Infinite barrier model

This result is a definitive counter-example to the short wavelength hypothesis. However, note that these values for a are not very much different numerically, suggesting that the short wavelength hypothesis may not be too bad in practice. (We have also found that, for the metal surface energy contribution in the semiclassical infinite barrier model, o = 0 both exactly and in the LSD.)... 

The first, most primitive, model is the infinite barrier model (IBM). Here the electronic motion is confined by a spherical potential hole with infinitely high barriers. Once the electronic wave functions (spherical Bessel functions) and eigenvalues are known, one can proceed and calculate the dynamic polarizability a co). From this quantity the collective excitations are determined in a straightforward manner (see below). The theoretical prediction [50], shown in Figure 1.2, matches the experimental data (indicated by dots) rather well from very small to mesoscopic particle sizes. The result obtained shows that the IBM, which models the kinetic repulsion of the occupied 4d-shell of atomic Xe, works surprisingly well. This repulsion causes an enhanced electronic density, leading to the blue-shift of the surface-plasmon line. 

Figure 1.2 Crude theoretical interpretation of the experimental data of Figure 1.1. Here the electronic motion is confined by the IBM (infinite barrier model). For details see the original work by Ekardt et al. [50]. Reprinted by permission of Elsevier Science Publishers...

Why Do Microphases Form The most simple quantum mechanical model of a metal is a box with only one finite dimension. The walls are infinite barriers, so an electron inside has no chance to escape. This model accepts only those electron energies which correspond to electronhalf-wavelengths which are simple fractions of the box size. Metal electrons may fill part of these energy levels with two electrons per level. The upper level of energy reached is called the Fermi level. A smooth U-shaped curve of energy vs. wave number includes all acceptable energies in this model. 

The usual treatment of confinement is where the charge p = 37 (r) 2 is essentially zero on the sphere (or hypersphere) r = R and the system may therefore be isolated. For example, if we envisage a large repulsive potential in r > R, then we may model this as an infinite barrier on r = R and consequently F(TJ) = 0. This model isolates the wavefunction completely in r < R and, for the examples considered here, leads to the general form... 

Now we check the above general results with Winter s delta-barrier model [63], see Figure 9.1. The initial state is an eigenstate of the infinite square well potential. 

Fig. 1 Exact and Thomas-Fermi electron density n as a function of position z for the Airy gas model with force F = 0.10. The scaling length is 1 = 1.71. The edge region is —/ < z < / and the Thomas-Fermi density is reasonably accurate for z > I- The infinite barrier is at z = 201 = 34.2. The magnitudes of the densities in this figure are valence-electron-like the density parameter Ts (the radius of a sphere containing on average one electron) is about 3.3 at z = I and about 1.3 at z = 10 (atomic units)...

Airy gas and the Airy gas model (with an infinite barrier at z = 201) suggests that the ratio plotted here is less nearsighted than the density itself (atomic units)... 

A very general effect of quantum confinement in semiconductors is a widening of their optical band gap. For the model system of an infinite-barrier 2-D quantum well, (3.1) shows that the lowest energy N = 1) of a con-... 

In Eq. (5.63), kp is the Fermi wavevector. This dielectric constant can be used in the framework of the specular scattering or semiclassical infinite-barrier (SCIB) model for planar metal surface [88]. Within the framework of such approach, the different descriptions of the metal response discussed so far (local dielectrics, hydrod3mamic dielectric function and Lindhard-Mermin dielectric function) can be compared. This has been done in Ref. [89] for the metal-induced non radiative rate of a molecule close to an Ag surface. The results are summarized in Fig. 5.9. 

Let us first assume that the center of mass of the exciton is only laterally confined by the AlN barrier (in other words, no supplemental localization inside the QD occurs). In that case, we need to estimate the exciton center of mass extension parameter. In a simple approach, we can consider that the exciton is laterally confined by infinite barriers in a cylinder that has the height of the QD and a diameter corresponding to the diameter of the top facet of the QD. Let us note that this model tends to underestimate the lateral extent of the confined wavefunction. For the center of the QD size distribution (height 1.6 nm, diameter 19 nm), the parameter deduced from this simple approach is 3.4 nm. Equations 3 and 4 thus yield/ = 85 and r = 24 ps. This is clearly a much faster decay than the experimentally measured decay, and thus shows that the expected giant oscillator strength regime is not reached in our structures. 

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

Studies based on the Frenkel-Kontorova model reveal that static friction depends on the strength of interactions and structural commensurability between the surfaces in contact. For surfaces in incommensurate contact, there is a critical strength, b, below which the depinning force becomes zero and static friction disappears, i.e., the chain starts to slide if an infinitely small force F is applied (cf. Section 3). This is understandable from the energetic point of view that the interfacial atoms in an incommensurate system can hardly settle in any potential minimum, or the energy barrier, which prevents the object from moving, can be almost zero. 

Figure 1. The tunneling of a single electron (SE) between two metal electrodes through an intermediate island (quantum dot) can be blocked of the electrostatic energy of a single excess electron trapped on the central island. In case of non-symmetric tunneling barriers (e.g. tunneling junction on the left, and ideal (infinite-resistance) capacitor on the right), this device model describes a SE box .

Another model which combined a model for the solvent with a jellium-type model for the metal electrons was given by Badiali et a/.83 The metal electrons were supposed to be in the potential of a jellium background, plus a repulsive pseudopotential averaged over the jellium profile. The solvent was modeled as a collection of equal-sized hard spheres, charged and dipolar. In this model, the distance of closest approach of ions and molecules to the metal surface at z = 0 is fixed in terms of the molecular and ionic radii. The effect of the metal on the solution is thus that of an infinitely smooth, infinitely high barrier, as well as charged surface. The solution species are also under the influence of the electronic tail of the metal, represented by an exponential profile. 

A very crude model to calculate the increase in bandgap energy is the effective-mass particle-in-a-box approximation. Assuming parabolic bands and infinitely high barriers the lowest conduction band (CB) level of a quantum wire with a square cross-section of side length w is shifted by AEC compared to the value Ec of the bulk crystal [Lei, Ho3] ... 

The percolation model, which can be applied to any disordered system, is used for an explanation of the charge transfer in semiconductors with various potential barriers [4, 14]. The percolation threshold is realized when the minimum molar concentration of the other phase is sufficient for the creation of an infinite impurity cluster. The classical percolation model deals with the percolation ways and is not concerned with the lifetime of the carriers. In real systems the lifetime defines the charge transfer distance and maximum value of the possible jumps. Dynamic percolation theory deals with such case. The nonlinear percolation model can be applied when the statistical disorder of the system leads to the dependence of the system s parameters on the electrical field strength. 

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