Fermi, wave vector conductivity

Transport of electrons along conducting wires surrounded by insulators have been studied for several decades mechanisms of the transport phenomena involved are nowadays well understood (see [1, 2, 3] for review). In the ballistic regime where the mean free path is much longer than the wire lengths, l 3> d, the conductance is given by the Sharvin expression, G = (e2/-jrh)N, where N (kpa)2 is the number of transverse modes, a, is the wire radius, a Fermi wave vector. For a shorter mean free path diffusion controlled transport is obtained with the ohmic behavior of the conductance, G (e2/ph)N /d, neglecting the weak localization interference between scattered electronic waves. With a further decrease in the ratio /d, the ohmic behavior breaks down due to the localization effects when /d < N-1 the conductance appears to decay exponentially [4]. 

Within a one-electron description (i.e., U = 0, U being the on-site Coulomb repulsion [2,3], regular conducting TCNQ chains with p = electron per molecular site correspond to quarter-filled electronic bands. Consequently, the Fermi wave vector is in this case kF = n/4d, d being the spacing parameter between adjacent sites, and the chains are metallic. This is the case, for instance, for MEM(TCNQ)2 and TEA(TCNQ)2. Note that in these two salts the cations MEM+ and TEA+ are diamagnetic and do not participate in electrical conduction. 

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 electrical conductivity is mainly determined by the carrier density (n), relaxation time (r), and effective mass (m) of the carrier (electrical conductivity, cr = ne T/m). According to the loffe-Regel criterion, the interatomic distance is considered as the lower limit to the mean free path (A) in a metallic system. Hence, for a metallic system, kfX > 1, where kfX = [/j(37r ) / ]/(c pn / ), kfis the Fermi wave vector, and p is the electrical resistivity [1125, 1126]. In highly doped conducting polymers, n 10 per unit volume, m is nearly the electron mass, A is a few tens of angstroms, and kfk 1-10 at room temperature. The mean free path is limited by both the interchain transport and the extent of disorder present in the system. The details about metallic conducting polymers are shown in Table VI [1127]. 

Fig. 2. (a) Energy, E, versus wave vector, k, for free particle-like conduction band and valence band electrons (b) the corresponding density of available electron states, DOS, where Ep is Fermi energy (c) the Fermi-Dirac distribution, ie, the probabiUty P(E) that a state is occupied, where Kis the Boltzmann constant and Tis absolute temperature ia Kelvin. The tails of this distribution are exponential. The product of P(E) and DOS yields the energy distribution... 

In conducting solids, the conduction electron density is spatially modulated, forming charge density waves (CDW) the periodic distortion accompanying the CDW (due to interaction between the conduction electron and the lattice) is responsible for the incommensurate phase (Overhauser, 1962 Di Salvo Rice, 1979 Riste, 1977). The occurrence of CDW and the periodic distortion can be understood in terms of the model proposed by Peierls and Frdhlich for one-dimensional metals. Let us consider a row of uniformly spaced chain of ions (spacing = a) associated with conduction electrons of energy E k) and a wave vector k. At 0 K, all the states are filled up to the Fermi energy, = E(kp). If the electron density is sinusoidally modulated as in Fig. 4.15 such that... 

An impurity atom in a solid induces a variation in the potential acting on the host conduction electrons, which they screen by oscillations in their density. Friedel introduced such oscillations with wave vector 2kp to calculate the conductivity of dilute metallic alloys [10]. In addition to the pronounced effect on the relaxation time of conduction electrons, Friedel oscillations may also be a source of mutual interactions between impurity atoms through the fact that the binding energy of one such atom in the solid depends on the electron density into which it is embedded, and this quantity oscillates around another impurity atom. Lau and Kohn predicted such interactions to depend on distance as cos(2A pr)/r5 [11]. We note that for isotropic Fermi surfaces there is a single kp-value, whereas in the general case one has to insert the Fermi vector pointing into the direction of the interaction [12,13]. The electronic interactions are oscillatory, and their 1 /r5-decay is steeper than the monotonic 1 /r3-decay of elastic interactions [14]. Therefore elastic interactions between bulk impurities dominate the electronic ones from relatively short distances on. 

The energy of the conduction electrons is given by h2k2/2 x, where k is the wave vector number and p, is the effective mass of the electron-nucleus. In a real space of Cartesian coordinates k = [kx, ky, fcj, a Fermi sphere can be constructed with radius k = (2 lE )x 2/h. The shape of this sphere is a clearly defined by the electrical properties of the metal. The current density obeys the change in the occupancy of states near the Fermi level, which separates the unfilled orbitals in the metal from the filled ones in the linear momentum space p = hk. 

Intertubular interactions occur as well in bundles of SWNTs. They reduce the symmetry of the system and thus influence the band structure. For metaUic tubes this results in the opening of a bandgap because the valence and conduction bands for wave vectors in parallel with the tube s axis do no longer touch on the Fermi level. Still the bands overlap in another manner due to their dispersion rectangular with respect to the tube axes, so actually this is a pseudo-bandgap. 

The original work of Ruderman and Kittel is concerned with the interaction between nuclear spins due to the indirect interaction of the conduction electrons in simple s band metals. There the interaction between the electrons and the spins comes from the Fermi contact hyperfine interaction, so the matrix elements are independent of the initial and final wave vectors. The band structure is parabolic. If the same assumptions are made for the rare earths, the indirect coupling energy in eq. (3.56) has the form... 

We now consider the EET from fluorophores to doped graphene. We imagine that the Fermi level is shifted into the conduction band to a level with magnitude of wave vector kp. Thus, the rate of energy transfer has contributions from two different sets of transitions in graphene. In the first set, k lies in the valence band, with 0 < k, < o , and lies in the conduction band, with < k < oo. in the second set, both k, and ky lie in the conduction band, with 0 < k and < k < < (see Figure 10.6). The total rate can thus be written as the sum total of both contributions, i.e., k = k + ki-... 

---