Space Charge Region and Poisson Equation

We start from a (constant) distribution of positive charges (for n-doped semiconductors) in the space charge region.
The corresponding negative charges are all on the surface; the charge distribution is shown in the first frame of the illustration.
Poisson's equation states that (for the one-dimensional case).
ee0 d2V(x)
 =  –   r(x)  =  e · ND
For 0 < x < dSCR and = 0 everywhere else. (We can also use the voltage U(x) instead of V(x) if we think as V(x = ¥) = 0). That will also be reflected in the choice of boundary conditions made below.
The drawing below shows the situation, including the slight approximation implicit in our choice of r(x). Note that the x-direction ist to the left in this case.
SCR and POissOn equatiOn
The first straight-forward integration yields dU/d(x) which is the electrical field strength Ex = –dU/dx, or
ee0 dV(x)
 =  –  ee0 Ex  =  e · ND · x + const.
With the boundary condition Ex(x = dSCR) = 0, we obtain (always for the interval x = 0 and x = dSCR, of course):
e ·ND · dSCR + const  =  0
const  =  – e ·ND · dSCR

Ex  =  1 
  ·   (e · NDdSCR  –  e · ND · x)
The second integration yields
ee0 · U(x)  =  e · ND · x2
  –  e · ND · dSCR · x  +  const.
With the boundary condition U(dSCR) = 0, we obtain .
 – e · ND · d 2SCR
 +  const.  =  0
  const.  =  e · ND · d 2SCR
Using the proper expression for the integration constant gives gives us the complete voltage function or the shape of the band bending
ee0 · U(x)  =  e · ND · x2
 –  e · ND · dSCR · x  +  e · ND · d 2SCR
The width of the space charge region can be obtained by considering the voltage at x = 0, where we have U(x = 0) = DEF/e.Using this we obtain
 · DEF  =  e · ND · d 2SCR
This gives us the final result for the width of the space charge region

dSCR  =  1
 ·  æ
2DEF · e e0

The corresponding curves are shown in the drawing above. We obtained the same formula as before, but now we have a better awareness of the approximations it contains.
The positive charge distribution was assumed to be box-shaped and uniform. This is a rather good approximation; the drawing indicates the precise shape of the charge distribution for comparison.
The counter charges are described by a d-function at the surface; these charges only enter the calculation in the indirect form of a boundary condition.

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© H. Föll (Semiconductor - Script)