in the applied electric field (the regime where Ohms law holds) for two physical situations: (i)when ot<< 1 Hurd, 1972, P. 69] and (ii)when ot >>1[Hurd, 1972; Lifshitz et al, 1956] where O- Belm is the cyclotron frequency. Situation(ii)means the electron is able to complete many cyclotron orbits under the influence of B in the time between scatterings and is called the high(magnetic) field limit. Conversely, situation (i)is obtained when the electron is scattered in a short time compared to the time necessary to complete one cyclotron orbit and is known as the low field limit. In effect, the solution to Eq. (52.7) is obtained by expanding g(k)in a power series in o t or 1/ot for (i) and(ii), respectively. Given g(k) the current vector, h(l=x,) z)can be calculated from [Blatt, 1957] (4r3月v(kg(k)(606)dk (528) where v, (k)is the velocity of the electron with wave vector k. Every term in the series defining J is linear in the applied electric field, E, so that the conductivity tensor om is readily obtained from J=O mEm [Hurd, 1972, 9] This matrix equation can be inverted to give E1=PlmM. For the same geometry used in defining Eq (52.1) Ey= EH=Pxx/x (529) where p2 is a component of the resistivity tensor sometimes called the Hall resistivity. Comparing Eqs. (52.1) and(52.9)it is clear that the B dependence of Eu is contained in P,2. However, nothing in the derivation of p excludes the possibility of terms to the second or higher powers in B. Although these are usually small, this is one of the reasons that experimentally one usually obtains R from the measured transverse voltage by reversing magnetic fields and averaging the measured EH by calculating(1/2)[EH(B)-EH(B)]. This eliminates the second-order term in B and in fact all even power terms contributing to the ER Using the Onsager relation Smith and Jensen, 1989, P 60]P2 (B)=P2(B), it is also easy to show that in terms of the Hall resistivity R p12(B)+p21(B) (52.10) 2 B Strictly speaking, in a single crystal the electric field resulting from an applied electric current and magnetic field, both of arbitrary direction relative to crystal axes and each other, cannot be fully described in terms of a second-order resistivity tensor. [Hurd, 1972, p. 71] On the other hand, Eqs. (52.1),52.9), and(52.10)do define the Hall coefficient in terms of a second-order resistivity tensor for a polycrystalline(assumed isotropic)sample or for a cubic single crystal or for a lower symmetry crystal when the applied fields are oriented along major symmetry directions. In real world applications the Hall effect is always treated in this manner. 52.3 Relation to the Electronic Structure -(i)oct < 1 General expressions for R in terms of the parameters that describe the electronic structure can be obtained using Eqs. (52.7)-(52.10)and have been given by Blatt [Blatt, 1957] for the case of crystals having cubi symmetry. An even more general treatment has been given by McClure [McClure, 1956]. Here the discussion of specific results will be restricted to the free electron model wherein the material is assumed to have one or more conducting bands, each of which has a quadratic dispersion relationship connecting e and k; that is 力2k2 (52.11) 2m where the subscript specifies the band number and m, the effective mass for each band. These masses need not be equal nor the same as the free electron mass. In effect, some of the features lost in the free electron c 2000 by CRC Press LLC© 2000 by CRC Press LLC in the applied electric field (the regime where Ohm’s law holds) for two physical situations: (i) when wct << 1 [Hurd, 1972, p. 69] and (ii) when wct >> 1 [Hurd, 1972; Lifshitz et al., 1956] where wc = Be/m is the cyclotron frequency. Situation (ii) means the electron is able to complete many cyclotron orbits under the influence of B in the time between scatterings and is called the high (magnetic) field limit. Conversely, situation (i) is obtained when the electron is scattered in a short time compared to the time necessary to complete one cyclotron orbit and is known as the low field limit. In effect, the solution to Eq. (52.7) is obtained by expanding g(k) in a power series in wct or 1/wct for (i) and (ii), respectively. Given g(k) the current vector, Jl (l = x,y,z) can be calculated from [Blatt, 1957] (52.8) where vl(k) is the velocity of the electron with wave vector k. Every term in the series defining Jl is linear in the applied electric field, E, so that the conductivity tensor slm is readily obtained from Jl= slm Em [Hurd, 1972, p. 9] This matrix equation can be inverted to give El = rlm Jm . For the same geometry used in defining Eq. (52.1) Ey = EH = ryx Jx (52.9) where r21 is a component of the resistivity tensor sometimes called the Hall resistivity. Comparing Eqs. (52.1) and (52.9) it is clear that the B dependence of EH is contained in r12. However, nothing in the derivation of r12 excludes the possibility of terms to the second or higher powers in B. Although these are usually small, this is one of the reasons that experimentally one usually obtains R from the measured transverse voltage by reversing magnetic fields and averaging the measured EH by calculating (1/2)[EH (B) – EH (–B)]. This eliminates the second-order term in B and in fact all even power terms contributing to the EH. Using the Onsager relation [Smith and Jensen, 1989, p. 60] r12(B) = r21(–B), it is also easy to show that in terms of the Hall resistivity (52.10) Strictly speaking, in a single crystal the electric field resulting from an applied electric current and magnetic field, both of arbitrary direction relative to crystal axes and each other, cannot be fully described in terms of a second-order resistivity tensor. [Hurd, 1972, p. 71] On the other hand, Eqs. (52.1), (52.9), and (52.10) do define the Hall coefficient in terms of a second-order resistivity tensor for a polycrystalline (assumed isotropic) sample or for a cubic single crystal or for a lower symmetry crystal when the applied fields are oriented along major symmetry directions. In real world applications the Hall effect is always treated in this manner. 52.3 Relation to the Electronic Structure — (i) vct << 1 General expressions for R in terms of the parameters that describe the electronic structure can be obtained using Eqs. (52.7)–(52.10) and have been given by Blatt [Blatt, 1957] for the case of crystals having cubic symmetry. An even more general treatment has been given by McClure [McClure, 1956]. Here the discussion of specific results will be restricted to the free electron model wherein the material is assumed to have one or more conducting bands, each of which has a quadratic dispersion relationship connecting E and k; that is (52.11) where the subscript specifies the band number and mi , the effective mass for each band. These masses need not be equal nor the same as the free electron mass. In effect, some of the features lost in the free electron J e v g f d k l l = Ê Ë Á ˆ ¯ ˜ 4 3 Ú 0 3 p (k) (k)(¶ /¶E ) R B = + 1 2 1 12 21 [r (B B ) r ( )] E i i i k m = h2 2 2