9780030839931, Chapter 12, Problem 4P

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Chapter 12, Problem 4P Problem The free electron result (1 19) for the current induced by an electric field perpendicular to auniform magnetic field can be written in the form (12.69) where the resistivity tensor P has the form RH (12.70) (It follows from the definitions (1 14) and (1 15) that the magnetoresistuncc and Hall coefficient.) (a) Consider metal with several partially filled bands, in each of which the induced current is related to the field by En pnjn, where the pn have the form (12.70): Pn (12.71) Show that the total induced current is given by E with (12.72) (b) Show that there are only two bands, then the Hall coefficient and magnetoresistance are given by: R = + (12.73) = (12.74) Note the explicit dependence on magnetic field strength even the R/, and pi, are field- indepeadent, (as they are for free electron (c) Deduce directly from (12 73) the form (12.55) of the high-field Hall coefficient when both bands have closed orbits, and discuss the limiting high-field behavior in the case neff for compensated band metal). Show that in this case the magnetoresistance increases as H2 with increasing Step-by-step solution Step (a) The relation between current density (induced current) and electric field E is, E (1) Here is the conductivity From equation (1), the current density will be, Thus the total current density will be, So, Comparing equation (1) and (2) we get, Step (b) The resistivity tensor has the form, So, R,H And similarly R2H Step So the inverse of P1 and P2 will P1 R,H (3) -R,H And (4) -R2H Step Adding equation (4) and taking the inverse, we get, -RH If we write the above equation in the form then get, RH p R And Step (c) In accordance to the equation of Hall as lim R then we get, the high field hall coefficient has =0. 1 Since, and therefore the high field Hall coefficient R will tend to Again, since lim R,R, and R thus R1 R2 should be zero. So, We know that that, (5) Substitute for and -R1 for R2 in the equation we get, (6) From the above observation we conclude that, the magnetoresistance increases as with increasing field.