The inelastic diffusion length

The concept of the inelastic diffusion length was introduced in Section 8, and is of major importance in discussions of quantum interference. It is defined by the equation 

In Chapter 2, Section 6 we point out that, according to the calculations of Kaveh and Wiser (1986), for an amorphous material the index v may be between 1 and J. 

At higher temperatures, scattering by phonons, always an inelastic process, will in most cases determine L. For temperatures above the Debye temperature 

Through the term l/L, in (52) for the conductivity, an increase A r in a with ASoc Tlf2 is predicted this has been observed for amorphous metals (Chapter 10). This arises because the probability of scattering by a phonon of energy hco should be proportional to co, so for T D we expect 

That all scattering by phonons is inelastic has been disproved by Afonin and Schmidt (1986) see Chapter 10, where we maintain none the less that L-,=a in liquids, all collisions being inelastic, so the quantum-interference term is absent. 

---

The constant C is uncertain, but we give evidence that it is unity. However, if some collisions are inelastic, this interference does not take place. We therefore introduce the inelastic diffusion length... 

However, for the resistivity of non-crystalline materials at low temperatures, the most important effect is on Lb the inelastic diffusion length. We have seen (Chapter 1, Section 10) that thisisgivenbyLi=(DTi)1/2 and that there is a term in the conductivity proportional to //Lj, so that electron-electron scattering gives a negative term in the resistivity proportional to T with v between 1 and J. 

The inelastic diffusion length decreases, and the contribution due to localization increases near the M-I transition. 

An entirely new situation arises for a semiconductor with NEA. Here the threshold for photoemission of electrons is the band gap energy, that is, a bulk rather than a surface property, and novel phenomena are to be expected. Tbis is indeed the case, and the most spectacular of these phenomena is certainly the contribution of bulk excitons to the photoelectron yield of diamond surfaces with NEA as first reported by Bandis and Pate [73, 107]. In addition, the depth from which electrons contribute to the yield is no longer limited by the inelastic mean free path of some tens of Angstroms but by the diffusion length of electrons and excitons of the order of micrometers, a fact that is responsible for the near 100% quantum efficiency of NEA diamond surfaces alluded to earlier (Section 10.3). 

In all substances, at high temperatures, the electrical resistivity is dominated by inelastic scattering of the electrons by phonons, and other electrons. As classical particles, the electrons travel on trajectories that resemble random walks, but their apparent motion is diffusive over large-length scales because there is enough constructive interference to allow propagation to continue. Ohm s law holds and with increasing numbers of inelastic... 

In Fig. 3.1-9 on the left side a tube with diameter d and length L is shown where impinging molecules are reflected by the wall due to elastic impacts. On the right side a diffuse reflexion is illustrated as a result of rough walls and partially inelastic... 

Crystal as proposed by Slack. The material should have lattice thermal conductivities close to those of amorphous materials, but electronic properties associated with crystalline materials. As the characteristic length scales for phonon and electron diffusion can be different, it could in principle be possible to independently optimise these two properties. Some clathrates indeed show a glass-like thermal conductivity. Also, in the case of skutterudites, it has been proposed that rattling by introducing one heavy atom in vacant sites could reduce the thermal conductivity. Thermal conductivity is indeed reduced, even if the rattling mechanism is not at the origin of this phenomenon as shown recently by inelastic neutron diffraction. 

---