Sunday, May 20, 2012

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 VTH.
       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 = VH/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 VH = Ed = B v d                                 .............  Using equation (1)
                      = B (I / (ρ w d)) . d                       ............... Using equation (5)
...              VH  = BI / ρ w                                                            ............. (6)
Note : Thus if VH , B, I and W are measured, the charge density can be determined.
1.1 Measurement of Mobility and Conductivity
       If the polarity of VH 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 RH called Hall coefficient is defined as,
                            RH = 1(n q) = 1/ρ                                             ...............(9)
       Substituting in the equation (6),
               VH  = ( RH B I)/w
...                           RH = (VH  w)/ (BI)                                          ............. (10)
       The conductivity for extrinsic semiconductor is given by,
                              σ = μ n q = μ/RH                                          ................ (11)
       where μ = Mobility of carriers in m2/V-s
                              μ = σ RH = (σ VH  w) / (BI)                           .............. (12)
       But                  σ = Conductivity = 1/Resistivity                     ............... (13)
Note : If the specimen is n-type, μ gives μn which is mobility of electrons while for p-type specimen is which μ is μp 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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