Lecture 10 - Carrier Flow (cont.) February 28, 2007

Transcription

1 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 10-1 Lecture 10 - Carrier Flow (cont.) February 28, 2007 Contents: 1. Minority-carrier type situations Reading assignment: del Alamo, C. 5, 5.6

2 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 10-2 Key questions Wat caracterizes minority-carrier type situations? Wat is te lengt scale for minority-carrier type situations? Wat do majority carriers do in minority-carrier type situations?

3 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 10-3 Overview of simplified carrier flow formulations General drift-diffusion situation (Sockley's equations) 1D approx. Quasi-neutral situation (negligible volume carge) Space-carge situation (field independent of n, p) Majority-carrier type situation (V=0, n'=p'=0) Minority-carrier type situation (V=0, n'=p'=0, LLI)

4 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 10-4 Simplified set of Sockley equations for 1D quasi-neutral situations n t = G ext U + p n + N D N A 0 J e = qnv drift + qd e n e drift p J = qpv qd 1 J e p 1 J or = G q ext U t q J t 0 J t = J e + J

5 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture Minority-carrier type situations Situations caracterized by: excess carriers over TE no external electric field applied (but small internal field generated by carrier injection: E = E o + E ) Example: electron transport troug base of npn BJT. Two approximations: 1. E small v drift E 2. Low-level injection for n-type: n n o p p U p τ negligible minority carrier drift due to E (but can t say te same about majority carriers)

6 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 10-6 Sockley equations for 1D quasi-neutral situations p n + N D N A 0 J e = qnv drift + qd e n e drift p J = qpv qd n 1 J e p 1 J = G ext U + or = G ext U t q t q J t 0 J t = J e + J Furter simplifications for n-type minority-carrier-type situations Majority-carrier current equation: but in TE: n o n J e q(n o + n )µ e (E o + E ) + qd e ( + ) n o J eo = qn o µ e E o + qd e = 0 Ten: n J e qn o µ e E + qn µ e E o + qd e

7 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 10-7 Minority-carrier current equation: J q(p o + p )µ (E o + E ) qd ( p o + p ) In TE, J o = 0, and: p p J qp µ E o + qp µ E qd qp µ E o qd Minority-carrier continuity equation: Now plug in J from above: p p 1 J = G ext t τ q 2 p p p p D 2 µ E o τ + G ext = t One differential equation wit one unknown: p. If G ext and BC s are specified, problem can be solved.

8 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 10-8 Sockley equations for 1D minority-carrier type situations n-type p-type p o n o + N D N A 0 p n n J e = qn o µ e E + qn µ e E o + qd e J = qp µ E o qd p 2 p p p p D 2 µ E o τ + G ext = t n J e = qn µ e E o + qd e J = qp o µ E + qp µ E o qd p 2 n n n n D e 2 + µ e E o τ + G ext = t J t 0 J t = J e + J

9 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 10-9 Example 1: Diffusion and bulk recombination in a long bar Uniform doping: E o = 0; static conditions: t = 0 υ n g l 0 x Minority carrier profile: p' p'(0) 0 x Majority carrier profile? n = p exactly?

10 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture p' p'(0) 0 x Far away J t = 0 J t = 0 everywere. J t = J e + J qn o µ e E + q(d e D ) dp = 0 dx If D e = D diffusion term = 0 drift term = 0 E = 0 n = p But, typically D e > D diffusion term < 0 (for x > 0) drift term > 0 E > 0 (for x > 0) and E D e D n p (but still n p ) and n p D e D

11 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture υ n 0 x p',n' ε' p' ε' + + n' - J e, J 0 x - J (diff) J e (drift) J t =0 everywere 0 J e (diff) x

12 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture Solution Step 1. Minority carrier flow problem (for x > 0): d 2 p p dx 2 L 2 = 0 wit L = D τ solution of te form: p = A exp x L x + B exp L B.C. at x = 0: g l 1 dp 2 = q J (0) = D dx x=0 Ten: p = g l L x exp 2D L Tis works for x 0 because p as to be continuous.

13 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture Step 2. Hole current: Assuming J (drift) J (diff) dp J qd dx = qg l 2 exp x L Step 3. Total current: J t = 0 everywere Step 4. Electron current: J e = J = qgl 2 x exp L Step 5. Electron profile: n p = gll 2D exp x L

14 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture Step 6. Electron diffusion current: dn J e (diff) = qd e = dx qg l D e x exp 2 D L Step 7. Electron drift current: J e (drift) = J e J e (diff) qg l D e D x = exp 2 D L Note: if D e = D J e (drift) = 0 Step 8. Average velocity of ole diffusion: diff diff J (x) J (x) D v = = qp(x) qp (x) independent of x. [will use wen deriving I-V caracteristics of pn junction diode] L

15 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture Now verify assumptions Step 9. Verify quasi-neutrality: n p p 1 Compute E from J e (drift): E = J e (drift) kt g l D e D x = exp qµ e n o q 2n o D e D L From Gauss law, get difference between n and p : Ten ɛkt g l D e D p n = q2 n o 2L D e D exp p n L = ( p L D ) 2D e D De x L If caracteristic lengt of problem is muc longer tan L D (Debye lengt), quasi-neutrality applies in minority-carrier-type situations. Put numbers: for N D = cm 3, L D 0.04 µm, L 400 µm, and (L D /L )

16 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture Step 10. Verify J (drift) J (diff) J (drift) qµ p E = J (diff) dp qd dx = 1 p D e D 2n o D e as good as low-level injection Step 11. Limit to injection to maintain LLI: p (0) n o g l 2D n o L Step 12. Verify linearity between v drift and E At x = 0 (worst point): µ e E gl D e D = 2 no D D e D L 1000 cm/s v sat.

17 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture Example 2: Diffusion and surface recombination in a sort or transparent bar Uniform doping: E o = 0; static conditions: = 0 t Bar lengt: L L ; S = at bar ends. υ S= S= n -L/2 0 L/2 x p' -L/2 0 J e, J L/2 x J (diff) J t =0 everywere J e (drift) -L/2 0 L/2 x J e (diff)

18 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture Lengt scales of minority-carrier situations Diffusion Lengt: mean distance tat a carrier diffuses in a bulk semiconductor before recombining L diff = Dτ L diff strong function of doping: 1E+0 Minority carrier diffusion lengt (cm) 1E-1 1E-2 1E-3 1E-4 L L e T=300 K 1E-5 1E+14 1E+15 1E+16 1E+17 1E+18 1E+19 1E+20 1E+21 doping level (cm -3 ) Sample size, L If L L diff, L diff is caracteristic lengt of problem If L L diff, L is caracteristic lengt of problem

19 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture Key conclusions Minority-carrier type situations dominated by beavior of minority carriers: diffusion, recombination and drift. Two caracteristic lengts in minority-carrier type situations dominated by diffusion and recombination: diffusion lengt, L = Dτ, average distance tat a carrier diffuses in a bulk semiconductor before recombining; sample size, L wicever one is smallest, L or L diff, dominates beavior of minority carriers. Minority-carrier type situations called tat way because: lengt and time scales of problem dominated by minority carrier beavior (diffusion, recombination, and drift) role of majority carriers is to preserve quasi-neutrality and total current continuity Order of magnitude of key parameters in Si at 300K: Diffusion lengt: L diff µm (depends on doping level).

20 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture Self-study Work out example 2: diffusion and surface recombination in a sort bar ( 5.6.2)