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1. Lecture #7 Quiz #1 Results (undergrad. scores only) N = 73; mean = 21.6; Пғ = 2.1; high = 25; low = 14 OUTLINE вҖ“ Continuity equations вҖ“ Minority carrier diffusion equations вҖ“ Quasi-Fermi levels Reading: Chapter 3.4, 3.5 Spring 2003 EE130 Lecture 7, Slide 1 Clarification: Direct vs. Indirect Band Gap Small change in momentum Large change in momentum required for recombination required for recombination ГҶ momentum is conserved by ГҶ momentum is conserved by photon emission phonon + photon emission Spring 2003 EE130 Lecture 7, Slide 2 1

2.Example: Relaxation to Equilibrium State Consider a semiconductor with no current flow in which thermal equilibrium is disturbed by the sudden creation of excess holes and electrons. The system will relax back to the equilibrium state via R-G mechanism: dn вҲҶn =вҲ’ for electrons in p-type material dt П„n dp вҲҶp =вҲ’ for holes in n-type material dt П„p Spring 2003 EE130 Lecture 7, Slide 3 Net Recombination Rate (General Case) вҖў For arbitrary injection levels and both carrier types in a non-degenerate semiconductor, the net rate of recombination is: dвҲҶn dвҲҶp pn вҲ’ ni2 = =вҲ’ dt dt П„ p (n + n1 ) + П„ n ( p + p1 ) where n1 вүЎ ni e( ET вҲ’ Ei ) / kT and p1 вүЎ ni e( Ei вҲ’ ET ) / kT Spring 2003 EE130 Lecture 7, Slide 4 2

3. Derivation of Continuity Equation вҖў Accounting of carrier-flux into/out-of an infinitesimal volume: Area A, volume Adx Jn(x) Jn(x+dx) dx пЈ« вҲӮn пЈ¶ 1 вҲҶn AdxпЈ¬ пЈ· = вҲ’ [J n ( x ) A вҲ’ J n ( x + dx ) A] вҲ’ Adx вҲӮ пЈӯ пЈёt q П„ n Spring 2003 EE130 Lecture 7, Slide 5 вҲӮJ n ( x ) J n ( x + dx ) = J n ( x ) + dx вҲӮx вҲӮn 1 вҲӮJ n ( x ) вҲҶn => = вҲ’ вҲӮt q вҲӮx П„n вҲӮn 1 вҲӮJ n ( x ) вҲҶn = вҲ’ + GL Continuity вҲӮt q вҲӮx П„n Equations: вҲӮp 1 вҲӮJ p ( x ) вҲҶp =вҲ’ вҲ’ + GL вҲӮt q вҲӮx П„p Spring 2003 EE130 Lecture 7, Slide 6 3

4.Derivation of Minority-Carrier Diffusion Equations вҖў Simplifying assumptions: вҖ“ 1-D вҖ“ negligible electric field вҖ“ n0, p0 are independent of x вҖ“ low-level injection conditions Spring 2003 EE130 Lecture 7, Slide 7 Minority Carrier Diffusion Length вҖў Consider the special case: вҖ“ Constant minority-carrier (hole) injection at x=0 вҖ“ Steady state, no light d 2 вҲҶp вҲҶp вҲҶp = = 2 dx 2 DPП„ p LP LP is the hole diffusion length L p вүЎ D pП„ p Spring 2003 EE130 Lecture 7, Slide 8 4

5.вҖў Physically, LP and LN represent the average distance that minority carriers can diffuse into a sea of majority carriers before being annihilated. вҖў Example: ND=1016 cm-3; П„p = 10-6 s Spring 2003 EE130 Lecture 7, Slide 9 Quasi-Fermi Levels вҖў Whenever вҲҶn = вҲҶp вү  0, np вү  ni2. However, we would like to preserve and use the relations: n = N c e вҲ’ ( Ec вҲ’ EF ) / kT p = N v e вҲ’ ( EF вҲ’ Ev ) / kT вҖў These equations imply np = ni2, however. The solution is to introduce two quasi-Fermi levels FN and FP such that n = N c e вҲ’ ( Ec вҲ’ FN ) / kT p = N v e вҲ’ ( FP вҲ’ Ev ) / kT Spring 2003 EE130 Lecture 7, Slide 10 5

6. Example: Quasi-Fermi Levels Consider a Si sample with ND = 1017 cm-3 and вҲҶn = вҲҶp = 1014 cm-3. (a) Find n: n = n0+ вҲҶn = ND + вҲҶn вүҲ 1017 cm-3 (b) Find p: p = p0+ вҲҶp = ( ni2 / ND ) + вҲҶp вүҲ 1014 cm-3 (c) Find the np product: np вүҲ 1017 Г— 1014 = 1031 cm-6 >> ni2 Spring 2003 EE130 Lecture 7, Slide 11 (d) Find FN: вҲ’ ( E вҲ’ F ) / kT n = 1017 cm-3 = N c e c N Ec- FN = kT Г— ln(Nc/1017) = 0.026 eV Г— ln(2.8Г—1019/1017) = 0.15 eV (e) Find FP: вҲ’ ( F вҲ’ E ) / kT p = 1014 cm-3 = N v e P v FPвҖ“Ev = kT Г— ln(Nv/1017) = 0.026 eV Г— ln(1019/1014) = 0.30 eV Spring 2003 EE130 Lecture 7, Slide 12 6

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