Transcript Chapter 4 Static Characteristics

Chapter 4
4.1
4.2
4.3
4.4
Static Characteristics
Intuitive Picture
Collector Current Density and Current Gain
Output Conductance
Equivalent Circuit Model
4.1 Intuitive Picture




Fig 4.1:Energy band diagram for a Si BJT and gradedbase SiGe HBT,both biased in forward active
mode in low-injection.
An ideal,graded-base SiGe HBT with
constant doping in the emitter, base,
and collector region.
Ge content is linear graded from 0%
near the metallurgical emitter-base
junction to some maximum value of
Ge content near the metallurgical
collector-base junction,and then rapidly
ramped back down to 0% Ge.
Observe in Fig 4.1 that a Ge-induced
reduction in base bandgap occurs at
EB edge of the quasi-neutral base, and
at the CB edge of the quasi-neutral base.
This grading of the Ge across the neutral
base induces a built-in quasi-drift field in
the neutral base that will impact minority
transport.




Fig 4.2:Illustration of bandgap changes induced by
the introduction of Ge into the base region
of an n-p-n SiGe HBT.
The Fermi level must realign itself such
that it is fixed in energy to its previous
(Si) value,and further, that it must be
constant (flat) if the system is in
equilibrium.
This grading of the Ge across the neutral
base induces a built-in quasi-drift field in
the neutral base that will impact minority
carrier transport
Induced drift field will positively influence
the minority electron transport through
the base.The injected electrons diffuse
across the base, and sweep into the
electric field of the CB junction ,yielding a
useful collector current.
The density of back-injected holes will be
small compared to the forward-injected
electron density ,and hence a finite
current gain ß∝n/p results.


The potential barrier to injection of electron from emitter into the base is
decrease.Intuitively, this will ylied expontentially more electron injection
for the same applied VBE ,tanslating into higher collector current and higher
current gain.We can trade the higher gain induced by the Ge band offset for
a higher base doping level.
The present of a finite Ge content in the CB junction will positively influence the
output conductance of the transistor,yielding higher Early voltage.
4.2 Collector Current Density and
Current Gain
4.2.1
4.2.2
4.2.3
4.2.4
Jc in SiGe HBT
Relevant Approximations
Other SiGe Profile Shapes
Implication and Optimization Issue
for ß
Jc in SiGe HBT
Fig 4.3 Schematic doping and Ge profile used
in the derivation.
Fig 4.4 Schematic base bandgap in a linearly
graded SiGe HBT.
Generalized Moll-Ross collector density relation:

Jc 
q VBE
 1)
kT
pb ( x )d x
2
Dn b ( x )nib
( x)
q( e
Wb
0
With the condition:
nib2 ( x)  ( Nc Nv )SiGe( x)e Egb( x) / kT
app
Egb ( x )  Egbo  Egb
 [Eg ,Ge (0)  Eg ,Ge (Wb )]
 Eg 0 / kT
ni20  Nc Nve

( N c N v ) SiGe
1
( N c N v ) Si
x
 Eg ,Ge (0)
Wb
We can get the expression
Jc 
~ qVBE / kT 1 ~ 2 Egbapp / kT Eg ,Ge ( 0) / kT
qDnb (e
) nioe
e
N

ab
Wb

0
e
[ Eg ,Ge (Wb )  Eg ,Ge ( 0 )]( x / WbkT )
With the condition:
dx
0 n ( x)
dx
Dnb ( x )nib2 ( x )
Wb
~
Dnb 

Wb

0
2
ib
Eg ,Ge ( grade)  Eg ,Ge (Wb )  Eg ,Ge (0)
dx
We obtain the final expression
~ qVBE / kT 1 ~ 2 Egbapp / kT Eg ,Ge ( 0) / kT
qDnb (e
) nioe
e
Jc , SiGe 
WbkT
 E
( grade ) / kT
N ab
{1  e g ,Ge
}
Eg ,Ge ( grade)
where
~
(
D
)
~  nb SiGe
( Dnb ) Si
Fig 4.5 Comparison of
current-voltage
characteristic of
a comparably
constructed
SiGe HBT and
Si BJT.
The current gain issue
 SiGe Jc, SiGe

 SiGe
Jc, Si
E g ,Ge ( 0 ) / kT
~~Eg ,Ge ( grade) / kTe
 SiGe
 {
 E
( grade ) / kT
 SiGe v
1 e
BE
g ,Ge
1000/T
Fig 4.6 Measure and calculated current
gain ratio as a function of
reciprocal temperature for
a comparably constructed
SiGe HBT and Si BJT.
}
Relevant Approximations
First:we can assume that △Eg,Ge(grade)kT
qDnb qVBE / kT 1 2 Egbapp / kT ~~Eg ,Ge ( grade) Eg ,Ge ( 0) / kT
Jc, SiGe   (e
)ni 0e
{
e
}
N abWb
kT
Second:weak Ge grading
Jc, SiG 
~~
app
qDnb qVBE / kT
2 Egb / kT   Eg ,Ge (0)
(e
 1)ni 0e
{
}

N abWb
kT
Fig 4.7 Theoretical calculation of the current gain ratio
as a function of Ge profile shape.
Other SiGe Profile Shapes
Trapezoids profile
Eg ,Ge ( 0)Eg ,Ge ( grade )(
Eg ,Ge ( x)  {Eg ,Ge (Wb )
x
)
xt
~ app
[ E g ,Ge ( grade )( )]
2 E gb / kT E g ,Ge ( 0 ) / kT

xt / kT

n
e
e
e
2
i0
nib ( x )   E~app / kT E (W ) / kT
ni20e gb
e g ,Ge b

Fig 4.8 schematic representation of
x
the hybrid Ge trapezoidal
profile.
Jc, SiGe

Jc, Si
kT
Eg ,Ge ( grade)
~~e
 {1   (1 
Eg ,Ge ( 0) / kT
kT
E ( grade) / kT
)}e g ,Ge
Eg ,Ge ( grade)
Fig 4.9 current gain ratio as a
function of reciprocal
temperature for varying
ξ values.Noted that the
integrated Ge content
is not fixed in this case.
Fig 4.10 current gain ratio as a
function of reciprocal
temperature for varying
ξ values.Noted that the
integrated Ge content
is held fixed in this case.
Implication and Optimization
issue for ß


The present of Ge in the base region will enhance JC at fixed
VBE over a comparably constructed Si BJT.
The JC enhancement depends exponentially ob the EB
boundary value of Ge-induced band offset,and linearly on the
Ge grading across the base.

A box Ge profile is better for current gain enhancement than a
triangular Ge profile.

The Ge-induced JC enhancement is thermally Activated
(exponentially dependent on the reciprocal temperature),and
thus cooling will produce a strong magnification of the
enhancement.
4.3 Output Conductance
4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
4.3.6
VA Trade-offs in Si BJTs
VA in SiGe HBTs
Relevant Approximations
Current Gain-Early Voltage Product
Other SiGe Profile Shapes
Implication and Optimization Issue
for VA and ßVA
VA Trade-offs in Si BJTs

Basically Definition
JC
JC
Wb 1
VA  JC (0){
}1  VBE  JC (0){
}
VCB VBE
Wb VBE VCB

And than
Wb
VA,Si


0
pb ( x )dx Wb
Qb(0)
{
}
pb (Wb ) VCB
Ccb
VA,Si  Wb(0){
Wb 1
}
VCB
Where
{
Where
2
N dc
Wb  Wm  ( )(bi  VCB ){ 
} Where
q
N ab ( N dc  N ab )
Qb(0) is the total base charge at VCB=0V
,Ccb is the base depletion capacitance.
Wb(0) is the neutral base width at VCB=0
{
Wm is the metallurgical base width
bi is the CB junction built-in voltage
Increasing collector concentration degrades VA.
Increasing base concentration increasing VA.
Fig 4.11 Schematic representation of
the Early effect.
Increasing base concentration degrades ß.
Increasing base concentration degrades fT due
to the reduction in the minority electron mobility.
It is inherently difficult to simultaneous obtain
high VA,high ß,and high fT.
Fig 4.12 Schematic representation
impact of the Early effect.
VA in SiGe HBTs
 Basically Definition for HBT
Wb
VA, SiGe 

0

VCB
pb( x )dx
Dnb( x )nib2 ( x )
pb( x )dx
Dnb( x )nib2 ( x )

 And than
If the nib is position-independent,there is no
VA enhancement due to Ge .If nib is positiondependent ,VA will depend exponentially on
the difference in bandgap between x=Wb and
that region in the base where nib is smallest.
Dnb(Wb)nib2 (Wb) Wb pb( x)dx
Wb 1
VA, SiG  {
}
[
0 Dnb( x)nib2 ( x) VCB ]
pb(Wb)
E
( grade) / kT
VA, SiGe
e g ,Ge
Eg ,Ge ( grade) / kT
  e
[
]
VA, Si VBE
Eg , Ge( grade) / kT
+
Fig 4.13 Dependence of VA on both Ndc
and Nab for a Si BJT.
Fig 4.14 Measured and calculated
Early voltage ratio as a
function of reciprocal
Temperature for a
comparably constructed
SiGe HBT and Si BJT.
Relevant Approximations
we can assume that △Eg,Ge(grade) ≫ kT
E
( grade) / kT
VA, SiGe
e g ,Ge

VA, Si VBE Eg , Ge( grade) / kT
we can assume that △Eg,Ge(grade)≪kT
Fig 4.15 Theoretical calculations
f
of the VA ratio and ßVA
ratio as a function of Ge
profile shape.
VA, SiGe
1
VA, Si VBE
Current Gain-Early Voltage
Product
 One conventionally defines a figure-of-merit
for analog circuit design:the so-call “ßVA”
product.In a conventional Si BJT,it is not
favorably impacted by conventional scaling.
For a SiGe HBT,however,both ß and VA are
decoupled from the base profile,and can be
Independent tuned by changing the Ge
profile shape.pppppppppppppppppppp
Fig 4.16
Current gain-Early voltage
product ratio as a function
for a comparable SiGe
HBT And Si BJT.
VA, SiGe ~~ E
 e
VA, Si
g ,Ge
( 0 ) / kT E g ,Ge ( grade) / kT
e
Other SiGe Profile Shapes
 Trapezoids profile
E
VA, SiGe
 (e g ,Ge
 1)
 1 
VA, SiGe VBE
E g ,Ge ( grade) / kT
( grade) / kT
Implication and Optimization
issue for VA and ßVA






Unlike JC ,only the presence of a large Ge content at the CB side of the neutral
base than at the EB side of the neutral base will enhance VA over a comparably
costructed Si BJT.
This VA enhancement depends exponentially on the Ge grading cross the base.
A triangular Ge profile is better for Early voltage enhancement than a box Ge
profile is,everything else being equal.
The Ge-induced VA enhancement is thermally activated,and thus cooling will
produce a strong magnification of the enhancement.
Putting any Ge into base region of a device will exponentially enhance this key
analog figure-of-merit,a highly favorable scenario given the discussion above of
inherent difficulties of achieving high ßVA in a Si BJT.
A reasonable compromise Ge profile design that balance the dc optimization
needs of ß,VA,and ßVA would be a Ge trapezoid,with a small Ge content at the
EB junction ,and a large Ge content at the CB junction.
4.4 Equivalent Circuit Model
4.4.1 Basic Ebers-Moll Model
4.4.2 Transport Version
4.4.3 Small-Signal Equivalent Circuit
Model
Basic Ebers-Moll Model

A fundamental assumption in this modal
is that the overall transistor operation can
be viewed as a superposition of both the
forward and the reverse mode operation.
I F  I F 0 ( e qVBE / kT  1)
I R  I R 0 ( e qVBC / kT  1)
Fig 18 The basic Ebers-Moll for a bipolar transisitor.
Transport Version
foward :
I CF  I S ( e qVBE / kT  1)
I BF 
I CC
F
inverse
I CR  I S ( e qVBC / kT  1)
I BR 
I CR
R
Re ciprocity
I S   F I F 0   R I R0
Small-Signal Equivalent
Circuit Model
4.5 Avalanche Multiplication
4.5.1 Carrier Transport and Terminal
Current
4.5.2 Forced-VBE Measurement of M-1
4.5.3 Forced-I E Measurement of M-1
4.5.4 Effects of Self-Heating
4.5.5 Impact of Current Density
4.5.6 Si Versus SiGe
Carrier Transport and Terminal
Current
Simple analysis shows that the minimum threshold
energy for impact ionization is 1.5Eg
M
I n ,out
I n ,in
I B  I p,e  ( M  1) I n,in
Fig 21 The avalanche multiplication process in a bipolar
transistor under normal operation. ……….
Forced-VBE Measurement of M-1
( M  1) I n,in  I B
I B  I B (VCB  0)  I B (VCB )
FEarly 
I n,in (VCB )
I (V )  I B
 C CB
I n,in (VCB0 )
IC (VCB  0)
I n.in  I C (VCB )  I B
M 1 
Fig 4.22 Force VBE setup for M-1 measurement.
I B
I C (VCB )  I B
FEarly 
I n,in (VCB )
I (V )  I B
 C CB
I n,in (VCB0 )
IC (VCB  0)
Forced-I E Measurement of M-1
At high JC and VCB,self-heating
and thermal runaway occur. .
I n,in  I E  I B (VBE ) VCB0
M 1 
IC
I E  I B (VBE ) V
1
CB 0
FEarly 
Fig 4.22 Force IE setup for M-1 measurement.
I E  I B (VEB ) V 0
I n ,in (VCB )
CB

I n ,in (VCB  0) I C (V BE ) VCB  0
IC (V BE )VCB  IC (V BE )VCB0  FEARLY  M
I B (V BE )VCB  I B (V BE )VCB0  Ic (V BE )VCB0  FEARLY  ( M  1)
V
CB 0
 FEarly  ( M  1)  1
Effects of Self-Heating
M
IC
I E  I B (VBE ) V
CB

IC
I E (1  1 /  (VBE ) V
0,T (VCB )
CB
0,T (VCB )
IC

(1  1 /  (VBE ) V 0,T (VCB )
CB
IE
Impact of Current Density
Si Versus SiGe
Fig 27 Simulated depth profile of carrier
temperature at VBE=0.7V and
VCB=5.
Fig 28 M-1 versus VCB comparison between
SiGe HBTs with different Ge profile
4.6 Breakdown Voltage
4.6.1 BVCBO
4.6.2 BVCEO
4.6.3 Circuit Implication
BVCBO
M
Fig 29 An example of M-1 versus VCB fitting
using the Miller equation.
.
1
1  (VCB / BVCBO )m
BVCEO
I E  IC
I n,in (1  1 /  )  MI n,in  I CBO
I n.in 
Fig 4.30 An illustration of the avalanche multiplication
process with an open base.
.
I CBO
1 /   ( M  1)
Circuit Implication