pproximation are recovered by The relaxation times, t will also be taken to be sotropic(not k dependent) within each band but can be different from band to band. Although extreme, these approximations are often qualitatively correct, particularly in polycrystalline materials, which are macroscop- ically isotropic. Further, in semiconductors these results will be strictly applicable only if t; is energy independent as well as isotropic. For a single spherical band, RH is a direct measure of the number of current carriers and turns out to be given by [Blatt, 1957] R (52.12) carriers being negative for electrons and positive for holes. This identification of the carrier sign is itsel where n is the number of conduction carriers/volume. RH depends on the sign of the charge of the curre matter of great importance, particularly in semiconductor physics. If more than one band is involved in electrical conduction, then by imposing the boundary condition required for the geometry of Fig. 52.1 that the total current in the y direction from all bands must vanish, J,=0, it is easy to show that [wilson, 1958] RH=(1/o)2o?RI (52.13) where R, and o, are the Hall coefficient and electrical conductivity, respectively, for the ith band(o net, /m) o=Eo, is the total conductivity of the material, and the summation is taken over all bands Using Eq (52.12), Eq(52. 13)can also be writte R (5214) where nef is the effective or apparent number of electrons determined by a Hall effect experiment. Note that ome workers prefer representing Eqs. (52.13)and(52. 14)in terms of the current carrier mobility for each band, u, defined by 0, =n;eu. The most commonly used version of Eq (52.14)is the so-called two-band model, which assumes that there are two spherical bands with one composed of electrons and the other of holes. Eq (52. 14)then takes the for From Eq. (52. 14)or(52. 15)it is clear that the Hall effect is dominated by the most highly conducting band Although for fundamental reasons it is often the case that ne= nh,(a so-called compensated material), RH would rarely vanish since the conductivities of the two bands would rarely be identical. It is also clear from any of Eqs. (52.12),(52.14), or(52. 15)that, in general, the Hall effect in semiconductors will be orders of magnitude larger than that in metals 52.4 Relation to the Electronic Structure -(ii)oct >>1 The high field limit can be achieved in metals only in pure, crystalographically well-ordered materials and at low temperatures, which circumstances limit the electron scattering rate from impurities, crystallographic c 2000 by CRC Press LLC© 2000 by CRC Press LLC approximation are recovered by allowing the masses to vary. The relaxation times, ti , will also be taken to be isotropic (not k dependent) within each band but can be different from band to band. Although extreme, these approximations are often qualitatively correct, particularly in polycrystalline materials, which are macroscop￾ically isotropic. Further, in semiconductors these results will be strictly applicable only if ti is energy independent as well as isotropic. For a single spherical band, RH is a direct measure of the number of current carriers and turns out to be given by [Blatt, 1957] (52.12) where n is the number of conduction carriers/volume. RH depends on the sign of the charge of the current carriers being negative for electrons and positive for holes. This identification of the carrier sign is itself a matter of great importance, particularly in semiconductor physics. If more than one band is involved in electrical conduction, then by imposing the boundary condition required for the geometry of Fig. 52.1 that the total current in the y direction from all bands must vanish, Jy = 0, it is easy to show that [Wilson, 1958] RH = (1/s)2 S[si 2 Ri ] (52.13) where Ri and si are the Hall coefficient and electrical conductivity, respectively, for the ith band (si = nie2ti/mi ), s = Ssi is the total conductivity of the material, and the summation is taken over all bands. Using Eq. (52.12), Eq. (52.13) can also be written (52.14) where neff is the effective or apparent number of electrons determined by a Hall effect experiment. (Note that some workers prefer representing Eqs. (52.13) and (52.14) in terms of the current carrier mobility for each band, mi , defined by si = niemi .) The most commonly used version of Eq. (52.14) is the so-called two-band model, which assumes that there are two spherical bands with one composed of electrons and the other of holes. Eq. (52.14) then takes the form (52.15) From Eq. (52.14) or (52.15) it is clear that the Hall effect is dominated by the most highly conducting band. Although for fundamental reasons it is often the case that ne = nh (a so-called compensated material), RH would rarely vanish since the conductivities of the two bands would rarely be identical. It is also clear from any of Eqs. (52.12), (52.14), or (52.15) that, in general, the Hall effect in semiconductors will be orders of magnitude larger than that in metals. 52.4 Relation to the Electronic Structure — (ii) vct >> 1 The high field limit can be achieved in metals only in pure, crystalographically well-ordered materials and at low temperatures, which circumstances limit the electron scattering rate from impurities, crystallographic R ne H = 1 R en e n H i i = = Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ Â ˙ 1 1 1 2 eff s s R e n n H e e h h = Ê Ë Á ˆ ¯ ˜ - Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ 1 1 1 2 2 s s s s