2D quantum wells

In an experiment of thoughts the 3D piece of metal shall be reduced in the z-direction to only a few nanometres, comparable with the electronic de Broglie wavelength (t4 = k), whereas in x- and y-direction it is kept infinite a 2D quantum well is formed. Compared with the former 3D situation, the electrons in x-and y-direction can still freely move in these directions, but not in the z-direction. Electrons in this direction are confined like in a box. The states are quantized, whereas in x- and y-direction the situation does not differ from that in the 3D case. Figure 5a shows the quantized situation in z-direction with well-defined Ak values n— I, 2, etc., and Figure 5b indicates the Ak values close to 0. 

The methodology used to answer these questions can be classified as either semi-empirical or based on first principles. The confined structure is assumed to be two-dimensional (2D = quantum well), one-dimensional (lD = quantum wire) or zero-dimensional (0D = quantum dot). 

The confinement model is also useful to systematically study effects on an atom or molecule trapped in a microscopic cavity or in fullerenes [72-78]. As mentioned above, some of the system observables undergo changes as a result of spatial confinement. The same situation is found at a nanoscopic scale in artificial systems constructed within semiconductors [79-87,172-188], such as two-dimensional quantum wells, quantum wires and quantum dots. Properties of a hydrogen-like impurity in a 2D quantum well have been investigated by several authors [172,173,185-188], who have concluded that particular features associated with the states, as well as the properties of an impurity, are determined, among other factors, by the size of the confining structure. Other applications of confined systems refer to Metal properties [147,148], astrophysical spectroscopic data [40,146], phase transitions [155], matter embedded in electrical fields [68], nuclear models [164], etc. For a detailed list of references, several review articles [25, 48,54,95,125,127] are available. 

Note, however, that the dependence of the polariton energy on the wavevec-tor, which arose when only retardation is taken into account, is correct only if we can neglect the dependence of the energy of the Coulomb exciton Etl on k, arising from instantaneous Coulomb interaction. For example, if we apply this theory for 2D quantum well polaritons, the linear term in the dispersion of po-laritons will be cancelled because in this case the linear term as a function of the energy of the quantum well exciton on the wavevector has the same value with opposite sign. 

A logical consequence of this trend is a quantum w ell laser in which tire active region is reduced furtlier, to less tlian 10 nm. The 2D carrier confinement in tire wells (fonned by tire CB and VB discontinuities) changes many basic semiconductor parameters, in particular tire density of states in tire CB and VB, which is greatly reduced in quantum well lasers. This makes it easier to achieve population inversion and results in a significant reduction in tire tlireshold carrier density. Indeed, quantum well lasers are characterized by tlireshold current densities lower tlian 100 A cm . ... 

Fig. 1 Schematic drawing to show the concept of system dimensionality (a) bulk semiconductors, 3D (b) thin film, layer structure, quantum well, 2D (c) linear chain structure, quantum wire, ID (d) cluster, colloid, nanocrystal, quantum dot, OD. In the bottom, it is shown the corresponding density of states [A( )] versus energy (E) diagram (for ideal cases).

It is known that for a set of ID and 2D quantum spin models the exact ground state of which can be represented in the RVB form [15, 6, 36, 13, 31]. It is natural to try to find electronic models with an exact ground state at half-filling formed by SB functions in the same manner as for above mentioned spin models. The electronic models of these types include the correlated hopping of electrons as well as the spin interactions and pair hopping terms. 

Though quantum dots are typically thought of as OD nanostructures, quantum confinement effects are also exhibited in ID nanowires and nanorods. Buhro and coworkers have studied the effect on both size and shape on quantum confinement (Yu, H. Li, J. Loomis, R. A. Wang, L.-W. Buhro, W. E. Nature Mater. 2003, 2, 517). Their work provides empirical data to back up the theoretical order of increasing quantum confinement effects dots (3D confinement) > rods > wires (2D confinement) > wells (ID confinement). For an example of an interesting nanostructure comprised of both a nanorod and nanodot, see Mokari, T Sztrum, C. G. Salant, A. Rabani, E. Banin, U. Nature Mater. 2005,4, 855. 

Numerical applications of the new formalism implied by (248) are under development [117]. Preliminary indications are that the ALDA, the Gross-Kohn approximation (191) and (248) will all give substantially different results for at least one of the plasmon modes of a low-density parabolic quantum well, say for = 6. (The modes in question are the HPT ( Kohn or sloshing ) mode, the standing plasmon modes [118], and also the 2D plasmon mode at... 

Figure 2. THz gain for a 5 nm (a) and a 10 nm (b) quantum well. Only the first two subbands are occupied with the electronic temperature Ti=T2 and the same global population density in both subbands. In both panels from bottom to top the 2D carrier density is increased by ni=n2=2, 4. 8 xlO" carriers/cm. ...

Dispersion Relation for Coupling of Surface and Adsorbed Layer 2D Plasmons Coupling of Surface and Quantum Well Plasmons Conclusions... 

Incidentally, it is also of interest to consider the case of a 2D sheet representing a 2D plasma in a quantum well within the semi-infinite bulk, Zq > 0. In this case Eq. (23) becomes... 

Charge carriers in semiconductors can be confined in one spatial dimension (ID), two spatial dimensions (2D), or three spatial dimensions (3D). These regimes are termed quantum films, quantum wires, and quantum dots as illustrated in Fig. 9.1. Quantum films are commonly referred to as single quantum wells, multiple quantum wells or superlattices, depending on the specific number, thickness, and configuration of the thin films. These structures are produced by molecular beam epitaxy (MBE) and metalorganic chemical vapor deposition (MOCVD) [2j. The three-dimensional quantum dots are usually produced through the synthesis of small colloidal particles. 

The picosecond time-scale observed in these experiments was the first example of superradiance of two-dimensional Frenkel excitons. Relative quantum yield measurements of the photoluminescence from bulk and the photoluminescence from the outermost monolayer indicate that the decay of excitons in the monolayer is purely radiative with a very small contribution from relaxation to the bulk. Later the same phenomenon for a 2D Wannier-Mott exciton in a semiconductor quantum well was observed by Deveaud et al. (4). 

Hybrid 2D Frenkel Wannier Mott excitons at the interface of organic and inorganic quantum wells. Strong coupling regime... 

As follows from eqn (13.48), the interwire coupling is suppressed exponentially for excitons with wavevectors k > 1/R, i.e. for a major part of the Brillouin zone. In contrast, coupling of excitons with relatively small wavevectors k < 1/R is quite efficient. This is different from the case of a 2D system of quantum wells where the coupling at small wavevectors is suppressed because the electric field outside of a uniformly polarized layer vanishes. 

Now we turn to the situation when the QW width fluctuations, alloy disorder or impurities localize the 2D exciton (such a situation is more frequent for II-VI semiconductor quantum wells than for III—V ones). Then, the wavefunction of the center-of-mass exciton motion (ry) is no longer just a plane wave, and the corresponding polarization is given by... 

EXCITON-PHONON COUPLING OF LOCALIZED QUASI-2D EXCITONS IN SEMICONDUCTOR QUANTUM WELL HETEROSTRUCTURES... 

EFFECTS OF SPATIAL REPRODUCTION AND MULTIPLICATION AT THE INTERFERENCE OF THE ELECTRON WAVES IN SEMICONDUCTOR 2D NANOSTRUCTURES WITH RECTANGULAR QUANTUM WELLS... 

Effects of spatial nonhomogeneity for the probability current density (or a quantum-mechanical current density) in the semiconductor 2D nanostructures in the form of joints in the direction of propagation of the electron wave of narrow and wide rectangular quantum wells have been theoretically studied. 

The second type of microlaser (Fig. 8b) utilizes a localized state defect mode as a laser cavity (Painter et al., 1999). Here, the localized electromagnetic mode is associated with a missing hole in the 2D triangular lattice. This particular structure has been proposed as the world s smallest microlaser with a cavity volume of 0.03 /xm. Spontaneous emission from electron-hole pair recombination in the multiple quantum well active region occurs preferentially into the localized state. Since the photonic crystal is two-dimensional, spontaneous emission is not exclusive to the lasing mode. This results in a finite pumping threshold before lasing occurs. 

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