No abstract
The lowest-order solution to the linearized Boltzmann equation is calculated for Nb and Cu with uniform external electric and magnetic fields. This solution corresponds to a rigid displacement of the Fermi surface, and should be accurate when the anisotropy of the electron scattering function is small. For Cu the result agrees very well with experiment. The agreement with experiment for Nb is within 14'.The Hall coefficient R~h as been measured in many metals. ' Interpretation of the coefficient is more difficult for metals than for semiconductors because of the more complicated Fermisurface topology. Relatively few calculations of the Hall coefficient have been made for transition metals. ' ' A method for calculation R~w as given by Jones and Zener, ' and more recently by Hasegawa and Kasuya' (HK). Jones and Zener assume no anisotropy of scattering; HK allow for anisotropy but find very little at room temperature for Cu. An expression for R~equivalent to the theory of Ref. 6 was given by Tsuji'. 688c g»~( -sf a k~a where (1/p)» is the mean curvature of the Fermi surface at the point k. The notation k is short for (k n), the wave-vector and band index of an electron of energy e» and velocity Iv» =. Be»/sk. The Fermi-occupation factor is denoted f. The interpretation of Eq. (1) is that R» is a measure of curvature. For a free-electron sphere (1/p} is I/O» and Eq. (1) yields B» = -1/nec. For nonspherical surfaces, Eq. (1) can differ strongly from -I/nec because regions with large e» or large curvature are more heavily weighted. The Fermi surface of Cu, for example, is an electron surface by any reasonable definition, but the curvature (1/p)» is negative over significant regions near the necks. These happen to be regions of relatively low v~so R~i s negative as expected, but in principle the result could have been positive if the v"'s had been different. The evaluation of Eq. (1) is straightforward, provided one has values of v~on a fine mesh of k points on the Fermi surface. In particular, it is not necessary to know components of V~vñ ormal to the Fermi surface as may be seen by writing R~a s "»p(v» x&»sa } &a i 2ekc g~» -8 fl A ) (2)Equation (2) shows that only those components of V~v~"normal to v~, that is only those components of V~v~lying on the Fermi surface, contribute to the sum. A derivation of Eqs. (1) and (2} is given in the Appendix. It is shown there that these equations follow from a very simple ansatz that the whole Fermi surface is displaced in a direction given by E + a (E x H), where a is a constant determined by the Boltzmann equation.A Korringa-Kohn-Rostoker (KKR) program in the constant energy mode was used to generate a mesh of k points on the Fermi surface. In the irreducible 48 th, 492 points were used for Cu and 1060 points for Nb. In contrast, HK used 576 points over the Fermi surface of Cu which corresponds to 12 points in the irreducible-', th. The reason we used such a fine mesh was to reduce the error arising from computing the derivatives in Eq.(2) and in(1/p)» via the central differ...
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