窄基线二极管和电荷控制模型

窄基线二极管;本文利用电荷控制模型通过连续性方程等来验证窄基线二极管的过剩载流分布。
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1. Lecture #12 OUTLINE – pn Junctions • narrow-base diode • charge-control model Reading: Finish Chapter 6 Spring 2003 EE130 Lecture 12, Slide 1 Carrier Concentration Profiles Spring 2003 EE130 Lecture 12, Slide 2 1

2. Narrow or Short-Base Diode • We have the following boundary conditions: ∆pn ( xn ) = pno ( e qVA / kT − 1) ∆pn ( x' = xc ' ) → 0 • With the following coordinate system: x’’ 0 0 x’ NEW: 0 x’c • Then, the solution is of the form: x′ / L p − x′ / L p ∆p ( x ) = A1e + A2 e Spring 2003 EE130 Lecture 12, Slide 3 Applying the boundary conditions, we have: ∆pn (0) = A1 + A2 xc' / L p − xc' / L p 0 = A1e + A2 e • So, we have:  e (xc − x ' )/ LP − e − (xc − x ' )/ LP  ' ' ∆p n ( x ' ) = p n 0 ( e qV A / kT − 1) , 0 < x' < xc'  e xc' / LP − e − xc' / LP    • Note: sinh (ξ ) = eξ − e − ξ 2 Spring 2003 EE130 Lecture 12, Slide 4 2

3.• So: ∆pn ( x ' ) = pn 0 ( e qV A / kT − 1) [  sinh (xc' − x ') / LP ], 0 < x' < x ' '  sinh xc / LP [ ]   c • If we write: I = I 0 ' ( e qVA kT − 1) • Then I 0' = qA DLPP ( ni 2 cosh xc' / LP ) ( N D sinh xc' / LP ) where cosh (ξ ) = eξ + e − ξ 2 Spring 2003 EE130 Lecture 12, Slide 5 Note: sinh (ξ ) → ξ as ξ → 0 and cosh(ξ ) → 1 + ξ 2 as ξ → 0 • If xc’ << LP: Spring 2003 EE130 Lecture 12, Slide 6 3

4. Narrow Base Diode: I-V Equation Let WN ≡ width of n - type region WP ≡ width of p - type region and WN′ ≡ WN − xn << LP WP′ ≡ WP − x p << LN 2  DP DN  qVA / kT Then, J = qni  + e −1 ( ) WN′ N D WP′ N A  2  DP DN  qVA / kT I = qAni  +  e ( − 1 = I 0 e qVA / kT − 1) ( ) WN′ N D WP′ N A  Spring 2003 EE130 Lecture 12, Slide 7 Current Flow in a One-Sided pn Junction • Note that the diode current is dominated by the term associated with the more lightly doped side: 2  DP  qAni   long n − side  LP N D  p+n diode: I 0 ≅ I P ( xn ) = 2 DP  qAni   short n − side  WN′ N D  2 D  qAni  N  long p − side pn+ diode: I 0 ≅ I N (− x p ) =  LN N A  2  DN  qAni   short p − side  WP′ N A  i.e. current flowing across junction is dominated by carriers injected from the more heavily doped side Spring 2003 EE130 Lecture 12, Slide 8 4

5.Excess Carrier Profiles: Limiting Cases Long base (xc’ → ∞):  e (xc − x ' )/ LP − e − (xc − x ' )/ LP  ' ' ∆p n ( x ' ) = p n 0 ( e qV A / kT − 1)   e xc' / LP − e − xc' / LP     e x c / L P e − x ' / L p − e − xc / L P e x ' / L p  ' ' = pn 0 (e qV A / kT − 1)   ' ' e xc / LP − e − xc / LP    − x '/ L p ≅ pn 0 (e qV A / kT − 1)e Spring 2003 EE130 Lecture 12, Slide 9 2. Short base (xc’ → 0): ∆p n ( x ' ) = p n 0 ( e qVA / kT − 1) [(  sinh xc' − x' / LP  ) ]   sinh xc ' / LP [ ]   = pn 0 (e qVA / kT − 1) ( ' ) xc − x' / LP   x'   = pn 0 (e qVA / kT − 1)1 − '  '  xc / LP   xc  ∆pn is a linear function of x Æ Jp is constant (no recombination) Spring 2003 EE130 Lecture 12, Slide 10 5

6. Minority-Carrier Charge Storage • When VA>0, excess minority carriers are stored in the quasi-neutral regions of a pn junction: −∞ ∞ QN = − qA∫ ∆n p ( x) dx QP = qA∫ ∆pn ( x)dx −xp xn = − qA∆n p ( − x p ) LN = qA∆pn ( xn ) LP Spring 2003 EE130 Lecture 12, Slide 11 Derivation of Charge Control Model • Consider a forward-biased pn junction. The total excess hole charge in the∞ n quasi-neutral region is: QP = qA ∫ ∆pn ( x, t )dx xn • The minority carrier diffusion equation is (without GL): ∂∆pn ∂ 2 ∆pn ∆pn = DP − ∂t ∂x 2 τp • Since the electric field is very small, ∂∆p n J P = − qDP ∂x • Therefore: ∂ ( q ∆p n ) ∂J =− P − q ∆p n ∂t ∂x τp Spring 2003 EE130 Lecture 12, Slide 12 6

7. (Long Base Diode) • Integrating over the n quasi-neutral region: ∂  1   ∞ J P (∞) ∞  ∫ n  qA ∆p dx = − A ∫ dJ P − qA ∫ ∆pn dx  ∂t  x n  J p ( xn ) τ p  xn  • Furthermore, in a p+n junction: J P (∞) −A ∫ dJ J p ( xn ) P = − AJ P (∞) + AJ P ( xn ) = AJ P ( xn ) ≅ iDIFF dQP Q • So: = iDIFF − P dt τp Spring 2003 EE130 Lecture 12, Slide 13 Charge Control Model We can calculate pn-junction current in 2 ways: 1. From slopes of ∆np(-xp) and ∆pn(xn) 2. From steady-state charges QN, QP stored in each excess-minority-charge distribution: dQP Q = AJ P ( xn ) − P = 0 dt τp QP ⇒ AJ P ( xn ) = I P ( xn ) = τp − QN Similarly, I N (− x p ) = τn Spring 2003 EE130 Lecture 12, Slide 14 7

8.Charge Control Model for Narrow Base • For a narrow-base diode, replace τp and/or τn by the minority-carrier transit time τtr – time required for minority carrier to travel across the quasi-neutral region – For holes on narrow n-side: WN 1 QP = qA∫ ∆pn ( x )dx = qA ∆pn ( xn )WN′ xn 2 d∆pn ∆p ( x ) I P = AJ P = − qADP = qADP n n dx WN′ Q ⇒ τ tr = P = N (W ′ ) 2 IP 2 DP τ = (WP′ )2 – Similarly, for electrons on narrow p-side: tr 2 DN Spring 2003 EE130 Lecture 12, Slide 15 Summary: Narrow (Short) Base Diode • If the width of the quasi-neutral region (e.g. xc’) is not much larger than the minority-carrier diffusion length (e.g. LP), then the solution to the minority-carrier diffusion equation is  x′ − x  sinh  c  L  P  ∆pn ( x' ) = ∆pn ( xn )  x′  sinh  c   LP  • If xc’ < 0.1LP:  x′  ∆pn ( x' ) ≅ ∆pn ( xn )1 −   xc′  Spring 2003 EE130 Lecture 12, Slide 16 8

9.• If LP >> xc’, negligible recombination occurs ⇒ hole current is constant throughout n-type region 2 1 J p = − qDP d∆pn = qDP n i qVA / kT e ( − 1   ) dx ND  xc′  • Compare this to the hole current contribution in the long-base diode: 2  1  n i qV / kT J p = qDP ND (e A ) − 1    LP  Spring 2003 EE130 Lecture 12, Slide 17 9