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Hall Effect in p-type semiconductors

02/01/2018

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HALL EFFECT IN p-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 ( $chart?cht=tx&chl=i %7B%7B%7D%7D$ )

$chart?cht=tx&chl=E$ = 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

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

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

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

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

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

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

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