# 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).
e e0 | d2 V(x) dx2 | = – | 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.
|  | The first straight-forward integration yields dU/d( x) which is the electrical field strength Ex = –dU/dx , or
e e0 | dV( x) dx | = – e e0 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 e 0 | · | (e · NDdSCR – e · ND · x)
--- | --- | --- | --- | ---
|  | The second integration yields
e e0 · U(x) | = | e · ND · x2 2 | – e · ND · dSCR · x + const.
--- | --- | --- | ---
|  | With the boundary condition U(dSCR) = 0, we obtain .
– | e · ND · d 2SCR 2 | + const. | = | 0 |
--- | --- | --- | --- | --- | ---
|  | const. | = | e · ND · d 2SCR 2
|  | Using the proper expression for the integration constant gives gives us the complete voltage function or the shape of the band bending
e e0 · U(x) | = | e · ND · x2 2 | – | e · ND · dSCR · x | + | e · ND · d 2SCR 2
--- | --- | --- | --- | --- | --- | ---
|  | The width of the space charge region can be obtained by considering the voltage at x = 0, where we have U( x = 0) = D E F/e.Using this we obtain
e e0 e | · D EF | = | e · ND · d 2SCR 2
--- | --- | --- | ---
| This gives us the final result for the width of the space charge region
dSCR | = | 1 e | · | æ ç è | 2D EF · e e0 ND | ö ÷ ø | 1/2
| 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.

With frame

2.2.4 Simple Junctions and Devices

Basic Equations

Depletion

© H. Föll (Semiconductors - Script)