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Lecture 6

Lecture 6. OUTLINE Semiconductor Fundamentals (cont’d) Continuity equations Minority carrier diffusion equations Minority carrier diffusion length Quasi-Fermi levels Poisson’s Equation Reading : Pierret 3.4-3.5, 5.1.2; Hu 4.7, 4.1.3. Derivation of Continuity Equation.

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Lecture 6

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  1. Lecture 6 OUTLINE • Semiconductor Fundamentals (cont’d) • Continuity equations • Minority carrier diffusion equations • Minority carrier diffusion length • Quasi-Fermi levels • Poisson’s Equation Reading: Pierret 3.4-3.5, 5.1.2; Hu 4.7, 4.1.3

  2. Derivation of Continuity Equation • Consider carrier-flux into/out-of an infinitesimal volume: Area A, volume Adx Jn(x) Jn(x+dx) dx EE130/230A Fall 2013 Lecture 6, Slide 2

  3. Continuity Equations: EE130/230A Fall 2013 Lecture 6, Slide 3

  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: 1. The electric field is small, such that in p-type material in n-type material 2. n0 and p0 are independent of x (i.e. uniform doping) 3. low-level injection conditions prevail EE130/230A Fall 2013 Lecture 6, Slide 4

  5. Starting with the continuity equation for electrons: EE130/230A Fall 2013 Lecture 6, Slide 5

  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 mat’l np is the electron (minority-carrier) concentration in n-type mat’l • Thus the minority carrier diffusion equations are EE130/230A Fall 2013 Lecture 6, Slide 6

  7. Simplifications (Special Cases) • Steady state: • No diffusion current: • No R-G: • No light: EE130/230A Fall 2013 Lecture 6, Slide 7

  8. Example • Consider an n-type Si sample illuminated at one end: • constant minority-carrier injection at x = 0 • steady state; no light absorption for x > 0 Lp is the hole diffusion length: EE130/230A Fall 2013 Lecture 6, Slide 8

  9. The general solution to the equation is where A,B are constants determined by boundary conditions: Therefore, the solution is EE130/230A Fall 2013 Lecture 6, Slide 9

  10. 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 EE130/230A Fall 2013 Lecture 6, Slide 10

  11. Summary: Continuity Equations • The continuity equations are established based on conservation of carriers, and therefore hold generally: • 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): EE130/230A Fall 2013 Lecture 6, Slide 11

  12. 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 EE130/230A Fall 2013 Lecture 6, Slide 12

  13. 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 ? EE130/230A Fall 2013 Lecture 6, Slide 13

  14. Find FN and FP: EE130/230A Fall 2013 Lecture 6, Slide 14

  15. Poisson’s Equation area A Gauss’ Law: E(x) E(x+Dx) Dx s :permittivity (F/cm)  :charge density (C/cm3) EE130/230A Fall 2013 Lecture 6, Slide 15

  16. Charge Density in a Semiconductor • Assuming the dopants are completely ionized: r = q (p – n + ND – NA) EE130/230A Fall 2013 Lecture 6, Slide 16

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