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Determination of Small-Signal Equivalent Circuit Elements and
Large-Signal Model Parameters of Si/SiGe HBTs up to 50 GHz
Peter Heymann, Ralf Doerner, and Frank Schnieder
Ferdinand-Braun-Institut für Höchstfrequenztechnik Berlin
D-12489 Berlin, Rudower Chaussee 5
ABSTRACT
A successive method is presented suitable to obtain the small-signal HBT hybrid-T equivalent
circuit elements. The applied equivalent circuit consists of an intrinsic transistor embedded in
parasitic elements. This new method is a deembedding procedure of that intrinsic transistor and
an analytical solution for its elements. The parasitics are determined by special measurements
and subtracted stepwise. In conjunction with the extracted bias independent elements, the
large-signal model parameters of Si/SiGe HBTs are obtained from additional DC measurements.
INTRODUCTION
This paper is focused on the small-signal equivalent circuit and the parameters of a largesignal model implemented in LIBRA for CAD purposes. Most of the methods applied to Si
BJTs [1] recommended for the determination of the parameters of the Ebers-Moll model
type 3 or the Gummel-Poon model for the lower GHz range fail for HBT microwave applications. Several ways to extract the parameters of AlGaAs/GaAs HBTs have been reported [2][7]. These methods also need modifications for the application to the present Si/SiGe HBTs
under test. The direct extraction of the circuit parameters is the approved method for
MESFETs and HEMTs [8][9] and a similar procedure should be developed for Si/SiGe HBTs.
We combine measurements of test patterns and the device itself at different bias points for
this task. The most serious problem arises from the distributed nature of the base electrode
which results in the feedback capacitance Cfb. There are various approaches to deal with this
capacitance. The value can be obtained by optimization [3], or from the non-conducting state
[5], or from an analytical solution of the whole network of the active transistor [7], sometimes
the existence of Cfb is ignored [10]. In the present paper the non-conducting state (off-mode)
is used.
For our investigations, Si/SiGe HBTs developed at the Daimler-Benz Research Center, Ulm
have been used. The on-wafer microwave measurements were performed with a VNA
HP 8510C with Picoprobe 67A probe tips and a LRM calibration. The power of the probing
signal is as low as possible and the power sweep mode was used in order to keep the power
level nearly frequency independent. The DC measurements were done in the same probing
configuration with a HP 4145B semiconductor parameter analyzer. The parameters for the
LIBRA model were taken from the forward and reverse Gummel plots and from the output
current voltage characteristics.
THE EQUIVALENT CIRCUIT
This work is based on the hybrid-T equivalent circuit of the HBT in common emitter configuration (Fig. 1). It contains some additional elements which are not necessary for a general
description of the transistor function but they are important for a close fit of the S parameters
in the lower and upper frequency ranges. The pad capacitances CBE, CCE, and CBC are only a
few Femtofarad but they are effective above 30 GHz. The parallel RBP and CBP are effective
only in the lower GHz range. They arise presumably from non-ideal base contact behaviour
and are introduced to explain the Z11 dependence.
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Fig. 1: Small-signal equivalent circuit of the Si/SiGe HBT.
DEEMBEDDING OF THE INTRINSIC TRANSISTOR
All external parasitic elements are subtracted until the remaining intrinsic elements can be
obtained by an analytical solution. The procedure of deembedding is explained in Fig. 2.
In the first step, the S parameters measured in the frequency range 0.1-50 GHz are transformed to admittance parameters from which the pad capacitances CBE, CBC, and CCE are
removed. In the following steps successive transformations from an impedance matrix to an
admittance matrix and reverse are carried out removing the right elements after each transformation as shown in Fig. 2. After removing RC2, the last external element, the intrinsic transistor
is obtained analytically.
Fig. 2: Deembedding procedure for the intrinsic HBT.
DETERMINATION OF THE EXTERNAL ELEMENTS
1) Pad capacitances CBC, CBE, CCE
The pad capacitances are estimated from S parameter measurements of a test structure consisting of the contact areas and conducting leads only. The active layers are left out in this
structures. The value of CCE is smaller than in the complete transistor because of the absence
of the subcollector in the test structure.
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2) Parasitic inductances LB, LE, LC
The inductances are estimated from a similar test structure with shorts instead of the
transistor.
3) Emitter resistance RE
The external emitter resistance RE can be determined very exactly from the current dependence of Re(Z12) [4]. The measured values at different emitter currents IE are extrapolated to
1/IE approaching zero. The Y-axis intercept point gives the resistance RE. The frequency range
0.1-10 GHz is recommended.
4) Elements of the base-emitter diode RB1, RBP, CBP
These elements are parts of the base-emitter diode. One-port measurements of the forward
biased diode give informations of these quantities and of the base-emitter junction
capacitance Cje, too. The boundary value of Re(Z11) at the highest frequencies (50 GHz) leads
to RB1+RE. RBP and CBP are obtained from both Re(Z11) and Im(Z11) in the lower GHz range
(0.1-10 GHz).
5) Collector resistances RC1, RC2
The sum of these resistances together with RE determines the slope of the current-voltage
characteristics in the saturation range. The individual values of RC1 and RC2 cannot be obtained. We observed by subsequent S parameter comparisons of the active transistor that in
the present case RC2 can be neglected.
6) Feedback capacitance Cfb
This element is the most difficult to determine. It must be assumed to be slightly bias
dependent. We have determined Cfb from a S parameter measurement in the off-mode, i. e.
with no base and collector currents but with a collector voltage applied. The equivalent circuit
of the off-mode is shown in Fig. 3 [5].
Fig. 3: Small-signal equivalent circuit of the off-mode.
The four capacitances in Fig. 3 cannot be calculated by an analytical solution since only three
equations can be derived from the impedance matrix. Due to the frequency independent
coupling, there is no further simplification applicable yielding additional equations.
We solved the problem by an initial estimation of C2 from the measured Im(Z12). This reduces
the number of unknowns and Cfb can now be calculated. A first proof of the usefulness of the
estimation of C2 is the frequency independence of the resulting Cfb. An example (Fig. 4)
shows that Cfb is nearly frequency independent up to 20 GHz. A second proof is the
frequency independence of Rb2 obtained from the analytical solution which is very sensitive
to Cfb.
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Fig. 4: Frequency dependence of Cfb.
THE ANALYTICAL SOLUTION FOR THE INTRINSIC TRANSISTOR
The elements of the impedance matrix of the intrinsic transistor are given by (1)-(4). After the
deembedding procedure including various transformations the actual values are now available
in form of the four Z parameters at each frequency.
Z 11 = Rb 2 + R E +
Z 12 = R E +
R je
1+ jω C je R je
R je
1+ jω C je R je
(2)
 1

+ jω C jc
R je - α 
 R jc

Z 21 = R E +
1+ jω C je R je
R je Z 22 = R E +
(1)
1
1
+ jω C jc
R jc
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+
-1
(3)
α
1
+ jω C jbc
R jc
1+ jω C je R je
(4)
The intrinsic base resistance follows from:
Rb 2 = Z 11 - Z 12
(5)
Rje can be obtained from Re(Z11) in the limit ω → 0.
R je = Re( Z 12 ) - R E
(6)
The base-emitter junction capacitance follows from Im(Z11) and Rje.
C je =
2
R2je - 4Im( Z 11 ) - R je
2ω R je Im( Z 11 )
(7)
This formula did not yield useful results in the present example. Therefore, we take the value
from the measurement of the base-emitter diode as mentioned above.
The current gain a results from Z21 and the quantities determined previously.
α=
R je - ( Z 21 - R E )(1+ jω C je R je )
Z 22 - Z 21
(8)
In the low frequency limit a can also be calculated from the formula α0 = (Z12 - Z21)/Rjc with
Rjc = Z22 - Z21.
The capacitance Cjc and the collector resistance Rjc can be obtained from the following
formula:
1
1
1
+ jω
=
.
R jc
C jc
Z 22 - Z 21
(9)
The procedure yields all elements of the small-signal equivalent circuit without the need of
optimization. Fig. 5 shows a comparison of calculated S parameters from the small-signal
model with measurements at the bias point of VCE = 2 V, IC = 7.5 mA, IB = 30 mA. The computed S parameters agree well with the experimental data throughout the 0.1-50 GHz range.
THE LARGE-SIGNAL MODEL PARAMETERS
The first problem is the mathematical description of the HBTs current voltage characteristics.
The Gummel-Poon model is utilized to model the collector and base current, IC and IB, in
dependence on the base collector voltage VBC and the base emitter voltage VBE.


 V BE 
 V BC   I S 
 V BC  
 V BC  
 exp
I C = I S  exp
 - exp
  - 1 - I SC  exp
 - 1 (10)
 nF V T 
 nRV T   β R 
 nRV T  
 nCL V T  


IB=


 V BE  
 V BC  
 V BE  
 V BC  
IS 
IS 
 exp
 exp
 - 1 +
 - 1 + I SE  exp
 - 1 + I SC  exp
 - 1 (11)
βF 
βR 
 nF V T  
 nRV T  
 n EL V T  
 nCL V T  


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Tab. 1: Elements of the equivalent circuit. Dimensions : fF, Ohm, ps. The external parasitics
are: LB = LE = 15 pH, LC = 30 pH, CBC = CBE = 5 fF, CCE = 25 fF.
Rb1
Rb2
RC1
RC2
Rjc
3
24
0
4
170k 7.9
Rje
RE
RBP
CBP
Cfb
Cjc
Cje
α0
τ
8
30
150
39
24
73
.996
.7
Fig. 5: Comparison of modeled and measured small-signal S parameters.
According to the procedure recommended for bipolar transistors [1][11][12] the parameters
describing the DC characteristics were determined. These are measurements of the transistor
at forward and reverse bias conditions, i. e. log IC(VBE) and log IB(VBE) for VBC = 0 (Fig. 6)
and log IE(VBC) and log IB(VBC) for VBE = 0, respectively. The collector current is only partly
described by (10). The low current part is covered by a substrate current which is not
included in (10). The current gain should be obtained from a part of the Gummel plot where
the ratio IC/IB is constant. Due to the large value of the base resistance this part does not exist.
1e4
IB, IC (µA) log. scale
12
IC
1e2
Collector current (mA)
10
IB=60µA
8
IB
40µA
6
1e0
4
20µA
2
1e-2
0
0
0.2
0.4
0.6
0.8
1
Base-emitter voltage (V)
1.2
0
0.5
1
1.5
2
Collector-emitter voltage (V)
2.5
Fig. 7: Output current-voltage characteristics. Solid lines - modeled.
Fig. 6: Forward Gummel plots.
Solid lines - modeled.
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So the current gain was taken from the small-signal model and the output characteristics,
respectively. The parasitic resistances are taken from the small-signal analysis.
In Fig. 7 measured and calculated output characteristics of a Si/SiGe HBT are shown. The
used DC parameters of the Gummel-Poon model determined as mentioned above are:
IS = 2.81×10-15 A
βF = 150
nF = 1.05
ISE = 9×10-9 A
nEL = 7.16
βR = 0.0313
nR = 1.06
ISC = 1.1×10-8 A
nCL = 3.19
RB = 43 Ω
RE = 8 Ω
RC = 4 Ω
The agreement between calculated and measured DC characteristics is sufficient for a first
step of nonlinear modeling. An improvement is possible by accounting for a current dependent βF and the introduction of a breakdown voltage.
The second problem in large-signal modeling is the correct description of the AC behaviour
at each bias point. First attempts of a linear analysis at some bias points of the HBT simulated
with the large-signal model show acceptable agreements between measured and computed S
parameters. The parameters describing the microwave behaviour are taken from
the small-signal analysis at several bias points.
A final proof has to be done by investigating other HBTs of this type.
CONCLUSION
A direct extraction method for the circuit elements of HBTs in hybrid-T configuration has
been developed. It requires microwave S parameter measurements of open and short teststructures, of the base-emitter diode and of the transistor in the off-mode and at several bias
points in th1e normal operation mode. The method has been successfully applied to Si/SiGe
HBTs produced by the Daimler-Benz Research Center, Ulm. A first attempt of a nonlinear
model for the use in LIBRA led to reasonable results in modeling the current voltage
characteristics of these HBTs.
ACKNOWLEDGMENT
The authors thank Dr. A. Schüppen and Dr. J.-F. Luy from Daimler-Benz Research Center,
Ulm for supplying the Si/SiGe HBTs.
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