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Learn about the derivation and application of minority carrier diffusion equations in semiconductor materials, including simplifying assumptions and special cases. Explore the concepts of carrier concentration notation, diffusion length, and quasi-Fermi levels.
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Lecture #9 OUTLINE Continuity equations Minority carrier diffusion 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 EE130 Lecture 9, Slide 2
Continuity Equations: EE130 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 • n0 and p0 are independent of x (uniform doping) • low-level injection conditions prevail EE130 Lecture 9, Slide 4
Starting with the continuity equation for electrons: EE130 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 n-type material np is the electron (minority-carrier) concentration in n-type material • Thus the minority carrier diffusion equations are EE130 Lecture 9, Slide 6
Simplifications (Special Cases) • Steady state: • No diffusion current: • No R-G: • No light: EE130 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: EE130 Lecture 9, Slide 8
The general solution to the equation is where A,B are constants determined by boundary conditions: Therefore, the solution is EE130 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 EE130 Lecture 9, Slide 10
Quasi-Fermi Levels • WheneverDn = 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 twoquasi-Fermi levels FNand FPsuch that EE130 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 ? EE130 Lecture 9, Slide 12
Find FN and FP: EE130 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 E-field, low-level injection, uniform doping profile): EE130 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: EE130 Lecture 9, Slide 15