Derivation of conductivity of Semiconductor


Derivation of conductivity of Semiconductor:



In last post Conductivity in Semiconductor , we knew that flow of current in any semiconductor is due to both electron and holes and we also knew conductivity of semiconductors 

in above figure let current i is flowing in x direction in semiconductor ( cylindrical form ) , this current is addition of two current , current chart?cht=tx&chl=i %7B%7Bh%7D%7D due to flow of holes and current chart?cht=tx&chl=i %7B%7Be%7D%7D due to flow of electron

so  chart?cht=tx&chl=i %7B%7B%7D%7D%3D%20i %7B%7Bh%7D%7D%2Bi %7B%7Be%7D%7D

first we will calculate conductivity due to flow electron only

E = electric field vector across the small  section ( in cylindrical form ) of semiconductors

N= number of electron in above semiconductor

A= cross sectional area

L= length of section of cylindrical semiconductor

let one electron contains charge q than total charge flow will be = Nq

so current flow due to electron will be

chart?cht=tx&chl=i %7B%7Be%7D%7D = (Total charge flow)/( time taken in flow of charge)

chart?cht=tx&chl=i %7B%7Be%7D%7D = chart?cht=tx&chl=%5Cfrac%7BNq%7D%7Bt%7D                                     (1)

time chart?cht=tx&chl=t = (Average distance covered by electron) / (drift velocity of electron )

chart?cht=tx&chl=t  = chart?cht=tx&chl=%5Cfrac%7BL%7D%7Bv %7Bd%7D%7D                                     (2)

From above equation (1) and (2)

chart?cht=tx&chl=i %7B%7Be%7D%7D  = chart?cht=tx&chl=%5Cfrac%7BNqv %7Bd%7D%7D%7BL%7D                                (3)

Let conduction current density due to flow of current ( due to flow of electron ) is chart?cht=tx&chl=J , and we know that

chart?cht=tx&chl=J = chart?cht=tx&chl=%5Cfrac%7Bi %7Be%7D%7D%7BA%7D                                       (4)

From above equation (3) and (4)

chart?cht=tx&chl=J%3D%20%5Cfrac%7BNqv %7Bd%7D%7D%7BLA%7D

=   chart?cht=tx&chl=(%5Cfrac%7BN%7D%7BLA%7D)qv %7B%7Bd%7D%7D

chart?cht=tx&chl=J =  chart?cht=tx&chl=nqv %7B%7Bd%7D%7D                                   (5)

where  chart?cht=tx&chl=n = chart?cht=tx&chl=%5Cfrac%7BN%7D%7BLA%7D  = number of electron per unit volume

Drift velocity of electron can be expressed as

chart?cht=tx&chl=v %7B%7Bd%7D%7D = chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7DE                                       (6)

where chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D is mobility of electron

From Equation (5) and (6)

chart?cht=tx&chl=J =  chart?cht=tx&chl=nq%5Cmu %7B%7Be%7D%7DE                                   (7)

from ohms low we know that

chart?cht=tx&chl=J = chart?cht=tx&chl=%5Csigma %7B%7Be%7D%7DE                                          (8)

where chart?cht=tx&chl=%5Csigma %7B%7Be%7D%7D is conductivity due to flow of  electron

by comparing equation (7) with equation (8) , we can concluded that

chart?cht=tx&chl=%5Csigma %7B%7Be%7D%7Dchart?cht=tx&chl=nq%5Cmu %7B%7Be%7D%7D                                        (9)

so equation (9) gives the expression of conductivity of semiconductor due to flow of electrons

In similar fashion we can calculate conductivity of semiconductor due to flow of holes

let chart?cht=tx&chl=p holes per unit volume flow in semiconductor bar above shown , and chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D is the mobility of holes , chart?cht=tx&chl=q is charge on hole,

so conductivity expression due to holes can be given by

chart?cht=tx&chl=%5Csigma %7B%7Bh%7D%7Dchart?cht=tx&chl=pq%5Cmu %7B%7Bh%7D%7D                                           (10)

So total conductivity chart?cht=tx&chl=%5Csigma %7B%7B%7D%7D of semiconductor (due to flow of electron as well as holes) can be given by

chart?cht=tx&chl=%5Csigma %7B%7B%7D%7D = chart?cht=tx&chl=%5Csigma %7B%7Be%7D%7D+chart?cht=tx&chl=%5Csigma %7B%7Bh%7D%7D

chart?cht=tx&chl=nq%5Cmu %7B%7Be%7D%7D + chart?cht=tx&chl=pq%5Cmu %7B%7Bh%7D%7D

chart?cht=tx&chl=%5Csigma%20 = chart?cht=tx&chl=q(n%5Cmu %7Be%7D%2Bp%5Cmu %7Bh%7D)

For intrinsic semiconductor

chart?cht=tx&chl=n%3Dp%3D%20n %7B%7Bi%7D%7D

where chart?cht=tx&chl=n %7B%7Bi%7D%7D is the intrinsic carrier concentration per unit volume of intrinsic semiconductor 

so conductivity of intrinsic semiconductor can be given by

chart?cht=tx&chl=%5Csigma = chart?cht=tx&chl=qn %7B%7Bi%7D%7D(chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D%2B%5Cmu %7B%7Bh%7D%7D)