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# Lecture 9

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1. Lecture #9 OUTLINE Continuity equations Minority carrier diffusion equations Minority carrier diffusion length Quasi-Fermi levels Read: Sections 3.4, 3.5

2. EE130 Lecture 9, Slide 2 Derivation of Continuity Equation Consider carrier-flux into/out-of an infinitesimal volume:

3. EE130 Lecture 9, Slide 3

4. EE130 Lecture 9, Slide 4 Derivation of Minority Carrier Diffusion Equation The minority carrier diffusion equations are derived from the general continuity equations, and are applicable only for minority carriers. Simplifying assumptions: The electric field is small, such that in p-type material in n-type material n0 and p0 are independent of x (uniform doping) low-level injection conditions prevail

5. EE130 Lecture 9, Slide 5 Starting with the continuity equation for electrons:

6. EE130 Lecture 9, Slide 6 Carrier Concentration Notation The subscript �n� or �p� is used to explicitly denote n-type or p-type material, e.g. pn is the hole (minority-carrier) concentration in n-type material np is the electron (minority-carrier) concentration in n-type material Thus the minority carrier diffusion equations are

7. EE130 Lecture 9, Slide 7 Simplifications (Special Cases) Steady state: No diffusion current: No R-G: No light:

8. EE130 Lecture 9, Slide 8 Example Consider the special case: constant minority-carrier (hole) injection at x=0 steady state; no light absorption for x>0

9. EE130 Lecture 9, Slide 9 The general solution to the equation is where A, B are constants determined by boundary conditions: Therefore, the solution is

10. EE130 Lecture 9, Slide 10 Physically, LP and LN represent the average distance that minority carriers can diffuse into a sea of majority carriers before being annihilated. Example: ND=1016 cm-3; tp = 10-6 s Minority Carrier Diffusion Length

11. EE130 Lecture 9, Slide 11 Whenever Dn = Dp ? 0, np ? ni2. However, we would like to preserve and use the relations: These equations imply np = ni2, however. The solution is to introduce two quasi-Fermi levels FN and FP such that Quasi-Fermi Levels

12. EE130 Lecture 9, Slide 12 Example: Quasi-Fermi Levels

13. EE130 Lecture 9, Slide 13 Find FN and FP :

14. EE130 Lecture 9, Slide 14 Summary

15. EE130 Lecture 9, Slide 15 The minority carrier diffusion length is the average distance that a minority carrier diffuses before it recombines with a majority carrier: The quasi-Fermi levels can be used to describe the carrier concentrations under non-equilibrium conditions: