; the three spatial directions are confined (see Fig. 4). The energy spectrum in the spatial direction of confinement is always discrete if the particle remains totally confined and discrete and continuous if the particle remains partially confined.

Recently, a great number of experimental type articles have been published which are related to the impurity states and atomic systems confined in different types of nanostructures; we will mention some of them. Cheong and Jeong [1] studied the GaAs/AlGaAs quantum wells

using reflectance-based optically detected resonance spec-troscopy. This technique consists of monitoring changes in the strength of the e1 h1 excitonic reflectance feature induced by the far-infrared laser beam. A reflectance signal is modulated when the far-infrared photon energy matches the energy difference between two electronic states of the system under study. The more conventional optically detected resonance spectroscopy with which the interband photoluminescence intensity is monitored will be referred to as photoluminescence based optically detected resonance [2-6]. By comparing photoluminescence optically detected resonance and reflectance-based optically detected resonance results from the same samples it was established that reflectance-based optically detected resonance was sensitive to both neutral and negatively charged donors. In this work they proved the utility of this new technique, reflectance-based optically detected resonance, for investigating the electronic states of neutral and negatively charged donors in several GaAs/AlGaAs multiple-quantum-well samples. The high sensitivity of reflectance-based optically detected resonance reveals new internal donor transitions of negatively charged donors. These results were agreed with theoretical calculations [6, 7].

Zhu and Xu [8] presented advances in nanoscopic probing techniques that have led to the development of direct spatial and spectroscopic methodologies, which have revealed a wealth of information about single quantum dots. These techniques, combined with the favorable properties of naturally formed quantum dots, have allowed one to

Figure 1. Types of semiconductor nanostructures. (a) Type I; (b) type II; (c) type III. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

treat a single exciton confined to a quantum dot as a single solid state quantum absorber in analogy to atomic and molecular systems. Direct measurements on these systems, including absorption and coherent nonlinear spectroscopy, have revealed dipole moments and dynamic time scales of the individual excitations. Combined with microscopy, these probes have revealed global properties of the quantum dot system.

Zhao et al. [9] experimentally and theoretically studied the electronic structures of acceptors confined at the center of InxGa1-xAs/Al03Ga0JAs quantum wells. The aim of this investigation was to explore the effects of biaxial deformation potential on acceptors confined in quantum well structures. Satellite peaks related to 2S and 2P of the confined acceptors were observed in selective photoluminescence spectra. The deduced energy separations between the acceptor heavy-hole-like ground state and different excited states were compared with the theoretical calculated results. The impurity states were calculated using a four-band effective mass theory, in which the valence band mixing as well as the mismatch of the band parameters and the dielectric constants between well and barrier materials were taken into account. An excellent agreement between experimental and theoretical data was found.

In [10] coherent transport in Si metal-oxide-semiconductor field-effect transistors with nominal gate lengths 50-100 nm and various widths at very low temperature were investigated. Independent of the geometry, localized states appear when G = e2/h and transport is dominated by resonant tunneling through a single quantum dot formed by an impurity potential. They found that the typical size of the relevant impurity quantum dot is comparable to the channel length and that the periodicity of the observed Coulomb blockade oscillations is roughly inversely proportional to the channel length. The spectrum of resonance and the nonlinear I-V curves allowed them to measure the charging energy and the mean level energy spacing for electrons in the localized state. Furthermore, they found that in the dielectric regime the variance var(ln g) of the logarithmic conductance ln g is proportional to its average value (ln g), consistent with one-electron scaling models.

On the other hand, there are many theoretical works carried out on the impurity states and atomic systems confined in the nanostructures. We will mention some of them. Thus, in [11] the effect of a spatially dependent effective mass in a finite GaAs/AlxGa1-xAs parabolic quantum well with a magnetic field on hydrogenic impurity ground state (1S) binding energies and lowly excited state (2p± -like) was calculated, respectively, as a function of well width and magnetic field and impurity position by using the one-dimensional method. They compared their results with square quantum wells and without the effect of spatially dependent effective mass result. The physical meaning of the square wavefunction was discussed.

Hsieh and Chuu [12] studied the binding energy of the ground state of a hydrogenic impurity located at the center of a multilayered quantum wire within of the effective mass approximation. The multilayered quantum wire consisted of a core wire (GaAs) coated by cylindrical shell (Ga1-xAlxAs) and then embedded in the bulk (Ga1-xAlxAs). The calculation was performed, by using a trial wavefunction. To make a comparison, the ground and excited state (1s, 2p, and 3d states) binding energies of a hydrogenic impurity located at the center of a single-layered quantum wire were also calculated. It was found for small wire radius that the ground state binding energy of the hydrogenic impurity located at the center of a multilayered quantum wire behaves very differently from that of a single-layered quantum wire.

Hsieh [13] calculated the ground state binding energy of the hydrogenic impurity located at the center and off center of a multilayered quantum wire in a constant magnetic field applied parallel to the wire axis by using the effective mass approximation. His system was constructed as a core wire made of GaAs surrounded by a cylindrical shell of Alx Ga1-xAs and then embedded in the bulk of Al^ Ga1-yAs. A variational trial wavefunction was proposed. It was found for a small wire radius that the ground state binding energy of a hydrogenic impurity located at the center of a mul-tilayered quantum wire behaves very differently from that of a single-layered quantum wire. The calculation showed that the binding energy depends on the potential profiles, potential barrier height, impurity position, shell thickness, magnetic field, and the difference between the Al concentration contained in the shell and bulk regions. His trial function was also able to reproduce the binding energies of a hydrogenic impurity located at the center of a single-layered quantum wire; a good agreement with the previous results was obtained.

In [14] the ground state energies of several interacting electrons confined in a parabolic dot in two dimensions were obtained by using hyperspherical coordinates and high order perturbation theory. The effect of a perpendicular magnetic field is to change the ground state discontinuously in orbital angular momentum L. The preferred values of L for the ground state and the associated electronic structures were studied in detail. It was found that the effective interaction between two electrons moving in different cyclic orbits is a short-range attraction matched a long-range repulsive tail. Because of this, electrons tend to fill adjacent cyclic orbits and form bunches in the ground states. The effects of an impurity ion were also considered.

Chen et al. [15] employed a simple approximation treatment, which combines the spirit of the variational principle and the perturbation approach to study the confined phonon effect on the binding energy of a hydrogenic impurity located inside a quantum well. The electron bulk longitudinal optical phonon and the electron surface phonon interactions were both taken into account in their calculation. The image charge effects may be significant if there is a large dielectric discontinuity between the well and the surrounding medium. However, since the purpose of this work is to concentrate on the confined phonon effects on the impurity binding energies, the image effects were not taken into account in their calculation.

Sali et al. [16] reported a calculation of the photoion-ization cross-section describing the excitation of a shallow bound electron from the hydrogenic donor impurity ground state to three of the conduction subbands nx, ny, and nz corresponding, respectively, to the x, y, and z directions of the dot in the case of an infinite well model. The photoion-ization cross-section associated with a hydrogenic shallow donor impurity was calculated as a function of the excitation energy hw in a GaAs quantum dot for several values of the dimensions of the dot and for an on-center donor impurity and off-center donor impurities.

Reyes-Gómez et al. [17] extended the fractional-dimensional space approach to study exciton and shallow-donor states in symmetric-coupled GaAs/Ga1-xAlx As multiple quantum wells. In this scheme, the real anisotropic "exciton (or shallow donor) plus multiple quantum well" semiconductor system is mapped, for each exciton (or donor) state, into an effective fractional-dimensional isotr-opic environment, and the fractional dimension is essentially related to the anisotropy of the actual semiconductor system. Moreover, the fractional-dimensional space approach was extended to include magnetic-field effects in the study of shallow-impurity states in GaAs/Ga1-xAlxAs quantum wells and superlattices. In their study, the magnetic field was applied along the growth direction of the semiconductor heterostructure and introduced an additional degree of confinement and anisotropy besides the one imposed by the heterostructure barrier potential. The fractional dimension was then related to the anisotropy introduced both by the heterostructure barrier potential and magnetic field. Calculations within the fractional-dimensional space scheme were performed for the binding energies of 1s-like heavy-hole direct exciton and shallow-donor states in symmetric-coupled semiconductor quantum wells, and for shallow-impurity states in semiconductor quantum wells and superlattices under growth-direction applied magnetic fields. Fractional-dimensional theoretical results were shown to be in good agreement with previous variational theoretical calculations and available experimental measurements.

Reference [18] presented that interelectron interactions and correlations in quantum dots can lead to spontaneous symmetry breaking of the self-consistent mean field, resulting in the formation of Wigner molecules. With the use of spin-and-space unrestricted Hartree-Fock calculations, such symmetry breaking was discussed for field-free conditions, as well as under the influence of an external magnetic field. Using as paradigms impurity-doped (as well as the limiting case of clean) two-electron quantum dots (which are analogs to heliumlike atoms), it was shown that the interplay between the interelectron repulsion and the electronic zero-point kinetic energy leads, for abroad range of impurity parameters, to the formation of a singlet ground state electron molecule, reminiscent of the molecular picture of doubly excited helium. A comparative analysis of the conditional probability distributions for the spin-and-space unrestricted Hartree-Fock and exact solutions for the ground state of two interacting electrons in a clean parabolic quantum dot revealed that both of them describe the formation of an electron molecule with similar characteristics. The self-consistent field associated with the triplet excited state of the two-electron quantum dot (clean as well as impurity doped) exhibits a symmetry breaking of Jahn-Teller type, similar to that underlying formation of nonspherical open-shell nuclei and metal clusters. Furthermore, they showed that impurity and/or magnetic field effects can be used to achieve controlled manipulation of the formation and pinning of the discrete orientations of the Wigner molecules. Impurity effects were further illustrated for the case of a quantum dot with more than two electrons.

In [19] Buonocore et al. presented the results of their theoretical studies on the influence of hydrogenic impurities on the electron localization in a deformed quantum wire. Although their primary interest was on porous silicon, it must be said that the calculation methods they illustrated are very general and can be applied to all those situations where the effects of the nanostructure geometry cannot be neglected.

Kayanuma [20] showed that the eigenvalue problem for bound states of hydrogenic donors located at the edge of a semiconductor surface could also be solved exactly for arbitrary values of the edge angle. Considering a neutral donor located at the edge of a crystal surface, which is formed by an intersection of two flat surfaces, he chose the edge line as the 2 axis. The two surfaces were assumed to be given by y = ± tan($0/2)x, with the angle between edges. For 0 < < ft, the crystal surface has a wedgelike shape, but for ft < < ft, it has a notch. The polar coordinates (r, 6, p) of the electron were defined in the usual way. It was assumed that the electron is completely confined in the region — /2 < $ < /2. Throughout his work, he adopted the effective mass approximation. The attractive potential due to the positive ion core was given by V(r) = —e2/Kr, in which k is the dielectric constant of the crystal. The penetration of the wavefunction outside the crystal and the effect of the image potential were both neglected. All these effects should be taken into account as a refinement of the model in actual analyses of experimental data [21]. Levine's model of the surface donor [22] was included in his model as a special case.

Kandemir and Cetin [23] studied the dependence of low-lying energy levels of impurity magnetopolaron subjected to a 3D parabolic potential on both magnetic field and spatial confinement length. To achieve this, they restricted to the case of bulk longitudinal optical (LO) phonons and introduced a trial wavefunction taken to be the direct product of an electronic part and a part of coherent phonons. They were also concerned with the study of the effects due to the electron-LO-phonon interaction, quantum confinement, and magnetic field on the cyclotron masses associated with the transitions between the ground and first-excited states of an electron bound to a hydrogenic impurity in a 3D parabolic potential. They found an analytical expression for the impurity magnetopolaron energy that allows them to perform a systematical analysis of the effects of both magnetic field and spatial confinement on the binding energies of impurity magnetopolaron in quantum dots, quantum well wires, and quantum wells. They established a unified treatment that allows them to make comparisons between the results of binding energies of impurity magnetopolarons in these three systems.

Schlottmann [24] studied an Anderson impurity placed at the center of a nanosized metallic sphere. The main difference with the standard mixed-valence problem is that the energy spectrum of the host is now discrete. Similar to the Kondo impurity [25], the problem could be mapped onto the Bethe ansatz solution of the Anderson model [26, 27]. The model and the Bethe ansatz equations diagonalizing were solve numerically for the ground state and the lowest energy charge and spin excitations. The energies of the states increase monotonically with the /-level energy. For an even number of electrons in the system (in s states and localized at the impurity), the impurity in the ground state is spin compensated into a spin singlet via the Kondo effect. The specific heat and the susceptibility are exponentially activated at low T due to the discreteness of the energy spectrum, with the gaps given by the lowest energy charge and spin excitations. The model also represented a quantum dot as a side branch to a short quantum wire.

Yen [28] theoretically studied resonant states of shallow donor arbitrarily lying in a two-dimensional quantum well structure based on the multisubband model. The density-of-states spectra of the resonant states were obtained using the resolvent operator technique. For illustration, he calculated the binding energy, the density of state spectrum width, and the resonance energy shift of the 2p0 state in a quantum well as a function of the impurity position. Considerable coupling was found. It caused a capture or escape time as short as «0.1 ps.

Apalkov and Chakraborty [29] considered a system of an incompressible quantum Hall liquid in close proximity to a parabolic quantum dot containing a few electrons. They observed a significant influence of the interacting electrons in the dot on the excitation spectrum of the incompressible state in the electron plane. Their calculated charge density indicates that, unlike the case of an impurity, interacting electrons in the dot seem to confine the fractionally charge excitations in the incompressible liquid.

Yang and Hsu [30] calculated the binding energy of the ground state of hydrogenic impurity seated at the center of quantum well GaAs/Ga1-xAlxAs with the inclusion of the effect due to the image charges (excluding the self-energy image potential). For a given depth of the quantum well, they found that the absolute value of the image potential energy without the metallic mirror is equal to that with a fixed metallic mirror while a certain width of the quantum well is met. Hence, it was proposed that the binding energy of the shallow impurity ground state in the quantum well with and without the metallic mirror could be separately measured by the variation of the width of quantum well. The contribution of the image potential energy to the binding energy of the shallow impurity ground state could then be deduced.

Essaoudi et al. [31] discussed the stability of an exciton bound to an ionized donor impurity in a GaAs/Ga1-xAlxAs semiconductor quantum well subjected to an external magnetic field for different values of the impurity location. The binding energy was calculated in the effective mass approximation by means of variational method. At zero magnetic field, the complex becomes unstable when the impurity is far away from the center of the well. When the magnetic field increases, the stability holds in all cases for an impurity located at the center of the well.

Brandi et al. [32] treated the interaction of light with a spherical GaAs/AlGaAs quantum dot within a dressed-band approach. The Kane band structure scheme was used to model the GaAs bulk semiconductor and the interaction with the laser field is treated through the renormalization of the semiconductor energy gap and conduction/valence effective masses. This approach, valid far from resonances, was used to investigate the light shifts induced in the electronic and shallow on-center donor states in semiconductor quantum dots, which were shown to be quite considerable. This model calculation could be extended to include magnetic-field effects, and it was suggested that the strong localization of the electronic and impurity states due to the quantum dot and enhanced by laser confinement could prove useful for manipulation of electronic and donor states in some proposed solid-state-based quantum computers.

Stebe et al. [33] concentrated their study on the (D+, X) complex resulting from the binding of an exciton X to an ionized hydrogenic donor impurity D+. Its possible existence was predicted in 1958 by Lampert [34]. Its stability and binding in 3D semiconductors was the subject of several theoretical studies within the effective-mass approximation as a function of the electron to hole effective mass ratio a = me/mh. As a result, it appeared [35] that the (D+, X) complex is stable if a < ac = 0.454. It was expected that due to the quantum confinement, the (D+, X) complex will be more stable in quantum wells than in 3D semiconductors. There are few measurements available [36-40] on an exciton bound to ionized donors in quantum wells. From the theoretical point of view, variational determinations of the ground state energies [41, 42] have first proven that the (D+, X) complex with the impurity located at the center of the well is stable in GaAs/Ga1-xAlxAs quantum wells with x = 0.15 and x = 0.30.

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