1. PN Junction Electrostatics
Chapter 5.
PN Junction Electrostatics
Sung June Kim

2. Subject Analysis of Field and potential distribution in the junction.
- Quantitative Electrostatic Relationships

3. Junction Terminology/Idealized Profiles
Preliminaries
Junction Terminology/Idealized Profiles
Net doping profile

4. Step (abrupt) junction
Linearly graded junction
The step junction is an acceptable approximation to an ion-implantation or shallow diffusion into a lightly doped starting wafer

5. Poisson’s Equation 1-Dimension
Ks is the semiconductor dielectric constant and 0 is the permittivity of free space.  is the charge density (charge/cm3)

6. Let us assume an equilibrium conditions
Qualitative Solution
Let us assume an equilibrium conditions
It is reasonable to expect regions far removed from the metallurgical junction to be identical to an isolated semiconductor.

7. Under equilibrium conditions, the Fermi level is a constant

8. P-N Diode Junction Energy BandEc Ef Ev Ec Ef Ev

9. Equilibrium P-N Junction
V=0 P N Ec Ef Ev Ec Ef Ev

10. Forward Biased P-N Junction
V>0 P N Ec Ef Ev Ec Ef Ev

11. Reverse Biased P-N JunctionEc Ef Ev Ec Ef Ev

12. V versus x relationship must have the same functional form as the “ upside-down” of Ec
Eref

13. The voltage drop across the junction under equilibrium conditions and the appearance of charge near the metallurgical boundary
Where does this charge come from?

14. Charge neutrality is assumed to prevail in the isolated, uniformly doped semiconductors
hole
electron
Charge redistribution
Space charge region or
depletion region
Charge density

15. The build-up of charge and the associated electric field continues until the diffusion is precisely balanced by the carrier drift
The individual carrier diffusion and drift components must of course cancel to make JN and JP separately zero

16. The Built-in Potential (Vbi)
Consider a nondegenerately-doped junction
Integrating
Solving for and making use of the Einstein relationship, we obtain

18. The Depletion Approximation
It is very hard to solve
The carrier concentrations are negligible in
The charge density outside the depletion region=0

19. Exact
Depletion Approximation

20. Assumptions/definitions
Quantitative Electrostatic Relationships
Assumptions/definitions

21. Step Junction with VA=0
Solution for 
Solution for

22. For the p-side of the depletion region
Similarly on the n-side

23. Solution for V ( )
With the arbitrary reference potential set equal to
zero at x=-xp and Vbi across the depletion region
equilibrium conditions

24. For the p-side of the depletion region
Similarly on the n-side of the junction
@ x=0

25. Solution for xn and xp
Depletion width

26. Step Junction with VA0
When VA > 0, the externally imposed voltage drop lowers the potential on the n-side relative to the p-side

27. The voltage drop across the depletion region, and hence the boundary condition at x=xn, becomes Vbi-VA

28. The voltage drop across the depletion region, and hence the boundary condition at x=xn, becomes Vbi-VA

29. Examination/Extrapolation of Results
Depletion widths decrease under forward biasing and increase under reverse biasing
A decreased depletion width when VA > 0 means less charge around the junction and a correspondingly smaller -field. Similarly, the potential decreases at all points when VA >0
The Fermi level is omitted from the depletion region

31. pn junction energy band diagrams.

32. Linearly Graded Junctions
The linearly graded profile is modeled by
The profile is continuous through x = 0
The profile is symmetrical about x = 0, all of the electrostatic variables exhibit a symmetry about x = 0
Although V is taken to be a constant outside of the depletion region, a modification of the equation for Vbi is still necessary

33. The charge density is symmetrical about x = 0,… …
The charge density is symmetrical about x = 0, … …

34. Where ,