If an electric current flows through a conductor in a magnetic field, the magnetic field exerts a transverse force on the moving charge carrier which tends to push them to one side of the conductor. A buildup of charge at the sides of the conductors will balance this magnetic influence, producing a measurable voltage between the two sides of the conductor. This presence of measurable transverse voltage is the Hall Effect. The Hall effect was discovered in 1879 by Edwin Herbert Hall while working on his doctoral degree at the Johns Hopkins University. The Hall effect is due to the nature of current in a conductor. Current consists of the movement of many small carriers(electrons, holes or both). Moving charges experience the Lorentz force when a magnetic field is present that is perpendicular to their motion. In the absence of the magnetic field, the charges follow ‘line of sight’ paths between collisions with impurities. On the other hand, when a perpendicular magnetic field is present, their paths are curved and moving charges accumulate on one face of the material. Equal and opposite charges are exposed on the other face, where there is a scarcity of mobile charges. The result is an asymmetric distribution of charge density across the element that is perpendicular to 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, and a steady electrical charge builds up while the current is flowing. The Hall effect offered the first proof that electric currents in metals are carried by moving electrons. The effect is very useful in measuring either the carrier density or the magnetic field.
HALL EFFECT IN SEMICONDUCTORS
A static magnetic field has no effect on a charged particle unless it is moving. When charges flow, a mutually perpendicular force (Lorentz force) is induced on the charge. A Hall voltage Vhall will measured perpendicular to B and j. Thus, an electrical field Ehall develops in y-direction which is already the essence of the Hall effect. The magnitude of the Hall voltage Uhall that is induced by the magnetic field B.
FL = q . ( vd x B)
where FL is the Lorentz force
Vd = μ . E where μ is the mobility of the carriers
Fy = – q · µ · Ex · Bz
This means that the current of carriers will be deflected from a straight line in y-direction. In other words, there is a component of the velocity in y-direction and the surfaces perpendicular to the y-direction will become charged as soon as the current (or the magnetic field) is switched on. The flow-lines of the carriers will look like this:
q · Ey
| – q · µ · Ex · Bz
| – µ · Ex · Bz
Hall coefficient Rhall:
Rhall = EyBz . jx
Ey = RHall · Bz · jx
| The Hall coefficient is a material parameter, indeed, because we will get different numbers for RHall if we do experiments with identical magnetic fields and current densities, but different materials. The Hall coefficient, as mentioned before, has interesting properties: * RHall will change its sign, if the sign of the carriers is changed because then Ey changes its sign, too. It thus indicates in the most unambiguous way imaginable if positive or negative charges carry the current. * RHall allows to obtain the mobility µ of the carriers, too, as we will see immediately.
* RHall is easily calculated: Using the equation for Ey from above, and the basic equation jx = s · Ex, we obtain for negatively charged carriers:Rhall= μ . Ex .Bzσ . Ex .Bz = -μσ= -μq . n . μ= -1q . n Material
| Semiconductors (e.g. Si, Ge,GaAs, InP,...)
| positive or negativevalues, depending on "doping"
1. the positive values for the metals were measured under somewhat special conditions (low...
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