1. ECE 4339: Physical Principles of Solid State Devices
Len Trombetta
Summer 2006
Chapter 5:
PN Junction Electrostatics
Goal: To understand these diagrams.
ECE L. Trombetta

2. Simplified fabrication of a pn junction diode
We start with a wafer uniformly doped n-type (ND donors), and we diffuse acceptors (NA) into it. Acceptors have a large concentration at the surface; their concentration decreases as we move below the surface. (We could of course have reversed the dopant types.)
Net dopant concentration, ND – NA, a function of distance below the surface.
There is a pn junction (the “metallurgical junction”) at the dashed line. To the left of the dashed line, the semiconductor is p-type, and to the right it is n-type.
ECE L. Trombetta

3. Approximations to junction doping profiles
We can approximate the actual doping profile with either (a) the “step junction”, or (b) the linearly graded junction. The point is that (as we shall see) it is the doping profile near the junction that is important for the electrical properties of the diode, so the details far from the junction don’t matter.
ECE L. Trombetta

4. We are assuming a step junction here (constant doping on either side).
The key point in constructing the energy bands is that, as we discussed earlier, the Fermi level must be constant in a device at equilibrium.
ECE L. Trombetta

5. ECE L. Trombetta

6. Equilibrium Diode qVbiEc EF Ei Ev
This figure shows the energy band diagram for an equilibrium diode calculated assuming Si at 300 K, ND = 1 x 1016 cm-3 and NA = 2 x 1016 cm-3. The built-in potential Vbi = V. The red curves indicate where the Si is p-type, and the blue where it is n-type.
ECE L. Trombetta

7. Here a coordinate system has been added
Here a coordinate system has been added. The coordinates xn and -xp indicate the extent of the depletion region on either side of x = 0. Note that xn and xp are not equal because the doping densities are not the same; this can be observed from the position of the Fermi level relative to the band edges on either side.
qVbi x
-xp
x = 0 xn
ECE L. Trombetta

8. Here is a qualitative look at the potential, electric field, and charge distribution. We will solve Poisson’s equation later to find the potential and electric field profiles.
ECE L. Trombetta

9. ECE L. Trombetta

10. This figure explains conceptually how the junction is formed
This figure explains conceptually how the junction is formed. Electrons and holes diffuse to the other side of the junction (b), but this leaves behind ionized donors and acceptors (which cannot move), which create an electric field (c).
The electric field acts to pull electrons and holes back to where they came from. When this force balances the tendency to diffuse, we have reached equilibrium.
ECE L. Trombetta

11. ECE L. Trombetta

12. e fn Drift fn Diff Jn Drift Jn Diff fp Drift Jp Drift Jp Diff fp Diff
This figure provides more detail on the balance between diffusion and drift components. e
fn Drift
fn Diff
Jn Drift
Jn Diff + + + + + + + + + + - - - - - - - - - - -
fp Drift
fp Diff
Jp Drift
Jp Diff
The presence of ionized, uncompensated donors and acceptors has been indicated schematically. These are responsible for the “space charge” that creates the electric field. Also shown are diffusion and drift currents for both holes and electrons, as well as the movement of each of these carriers (indicated as “flux” f). For each carrier type, Jdiff = Jdiff, so the net current is 0.
ECE L. Trombetta

13. x -xp x = 0 xn qVbi pp = p(-xp) pn = p(xn)
Here we note that the concentration of holes and electrons is indicated by the distance from the Fermi level to the band edges, so these concentrations are changing within the depletion region. For future reference, note that the hole concentration at x = -xp is pp; the hole concentration at x = xn is pn. We can take pp = NA and pn = ni2/ND.
ECE L. Trombetta

14. Depletion Approximation
As shown in the last slide, the electron and hole concentrations vary gradually across the depletion region. The depletion approximation says they are 0 in this region: p = n = 0 for –xp < x < xn. In that case the charge density r (Coul/cm3) is as shown below. + -
r = 0
r = -qNa
r = qNd
r = 0
x £ -xp
-xp £ x £ 0
0 £ x £ xn
xn £ x x
-xp
x = 0 xn
ECE L. Trombetta

15. The “Depletion Approximation” says that there are no free carriers (electrons or holes) in the region between –xp and xn. This is not strictly correct, but it is a good approximation.
ECE L. Trombetta

16. ECE L. Trombetta

17. These figures show the results of calculation of the electric field and potential within the depletion region.
ECE L. Trombetta

18. ECE L. Trombetta

19. Because the depletion region has few mobile carriers (compared with the neutral regions), we make the approximation that any applied voltage is dropped there. Thus, in our equations, we make the substitution Vbi  Vbi – VA, where VA is the applied voltage. Note that VA is applied at the p-type side relative to the n-type side, as indicated in Fig. 5-8b.
ECE L. Trombetta

20. ECE L. Trombetta

21. ECE L. Trombetta

22. Reverse Bias Energy Bands
q(Vbi – Va) Va
q(– Va)
This figure shows the energy bands calculated assuming the same conditions as for the equilibrium diode presented earlier, but with an applied reverse bias of Va = -1 V. The Fermi levels on either side are separated by –qVa.
ECE L. Trombetta

23. Forward Bias Energy Bands
q(Vbi – Va) Va
q(– Va)
This figure shows the energy bands calculated assuming the same conditions as for the equilibrium diode presented earlier, but with an applied forward bias of Va = +0.5 V. Again the Fermi levels are separated by –qVa.
ECE L. Trombetta

24. This figure shows what happens to the charge density, electric field, and potential if instead of the constant doping assumption, we assume that the doping density varies linearly across the junction. Fig. 5.2 suggested that this might be the appropriate approximation in some circumstances.
ECE L. Trombetta

25. ECE L. Trombetta