Lecture 25
OUTLINE
The BJT (cont’d)
• Ideal transistor analysis
• Narrow base and narrow emitter
• Ebers-Moll model
• Base-width modulation
Reading: Pierret 11.1-11.2; Hu 8.2-8.6
Notation (PNP BJT)
NE  NAE
NB  NDB
NC  NAC
DE  DN
DB  DP
DC  DN
tE  tn
tB  tp
tC  tn
LE  LN
LB  L P
LC  LN
nE0  np0 = ni2/NE pB0  pn0  ni2/NB nC0  np0  ni2/NC
EE130/230M Spring 2013
Lecture 25, Slide 2
“Game Plan” for I-V Derivation
• Solve the minority-carrier diffusion equation in each quasineutral region to obtain excess minority-carrier profiles
– different set of boundary conditions for each region
• Find minority-carrier diffusion currents at depletion region edges
dnE
E dx"
x"0
I En  qAD
dnC
C dx '
x ' 0
I Cn  qAD
dp B
B dx
x 0
I Ep  qAD
dp B
B dx
x W
I Cp  qAD
• Add hole & electron components together  terminal currents
EE130/230M Spring 2013
Lecture 25, Slide 3
Emitter Region Analysis
d 2 nE
 tnEE
• Diffusion equation:
0  DE
• General solution:
nE ( x" )  A1e  x"/ LE  A2e x"/ LE
• Boundary conditions:
dx"2
nE ( x"  )  0
nE ( x"  0)  nE 0 (e qVEB / kT  1)
• Solution:
nE ( x" )  nE 0 (e qVEB / kT  1)e  x"/ LE
nE
I En  qADE ddx
"
EE130/230M Spring 2013
x"0
 qA DLEE nE 0 (e qVEB / kT  1)
Lecture 25, Slide 4
Collector Region Analysis
0  DC
• Diffusion equation:
• General solution:
d 2 nC
dx'2
 tnCC
nC ( x' )  A1e  x '/ LC  A2 e x '/ LC
• Boundary conditions:
nC ( x'  )  0
nC ( x'  0)  nC 0 (e qVCB / kT  1)
• Solution:
nC ( x' )  nC 0 (e qVCB / kT  1)e  x '/ LC
I Cn  qADC ddxn'C
EE130/230M Spring 2013
x ' 0
 qA DLCC nC 0 (e qVCB / kT  1)
Lecture 25, Slide 5
Base Region Analysis
• Diffusion equation: 0  DB
• General solution:
d 2 nB
dx2
 tpBB
pB ( x)  A1e  x / LB  A2e x / LB
• Boundary conditions:
pB (0)  pB 0 (e qVEB / kT  1)
pB (W )  pB 0 (e qVCB / kT  1)
• Solution:
pB ( x )  pB 0 (e
 pB 0 ( e
EE130/230M Spring 2013
qVCB / kT
qVEB / kT
 1)

Lecture 25, Slide 6
 1)

e ( W  x ) / LB  e  ( W  x ) / LB
eW / LB  e W / LB
e x / LB  e  x / LB
eW / LB  e W / LB


Since
sinh   
e e 
2
we can write
pB ( x)  pB 0 (e
 p B 0 (e
as
qVCB / kT
qVEB / kT
 1)

 p B 0 (e
EE130/230M Spring 2013

e x / LB  e  x / LB
eW / LB  e W / LB
pB ( x)  pB 0 (e qVEB / kT
qVCB / kT
 1)
e (W  x ) / LB e  (W  x ) / LB
eW / LB e W / LB

sinh W  x  LB 
 1)
sinh W LB 
sinh  x LB 
 1)
sinh W LB 
Lecture 25, Slide 7

d
d  e  e 

sinh   
d
d  2
pB ( x)  pB 0 (e qVEB / kT
 e  e 
 
 cosh  
2

sinh W  x  LB 
sinh  x LB 
qVCB / kT
 1)
 p B 0 (e
 1)
W
sinh  LB 
sinh W LB 
dp B
B dx
x 0
I Ep  qAD
 qA
DB
LB
p

cosh(W / LB )
B 0 sinh(W / LB )
I Cp  qADB ddxpB
 qA
DB
LB
p

qVEB / kT
 1) 
1
sinh(W / LB )
e
 1) 
cosh(W / LB )
sinh(W / LB )
e
x W
1
B 0 sinh(W / LB )
EE130/230M Spring 2013
(e
qVEB / kT
(e
Lecture 25, Slide 8
qVCB / kT
qVCB / kT

1

1
BJT Terminal Currents
• We know:
I En  qA
DE
LE
I Ep  qA
DB
LB
I Cp  qA
DB
LB
I Cn  qA
nE 0 (e qVEB / kT  1)
pB0
p
DC
LC


1
B 0 sinh(W / LB )
nC 0 (e
qVCB / kT
• Therefore:

 qA
(e
qVEB / kT

DE
LE
nE 0 
IC
DB
LB
pB 0 sinh(W1 / LB ) (eqVEB / kT  1) 
EE130/230M Spring 2013

1) 
pB 0 sinh(W / LBB )) (eqVEB / kT  1) 
cosh(W / L

cosh(W / LB )
sinh(W / LB )
e
qVCB / kT

1
 1)
I E  qA
DB
LB

qVCB / kT
qVEB / kT
1
(
e

1
)

e
1
sinh(W / LB )
sinh(W / LB )
cosh(W / LB )

DC
LC
Lecture 25, Slide 9

nC 0 




DB
LB
pB 0 sinh(W1 / LB ) eqVCB / kT  1
DB
LB
pB 0 sinh(W / LBB )) eqVCB / kT  1
cosh(W / L
BJT with Narrow Base
• In practice, we make W << LB to achieve high current gain.
Then, since
sinh     for   1
cosh    1 
2
for   1
2
we have:
pB ( x)  pB 0 (e qVEB / kT  1)1  Wx 
 pB 0 (e qVCB / kT  1)Wx 
EE130/230M Spring 2013
Lecture 25, Slide 10
BJT Performance Parameters
 
1
1
T 
 dc 
 dc 
Assumptions:
• emitter junction forward
biased, collector junction
reverse biased
• W << LB
ni E 2 D N W
E
B
ni B 2 DB N E LE
1
1
1
2
 
W 2
LB
1
1
ni E 2 D N W
E
B
ni B 2 DB N E LE
1
ni E 2 D N W
E
B
ni B 2 DB N E LE
EE130/230M Spring 2013
 12

1
2
 
W 2
LB
 
W 2
LB
Lecture 25, Slide 11
BJT with Narrow Emitter
 
1
1
T 
 dc 
 dc 
ni E 2 D N W
E
B
Replace with WE’ if emitter is narrow
ni B 2 DB N E LE
1
1
1
2
 
W 2
LB
1
1
ni E 2 D N W
E
B
ni B 2 DB N E LE
1
ni E 2 D N W
E
B
ni B 2 DB N E LE
EE130/230M Spring 2013
 12

1
2
 
W 2
LB
 
W 2
LB
Lecture 25, Slide 12
Ebers-Moll Model
increasing
(npn) or VEC (pnp)
The Ebers-Moll model is a large-signal equivalent circuit which
describes both the active and saturation regions of BJT operation.
• Use this model to calculate IB and IC given VBE and VBC
EE130/230M Spring 2013
Lecture 25, Slide 13

 qA
I E  qA
DE
LE
IC
DB
LB
nE 0 
DB
LB
1
B 0 sinh(W / LB )
p

pB 0 sinh(W / LB ) (eqVEB / kT  1) 
cosh(W / LB )
(e
qVEB / kT
 1) 

DC
LC

nC 0 
If only VEB is applied (VCB = 0):
DB
LB
DB
LB

e

 1
pB 0 sinh(W1 / LB ) eqVCB / kT  1
cosh(W / LB )
pB 0 sinh(W / LB )
qVCB / kT
V EB
V CB
I E  I F 0 ( e qVEB / kT  1)
IB
I C   F I F 0 ( e qVEB / kT  1)
I B  1   F I F 0 ( e qVEB / kT  1)
If only VCB is applied (VEB = 0): :
I C   I R 0 (e qVCB / kT  1)
I E   R I R 0 (e qVCB / kT  1)
I B  I R 0 (1   R )(e
EE130/230M Spring 2013
qVCB / kT
 1)
E
B
C
IC
aR : reverse common base gain
aF : forward common base gain
Reciprocity relationship:
DB
pB 0
 F I F 0   R I R 0  qA
LB sinh( W / LB )
Lecture 25, Slide 14
In the general case, both VEB and VCB are non-zero:
I C   F I F 0 (e qVEB / kT  1)  I R 0 (e qVCB / kT  1)
IC: C-B diode current + fraction of E-B diode current that makes it to the C-B junction
I E  I F 0 (e qVEB / kT  1)   R I R 0 (e qVCB / kT  1)
IE: E-B diode current + fraction of C-B diode current that makes it to the E-B junction
Large-signal equivalent circuit for a pnp BJT
EE130/230M Spring 2013
Lecture 25, Slide 15
Base-Width Modulation
Common Emitter Configuration, Active Mode Operation
W
IE
P
+
N
P
IC
IC
  dc 
IB
1
niE 2 DE N B W
n 2 DB N E LE
iB
2
+
VEB
niB DB N E LE

2
niE DE N BW

pB(x)


pB 0 e qVEB / kT  1
IC
(VCB=0)
x
0
VEC
W(VBC)
EE130/230M Spring 2013
Lecture 25, Slide 16
 12
 
W 2
LB
Ways to Reduce Base-Width Modulation
1. Increase the base width, W
2. Increase the base dopant concentration NB
3. Decrease the collector dopant concentration NC
Which of the above is the most acceptable action?
EE130/230M Spring 2013
Lecture 25, Slide 17
Early Voltage, VA
1
Output resistance:
 I C 
VA


r0  


IC
 VEC 
A large VA (i.e. a large ro ) is desirable
IC
IB3
IB2
IB1
VA
EE130/230M Spring 2013
0
Lecture 25, Slide 18
VEC
Derivation of Formula for VA
Output conductance: g 0 
VEC  VEB  VBC
dI C
I
 C
dVEC VA
dI C
dI C
so g o 

dVEC dVBC
dI C dW
dI C  dxnC 

go 


  
dW dVBC dW  dVBC 
EE130/230M Spring 2013
N
VA 
IC
g0
for fixed VEB
where xnC is the width of the
collector-junction depletion region
on the base side
xnC
P+

P
Lecture 25, Slide 19

IC  qA
DB
LB

pB 0 sinh(W1 / LB ) (eqVEB / kT  1) 


DC
LC
nC 0 
DB
LB
cosh(W / L

qAni2 DB qVEB / kT
IC 
e
1
WN B


dI C
qAni2 DB qVEB / kT
IC
 2
e
1  
dW
W NB
W
C JC 
dQdepC
dVBC
d (qN B xnC )
dxnC

 qN B
dVBC
dVBC
dxnC C JC


dVBC qN B
VA 
IC

g 0 dI C
dW
EE130/230M Spring 2013
IC

 dxnC   I C
  
  
 dVBC   W
Lecture 25, Slide 20


pB 0 sinh(W / LBB )) eqVCB / kT  1
IC
qN BW

C JC
  C JC 

   
  qN B 
Summary: BJT Performance Requirements
• High gain (dc >> 1)
 One-sided emitter junction, so emitter efficiency   1
• Emitter doped much more heavily than base (NE >> NB)
 Narrow base, so base transport factor T  1
• Quasi-neutral base width << minority-carrier diffusion length (W << LB)
• IC determined only by IB (IC  function of VCE,VCB)
 One-sided collector junction, so quasi-neutral base width W does
not change drastically with changes in VCE (VCB)
• Based doped more heavily than collector (NB > NC)
(W = WB – xnEB – xnCB for PNP BJT)
EE130/230M Spring 2013
Lecture 25, Slide 21