Ehrlich a c. The Hall Effect The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Ehrlich A.C. “The Hall Effect” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
52 The Hall Effect 52.1 Introduction 52.2 Theoretical Background Alexander C. ehrlich 52.3 Relation to the Electronic Structure-i)OT<<1 U.S. Naval Research Laboratory 52.4 Relation to the Electronic Structure-(ii)ot>>1 52.1 Introduction The Hall effect is a phenomenon that arises when an electric current and magnetic field are simultaneously imposed on a conducting material. Specifically, in a flat plate conductor, if a current density, J,, is applied in the x direction and(a component of )a magnetic field, B,, in the z direction, then the resulting electric field, Ey transverse to /, and B, is known as the Hall electric field EH (see Fig. 52. 1)and is given by Ey=R,B (521) where r is known as the hall coefficient The hall coefficient can be related to the electronic structure and properties of the conduction bands in metals and semiconductors and historically has probably been the most important single parameter in the characterization of the latter. Some authors choose to discuss the Hall effect in terms of the Hall angle, shown in Fig. 52.1, which is the angle between the net electric field and the imposed current. Thus tan O= EHE (522) For the vast majority of Hall effect studies that have been carried out, the origin of EH is the Lorentz force, FL that is exerted on a charged particle as it moves in a magnetic field. For an electron of charge e with velocity , Ft is proportional to the vector product of v and B; that is, FL= evxB In these circumstances a semiclassical description of the phenomenon is usually adequate. This description combines the classical Boltzmann transport equation with the Fermi-Dirac distribution function for the charge carriers(electrons or holes)[Ziman, 1960], and this is the point of view that will be taken in this chapter Examples of Hall effect that cannot be treated semiclassically are the spontaneous(or extraordinary)Hall effect that occurs in ferromagnetic conductors [Berger and Bergmann, 1980], the quantum Hall effect [Prange and Girvin, 1990], and the Hall effect that arises in conjuction with hopping conductivity [Emin, 1977] In addition to its use as an important tool in the study of the nature of electrically conducting materials, the Hall effect has a number of direct practical applications. For example, the sensor in some commercial levies for measuring the magnitude and orientation of magnetic fields is a Hall sensor. The spontaneous Hall ffect has been used as a nondestructive method for exploring the presence of defects in steel structures. The quantum Hall effect has been used to refine our knowledge of the magnitudes of certain fundamental constants such as the ratio of e /h where h is Planck's constant. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 52 The Hall Effect 52.1 Introduction 52.2 Theoretical Background 52.3 Relation to the Electronic Structure—(i) wct << 1 52.4 Relation to the Electronic Structure—(ii) wct >> 1 52.1 Introduction The Hall effect is a phenomenon that arises when an electric current and magnetic field are simultaneously imposed on a conducting material. Specifically, in a flat plate conductor, if a current density, Jx , is applied in the x direction and (a component of) a magnetic field, Bz, in the z direction, then the resulting electric field, Ey , transverse to Jx and Bz is known as the Hall electric field EH (see Fig. 52.1) and is given by Ey = RJxBz (52.1) where R is known as the Hall coefficient. The Hall coefficient can be related to the electronic structure and properties of the conduction bands in metals and semiconductors and historically has probably been the most important single parameter in the characterization of the latter. Some authors choose to discuss the Hall effect in terms of the Hall angle, f, shown in Fig. 52.1, which is the angle between the net electric field and the imposed current. Thus, tan f = E H/Ex (52.2) For the vast majority of Hall effect studies that have been carried out, the origin of EH is the Lorentz force, FL , that is exerted on a charged particle as it moves in a magnetic field. For an electron of charge e with velocity v, FL is proportional to the vector product of v and B; that is, FL = evxB (52.3) In these circumstances a semiclassical description of the phenomenon is usually adequate. This description combines the classical Boltzmann transport equation with the Fermi–Dirac distribution function for the charge carriers (electrons or holes) [Ziman, 1960], and this is the point of view that will be taken in this chapter. Examples of Hall effect that cannot be treated semiclassically are the spontaneous (or extraordinary) Hall effect that occurs in ferromagnetic conductors [Berger and Bergmann, 1980], the quantum Hall effect [Prange and Girvin, 1990], and the Hall effect that arises in conjuction with hopping conductivity [Emin, 1977]. In addition to its use as an important tool in the study of the nature of electrically conducting materials, the Hall effect has a number of direct practical applications. For example, the sensor in some commercial devices for measuring the magnitude and orientation of magnetic fields is a Hall sensor. The spontaneous Hall effect has been used as a nondestructive method for exploring the presence of defects in steel structures. The quantum Hall effect has been used to refine our knowledge of the magnitudes of certain fundamental constants such as the ratio of e2 /h where h is Planck’s constant. Alexander C. Ehrlich U.S. Naval Research Laboratory
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
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
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 macroscopically 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