SPICE Gummel-Poon (SGP) BJT Model
HO #8: ELEN 251 - SGP BJT Model Page 1S. Saha
• SPICE Gummel-Poon (SGP) model is most widely used in the semiconductor industry.
• SGP model improves dc characterization of EM3 model by a unified approach.
• The SGP unified model was developed to improve:– base-width modulation – high-injection effects– base-widening effect resulting in τF vs. IC.
• The starting point of SGP model is:– EM2-model– two additional diodes in EM2 representing the extra
component of IB for β roll-off at low IC.
SGP BJT Model: Starting Point
HO #8: ELEN 251 - SGP BJT Model Page 2S. Saha
E
r'c
r'e
C
C'
E'
Csub
r'bB B'
CjC
CjE
CDC
CDE
ICT = ICC - IEC
IEC/βR
ICC/βF
⎟⎠⎞⎜
⎝⎛ −
′′
1)0(2 eIC kTEEB
nVq
S
⎟⎠⎞⎜
⎝⎛ −
′′
1)0(4 eIC kTCCB
nVq
S
⎟⎠⎞⎜
⎝⎛ −=
⎟⎠⎞⎜
⎝⎛ −=
′′
′′
1
1
eII
eII
kTV CBq
SEC
kTV EBq
SCC
The starting point of SGP model is EM2-model with two extra diodes to account for β roll-off at low current level.
SGP BJT Model: Model Parameters
HO #8: ELEN 251 - SGP BJT Model Page 3S. Saha
• EM2-parameter set: dc (EM1) - βF, βR, Tref, Eg (re-define IS in SGP) bulk-ohmic resistors - r'c, r'e, r'b, charge storage effects - CjE0, φE, mE,CjC0, φC, mC, τF,τR Csub
• Extra model parameters: transistor sat. current - ISS (replacement of IS) low-current β roll-off - C2, nE, C4, nC
forward Early voltage - VA
inverse Early voltage - VB
knee current in ln(IC) vs. VBE - IK
inverse knee current - IKR
τF vs. IC model - B
Derivation of ISS
HO #8: ELEN 251 - SGP BJT Model Page 4S. Saha
xjE xjCxE xC
BaseEmitter CollectorSpace-chargelayer
Space-chargelayer
)(xε←)(xp
)(xn
x
• Assumptions:– one-dimensional current equations hold– npn-BJT with EB junction forward biased and BC reverse biased– depletion approximation, that is, no mobile charge inside the
depletion region
– BJT gain is high, that is IB ≅ 0.
Derivation of ISS
HO #8: ELEN 251 - SGP BJT Model Page 5S. Saha
One-dimensional current equations (HO #2, slide #36) are: Jn = qµnn(x)ε(x) + qDn(dn(x)/dx) (1) Jp = qµpp(x)ε(x) − qDp(dp(x)/dx) (2)
Since, we assume IB ≅ 0,
∴ Jp = hole current in base ≅ 0 and from (2) we get,
or, qµpp(x)ε(x) − qDp(dp(x)/dx) = 0 (3)
here we used, Dn/µn = Dp/µp = kT/q
The direction of the ε-field in (4) aids e- flow from E → C and retards e- flow from C → E.
dxxdp
xpqkT
dxxdp
xpD
p
p )()(
1)()(
1)x( ==∴µ
ε (4)
Derivation of ISS
HO #8: ELEN 251 - SGP BJT Model Page 6S. Saha
The e- flow between E and C is given by (1):
Jn = qµnn(x)ε(x) + qDn(dn(x)/dx) (1)
Using (4) in (1) we get:
We integrate (6) over the neutral base width WB from xE to xC .
⎥⎦⎤
⎢⎣⎡ +=∴
dxxdnxp
dxxdpxnxp
DqJ n
n)()()()()(
dxdn
Dqdx
dppnkTJ nnn
)x()x()x()x(
+= µ (5)
[ ])()()( xpxndxd
xpDq
J nn = (6)or,
Derivation of ISS
HO #8: ELEN 251 - SGP BJT Model Page 7S. Saha
xjE xjCxE xC
BaseEmitter CollectorSpace-chargelayer
Space-chargelayer
)(xε←)(xp
)(xn
x
[ ]dxxpxndxdqDdxxpJ
C
E
C
E
X
Xn
X
Xn ∫∫ =∴ )()()( (7)
( ) ( )[ ]
∫
−=∴
C
E
X
X
EECCnn
dxxp
xpxnxpxnqDJ
)(
)()((8)
Derivation of ISS
HO #8: ELEN 251 - SGP BJT Model Page 8S. Saha
From PN-junction analysis, we know that the pn-product at the edge of the depletion regions are:
Substituting (9) and (10) in (8) we get:
enxnxp
enxnxp
kTV EBq
iEE
kTV CBq
iCC
′′
′′
=
=
2
2
)()(
)()( (9)
(10)
(11)
∫
⎥⎦⎤
⎢⎣⎡ −
=∴
′′′′
C
E
x
x
kTV EBq
kTV CBq
n i
n
dxxp
eenDqJ
)(
2
Derivation of ISS
HO #8: ELEN 251 - SGP BJT Model Page 9S. Saha
If A = cross-sectional area of the emitter, then from (11) we can show that:
∫
⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛ −−⎟
⎠⎞⎜
⎝⎛ −−
=
′′′′
C
E
x
x
kTV CBq
kTV EBq
n i
n
dxxp
eenADqI
)(
112
(12)
Where In = total dc minority injection current from E → B in the positive x-direction. We have shown in EM1-model:
⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛ −−⎟
⎠⎞⎜
⎝⎛ −=
−≡
′′′′
−
11
)modelEM1(
eeI
III
kTV CBq
kTV EBq
S
ECCCCT
(13)
Derivation of ISS
HO #8: ELEN 251 - SGP BJT Model Page 10S. Saha
At low level injection, p(x) ≅ NA(x) in the neutral base region where xE ≤ x ≤ xc. Then we can write (12) as:
⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛ −−⎟
⎠⎞⎜
⎝⎛ −=
′′′′−
∫11
)(
2
)levellow( eedxxN
nADqI kT
V CBqkT
V EBq
x
xA
n iCT
C
E
(14)
Since xE and xC depend on applied voltages, we define the fundamental constant, ISS @ VBE = 0 = VBC.
Where xE0 and xC0 are the values of xE and xC with applied zero voltages.
(15)∫
≡0
0
)(
2
C
E
x
xA
n iSS
dxxN
nADqI
Derivation of ISS - Base Charge, QB
HO #8: ELEN 251 - SGP BJT Model Page 11S. Saha
Again, (14) can be expressed as:
(16)
⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛ −−⎟
⎠⎞⎜
⎝⎛ −=
⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛ −−⎟
⎠⎞⎜
⎝⎛ −
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
=
′′′′
′′′′
∫
∫
∫
∫
∫
∫
11)(
)(
)(
11)(
)(
)(
0
0
0
0
0
0
0
0
2
2
eedxxp
dxxNqA
dxxNqA
nADq
eedxxNqA
dxxNqA
dxxp
nADqI
kTV CBq
kTV EBq
x
x
x
xA
x
xA
n i
kTV CBq
kTV EBq
x
xA
x
xA
x
x
n iCT
C
E
C
E
C
E
C
E
C
E
C
E
Derivation of ISS - Normalized Base Charge, qb
HO #8: ELEN 251 - SGP BJT Model Page 12S. Saha
Defining:
∫
∫
≡
≡′′
′′
0
0
)(
)(
0
)(
)(
C
E
CBC
EBE
x
xAB
Vx
VxB
dxxNqAQ
dxxpqAQ (17)
(18)
⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛ −−⎟
⎠⎞⎜
⎝⎛ −=
⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛ −−⎟
⎠⎞⎜
⎝⎛ −=
′′′′
′′′′
11
110
eeqI
eeQQII
kTV CBq
kTV EBq
b
SS
kTV CBq
kTV EBq
B
BSSCT
(19)
where qb ≡ QB/QB0 (20)
We get:
Saturation Current ISS - Summary
HO #8: ELEN 251 - SGP BJT Model Page 13S. Saha
⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛ −−⎟
⎠⎞⎜
⎝⎛ −=
′′′′ 11 eeqII kT
V CBqkT
V EBq
b
SSCT (19)
∫≡
0
0
)(
2
C
E
x
xA
n iSS
dxxN
nADqI
• Eq. (19) is the generalized expression for current source at allinjection levels.
• is a fundamental constant @ VBE = 0 = VBC.
• qb ≡ QB/QB0 is the normalized majority charge in the neutral base region and accurately models base width modulation:– QB0 = majority carrier charge @ VBE = 0 = VBC
– QB = majority carrier charge under applied voltages.
• qb is expressed as bias-dependent measurable parameters in SGP.
Derivation of Base Charge, QB
HO #8: ELEN 251 - SGP BJT Model Page 14S. Saha
• In order to evaluate qb, we first determine the components of QB.
• For the simplicity of QB analysis, we assume:– npn-BJT is in saturation, that is, VB'E' > 0 and VB'C' > 0, then
♦ minority carriers are injected into the Base both from Emitter and Collector
♦ from the charge neutrality, total increase in majority carriers in Base = total increase in minority concentration
– superposition of carriers in different regions hold♦ total excess majority carrier density = sum of the excess majority
carrier density due to each junction separately∴ excess majority carrier concentration in base = excess carriers
due to forward voltage [VB'E' + VB'C']
− depletion approximation holds.
Components of Base Charge, QB
HO #8: ELEN 251 - SGP BJT Model Page 15S. Saha
If pF(x) = majority carrier concentration in base @ VB'C' = 0 pR(x) = majority carrier concentration in base @ VB'E' = 0 NA(x) = base doping concentration
Then the excess majority carrier concentration in the base is given by: p'(x) = [pF(x) − NA(x)] + [pR(x) − NA(x)] (21)
xjE xjCxE0 xC0Base
Emitt
er
Col
lect
or
↓)(xpF
xxE(VB'E') xC(VB'C')
QEQC
QF
QR
↓)(xp
↓)(xpR
QB0
)(xN A
↑
Components of Base Charge, QB
HO #8: ELEN 251 - SGP BJT Model Page 16S. Saha
From (17), the total majority charge in the base is given by:
(22)∫∫
∫′′
′′
′′
′′
′′
′′
′+=
≡
)(
)(
)(
)(
)(
)(
)()(
)(
CBC
EBE
CBC
EBE
CBC
EBE
Vx
Vx
Vx
VxA
Vx
VxB
dxxpqAdxxqAN
dxxqApQ
equilibrium component excess component
The equilibrium component of QB can be split into three-components:
∫∫∫∫′′
′′
′′
′′
++=)(
)(
)(
)( 0
0
0
0
)()()()(CBC
C
C
E
E
EBE
CBC
EBE
Vx
xA
x
xA
x
VxA
Vx
VxA dxxqANdxxqANdxxqANdxxqAN
QE QB0 QC
Components of Base Charge, QB
HO #8: ELEN 251 - SGP BJT Model Page 17S. Saha
So that:
∫∫∫∫′′
′′
′′
′′
′+++=)(
)(
)(
)(
)()()()(0
0
0
0 CBC
EBE
CBC
C
C
E
E
EBE
Vx
Vx
Vx
xA
x
xA
x
VxAB dxxpqAdxxqANdxxqANdxxqANQ
QE QB0 QC
(23)∫′′
′′
′+++=∴)(
)(0 )(
CBC
EBE
Vx
VxCBEB dxxpqAQQQQ
xjE xjCxE0 xC0Base
Emitt
er
Col
lect
or
↓)(xpF
xxE(VB'E') xC(VB'C')
QEQC
QF
QR
↓)(xp
↓)(xpR
QB0
)(xN A
↑
Components of Base Charge, QB
HO #8: ELEN 251 - SGP BJT Model Page 18S. Saha
From (21) and (23),
(24)RFCBEB QQQQQQ ++++=∴ 0
QF
[ ]
[ ]∫
∫′′
′′
′′
′′
−+
−+++=
)(
)(
)(
)(0
)()(
)()(
CBC
EBE
CBC
EBE
Vx
VxAR
Vx
VxAFCBEB
dxxNxpqA
dxxNxpqAQQQQ
QR
xjE xjCxE0 xC0Base
Emitt
er
Col
lect
or
↓)(xpF
xxE(VB'E') xC(VB'C')
QEQC
QF
QR
↓)(xp
↓)(xpR
QB0
)(xN A
↑
Components of Base Charge, QB
HO #8: ELEN 251 - SGP BJT Model Page 19S. Saha
xjE xjCxE0 xC0Base
Emitt
er
Col
lect
or
↓)(xpF
xxE(VB'E') xC(VB'C')
QEQC
QF
QR
↓)(xp
↓)(xpR
QB0
)(xN A
↑
QB0 = charge in the neutral base under VB'E' = 0 = VB'C'. QE = increase in QB under VB'E' and is only a mathematical entity. QC = increase in QB under VB'C' and is only a mathematical entity. QF = excess majority charge in the biased-device with VB'C' = 0. It is
only a mathematical entity and important under high level injection. QR = excess majority charge in the biased-device with VB'E' = 0. It is
only a mathematical entity and important under high level injection.
Components of Normalized Base Charge, qb
HO #8: ELEN 251 - SGP BJT Model Page 20S. Saha
From Eq. (24), we get the normalized components of base charge:
rfceb
B
R
B
F
B
C
B
B
B
E
B
B
qqqqqQQ
++++=
++++=
10000
0
00
(25)rfceb qqqqq ++++=∴ 1
0
0
0
0
Where,
B
Rr
B
Ff
B
Cc
B
Ee
QQq
QQq
QQq
QQq
≡
≡
≡
≡
Evaluation of qb: Component qe
HO #8: ELEN 251 - SGP BJT Model Page 21S. Saha
We defined, QE = increase in the majority charge due to VB'E'.
(26)
∫
∫′′
′′
=
=∴
EB
EB
V
jEB
e
V
jEE
dVVCQ
q
dVVCQ
00
0
)(1and,
)(
(27)
jE
BV
jEEB
BB
B
B
EB
B
EBjEe
jEjEjE
CQ
dVVCV
QV
sVV
VQ
VCq
CCC
EB
0
0
0
0
)(1
:adefinedvoltageEarlyinversewhere
then,
ofvalueaverageconstantAssume,
=≡
=
≡=
=≡=
∫′′
′′
′′′′(28)
(29)
Evaluation of qe
HO #8: ELEN 251 - SGP BJT Model Page 22S. Saha
• VB in Eq. (29) models base-width modulation due to the variation of E-B junction depletion layer under VB'E'.
• VB due to VB'E' is the inverse of Early voltage due to VB'C'.
• In Eq (29), VB = constant ⇒ CjE = constant independent of VB'E'.
• Constant CjE is justified as:since QE << QB0
∴ qe << 1 and is not a dominant components of qb.
• VB = constant may cause large error in qe, especially, @ VB'E' > 0.
• The error in (29) due to qe for VB'E' > 0 can be eliminated by:– integrating CjE over the bias range– extracting VB from the slope of ln(IC) vs. qVB'E'/kT plot.
Evaluation of qe
HO #8: ELEN 251 - SGP BJT Model Page 23S. Saha
• In order to determine the effect of qe accurately, we set: qc = qr = qf = 0 ⇒ qb = 1 + qe,
• Then from (19) in the normal active region, we have:
• Thus, the slope of IC vs. VB'E' plot is given by:
( ) ⎟⎠⎞⎜
⎝⎛ −
+=
′′ 11 eq
II kTV EBq
e
SSC (30)
( )
( )( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=⎟⎠⎞
⎜⎝⎛
=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=
′′
′′′′
′′
′′
0
0
1)(
1ln1
1)(
1
Be
EBjEC
EBEB
C
C
Be
EBjEC
EB
C
QqVC
qkTI
kTqVd
ddVdI
IqkT
QqVC
qkT
kTqI
dVdI
(31)
Evaluation of qe
HO #8: ELEN 251 - SGP BJT Model Page 24S. Saha
• From (31), the slope of ln(IC) vs. qVB'E'/kT plot is given by:
• When CjE = constant ≡ <CjE>, then from (28), qe = VB'E'/VB; and from (29), <CjE> = QB0/VB
therefore, from (32):
(32)
( )
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=∴
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=≡
′′
′′
′′=′′
0
0
1)(
1
11
)(111
0
Be
EBjEE
Be
EBjEV
EB
C
CE
QqVC
qkT
n
QqVC
qkT
dVdI
IqkT
n CB
(33)( )⎟⎟
⎠
⎞⎜⎜⎝
⎛+
−≅
′′EBB
E
VVqkT
n11
1
Evaluation of qe: Summary
HO #8: ELEN 251 - SGP BJT Model Page 25S. Saha
• For most transistors:– qe = VB'E'/VB
– VB = QB0/<CjE> is the inverse of Early voltage = constant.
• VB = constant ⇒ CjE = constant independent of VB'E'.
• Constant CjE is justified:since QE << QB0
∴ qe << 1 and is not a dominant component of qb
• VB = constant may cause large error in qe, especially, @ VB'E' > 0.
• The error due to qe for VB'E' > 0 can be eliminated by:– integrating CjE over the bias range– extracting VB from the slope of ln(IC) vs. qVB'E'/kT plot.
Evaluation of qb: Component qc
HO #8: ELEN 251 - SGP BJT Model Page 26S. Saha
• qc models base-width modulation and therefore, Early voltage by changing depletion layer-width due to VB'C' at low current level.
• Using the procedure used for qe, we can show:
∫′′
=CBV
jCB
c dVVCQ
q00
)(1(34)
jC
BV
jECB
BA
A
A
CB
B
CBjCc
jCjCjC
CQ
dVVCV
QV
VV
VQ
VCq
CCC
CB
0
0
0
0
)(1
bydefinedvoltageEarlywhere
then,
ofvalueaverageconstantAssume,
=≡
=
≡=
=≡=
∫′′
′′
′′′′(35)
(36)
Evaluation of qc
HO #8: ELEN 251 - SGP BJT Model Page 27S. Saha
• VA in Eq. (36) models base-width modulation due to the variation of C-B junction depletion layer under VB'C'.
• In Eq (36), VA = constant ⇒ CjC = constant independent of VB'C'.• Constant CjC is justified in the normal active region when B-E
junction is reversed biased, that is, VC'B' < 1. • VA = constant may cause large error in qc, when B-C junction is
forward biased, that is, the device is in:– inverse region– saturation region.
• A more accurate expression for qc is required for accurate modeling of Early voltage in the:– inverse region– saturation region.
Effect of qc on Ic
HO #8: ELEN 251 - SGP BJT Model Page 28S. Saha
• The effect of qc on BJT device characteristics in the normal active region is finite output conductance g0.
• In order to determine g0, we set: qe = qr = qf = 0 ⇒ qb = 1 + qc, • Then neglecting bulk-ohmic resistors, we get:
• From (37), we can show:
( )
⎟⎠⎞⎜
⎝⎛ −
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
⎟⎠⎞⎜
⎝⎛ −
+=
11
11
e
VV
I
eq
II
kTV BEq
A
BC
SS
kTV BEq
c
SSC
(37)
A
C
BECE
C
VI
VdVdIg )0(
constant0 ≅
== (38)
Evaluation of qb: Component qf
HO #8: ELEN 251 - SGP BJT Model Page 29S. Saha
• qf can be considered as the normalized excess carrier density in the base with E-B bias VB'E' and models high level injection.
• From charge neutrality: total excess majority carriers = total excess minority carriers.
Therefore, for an npn transistor with |VB'E'| > 0; VB'C' = 0
• QF in (39) (denoted by QB) in EM2 model is given by: QF = QB
EM2 = τBICC (40)
[ ] ∫∫ ⎥⎦
⎤⎢⎣
⎡−=−=
C
E
C
E
x
x A
iF
x
xAFF dx
xNnxnqAdxxNxpqAQ
)()()()(
2
(39)
⎟⎠⎞⎜
⎝⎛ −==∴
′′ 100
eqI
QQIq kT
V EBq
b
SS
B
B
B
CCBf
ττ(41)
Evaluation of qb: Component qr
HO #8: ELEN 251 - SGP BJT Model Page 30S. Saha
• qr can be considered as the normalized excess carrier density in the base with C-B bias VB'C' and models high level injection.
• From charge neutrality: total excess majority carriers = total excess minority carriers.
Therefore, for an npn transistor with |VB'C'| > 0; VB'E' = 0
• QR in (42) (denoted by QBR) in EM2 model is given by: QR = QBR
EM2 = τBRIEC (43)
[ ] ∫∫ ⎥⎦
⎤⎢⎣
⎡−=−=
C
E
C
E
x
x A
iR
x
xARR dx
xNnxnqAdxxNxpqAQ
)()()()(
2
(42)
⎟⎠⎞⎜
⎝⎛ −==∴
′′1
00eq
IQQ
Iq kTV CBq
b
SS
B
BR
B
ECBRr
ττ(44)
Solution for qb
HO #8: ELEN 251 - SGP BJT Model Page 31S. Saha
• We know:
• Then substituting (28), (35), (41), (44) in (25) we get:
Where τBdc ≡ modified forward base transit time due to mobile charge in
the depletion region (i.e., without depletion approximation).
τBRdc ≡ modified reverse base transit time due to mobile charge in the depletion region (i.e., without depletion approximation).
(25)rfceb qqqqq ++++= 1
⎟⎠⎞⎜
⎝⎛ −+
⎟⎠⎞⎜
⎝⎛ −+++=
′′
′′′′′′
1
11
0
0
eqI
Q
eqI
QVV
VVq
kTV CBq
b
SS
B
BR
kTV EBq
b
SS
B
B
A
CB
B
EBb
dc
dc
τ
τ
(45)q1
q2/qb
Solution for qb
HO #8: ELEN 251 - SGP BJT Model Page 32S. Saha
We can simplify (45) to get:
Here
Where q1 in (47) models the base width modulation q2 in (48) models the effect of high level injection.
A
CB
B
EB
VV
VVq ′′′′ ++≡ 11 (47)
(46)b
b qqqq 2
1 +=
02
11
B
kTV CBq
SSBRkTV EBq
SSB
Q
eIeIq
dcdc⎟⎠⎞⎜
⎝⎛ −+⎟
⎠⎞⎜
⎝⎛ −
≡
′′′′ττ
(48)
Solution for qb
HO #8: ELEN 251 - SGP BJT Model Page 33S. Saha
• From (45) we get, qb
2 − qbq1 − q2 = 0 (49)
since qb > 0, (50) is obtained taking the positive solution only.
• Equation (50):– offers solution for IC
– defines injection level♦ if q2 << q1
2/4, qb = q1, then qf = qr = 0 ⇒ low level injection♦ if q2 >> q1
2/4 ⇒ high level injection (51) ∴ qb = (q2)1/2 (52)
(50)2
211
22qqqqb +⎟
⎠⎞
⎜⎝⎛+=∴
Solution for qb: High Level Injection
HO #8: ELEN 251 - SGP BJT Model Page 34S. Saha
For simplicity, we assume VB'C' = 0 (i.e., qr = 0). Then from (48) and (52), for high level injection in the forward active region:
∴ Substituting for qb in (19), we get for high level injection @ VB'C'= 0 and VB'E' >> kT/q:
(53)kTqV
B
SSB
kTqV
B
SSBb
EBdc
EBdc
eQ
I
eQ
Iqq
2
0
02
′′
′′
=
≅=
τ
τ
kTqV
B
SSB
kTqV
B
SSB
kTV EBq
SS
CTC
EB
dcEB
dc
eIQ
eQ
I
eIII 20
2
0
1 ′′
′′≅
⎟⎠⎞⎜
⎝⎛ −
=≅
′′
ττ(54)
High Level Injection: Knee Current, IK
HO #8: ELEN 251 - SGP BJT Model Page 35S. Saha
• For low level injection, if we assume, qe = qc = 0, then qb = 1:
• From (19), we get for low level injection @ VB'C' = 0 and VB'E' >> kT/q:
• The intersection of high current and low current asymptote is given by (IK,VK) where IC(high-level) = IC(low-level).
• Therefore, from (54) and (55):
(55)eII kTV EBq
SSlevellowC′′
− ≅)(
(56)
kTqV
SSK
kTqV
B
SSBK
K
K
dc
eII
eIQI
=
⎟⎟⎠
⎞⎜⎜⎝
⎛= 20
τ
(57)
High Level Injection: Knee Current, IK
HO #8: ELEN 251 - SGP BJT Model Page 36S. Saha
• From (56) and (57), we get:
• Similarly, for inverse region:
• Model parameters: (VA, VB, ISS, IK, IKR) are extracted from – ln(IC) vs. VB'E' plot– ln(IE) vs. VB'C' plot.
(58)dcB
BK
QIτ
0= ln(IC)
qVB'E'/kT
VBC = 0
( )KI
slope = 1
slope = 1/2
qVK/kT
ln
(59)dcBR
BKR
QIτ
0=
SGP Model: Summary
HO #8: ELEN 251 - SGP BJT Model Page 37S. Saha
(47)
⎟⎠⎞⎜
⎝⎛ −+⎟
⎠⎞⎜
⎝⎛ −=
⎟⎠⎞⎜
⎝⎛ −+⎟
⎠⎞⎜
⎝⎛ −≡
′′′′
′′′′
11
1100
2
eII
eII
eIQ
eIQ
q
kTV CBq
KR
SSkT
V EBq
K
SS
kTV CBq
SSB
BRkT
V EBq
SSB
B dcdcττ
(60)
From (58) and (59)
(50)2
211
22qqqqb +⎟
⎠⎞
⎜⎝⎛+=∴
A
CB
B
EB
VV
VVq ′′′′ ++≡11
where
⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛ −−⎟
⎠⎞⎜
⎝⎛ −=
′′′′ 11 eeqII kT
V CBqkT
V EBq
b
SSCT (19)
SGP Model: Summary
HO #8: ELEN 251 - SGP BJT Model Page 38S. Saha
• q1 models base-width modulation• q2 models high level injection
– the condition for high level injection:
4
21
2qq ≥
• The model parameters: (VA, VB, ISS, IK, IKR) are extracted from – ln(IC) vs. VB'E' plot in the normal mode of BJT operation– ln(IE) vs. VB'C' plot in the inverse mode of BJT operation– IC vs. VCE characteristics in the normal mode of BJT operation– IE vs. VEC characteristics in the inverse mode of BJT operation.
• A parameter, B is used to model base widening (base push-out) effect at high currents.
Limitations of SGP
HO #8: ELEN 251 - SGP BJT Model Page 39S. Saha
Trench-Isolated double-poly NPN• Inability to model collector resistance modulation (quasi-saturation).
• Inability to consider parasitic substrate transistor action.
• Inaccurate Early effect formulation to model output resistance go in narrow-base BJTs.
• Its inability to consider fixed (i.e. bias independent) dielectric capacitances of double poly BJTs for accurate capacitance modeling.
Features of VBIC Model
HO #8: ELEN 251 - SGP BJT Model Page 40S. Saha
• Vertical Bipolar Inter-Company (VBIC) model was developed to address the limitations of SGP model.
• Features of VBIC (http://www.designers-guide.com/VBIC/) model: – improved Early effect (go) modeling– quasi-saturation modeling– parasitic substrate transistor modeling– parasitic fixed (oxide) capacitance modeling– avalanche multiplication modeling– improved temperature dependence modeling– de-coupling of base and collector currents– electrothermal (self-heating) modeling– improved heterojunction bipolar transistor (HBT) modeling.
Features of VBIC Model
HO #8: ELEN 251 - SGP BJT Model Page 41S. Saha
• VBIC was developed to model:– vertical npn-BJTs– can, also, be used to model vertical pnp-BJTs– can model junction-isolated as well as trench-isolated BJTs.
Junction isolated diffused npn-BJT
VBIC Model - Equivalent Network
HO #8: ELEN 251 - SGP BJT Model Page 42S. Saha
Home Work 2: Due April 14, 2005
HO #8: ELEN 251 - SGP BJT Model Page 43S. Saha
1) Consider an npn-BJT in the normal active mode of operation.(a) Schematically show the components of base charge responsible for base-width
modulation. Label your plot, show the integration limits to compute charge components, and state your assumptions.
(b) Write down the expression for the normalized base-charge responsible for base-width modulation.
(c) Write down the expression for SGP-model current source for base-width modulation in terms of forward and reverse Early voltages only.
(d) Write down the expression for current source without the base-width modulation. Assume low-level injection.
(e) Show the effect of Eq. in part (c) and (d) on IC vs. VCE plots. Explain.
2) Describe the procedure to extract the following model parameters of an npn-BJT:
(a) transistor saturation current IS, (b) DC current gain βFM, (c) bulk collector ohmic resistor rc', (d) forward knee current IKF.
Home Work 2: Due April 14, 2005
HO #8: ELEN 251 - SGP BJT Model Page 44S. Saha
3) For an accurate modeling of BJTs, VBIC-BJT-model was developed to include the parasitic effects. In this problem, you will modify SGP-model for vertical npn-BJTs to include the parasitic vertical pnp-BJT for the advanced BJT structure shown below.
(a) Draw the SGP-equivalent network for the parasitic pnp-BJT.
(b) Use block diagrams to show the effect of parasitic pnp device in the intrinsic npn device model.
(c) Draw the equivalent network of the modified model in part (b) showing all components (ohmic resistors, charge storage elements, etc.).
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