HALL Effect in n-type Semiconductor

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HALL EFFECT IN n-TYPE SEMICONDUCTOR

THEORY :- If a current carrying semiconductor specimen  is placed in a magnetic field , then an induced Electric field (chart?cht=tx&chl=E %7B%7B%7D%7D) is generated , which will produced potential difference between two surfaces of semiconductor . This potential difference is known as “Hall Voltage” (chart?cht=tx&chl=V H) and is proportional to magnetic field (chart?cht=tx&chl=B %7B%7B%7D%7D) and current (chart?cht=tx&chl=i %7B%7B%7D%7D)

HALL effect

                                     n-type Semiconductor specimen figure

chart?cht=tx&chl=E=    Electric field

chart?cht=tx&chl=F %7B%7BE%7D%7D= Electric force

chart?cht=tx&chl=B  = Magnetic field

chart?cht=tx&chl=F %7B%7Bm%7D%7D = magnetic force

chart?cht=tx&chl=a %7B%7Bx%7D%7D%20%20a %7B%7By%7D%7D%20%20a %7B%7Bz%7D%7D = Unit vector along x,y and z direction 

chart?cht=tx&chl=i = current flow in semiconductor specimen along x direction 

chart?cht=tx&chl=A%3D%20wd  → Cross sectional area of surface perpendicular to direction of flow of current chart?cht=tx&chl=i

 

ASSUME : According to figure shown above :

(1) Current (chart?cht=tx&chl=i %7B%7B%7D%7D ) flow in Semiconductor towards X- direction (chart?cht=tx&chl=a %7B%7Bx%7D%7D) so motion of electron will be in (-X)-direction (chart?cht=tx&chl= a %7B%7Bx%7D%7D)

(2) Magnetic field (chart?cht=tx&chl=B %7B%7B%7D%7D) is in z-direction (chart?cht=tx&chl=a %7B%7Bz%7D%7D), represented as chart?cht=tx&chl=B %7B%7Bz%7D%7D

(3) Here we taken  n-type Semiconductor 

(4) In n-type semiconductor electrons are  majority carriers and holes is minority carriers 

Analysis :

If the electron are moving in a magnetic field then it acted by a magnetic force (chart?cht=tx&chl=F %7B%7Bm%7D%7D)

Vector chart?cht=tx&chl=F %7B%7Bm%7D%7D         = q(vector v× vector chart?cht=tx&chl=B %7B%7B%7D%7D)            q represent charge on electron = -e

                                                                              v represent drift velocity of electron in -x direction

                            = chart?cht=tx&chl= e %7B%7B%7D%7D( chart?cht=tx&chl= v %7B%7Bx%7D%7Da %7B%7Bx%7D%7D%5Ctimchart?cht=tx&chl=B %7B%7Bz%7D%7Da %7B%7Bz%7D%7D)

                            =     chart?cht=tx&chl=e %7B%7B%7D%7Dchart?cht=tx&chl=v %7B%7Bx%7D%7DB %7B%7Bz%7D%7D(chart?cht=tx&chl=a %7B%7Bx%7D%7D%20%5Ctimes%20a %7B%7Bz%7D%7D)

                            =      chart?cht=tx&chl=e %7B%7B%7D%7D%20%20v %7B%7Bx%7D%7DB %7B%7Bz%7D%7D(%20a %7Bx%7D%20%5Ctimes%20a %7Bz%7D%20)

                            =      chart?cht=tx&chl=e %7B%7B%7D%7D%20%20v %7B%7Bx%7D%7DB %7B%7Bz%7D%7D(%20 a %7By%7D%20)

 Vector chart?cht=tx&chl=F %7B%7Bm%7D%7D%20%20         =     chart?cht=tx&chl= e %7B%7B%7D%7D%20%20v %7B%7Bx%7D%7DB %7B%7Bz%7D%7D(%20a %7By%7D%20) 

So magnitude of magnetic force vector will be 

              chart?cht=tx&chl=F %7B%7Bm%7D%7D%20%20           =   chart?cht=tx&chl= e %7B%7B%7D%7D%20%20v %7B%7Bx%7D%7DB %7B%7Bz%7D%7D ( this is the force acted on electron in -y direction )

Due to this magnetic force, electron start to  accumulate towards -y direction ( at surface 2) and holes start to accumulate towards +y direction ( at surface 1) to maintain the charge neutrality . so surface 2 get negative charge (due to -ve charge on electron ) and surface 1 get positive charge ( due to +ve charge on holes ) 

If this process of accumulation of electron and holes continue , charge density on surface 1 and surface 2 increases and due to positive ( at surface 1) and negative charge ( at surface 2) , an Electric field (chart?cht=tx&chl=E %7B%7B%7D%7D%20%20) is developed between surface 1 and surface 2 of semiconductor , 

So a potential difference between surface 1 and surface 2 is develop , this potential difference  is called Hall potential or Hall voltage (chart?cht=tx&chl=V %7B%7BH%7D%7D)

Direction of electric field (chart?cht=tx&chl=E %7B%7B%7D%7D%20%20) exist from surface 1 to surface 2 ( towards -y direction )

Electric field always start from positive charge and ends at negative charge

This electric field (chart?cht=tx&chl=E %7B%7B%7D%7D%20%20) act an electric force (chart?cht=tx&chl=F %7B%7BE%7D%7D%20%20) on moving electron  and direction of this electric force will be opposite to the direction of flow of electron i.e.opposite to electric field direction  (towards +y direction )

value of electric force vector will be 

 Vector chart?cht=tx&chl=F %7B%7BE%7D%7D%20%20   = q( vector chart?cht=tx&chl=E %7B%7B%7D%7D%20%20)                                                               q= charge on electron  =-e

                        = chart?cht=tx&chl= e %7B%7B%7D%7D%20(%20E %7By%7D%20)%20(%20 a %7By%7D%20)

                        =      chart?cht=tx&chl=e %7B%7B%7D%7D%20(%20E %7By%7D%20)%20(%20a %7By%7D%20)

  Magnitude of electric force chart?cht=tx&chl=F %7BE%7D%7D   is 

                        chart?cht=tx&chl=F %7BE%7D%7D   =     chart?cht=tx&chl=e %7B%7B%7D%7D%20(%20E %7By%7D%20)%20

At equilibrium electric force is equal to magnetic force 

                        chart?cht=tx&chl=F %7BE%7D%7D%20%20%20  = chart?cht=tx&chl=F %7Bm%7D%7D%20%20%20

                        chart?cht=tx&chl=e %7B%7D%7D%20%20%20(chart?cht=tx&chl=E %7By%7D%7D%20%20%20) = chart?cht=tx&chl=e %7B%7D%7D%20%20%20v %7B%7Bx%7D%7DB %7B%7Bz%7D%7D

                       chart?cht=tx&chl=E %7B%7By%7D%7D  = chart?cht=tx&chl=v %7B%7Bx%7D%7D%20%20B %7B%7Bz%7D%7D%20%20

 or we can simply write 

                       chart?cht=tx&chl=E %7B%7B%7D%7D   =  chart?cht=tx&chl=v %7B%7B%7D%7D%20%20B %7B%7B%7D%7D%20%20                eq (1) 

CALCULATION OF HALL VOLTAGE (chart?cht=tx&chl=V %7B%7BH%7D%7D):

Hall voltage chart?cht=tx&chl=V %7B%7BH%7D%7D is the potential difference between the surface 1 and surface 2

let the voltage on the surface 1 is chart?cht=tx&chl=V %7B%7B1%7D%7D and voltage on the surface 2 is chart?cht=tx&chl=V %7B%7B2%7D%7D

so               chart?cht=tx&chl=V %7B%7B1%7D%7Dchart?cht=tx&chl=V %7B%7B2%7D%7Dchart?cht=tx&chl=V %7B%7BH%7D%7D > chart?cht=tx&chl=0 %7B%7B%7D%7D 

from above figure 

                  chart?cht=tx&chl=E %7B%7B%7D%7D    =       chart?cht=tx&chl=%5Cfrac%7BV %7BH%7D%7D%7Bd%7D                 eq (2)                    d= distance between surface 1 and surface 2

from eq(1) and eq(2) 

                  chart?cht=tx&chl=%5Cfrac%7BV %7BH%7D%7D%7Bd%7D = chart?cht=tx&chl=v %7B%7B%7D%7D%20%20B %7B%7B%7D%7D%20%20

                  chart?cht=tx&chl=V %7B%7BH%7D%7D =  chart?cht=tx&chl=d %7B%7Dv %7B%7B%7D%7D%20%20B %7B%7B%7D%7D%20%20                     eq(3)

we know that conduction current density chart?cht=tx&chl=J is   

                  chart?cht=tx&chl=J %7B%7B%7D%7D   =  chart?cht=tx&chl=%5Csigma %7B%7B%7D%7D%20%20E %7B%7B%7D%7D%20%20                         eq(4)                                                           chart?cht=tx&chl=%5Csigma%20 = conductivity 

we know that conductivity chart?cht=tx&chl=%5Csigma is 

                 chart?cht=tx&chl=%5Csigma    =   chart?cht=tx&chl=%20n %7B%7B%7D%7D%20%20q %7B%7B%7D%7D%5Cmu %7B%7B%7D%7D                      eq(5)                                                            chart?cht=tx&chl=n = electron density 

                                                                                                                                  chart?cht=tx&chl=%5Cmu = mobility of electron 

   

from eq(4) and eq(5)      

                 chart?cht=tx&chl=J  =   chart?cht=tx&chl=nq%5Cmu%20E                    eq(6)

and drift velocity chart?cht=tx&chl=v of electron is 

 

                   chart?cht=tx&chl=v =  chart?cht=tx&chl=%20%5Cmu %7B%7B%7D%7D%20%20E %7B%7B%7D%7D                        eq(7)

from eq(6) and eq(7) 

                  chart?cht=tx&chl=J  = chart?cht=tx&chl=nqv                      

                   chart?cht=tx&chl=v   = chart?cht=tx&chl=%5Cfrac%7BJ%7D%7Bnq%7D                        eq (8)

from eq(3) and eq(8)

                  chart?cht=tx&chl=V %7B%7BH%7D%7D  = chart?cht=tx&chl=%5Cfrac%7BJBd%7D%7Bnq%7D                  eq(9)

we know  chart?cht=tx&chl=J   =    chart?cht=tx&chl=%5Cfrac%7Bi%7D%7BA%7D                     eq(10)                                 i = current flowing in semiconductor 

                                                                                                       chart?cht=tx&chl=A = area semiconductor surface = chart?cht=tx&chl=wd

     from eq(9) and eq(10) 

                 chart?cht=tx&chl=V %7B%7BH%7D  =   chart?cht=tx&chl=%5Cfrac%7BiBd%7D%7BAnq%7D 

                          =  chart?cht=tx&chl=%5Cfrac%7BiBd%7D%7Bwdnq%7D

                          =   chart?cht=tx&chl=%5Cfrac%7BBi%7D%7Bwnq%7D                                                                                    chart?cht=tx&chl=nq = charge density = chart?cht=tx&chl=%5Crho %7B%7Bc%7D%7D

          

                 chart?cht=tx&chl=V %7B%7BH%7D%7D =   chart?cht=tx&chl=%5Cfrac%7BBiR %7BH%7D%7D%7Bw%7D                                                                                 chart?cht=tx&chl=R %7B%7BH%7D%7D  =chart?cht=tx&chl=%5Cfrac%7B1%7D%7Bnq%7D = Hall coefficient 

 

NOTE: Conclusion from Hall effect analysis on a semiconductor specimen 

(1) For any semiconductor  specimen chart?cht=tx&chl=w and chart?cht=tx&chl=R %7B%7BH%7D%7D is a constant , Hall Voltage chart?cht=tx&chl=V %7B%7BH%7D%7D is proportional to magnetic field chart?cht=tx&chl=B and current flowing in the semiconductor specimen 

(2) Current flow is due to electron only not due to Proton

(3) Hall gives idea of hole current or simply hole