E FIGURE 52.1 Typical Hall effect experimental arrangement in a flat plate conductor with current J and magnetic field B2. The Hall electric field EH=E, in this geometry arises because of the Lorentz force on the conducting charges and is of just such a magnitude that in combination with the Lorentz force there is no net current in the y direction. The angle o between the current and net electric field is called the hall angle 52.2 Theoretical background The Boltzmann equation for an electron gas in a homogeneous, isothermal material that is subject to constant electric and magnetic fields is [Ziman, 1960 4E+以XB五Ff(k)(a (524) Here k is the quantum mechanical wave vector, h is Plancks constant divided by 2T, t is the time, f is the electron distribution function and "s" is meant to indicate that the time de scattering of the electrons. In static equilibrium(E=0,B=0)f is equal to fo and fo is the Fermi-Dirac distribution function f6 e(k)-/Kr+1 (525) where &(k)is the energy, s is the chemical potential, Kis Boltzmann's constant, and Tis the temperature. Each term in Eq (52.4)represents a time rate of change of f and in dynamic equilibrium their sum has to be zero The last term represents the effect of collisions of the electrons with any obstructions to their free movement such as lattice vibrations, crystallographic imperfections, and impurities. These collisions are usually assumed to be representable by a relaxation time, t(k), that -(f-f0) at t(k) (526) τ(k) where f-fo is written as(df/de)g(k), which is essentially the first term in an expansion of the deviation of f from its equilibrium value, fo Eqs. (52.6)and(52. 4)can be combined to give eE + vXBVif(k)= df /do)g(k) h τ(k) If Eq (52.7)can be solved for g(k), then expressions can be obtained for both the eHand the magnetoresistance ( the electrical resistance in the presence of a magnetic field). Solutions can in fact be developed that are linear c 2000 by CRC Press LLC© 2000 by CRC Press LLC 52.2 Theoretical Background The Boltzmann equation for an electron gas in a homogeneous, isothermal material that is subject to constant electric and magnetic fields is [Ziman, 1960] (52.4) Here k is the quantum mechanical wave vector, h is Planck’s constant divided by 2p, t is the time, f is the electron distribution function, and “s” is meant to indicate that the time derivative of f is a consequence of scattering of the electrons. In static equilibrium (E = 0, B = 0) f is equal to f0 and f0 is the Fermi–Dirac distribution function (52.5) where E(k) is the energy, z is the chemical potential, K is Boltzmann’s constant, and T is the temperature. Each term in Eq. (52.4) represents a time rate of change of f and in dynamic equilibrium their sum has to be zero. The last term represents the effect of collisions of the electrons with any obstructions to their free movement such as lattice vibrations, crystallographic imperfections, and impurities. These collisions are usually assumed to be representable by a relaxation time, t(k), that is (52.6) where f – f0 is written as (¶f0/¶e)g(k), which is essentially the first term in an expansion of the deviation of f from its equilibrium value, f0. Eqs. (52.6) and (52.4) can be combined to give (52.7) If Eq. (52.7) can be solved for g(k), then expressions can be obtained for both the EH and the magnetoresistance (the electrical resistance in the presence of a magnetic field). Solutions can in fact be developed that are linear FIGURE 52.1 Typical Hall effect experimental arrangement in a flat plate conductor with current Jx and magnetic field Bz . The Hall electric field EH = Ey in this geometry arises because of the Lorentz force on the conducting charges and is of just such a magnitude that in combination with the Lorentz force there is no net current in the y direction. The angle f between the current and net electric field is called the Hall angle. e f f t s [ =0 E vX B k k + Ê Ë Á ˆ ¯ ˜— - Ê Ë Á ˆ ¯ ] () ˜ 1 h ¶ ¶ f e 0 KT 1 1 = + ( ) E ( ) k – / z ¶ ¶ t ¶ ¶ t f t f f f g c Ê Ë Á ˆ ¯ ˜ = ( ) ( ) = – – ( / )() ( ) 0 0 k k k E e f f g [ ] () ( )( ) ( ) E vB k k k k + —= X 1 0 h ¶ ¶ t / E