Minimum value of conductivity of a Semiconductor Sample


Minimum value of conductivity of a Semiconductor:

In previous post Derivation of conductivity of Semiconductor we derived the expression of conductivity of a semiconductor , that is 

chart?cht=tx&chl=%5Csigma%20 = chart?cht=tx&chl=q(n%5Cmu %7Be%7D%2Bp%5Cmu %7Bh%7D)                             (1) 


chart?cht=tx&chl=q= charge on carrier 

chart?cht=tx&chl=n=  electron concentration per unit volume 

chart?cht=tx&chl=p= hole concentration per unit volume 

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

chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D = mobility of holes 

From “Mass Action” law we know that 


chart?cht=tx&chl=p = chart?cht=tx&chl=%20n %7Bi%7D%5E%7B2%7D/chart?cht=tx&chl=n                                                  (2) 

from equation (1) and (2) 

chart?cht=tx&chl=%5Csigma = chart?cht=tx&chl=q(chart?cht=tx&chl=n%5Cmu %7B%7Be%7D%7D+(chart?cht=tx&chl=%20n %7Bi%7D%5E%7B2%7D/chart?cht=tx&chl=n )chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D)                            (3) 

on  differentiating equation (3) with respect to chart?cht=tx&chl=n

chart?cht=tx&chl=(d%5Csigma%2Fdn) = chart?cht=tx&chl=q(chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D-(chart?cht=tx&chl=%20n %7Bi%7D%5E%7B2%7D/chart?cht=tx&chl=%20n %7B%7D%5E%7B2%7D)chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D)                 (4)

For maximum or minimum of conductivity (chart?cht=tx&chl=%5Csigma)

put  chart?cht=tx&chl=(d%5Csigma%2Fdn)  = chart?cht=tx&chl=0 

from equation (4) 

chart?cht=tx&chl=q(chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D-(chart?cht=tx&chl=%20n %7Bi%7D%5E%7B2%7D/chart?cht=tx&chl=%20n %7B%7D%5E%7B2%7D)chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D) =chart?cht=tx&chl=0 

after calculation we will get 

chart?cht=tx&chl=n = chart?cht=tx&chl=n %7B%7Bi%7D%7D√(chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D/chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D)  

Again differentiate equation (4) with respect to chart?cht=tx&chl=n to know occurrence of minimum or maximum at 

chart?cht=tx&chl=n = chart?cht=tx&chl=n %7B%7Bi%7D%7D√(chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D/chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D)  .

chart?cht=tx&chl=%20d%5E%7B2%7D%5Csigma/chart?cht=tx&chl=%20d%20n%5E%7B2%7D = chart?cht=tx&chl=%202n %7Bi%7D%5E%7B2%7Dq%5Cmu %7B%7Bh%7D%7D/chart?cht=tx&chl=%20n%5E%7B3%7D                      (5)

After putting the value of chart?cht=tx&chl=n = chart?cht=tx&chl=n %7B%7Bi%7D%7D√(chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D/chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D)  in equation (5) we will get chart?cht=tx&chl=%20d%5E%7B2%7D%5Csigma/chart?cht=tx&chl=%20d%20n%5E%7B2%7D  =   a positive quantity 

that means at chart?cht=tx&chl=n = chart?cht=tx&chl=n %7B%7Bi%7D%7D√(chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D/chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D) minimum  conductivity occurs 

and minimum value of conductivity (chart?cht=tx&chl=%5Csigma %7Bmin%7D%7D) can be calculated by putting the  value of chart?cht=tx&chl=n = chart?cht=tx&chl=n %7B%7Bi%7D%7D√(chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D/chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D) in equation (3) 

chart?cht=tx&chl=%5Csigma %7Bmin%7D%7D = chart?cht=tx&chl=2%20n %7B%7Bi%7D%7Dq√(chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D%5Cmu %7B%7Bh%7D%7D)  

Try to solve these Important Problems 

Q1. Consider a piece of pure silicon 100 µm long with a cross-sectional area of 1 µm2. How much current would flow through this “resistor” at room temperature in response to an applied voltage of 1 volt?

Q2 (a) Show that the minimum conductivity of a semiconductor sample occurs when 

chart?cht=tx&chl=n = chart?cht=tx&chl=n %7B%7Bi%7D%7D√(chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D/chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D)

(b) What is the expression for the minimum conductivity?

(c) Is this value greatly different than the value of intrinsic conductivity chart?cht=tx&chl=%5Csigma %7B%7Bi%7D%7D? given that

chart?cht=tx&chl=n %7B%7Bi%7D%7D = 1.45 x 1010 cm-3, chart?cht=tx&chl=%5Cmu %7B%7Be%7D%7D = 1500 cm2 volt-1 sec-1 and , chart?cht=tx&chl=%5Cmu %7B%7Bh%7D%7D = 500 cm2 volt-1 sec-1