Mextram 504 BJT model
F. Yuan
Advisor： Prof. C. W. Liu
Graduate Institute of Electronics Engineering and
Department of Electrical Engineering,
National Taiwan University, Taipei, Taiwan
Outline








Charge modeling
Collector current
Base current
series resistance, epilayer resistance
Avalanche multiplication
Extrinsic region
AC small-signal model
Noise and temperature effect
Depletion charge (Qte,Qtc)
C je
Cte  (1  XC je )
(1 
2 1
VdE
) pE
C je
CteS  XC je
(1 
 B2E1
B E
B E
2 1
VdE
) pE

 B2 E1 1 pE 
C jeVdE 
Qte   CtedV  (1  XC je )
1

(
1

) 

1  pE 
VdE
 
0

pE
 (1  K ) 2 (1   B2 E1 )
(1  K )C jeVdE 
VdE
Qte  (1  XC je )
1 
pE
1  pE  K 
 B2 E1 2

2
 (1 
)  K
V
 
dE








Set Q=0 at V=0
Change function to
a smooth one to
prevent the value
become infinite at
V=Vd
Base diffusion charge (QBE,QBC)

Injected n  p , so we caculate injected eDefine base charge at zero bias Q   p dx  N W
WB

B0
p
0
WB
QBE   n( x)dx 
0

QBC

n(0) '
WB
2
Assumed linear
1 n(0)
1
N BWB'  n0 q1QB 0
2 NB
2
1
 nB q1QB 0
2
QBE+QBC=all diffusion charge in base
B
B
Main current (IN)
I C  WB
D
0
 BE
q
pp
AE (e VT  e VT )
2
nB ieB
n
dx
 B2E1
I N  I S (e
VT
 BC
 B2C2
e
VT
)
1
qB
WB
qB 
q1 
 p( x)dx
0
QB 0

QB 0  Qte  Qtc  QBE  QBC
QB 0
QB 0  Qte  Qtc
V
V
 1  tE  tC
QB 0
Ver Vef
q1=1 means no early effect
Total base charge (qB)

Early effect (base width modulation)
Qte,Qtc

High level injection
QBE,QBC
Base current (IB1,IB2)
I B1  (1  XI B1 )
I B1  XI B1
S
IS
f
IS
f
 B2E1
(e
VT
 1)
 B1E1
(e
VT
 1)

 B2E1
I B 2  I Bf (e

mLf VT
 1) , mLf  2

IB1 is ideal forward base
current
IB2 is non-ideal forward
base current
(2kT current at low bias)
S means sidewall
SiGe HBT



qB is modified by the bandgap difference of the
base region
Only considered the linear graded Ge profile
If there are a lot of defects in SiGe base, there is
neutral base recombination current (1kT current)
q1  1 
qBI 
e
VtE VtC

Ver Vef
  VtE  dE g

 V 1 V
  er  T




 dE g

 VT
e



 VtC dE g 


 Vef VT 


1
 B2E1
 B2E1
 B2C2
IS 
VtC 
VT
VT
VT
(1  X rec )(e
I B1  (1  XI B1 )
 1)  X rec (e
e
 2)(1 
)
f 
Vef 


e
Diffusion charge (QE, Qepi)



Emitter diffusion charge QE
Collector epilayer diffusion charge Qepi
dQDiffusion
When I  I k , Let  QE (min)   E
dI
 B2E1


I 
m VT


QE  QE 0 e
 1  QE 0  


 Is 


Q 
E
dQE QE 0

dI
m I s
I 
 
 Is 
1
1
m
1
m
1
1
m
B
2 1
 IS 
 QE   E  
I S (e m VT  1)
 Ik 
2VT xi
Qepi   epi
( p 0  p w  2)
RCv Wepi
E
Base capacitance


Base current is injected from side, the
voltage on B1 and B2 may be different
We must compensate the charge
Q  CV
VB1  VB2
1
QB1B2   B1B2 (CtE  C BE  C E )
5
Base resistance

DC crowding effect
B2
B
Rb 
B1
RBc
R B  RBC  Rb
RBv
RBv
qB
I B  eVBE
I B1B2
2V
 T (e
3Rb
 B1B2
VT
 1) 
B B
1 2
3Rb
Collector resistance


Buried layer to collector electrode resistance
is constant RCC
Epilayer resistance is a variable
Collector resistance

When IC large, RC ：small to high to small
I hc  qNepivsat Aem
Collector resistance

Kull, TED vol.32, no.6, p1103, 1985
N epi  const.
EC  C1C2
 f ( B2C2 , B2C1 )
n  p  N epi
I C1C2 
Jp  0

 p0  1  

EC  VT 2 p0  2 pw  ln
 pw  1 

d p
dx
n 
0
1
 n0
 n 0  n
vsat x
J n  const.
RCv
1
p0 
1  4e
2
pw 
 B2C2 VdC
1
1  4e
2
VT
1

2
 B2C1 VdC
VT

1
2
Collector resistance

Jeroen, SSC vol.36, no.9, p1390, 2001

Also considered the high current base
push-out (Kirk effect)
Velocity saturation
Final equation is


xi
2VT
 p 0  p w  p0  p w  1

Wepi I C1C2 RCv
p0  p w  2
I C1C2 
VdC   B2C1

xi 2
SCRCv (1 
)
Wepi
VdC   B2C1  SCRCv I hc (1 
VdC   B2C1  RCv I hc
xi
)
Wepi
RF performance



fT roll-off at high IC, IC1C2 is the key
When IC get large enough, base push-out
occurs,  F increase and makes fT roll-off
Mextram model based on more physical
parameters
1
kT
CBE  CBC   CBC re  rc 
F 
2fT
qIC
Avalanche multiplication

Weak avalanche effect
Valid only for IC1C2 < Ihc

Kloosterman, p172, BCTM 2000

I avl  I C1C2
WtC


An e
Bn
E ( x)
dx
0
x
E
E ( x)  EM (1  )  M
 1 x

I avl  I C1C2
B
x
 n (1 d ) 
  EBn
An

EM  e M  e EM  
Bn


Extrinsic region



Base-SIC：intrinsic
Base-epilayer-buried layer：extrinsic
Base-(p-poly)-buried layer：external
Reverse base current (Iex,IB3)
I B1  (1  XI B1 )
 B2E1
1
f
I S (e
VT
 1)

1 1
I ex  (1  X ext )
( I k nBex ( B1C1 )  I S )
 ri 2

 B1C1
I B 3  I Br
e
VT
 B1C1
1
VLr
, VLr  2
e 2VT  e 2VT
I B3
 B1C1

 I Br e 2VT ,  B1C1  VLr

 B1C1

V
 I Br e T ,  B1C1  VLr

Iex is ideal reverse base
current
IB3 is non-ideal reverse
base current
(2kT current at low bias)
Xext is partitioning factor
Extrinsic region





External reverse base current, XIex
Extrinsic depletion charge, Qtex
External depletion charge, XQtex
Extrinsic diffusion charge, Qex
External diffusion charge, XQex
Parasitic PNP


Base-Collector-Substrate：parasitic PNP
Only for it’s main current
 B1C1
I sub 
2 I ss (e
VT
1 1 4
I sub
 1)
IS
e
I kS
 B1C1
VT
 B1C1
 I
 ss e 2VT ,  B1C1 is big
 I S
  I kS

 B1C1

V
 I ss (e T  1) ,  B1C1 is sm all
Others



Collector-Substrate depletion capacitance
Reverse substrate current
Constant B-E, B-C overlap capacitance
 SC 1
I sf  I SS (e VT  1)
Cts
C BE 0
C BC 0
Small-signal equivalent circuit
Small-signal equivalent circuit
I N
I
I
g y  N gz  N
x
y
z
I C1C2
I C1C2
I C1C2
g RCv , x 
g RCv , y 
g RCv , z 
x
y
z
I
I
I
g , x  BE g , y  BE g , z  BE
x
y
z
I
I
I
g  , x  BC g  , y  BC g  , z  BC
x
y
z
gx 
x：VB2E1
y：VB2C2
z：VB2C1



I BC   I avl
I N  I BC  I C1C2

I N   I BC  I C1C2







g x dx  g y dy  g z dz  g RCv , x  g  , x dx  g RCv , y  g  , y dy  g RCv , z  g  , z dz
Small-signal equivalent circuit
g x  g RCv , x  g  , x
dy  y 
  
dx  x  z g RCv , y  g  , y  g y
g z  g RCv , z  g  , z
dy  y 
  
dz  z  x g RCv , y  g  , y  g y
 I C1C2
g m  
 x
 I C1C2
 
 x

 I C1C2

 
vC1E1  x
  x 
 I
     C1C2
 z  x vC1E1  z
  I C1C2 
  

 z  z  x
dy
dy
 g RCv , x  g RCv , y
 g RCv , z  g RCv , y
dx
dz



x：VB2E1
y：VB2C2
z：VB2C1
  z 
  
 x  x  vC1E1
Small-signal equivalent circuit
  I  I  
g   BE BC 
x

 vC1E1
  I  I     I  I  
  BE BC    BE BC 
x
z

z 
x
 dy dy 
 g , x  g  , x  g , z  g  , z  g , y  g  , y    
 dx dz 
 I C1C2   I C1C2 
 I C1C2 





g out 

 
 vC E    ( x  z ) 
 z  x
x
 1 1 x
  g RCv , z  g RCv , y
dy
dz



x：VB2E1
y：VB2C2
z：VB2C1
Small-signal equivalent circuit
  I BE  I BC     I BE  I BC  
 
g   

 vC E
 
z
x
1 1

x
dy
 g , z  g  , z  g , y  g  , y 
dz
dy
C BE  C BE , x  C BC , x  (C BE , y  C BC , y )
dx
dy
C BC  C BC , z  (C BE , y  C BC , y )
dz



x：VB2E1
y：VB2C2
z：VB2C1
Small-signal equivalent circuit
 dy dy 
g  g S  g , x  g , z  g  , x  g  , z  ( g , y  g  , y )   
 dx dz  
 dy dy 
g m  g RCv , z  g RCv , y   

 dx dz 
g
 m

g
g   g , z  g  , z  ( g , y  g  , y )
dy
 g ex  Xgex
dz
rB  RBcT  rbv

S
C BE  C BE , x  C BE
 C BC , x  (C BE , y  C BC , y )
C BC  C BC , z  (C BE , y  C BC , y )
x：VB2E1
y：VB2C2
z：VB2C1
dy
 C BE 0
dx
dy
 C BC 0  C BCex  XCBCex
dz

Can get more
precise
parameters
Extrinsic added
Hybrid-π model

Let the equivalent circuit has only One
current source
g m'  g m  g 
g'  g  g 
'
g out
 g out  g 
g '  g 
'
g
 '  m'
g
B2-E1-(C1-E1)=B2-C1
Cutoff frequency fT
T 
1
2fT
Q
T 
I C1C2
VCE  0
VCE is const.
Qtot vi
vi
T  
  Ci
vi I C1C2
I C1C2
i
i


S
 C BE , x  C BE
 C BC , x rx  C BE , y  C BC , y ry  C BE , z  C BC , z rz
 C BCex rex  XCBCex Xrex  C BEO  C BCO  Xrex  RCc 
Cutoff frequency fT
Noise (for AC)


v2
 4kTR
f
Thermal noise
-- consider variable resistance
Shot noise
i
2
f

 2qID
Flicker noise (1/f noise)
-- non-ideal base current use KfN
A
i2
I f
 K f b , b 1
f
f
Temperature


Temperature rules are applied to various
parameter
Self-Heating is considered
Comparison to GP





fT-IC is more accurate
Mextram parameters are base on more
physical way
Noise is considered more accurate because
the variable resistance
Linear graded SiGe HBT model in
Mextram 504
Weak avalanche breakdown
Still unconsidered


B-E junction breakdown
High injection current breakdown