Diffusion in Semiconductors

DIFFUSION: 

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 l between collision is a small incremental distance , we can divide x into segment 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 x_{{o}} in above figure have equal chance to moving left or right and in a mean free time t 

it is a chance to move half of electron [ \frac{n_{1}}{2} per unit volume] of segment (1)  into the segment (2) and half of the electron [ \frac{n_{1}}{2} 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 [\frac{n_{2}}{2}per unit volume] of segment (2) move into the segment (1) and half of the electron  [\frac{n_{2}}{2}per 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 x_{{o}} in one mean free time t is 

= (\frac{n_{1}}{2}-\frac{n_{2}}{2}) per unit volume 

= (\frac{n_{1}}{2}-\frac{n_{2}}{2})lA 

where l= mean free path between collision 

          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[\phi_{{n}}(x_{o})] crossing point x_{{o}}

\phi_{{n}}(x_{o}) = \frac{l}{2t}(n_{1}-n_{2})                                         (1)

since l is very small differential length so difference in electron concentration can be written as 

n_{{1}}-n_{{2}} = \frac{n(x)-n(x+\Delta x)}{\Delta x}.l

diffussion

and           lim_{{\Delta x\rightarrow0}}\frac{n(x)-n(x+\Delta x)}{\Delta x}.l] =-– \frac{dn(x)}{dx}.l     (2)

from equation (1) and equation(2) 

\phi_{{n}}(x)= – (l^{2}/2t) \frac{dn(x)}{dx}                                                     (3)

here  (l^{2}/2t) = is called electron diffusion coefficient D_{{n}} 

\phi_{{n}}(x)=-D_{{n}} \frac{dn(x)}{dx}                                                            (4) 

in similar fashion holes flux density will be 

   \phi_{{p}}(x) =-D_{{p}} \frac{dp(x)}{dx}                                                         (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 

J_{{n}}(diff.) = (-q)(-D_{{n}} \frac{dn(x)}{dx} ) = q D_{{n}}\frac{dn(x)}{dx}              (6)

diffusion current density due to diffusion of holes will be 

J_{{p}}(diff.)   =  (q)(-D_{{p}} \frac{dp(x)}{dx} )    = -q D_{{p}}\frac{dp(x)}{dx}        (7) 

diffusion current(I) = diffusion current density(J) × cross sectional area (A)

so diffusion current due to electron will be 

I_{{n}} = J_{{n}}(diff.).A = qAD_{{n}} \frac{dn(x)}{dx}                          (8) 

diffusion current due to holes will be 

I_{{p}} =  J_{{p}}(diff.).A   =  -qAD_{{p}} \frac{dp(x)}{dx}                    (9) 

note that if electric field 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

Jn(x)_{{total}} = Jn(x)_{{diffusion}} +Jn(x)_{{drift}}

= nq\mu_{{n}}E+qD_{{n}} \frac{dn(x)}{dx}                        (10)

total current density due to flow of holes will be

Jp(x)_{{total}} = Jp(x)_{{diffusion}}+Jp(x)_{{drift}}

= pq\mu_{{p}}E-qD_{{p}} \frac{dp(x)}{dx}                         (11)

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

J(x)_{{total}} = Jn(x)_{{total}}+Jp(x)_{{total}}                     (12)

at equilibrium

Jn(x)_{{total}} =0

this will give

E=-\frac{Dn}{n \mu_{n}}.\frac{dn(x)}{dx}                                                        (13)

similarly at equilibrium

Jp(x)_{{total}} =0

this will give

E = \frac{Dp}{ p \mu_{p}}. \frac{dp(x)}{dx}                                                             (14)

 

 

 

 

 

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