Quantum critical dimensions

It is generally assumed that the electronic properties of a semicondnctor are independent of crystal size. However, recent studies have shown that if the particle size of the semiconductor is less than approximately 10 nm, then many of their physicochemical properties appear to be substantially different from analogous properties of a bulk material. This is because the electrons and holes are confined in the region of space defined by potential barriers that are comparable to or smaller than their respective de Broglie wavelengths so that allowed energy states become discrete (or quantized). This effect is referred to as the size quantization effect (SQE) or the quantum size effect. SQE is also observed in the noble metals (4). Eor semiconductors, the critical dimension for SQE depends upon the effective masses of the electrons () and the holes (m ). For example, for m 0.05, the critical dimension is approximately 30 mn. Thus, for semiconductors, SQE is observed in... 

Figure 2 EL junction of two semiconducting polymer films. The frequencyy V, of the luminescence is found to be dependent on the bias voltage, V, for films with critical thickness dj=40 nm and d2=20 nm, respectively (see Figure J). Note that the finite size distance is outside the expected quantum-well dimension (-10 nm) asking for mesoscale interpretations.

One kind of a multicritical point is a point over a critical line where more than two different states coalesce. The common multicritical points in statistical mechanics theory of phase transition are tricritical points (the point that separates a first order and a continuous line) or bicritical points (two continuous lines merge in a first order line) (see, for example, Ref. 166). These multicritical points were observed in quantum few-body systems only in the large dimension limit approximation for small molecules [10,32]. For three-dimensional systems, this kind of multicritical points was not reported yet. 

As device dimensions continue to shrink, pattern resolution becomes of critical importance. In this regard, the use of radiation sources with maximized output in the mid-UV region (300-350 nm) would permit increased resolution if an appropriate resist could be utilized. In this regard, commercial resist formulations function inefficiently in this region for a variety of reasons. Accordingly, we have used semiemipirical quantum mechanical techniques to calculate the electronic absorption spectra of two of the most commonly employed chromophores and to computationally assess the effect of a variety of substituents. The results of these calculations have been used to drive a synthetic program designed to produce an efficient sensitizer for use in the mid-UV. 

In 1944 Landau published the theory [23c] that it should be possible to observe quantum-mechanical effects in metals if the dimensions of the metal were in the region of, or less than the wavelength of the electrons. For many years attempts were made to confirm or refute this theory by creating thin layers of metals. From today s viewpoint these experiments were unsuitable because only one of three dimensions of the metal was brought below the critical limit, with the result that the electrons were still able to spread unhindered. 

It may be, also, that orbits that are not circular would give better values than circular orbits. Computations of the frequences on this basis present formidable difficulties. The fact, however, that the two quantum and three quantum orbits lie not in a plane, but in space of three dimensions may explain the appearance of three critical absorption wave-lengths in the L series, and six critical absorption wave-lengths in the M series, etc. 

According to quantum mechanics, isolated molecules do not have a finite boundary, but rather fade away into the regions of low electron density. It has been well established, however, from properties of condensed matter and molecular interactions, that individual molecules occupy a finite and measurable volume. This notion is at the core of the concept of molecular structure. 33 A number of physical methods yield estimations of molecular dimensions. These methods include measurements of molar volumes in condensed phases, critical parameters (lattice spacings and bond distances), and collision diameters in the gas phase. 34 From these results, one derives values of atomic radii from which a number of empirical molecular surfaces can be built. Note that the values of the atomic radii depend on the physical measurement chosen. 35-i37... 

The electronic structure of a nanocrystal critically depends on its very size. For small particles, the electronic energy levels are not continuous as in bulk materials, but discrete, due to the confinement of the electron wavefunction because of the physical dimensions of the particles (see Figure 1.4). The average electronic energy level spacing of successive quantum levels, 6, known as the so-called Kubo gap, is given by... 

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