Hall effect

The Hall effect is the production of a voltage difference (the Hall voltage) across an electrical conductor, transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was discovered by Edwin Hall in 1879. For clarity, the original effect is sometimes called the ordinary Hall effect to distinguish it from other 'Hall effects' which have different physical mechanisms. The Hall effect is the production of a voltage difference (the Hall voltage) across an electrical conductor, transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was discovered by Edwin Hall in 1879. For clarity, the original effect is sometimes called the ordinary Hall effect to distinguish it from other 'Hall effects' which have different physical mechanisms. The Hall coefficient is defined as the ratio of the induced electric field to the product of the current density and the applied magnetic field. It is a characteristic of the material from which the conductor is made, since its value depends on the type, number, and properties of the charge carriers that constitute the current. The modern theory of electromagnetism was systematized by James Clerk Maxwell in the paper On Physical Lines of Force, which was published in four parts between 1861-1862. While Maxwell's paper established a solid mathematical basis for electromagnetic theory, the detailed mechanisms of the theory were still being explored. One such question was about the details of the interaction between magnets and electric current, including whether magnetic fields interacted with the conductors or the electric current itself. In 1879 Edwin Hall was exploring this interaction, and discovered the Hall effect while he was working on his doctoral degree at Johns Hopkins University in Baltimore, Maryland. Eighteen years before the electron was discovered, his measurements of the tiny effect produced in the apparatus he used were an experimental tour de force, published under the name 'On a New Action of the Magnet on Electric Currents'. The Hall effect is due to the nature of the current in a conductor. Current consists of the movement of many small charge carriers, typically electrons, holes, ions (see Electromigration) or all three. When a magnetic field is present, these charges experience a force, called the Lorentz force. When such a magnetic field is absent, the charges follow approximately straight, 'line of sight' paths between collisions with impurities, phonons, etc. However, when a magnetic field with a perpendicular component is applied, their paths between collisions are curved, thus moving charges accumulate on one face of the material. This leaves equal and opposite charges exposed on the other face, where there is a scarcity of mobile charges. The result is an asymmetric distribution of charge density across the Hall element, arising from a force that is perpendicular to both the 'line of sight' path and the applied magnetic field. The separation of charge establishes an electric field that opposes the migration of further charge, so a steady electric potential is established for as long as the charge is flowing. In classical electromagnetism electrons move in the opposite direction of the current I (by convention 'current' describes a theoretical 'hole flow'). In some semiconductors it appears 'holes' are actually flowing because the direction of the voltage is opposite to the derivation below. For a simple metal where there is only one type of charge carrier (electrons), the Hall voltage VH can be derived by using the Lorentz force and seeing that, in the steady-state condition, charges are not moving in the y-axis direction. Thus, the magnetic force on each electron in the y-axis direction is cancelled by a y-axis electrical force due to the buildup of charges. The vx term is the drift velocity of the current which is assumed at this point to be holes by convention. The vxBz term is negative in the y-axis direction by the right hand rule. In steady state, F = 0, so 0 = Ey − vxBz, where Ey is assigned in the direction of the y-axis, (and not with the arrow of the induced electric field ξy as in the image (pointing in the −y direction), which tells you where the field caused by the electrons is pointing). In wires, electrons instead of holes are flowing, so vx → −vx and q → −q. Also Ey = −VH/w. Substituting these changes gives The conventional 'hole' current is in the negative direction of the electron current and the negative of the electrical charge which gives Ix = ntw(−vx)(−e) where n is charge carrier density, tw is the cross-sectional area, and −e is the charge of each electron. Solving for w {displaystyle w} and plugging into the above gives the Hall voltage: