Hall Effect

       If a specimen of metal or semiconductor, carrying current I is placed in a transverse magnetic field with flux density B then an electric field E is induced in the direction perpendicular to both I and B. This phenomenon is called Hall effect.

Fig. 1  Hall effect

       Consider a semiconductor strip carrying current I as shown in the Fig. 1. The current I is flowing in opposite X direction. It is placed in the magnetic field with density B acting in positive Z direction.

       In such conditions, the force is exerted on the current carriers in the negative Y direction. This force is called Lorentz force. Thus if semiconductor used is n type, all the electrons, which are majority carriers will be forced towards side 1 of the strip. Hence side 1 electron will become negatively charged with respect to side 2.

Note : Thus there exists a potential difference across the sides 1 and 2. This voltage is called Hall voltage denoted as V_TH.

       In the equilibrium condition, the electric field intensity E due to the Hall effect must exert a force on the carrier which just balances the force exerted by the magnetic field.

.^..                  qE = B qv                                                           .............. (1)

       where q = Magnetic of charge on the carrier

       v = Drift speed

       Now       E = V_H/d                                                             ............ (2)

       where       d = Distance between the surfaces 1 and 2.

       The current density J is given by,

                       J = I/(wd)              A/m²                                     ............. (3)

       While the current density can be expressed interns of charge density as,

                        J = ρ v                                                               ............ (4)

       where ρ = Charge density in C/m³

       v = Speed in m/s

       w = Width of the strip in the direction of B

       Equating (3) and (4),

                         I/(wd) = ρ v                                                     ............. (5)

       Now V_H = Ed = B v d                                 .............  Using equation (1)

                      = B (I / (ρ w d)) . d                       ............... Using equation (5)

.^..              V_H  = BI / ρ w                                                            ............. (6)

Note : Thus if V_H , B, I and W are measured, the charge density can be determined.

1.1 Measurement of Mobility and Conductivity

       If the polarity of V_H is such that the surface 2 is positive then the carriers are the electrons and we can write,

                             ρ = nq                                                          ............. (7)

       where n = Electron concentration 

       While if the surface 1 is positive then the carriers are holes and we can write,

                             ρ = p q                                                         .............. (8)

       where  p = Hole concentration

       Practically a constant R_H called Hall coefficient is defined as,

                            R_H = 1(n q) = 1/ρ                                             ...............(9)

       Substituting in the equation (6),

               V_H  = ( R_H B I)/w
.^..                           R_H = (V_H  w)/ (BI)                                          ............. (10)

       The conductivity for extrinsic semiconductor is given by,

                              σ = μ n q = μ/R_H                                          ................ (11)

       where μ = Mobility of carriers in m²/V-s

                              μ = σ R_H = (σ V_H  w) / (BI)                           .............. (12)

       But                  σ = Conductivity = 1/Resistivity                     ............... (13)

Note : If the specimen is n-type, μ gives μ_n which is mobility of electrons while for p-type specimen is which μ is μ_p mobility of holes.

       Thus Hall effect can be used to determine whether a semiconductor is n-type or p-type, to find carrier concentration and also to calculate mobility, measuring conductivity.

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