Conduction electrons mean free path

This use of bulk dielectric constants for nanoparticle calculations is appropriate for particles that are large enough (larger than the conduction electron mean free path), such as particles having radii > 20 nm considered in Fig. 4.2. For smaller particles, one needs to correct the dielectric constant for the effect of scattering of the conduction electrons from the particle surfaces. Procedures for doing this have been studied in several places, as recently reviewed by Coronado and Schatz [37]. 

Fig. 4.6a considers a spherical core-shell particle in which the core is taken to be vacuum and the shell is silver. The particle radius is 50 nm, so when the shell thickness is 50 nm we recover the solid particle result. As the shell becomes thinner, the plasmon resonance red-shifts considerably, very much like we see for highly oblate spheroids. Fig. 4.6a assumes that the dielectric constant of silver is independent of shell thickness, so the resonance width does not change much when the shell becomes thin. However, the correct dielectric response needs to include for finite size effects (as noted above) when the shell thickness is smaller than the conduction electron mean free path. Fig. 4.6b shows what happens to the spectrum in Fig. 4.6a when the finite size effect is incorporated, and we see that it has a significant effect for shells below 10 nm thickness, leading to much broader plasmon lineshapes. 

The above measurements all rely on force and displacement data to evaluate adhesion and mechanical properties. As mentioned in the introduction, a very useful piece of information to have about a nanoscale contact would be its area (or radius). Since the scale of the contacts is below the optical limit, the techniques available are somewhat limited. Electrical resistance has been used in early contact studies on clean metal surfaces [62], but is limited to conducting interfaces. Recently, Enachescu et al. [63] used conductance measurements to examine adhesion in an ideally hard contact (diamond vs. tungsten carbide). In the limit of contact size below the electronic mean free path, but above that of quantized conductance, the contact area scales linearly with contact conductance. They used these measurements to demonstrate that friction was proportional to contact area, and the area vs. load data were best-fit to a DMT model. 

Two mechanisms which contribute to GMR have been identified, a "non-local" mechanism and a "quantum" mechanism. To understand the first or non-local, mechanism it is necessary to understand that on the scale of the electron mean free path (possibly 10 to 20 nanometers at room temperature) electrical conduction is a non-local phenomenon. Electrons may be accelerated by an electric field in one region and contribute to the current in other regions. To a good approximation they may viewed as contributing to the current until they are scattered. 

When the size of metals is comparable or smaller than the electron mean free path, for example in metal nanoparticles, then the motion of electrons becomes limited by the size of the nanoparticle and interactions are expected to be mostly with the surface. This gives rise to surface plasmon resonance effects, in which the optical properties are determined by the collective oscillation of conduction electrons resulting from the interaction with light. Plasmonic metal nanoparticles and nanostructures are known to absorb light strongly, but they typically are not or only weakly luminescent [22-24]. 

Abstract In strong-coupling superconductors with a short electron mean free path the self-energy effects in the superconducting order parameter play a major role in the phonon manifestation of the point-contact spectra at above-gap energies. We compare the expressions for the nonlinear conductivity of tunnel, ballistic, and diffusive point-contacts and show that these expression are similar and correspond to the measurements of the phonon structure in the point-contact spectra for the 7r-band of MgB2. 

Equation 6.22 predicts that electronic conductivity is dependent on the electron relaxation time. However, it suggests no physical mechanisms responsible for controlling this parameter. Since electrons exhibit wave-particle duahty, scattering events could be suspected to play a part. In a perfect crystal, the atoms of the lattice scatter electrons coherently so that the mean-free path of an electron is infinite. However, in real crystals there exist different types of electron scattering processes that can limit the electron mean-free path and, hence, conductivity. These include the collision of an electron with other electrons (electron-electron scattering), lattice vibrations, or phonons (electron-phonon scattering), and impurities (electron-impurity scattering). 

We have seen in the previous section that disorder results in the localisation of charge carriers and that the conductivity will fall as a consequence of this. There is a minimum metallic conductivity, which corresponds to the electron mean free path being equal to the lattice repeat distance. The occurrence of the mobility edge means that in an amorphous metal the conductivity can switch... 

Recently, it has been predicted for armchair SWNTs that the electron mean free path should increase with increasing nanotube diameter, leading to exceptional ballistic transport properties and localization lengths of 10 pm or more [149]. The effect arises because the conductance is independent of the tube diameter (i.e. 2Go) and the electrons experience an effective disorder which is the real disorder averaged over the circumference of the tube [144,149]. The effective disorder then reduces as the tube diameter increases so that scattering becomes less effective. 

The lattice constant of the x = 3 face-centered cubic unit cell is 14.28 A (4). Accordingly, the carrier density is 4.1 x 10 cm , with four C o molecules and twelve donated electrons p>er unit cell. This charge density corresponds to a Fermi wave vector kf = 0.50 A which, when substituted into a Boltzmann equation description of the minimum resistivity gives = 2.3 A for the electronic mean free path. This unphysically small implies that, even at X = 3, the Boltzmann equation is inadequate for describing a system where intergranular transport may still be limiting the conductivity. 

The Fermi surface is assumed to be spherical. In the above equations, is the Fermi wave vector, / is the electron mean free path, m is the electron mass and x is the relaxation time, x = ml/Pikp. As the disorder increases, more and more states get localized and Ec and Ec move toward the centers of the respective bands. The mean free path (/) also decreases and in the limit, I = a which is the lattice distance (loffe-Regel limit). The conductivity also reaches the limit and is e 3nha), since kfl becomes approximately equal to I. Introduction of any further disorder only broadens the band and does not affect /, it alters N ( ). The minimum metallic conductivity, csm (all a values like am, , a, afO) etc. refer only to d.c. conductivities in this chapter the subscript d.c. is dropped to make the notation less cumbersome. A.c. conductivities will be referred to as cr(eo)), before the disorder localizes all the states and the conductivity drops to zero for the three dimensional problem may be approximated as 2... 

The velocity relevant for transport is the Fermi velocity of electrons. This is typically on the order of 106 m/s for most metals and is independent of temperature [2], The mean free path can be calculated from i = iyx where x is the mean free time between collisions. At low temperature, the electron mean free path is determined mainly by scattering due to crystal imperfections such as defects, dislocations, grain boundaries, and surfaces. Electron-phonon scattering is frozen out at low temperatures. Since the defect concentration is largely temperature independent, the mean free path is a constant in this range. Therefore, the only temperature dependence in the thermal conductivity at low temperature arises from the heat capacity which varies as C T. Under these conditions, the thermal conductivity varies linearly with temperature as shown in Fig. 8.2. The value of k, though, is sample-specific since the mean free path depends on the defect density. Figure 8.2 plots the thermal conductivities of two metals. The data are the best recommended values based on a combination of experimental and theoretical studies [3],... 

When the concentration of chain interruptions is sufficiently high such that the left hand side of equation 1.3 is small, then the wave function will be localised. The possible limits for the conductivity arise from the chain interruptions and/or phonon scattering. All the above factors suggest that in high-quality conducting polymers the electronic mean free path could be much larger than the structural coherence length and real metallic features could be observed. 

At the beginning of Section 7.1.5., reference was made to the classification scheme introduced by Mott (see especially Mott (1971)). The significant parameter in Mott s approach is the d.c. conductivity because this quantity enables the magnitude of A, the electronic mean free path, to be estimated. If d represents the average interatomic spacing then the three cases of interest are ... 

---