Fermi, wave vector

Following Pauli s exclusion principle, each state corresponding to a given can contain at most two electrons of opposite spin. Therefore, at the absolute zero of temperature aU the states, k, will be occupied within a sphere of radius, kF, the so-called Fermi sphere because these correspond to the states of lowest energy, as can be seen from Fig. 2.9(a). The magnitude of the Fermi wave vector, kF, may be related to the total number of valence electrons N by... 

Here k is the Fermi wave vector determined from the value of the hole concentration p assuming a spherical Fermi surface, m is the hole effective mass taken as 0.5/no (mo is the free electron mass), is the exchange integral between the holes and the Mn spins, and h is Planck s constant. The transverse and longitudinal magnetic susceptibilities are determined from the magnetotransport data according to x = 3M/dB and xh = M/B. 

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 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]. 

On the contrary, the Fourier transform of an image such as shown in Fig. 6 contains an internal calibration, because the distance separating the lattice spots in the Fourier transform is related to the lateral lattice parameter of Cu(lll). In this case one can determine with high precision the size of the surface Fermi wave vector, which turns out to be kp = 0.205 0.02 A, i.e. a Fermi wavelength of 30 3 A for Cu(lll), in nice agreement with... 

The required 2D nearly free electron gas is realized in Shockley type surface states of close-packed surfaces of noble metals. These states are located in narrow band gaps in the center of the first Brillouin zone of the (lll)-projected bulk band structure. The fact that their occupied bands are entirely in bulk band gaps separates the electrons in the 2D surface state from those in the underlying bulk. Only at structural defects, such as steps or adsorbates, is there an overlap of the wave functions, opening a finite transmission between the 2D and the 3D system. The fact that the surface state band is narrow implies extremely small Fermi wave vectors and consequently the Friedel oscillations of the surface state have a significantly larger wave length than those of bulk states. 

Commensurability. Incommensurate lattice distortions and commensurate-incommensurate phase transitions are often observed in these materials. The incommensurability comes either from an incommensurate Fermi wave vector (2A F, 4kF scattering in charge-transfer salts) or from the counterion stacks (e.g., triiodide-containing materials). 

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. 

A weak 2kF diffuse x-ray scattering [58], seen only at low temperature, confirms that p = in this salt and indicates that the Fermi wave vector is rather well defined despite structural cation disorder. A stronger 4kF scattering [58] observed from 25 K to 300 K is an indication of large Coulomb effects in this salt, with U t, in agreement with other known magnetic and thermopower data [57,58]. 

However, as discussed extensively in review articles by Pouget [9] and by Barisic and Bjelis [10], the presence of both 2kF and 4kF anomalies, where kF is the Fermi wave vector of the quasi-one-dimensional electron gas, the fact that phonon softening at 2kF is relatively small, combined with theoretical considerations, have lead to the present-day viewpoint that electron-electron Coulomb interactions play an important role. 

Figure 4.3. For the monovalent FCC alkali metals (a) with a low electron density, the Fermi-wave vector (the radius of the Fermi sphere) lies well below the first BZ boundary. The Fermi surface is unperturbed. For monovalent FCC copper (b), the increased electron density forces the Fermi wave vector to terminate very near the L-point. The electron energy is lowered and the Fermi sphere necks outwards towards that face of the BZ boundary.

The connection follows the line of reasoning just presented. As a polyvalent metal is dissolved in a monovalent metal, the electron density increases, as does the Fermi energy and Fermi-wave vector. Eventually, the Fermi sphere touches the BZ boundary and the crystal stmcture becomes unstable with respect to alternative structures (Ra5mor, 1947 Pettifor, 2000). Subsequent work has been carried out confimiing that the structures of Hume-Rothery s alloys (alloys comprised of the noble metals with elements to the right on the periodic table) do indeed depend only on their electron per atom ratio (Stroud and Ashcroft, 1971 Pettifor and Ward, 1984 Pettifor, 2000). Unfortunately, the importance of the e/a ratio on phase equilibria is much less clear when it does not correspond precisely to BZ touching. 

Fig. 2.2. (a) Schematic illustration of the free-electron-like ID dispersion relation. f is the Fermi energy, fcp is the Fermi wave vector, and a is the lattice constant, (b) 3D schematic view of the resulting Fermi surface. Dashed line without interstack overlaps. Solid lines with small transfer integrals... 

Successive high-resolution AMRO experiments shown in Fig. 4.39 verified the proposed FS in an impressive way [376]. As mentioned in Sect. 3.3, 0-(ET)2l3 was one of the first compounds where AMRO, i. e., resistance oscillations periodic in tan (9, were observed [258]. These results, which were reproduced later [377], are understood by the warped FS model explained above. The period of the oscillations is related to the Fermi wave vector via (3.18). In the experimental data shown in Fig. 4.39 not only the previously reported fast AMROs but also slow ones (indicated by small dashes) were observed [376]. The insets of Fig. 4.39 show the peak numbers of the (a) fast and (b) slow oscillation frequency vs tan O. From the slopes for different field rotation planes fcr(0) could be constructed. The resulting two ellipsoidal FSs are in good agreement with the proposed topology of Fig. 4.37c with respect to both form and area. 

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