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

Spring 2007EE130 Lecture 9, Slide 2 Derivation of Continuity Equation Consider carrier-flux into/out-of an infinitesimal volume: JN(x)JN(x)J N (x+dx) dx Area A, volume Adx

Spring 2007EE130 Lecture 9, Slide 3 Continuity Equations:

Spring 2007EE130 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 –n 0 and p 0 are independent of x (uniform doping) –low-level injection conditions prevail

Spring 2007EE130 Lecture 9, Slide 5 Starting with the continuity equation for electrons:

Spring 2007EE130 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. p n is the hole (minority-carrier) concentration in n- type material n p is the electron (minority-carrier) concentration in n-type material Thus the minority carrier diffusion equations are

Spring 2007EE130 Lecture 9, Slide 7 Simplifications (Special Cases) Steady state: No diffusion current: No R-G: No light:

Spring 2007EE130 Lecture 9, Slide 8 L P is the hole diffusion length: Example Consider the special case: –constant minority-carrier (hole) injection at x=0 –steady state; no light absorption for x>0

Spring 2007EE130 Lecture 9, Slide 9 The general solution to the equation is where A, B are constants determined by boundary conditions: Therefore, the solution is

Spring 2007EE130 Lecture 9, Slide 10 Physically, L P and L N represent the average distance that minority carriers can diffuse into a sea of majority carriers before being annihilated. Example: N D =10 16 cm -3 ;  p = s Minority Carrier Diffusion Length

Spring 2007EE130 Lecture 9, Slide 11 Whenever  n =  p  0, np  n i 2. However, we would like to preserve and use the relations: These equations imply np = n i 2, however. The solution is to introduce two quasi-Fermi levels F N and F P such that Quasi-Fermi Levels

Spring 2007EE130 Lecture 9, Slide 12 Example: Quasi-Fermi Levels Consider a Si sample with N D = cm -3 and  n =  p = cm -3. What are p and n ? What is the np product ?

Spring 2007EE130 Lecture 9, Slide 13 Find F N and F P :

Spring 2007EE130 Lecture 9, Slide 14 Summary The continuity equations are established based on conservation of carriers, and therefore are general: The minority carrier diffusion equations are derived from the continuity equations, specifically for minority carriers under certain conditions (small E- field, low-level injection, uniform doping profile):

Spring 2007EE130 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: