NOISE MODELLING OF SILICON GERMANIUM HETEROJUNCTION BIPOLAR TRANSISTORS AT MILLIMETRE-WAVE FREQUENCIES BY KENNETH HOI KAN Y AU A THESIS SUBMITTED IN CONFORMITY WITH THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE GRADUATE DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING UNIVERSITY OF TORONTO c KENNETH HOI KAN Y AU, 2006
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NOISE MODELLING OF SILICON GERMANIUMHETEROJUNCTION BIPOLAR TRANSISTORS AT
MILLIMETRE-WAVE FREQUENCIES
BY
KENNETH HOI KAN YAU
A THESIS SUBMITTED IN CONFORMITY WITH THE REQUIREMENTS
FOR THE DEGREE OFMASTER OFAPPLIED SCIENCE
GRADUATE DEPARTMENT OFELECTRICAL AND COMPUTER ENGINEERING
it can be seen that the variation in〈i∗nBinC〉 with frequency is mainly captured in the phase
of gm (ω). Recently published compact shot noise models introduced the noise transit time
parameterτn and expressed〈i∗nBinC〉 throughτn as [20]
〈i∗nBinC〉 = 2qIC [exp (−jωτn) − 1] , (4.58)
which is used in this thesis. The above equation assumes that|gm (ω)| = gm0 and the phase
of gm (ω) is given byωτn. Applicable only in low to moderate injection levels,τn is assumed
to be bias and frequency independent. At present,τn is extracted by fitting the noise model
1Low to moderate of injection is also assumed in the shot noiseequations for⟨i2nB
⟩and
⟨i2nC
⟩presented as
equations (4.1) and (4.2) which are repeated in (4.47) and (4.48).
4.4 Noise Transit Time 31
equations to the measured noise parameters as in [6]. This prevents noise parameters that
account for noise correlation be calculated solely fromS-parameters andτn has to be extracted
by fitting to measured high frequency noise parameters, which are more noisy compared to
S-parameters measured at the same frequency.
High injection effects are assumed to be insignificant up to the peakfT/fMAX bias. It is
proposed thatτn be extracted fromS-parameters as follows. The series resistancesRBX (IC),
RE andRC are extracted and de-embedded to obtain the intrinsicy-parameters using equa-
tions (B.22)–(B.25). It is assumed that the resulting intrinsic transistor can be adequately de-
scribed by a generalizedπ network as shown in Fig. 4.8. The general impedancesZ1, Z2 and
Z3 can be complex and frequency dependent. It can be shown that
yINT
21= gm0 exp (−jωτn) − 1
Z2
(4.59)
yINT
12= − 1
Z2
, (4.60)
whereyINTij are they-parameters of theπ-network shown in Fig. 4.8. The high frequency
transconductance is extracted asyINT21
− yINT12
and the noise transit time is extracted from the
y-parameters corresponding to peakfMAX bias as
τn = − ∂
∂ωphase
(yINT
21− yINT
12
)(4.61)
in the high frequency domain where the phase of the transconductance is linear. At first glance,
the proposed method has the same problem as equation (4.56) as τn is computed the phase
of gm whose high pass characteristic is problematic in practice.However, sinceτn is now
assumed to be frequency and bias independent, linear regression is performed in the domain
Z1
Z2
Z3vbe gm0e−jωτnvbe
+
−
Fig. 4.8: A generalized π network for the extraction of gm0 exp (−jωτn).
32 SiGe HBT Noise Modelling at Millimetre-Wave Frequencies
where the phase is linear with frequency andτn is then extracted as the negative of the slope of
the regression line. Measurement error is minimized through the linear regression technique.
5 Verification by Device
Simulations
V ERIFICATION of the derived noise parameter equations is first provided by device sim-
ulations. A 2-D SiGe HBT structure is constructed using a TCAD process simulator. Its
y-parameters are obtained from a device simulator. The derived noise equations and noise tran-
sit time extraction technique are applied to the simulated data to calculate the noise parameters.
These values are then compared with those that are directly calculated by the noise simulation
module in the device simulator using the impedance field method.
Note that the 2-D SiGe HBT simulated does not correspond exactly to fabricated device
that will be discussed in Chapters 6 and 7. This is because theexact geometry and doping
profiles of a SiGe HBT are considered proprietary information and hence unavailable. However
sometimes partial information such as the integrated dose in the base and germanium profiles
are available in the literature, such as in [23, 24]. Based onthese publications, the 2-D SiGe
HBT structure is adjusted to have similarfT/fMAX characteristics as the fabricated device.
5.1. SiGe HBT Process Simulation
This section describes the simulation of the process flow of a160 GHz SiGe HBT using Athena,
the process simulator within the Silvaco TCAD simulators suite. The process started with the
formation of then+ buried layer by ion implantation of Arsenic ions at50 keV at a dose of
7×17 cm−3. This was followed by a diffusion drive-in at1100C for two minutes. An Arsenic
doped,0.1 µm thick collector epitaxial layer was formed next, followed by the formation of the
shallow trench isolation (STI) regions and then+ collector reach through by ion implantation
using Arsenic. Geometrical etching was used to form the STI regions, which implied that the
silicon etched was merely removed from the structure [25]. In addition, geometrical etching,
which is simulated as a low-temperature process, ignores impurity redistribution [25]. Note
that geometrical etching was used to simulate all etching steps in this SiGe HBT fabrication
process.
33
34 Verification by Device Simulations
The base region was fabricated next. One common way of forming the base region in
commercial processes is as follows [26]. The extrinsicp+ base polysilicon and appropriate etch
stop layers are deposited and the emitter window is opened. Next, the selectively implanted
collector (SIC) is fabricated using the extrinsic base polysilicon as a self-aligned mask. The
purpose of the SIC is described in detail in Appendix C. An underetch is performed to remove
a thin layer of silicon under the emitter window. Finally, this void is filled by the intrinsic SiGe
base, which is in-situ doped and formed by selective epitaxy. Fig. 5.1 summarizes this SiGe
HBT fabrication method.
Fig. 5.1: SiGe HBT Process Flow Employing Selective SiGe Base Epitaxy: (a) emitterwindow, SIC implantation, (b) underetch, (c) selective SiGe base epitaxy and (d) emitterformation [26].
The problem in implementing this sequence in the Athena simulator is mainly in the SiGe
base epitaxial step to fill the void left by the underetch. In Athena, all deposition steps are
conformal, meaning that the same thickness is deposited on all surfaces, including the regions
underneath the base poly that are exposed by the underetch step. This often causes convergence
problems in subsequent simulations.
Therefore, the base processing steps were modified as follow. After the collector reach
through and the STI regions were formed, the SIC was implanted using an oxide layer as a
mask. The intrinsic SiGe base was then deposited, followed by the deposition of the extrin-
sic base polysilicon and the opening of the emitter window. This process flow avoided an
5.1 SiGe HBT Process Simulation 35
underetching step and hence eliminated the associated numerical convergence problems. The
modified flow is summarized in Fig. 5.2.
Fig. 5.2: Modified SiGe Process Flow: (a) SIC implantation, (b) SiGe Base, (c) p+-polyextrinsic base and (d) emitter formation.
The intrinsic base had a thickness of 20 nm and a uniform borondoping profile of1 ×19 cm−3. The germanium content was graded from 10% at the emitter side to 30% at the
collector side. To reduce boron out diffusion during subsequent thermal cycles, a boxed carbon
profile of 2 × 20 cm−3, which corresponds to 0.4% concentration, was also incorporated into
the SiGe base. The base doping, carbon content, germanium profile and thickness are similar to
values published in recent literature on SiGe HBTs that haveafT of about 230 GHz [23,24]. To
account for the out diffusion of boron in the presence of carbon in the SiGe layer, an empirical
boron diffusion model provided by Athena [25] was used in thesimulations.
The emitter polysilicon was deposited followed by a thermalanneal. Out diffusion of
arsenic from the emitter polysilicon formed the mono-emitter region. Note that in latest fabri-
cation processes, both the emitter and base polysilicon layers are silicided with titanium, cobalt
or nickel to reduce their sheet resistivity [27]. Although the silicidation process can be simu-
lated in Athena, the device simulator Atlas regards silicided layers as metals unless the sheet
resistance or resistivity of the layers are specified explicitly. Since the sheet resistance of these
36 Verification by Device Simulations
polysilicon layers is neither measured experimentally norprovided by the foundry that fabri-
cated the 160 GHz SiGe HBTs used to experimentally verify thederived noise equations and
parameter extraction technique, therefore, all polysilicon layers simulated were not silicided to
avoid overly optimistic results. The cross section of the constructed device is shown in Fig. 5.3.
The doping profile and the germanium content along a line through the centre of the emitter is
whereZHBT is thez-parameter matrix of the SiGe HBT obtained by de-embedding the para-
sitic elements.
Although [33] is simple and easy to implement, it is not without its shortcomings. The
most important one lies in its use of lumped elements to modeland de-embed the parasitics. In
the millimetre-wave regime, the lumped element approximation starts to fail, directly affecting
the validity of [33]. Advances have been made such as in [38] and [39] to attempt to resolve
this problem. However, the required de-embedding structures were not included in this work.
6.3 Modelling of Parasitic Elements 55
6.3.2. Noise Parameter De-embedding
In contrast toS-parameters, noise parameters are more complicated to de-embed. Rather than
de-embedding the measured data, the parasitics are embedded into the noise parameter equa-
tions and the results are compared with the experimental data. This is opposite to the method
employed in theS-parameter experiment where the parasitics are mathematically removed
from the measured values. The approach taken is a direct application of [10, 11]. The follow-
ing shall first present a general methodology to embed parasitics. A specific case applicable to
this work is presented afterwards.
For noise parameter analysis purposes, the SiGe HBT test structures are modelled as shown
in Fig. 6.7.〈v2n〉 and〈i2n〉, which may be calculated from equations (4.42)–(4.44), arethe power
SiGe HBT OutputNetwork
Input Network
[ASiGe]
〈v2n〉
〈i2n〉〈i2IN1〉 〈i2IN2
〉 〈i2OUT1〉 〈i2OUT2
〉[YIN ] [YOUT ]
︸ ︷︷ ︸
CA1−2, A1−2
︸ ︷︷ ︸
CA, A
Fig. 6.7: A model of SiGe HBT test structures for noise parameter analysis
spectral densities of the input referred noise voltage and current, respectively, of the SiGe HBT.
Represented by theiry-parameter matrix,YIN andYOUT , respectively, the input and output
network are assumed to be passive and non-interacting except through the SiGe HBT. The noise
properties of each of the two networks are modelled in admittance formalism by two noise
current sources. The case where the input and output networks interact is solved in [38,40] by
performing four-port network analysis. In the following, the two equations which serve as the
foundation of the embedding technique are presented.
The first equation relates the noise correlation matrix of a passive network to its two-port
network parameters. Derived from thermodynamics, the admittance correlation matrix whose
elements are the ensemble averages of the terminal noise currents of a passive network is given
by [41]
CY = kBT(Y + Y
†), (6.11)
wherekB is the Boltzmann’s constant,T is the absolute temperature in kelvin,Y is they-
parameter matrix of the passive two port andY† is the adjoint ofY. The second equation gives
56 Experimental Procedure and De-embedding Techniques
the noise correlation matrix in the chain representation ofa cascade of two noisy two-port
networks as [10]
CA = A1CA2A†1+ CA1 (6.12)
whereCA is the overall chain noise correlation matrix,A1 is the ABCD matrix of the first
two-port andCA1 andCA2 are the chain correlation matrix of the first and second two-port
networks, respectively.
The chain noise correlation matrix of the overall cascade isobtained by utilizing the equa-
tions (6.11) and (6.12) as shown below. From equation (6.11), the noise correlation matrix in
admittance representation of the input network is
CY IN = kBT(
YIN + Y†IN
)
. (6.13)
From the admittance representation, the matrix may be converted to its chain representation as
required by equation (6.12) using [10]
CAIN = TCY INT†, (6.14)
whereT is the matrix that transformsCY IN from the admittance representation to the chain
representation. It is given by [10]
T =
[
0 a12
1 a22
]
, (6.15)
wherea12 anda22 are the elements of the ABCD matrix of the input network. The enumeration
of the matrix elements follows the row-column convention. The chain noise correlation matrix,
CA1−2, and the ABCD matrix,A1−2 of the input and SiGe HBT cascade are given by
CA1−2 = AINCA,SiGeA†IN + CAIN (6.16)
A1−2 = AINASiGe (6.17)
whereCA,SiGe is the chain correlation matrix of the SiGe HBT, obtainable from the noise
parameters as
CA,SiGe = 2kBT
[
RnFMIN − 1
2 − RnY ∗OPT
FMIN − 12 − RnYOPT Rn |YOPT|2
]
, (6.18)
whereRn, YOPT andFMIN are the noise parameters of the SiGe HBT as calculated from the
derived equations [10,11]. The above procedure is then repeated to cascade the input and SiGe
6.3 Modelling of Parasitic Elements 57
HBT networks with the test structure’s output network, resulting in
CA = A1−2CAOUTA†1−2
+ CA1−2
= AINASiGeCAOUTA†SiGeA
†IN + AINCA,SiGeA
†IN + CAIN (6.19)
A = A1−2AOUT
= AINASiGeAOUT , (6.20)
whereAOUT is the ABCD matrix of the output network.CAOUT is the noise correlation
matrix of the output network obtained by first calculating itin admittance representation by
equation (6.11) and then converting the result to chain representation by equation (6.14). The
noise parameters of the SiGe HBT test structure are calculated from its noise correlation matrix
derived above using equations (2.3)–(2.5).
Remaining to be determined are the ABCD matrices of the inputand output networks,
AIN andAOUT , respectively. However, because structures that allowAIN andAOUT to be
measured directly, such as those in [42], are unavailable, further assumptions have to be made
about the structure of the input and output networks.
It is assumed that the input and output network can be adequately described up to 18 GHz,
which is the upper limit on the noise parameter experiment, by the equivalent circuits shown
in Fig. 6.8, where the parasitics in each of the networks are lumped into two general impedances.
The idea is that the shunt elements capture the parasitics between the signal pads and the sub-
(a) (b)
ZSI
ZPI
ZSO
ZPO
Fig. 6.8: Lumped models for the (a) input and (b) output networks of SiGe HBT teststructures
strate while the series elements capture the resistance andinductance of the interconnects from
the signal pads to the SiGe HBT device. With reference to Fig.6.6, the four impedances are
obtained as
ZPI = ZP1 =(yOPEN
11+ yOPEN
12
)−1(6.21)
ZSI = ZS1 = zS−O11 − zS−O
12 (6.22)
58 Experimental Procedure and De-embedding Techniques
ZPO = ZP2 =(yOPEN
22+ yOPEN
21
)−1(6.23)
ZSO = ZS2 = zS−O22 − zS−O
21 . (6.24)
To obtain the ABCD matrices of the equivalent circuits in Fig. 6.8, first consider the prob-
lem of determining the ABCD matrices of Fig. 6.9. By applyingthe definition of the ABCD
parameters, it can be shown that the matrices are
AP (ZP ) =
[
1 01
ZP1
]
(6.25)
AS (ZS) =
[
1 ZS
0 1
]
(6.26)
whereAP andAS are the ABCD matrices of the circuit in Fig. 6.9(a) and Fig. 6.9(b), respec-
tively. The ABCD matrices of the input and output equivalentcircuits are then obtained as
AIN = AP (ZPI)AS (ZSI) =
[
1 ZSI
1ZPI
ZSI
ZPI+ 1
]
(6.27)
AOUT = AS (ZSO)AP (ZPO) =
1 + ZSO
ZPOZSO
1ZPO
1
. (6.28)
In reality, due to measurement errors, the admittance of thepads,Z−1
PI andZ−1
PO, may ex-
hibit a small negative real part at a few frequency points. This is due to the real part of the
admittances of the dummy open structures being close to zero, especially at low frequencies.
Since the noise correlation matrix of a passive two-port network is given by equation (6.11),
an unphysical negative real admittance implies that the two-port enhances the output signal-to-
noise ratio. This problem is resolved by data fitting a pad lumped element model, as shown in
Fig. 6.10, to the data obtained from equations (6.21) and (6.23). The admittance across the pad
model can be written as
YPAD =
[(
jωCSUB +1
RSUB
)−1
+1
jωCPAD
]−1
=ω2C2
PADRSUB
1 + ω2R2
SUB (CSUB + CPAD)2+ jωCPAD
[
1 − ω2R2SUBCPAD (CPAD + CSUB)
1 + ω2R2
SUB (CSUB + CPAD)2
]
.
(6.29)
In the frequency range of interest,ω is on the order of1010 rad/s and for these test structures,
the capacitancesCPAD, CSUB are on the order of 10 fF.RSUB is on the order of10 Ω. With
6.3 Modelling of Parasitic Elements 59
(a) (b)
ZS
ZP
Fig. 6.9: Building blocks of the input and output equivalent circuits
these assumptionsω2R2
SUB (CSUB + CPAD)2 ∼ 10−6. Hence,
1 + ω2R2
SUB (CSUB + CPAD)2 ≈ 1. (6.30)
Similarly,ω2R2
SUBCPAD (CPAD + CSUB) ∼ 10−6. Therefore, the admittanceYPAD is approx-
imately
YPAD ≈ ω2C2
PADRSUB + jωCPAD. (6.31)
The above equation suggests that the three parametersCPAD, CSUB andRSUB may be found
as follows:
1. CalculateZ−1
PI andZ−1
PO from equations (6.21) and (6.23), respectively.
2. ObtainCPAD from the slope of the measured admittances versus angular frequency.
3. Plot the real part of the measured admittances versus angular frequency. ObtainRSUB
by fitting to the curvature of the parabola.
CPAD
CSUBRSUB
Fig. 6.10: Signal pad lumped element model
60 Experimental Procedure and De-embedding Techniques
4. UseCSUB to obtain a better fit at high frequencies.
This approach however is not without limitations. First, athigh enough frequencies, the
lumped element approach taken in Fig. 6.8 may not be valid. Second, a more involved model
or equivalent circuit is required to properly model the substrate noise as frequency increases.
Both of these limitations point to a need for a better characterisation technique for the parasitic
input and output networks.
7 Verification by Experiments
T HE experimental verification results of the derived equations and noise transit time ex-
traction technique is presented in this chapter. Both theS-parameters and noise param-
eters of a SiGe HBT were measured using the experiments described in the previous chapter
to verify the developed technique. The transistor characterized had 2-emitter, 3-base and 2-
collector contacts and the emitter length was5.46µm long. The derived equations were applied
to the measuredS-parameters of the device to calculate its noise parameters. The calculated
results were compared with those measured by the noise parameter experiment.
The organization of this chapter is as follows. The extracted lumped element model for the
pads as described in section 6.3.2 is presented first. Then, acomparison is made between the
measured and calculated noise parameters of the SiGe HBT.
7.1. Model Extraction for the Pads
Using the methodology derived in section 6.3.2, the parameters required by the lumped pad
model are extracted and summarized in Table 7.1. The measured and modelled values are
CPAD
CSUBRSUB
Fig. 7.1: Signal pad lumped elementmodel
Table 7.1: Parameter values for lumpedpad model
Parameter Value
CPAD 10 fFCSUB 5 fFRSUB 60Ω
61
62 Verification by Experiments
plotted in Figs. 7.2 and 7.3.
800
600
400
200
0
RE
AL
(Y11+
Y12)
(mS
)
605040302010FREQUENCY (GHz)
Measured
Modelled
Fig. 7.2: Measured and Modelledℜ (y11 + y12) vs. frequency.
3.5
3.0
2.5
2.0
1.5
1.0
0.5
IMA
G(Y
11+
Y12)
(mS
)
605040302010
FREQUENCY (GHz)
Measured Modelled
Fig. 7.3: Measured and Modelledℑ (y11 + y12) vs. frequency.
7.2. Device Parameter Extraction
7.2.1. Unity Gain Frequencies
From the de-embeddedS-parameters up to 65 GHz, thefT and fMAX of the transistor are
extracted and plotted in Fig. 7.4 versusJC = IC/AE, whereAE is the total emitter area.fT
andfMAX versusVCE at the peakfT bias are plotted in Fig. 7.5.
10-1
100 10
1
COLLECTOR CURRENT DENSITY [I C/AE] (mA/ µm2)
0
50
100
150
200
FR
EQ
UE
NC
Y (
GH
z)
fTfMAX
Fig. 7.4: fT and fMAX vs. collector cur-rent density at VCE = 1.5 V.
0.5 1.0 1.5COLLECTOR-EMITTER VOLTAGE (V CE)
0
50
100
150
200
FR
EQ
UE
NC
Y (
GH
z)
fTfMAX
Fig. 7.5: fT and fMAX vs. VCE at peakfT bias.
7.2 Device Parameter Extraction 63
7.2.2. Emitter Resistance
Fig. 7.6 plotsℜz12 versus frequency for all bias points measured. To avoid self-heating
effects, the emitter resistance is extracted atVCE = 1 V [1]. The low frequency values of
ℜz12 are averaged and plotted against the DC emitter bias currentin Fig. 7.7. The bias-
independent value forRE is extracted as they-intercept of the extrapolation to be2.08Ω, which
is reasonably close to the value extracted by the foundry.
30
25
20
15
10
5
0
RE
AL
(Z1
2)
(W)
605040302010
FREQUENCY (GHz)
Fig. 7.6: ℜz12 vs. frequency charac-teristics.
0.00 0.10 0.20 0.30 0.40IE
-1(mA
-1)
0
5
10
15
Re(
Z 12)
(Ω)
Slope: 25.6 mV
Intercept (R E) = 2.08Ω
Fig. 7.7: Extraction of RE from ℜz12by extrapolation.
7.2.3. Base Resistance
Theℜz11 − z12 characteristics of the transistor is plotted versus frequency in Fig. 7.8 for
all the bias points measured. In the high frequency domain, abias-dependentRBX is ex-
tracted as presented in section 3.2. The intrinsic base resistance is extracted using the modified
impedance circle method. A representative plot of[yINT
11+ yINT
12
]−1of one of the bias points
on the complex plane is shown in Fig. 7.9. The extractedRBX andRBI versus bias are plotted
in Fig. 7.10.
Upon comparison with the foundry values, unfortunately, ithas been found that the ex-
tracted intrinsic base resistance values experience a large error while the extrinsic base resis-
tance is relatively close. This is because the extrinsic base resistance is mostly due to the sheet
resistance of the base polysilicon and the base contacts andit is captured relatively well by the
extraction method used. However, the extraction of the intrinsic base resistance is more com-
plicated, as it captures various effects such as the distributive base current and emitter current
crowding. An alternate extraction methodology is available in [43]. However, it uses a separate
test structure that is not available in this work to measure the sheet resistance of the SiGe base.
64 Verification by Experiments
50
40
30
20
10
RE
AL
(Z1
1-Z
12)
(W)
605040302010
FREQUENCY (GHz)
Fig. 7.8: Extraction of RBX fromℜz11 − z12.
-160
-140
-120
-100
-80
-60
-40
-20
IMA
G([
y1
1+
y1
2]-1
) (W
)
250200150100500
REAL([y11+y12]-1
) (W)
FITTED RESULTS MEASURED RESULTS
Fig. 7.9: Extraction of RBI using themodified impedance circle method.
25
20
15
10
5
0
RE
SIS
TA
NC
E (W
)
5 6 7
12 3 4 5 6 7
102 3 4 5
CURRENT DENSITY [IC/AE] (mA/mm2)
RBX
RBI
Fig. 7.10: Extracted base resistance vs. bias.
The suspicion that the intrinsic base resistance plotted inFig. 7.10 experiences a large error
is confirmed by noise parameter measurements results shown below. Unless the values from
the foundry are used, the measured and modelled equivalent noise resistanceRn experience
a significant deviation from one other, even in the low frequencies where correlation may be
7.2 Device Parameter Extraction 65
ignored. Ignoring correlation,Rn may be approximated by [1]
Rn =IC
2VT |y21|2+ RE + RB, (7.1)
whereRB = RBX + RBI and VT is the thermal voltage of the device. SinceIC may be
accurately measured and the extractedRE andRBX are verified against the model file, it is
reasonable to assume the deviation between measured and modelled values inRn at low fre-
quencies is due to the inaccuracy in extractingRBI .
Therefore, only the extractedRBX values are used in this work.RBI is taken from the
model file provided by the foundry under the assumption that those models are verified against
measured data.
7.2.4. Noise Transit Time
The noise transit timeτn is extracted from the negative of the phase of the high frequency
transconductance of the device. The negative phase ofgm at the minimum noise bias is plotted
against the frequency in Fig. 7.11. From the high frequency domain, a value of 0.28 ps is
extracted forτn.
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-PH
AS
E(G
M)
(RA
DIA
NS
)
6050403020100
FREQUENCY (GHz)
MEASURED LINEAR REGRESSION
tn=0.28ps
Fig. 7.11: Phase of gm (ω) at minimum noise bias.
66 Verification by Experiments
7.3. Noise Parameters vs. Bias
The noise parameters with and without accounting for noise correlation are compared against
the measured values at 3 different frequencies, 2, 10 and 18 GHz, as functions of bias. In
the plots below, the results indicate that noise correlation is insignificant for these 160-GHz
SiGe HBTs at frequencies below 18 GHz for all bias points. Qualitatively, the scatter in the
measured data is larger than the difference between the values calculated with and without
noise correlation.
2 GHz5
4
3
2
1
0
NF
MIN
(dB
)
0.1 1 10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
WITHOUT CORRELATION WITH CORRELATION MEASURED
Fig. 7.12: Comparison between mea-sured and modelled NFMIN at 2 GHz vs.bias (with pad parasitics).
120
100
80
60
40
20
Rn(W
)
0.12 3 4 5 6 7
12 3 4 5 6 7
10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
WITHOUT CORRELATION WITH CORRELATION MEASURED
Fig. 7.13: Comparison between mea-sured and modelled Rn at 2 GHz vs. bias(with pad parasitics).
20
15
10
5
0
RE
AL
(YO
PT)
(mS
)
0.12 3 4 5 6 7
12 3 4 5 6 7
10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
MEASURED WITHOUT CORRELATION WITH CORRELATION
Fig. 7.14: Comparison between mea-sured and modelled ℜYOPT at 2 GHzvs. bias (with pad parasitics).
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
IMA
G(Y
OP
T)
(mS
)
0.12 3 4 5 6 7
12 3 4 5 6 7
10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
MEASURED
WITHOUT CORRELATION WITH CORRELATION
Fig. 7.15: Comparison between mea-sured and modelled ℑYOPT at 2 GHzvs. bias (with pad parasitics).
7.3 Noise Parameters vs. Bias 67
10 GHz8
6
4
2
0
NF
MIN
(dB
)
0.1 1 10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
MEASURED WITHOUT CORRELATION WITH CORRELATION
Fig. 7.16: Comparison between mea-sured and modelled NFMIN at 10 GHzvs. bias (with pad parasitics).
120
100
80
60
40
20R
n(W
)0.1
2 3 4 5 6 7
12 3 4 5 6 7
10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
MEASURED WITHOUT CORRELATION WITH CORRELATION
Fig. 7.17: Comparison between mea-sured and modelled Rn at 10 GHz vs.bias (with pad parasitics).
20
15
10
5
0
RE
AL
(YO
PT)
(mS
)
0.12 3 4 5 6 7
12 3 4 5 6 7
10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
MEASURED WITHOUT CORRELATION WITH CORRELATION
Fig. 7.18: Comparison between mea-sured and modelled ℜYOPT at 10 GHzvs. bias (with pad parasitics).
-8
-6
-4
-2
0
2
IMA
G(Y
OP
T)
(mS
)
0.12 3 4 5 6 7
12 3 4 5 6 7
10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
WITHOUT CORRELATION WITH CORRELATION MEASURED
Fig. 7.19: Comparison between mea-sured and modelled ℑYOPT at 10 GHzvs. bias (with pad parasitics).
68 Verification by Experiments
18 GHz8
6
4
2
0
NF
MIN
(dB
)
0.1 1 10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
WITHOUT CORRELATION WITH CORRELATION MEASURED
Fig. 7.20: Comparison between mea-sured and modelled NFMIN at 18 GHzvs. bias (with pad parasitics).
140
120
100
80
60
40
20
Rn(W
)
0.12 3 4 5 6 7
12 3 4 5 6 7
10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
WITHOUT CORRELATION WITH CORRELATION MEASURED
Fig. 7.21: Comparison between mea-sured and modelled Rn at 18 GHz vs.bias (with pad parasitics).
20
15
10
5
0
RE
AL
(YO
PT)
(mS
)
0.12 3 4 5 6 7
12 3 4 5 6 7
10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
WITHOUT CORRELATION WITH CORRELATION MEASURED
Fig. 7.22: Comparison between mea-sured and modelled ℜYOPT at 18 GHzvs. bias (with pad parasitics).
-8
-6
-4
-2
0
2
4
6
8
IMA
G(Y
OP
T)
(mS
)
0.12 3 4 5 6 7
12 3 4 5 6 7
10
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
MEASURED WITHOUT CORRELATION WITH CORRELATION
Fig. 7.23: Comparison between mea-sured and modelled ℑYOPT at 18 GHzvs. bias (with pad parasitics).
7.4 Noise Parameters vs. Frequency 69
7.4. Noise Parameters vs. Frequency
Another verification is performed by comparing the noise parameters at the minimum noise
bias of JC = 1.34 mA/µm2. Plotted in the figures below are the noise parameters versus
frequency at this bias point. These results also indicate that noise correlation is insignificant up
to 18 GHz compared to the scatter present in the measured data, in agreement with the results
presented in the previous section.
8
6
4
2
0
NF
MIN
(dB
)
18161412108642
FREQUENCY (GHz)
WITHOUT CORRELATION WITH CORRELATION MEASURED
Fig. 7.24: Comparison between mea-sured and modelled NFMIN vs. fre-quency at minimum noise bias (with padparasitics).
70
60
50
40
30
Rn(W
)
18161412108642
FREQUENCY (GHz)
WITHOUT CORRELATION WITH CORRELATION MEASURED
Fig. 7.25: Comparison between mea-sured and modelled Rn vs. frequency atminimum noise bias (with pad parasitics).
8
6
4
2
0
RE
AL
(YO
PT)
(mS
)
18161412108642
FREQUENCY (GHz)
WITHOUT CORRELATION WITH CORRELATION MEASURED
Fig. 7.26: Comparison between mea-sured and modelled ℜYOPT vs. fre-quency at minimum noise bias (with padparasitics).
-4
-3
-2
-1
0
IMA
G(Y
OP
T)
(mS
)
18161412108642
FREQUENCY (GHz)
WITHOUT CORRELATION WITH CORRELATION MEASURED
Fig. 7.27: Comparison between mea-sured and modelled ℑYOPT vs. fre-quency at minimum noise bias (with padparasitics).
70 Verification by Experiments
7.5. Impact of Correlation at Millimetre-Wave Frequencies
Having verified the derived equations and noise transit timeextract technique, the equations
are applied to measured and de-embeddedS-parameters up to 65 GHz to predict the noise
parameters at millimetre-wave frequencies. Fig. 7.28 is the NFMIN as a function ofJC at
60 GHz.
10
9
8
7
6
5
4
3
2
NF
MIN
(d
B)
0.12 4 6 8
12 4 6 8
102
COLLECTOR CURRENT DENSITY [IC/AE] (mA/mm2)
WITHOUT CORRELATION WITH CORRELATION
Fig. 7.28: Comparison between modelled NFMIN with and without correlation at 60 GHz(without pad parasitics).
The technique predicts a minimumNFMIN that is approximately 1.5 dB lower when noise
correlation is accounted for. Because of equipment limitations, this prediction cannot be ver-
ified directly by experiments. However, recent publications on 60-GHz SiGe HBT circuits
reported that the measured phase noise of VCOs [3] and noise figure of LNAs [4] are system-
atically lower than simulated values.
Especially significant are the results reported on VCOs in [3]. The measured data are obtain
from millimetre-wave VCOs fabricated using the same SiGe BiCMOS technology on the same
tapeout. Since VCO phase noise is directly proportional to the noise figure of the transistors, a
lower device noise figure translates directly to lower VCO phase noise. These results reported
by others indicate qualitatively that noise correlation issignificant at 60 GHz, consistent with
Fig. 7.28.
8 Conclusion
8.1. Summary
The focus of this thesis is to extend the originaly-parameter based technique in [1] to extract
the noise parameters of bipolar transistors to account for the correlation between the base and
collector shot noise currents. The contributions are a set of noise equations that account for
the correlation between base and collector shot noise currents and a new technique to extract
the noise transit timeτn from measuredy-parameters. Verification of the equations andτn
extraction technique is provided by device simulations andmeasurements up to 18 GHz.
Based on a noise equivalent circuit that considers the intrinsic transistor as a black box, a
set of new equations for the power spectral densities of the input-referred noise sources of a
bipolar transistor is systematically derived in Chapter 4.A technique has also been developed
to extractτn from the high frequency transconductance of the transistor, without fitting to noise
data.
Verification of the derived equations andτn extraction technique is provided initially by
device simulations using 2-D TCAD simulations. The technique is applied to the simulated
y-parameters of a two-dimensional SiGe HBT to calculate its noise parameters. The values
calculated from these equations are then compared with those calculated directly by the sim-
ulator using the impedance field method. From the results presented in Chapter 5, it can be
seen that the equations have reasonable agreement with the noise parameters calculated by the
simulator, even at millimetre-wave frequencies, when noise correlation is accounted for.
Further verification of the technique is provided experimentally as described in Chapters 6
and 7. TheS-parameters and noise parameters of a commercial 160-GHz SiGe HBT are mea-
sured up to 65 GHz and 18 GHz, respectively. The noise parameters calculated by applying the
equations to the measuredS-parameters of the device are compared with those measured using
the noise parameter measurement system. Experimental results indicate that noise correlation
is insignificant up to 18 GHz for these devices. Finally, thistechnique predicts a minimum
71
72 Conclusion
NFMIN at 60 GHz that is approximately 1.5 dB lower when correlationis accounted for, in
agreement with published VCO and LNA results.
8.2. Future Work
Future work can be grouped into two major areas: experimental verification and development.
On the verification side, it is necessary to provide experimental proof that this technique is
able to provide accurate results at millimetre-wave frequencies in addition to the simulation
results presented. This points to a need to perform noise parameter characterization at these
frequencies. The required equipment, however, is not currently available at the University
of Toronto. Although there is commercially available noisecharacterization equipment up to
110 GHz, the experimental uncertainly may be larger than thedifference between uncorrelated
and correlated values. This is especially true for device noise characterization, as the test
structures are impedance mismatched to 50Ω and most of the noise power is reflected at the
interfaces. One possible solution is to develop a mathematical link between the device noise
figure and some figure of merit of simple millimetre-wave circuits, such as the phase noise of
a VCO.
Also, the accuracy of verification may be improved through the use of more advanced
de-embedding techniques. In particular, the transmissionline based de-embedding technique
that was developed and presented in [39] is a suitable candidate. Unlike the lumped-element
based open-short de-embedding used in this work, the transmission-line based method ac-
counts for the distributive nature of the parasitics, whichis important at the millimetre-wave
regime. This technique can also be applied effectively to de-embed the noise parameters, as
the line itself is a passive device and hence its noise is completely characterized by itsY/Z-
parameters [41]. Since the noise contributed by the transmission lines leading from the pads to
the device is known, it can be de-embedded from the measured noise parameters using matrix
techniques [10,11].
On the development side, the error introduced by ignoring the distributed nature of the base
resistance should be investigated. This involves derivinganother set of equations that splits
RB into RBX andRBI . The equations derived in this thesis may overestimate the noise of the
device, since at sufficiently high frequencies, the noise contributed byRBI may be shunted out
by the extrinsic base-collector capacitanceCBCX and therefore reducing the noise figure of the
device.
ADetailed Derivation of SiGe
HBT Noise Parameter
Equations
A.1. Input Referred Noise Voltage
In section 4.2.1, it was shown that a system of seven equations with seven unknowns has to
be solved in order to obtain an expression for the output short-circuit current of the SiGe HBT
noise equivalent circuit. The algebraic details omitted inthat section are present here.
For completeness, the SiGe HBT noise equivalent circuit is reproduced in Fig. A.1. The
system of seven equations are
ISC + I INT
2+ inC = 0 (A.1)
IRE− I INT
1− inB − I INT
2− inC = 0 (A.2)
vnB −(I INT
1 + inB
)RB − vINT
1 − vX = 0 (A.3)
vINT
2+ vX = 0 (A.4)
vnE + IRERE = vX (A.5)
[
I INT1
I INT2
]
=
[
yINT11
yINT12
yINT21
yINT22
][
vINT1
vINT2
]
. (A.6)
The expression for the input referred noise voltage,vn, is obtained by equating the output
short-circuit currents of the SiGe HBT noise equivalent circuit and the chain representation of
a noisy two-port network.
The strategy for solvingISC from the seven equations is as follows:
1. Solve equation (A.2) forIRE.
2. Substitute the result from above into equation (A.5) to obtainvX = f(I INT1
, I INT2
).
3. SubstitutevX = f(I INT1
, I INT2
)into equations (A.3) and (A.4) to obtain two equations,
vINT1 = f1
(I INT1 , I INT
2
)andvINT
2 = f2
(I INT1 , I INT
2
).
73
74 Detailed Derivation of SiGe HBT Noise Parameter Equations
inCinB
RB
RE
vnE
vnB YINT
Y
EE
CB+ +
+
+
− −
−
−
I INT1
I INT2
IRE
ISC
L1
L2
vX
vINT1
vINT2
Fig. A.1: Schematic of Noise Equivalent Circuit Defining Symbols used in Deriving vn
4. SubstitutevINT1
= f1
(I INT1
, I INT2
)andvINT
2= f2
(I INT1
, I INT2
)into the first matrix equa-
tion of (A.6) and obtainI INT1 = f3
(I INT2
).
5. Repeat the above step for the second matrix equations of (A.6) and obtainI INT1
=
f(I INT2
).
6. Equate the equations obtained from steps 4 and 5 to obtain an expression forI INT2 .
7. Using the above expression, obtain an equation forISC from equation (A.1).
Substituting the expression forIREobtained from equation (A.2) into equation (A.5), the
equation forvX is
vX = vnE +(I INT
1+ I INT
2+ inB + inC
)RE . (A.7)
vINT1
= f(I INT1
, I INT2
)is obtained by solving equation (A.3) forvINT
1and substituting the
expression ofvX from above.
vINT
1= vnB −
(I INT
1+ inB
)RB − vnE −
(I INT
1+ I INT
2+ inB + inC
)RE (A.8)
From the first matrix equation (A.6), upon substituting expressions forvINT1
, vINT2
and using
equation (A.4),I INT1
is obtained as
I INT
1= yINT
11
[vnB −
(I INT
1+ inB
)RB − vnE −
(I INT
1+ I INT
2+ inB + inC
)RE
]
+ yINT
12
[−vnE −
(I INT
1 + I INT
2 + inB + inC
)RE
].
(A.9)
A.1 Input Referred Noise Voltage 75
IsolatingI INT1
in the above equation gives
I INT
1
[1 + yINT
11 (RB + RE) + yINT
12 RE
]= I INT
2
(−yINT
11 RE − yINT
12 RE
)
+ yINT
11 (vnB − inBRB − vnE − (inB + inC)RE) − yINT
12 (vnE + (inB + inC)RE) .
(A.10)
Similarly, by using the second equation from the matrix equation (A.6) and substituting expres-
sions forvINT1 andvINT
2 from, equations (A.8) and (A.4), respectively, gives a second relation
betweenI INT1
andI INT2
.
I INT
2= yINT
21
(vnB −
(I INT
1+ I INT
2
)RB − vnE −
(I INT
1+ I INT
2+ inB + inC
)RE
)
− yINT
22
(vnE +
(I INT
1+ I INT
2+ inB + inC
)RE
) (A.11)
SeparatingI INT1 andI INT
2 in the above equation gives
I INT
2
(1 + yINT
21RE + yINT
22RE
)= I INT
1
(−yINT
21RB − yINT
21RE − yINT
22RE
)
+ yINT
21(vnB − inBRB − vnE − (inB + inC)RE) − yINT
22(vnE + (inB + inC)RE) .
(A.12)
Substituting the expression forI INT1
from equation (A.10) results in
I INT
2
(1 +
[yINT
21+ yINT
22
]RE
)= yINT
21(vnB − inBRB − vnE − (inB + inC)RE)
− yINT
22 (vnE + (inB + inC) RE) −(
yINT21
RB +(yINT
21+ yINT
22
)RE
1 + yINT11
(RB + RE) + yINT12
RE
)
×[−I INT
2
(yINT
11+ yINT
12
)RE + yINT
11(vnB − inBRB − vnE − (inB + inC) RE)
−yINT
12(vnE + (inB + inC) RE)
].
(A.13)
FactoringI INT2 in the above equation gives
I INT
2
1 +[yINT
21 + yINT
22
]RE −
(
yINT21 RB +
(yINT
21 + yINT22
)RE
1 + yINT11
(RB + RE) + yINT12
RE
)
(yINT
11 + yINT
12
)RE
= − yINT21
RB +(yINT
21+ yINT
22
)RE
1 + yINT11
(RB + RE) + yINT12
RE
[yINT
11 (vnB − inBRB − vnE − (inB + inC) RE)
− yINT
12(vnE + (inB + inC)RE)
]+ yINT
21(vnB − inBRB − vnE − (inB + inC) RE)
− yINT
22(vnE + (inB + inC)RB) .
(A.14)
76 Detailed Derivation of SiGe HBT Noise Parameter Equations
It would be more convenient at this point to assign the factormultiplying I INT2
in the above
equation to a variableΓ and simplify it separately.
Γ = 1 +[yINT
21 + yINT
22
]RE −
(
yINT21
RB +(yINT
21+ yINT
22
)RE
1 + yINT11
(RB + RE) + yINT12
RE
)
(yINT
11 + yINT
12
)RE
=
(1 + yINT
11RB
) (1 +
(yINT
21+ yINT
22
)RE
)+(yINT
11+ yINT
12
) (1 − yINT
21RB
)RE
1 + yINT11 RB + (yINT
11 + yINT12 ) RE
(A.15)
The above expression forΓ can then be substituted back into equation (A.14) to obtain another
expression forI INT2 . ζ is defined to be the numerator ofΓ and simplified as
ζ =(1 + yINT
11 RB
) (1 +
(yINT
21 + yINT
22
)RE
)+(yINT
11 + yINT
12
) (1 − yINT
21 RB
)RE (A.16)
= 1 + yINT
11 RB + RE
∑
i,j
yINT
ij + RBRE det [YINT] .
ζI INT
2= −yINT
22(vnE + (inB + inC)RE)
(1 + yINT
11RB +
(yINT
11+ yINT
12
)RE
)
+ yINT
21 (vnB − inBRB − vnE − (inB + inC) RE)(1 + yINT
11 RB +(yINT
11 + yINT
12
)RE
)
−(yINT
21 RB +(yINT
21 + yINT
22
)RE
) [yINT
11 (vnB − inBRB − vnE − (inB + inC) RE)
− yINT
12(vnE + (inB + inC) RE)
]
(A.17)
The above expression can be rearranged and simplified to obtain the final expression forI INT2
.
Finally, by equation (A.1), the output short-circuit current is given by
ISC = −I INT
2 − inC . (A.18)
The equivalent circuit for the chain representation is reproduced in Fig. A.2. It was shown in
section 4.2.1 that the output short-circuit current can be obtained directly from equation (A.18)
by takingvnB → vn and setting all other noise sources to zero. The result is
ISC =vn
(yINT
21 − det [YINT] RE
)
ζ. (A.19)
By equating equations (A.18) and (A.19), the expression forvn is obtained as
vn = vnB +1
C(DvnE + EinB + FinC) (A.20)
A.2 Input Referred Noise Current 77
RB
RE
YINT
Y
EE
CB
+ −
ISC
vn
Fig. A.2: Schematic for Deriving Output Short-Circuit Current of Chain Representation ofNoisy Two-Port
where
C = yINT
21− RE det [YINT] (A.21)
D = −yINT
21− yINT
22− RB det [YINT] (A.22)
E = −RByINT
21 − RE
(yINT
21 + yINT
22
)(A.23)
F = ζ − RBRE det [YINT] − RE
(yINT
21+ yINT
22
). (A.24)
A.2. Input Referred Noise Current
The output open-circuit voltage of the SiGe HBT noise equivalent circuit is obtained by solving
a system of equations, as shown in section 4.2.2. Following is the algebraic details that were
omitted in that section.
Reproduced in Fig. A.3 is the open-circuited noise equivalent circuit. The equations to be
solved are
I INT
1+ inB = 0 (A.25)
I INT
2 + inC = 0 (A.26)
vo = vINT
2 + vnE (A.27)[
vINT1
vINT2
]
= ZINT
[
I INT1
I INT2
]
(A.28)
78 Detailed Derivation of SiGe HBT Noise Parameter Equations
RB
RE
YINT
Y
E E
CB+ + ++
+
− −
−−
−
I INT1
I INT2
vINT1
vINT2
vo
vnB
L1
vnE
inB inC
Fig. A.3: Schematic for Deriving Output Open-Circuit Voltage of SiGe HBT Noise Equiva-lent Circuit
where,
ZINT =1
det [YINT]
[
yINT22
−yINT12
−yINT21 yINT
11
]
. (A.29)
Solving the above equations forin is considerably simpler than in thevn case. From equa-
tions (A.28) and (A.29),
vINT
2=
1
det [YINT]
(−yINT
21I INT
1+ yINT
11I INT
2
)(A.30)
=1
det [YINT]
(yINT
21inB − yINT
11inC
).
The last step is obtained by applying equations (A.25) and (A.26). Hence, from equation (A.27),
vo =1
det [YINT]
(yINT
21inB − yINT
11inC
)+ vnE. (A.31)
Fig. A.4 reproduces the equivalent circuit of the chain representation with its inputs and
outputs open-circuited for completeness. The equations that need to be solved to obtain the
open-circuit voltage are
I INT
2= 0 (A.32)
I INT
1= −in (A.33)
vo = vINT
2 + vX (A.34)
A.2 Input Referred Noise Current 79
RB
RE
YINT
Y
E E
CB
+
−
I INT1
I INT2
voL1
invX
Fig. A.4: Schematic for Deriving Output Open-Circuit Voltage of SiGe HBT Chain Repre-sentation
vX = −inRE = I INT
1RE (A.35)
vINT
2 =1
det [YINT]
(−yINT
21 I INT
1 + yINT
11 I INT
2
). (A.36)
By using equations (A.32) and (A.33),
vINT
2=
yINT21 in
det [YINT]. (A.37)
Finally, from equations (A.34) and (A.35),
vo =yINT
21in
det [YINT]− inRE . (A.38)
By equating equation (A.31) with (A.38), the input-referred noise current is given by
in
(yINT
21
det [YINT]− RE
)
=yINT
21inB − yINT
11inC
det [YINT]+ vnE (A.39)
in =yINT
21inB − yINT
11inC + vnE det [YINT]
J(A.40)
where
J ≡ yINT
21 − RE det [YINT] . (A.41)
80 Detailed Derivation of SiGe HBT Noise Parameter Equations
A.3. Transforming the Noise Power Spectral Densities to Ext rinsic
Y-Parameters
In section 4.3 that the input referred noise power spectral densities are given by
⟨v2
n
⟩≡ 〈v∗
nvn〉
=⟨v2
nB
⟩+
∣∣∣∣
D
C
∣∣∣∣
2⟨v2
nE
⟩+
∣∣∣∣
E
C
∣∣∣∣
2⟨i2nB
⟩+
∣∣∣∣
F
C
∣∣∣∣
2⟨i2nC
⟩+
2
|C|2ℜ (EF ∗ 〈inBi∗nC〉)
(A.42)⟨i2n⟩≡ 〈i∗nin〉
=
∣∣∣∣
yINT21
J
∣∣∣∣
2⟨i2nB
⟩+
∣∣∣∣
yINT11
J
∣∣∣∣
2⟨i2nC
⟩+
∣∣∣∣
det [YINT]
J
∣∣∣∣
2⟨v2
nE
⟩(A.43)
− 2
|J |2ℜ(yINT
21
(yINT
11
)∗ 〈inBi∗nC〉)
〈v∗nin〉 =
det [YINT]
J
D∗
C∗
⟨v2
nE
⟩+
yINT21
J
E∗
C∗
⟨i2nB
⟩− yINT
11
J
E∗
C∗〈i∗nBinC〉 (A.44)
+yINT
21
J
F ∗
C∗〈inBi∗nC〉 −
yINT11
J
F ∗
C∗
⟨i2nC
⟩.
In this section, the above equations will be rewritten in terms of the extrinsicy-parameters of
the SiGe HBT. From appendix B, the intrinsicy-parameters of the SiGe HBT are related to its
extrinsicy-parameters by
yINT
11=
y11 − det [Y] RE
1 − y11RB −∑
ij yijRE + RBRE det [Y](A.45)
yINT
12=
y12 + det [Y]RE
1 − y11RB −∑ij yijRE + RBRE det [Y](A.46)
yINT
21 =y21 + det [Y]RE
1 − y11RB −∑
ij yijRE + RBRE det [Y](A.47)
yINT
22=
y22 − (RB + RE) det [Y]
1 − y11RB −∑
ij yijRE + RBRE det [Y]. (A.48)
The determinantdet [YINT] is first compute as follows. Since
yINT
11 yINT
22 =y11y22 − det [Y] [y11 (RE + RB) + y22RE] + RE (RE + RB) det2 [Y]
(
1 − y11RB −∑
ij yijRE + RBRE det [Y])2
(A.49)
yINT
12yINT
21=
y12y21 + y12RE det [Y] + y21RE det [Y] + R2
E det2 [Y](
1 − y11RB −∑
ij yijRE + RBRE det [Y])2
, (A.50)
A.3 Transforming the Noise Power Spectral Densities to Extrinsic Y-Parameters 81
then
det [YINT] =det [Y] − det [Y] y11RB − det [Y] RE (y11 + y22 + y12 + y21) + RERB det2 [Y]
(1 − y11RB −∑
yijRE + RBRE det [Y])2
=det [Y]
1 − y11RB −∑ yijRE + RBRE det [Y]. (A.51)
〈v2n〉 is expressed as a function of the extrinsicy-parameters first. The variablesC, D and
E are expressed in terms of extrinsicy-parameters as detailed below.
C ≡ yINT
21− RE det [YINT]
=y21 + det [Y] RE
1 − y11RB −∑
yijRE + RBRE det [Y]− RE det [Y]
1 − y11RB − RE
∑yij + RBRE det [Y]
=y21
1 − y11RB − RE
∑yij + RBRE det [Y]
(A.52)
D ≡ −RB det [YINT] − yINT
21− yINT
22
= − RB det [Y]
1 − y11RB − RE
∑yij + RBRE det [Y]
− y21 + RE det [Y]
1 − y11RB − RE
∑yij + RBRE det [Y]
− y22 − (RB + RE) det [Y]
1 − y11RB − RE
∑yij + RBRE det [Y]
= − y21 + y22
1 − y11RB − RE
∑yij + RERB det [Y]
(A.53)
E ≡ −yINT
21 RB − RE
(yINT
21 + yINT
22
)
= − (y21 + RE det [Y]) RB
1 − y11RB − RE
∑yij + RBRE det [Y]
− RE
y21 + RE det [Y] + y22 − (RB + RE) det [Y]
1 − y11RB − RE
∑yij + RBRE det [Y]
= − y21RB + RE (y21 + y22)
1 − y11RB − RE
∑yij + RBRE det [Y]
(A.54)
The summation∑
ij yINTij can be expressed in terms of the extrinsicy-parameters as
∑
ij
yINT
ij =1
1 − y11RB −∑
ij yijRE + RBRE det [Y]×
y11 − RE det [Y] + y12 + RE det [Y]
+y21 + RE det [Y] + y22 − (RB + RE) det [Y]
=
∑
ij yij − RB det [Y]
1 − y11RB −∑
ij yijRE + RBRE det [Y]. (A.55)
82 Detailed Derivation of SiGe HBT Noise Parameter Equations
This result is used in conjunction with equations (A.45)– (A.48) to simplifyζ = 1+RE
∑
ij yINTij +
RBy11 + RBRE det [YINT].
ζ = 1 + RE
∑
ij yij − RB det [Y]
1 − y11RB − RE
∑
ij yij + RBRE det [Y]
+ RB
y11 − RE det [Y]
1 − y11RB − RE
∑
ij yij + RBRE det [Y]
+ RBRE
det [Y]
1 − y11RB −∑
ij yijRE + RBRE det [Y]
= 1 −1 − y11RB − RE
∑
ij yij + RBRE det [Y] − 1
1 − y11RB −∑
ij yijRE + RBRE det [Y]
=1
1 − y11RB −∑
ij yijRE + RBRE det [Y](A.56)
F is then expressed in terms of the extrinsicy-parameters as
F = ζ − RBRE det [YINT] − RE
(yINT
21+ yINT
22
)
=1
1 − y11RB − RE
∑
ij yij + RBRE det [Y]− RBRE det [Y]
1 − y11RB − RE
∑
ij yij + RBRE det [Y]
− RE
y21 + RE det [Y] + y22 − (RB + RE) det [Y]
1 − y11RB − RE
∑
ij yij + RBRE det [Y]
=1 − RE (y21 + y22)
1 − y11RB − RE
∑
ij yij + RBRE det [Y]. (A.57)
The ratiosD/C, E/C andF/C and the productEF ∗ are computed in terms of extrinsicy-
parameters.
D
C= −y21 + y22
y21
= −(
1 +y22
y21
)
(A.58)
E
C= −y21RB + RE (y21 + y22)
y21
= −(
RB + RE
[
1 +y22
y21
])
(A.59)
F
C=
1 − RE (y21 + y22)
y21
=1
y21
− RE
[
1 +y22
y21
]
(A.60)
EF ∗ = −(y21RB + RE [y21 + y22]) (1 − RE [y21 + y22])∗
∣∣∣1 − y11RB − RE
∑
ij yij + RBRE det [Y]∣∣∣
2(A.61)
A.3 Transforming the Noise Power Spectral Densities to Extrinsic Y-Parameters 83
Using the above results,〈v2n〉 as a function of the extrinsicy-parameters of the SiGe HBT is
⟨v2
n
⟩=⟨v2
nB
⟩+
∣∣∣∣1 +
y22
y21
∣∣∣∣
2⟨v2
nE
⟩
+
∣∣∣∣RB + RE
[
1 +y22
y21
]∣∣∣∣
2⟨i2nB
⟩+
∣∣∣∣y−1
21− RE
[
1 +y22
y21
]∣∣∣∣
2⟨i2nC
⟩
− 2
|y21|2ℜ [(y21RB + RE [y21 + y22]) (1 − RE [y21 + y22])
∗ 〈inBi∗nC〉] .
(A.62)
Following the transformation of〈v2n〉, the power spectral density of the input referred noise
current, 〈i2n〉 is expressed in terms of the extrinsicy-parameters. First, equation (A.51) is
applied to simplifyJ as follows.
J ≡ yINT
21− RE det [YINT]
=y21 + det [Y] RE
1 − y11RB −∑
yijRE + RBRE det [Y]− RE det [Y]
1 − y11RB −∑
yijRE + RBRE det [Y]
=y21
1 − y11RB −∑ yijRE + RBRE det [Y]. (A.63)
The multiplying factors in the equation of〈i2n〉, namelyyINT21 /J, yINT
11 /J anddet [YINT] /J , are
transformed into extrinsicy-parameters. Using the above equation in conjunction with (A.45)
and (A.51) yields
yINT11
J=
y11 − det [Y]RE
1 − y11RB −∑ yijRE + RBRE det [Y]× 1 − y11RB −
∑yijRE + RBRE det [Y]
y21
=y11 − RE det [Y]
y21
(A.64)
det [YINT]
J=
det [Y]
1 − y11RB −∑ yijRE + RBRE det [Y]
× 1 − y11RB −∑
yijRE + RBRE det [Y]
y21
=det [Y]
y21
. (A.65)
Re-arranging the terms in
J ≡ yINT
21− RE det [YINT] (A.66)
asyINT
21
J= 1 +
RE det [YINT]
J(A.67)
84 Detailed Derivation of SiGe HBT Noise Parameter Equations
and observing that
RE det [YINT]
J=
RE det [Y]
1 − y11RB − RE
∑
ij yij + RERB det [Y]
×1 − y11RB − RE
∑
ij yij + RERB det [Y]
y21
=RE det [Y]
y21
,
(A.68)
therefore,yINT
21
J= 1 +
RE det [Y]
y21
. (A.69)
By using the above equations,〈i2n〉 as a function of extrinsicy-parameters is
⟨i2n⟩
=
∣∣∣∣1 +
RE det [Y]
y21
∣∣∣∣
2⟨i2nB
⟩+
∣∣∣∣
y11 − RE det [Y]
y21
∣∣∣∣
2⟨i2nC
⟩+
∣∣∣∣
det [Y]
y21
∣∣∣∣
2⟨v2
nE
⟩
− 2
|y21|2ℜ [(y21 + RE det [Y]) (y11 − RE det [Y])∗ 〈inBi∗nC〉]
(A.70)
and the cross power spectral density is given by
〈v∗nin〉 =
det [Y]
y21
(
1 +y22
y21
)∗⟨v2
nE
⟩+
(RE det [Y]
y21
+ 1
)(
RB + RE
[
1 +y22
y21
])∗⟨i2nB
⟩
−(
y11 − RE det [Y]
y21
)(
RB + RE
[
1 +y22
y21
])∗
〈i∗nBinC〉
−(
RE det [Y]
y21
+ 1
)(
y−1
21 − RE
[
1 +y22
y21
])∗
〈inBi∗nC〉
+
(y11 − RE det [Y]
y21
)(
y−1
21 − RE
[
1 +y22
y21
])∗⟨i2nC
⟩. (A.71)
BConversion Between
Intrinsic and Extrinsic
Y-Parameters
W HEN measuring or simulating the high frequency performanceof a SiGe HBT the
extrinsic two-port parameters of the device for each bias point are readily available1.
It is therefore convenient to express the input referred noise voltage and noise current in terms
of the extrinsicy-parameters of the device instead of intrinsicy-parameters.
Equations are derived and presented in this appendix for theconversion between intrinsic
and extrinsicy-parameters, as they are defined in Fig. B.1. These equationsare employed to
convert the expressions of the input referred noise voltageand noise current in terms of intrinsic
y-parameters to extrinsicy-parameters.
B.1. Converting from Intrinsic to Extrinsic Y-Parameters
Intrinsic y-parameters,yINTij , as functions of the extrinsicy-parameters,yij, are derived first.
By definition,
I1 = y11v1 + y12v2 (B.1)
I2 = y21v1 + y22v2. (B.2)
Also, by observation,
I1 = I INT
1 (B.3)
I2 = I INT
2. (B.4)
1Technically speaking, some form of parasitic de-embeedingis required for the experimental data to removethe effects of the pad and interconnect parasitics. De-embedding techniques are presented in section 6.3.1. Toavoid confusion, in this thesis, the term “extrinsicy-parameters” refers to they-parameters of the device, excludingthe effects of pads and interconnects.
85
86 Conversion Between Intrinsic and Extrinsic Y-Parameters
++ ++
−
− −
−
RB
RE
RCYINT
Y
vINT1
vINT2
v1 v2
I INT1
I INT2I1 I2
EE
B C
vX
Fig. B.1: Equivalent circuit relating intrinsic and extrinsic y-parameters of SiGe HBT
From KVL,
v1 − I1RB − vINT
1 − vX = 0 (B.5)
v2 − I2RC − vINT
2− vX = 0 (B.6)
vX − (I1 + I2) RE = 0. (B.7)
Using equations (B.3)–(B.7), equations (B.1) and (B.2) arerewritten as
I INT
1= y11
(I INT
1RB + vINT
1+[I INT
1+ I INT
2
]RE
)
+ y12
(I INT
2 RC + vINT
2 +[I INT
1 + I INT
2
]RE
)(B.8)
I INT
2= y21
(I INT
1RB + vINT
1+[I INT
1+ I INT
2
]RE
)
+ y22
(I INT
2RC + vINT
2+[I INT
1+ I INT
2
]RE
). (B.9)
Solving forI INT1
in both of the above equations,
I INT
1=
y11
(vINT1 + I INT
2 RE
)
1 − y11RB − y11RE − y12RE
+y12
(I INT2 RC + vINT
2 + I INT2 RE
)
1 − y11RB − y11RE − y12RE
(B.10)
I INT
1 =I INT2
y21RB + y21RE + y22RE
− y21
(vINT1
+ I INT2
RE
)
y21RB + y21RE + y22RE
− y22
(I INT2
RC + vINT2
+ I INT2
RE
)
y21RB + y21RE + y22RE
. (B.11)
B.1 Converting from Intrinsic to Extrinsic Y-Parameters 87
I INT1
is then eliminated by equating the two equations above. Rearranging the result yields
where the summation is to be taken over the entire range of indices. From the above equation,
it can be deduced that
yINT
21 =y21 + RE det [Y]
1 − y11RB − y22RC − RE
∑
ij yij + (RBRC + RBRE + RERC) det [Y](B.15)
yINT
22=
y22 − (RB + RE) det [Y]
1 − y11RB − y22RC − RE
∑
ij yij + (RBRC + RBRE + RERC) det [Y]. (B.16)
The equations ofyINT11
andyINT12
are obtained using the same procedure as above, but equa-
tions (B.8) and (B.9) are solved forI INT2
instead ofI INT1
. The following is obtained when the
two resultant equations are equated to eliminateI INT2 .
I INT
1Γ = vINT
1
[y11
y11RE + y12RC + y12RE
+y21
1 − y21RE − y22RC − y22RE
]
+ vINT
2
[y12
y11RE + y12RC + y12RE
+y22
1 − y21RE − y22RC − y22RE
]
, (B.17)
where
Γ = −y11RB + y11RE + y12RE − 1
y11RE + y12RC + y12RE
− y21 (RB + RE) + y22RE
1 − y21RE − y22RC − y22RE
. (B.18)
88 Conversion Between Intrinsic and Extrinsic Y-Parameters
Upon simplification,
I INT
1= vINT
1
(
y11 − det [Y] (RE + RC)
1 − y11RB − y22RC − RE
∑
ij yij + (RBRC + RBRE + RERC) det [Y]
)
+ vINT
2
(
y12 + RE det [Y]
1 − y11RB − y22RC − RE
∑
ij yij + (RBRC + RBRE + RERC) det [Y]
)
,
(B.19)
from whichyINT11
andyINT12
can be deduced as
yINT
11 =y11 − det [Y] (RE + RC)
1 − y11RB − y22RC − RE
∑
ij yij + (RBRC + RBRE + RERC) det [Y](B.20)
yINT
12=
y12 + RE det [Y]
1 − y11RB − y22RC − RE
∑
ij yij + (RBRC + RBRE + RERC) det [Y], (B.21)
where the summation is to be taken over the entire range of indices.
To summarize, it has been shown that for the equivalent circuit in Fig. B.1, the intrinsic and
extrinsicy-parameters are related by
yINT
11 =y11 − det [Y] (RE + RC)
1 − y11RB − y22RC − RE
∑
ij yij + (RBRC + RBRE + RERC) det [Y](B.22)
yINT
12=
y12 + RE det [Y]
1 − y11RB − y22RC − RE
∑
ij yij + (RBRC + RBRE + RERC) det [Y](B.23)
yINT
21=
y21 + RE det [Y]
1 − y11RB − y22RC − RE
∑
ij yij + (RBRC + RBRE + RERC) det [Y](B.24)
yINT
22 =y22 − (RB + RE) det [Y]
1 − y11RB − y22RC − RE
∑
ij yij + (RBRC + RBRE + RERC) det [Y]. (B.25)
B.2. Converting from Extrinsic to Intrinsic Y-Parameters
Following a procedure similar to the previous section, equations are derived relating the extrin-
sic y-parameters,yij, in terms of the intrinsicy-parameters,yINTij , and the series resistances,
RB, RE andRC . In terms ofyINTij ,
I INT
1 = yINT
11 vINT
1 + yINT
12 vINT
2 (B.26)
I INT
2= yINT
21vINT
1+ yINT
22vINT
2. (B.27)
B.2 Converting from Extrinsic to Intrinsic Y-Parameters 89
Equations (B.5)–(B.7) are then used to eliminate the currents and voltages of the intrinsic
device to arrive at the following two equations.
I1 = yINT
11 (v1 − I1RB − [I1 + I2] RE) + yINT
12 (v2 − I2RC − [I1 + I2] RE) (B.28)
I2 = yINT
21(v1 − I1RB − [I1 + I2] RE) + yINT
22(v2 − I2RC − [I1 + I2] RE) (B.29)
The same procedure as in the derivation ofyINTij in terms ofyij is used to eliminateI1 andI2
one at a time to determine the expressions ofyij. The results are
y11 =yINT
11 + (RC + RE) det [YINT]
Φ(B.30)
y12 =yINT
12− RE det [YINT]
Φ(B.31)
y21 =yINT
21 − RE det [YINT]
Φ(B.32)
y22 =yINT
22+ (RB + RE) det [YINT]
Φ(B.33)
where
Φ = 1 + yINT
11 RB + yINT
22 RC + RE
∑
yINT
ij + (RCRB + RCRE + RERB) det [YINT] (B.34)
90 Conversion Between Intrinsic and Extrinsic Y-Parameters
C The Selectively Implanted
Collector
T HE selectively implanted collector (SIC) is a region of the collector that is engineered
to optimise the performance of high-speed SiGe HBTs. The role of the SIC in the op-
timization of the high frequency performance of SiGe HBTs isqualitatively described in this
appendix.
C.1. Unity Gain Frequency Revisited
It can be shown that the unity gain frequency of a bipolar transistor is given by
1
2πfT
= τF +kBT
qIC
(CdBE + CdBC) + CdBC (RE + RC) , (C.1)
whereCdBE andCdBC are the base-emitter and base-collector depletion capacitances andRE
andRC are the emitter and collector series resistances [8].
ThefT of a transistor is increased when the forward transit timeτF is decreased. This can
be achieved by device scaling, which reduces the carrier transit time by reducing the device
dimensions. However,τF is not the only component in the above equation. In modern high-
speed SiGe HBTs, SICs are introduced to reduce theCdBCRC product to further increase the
fT’s of the transistors.
C.2. The Role of the SIC
One way to reduce theCdBCRC product is to reduce the collector resistance by increasingthe
collector doping. As with any design problems, there are tradeoffs that have to be considered.
There are two disadvantages to increasing the collector doping. First, as the doping of the
base region cannot be reduced without degrading the base resistance, increasing the collector
doping increases the base-collector depletion capacitance CdBC , as the capacitance per unit
area is inversely proportional to the depletion layer width[44]. For transistors in the common
91
92 The Selectively Implanted Collector
emitter configuration, this capacitance is multiplied by the Miller effect, which further degrades
the gain at high frequencies.
The second disadvantage is that if both the base and collector regions are highly doped, the
breakdown voltage decreases. The electric field within the base-collector space-charge region
is increased since the width of the depletion layer is reduced.
The current flow in present SiGe HBTs are predominantly vertical and directly under the
emitter, as indicated in Fig. C.1. The collector region outside the shaded arrow does not carry
significant current. Increasing the collector doping outside the emitter window does little in
reducingRC , but increases theCdBC .
Fig. C.1: Current Flow in Modern Vertical SiGe HBTs.
In modern high-speed SiGe HBTs, the collector region directly under the emitter has a
retrograded doping profile, similar to Fig. 5.4. The retrograded doping profile reduces the
electric field in the base-collector space charge region andhence improves the breakdown
voltage of the device. This region of increased doping concentration is known as the selectively
implanted collector (SIC). Since the SIC is directly located underneath the emitter, it can be
implanted using the emitter opening as a self-aligned mask,as shown in Fig. 5.1.
D Simulation Decks
D.1. Process Simulation (ATHENA)# This file simulates a 2-dimensional npn122 device with# collector reach through
go athena
method model.sigec model.sige back=6 min.temp=600
# create half device# put a vertical line at each location where etching will# occur in subsequent stepsline x loc=0.00 spac=0.01line x loc=0.10 spac=0.01line x loc=0.185 spac=0.025line x loc=0.435 spac=0.05line x loc=0.55 spac=0.05line x loc=0.77 spac=0.05line x loc=0.87 spac=0.05line x loc=1.17 spac=0.05line x loc=2.36 spac=0.05line x loc=2.5 spac=0.05
line y loc=0.00 spac=0.005line y loc=0.10 spac=0.005line y loc=0.4 spac=0.01line y loc=1.8 spac=0.2
# Specify the solver methodmethod block newton trap autonr
# Solve Gummel at Vce=1.5V, first ramp up to Vce=1.5solve initsolve vcollector=0.05solve prev vcollector=0.1solve proj vcollector=0.15 vstep=0.05 name=collector vfinal=1.5