The electrical engineering handbook
Ehrlich A.C. “The Hall Effect” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 © 2000 by CRC Press LLC 52 The Hall Effect 52.1Introduction 52.2Theoretical Background 52.3Relation to the Electronic Structure—(i) w c t << 1 52.4Relation to the Electronic Structure—(ii) w c t >> 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 x , is applied in the x direction and (a component of) a magnetic field, B z , in the z direction, then the resulting electric field, E y , transverse to J x and B z is known as the Hall electric field E H (see Fig. 52.1) and is given by E y = RJ x B z (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 / E x (52.2) For the vast majority of Hall effect studies that have been carried out, the origin of E H is the Lorentz force, F L , that is exerted on a charged particle as it moves in a magnetic field. For an electron of charge e with velocity v, F L is proportional to the vector product of v and B ; that is, F L = e vxB (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 e 2 / h where h is Planck’s constant. Alexander C. Ehrlich U.S. Naval Research Laboratory © 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 2 p , 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 f 0 and f 0 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 – f 0 is written as ( ¶ f 0 / ¶ e ) g ( k ), which is essentially the first term in an expansion of the deviation of f from its equilibrium value, f 0 . 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 E H 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 J x and magnetic field B z . The Hall electric field E H = E y 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. ef f t s [=0EvXB k k + æ è ç ö ø ÷ Ñ- æ è ç ö ø ÷ ]() 1 h ¶ ¶ f e KT 0 1 1 = + () E ()k –/z ¶ ¶ t ¶¶ t f t ff fg c æ è ç ö ø ÷ = ( ) ( ) = –– ( / )() () 0 0 k k k E ef fg []() ()() () EvB k k k k +Ñ=X 1 0 h ¶¶ t / E © 2000 by CRC Press LLC in the applied electric field (the regime where Ohm’s law holds) for two physical situations: (i) when w c t << 1 [Hurd, 1972, p. 69] and (ii) when w c t >> 1 [Hurd, 1972; Lifshitz et al., 1956] where w c = 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 w c t or 1/ w c t for (i) and (ii), respectively. Given g ( k ) the current vector, J l ( l = x,y,z ) can be calculated from [Blatt, 1957] (52.8) where v l (k) is the velocity of the electron with wave vector k. Every term in the series defining J l is linear in the applied electric field, E, so that the conductivity tensor s lm is readily obtained from J l = s lm E m [Hurd, 1972, p. 9] This matrix equation can be inverted to give E l = r lm J m . For the same geometry used in defining Eq. (52.1) E y = E H = r yx J x (52.9) where r 21 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 E H is contained in r 12 . However, nothing in the derivation of r 12 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 E H by calculating (1/2)[E H (B) – E H (–B)]. This eliminates the second-order term in B and in fact all even power terms contributing to the E H . Using the Onsager relation [Smith and Jensen, 1989, p. 60] r 12 (B) = r 21 (–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) c t << 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 m i , 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 vgfdk l l = æ è ç ö ø ÷ ò 4 3 0 3 p ¶¶()()( )kk/E R B =+ 1 2 1 12 21 [() ()]rrBB E i i i k m = h 22 2 © 2000 by CRC Press LLC approximation are recovered by allowing the masses to vary. The relaxation times, t i , 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 t i is energy independent as well as isotropic. For a single spherical band, R H 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. R H 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, J y = 0, it is easy to show that [Wilson, 1958] R H = (1/s) 2 S[s i 2 R i ] (52.13) where R i and s i are the Hall coefficient and electrical conductivity, respectively, for the ith band (s i = n i e 2 t i /m i ), s = Ss i 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 n eff 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, m i , defined by s i = n i em i .) 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 n e = n h (a so-called compensated material), R H 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) c t >> 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 == æ è ç ö ø ÷ é ë ê ê ù û ú ú å 111 2 eff s s R en n H e e h h = æ è ç ö ø ÷ - æ è ç ö ø ÷ é ë ê ê ù û ú ú 11 1 22 s s s s © 2000 by CRC Press LLC imperfections, and lattice vibrations, respectively. In semiconductors, the much longer relaxation time and smaller effective mass of the electrons makes it much easier to achieve the high field limit. In this limit the result analogous to Eq. (52.15) is [Blatt, 1968, p. 290] (52.16) Note that the individual band conductivities do not enter in Eq. (52.16). Eq. (52.16) is valid provided the cyclotron orbits of the electrons are closed for the particular direction of B used. It is not necessary that the bands be spherical or the t’s isotropic. Also, for more than two bands R H depends only on the net difference between the number of electrons and the number of holes. For the case where n e = n h , in general, the lowest order dependence of the Hall electric field on B is B 2 and there is no simple relationship of R H to the number of current carriers. For the special case of the two-band model, however, R H is a constant and is of the same form as Eq. (52.15) [Fawcett, 1964]. Metals can have geometrically complicated Fermi surfaces wherein the Fermi surface contacts the Brillouin zone boundary as well as encloses the center of the zone. This leads to the possibility of open electron orbits in place of the closed cyclotron orbits for certain orientations of B. In these circumstances R can have a variety of dependencies on the magnitude of B and in single crystals will generally be dependent on the exact orientation of B relative to the crystalline axes [Hurd, 1972, p. 51; Fawcett, 1964]. R will not, however, have any simple relationship to the number of current carriers in the material. Semiconductors have too few electrons to have open orbits but can manifest complicated behavior of their Hall coefficient as a function of the magnitude of B. This occurs because of the relative ease with which one can pass from the low field limit to the high field limit and even on to the so-called quantum limit with currently attainable magnetic fields. (The latter has not been discussed here.) In general, these different regimes of B will not occur at the same magnitude of B for all the bands in a given semiconductor, further complicating the dependence of R on B. Defining Terms Conducting band: The band in which the electrons primarily responsible for the electric current are found. Effective mass: An electron in a lattice responds differently to applied fields than would a free electron or a classical particle. One can, however, often describe a particular response using classical equations by defining an effective mass whose value differs from the actual mass. For the same material the effective mass may be different for different phenomena; e.g., electrical conductivity and cyclotron resonance. Electron band: A range or band of energies in which there is a continuum (rather than a discrete set as in, for example, the hydrogen atom) of allowed quantum mechanical states partially or fully occupied by electrons. It is the continuous nature of these states that permits them to respond almost classically to an applied electric field. Hole or hole state: When a conducting band, which can hold two electrons/unit cell, is more than half full, the remaining unfilled states are called holes. Such a band responds to electric and magnetic fields as if it contained positively charged carriers equal in number to the number of holes in the band. Relaxation time: The time for a distribution of particles, out of equilibrium by a measure F, to return exponentially toward equilibrium to a measure F/e out of equilibrium when the disequilibrating fields are removed (e is the natural logarithm base). Related Topic 22.1 Physical Properties R en n H eh = - 11 © 2000 by CRC Press LLC References L. Berger and G. Bergmann, in The Hall Effect and Its Applications, C. L. Chien and C. R. Westlake, Eds., New York: Plenum Press, 1980, p. 55. F. L. Blatt in Solid State Physics, vol. 4, F. Seitz and D. Turnbull, Eds., New York: Academic Press, 1957, p. 199. F. L. Blatt, Physics of Electronic Conduction in Solids, New York: McGraw-Hill, 1968, p. 290. See also N. W. Ashcroft and N. D. Mermin in Solid State Physics, New York: Holt, Rinehart and Winston, 1976, p. 236. D. Emin, Phil. Mag., vol. 35, p. 1189, 1977. E. Fawcett, Adv. Phys. vol. 13, p. 139, 1964. C. M. Hurd, The Hall Effect in Metals and Alloys, New York: Plenum Press, 1972, p. 69. I. M. Lifshitz, M. I. Azbel, and M. I. Kaganov, Zh. Eksp. Teor. Fiz., vol. 31, p. 63, 1956 [Soviet Phys. JETP (Engl. Trans.), vol. 4, p. 41, 1956]. J. W. McClure, Phys. Rev., vol. 101, p. 1642, 1956. R. E. Prange and S. M. Girvin, Eds., The Quantum Hall Effect, New York: Springer-Verlag, 1990. H. Smith, and H. H. Jensen, Transport Phenomena, Oxford: Oxford University Press, 1989, p. 60. A. H. Wilson, The Theory of Metals, London: Cambridge University Press, 1958, p. 212. J. M. Ziman, Electrons and Phonons, London: Oxford University Press, 1960. See also N. W. Ashcroft and N. D. Mermin in Solid State Physics, New York: Holt, Rinehart and Winston, 1976, chapters 12 and 16. Further Information In addition to the texts and review article cited in the references, an older but still valid article by J. P. Jan, in Solid State Physics (edited by F. Seitz and D. Turnbull, New York: Academic Press, 1957, p. 1) can provide a background in the various thermomagnetic and galvanomagnetic properties in metals. A parallel background for semiconductors can be found in the monograph by E. H. Putley, The Hall Effect and Related Phenomena (Boston: Butterworths, 1960). Examples of applications of the Hall effect can be found in the book Hall Generators and Magnetoresistors, by H. H. Wieder, edited by H. J. Goldsmid (London: Pion Limited, 1971). An index to the most recent work on or using any aspect of the Hall effect reported in the major technical journals can be found in Physics Abstracts (Science Abstracts Series A). . Ehrlich A.C. The Hall Effect” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 © 2000 by CRC Press LLC 52 The Hall. be solved for g ( k ), then expressions can be obtained for both the E H and the magnetoresistance (the electrical resistance in the presence of a magnetic