Semiconductor quantum dots towards a new generation of semiconductor devices

Transcription

1 Eur. J. Phys. 21 (2000) Printed in the UK PII: S (00) Semiconductor quantum dots towards a new generation of semiconductor devices Lucjan Jacak Institute of Physics, Technical University of Wrocław, Wyb. Wyspiańskiego 27, Wrocław, Poland Received 22 March 2000 Abstract. A short overview of the technology and physics of semiconductor quantum dots is given. Different methods of creation of quantum dots and mechanisms of carrier confinements are described. The fundamental properties of these systems are discussed including current attempts for applications in new ultra-small opto-electronic semiconductor devices. 1. Introduction The rapid development of semiconductor technology, including various epitaxy techniques for preparation of even single-molecular layers of materials, and lithography methods for fabrication of very densely packed electrical circuits, has resulted in quite new possibilities for artificial creation of ultra-small physical systems with controlled properties. In this unique situation the technical manipulation of materials meets, however, with limitations imposed on small systems by quantum mechanics. Description of extremely small systems taken from macroscopic physics is irrelevant, as regards both opto-electronic properties and their mechanical behaviour. This scale of miniaturization seems to be the natural limit for microelectronics or, more generally, nanotechnology. The microchips for computers currently produced by photolithography techniques have their smallest elements of dimension usually not less than 380 nm which corresponds to the lowest wavelength of visible light. Lithography using ultraviolet or x-ray radiation still meets with strong technical complications and is not yet acceptable for wide commercial use. Furthermore, electron-beam and ion-beam lithography, which allow for far higher precision, are inconvenient for industrial production since they can be thought of as resembling rewriting by pencil as opposed to the xerox -like copying performance of photolithography. Thus, typical single microstructures (e.g. transistors in microchips) are still rather far from the range of quantum mechanics, and classical intuitions about physical mechanisms are roughly applicable for their description. The same holds for elements currently fabricated for so-called nanotechnology (news on nano-engines and other ultra-precise mechanical devices tends to concern objects of micrometre rather than true nanometre scale) and hence they are again in the classical range. However, if the dimensions of systems reach the scale of a few nanometres, the classical description of their properties will be very misleading. Indeed, the practical realization of a quasi-two-dimensional electron gas in semiconductor heterostructures, with electrons confined in a layer of width a few nanometres (thickness of the order of the electron de Broglie /00/ $ IOP Publishing Ltd 487

2 488 L Jacak wavelength) resulted in discoveries such as the quantum Hall effect, which revealed sharply the unusual quantum nature of geometrically confined systems. The fractional quantum Hall effect and further experiments with so-called Hall metal [9] even suggest an essential development of quantum theory and lead to exotic new 2D particles: composite fermions instead of simple 3D electrons. Since the early 1980s the rapid progress in laboratory techniques of confinement of electrons, first in a plane, as in Hall configurations, and, subsequently, in other directions, has led to the creation of thinner objects called quantum wires [40] in which electrons move freely in one direction only and, finally, quantum dots (QD) [8, 30, 44, 52] with electrons enclosed in small boxes with dimensions of the order of tens or even a few nanometres. These completely confined electrons in artificially constructed boxes resemble ordinary atoms; hence, frequently, quantum dots are called artificial atoms, and they reproduce many properties from atomic physics. Quantum dots seem, however, to be much more flexible than atoms: QD are without nuclei inside, so they can contain an arbitrary number of confined electrons, from one to hundreds or thousands, enclosed in a small spatial volume. Their shapes and various other properties can be modelled on the stage of preparation, and in this they also differ from atoms. Currently there are many methods for manufacturing quantum dots [24]. Let us briefly describe them. 2. Methods for creating quantum dots Unlike quantum wells, where the motion of carriers is restricted to a plane through the crystallization of thin epitaxial layers, the creation of quantum wires or dots, which confine the carriers in a space with at least two of the three dimensions limited to the range of the de Broglie wavelength, requires far more advanced technology. Etching The earliest method of manufacturing quantum dots was implemented by Reed et al [44], who etched them in a structure containing a two-dimensional electron gas. Subsequent steps of this process are shown in figure 1(a). The surface of a sample containing one or more quantum Figure 1. (a) Process of quantum dot etching [45]. (b) Etched quantum dots in a GaAs/AlGaAs well, electron scanning microscope picture [48].

3 Semiconductor quantum dots 489 wells is covered with a polymer mask, and then partly exposed (1). The exposed pattern corresponds to the shape of the created nanostructure. Owing to the required high resolution, the mask is not exposed to visible light, but to the electron or ion beam (electron/ion-beam lithography). In the exposed areas the mask is removed (2). Later, the entire surface is covered with a thin metal layer (3). Using a special solution, the polymer film and the protective metal layer are removed, and a clean surface of the sample is obtained, except for the previously exposed areas, where the metal layer remains (4). Next, by chemically etching the areas not protected by the metal mask (5), slim pillars are created, containing the cut-out fragments of quantum wells (6). In this way, the motion of electrons, which is initially confined in the plane of the quantum well, is further restricted to a small pillar with a diameter of the order of nm. Owing to the simplicity of the production of thin, homogeneous quantum wells, GaAs is the most commonly used material for creating dots by means of etching. Figure 1(b) presents a picture of real dots obtained using this method. Modulated electric field Another method involves the creation of small electrodes over the surface of a quantum well by means of lithographic techniques. Application of an appropriate voltage to the electrodes produces a spatially modulated electric field, which localizes the electrons within a small area. The lateral confinement created in this way has no edge defects, which are characteristic of etched structures. The process of spreading a thin electrode over the surface of a quantum well may produce both single quantum dots [2] and large arrays (matrices) of dots [19, 34, 39, 47]. Modulation of the electric potential, produced by applying a voltage to an electrode, can be realized by the previous preparation (using a lithographic technique) of a regular array of islets of non-metallic material (e.g. of the barrier material) on the surface of the sample. As a result, the distance between the electrode (overlaying the surface with the islets) and the Figure 2. (a) Quantum dots produced on InSb; electrons confined by the electric field (electron scanning microscope picture). Bottom: shape of the electrode and the configuration of band edges (valence and conduction bands) [47]. (b) Quantum dots created on the surface of GaAs in the selective MOCVD growth (scanning electron microscope pictures); the configuration of layers in a single dot is given; the width of the electron localization area at the top of the pyramid is 100 nm [16].

4 490 L Jacak plane of the quantum well is modulated, and the electrons are bound in small areas under the prepared islets. A photograph of a matrix of such dots, together with the profile of the confining potential, is shown in figure 2(a). Instead of modulating the distance between the electrode and the well, it is also possible to build a pair of parallel, thin electrodes above the well. The lower one can have regularly placed holes which is where quantum dots are to be created [1, 20]. A voltage applied to the pair of electrodes results in a change in both the dot size and the depth of the confining potential. The potential depth influences the number of confined electrons. However, when an additional electrode is introduced between the quantum-well layer and the doped layer, the number of electrons and the potential depth can be changed independently. Even more complicated systems of micro-electrodes are currently also applied [53]. Inter-diffusion between the barrier and the quantum well Quantum dots can also be obtained by local heating of a quantum well sample with a laser beam [6]. The laser beam guided along a rectangular contour surrounding an unilluminated area of a diameter of nm causes a rapid inter-diffusion of atoms between the well and the barriers, which creates a local modulation of the material band structure, i.e. the potential barrier, which surrounds the unilluminated interior of the rectangle. For larger dimensions of the illuminated rectangle the obtained effective potential which confines the electrons is flat inside the rectangle, and steep near the edge. With the decrease of the illuminated rectangle, the confined area shrinks. According to the authors of [6], for dimensions near 450 nm the effective potential confining electrons is close to an isotropic parabola. However, it should be mentioned that details of the electron confining potential in a quantum dot of any type cannot be measured directly (except for the geometric dimensions) and are alternatively obtained through an interpretation of various indirect effects, which are related to the electronic structure of QD. Semiconductor microcrystals It is also possible to create quantum dots in the form of semiconductor microcrystals. In the first experiment based on this idea, carried out by Ekimov et al [11], silicate glass with 1% addition of the semiconducting phase (CdS, CuCl, CdSe, CuBr) was heated for several hours at high temperatures, which led to the formation of appropriate microcrystals of almost equal sizes, depending on the temperature and heating time. The radii of these dots, measured in different samples, varied in the range nm. Selective growth The next method is the selective growth of a semiconducting compound with a narrower band gap (e.g. GaAs) on the surface of another compound with a wider band gap (e.g. AlGaAs) [16]. Restriction of growth to chosen areas is obtained by covering the surface of the sample with a mask (SiO 2 ) and etching miniature triangles on it. On the surface not covered by the mask the growth is then carried out with the metal organic chemical vapour deposition method (MOCVD), at a temperature of C. The crystals created have the shape of tetrahedral pyramids and hence, when the first crystallized layers are the layers of the substrate compound (AlGaAs), and only the top of the pyramid is created of GaAs, it is possible to obtain a dot with an effective size under 100 nm. Pictures of such dots, and the configuration of GaAs/AlGaAs layers, are shown in figure 2(b). Self-organized growth Another very promising method is the self-crystallization of quantum dots [41]. When the lattice constants of the substrate and the crystallized material differ considerably (7% in the

5 Semiconductor quantum dots 491 Figure 3. Evolution of islets self-assembled quantum dots (white circles), in the molecular beam epitaxy (MBE) growth of InAs on a GaAs surface; subsequent pictures correspond to the increasing coverage of 1.6, 1.7, and 1.8 monolayer; size of presented areas: 1 1 µm 2 [41]. case of GaAs and InAs, the most commonly used pair of compounds), only the first deposited monolayers crystallize in the form of epitaxial, strained layers with lattice constant equal to that of the substrate. When the critical thickness is exceeded, a significant strain which occurs in the layer leads to the breakdown of such an ordered structure and to the spontaneous formation of randomly distributed islets of regular shape and similar size. The shape and average size of the islets depend on the strain intensity in the layer related to the misfit of lattice constants, the temperature at which the growth occurs, and the growth rate. The phase transition from the epitaxial structure to the random arrangement of islets is called the Stranski Krastanow transition [49]. Figure 3 presents subsequent phases of the creation of islets for a pair of compounds with a misfit of lattice constants: InAs and GaAs, where the transition occurs at the 1.8 monolayer deposition. When the process of crystallization is terminated shortly after reaching the phase transition, the islets evolve to a quasi-equilibrium state, in which they assume the shape of pyramids [17, 31, 38], or flat, circular lenses [14, 43] formed on a thin layer of InGaAs (the wetting layer). Raymond et al [43] report the growth of self-assembled dots in the shape of lenses with 36 nm diameter and 4.4 nm height (with fluctuations of 5 10%). Marzin et al [38] obtained dots in the shape of regular pyramids with a square base of 24 nm side length and 2.8 nm height (with fluctuations of 15%), and a distance between neighbouring dots of 55 nm. Ultra-small pyramidal dots of dimensions less than 10 nm were also obtained in an InAs/GaAs interface [17]. Quantum dots formed in the Stranski Krastanow phase transition are called self-assembled dots (SAD). The small sizes of the self-assembled quantum dots, the homogeneity of their shapes and sizes in a macroscopic sample, the perfect crystal structure without edge defects, as well as the convenient growth process, without the need for any precise deposition of electrodes or etching, are among their greatest advantages, which stimulate hopes regarding their future application in electronics and opto-electronics. 3. Properties of quantum dots and prospects for their application Because of the full confinement of carriers inside the dots, their properties correspond to discrete quantum levels separated typically by up to 50 mev in energy. The discretization of the energy spectrum enhances the sharpness of the density of states. In figure 4, the densities of states corresponding to the reduction of the dimensionality of the systems are presented. The sharp density of states of QD makes these systems very convenient for laser applications. Indeed, the first reported quantum dot laser constructions [31] applying pyramidshaped InAs/GaAs ultra-small self-assembled dots and [12] with lens-shaped InAs/GaInAs

6 492 L Jacak Figure 4. Density of states as a function of energy in systems with different numbers of spatial dimensions: 3D, bulk material; 2D, quantum well; 1D, quantum wire; 0D, quantum dot. self-assembled dots, revealed better parameters in comparison with even the best lasers built on the basis of quantum wells. This related to both lower threshold injection current and higher operating temperatures, due to the incommensurability of long-wave phonons with the small dimensions of the dots (the so-called bottleneck effect). At sufficiently low temperatures (several K) the energy of the phonons is too low to excite the electrons in QD and the strong quantization of energy determines the electronic properties of these systems. The confining potential of QD is not singular, unlike the Coulomb potential in atoms, and could be expanded in a series with respect to radius: the first term of this expansion is the parabolic potential which is commonly applied for model investigation of QD. The advantage of this potential lies in the possibility of analytical solution of the single-electron problem in 2D parabolic confinement even in the presence of perpendicular static magnetic fields. The resulting one-particle states, known as Fock Darwin levels [15], are presented in figure 5. In the limit of high magnetic fields they transform into ordinary Landau levels of free 2D electrons in the magnetic field, but for lower fields the modifications of Landau levels due to confinement are significant. Figure 5. Evolution of the Fock Darwin energy levels in a magnetic field; dashed lines, the Landau energy levels; vertical arrows, allowed optical transitions.

7 Semiconductor quantum dots 493 Inclusion of the electric interaction for a many electron system in QD strongly affects the energy levels. Taking account of the fact that the dimensions of quantum dots are visibly larger than those of atoms, the Coulomb interaction is of much greater importance here. According to the Heisenberg uncertainty relation the single-particle excitation energy depends on the size L as ε 1/L 2, while the Coulomb interaction energy behaves as: V C 1/L. In small quantum dots (e.g. self-assembled dots, L nm) the relation between ε and V C is similar to that in atoms, and the ground state satisfies the (slightly modified) Hund principles. On the other hand, as shown by Bryant [7], and later by other authors [21,37], in larger dots the interaction between electrons determines the ground state of the system. The ground state is a result of the competition between two opposite effects: the Coulomb repulsion favours pushing the electrons apart from one another, whereas the external confinement squeezes electrons in the centre of the dot and prevents a large separation between them. Under the influence of a magnetic field the electrons are subject to an additional squeezing in the plane perpendicular to the field, since the magnetic-field term appearing in the Hamiltonian also has the form of parabolic confinement: 1 8 mω2 c r2, where ω c is the cyclotronic frequency. Thorough theoretical analysis of energy levels, including interaction, is performed by applying numerical procedures for quasi-exact diagonalization of relevant Hamiltonians or using various versions of the Hartree Fock calculus. For QD one can observe a realization of the generalized Kohn theorem (independence of resonant electromagnetic attenuation from electron interaction and electron number in QD). This property corresponds to the separation of centre of mass and relative motion for the electron system in QD with a parabolic confining potential. For far-infrared radiation (FIR) with energy corresponding to inter-level distances in QD (of the order of mev), the wavelength (of the order of mm) is far larger than the dot diameter, hence only the centre-of-mass dynamics is involved in system response, exactly as for a uniform field. Hence, the resonant FIR radiation frequencies correspond only to transitions in the centre-of-mass quantum levels of the dot and, thus, are independent of both the interaction and the number of electrons. The fair coincidence of experimental data and theoretical predictions for FIR spectroscopy [10, 23], as presented in figure 6, is also evidence that the real confinement in QD is actually close to the parabolic type, for which the separation of centre-of-mass and relative dynamics occurs. Some special features of FIR spectroscopy data, namely the double splitting of frequencies and characteristic anticrossing of resonant lines (see figure 6(a), (b) and (d)) occurring for QD with a large number of electrons confined, indicate a small violation of the Kohn theorem (inclusion of spin orbit interaction can explain these effects [27]). The electric interaction between electrons in QD significantly modifies the relative dynamics of electrons. Very spectacular, albeit not yet confirmed experimentally, are the so-called magic states of QD [33, 37], revealing the composite fermion nature of 2D electrons in planar QD in the presence of strong magnetic fields, the same phenomenon which is crucial for the fractional quantum Hall effect. The magic states of strongly interacting electrons in QD were identified there with the so-called compact states of weakly interacting composite fermions. The transformation of an electron system into a composite-fermion system is performed by attaching to each electron an equal and even number of elementary fluxes of a magnetic field. As shown in the exact numerical calculations [28, 29], the eigenstates of the interacting electron system are very close to the respective compact states of non-interacting composite fermions. The electronic system described in terms of composite fermions is thus an effectively free (non-interacting) system. It should, however, be stressed that the system of non-interacting composite fermions is a many-body system. Each of its quantum states is non-local, and differs strongly from the antisymmetrized product of single-particle electronic states, i.e. the appropriate Slater determinant. Very important for future opto-electronic application of QD are their photoluminescence (PL) properties. The PL radiation from QD is caused by recombination of electron hole pairs (i.e. excitons) trapped by QD. The PL spectrum of QD is much more complicated than for

8 494 L Jacak Figure 6. Evolution of the FIR resonance energies for QD in a magnetic field: (a) 210 electrons; (b) 25 electrons [10]; (c) 4 electrons [39]; (d) Dependence of the FIR absorption spectrum of QD containing 25 electrons on the magnetic field: squares, the experiment [10]; lines, the model with spin orbit interaction [27]. bulk systems or quantum wells. The PL properties of QD are analysed very intensively both experimentally and theoretically for various types of QD, since these phenomena are utilized, e.g., in laser constructions [12, 31]. For QD, instead of a single PL peak as for quantum wells (or bulk structures), a multipeak structure is usually observed both in the presence and in the absence of a magnetic field [4, 6, 14]. Even for very weakly activated QD the doublet of PL features occurs. The relative positions and intensity of particular PL peaks depend strongly on the dot diameter and the magnetic field [5, 6, 13]. There are many approaches for theoretical investigation of these phenomena [24]. A simplified theoretical model in terms of metastable states can be formulated within the Hartree approach for electric-field-defined dots [25]. The simultaneous presence of both types of carriers in a small region leads to strong mutual modifications of the effective confinement for electrons and holes in these QD: they can attain the double-well shape, giving rise to a splitting of the excitonic ground state. The special features of the effective potentials for electron and hole in QD depend on the geometrical dimension of QD (see figure 7) (and also depend on the external magnetic field and the activation level), similarly to what is is observed in experiments (see figure 8). The opportunity for qualitative modifications of the confining potential originates from the non-singular nature of the bare lateral confinement in QD and it distinguishes these structures from ordinary atoms and ionized donors or acceptors in semiconductors. Long-lived metastable states in QD are very desirable as candidates for future applications as memory elements, especially combined with ultra-fast optical coherent switching techniques [3]. A very interesting, long-lived (even for seconds) exciton state in a pair of self-assembled

9 Semiconductor quantum dots 495 Figure 7. Effective self-consistent potential which acts on the electron U e (left), and approximated photoluminescence spectrum (right), for three values of κ, i.e. the dot sizes: (a) large dot, κ = 0.4, (b) medium dot, κ = 0.7 and (c) small dot, κ = 0.9 (bare lateral confinement for an electron V exp( r 2 /L 2 ), κ 1/L 2 ). Figure 8. Photoluminescence spectra of single dots with three different diameters [5]; splitting of the PL peak behaves similar to the scheme in figure 7. QD was demonstrated recently at the University of California in Santa Barbara [36]. Even though in single self-assembled QD excitons live only for nanoseconds, in QD-pairs obtained

10 496 L Jacak by the self-assembled strain-induced technique, the electron hole couple can be stored up to 10 9 times longer, provided that the electron and the hole are spatially separated, i.e. they are located in distinct dots of the pair. By application of a lateral electrical field it is possible to manipulate this exciton by changing the localization of both carriers. Very sophisticated experiments with QD created by an electric field in a narrow quantum well were performed by Ashoori et al [2,53]. Their method, called single-electron capacitance spectroscopy, allows the observation of electrons added one by one to the initially empty dot and also the measurement of the single-electron energy. In the recent version of this experiment [53], modified in order to examine the so-called Coulomb blockade, new and very interesting features, which have not yet been entirely elucidated, were indicated. QD created by electric fields are also promising from the point of view of applications. In the rapidly developing new domain of quantum computing (see, e.g., [42]), these dots could play an important role. They are relatively flexible and small quantum objects which could be used as so called qubits, quantum counterparts of classical bits. The two-level, ultra-small, quantum dots created by the electric field, which are easily rearrangeable with field magnitude, and are additionally susceptible to magnetic field effects, are very interesting objects for investigation as candidates for hardware elements for a future realistic quantum computer. Recently there has appeared a number of proposals on how to implement the qubit on quantum dot systems [35,46,50]. Pairs of closely coupled dots play the role of pairs of twolevel systems on which the entanglement of quantum states could be realized in a controlled manner. Such states, being non-direct tensor products of states of subsystems, are crucial for quantum computing. Under consideration are electric-field-defined dot systems [50] as well as other constructions: dots built one above the other by self-assembly strain-induced growth [36], or multi-layered dot systems obtained by the etching technique. Despite the very rapid progress in nanotechnology the realization of even a small quantum computer is hampered by severe obstacles due mainly to decoherence processes [32]. Other ideas concerning QD as single-electron transistors or single-electron memory are also currently being investigated [18, 51]. In [51] the authors reported a charge storage in nanocrystals of silicon of 5 nm in size with one electron per nanocrystal. One other new proposal is for a version of the infrared QD laser working on transitions between solely electronic states of dots, similar to in ordinary molecular lasers. The novelty consists of the possibility of pumping such a laser by rapid changes of the dot dimension via oscillations of the field which create the dot array in the quantum well [26]. The inverse occupation can be achieved by virtue of the distinct probabilities of filling with electrons of upper and lower dot levels. Dots should, however, be created very rapidly, non-adiabatically, which is beyond the current state-of-the-art technology; however, the simultaneous time-modulation of carrier densities in a quantum well (e.g. by application of ultra-fast resonant switching techniques [3]) can permit the creation of empty dots using a slower signal. Even though the practical applications and even demonstrations of all these novel constructions and ideas require further technological and theoretical progress, the advantages of the next step of miniaturization are very attractive. As was noticed in [36], currently the storage of a single bit of information involves millions of atoms, while application of quantum dots would decrease this number to thousands or even hundreds. References [1] Alsmeier J, Batke E and Kotthaus J P 1990 Phys. Rev. B [2] Ashoori R C 1996 Nature Ashoori R C, Störmer H L, Weiner J S, Pfeifer L N, Baldwin K W and West K 1993 Phys. Rev. Lett Ashoori R C, Störmer H L, Weiner J S, Pfeifer L N, Pearton S J, Baldwin K W and West K 1992 Phys. Rev. Lett [3] Awschalom D D and Kikkawa J M 1999 Phys. Today (June) 33 [4] Bayer M, Schmidt A, Forchel A, Faller F, Reinecke T L, Knipp P A, Dremin A A and Kulakovskii V D 1995 Phys. Rev. Lett

11 Semiconductor quantum dots 497 [5] Bockelmann U, Brunner K and Abstreiter G 1994 Solid-State Electron [6] Brunner K, Bockelmann U, Abstreiter G, Walther M, Böhm G, Tränkle G and Weimann G 1992 Phys. Rev. Lett [7] Bryant G W 1987 Phys. Rev. Lett [8] Cibert J, Petroff P M, Dolan G J, Pearton S J, Gossard A C and English J H 1986 Appl. Phys. Lett [9] Das Sarma S and Pinczuk A 1997 Perspectives in Quantum Hall Effects (Chichester: Wiley) [10] Demel T, Heitmann D, Grambow P and Ploog K 1990 Phys. Rev. Lett [11] Ekimov A I, Efros A L and Onushchenko A A 1985 Solid-State Commun [12] Fafard S, Hinzer K, Raymond S, Dion M, McCaffrey J and Charbonneau S 1996 Science [13] Fafard S, Leonard D, Merz J L and Petroff P M 1994 Appl. Phys. Lett [14] Fafard S, Leon R, Leonard D, Merz J L and Petroff P M 1994 Phys. Rev. B Fafard S, Leon R, Leonard D, Merz J L and Petroff P M 1995 Phys. Rev. B [15] Fock V 1928 Z. Phys Darwin C G 1930 Proc. Camb. Phil. Soc [16] Fukui T, Ando S and Tokura Y 1991 Appl. Phys. Lett [17] Grundmann M, Stier O and Bimberg D 1995 Phys. Rev. B [18] Guo L, Leobandung E and Chou S Y 1997 Science [19] Hansen W, Smith T P, Lee K Y, Brum J A, Knoedler C M, Hong J M and Kern D P 1989 Phys. Rev. Lett [20] Hansen W, Smith T P, Lee K Y, Hong J M and Knoedler C M 1990 Appl. Phys. Lett [21] Hawrylak P and Pfannkuche D 1993 Phys. Rev. Lett Hawrylak P and Pfannkuche D 1990 Appl. Phys. Lett. 56 (2) 168 [22] Heitmann D and Kotthaus J P 1993 Phys. Today (January) 56 [23] Heitmann D 1995 Physica B [24] Jacak L, Hawrylak P and Wójs A 1998 Quantum Dots (Berlin: Springer) [25] Jacak L, Krasnyj J, Korkusiński M and Wójs A 1998 Phys. Rev. B [26] Jacak L, Krasnyj J Jacak D and Bujkiewicz L 1999 Far-infrared laser on electric field created quantum dot matrix PRE 147/99 (Institute of Physics, Technical University of Wrocław) [27] Jacak L, Krasnyj J and Wójs A 1997 Physica B [28] Jain J K and Kawamura T 1995 Europhys. Lett [29] Kamilla R K and Jain J K 1995 Phys. Rev. B [30] Kash K, Scherer A, Worlock J M, Craighead H G and Tamargo M C 1986 Appl. Phys. Lett [31] Kirstaedter N, Ledentsov N N, Grundmann M, Bimberg D, Ustinov V M, Ruvimov S S, Maximov M V, Kop ev P S, Alferov Zh I, Richter U, Werner P, Gösele U and Heydenreich J 1994 Electron. Lett Grundmann M 2000 Physica E [32] Landauer R 1995 Phil. Trans. R. Soc [33] Laughlin R B 1981 Phys. Rev. B Laughlin R B 1983 Phys. Rev. B Laughlin R B 1983 Phys. Rev. Lett [34] Lorke A, Kotthaus J P and Ploog K 1990 Phys. Rev. Lett [35] Loss D and DiVincenzo D P 1998 Phys. Rev. A Burkard G, Loss D and DiVincenzo D P 1999 Phys. Rev. B [36] Lundstrom T, Schoenfeld W, Lee H and Petroff P M 1999 Science [37] Maksym P A and Chakraborty T 1990 Phys. Rev. Lett Maksym P A and Chakraborty T 1992 Phys. Rev. B [38] Marzin J Y, Gérard J M, Izraël A, Barrier D and Bastard G 1994 Phys. Rev. Lett [39] Meurer B, Heitmann D and Ploog K 1992 Phys. Rev. Lett [40] Petroff P M, Gossard A C, Logan R A and Wiegmann W 1982 Appl. Phys. Lett [41] Petroff P M and Denbaars S P 1994 Superlattices and Microstructures [42] Preskill J 1998 Lecture notes on Advanced Mathematical Methods of Physics: Quantum Information and Quantum Computation (California Institute of Technology) preskill/ph229 [43] Raymond S, Fafard S, Poole P J, Wójs A, Hawrylak P, Charbonneau S, Leonard D, Leon R, Petroff P M and Merz J L 1996 Phys. Rev. B [44] Reed M A, Bate R T, Bradshaw K, Duncan W M, Frensley W M, Lee J W and Smith H D 1986 J. Vac. Sci. Technol. B4 358 [45] Reed M A 1993 Sci. Am. (March) 40 [46] Sherwin M, Imamoglu A and Montroy T 1999 Phys. Rev. A [47] Sikorski C and Merkt U 1989 Phys. Rev. Lett [48] Smith T P, Lee K Y, Knoedler C M, Hong J M and Kern D P 1988 Phys. Rev. B [49] Stranski I N and von Krastanow L 1939 Akad. Wiss. Let. Mainz Math Natur K1 IIb [50] Tanamoto T 2000 Phys. Rev. A /1 [51] Tiwari S, Rana F, Hanafi H, Harstein A and Crabbe E F 1996 Appl. Phys. Lett [52] Temkin H, Dolan G J, Panish M B and Chu SNG1987 Appl. Phys. Lett [53] Zhitenev N B, Brodsky M, Ashoori R C, Pfeiffer L N and West K W 1999 Science