Drift Velocity

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Drift Velocity of charge carrier in semiconductors:

 

zigzagmotion

              Zig Zag Randomness of electron

We know that in absence of electric field in semiconductor electron moves randomly ( zig zag randomness) to maintain thermal equilibrium, as shown in above figure (a). the velocity with by they move is called random velocity(chart?cht=tx&chl=v %7B%7Br%7D%7D) and it is in order of chart?cht=tx&chl=2

Average value of random velocity at each point in a semiconductor will always be zero ( vector addition of random velocity of all electron at a particular point will be cancel  out to each other )

Average value of random velocity <chart?cht=tx&chl=v %7B%7Br%7D%7D> = 0

After application of electric field E > chart?cht=tx&chl=%2010%5E%7B4%7D%5Cfrac%7BV%7D%7Bcm%7D , due to electric force all electron will aligned in a particular direction that is opposite to the direction of electric field and moves with a particular velocity .average value of this velocity is called drift velocity (chart?cht=tx&chl=v %7B%7Bd%7D%7D)  as shown in figure (b) below

 

driftmotionelectron

or 

drift motion

Let m = mass of electron

a   = acceleration of electron

chart?cht=tx&chl=F %7B%7Be%7D%7D = Electric force on electron

chart?cht=tx&chl=%20m%5E%7B*%7D = Effective mass of electron in semiconductor lattice

e= charge on electron

so Electric force acted on electron due to electric field can be given by

chart?cht=tx&chl=%20F %7B%7Be%7D%7D =chart?cht=tx&chl=eE            (1)            

where e is the charge on electron 

suppose that electron is moving with acceleration “a”, so force due to mass and its acceleration 

chart?cht=tx&chl=F %7B%7Bm%7D%7D = chart?cht=tx&chl=ma         (2) 

at equilibrium 

chart?cht=tx&chl=F %7B%7Be%7D%7D +chart?cht=tx&chl=F %7B%7Bm%7D%7D%20 =chart?cht=tx&chl=0

chart?cht=tx&chl= eE%20%3D%20ma 

chart?cht=tx&chl=a%20%3D -( chart?cht=tx&chl=%5Cfrac%7BeE%7D%7Bm%7D

acceleration “a” can be expressed in terms of average distance (chart?cht=tx&chl=x) traveled by electron and time(chart?cht=tx&chl=t) taken to cover this average distance

i.e. chart?cht=tx&chl=a = chart?cht=tx&chl=%20d%5E%7B%7B2%7D%7Dx%2Fd%20t%5E%7B2%7D  

 chart?cht=tx&chl=%20d%5E%7B%7B2%7D%7Dx%2Fd%20t%5E%7B2%7D  = -( chart?cht=tx&chl=%5Cfrac%7BeE%7D%7Bm%7D)           (3) 

integrating both side with respect to time (t)

∫(chart?cht=tx&chl=%20d%5E%7B%7B2%7D%7Dx%2Fd%20t%5E%7B2%7D )chart?cht=tx&chl=dt = ∫-( chart?cht=tx&chl=%5Cfrac%7BeE%7D%7Bm%7D)chart?cht=tx&chl=dt  

chart?cht=tx&chl=dx%2Fdt = -( chart?cht=tx&chl=%5Cfrac%7BeE%7D%7Bm%7D) chart?cht=tx&chl=t + chart?cht=tx&chl=k         (4) 

 at chart?cht=tx&chl=t = chart?cht=tx&chl=o , chart?cht=tx&chl=E%3D0 and chart?cht=tx&chl=dx%2Fdt%20 = chart?cht=tx&chl=v %7B%7Br%7D%7D ( random velocity )

after putting these above value in equation (4) we can get 

 chart?cht=tx&chl=k=chart?cht=tx&chl=v %7B%7Br%7D%7D  

so

 chart?cht=tx&chl=dx%2Fdt = -( chart?cht=tx&chl=%5Cfrac%7BeE%7D%7Bm%7D) chart?cht=tx&chl=t +chart?cht=tx&chl=v %7B%7Br%7D%7D        (5) 

taking average on both side 

<chart?cht=tx&chl=dx%2Fdt > = <-( chart?cht=tx&chl=%5Cfrac%7BeE%7D%7Bm%7D) chart?cht=tx&chl=t > + <chart?cht=tx&chl=v %7B%7Br%7D%7D 

  chart?cht=tx&chl=v %7B%7Bd%7D%7D = chart?cht=tx&chl=%5Cfrac%7B eE%7D%7B%3Cm%3E%7D%3Ct%3E  

<t> = average value of time = average time of collision =τ 

<m> = average value of mass of electron in semiconductor lattice =effective mass of electron =chart?cht=tx&chl=%20m%5E%7B*%7D

 chart?cht=tx&chl=v %7B%7Bd%7D%7D = (-eτ/chart?cht=tx&chl=%20m%5E%7B*%7D)E              (6)

 we know that drift velocity chart?cht=tx&chl=v %7B%7Bd%7D%7D is 

 chart?cht=tx&chl=v %7B%7Bd%7D%7D  = chart?cht=tx&chl=%5Cmu%20E                          (7)

chart?cht=tx&chl=%5Cmu%20     = (-eτ/chart?cht=tx&chl=%20m%5E%7B*%7D

where chart?cht=tx&chl=%5Cmu = mobility of electron 

So from above calculation , it is clear that mobilitychart?cht=tx&chl=%5Cmu depends on 

(1) Effective mass (chart?cht=tx&chl=%20m%5E%7B*%7D) of charge carrier 

(2) Time of collision of charge carrier (τ)

(3) Type of material , on large doping time of collision (τ) of charge carrier will reduced