1. Introduction Present semiconductor physics appears to be the physics of systems of redu- ced dimensionality. Artificially made semiconductor structures show a surpri- sing variety of new interesting properties that are completely different from solid-state bulk materials and have never been observed there. The fabrica- tion of single or periodic potential wells by simply combining two semicon- ductor materials of different bandgap energies and with spatial dimensions confining the motion of electrons and holes, results in many impressive pos- sibilities for engineering of the semiconductor properties. Two-dimensionally layered material systems exhibiting quantum confinement in the direction of the growth axis are widely investigated and their study represents a new, rapidly developing field in solid-state physics. It is understandable that scientists' efforts are directed to further decrease the dimensions to quasi-one-dimensional or zero-dimensional structures. One possibility for obtaining zero-dimensional structures is the inclusion of sphe- rical semiconductor particles in a dielectric, transparent matrix. Such a struc- ture is mesoscopic in all three dimensions, i.e. its radius is large compared to the lattice constant, but comparable to the spatial extension of the wave func- tions of excitons, electrons or holes in the corresponding bulk semiconductor material. To define these particles many different terms have been used such as quantum dots (QD), nanocrystals, microcrystallites (MC), Q-particles, or nanoclusters. The small semiconductor spheres of, for example, II-VI or I-VII compounds can be grown in different matrices, such as glasses, solu- tions, polymers or even cavities of zeoliths, and by different manufacturing processes, for example by melting and annealing processes, by organometal- lic chemistry or by sol-gel techniques. Evidence for quasi zero-dimensional structures has likewise been obtained by investigating the epitaxial growth on highly mismatched substrates. It results in the development of small is- lands on the substrate surface with high regularity and sufficiently small sizes to show quantum confinement. Quantum dots are very attractive and interesting objects for scientific re- search of three-dimensionally confined systems. Their investigation requires an insight into many different branches of knowledge such as solid-state phy- sics, molecular physics, photochemistry, nonlinear optics and ultrafast spec- troscopy, materials sciences and structural analysis. The simplest, naturally 2 .1 Introduction Wavelength (nm) 057 650 600 055 005 450 400 4 CdSe QDs /'-" T=4-2 K/ R = 83 mn f_._~ 3 2 Fig. 1.1. Spectra of linear absorption of CdSe na- nocrystals with different 1 radii R = 1.2 nm, 2.6 nm and 83 nm, embedded in a 0 borosilicate glass matrix, 1.8 2.0 2.2 2.4 2.6 11.2 3.0 data taken from Ekimov Photon Energy (eV) et .la (1985a, 1993) given zero-dimensional systems are spherical semiconductor nanocrystals em- bedded in glasses or organic matrices. This has proved to be the ideal model system for the study of basic questions of three-dimensional confinement in semiconductors. Beyond basic research there is always the question of pos- sible applications. Promising ideas exist regarding the fields of integrated optics (active elements in waveguide structures, fast switching devices, light- emitting and laser diodes) and of physical chemistry (solar energy conversion, photocatalysis). The growth of quantum dots in glasses is one of the oldest and most frequently used techniques. The first hints of the existence of small inclu- sions of, for example, CdSe and CdS in silicate glasses causing its yellow to red color were connected to the development of X-ray analysis and pu- blished in the early 1930s by Rocksby (1932). Since the 1960s, semiconduc- tor doped glasses have been widely applied as sharp-cut color filter glas- ses in optics. The concept of quantum confinement, and by that the di- stinction between the coloring of the glasses by changes in stoichiometry of CdSxSel-x mixed crystals or by size changes of the binary nanocrystals, was introduced by Efros and Efros (1982), and confirmed experimentally by Ekimov and Onushenko (1984). At the same time, the change of color of se- miconductor colloidal solutions has been discussed in the context of quan- tum confinement effects by Rosetti et al. (1984). A period of control of the growth process of these nanocrystalline semiconductors followed in the 1980s combined with a detailed investigation of their linear and nonlinear optical behavior. As demonstrated in Fig. 1.1, a first approach in understanding the beha- vior of quantum dots is mainly the investigation of their optical properties, in particular of their absorption spectra. Using the data first reported by Ekimov et al. (1985a, 1993) the size-dependent change of the absorption spectra is plotted in Fig. 1.1 for CdSe-doped glasses. Nanocrystals of large radii (e.g. R = 38 nm) show the typical spectrum of bulk CdSe. The spec- .1 Introduction 3 trum is characterized by the sharp band-edge and, close to that, by the series of exciton states. Three peaks are located at the energy of the A-exciton (E = 1.826eV), B-exciton (E = 1.85eV) and C-exciton transitions (E = 2.26 eV, T -- 8.1 K, (Landolt-BSrstein 1982) originating from the spin-orbit and crystal-field splitting of the valence-band states. The spectral positions of the absorption peaks shift to higher energies and the lines become broader with decreasing sizes of the nanocrystals. The explanation for these spectral tnedneped-ezis changes in the absorption spectra by a effect was the begin- ning of the intensive research on three-dimensional quantum confinement in solid-state semiconductor composite materials. The main intention of this book is an approach to the three-dimensional quantum confinement in semiconductors by analyzing the optical properties of the corresponding semiconductor structures. To do this, one has to consider the absorption coefficient of an ensemble of quantum dots inside a transparent matrix. The averaged absorption spectrum ~(w) can then be expressed by )w(~5 P /dR ~4 3R P(R) ,3O(DQa )R (1.1) - 2- with p the volume fraction of the semiconductor material, DQV the average P(R) quantum dot volume, R the radius and a characteristic distribution function for the dot sizes, as well as O/QD(0) , R) the absorption coefficient of a single quantum dot. As a result of the quantum confinement effect, the absorption coefficient DQ~O is strongly dependent on the radius R of the dot. The absorption spectrum is given by a series of Lorentzian lines for the ground and excited states at energies E~D = hwj, with homogeneous line widths Fj and oscillator strengths .fj h G (R) R) ~ j(R) 2 (1.2) J (E~D(R)-hw)2+(hF2(R))2 " Equations (1.1) and (1.2) show that one needs information about the radius P(R), R, the size distribution and the semiconductor volume fraction p to correlate the appearance of structures in the absorption spectrum to, for ex- ample, electronic states of the quantum dots, as well as suitable relations for E(R), F(R) f(R), the size dependence of and the energy, homogeneous line broadening and oscillator strength, respectively. When the quantum dots are additionally exposed to, for example, electric fields, pressure or high photon densities, further interesting problems appear since then quantum confine- ment and external forces combine and influence the energy, line broadening and oscillator strength of the optical transitions. The size-dependence of the optical properties of quantum dots has been one of the main subjects of research work over the last decade. The enormous growth of experimental data (e.g. Ekimov et al. 1985a, 1991; Borelli et al. 1987; Brus 1991; Bawendi et al. 1990a; Henglein 1988; Wang 1991a), and the 4 .1 Introduction increasing exactness of the theoretical concepts (e.g. Banyai and Koch 1993; Hang and Banyai 1989; D'Andrea et al. 1992) resulted in a steadily growing understanding of the electronic and optical properties of quantum dots. In the present book an attempt has been made to provide an overview of the variety of phenomena which can influence the physical properties of quantum dots, starting with the matrix material, size, structure, and interfaces, but considering also excitation densities, external and internal fields, lattice pro- perties etc. The choice of topics covered here is determined by experimental points of view. However, this work is being written in a period of rapid deve- lopment and therefore without the claim of presenting a complete discussion of all experiments carried out in this field during the last few years. We will mainly concentrate on quantum dots of II-VI semiconductors em- bedded in glass matrices, But consider also other semiconductor compounds as well as other types of matrix materials. The widespread quantum dot sy- stems based on II-VI materials show absorption structures in the visible and near-ultraviolet part of the spectrum and are therefore compatible with a great number of laser sources used for experiments. The experimental tech- niques applied comprise almost all standard experiments of linear and nonli- near optics such as absorption, steady-state and time-resolved luminescence, pump-and-probe spectroscopy, and degenerate and non-degenerate four wave mixing. In this work I start with the growth process of the nanocrystals and then consider under which conditions a nanocrystal can be defined as a quan- tum dot. I will demonstrate that the application of experimental methods ta- ken from nonlinear optics of bulk materials is a powerful tool for identifying energy states and homogeneous line broadening. As a main topic, the new, specific properties related to excitons and biexcitons in three-dimensionally confined systems will be discussed. I will deal with the different mechanisms of phase relaxation and their influence on three-dimensional confinement. The interaction with the matrix, the specific aspect of optics of composite materials and the role of interfaces will be further topics. Furthermore, I will give a short overview of experimental results which have been obtained when applying external fields. Possible applications will be discussed as well. Currently, quantum dots derived from III-V compounds are also being studied intensively. They are prepared by etching techniques of two-dimen- sionally confined layered structures, ion implantation or island-like epitaxial growth. The detailed analysis of the properties of these structures is beyond the scope of this work. However, a brief survey will be given accompanied by a list of corresponding references. Recent summaries of the development in the field of interesting transport properties of quantum dots, such as sin- gle electron transport and Coulomb blockade, can be found, for example, in Kuchar et al. (1990), Merkt (1990), Reed (1993), Geerligs et al. (1993) and in the references therein. Over the last three years enormous progress could be observed in the investigation of quantum dots obtained from indirect-gap semiconductor ma- .1 Introduction 5 terials, for example Si-nanoclusters. Although we cannot provide a com- prehensive presentation of this field, a short summary of the first stu- dies will be given, taking into account that indirect-gap quantum dots are of some specific interest. For more information I refer, for example, to Littau et al. (1993), Takagahara and Takeda (1992), Brus (1994) and refe- rences therein. In the field of theory the influence of Coulomb interaction, the description of the electron and hole states, the electron-phonon coupling, the problem of the dielectric confinement and of the interface, high-density phenomena, field action etc. have been treated during the last five years. A comprehensive representation of the theory of semiconductor quantum dots may be found in the book of Banyai and Koch (1993). Therefore a detailed review of the results of theory of quantum dots has been omitted. I refer to theory only in the case where the theoretical results are complemented or confirmed by the presented experiments. Finally, I should mention that the concept of quantum dots is already included in modern textbooks (Peyghambarian et al. 1993; Haug and Koch 1993; Klingshirn 1995). For further information about quantum confinement, the book by Bastard (1988) provides a very good comparison of the physics of two-dimensionally confined layered systems and three-dimensionally confined quantum dots. 2. Growth of Nanocrystals This chapter deals with the problems of the growth process of nanocrystals, i.e. with the analysis of the laws of growth, the final sizes of the nanocrystals and expected size distributions. Different procedures have been reported for the growth of nanocrystals, such as the growth inside the cavities of a zeolith, or the growth in an organic environment by stabilization of the nanocrystals by organo-metallic ligand molecules at the surface (Chestnoy et al. 1986; Henglein 1988; Wang et al. 1989, 1995; Bawendi et al. 1990a; Bagnall and Zarzycki 1990; Spanhel and Anderson 1991). In glasses, it is common practice to describe the growth of nanocrystals by the model of condensation from a supersaturated solid solution (Lifshitz and Slezov 1961; Ekimov et al. 1985a, .)1991 The understanding of the growth process is a prerequisite for the under- standing of all basic properties of three-dimensional confinement in semicon- ductor doped glasses. Often, the description of quantum dots starts with the introduction of different classification schemes, for example the characteriza- tion of the confinement range with respect to the ratio of the radius R of the nanocrystal to the Bohr radius as of the exciton in the corresponding bulk material (strong, medium or weak confinement). Further classification is possible with respect to the treatment within the frame of the effective mass approximation or within the concepts of cluster physics. Other interesting parameters are the volume-to-surface ratio, the ratio between the confine- ment energy and the phonon energies, and the differences in the dielectric constants between the semiconductor and the matrix material. Thus, the growth and the following characterization procedure have to provide a minimum of data before the study of basic electronic and optical properties of quantum dots becomes meaningful. The most important in- formation to be obtained from growth analysis concerns the sizes, the size distribution, stoichiometry, structure and the interface configuration of the nanocrystals. 8 .2 Growth of Nanocrystals 2.1 Growth of Nanocrystals in Glass Matrices 2.1.1 The Diffusion-Controlled Growth Process How big will nanocrystals be if they are grown in a certain matrix, over a fixed time and within a given range of temperature? How much time is needed to achieve a certain mean radius and what size distribution is then obtained? To describe the growth process it is interesting to look for models giving answers to these questions. In the course of a real growth process, however, several growth stages may occur and sometimes even coexist. For example, we know the nucleation process with the formation of stable nuclei, the normal growth stage where these nuclei grow from the supersaturated solution and, at low values of su- persaturation, the onset of the competitive growth where the larger particles grow due to the dissolution of smaller particles. For simplification, theoreti- cal description commonly starts with the separate analysis of these different stages of growth to obtain simple analytical expressions. The attempts to find a functional description of the growth process, or generally of phase transitions, go back to the early 1930s (Becker and DSring 1935). The main idea of the classical homogeneous nucleation theory consists in the ansatz that the phase transition proceeds (at constant temperature) between a supersaturated vapor of n monomers (molecules) and droplets containing in (i > )2 monomers. The starting point is the rate equation determining the number of monomers n inside the droplets (clusters). This equation relates the gain gn and the losses In, where the size of the clusters changes by the gain or the loss of one monomer dn d-7 = gn - In. (2.1) The resulting change in the monomer density dn/dt is determined by the difference between the number of monomers impinging on and evaporating from the cluster through the surface into the supersaturated vapor phase. The variety of the occurring growth mechanisms is expressed by introducing the explicit terms for gn and In. For instance, for the growth of thin films the gain gn can be determined by surface diffusion, interface transfer (nonthermic transport), or cluster movement and unification (Turnbull 1956; Chakraverty 1967; Abraham 1974; Landau and Lifshitz 1974; Koch 1984). The losses In were derived by analyzing the energy balance between surface and volume energy contributions of the condensed phase. The primary aim of growth analysis is to evaluate asymptotic functions for the cluster size distribution. Involving detailed expressions for g~ and In in (2.1) and combining this equation with the matter conservation law and the equation of continuity, the cluster growth laws have been calculated. One of the most famous models is the Lifshitz-Slezov model (Lifshitz and Slezov 1961). It is often used to fit the growth process of semiconductor nanocrystals in glasses and is therefore presented here. 1.2 Growth of Nanocrystals in Glass Matrices 9 A search is underway for expressions for f(R, t), the size distribution for clusters with radius R at the time t. The continuity equation of the probability density in cluster space coordinates is Of(R ,3 t) O~ tO + f(R 3, t)./~3 = 0. (2.2) It = dR~dr is the cluster growth velocity and has to be determined from the rate equation (2.1). This can be done by the transformation of the average number of monomers per cluster to the average cluster size (R) and the consideration of the conservation of the total amount of matter. The solution of (2.2), and thus f(R3,t), depends on the different mechanisms considered for gn and In and leads to different expressions for/~. The Lifshitz-Slezov model assumes that the transport of monomers to the cluster surface is realized by diffusion in a supersaturated solution. The mo- nomer distribution in time and space is obtained from the diffusion equation which yields for/~ dR _ R A_ :a (2.3) dt D /~" with D the diffusion coefficient, A the supersaturation A = C - Coo (2.4) and 2a = v coo. (2.5) C~ is the concentration of the saturated solution in equilibrium, C the ave- raged concentration of the supersaturated solution, v the volume of one mo- nomer in the solution and a the surface energy (see for more details Lifshitz and Slezov 1961; Abraham 1974; Koch 1984). The critical droplet radius is defined by R~ = ~ (2.6) A0 The supersaturation is a function of time. Thus, for every value IA of the supersaturation there exists a critical cluster radius Rcr = Rcr(t), for which the cluster is in equilibrium with the solute. If R > Rcr, the cluster will grow; in the case of R < ~cR it will dissolve. For illustration, a scheme of the growth process of a supersaturated solu- tion is given in Fig. 2.1. A small cluster is instable because the concentration around it is too high, and a large cluster is instable because there is a very low concentration and a depletion region around it. The monomers feel a concentration gradient driving the diffusion of matter from smaller to larger clusters. A diffusion-controlled competitive growth of the nanoclusters is the result. In the framework of the Lifshitz-Slezov analysis, the asymptotically 01 .2 Growth of Nanocrystals ytisned latsyrconan Fig. 2.1. Concentra- noitautculf R < rcR R = rcR R > reR tion profile around the oC nanocluster during the Q growth process. ~' is the average concentra- r-, r tion within the supersa- Q turated solution, Co is ).C the concentration inside spatial coordinate the nanocluster stable size is the one for which the concentration directly near the surface (JR is equal to the average concentration of the matrix (CR = C). Applying the Lifshitz-Slezov model to the growth process in glassy ma- trices, the surface free energy, one of the parameters determining Rcr, will depend on both matrix environment and semiconductor material. Further- more, in the Lifshitz-Slezov analysis the number of nuclei is considered to be constant. To avoid a high formation rate of nuclei, a low value of supersatu- ration is required. When we carry out the asymptotic analysis we find that the final size distribution evaluated by Lifshitz and Slezov (1961) is asym- metrical (see below, Fig. 2.3) and independent of the initial distribution. Its maximum value does not correspond to the critical radius and is slightly shifted to larger radii R > Rcr. The slope to larger sizes is abrupt and na- noclusters of sizes larger than twice the critical radius do not exist. However, the function gradually decreases toward smaller sizes with a long tall. The volume of the cluster grows linearly in time corresponding to Rav= t (2.7) It depends on the diffusion coefficient D and, via a (2.5), on temperature and surface energy. The asymptotic limit of the classical Lifshitz-Slezov analysis describes the growth process in the case of long heat treatment times. But what determines the initial radii and the growth process at very early stages? Let us consider the thermodynamics of nucleation without the problem of long-time kinetics. In the framework of steady-state homogeneous nucleation theory, condensation is driven in analogy to the simple problem of liquid droplets in water vapor (Abraham 1974; Turnbull 1956) by the condition of minimization of free energy. In equilibrium the number of condensed droplets N can be estimated by an Arrhenius ansatz N = No exp(-AF/kBT). (2.8) AF is the free energy necessary for the formation of the nuclei. Thus, to become stable, a cluster must acquire an excess of free energy compared to 1.2 Growth of Nanocrystals in Glass Matrices 11 the single vapor molecules. In the vapor/droplet system the free energy is given by r74 R 3 AF = r74 R 2 a - - kB TIn p (2.9) 3v Poo The first term is the contribution of the surface free energy, the second term represents the contribution to AF caused by the bulk free energy change, p/poo where is the vapor supersaturation. If R is the droplet radius, the posi- tive term varies in proportion to R 2, while the opposing negative term varies in proportion to R 3. Thus, regardless of the magnitude of the coefficients of these terms, in some sufficiently small interval of R above zero the positive R 2 term dominates the R 3 term. We see that the number of nuclei is strongly influenced by the value of p/poo the supersaturation. For any case > 1 the free energy AF exhibits a maximum. The radius at which AF attains its maximum is obtained from the condition d AF - 0 (2.10) dR and again corresponds to the critical nucleus radius 2a v Rcr - kB T In _2_ " (2.11) Por The larger the supersaturation, the smaller the critical radius Rcr and even small clusters can grow. Low values of supersaturation lead to a large critical cluster radius Rcr. Using (2.11) and (2.9), the height of the energy barrier for the nucleation AFcr can be evaluated as 167ra 3 v 2 r74 2 (2.12) AFc~ = 3 (kB T In _2_)2 = -~- Rcr a . oo91 When the degree of supersaturation is increased gradually, the number of im- pinging monomers grows and the probability increases that more and more monomers will overcome the top of the activation barrier in a given time. Additionally, the increase of supersaturation reduces the height of the energy barrier to such an extent that the probability that some of the subcritical clusters will grow accidentally to supercritical size nearly approaches unity. At that value of supersaturation, homogeneous nucleation becomes an ef- fective process and the phase transition sets in. Nucleation is intrinsically a probabilistic event. The radius fluctuation probability around the critical radius has to be described by statistics. In this simple consideration, based on homogeneous nucleation theory, one would expect a Ganssian symmetric size distribution around Rcr. This model from the system water/vapor can be transferred to more gene- ral cases by introducing the expression Ao for an universal supersaturation. Also nanocrystals in glassy matrices can be grown near nucleation to obtain