Diffusion in Semiconductors

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when excess carriers are created non uniformly in a semiconductor , the electron and holes concentration varies with position in the sample . due to this process concentration gradient is formed and to maintain thermal equilibrium ,net motion of charge carriers from region of higher concentration to lower concentration takes place ,this is the natural phenomenon and this type of motion is called diffusion . and a net diffusion current will flow in semiconductor material 

 

diffusion

NOTE:

(1) Keep in mind that diffusion process occurs not due to mutual attraction or repulsion among the charge carriers , this is a natural phenomenon and occurs due to formation of concentration gradient of charge carriers in semiconductor sample to maintain thermal equilibrium 

(2) Keep in mind that diffusion process occurs in absence of electric field and drift occurs in presence of electric field 

Consider an n-type semiconductor in which excess charge carriers injected from left side as shown in below figure. initially  more number of electrons is present at left side whereas lesser number of electrons is present at right side , so due to non uniformity in charge carrier concentration a concentration gradient will create and electron starts to move from higher concentration to lower concentration(from left to right) in the semiconductor sample and a current starts to flow in opposite direction of electron flow this current is called diffusion current. after some time at thermal equilibrium concentration of electron will be same at everywhere in semiconductor sample 

 

diffusion1

Now we are going to calculate some important parameters and quantities as listed below 

(1) We can calculate rate at which charge carriers diffuse in semiconductor sample 

(2) Diffusion current in semiconductor sample 

(3) Diffusion current density in semiconductor sample 

Let excess electron injected at x=0 and at time t=0 will spread out in time as figure shown below . Initially , the excess electron are concentrated at x=0 , as time passes electrons diffuse to region of low electron concentration until finally n(x) will get constant 

Assume here diffusion in  one dimension  only 

 

diffusion profile

we can calculate the rate at which the electron diffuse in one dimensional problem by considering an arbitrary distribution n(x) as shown in figure below , since mean free path chart?cht=tx&chl=l between collision is a small incremental distance , we can divide x into segment chart?cht=tx&chl=l wide , with n(x) evaluated at the center of each segment 

 diffusion segment

consider two segment from above figure for the analysis 

 

diffusion electron1

consider segment (1) and segment (2) 

the electron in segment (1) to the left of chart?cht=tx&chl=x %7B%7Bo%7D%7D in above figure have equal chance to moving left or right and in a mean free time chart?cht=tx&chl=t 

it is a chance to move half of electron [ chart?cht=tx&chl=%5Cfrac%7Bn %7B1%7D%7D%7B2%7D per unit volume] of segment (1)  into the segment (2) and half of the electron [ chart?cht=tx&chl=%5Cfrac%7Bn %7B1%7D%7D%7B2%7D per unit volume] to the back side of segment (1) in one mean free time , same chance of movement of electron will possible for segment (2) means half of electron [chart?cht=tx&chl=%5Cfrac%7Bn %7B2%7D%7D%7B2%7Dper unit volume] of segment (2) move into the segment (1) and half of the electron  [chart?cht=tx&chl=%5Cfrac%7Bn %7B2%7D%7D%7B2%7Dper unit volume] towards forward direction of segment(2) 

so net flow of electron from left to right ( from segment (1) to segment(2) ) crossing the point chart?cht=tx&chl=x %7B%7Bo%7D%7D in one mean free time t is 

= (chart?cht=tx&chl=%5Cfrac%7Bn %7B1%7D%7D%7B2%7D %5Cfrac%7Bn %7B2%7D%7D%7B2%7D) per unit volume 

= (chart?cht=tx&chl=%5Cfrac%7Bn %7B1%7D%7D%7B2%7D %5Cfrac%7Bn %7B2%7D%7D%7B2%7D)chart?cht=tx&chl=lA 

where chart?cht=tx&chl=l= mean free path between collision 

          chart?cht=tx&chl=A = area of cross section of semiconductor sample 

so rate of flow of electron in +x direction per unit area will be = electron flux density[chart?cht=tx&chl=%5Cphi %7B%7Bn%7D%7D(x %7Bo%7D)] crossing point chart?cht=tx&chl=x %7B%7Bo%7D%7D

chart?cht=tx&chl=%5Cphi %7B%7Bn%7D%7D(x %7Bo%7D) = chart?cht=tx&chl=%5Cfrac%7Bl%7D%7B2t%7D(n %7B1%7D n %7B2%7D)                                         (1)

since chart?cht=tx&chl=l is very small differential length so difference in electron concentration can be written as 

chart?cht=tx&chl=n %7B%7B1%7D%7D n %7B%7B2%7D%7D = chart?cht=tx&chl=%5Cfrac%7Bn(x) n(x%2B%5CDelta%20x)%7D%7B%5CDelta%20x%7D

 

diffussion

and           chart?cht=tx&chl=lim %7B%7B%5CDelta%20x%5Crightarrow0%7D%7Dchart?cht=tx&chl=%5Cfrac%7Bn(x) n(x%2B%5CDelta%20x)%7D%7B%5CDelta%20x%7D] =chart?cht=tx&chl=– chart?cht=tx&chl=%5Cfrac%7Bdn(x)%7D%7Bdx%7D.chart?cht=tx&chl=l     (2)

from equation (1) and equation(2) 

chart?cht=tx&chl=%5Cphi %7B%7Bn%7D%7D(x)= –chart?cht=tx&chl=%20(l%5E%7B2%7D%2F2t)%20%5Cfrac%7Bdn(x)%7D%7Bdx%7D                                                     (3)

here chart?cht=tx&chl=%20(l%5E%7B2%7D%2F2t) = is called electron diffusion coefficient chart?cht=tx&chl=D %7B%7Bn%7D%7D 

chart?cht=tx&chl=%5Cphi %7B%7Bn%7D%7D(x)=chart?cht=tx&chl= D %7B%7Bn%7D%7D%20%5Cfrac%7Bdn(x)%7D%7Bdx%7D                                                            (4) 

in similar fashion holes flux density will be 

   chart?cht=tx&chl=%5Cphi %7B%7Bp%7D%7D(x) =chart?cht=tx&chl= D %7B%7Bp%7D%7D%20%5Cfrac%7Bdp(x)%7D%7Bdx%7D                                                         (5)

diffusion current crossing per unit area ( diffusion current density) = charge on charge carrier × flux density of charge carrier 

so diffusion current density due to diffusion of electron will be 

chart?cht=tx&chl=J %7B%7Bn%7D%7D(diff = (chart?cht=tx&chl= q)(chart?cht=tx&chl= D %7B%7Bn%7D%7D%20%5Cfrac%7Bdn(x)%7D%7Bdx%7D ) = chart?cht=tx&chl=q%20D %7B%7Bn%7D%7D%5Cfrac%7Bdn(x)%7D%7Bdx%7D              (6)

diffusion current density due to diffusion of holes will be 

chart?cht=tx&chl=J %7B%7Bp%7D%7D(diff   =  (chart?cht=tx&chl=q)(chart?cht=tx&chl= D %7B%7Bp%7D%7D%20%5Cfrac%7Bdp(x)%7D%7Bdx%7D )    = chart?cht=tx&chl= q%20D %7B%7Bp%7D%7D%5Cfrac%7Bdp(x)%7D%7Bdx%7D        (7) 

diffusion current(chart?cht=tx&chl=I) = diffusion current density(chart?cht=tx&chl=J) × cross sectional area (chart?cht=tx&chl=A)

so diffusion current due to electron will be 

chart?cht=tx&chl=I %7B%7Bn%7D%7D = chart?cht=tx&chl=J %7B%7Bn%7D%7D(diffchart?cht=tx&chl= = chart?cht=tx&chl=qAD %7B%7Bn%7D%7D%20%5Cfrac%7Bdn(x)%7D%7Bdx%7D                          (8) 

diffusion current due to holes will be 

chart?cht=tx&chl=I %7B%7Bp%7D%7D =  chart?cht=tx&chl=J %7B%7Bp%7D%7D(diff.)   =  chart?cht=tx&chl= qAD %7B%7Bp%7D%7D%20%5Cfrac%7Bdp(x)%7D%7Bdx%7D                    (9) 

note that if electric field chart?cht=tx&chl=E is also present in addition to the carrier gradient , so current densities will be due to both diffusion component and drift component

so total current density due to flow of electron will be

chart?cht=tx&chl=Jn(x) %7B%7Btotal%7D%7D%20%3D%20Jn(x) %7B%7Bdiffusion%7D%7D%20%2BJn(x) %7B%7Bdrift%7D%7D

chart?cht=tx&chl=%3D%20nq%5Cmu %7B%7Bn%7D%7DE%2BqD %7B%7Bn%7D%7D%20%5Cfrac%7Bdn(x)%7D%7Bdx%7D                        (10)

total current density due to flow of holes will be

chart?cht=tx&chl=Jp(x) %7B%7Btotal%7D%7D%20%3D%20Jp(x) %7B%7Bdiffusion%7D%7D%2BJp(x) %7B%7Bdrift%7D%7D

chart?cht=tx&chl=%3D%20pq%5Cmu %7B%7Bp%7D%7DE qD %7B%7Bp%7D%7D%20%5Cfrac%7Bdp(x)%7D%7Bdx%7D                         (11)

so total current density due to flow of both electrons and holes will be

chart?cht=tx&chl=J(x) %7B%7Btotal%7D%7D%20%3D%20Jn(x) %7B%7Btotal%7D%7D%2BJp(x) %7B%7Btotal%7D%7D                     (12)

at equilibrium

chart?cht=tx&chl=Jn(x) %7B%7Btotal%7D%7D%20%3D0

this will give

chart?cht=tx&chl=E%3D %5Cfrac%7BDn%7D%7Bn%20%5Cmu %7Bn%7D%7D                                                        (13)

similarly at equilibrium

chart?cht=tx&chl=Jp(x) %7B%7Btotal%7D%7D%20%3D0

this will give

chart?cht=tx&chl=E%20%3D%20%5Cfrac%7BDp%7D%7B%20p%20%5Cmu %7Bp%7D%7D                                                             (14)