AN EXACT SOLUTION FOR DIFFUSION EQUATION IN SEMICONDUCTOR DEVICES 

Akrami.Seyed Mostafa 

smostafa.a@gmail.com 

An analitycal solution for Diffusion equation in semiconductor devices has been presented. The complete solution for electrons and holes has been considered in terms of following processes: Diffusion, Drift, Generation, Recombination  and Carrier Trapping, using Maple.
 

1. Diffusion Equation 

>

restart:

 

Considering Diffusion Equation  [1] 

>	DiffusionEQ:=-diff(p[n](y,t),t)+p[n](y,t)*mu[p]*diff(epsilon(y),y)-mu[p]*epsilon*diff(p[n](y,t),y)+D[p]*diff(p[n](y,t),y,y)+G[p]-(p[n]-p[po])*((tau[tp]+tau[p])/(tau[p]*tau[tp]));

 

(1.1)

 

Where is density of holes for n-type semiconductor material, μ[p] is mobility of holes, ε is electric field, D[p] is diffusion coffiecent of holes, G[p] is the generation rate of  of holes, p[po] is density of holes for p-type in equilibrium conditions, τ[p] is lifetime for holes and τ[tp] is the time of trapping for holes. 

Putting: 

>	G[p]:=delta(t)*delta(y-y[0]);

 

(1.2)

 

It can be found that the generation of carriers occurrs at t=0 and y=y[0] 

We have: 

>	DiffusionEQ:=DiffusionEQ;

 

(1.3)

 

And  ε = - ε j, then we have: 

>	DiffusionEQ:=-diff(p[n](y,t),t)+mu[p]*epsilon*diff(p[n](y,t),y)+D[p]*diff(p[n](y,t),y,y)-(p[n]-p[po])/tau[eff];

 

(1.4)

 

Where the τ[eff]= τ[p]*τ[tp]/τ[p]+τ[tp]. 

2. Solution of Diffusion Equation 

It is clear that the especial solution of this equation is P[no], therefore an attampt is made to find the homogeneous solution for this equation. 

We attain this equation: 

>	DiffusionEQ:=-diff(p[n](y,t),t)+mu[p]*epsilon*diff(p[n](y,t),y)+D[p]*diff(p[n](y,t),y,y)-(p[n])/tau[eff];

 

(2.1)

 

Putting: 

>	P[n](y,t):=Y(y)*T(t);

 

(2.2)

 

>	diff(P[n](y,t),t); diff(P[n](y,t),y); diff(P[n](y,t),y,y);

 

 

 

	(2.3)

 

(2.4)

 

for DiffusionEQ we have: 

>	1/T(t)*diff(T(t),t)=mu[p]*epsilon*1/Y(y)*diff(Y(y),y)+D[p]/Y(y)*diff(Y(y),y$2)-1/tau[eff];

 

(2.5)

 

L.H.S. is just dependent on t  and R.H.S. dependent on y, so we can equal each sides to a constant "-k".  

For L.H.S. we have: 

>	1/T(t)*diff(T(t),t)=-k;

 

(2.6)

 

 

>	dsolve(1/T(t)*diff(T(t),t)=-k,T(t));

 

(2.7)

 

Where C1 is a constant. 

For R.H.S. we apporoach to this equation: 

>	EQ1:=diff(Y(y),y$2)+(mu[p]*epsilon/D[p])*diff(Y(y),y)+((k-1/tau[eff])*Y(y)/D[p]);

 

(2.8)

 

Putting: 

>	Y(y):=_C2*exp(-i*omega*y);

 

(2.9)

 

or 

>	EQ2:=subs(Y(y)=_C2*exp(-i*omega*y),EQ1);

 

(2.10)

 

>	EQ2:=factor(%);

 

(2.11)

 

after simplification we have: 

>	EQ2:=(omega)^2-mu[p]*epsilon*i*omega/D[p]-(k-1/tau[eff])/D[p];

 

(2.12)

 

>	k:=solve(EQ2,k);

 

(2.13)

 

So we have: 

>	Y(y):=_C2*exp(-i*omega*y);

 

(2.14)

 

>	T(t) = _C1*exp(-k*t);

 

(2.15)

 

Finally we approach to this equation:  

>	P[n](y,t):=_C(omega)*exp(-i*omega*y)*exp(-k*t);

 

(2.16)

 

With superposition principle we have: 

>	P[n](y,t):=int(_C(omega)*exp(-i*omega*y)*exp(-k*t),omega=-infinity..infinity);

 

(2.17)

 

Where the C(ω) is an arbitrary function of ω. 

Considering the initial condition, we have: 

>	P[n](y,t):=subs(t=0,P[n](y,t));

 

(2.18)

 

>	P[n](y,t):=F(y)=simplify(P[n](y,t));

 

(2.19)

 

By definition of Fourier transformation, we see that the C(ω) is Fourier transform of F(y),  

so we have: 

>	_C(omega):=1/(2*pi)*int(F(y)*exp(i*omega*y),y=-infinity..infinity);

 

(2.20)

 

P[n](y,t) and C(ω) describe a homogeneous solution with initial conditions. 

We attain this equation: 

>	P[n](y,t):=1/(2*pi)*int(int(F(Upsilon)*(e)^(((-omega^2*tau[eff]*D[p]+mu[p]*epsilon*i*omega*tau[eff]-1)*t)/(tau[eff]))*exp(-i*omega*(y-Upsilon)),omega=-infinity..infinity),Upsilon=-infinity..infinity);

 

(2.21)

 

>	P[n](y,t):=simplify(%);

 

(2.22)

 

At t=0 a pair of carriers are generated in y[0]. So we have:  

>	F(Upsilon):=delta(Upsilon-y[0]);

 

(2.23)

 

By simplification we have: 

>	P[n](y,t):=(1/(4*pi*D[p]*t))^(1/2)*exp(-(((y-Upsilon+mu[p]*epsilon*t)^2)/(4*D[p]*t))-t/tau[eff]);

 

(2.24)

 

Finally we attain this equation: 

 

 

>	P[n](y,t):=P[no]+C*(1/(4*pi*D[p]*t))^(1/2)*exp(-(((y-y[0]+mu[p]*epsilon*t)^2)/(4*D[p]*t))-t/tau[eff]);

 

(2.25)

 

where C is constant . 

We can write a similar relation for electrons: 

 

 

>	n[n](y,t):=n[no]+C*(1/(4*pi*D[n]*t))^(1/2)*exp(-(((y-y[0]-mu[n]*epsilon*t)^2)/(4*D[n]*t))-t/tau[eff]);

 

(2.26)

 

3. Analysis of the solution 

We know that at t=0 a pair of electron-hole was generated so the value of integration (2.22), at t=0 is 1, 

therefore we have   

                                                C=1 

so when N carriers were generated we have: 

 

 

>	P[n](y,t):=P[no]+N*(1/(4*pi*D[p]*t))^(1/2)*exp(-(((y-y[0]+mu[p]*epsilon*t)^2)/(4*D[p]*t))-t/tau[eff]);

 

(3.1)

 

 

 

>	n[n](y,t):=n[no]+N*(1/(4*pi*D[n]*t))^(1/2)*exp(-(((y-y[0]-mu[n]*epsilon*t)^2)/(4*D[n]*t))-t/tau[eff]);

 

(3.2)

 

Now we can plot the density of electrons and holes for various time and electric field for CdZnTe semiconductor material: 

 

 

 

>

plot([1/.5699136777e-4*exp(-(z-0.003)^2/(4*25.86*0.01e-9)-(1/(3.6e-6))*0.01e-9),1/.1274365725e-3*exp(-(z-0.003)^2/(4*25.86*0.05e-9)-(1/(3.6e-6))*0.05e-9),1/.1507849860e-3*exp(-(z-0.003)^2/(4*25.86*0.07e-9)-(1/(3.6e-6))*0.07e-9)],z=0.0028..0.0035,linestyle=[1,3],color=[red,blue,green],title=`electron density without electeric field for cylindrical shape   times(ns)=0.01,0.05,0.07 & r=0.002`);

 

 

>

plot([1/.3596776334e-4*exp(-(z-0.003)^2/(4*2.06*0.05e-9)-(1/(5e-6))*0.05e-9),1/.4255763151e-4*exp(-(z-0.003)^2/(4*2.06*0.07e-9)-(1/(5e-6))*0.07e-9),1/.4825581830e-4*exp(-(z-0.003)^2/(4*2.06*0.09e-9)-(1/(5e-6))*0.09e-9)],z=0.0029..0.00306,linestyle=[1,3],color=[red,blue,green],title=`hole density without electeric field for cylindrical shape   times(ns)=0.05,0.07,0.09 & r=0.002`);

 

 

>

 

>

 

>

 

Refereces 

1. SZE S.M. Semiconductor Devices: Physics and Technology. 2nd Edition.Wiley.2001. 

 

 

Legal Notice: The copyright for this application is owned by the author. Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.