# Lecture

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Lecture #19

OUTLINE
• pn junctions (cont’d)
– Charge control model

Spring 2007         EE130 Lecture 19, Slide 1
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 2007                 EE130 Lecture 19, Slide 2
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   p

• Since the electric field is very small,
p n
J P  qDP       x

• Therefore:     ( qp n )   J   qp n
 P 
t        x   p

Spring 2007         EE130 Lecture 19, Slide 3
(Long Base Diode)
• Integrating over the n quasi-neutral region:
                               1        
              J P ()          

qA  pn dx   A  dJ P  qA  pn dx
t  x n
           
      J p ( xn )
 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          p

Spring 2007                        EE130 Lecture 19, Slide 4
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                 QP
 AJ P ( xn )     0
dt                 τp
QP
 AJ P ( xn )  I P ( xn ) 
τp
 QN
Similarly, I N ( x p ) 
τn

Spring 2007           EE130 Lecture 19, Slide 5
Charge Control Model for Narrow Base
•    For a narrow-base diode, replace p and/or
n by the minority-carrier transit time tr
–    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 )
dx            WN
QP WN          2
 τ tr , p     
IP    2 DP
WP 2
– Similarly, for electrons on narrow p-side: τ tr ,n 
2 DN
Spring 2007            EE130 Lecture 19, Slide 6
Summary
• Under forward bias, minority-carrier charge is stored
in the quasi-neutral regions of a pn diode.

– Long base: QN  qA
ni2 qVA / kT
NA
e        
 1 LN       
QP  qA
ni2 qVA / kT
ND
e       1 LP     

– Short base: QN  qA
1 ni2 qVA / kT
2 NA
e            
 1 WP         
QP  qA
1 ni2 qVA / kT
2 ND
e            
 1 WN        
Spring 2007         EE130 Lecture 19, Slide 7
• The steady-state diode current can be viewed as the
charge supply required to compensate for charge
loss via recombination (long base) or collection at the
contacts (short base)
QN QP                                  LN D N      L   D
– Long base: I                              Note that        and P  P
τn τ p                                 τn   LN     τ p LP

QN       QP
– Short base: I         
τ tr ,n τ tr , p


WP 2                    
WN 2

where τ tr ,n                      τtr , p
2 DN                        2DP
Spring 2007              EE130 Lecture 19, Slide 8

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