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