Hall Effect in p-type semiconductors



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 in p type semiconductors



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 holes will  in (+X) direction (chart?cht=tx&chl=%2Ba %7B%7Bx%7D%7D) also

(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  p-type Semiconductor 

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

Analysis :

If the holes 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 holes = e =chart?cht=tx&chl=%2B1

                                                                                   v, represent drift velocity of holes in +x direction

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

                            =    chart?cht=tx&chl=echart?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 on holes 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 holes in -y direction )

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

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 2) and negative charge ( at surface 1) , an Electric field (chart?cht=tx&chl=E %7B%7B%7D%7D%20%20) is developed between surface 2 and surface 1 of semiconductor specimen , 

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 2 to surface 1 ( 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 holes  and direction of this electric force will be in opposite direction of flow of holes i.e. towards 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 holes  =+e

                        =  chart?cht=tx&chl=eE %7B%7By%7D%7Da %7B%7By%7D%7D

                        =   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     

            vector chart?cht=tx&chl=E    = vector chart?cht=tx&chl=v chart?cht=tx&chl=%5Ctimes vector chart?cht=tx&chl=B