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Unformatted text preview: 6.012  Electronic Devices and Circuits Lecture 4  pn Junctions: Electrostatics  Outline Review Poisson's equation for o (x) given N d (x) and N a (x): d 2 o (x)/dx 2 = q [n i {e q o (x)/kT e q o (x)/kT } N d (x) + N a (x)] Knowing o (x), we have n o (x) = n i e q o (x)/kT and p o (x) = n i e q o (x)/kT Slowly varying profiles: quasineutrality holds In ntype, for example, n o (x) N d (x) N a (x), p o (x) = n i 2 /n o (x) Given n o and/or p o , o (x) = (kT/q)ln[n o (x)/n i ] = (kT/q)ln[p o (x)/n i ] Abrupt pn junction in TE (electrostatics) Abrupt profile: Take as an example an abrupt pn junction with N a (x) N d (x) N Ap for x < 0 and N d (x) N a (x) N Dn for x > 0 Observe: 1. n o (x) and p o (x) depend exponentially on o (x) 2. o (x) is insensitive to the details of the charge profile, (x) Depletion approximation: 0 for x < x p and x > x n Approximate net charge, (x) q N Ap for x p < x < 0 q N Dn for 0 < x < x n Integrate once to get E(x), and again to get (x) Find x p and x n by fitting o (x) to known crossing junction Applying bias to a pn junction (what happens?) Clif Fonstad, 9/22/09 Lecture 4  Slide 1 Nonuniform doping in thermal equilibrium Reviewing from Lecture 3: In a nonuniformly doped sample in TE we have: g L (x,t) = 0, J e (x) = 0, J h (x) = 0, and / dt = 0. Also: n(x) = n o (x) and p(x) = p o (x) . Applying these conditions to the two current density equations gave: and = q e n o ( x ) E ( x ) + qD e dn o ( x ) dx " d # dx = D e e 1 n o ( x ) dn o ( x ) dx = q h p o ( x ) E ( x ) " qD h dp o ( x ) dx # d $ dx = " D h h 1 p o ( x ) dp o ( x ) dx And Poisson s equation became: " d 2 # ( x ) dx 2 = dE ( x ) dx = $ ( x ) % = q % p o ( x ) " n o ( x ) + N d ( x ) " N a ( x ) [ ] In the end, we had three equations in our three remaining unknowns, n o (x), p o (x), and (x). Clif Fonstad, 9/22/09 Lecture 4  Slide 2 Nonuniform doping in thermal equilibrium, cont. The first two equations can be solved by integrating to get: and Ref : " ( x ) = 0 at all x where p o ( x ) = n o ( x ) = n i n o ( x ) = n i e e D e " ( x ) p o ( x ) = n i e # h D h " ( x ) Next use the Einstein relation: h D h = e D e = q kT Note: @ R.T. q kT " 40 V # 1 and kT q " 25 mV Using the Einstein relation we have: Finally, putting these in Poissons equation, a single equation for (x) in a doped semiconductor in TE materializes: n o ( x ) = n i e q " ( x ) kT and p o ( x ) = n i e # q " ( x ) kT d 2 " ( x ) dx 2 = # q $ n i e # q " ( x )/ kT # e q " ( x )/ kT ( ) + N d ( x ) # N a ( x ) [ ] Clif Fonstad, 9/22/09 Lecture 4  Slide 3 Doing the numbers: I. D to conversions, and visa versa To convert between D and it is convenient to say 25 mV, kT/q in which case q/kT 40 V1 : 17 C/62F Example 1: e = 1600 cm 2 /Vs, h = 600 cm 2 /Vs D e = e q kT ( ) = 1600/40 = 40 cm 2 / s D h =...
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 Fall '09
 CharlesG.Sodini

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