Thermal and static disorder

Thermally-induced network vibrations broaden the absorption edge and shift the band gap of semiconductors. The thermal disorder couples to the optical transition through the deformation potential, which describes how the electronic energy varies with the displacement of the atoms. The bond strain in an amorphous material is also a displacement of atoms from their ideal position, and can be described by a similar approach. The description of static disorder in terms of frozen phonons is a helpful concept which goes back 20 years. Amorphous materials, of course, also have the additional disordering of the real phonon vibrations. 

According to the theory of the Urbach absorption edge in crystals, the slope E is proportional to the thermal displacement of atoms r(7). The frozen phonon model assumes that an amorphous semiconductor has an additional temperature independent term, r , representing the displacements which originate from the static disorder, so that 

35) relates the broadening of the Urbach edge and the shift of the band gap, both of which originate from the thermal and bonding disorder. 

The temperature dependence of the gap is next calculated, based on an Einstein model of the lattice vibrations, described by the Einstein temperature 0 which is of the Debye temperature, 0j, 

23 is obtained with a Debye temperature of 536 K, which is satisfactorily close to the value of 625 K in crystalline silicon. The same model gives for the temperature dependence of ,(7 ,/-jj) 

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Excitation spectroscopy Monitoring of the surface emission allows one to discriminate the upper excited surface states and their relaxation dynamics. Problems such as surface reconstruction, or quantum percolation of surface excitons upon thermal and static disorder, are connected with high accuracy to changes of the exciton spectra.61118,119,121... 

The local structure around zinc and manganese atoms in as-synthesised sample of MnZnAPO-34 was also characterised by means of EXAFS. This technique provides a description of the short-range order of selected atomic species in terms of the number of neighbours, distances, and thermal and static disorder within a range of those distances. 

If the difference in atomic number between the absorber element and the backscattering element is >10 and if only one kind of element backscatters, EXAFS spectra can be analyzed readily to provide local structural data on adsorbed species. However, because the electron mean free path, thermal and static disorder parameters (Debye-Waller factors), and coordination number for an absorber environment cannot be determined a priori with sufficient accuracy, EXAFS data for suitable reference compounds of known molecular structure must be used to help interpret the EXAFS spectrum for an interfaeial region. 

Another important factor is the Debye-Waller factor e-DW. This accounts for thermal and static disorder effects concerning the move-ment/position of atoms around their equilibrium/averaged position. A point to stress is that the nature of this term is different to the... 

Another important factor is the Debye-Waller factor e °. This accounts for thermal and static disorder effects concerning the movement/position of atoms around their equilibrium/averaged position. A point to stress is that the nature of this term is different to the counterpart term in XRD." Since vibrations increase with temperature, EXAFS spectra are usually acquired at low temperature (below 100 K) in order to maximise information. Spectra at different temperatures may, on the other hand, allow decouple thermal and static contributions to DW. The DW term smears the sharp interference pattern of the sinusoidal term and cuts off EXAFS at sufficiently... 

Photoionization and therefore EXAFS takes place on a time scale that is much shorter than that of atomic motions so the experiment samples an average configuration of the neighbors around the absorber. Therefore, we need to consider the effects of thermal vibration and static disorder, both of which will have the effect of reducing the EXAFS amplitude. These effects are considered in the so-called Debye-Waller factor which is included as... 

Whereas there is little that one can do to overcome the effects of static disorder, the effects of thermal vibration can be significantly decreased by performing experiments at low temperatures, and, in fact, many solid samples are typically run at liquid nitrogen temperatures just to minimize such effects. An example of the effect of thermal vibration can be ascertained in Fig. 8 A, where the EXAFS amplitude decreases precipitously due to the large vibrational amplitude of the Cu—O bond. In general, failure to consider the effects of thermal vibration and static disorder can result in large... 

Section IV is devoted to excitons in a disordered lattice. In the first subsection, restricted to the 2D radiant exciton, we study how the coherent emission is hampered by such disorder as thermal fluctuation, static disorder, or surface annihilation by surface-molecule photodimerization. A sharp transition is shown to take place between coherent emission at low temperature (or weak extended disorder) and incoherent emission of small excitonic coherence domains at high temperature (strong extended disorder). Whereas a mean-field theory correctly deals with the long-range forces involved in emission, these approximations are reviewed and tested on a simple model case the nondipolar triplet naphthalene exciton. The very strong disorder then makes the inclusion of aggregates in the theory compulsory. From all this study, our conclusion is that an effective-medium theory needs an effective interaction as well as an effective potential, as shown by the comparison of our theoretical results with exact numerical calculations, with very satisfactory agreement at all concentrations. Lastly, the 3D case of a dipolar exciton with disorder is discussed qualitatively. 

Because all of the coordinating atoms in a shell are not fixed in position at exactly R the term a2 in e accounts for the disorder in the interatomic distances. The term has contributions from dynamic (thermal) disorder and static disorder (structural). 

Modulations is a perturbation of the crystal lattice which, unlike random perturbations due to thermal motion and static disorder, has regular character and therefore creates sharp diffraction peaks, usually as satellites of ordinary reflections. The diffraction vector can be then expressed as (cf Section 2.2.1)... 

Peak width thermal or static disorder Atomic disorder in the form of thermal and zero-point motion of atoms, and any static displacements of atoms away from ideal lattice sites, gives rise to a distribution of atom-atom distances. The PDF peaks are therefore broadened resulting in Gaussian shaped peaks. The width and shape of the PDF peaks contain information about the real atomic probability distribution. For example, a non-Gaussian PDF peak may suggest an anharmonic crystal potential. 

The periodicity of this modulation of the absorption will be dependent upon the inteiatomic distance between the absorbing and back-scat tei i ng atoms, R, and the phase shifts, 5. , encountered when the photoelect i on expeiiences the potentials at these centres. Its intensity will be governed by the number of back-scatterers, Nj and their back-scattering amplitudes, F (k). Finally the amplitude is dampened by disorder (thermal and static), a in the interatomic distance and any inelastic piocesses (related to the mean free path of the election, j)- The recognition of the stiuctural information intrinsic to this phenomenon and the derivation of a tractable formula for the estimation of interatomic distances was the result of the work of Sayers, Lytle and... 

The inadequacy of the X-ray model for a straightforward evaluation of the relative contribution of the dynamic and static disorders has been emphasized in the structure analysis of TOT/(dl)-2-bromobutane. Suprisingly, the / -ratio test favored space group PS, as opposed to space group P3j21, in spite of the presence of pairs of TOT molecules shown to be identical within the limits of experimental error and symmetrically related by C2 axes. In this context it was demonstrated how static disorder could be mimicked by a magnified thermal motion of the Br atom. The Hamilton test gave opposite results with the TOT/(R)-2-butanol clathrate. 

Point defects, static disorder, and thermally induced displacements lead to an increase of the background intensity between the spots. Depending on the correlation between the scatters, the background is either homogeneous (no correlation) or... 

The outer layer (beyond the compact layer), referred to as the diffuse layer (or Gouy layer), is a three-dimensional region of scattered ions, which extends from the OHP into the bulk solution. Such an ionic distribution reflects the counterbalance between ordering forces of the electrical field and the disorder caused by a random thermal motion. Based on the equilibrium between these two opposing effects, the concentration of ionic species at a given distance from the surface, C(x), decays exponentially with the ratio between the electro static energy (zF) and the thermal energy (R 7). in accordance with the Boltzmann equation ... 

In figure 3 and show that the relative thermal motion of the surface atoms is significantly greater than in the bulk metal over the range from 100 - 800 K, This result is expected considering the partial coordination, hence lack of constraint of the surface atoms. A similar result has been found from LEED measurements on a Pt surface. ( ) Significantly, the surface atom disorder when extrapolated to 0 K remains sizable. This static disorder or strain appears to be a result of the interaction of the Ft atoms with the support, a kind of epitaxy to the oxygen (or hydroxyl) surface of the support. 

This can be separated into static disorder and thermal vibrational components ... 

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

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