Hall effect (redirected from Hall current)

Hall effect, experiment that shows the sign of the charge carriers in a conductor. In 1879 E. H. Hall discovered that when he placed a metal strip carrying a current in a magnetic field, a voltage difference was produced across the strip. The side of the strip that is at the higher voltage depends on the sign of the charge carrier; Hall's work demonstrated that in metals the charge carriers are negative. Today it is known that this negative charge carrier is the electron. The Hall effect has again become an active area of research with the discovery of the quantized Hall effect, for which Klaus von Klitzing was awarded the 1985 Nobel Prize in physics. Before von Klitzing's work it was thought that the amount of voltage difference across the strip varied in direct proportion to the strength of the magnetic field—the greater the magnetic field, the greater the voltage difference. Von Klitzing showed that under the special conditions of low temperature, high magnetic field, and two-dimensional electronic systems (in which the electrons are confined to move in planes), the voltage difference increases as a series of steps with increasing magnetic field.

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Hall effect

Development of a transverse electric field in a solid material carrying an electric current and placed in a magnetic field perpendicular to the current. Discovered in 1879 by Edwin H. Hall (1855–1938), the Hall field results from the force exerted by the magnetic field on the moving particles of the current. The Hall effect can be used to measure certain properties of current carriers as well as to detect the presence of a current on a magnetic field.

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Hall effect

An effect whereby a conductor carrying an electric current perpendicular to an applied magnetic field develops a voltage gradient which is transverse to both the current and the magnetic field. It was discovered by E. H. Hall in 1879. Important information about the nature of the conduction process in semiconductors and metals may be obtained through analysis of this effect.

A simple model which accounts for the phenomenon is the following. For a magnetic field of strength B in the z direction (see illustration), particles flowing with speed v in the x direction suffer a Lorentz force F_L in the y direction given by

(1) 

Eq. (1), where q is the charge of the particles. This force deflects the particles so that a charge imbalance develops between opposite sides of the conductor. Deflection continues until the electric field E_y resulting from this charge imbalance produces a force F_y = qE_y which cancels the Lorentz force. In practice, the equilibrium condition F_L + F_y = 0 is achieved almost instantaneously, giving a steady-state Hall field as in Eq. (2). The current density is J_x = nqv, where n is the carrier density. The Hall resistivity, defined by Eq. (3), is thus given by Eq. (4). The Hall coefficient, defined by Eq. (5),

(2) 

(3) 

(4) 

(5) 

satisfies Eq. (6)

(6) 

and thus R₀ provides a measure of the sign and magnitude of the mobile charge density in a conductor. Within the free-electron theory of simple metals, q is expected to be the electron charge -e, and n is taken to be n = Zn_A, where Z is the valence of the metal and n_A is the density of the atoms. This yields Eq. (7). (7)  See Free-electron theory of metals

Configuration of fields and currents in the Hall effect experiment

Equation (7) is approximately valid in simple monovalent metals but fails drastically for other materials, often even giving the wrong sign. The explanation of the failures of Eq. (7) was one of the great early triumphs of the quantum theory of solids. The theory of band structure shows how collisions with the periodic array of atoms in a crystal can cause the current carriers to be holes which have an effective positive charge which changes the sign of the Hall coefficient. Band structure theory also accounts for the observed dependence of R₀ on the orientation of the current and the magnetic field relative to the crystal axes, an effect which is very useful for studying the topology of the Fermi surface. See Band theory of solids, Fermi surface, Hole states in solids

In certain special field-effect transistors, it is possible to create an electron gas which is effectively two-dimensional. The Hall resistance for an idealized system in two dimensions is given by Eq. (8),

(8) 

where n_S is the density of electrons per unit area (rather than volume). However, if the measured value of &rgr;_xy for a high-quality (low-disorder) device is plotted as a function of B, the linear behavior predicted by Eq. (8) is observed only at low fields. At high fields the Hall resistance exhibits plateau regions in which it is a constant independent of B. Furthermore, the values of &rgr;_xy on these plateaus are given quite accurately by the universal relation of Eq. (9),

(9) 

where h is Planck's constant and &ngr; is an integer or simple rational fraction. The absolute accuracy with which Eq. (9) has been verified is better than 1 part in 10⁶.

This extremely accurate quantization of &rgr;_xy allows the realization of a new standard of resistance based solely on fundamental constants of nature. In addition, the quantum unit of Hall resistance, h/e² ≃ 25,812.80 ohms, determines the fine-structure constant. See Electrical units and standards, Fundamental constants

The explanation of this remarkable phenomenon involves several subtle quantum-mechanical effects. In the quantum regime (small &ngr;), &rgr;_xx, which is the dissipative (longitudinal) resistivity, approaches zero on the Hall plateaus. The quantization of the Hall resistance is intimately connected with this fact. It is speculated that at zero temperature the dissipation is zero and that Eq. (9) is then obeyed exactly. See Quantum mechanics

The nearly complete lack of dissipation in the quantum Hall regime is reminiscent of superconductivity. In both effects the ability of the current to flow without dissipation has its origin in the existence of a quantum-mechanical excitation gap, that is, a minimum threshold energy needed to disturb the special microscopic order in the system. See Entropy, Superconductivity

In the integer quantum Hall effect [where &ngr; in Eq. (9) in an integer], this excitation gap is a single-particle effect associated with the quantization by the strong magnetic field of the kinetic energy of the individual electrons into discrete states called Landau levels. In the fractional effect, the gap is associated with the highly collective, many-body ordering of the electrons into a quantum state which minimizes the strong Coulomb repulsion and hence lowers the overall energy. Thus, while the integer and fractional quantum Hall effects look superficially similar on a plot of resistivities versus magnetic field, their physical origins are actually quite different. See De Haas-van Alphen effect, Galvanomagnetic effects