Lecture 9 OUTLINE Continuity equations Minority carrier diffusion

Lecture #9 OUTLINE • Continuity equations • Minority carrier diffusion length • Quasi-Fermi levels Read: Sections 3. 4, 3. 5

Derivation of Continuity Equation • Consider carrier-flux into/out-of an infinitesimal volume: Area A, volume Adx JN(x) JN(x+dx) dx 2 Spring 2007 EE 130 Lecture 9, Slide 2

Continuity Equations: 3 Spring 2007 EE 130 Lecture 9, Slide 3

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 4 Spring 2007 EE 130 Lecture 9, Slide 4

• Starting with the continuity equation for electrons: 5 Spring 2007 EE 130 Lecture 9, Slide 5

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 ntype material np is the electron (minority-carrier) concentration in n -type material • Thus the minority carrier diffusion equations are 6 Spring 2007 EE 130 Lecture 9, Slide 6

Simplifications (Special Cases) • Steady state: • No diffusion current: • No R-G: • No light: 7 Spring 2007 EE 130 Lecture 9, Slide 7

Example • Consider the special case: – constant minority-carrier (hole) injection at x=0 – steady state; no light absorption for x>0 LP is the hole diffusion length: 8 Spring 2007 EE 130 Lecture 9, Slide 8

The general solution to the equation is where A, B are constants determined by boundary conditions: Therefore, the solution is 9 Spring 2007 EE 130 Lecture 9, Slide 9

Minority Carrier Diffusion Length • 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 10 Spring 2007 EE 130 Lecture 9, Slide 10

Quasi-Fermi Levels • Whenever Dn = Dp 0, np ni 2. However, we would like to preserve and use the relations: • These equations imply np = ni 2, however. The solution is to introduce two quasi-Fermi levels FN and FP such that 11 Spring 2007 EE 130 Lecture 9, Slide 11

Example: Quasi-Fermi Levels Consider a Si sample with ND = 1017 cm-3 and Dn = Dp = 1014 cm-3. What are p and n ? What is the np product ? 12 Spring 2007 EE 130 Lecture 9, Slide 12

• Find FN and FP : 13 Spring 2007 EE 130 Lecture 9, Slide 13

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 Efield, low-level injection, uniform doping profile): 14 Spring 2007 EE 130 Lecture 9, Slide 14

• 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: 15 Spring 2007 EE 130 Lecture 9, Slide 15