I. INTRODUCTION
Recently, heterojunction bipolar transistors (HBTs) have been widely applied to design
highspeed analog, digital and mixedsignal integrated circuits because of their highspeed
capability and highpower applications ^{(1}^{6)}. Consequently, an accurate smallsignal model for HBTs is very important tool used
to characterize transistors and improve fabrication process. There are two usuallyused
extraction methods for the determination of elements parameters in smallsignal models,
which are namely optimization method and direction method. However, direction method
is more accurate than optimization method, since the utility of the optimization method
is severely dependent on the initial value of elements. In the direct extraction methods,
one often adopted π ^{(7}^{9)} or T ^{(10}^{13)} topologies to represent the smallsignal equivalent circuits. Because the T topology
is developed by the physical mechanism of BJT devices, it has been widely adopted
over the last decade. However, a π topology is reduced rather than a T topology while
the common largesignal model (e.g., VBIC model, GummelPoon model) is linearized
^{(14)}.
In our previously published article ^{(15)}, a direct parameterextraction technique for InP HBT smallsignal model was presented,
which is based on peeling algorithm. In the parameterextraction technique, a πtype
smallsignal model was adopted. However, the smallsignal model only considered the
principle characteristics of HBTs, ignoring basecollector capacitance distributed
effect and AC current crowding effect. The model ignoring basecollector capacitance
distributed effect, presents a physical limitation to describe the microwave behavior
of the emitterup HBT structure. If the basecollector capacitance distributed effect
and AC current crowding effect are considered in the smallsignal equivalent circuit,
the method for direct parameter extraction would become different.
Most of the other published smallsignal equivalent circuits also account only for
pure base resistance to describe the base impedance ^{(7,}^{10}^{14,}^{16}^{19)}. Yet, a simply $R_{c}$ circuit could be adopted to model the AC current crowding,
which has been realized in ^{(20,}^{21)}. A. Oudir et al. ^{(20)} extracted the basespreading capacitance in a Ttype smallsignal model, however,
the smallsignal model also does not include the distributed basecollector capacitance
effect, which is valid to describe collectorup structure but appears a physical defect
for the configuration of emitterup devices. W. B. Tang et al. ^{(21)} considered both the AC current crowding effect and basecollector capacitance distributed
effect, however, the smallsignal model was developed based on a Ttype smallsignal
equivalent circuit, moreover, there are many simplified approximations in the process
of parameter extraction, which leads to inaccurate parameter extraction.
Fig. 1. 3D schematic view of an emitterup InP HBT structure.
In this study, to overcome the drawbacks mentioned above, a systematic and rigorous
extraction procedure for a complete InP HBT πtype smallsignal model parameters is
proposed, and there is no any simplified approximation step throughout the parameter
extraction process. This paper is organized as follows. The integral smallsignal
model, together with its twoport Yparameters, are derived and shown in Section II.
In order to accurately extract the model elements, the rigorous methodology of deriving
closedform equations is shown in Section III. In Section IV, the parameter extraction
results and verification for the smallsignal model are given and discussed. Finally,
the conclusions are summarized in Section V.
Fig. 2. Complete HBT smallsignal equivalentcircuit model.
Table 1. Symbols of all elements in the model
Module

Symbol

Description

Parasitic elements

$C_{pbc}$

Parasitic capacitance between base and collector

$C_{pbe}$

Parasitic capacitance between base and emitter

$C_{pce}$

Parasitic capacitance between collector and emitter

$L_{b}$

Lead inductance associated with base

$L_{c}$

Lead inductance associated with collector

$L_{e}$

Lead inductance associated with emitter

$R_{b}$

Series resistance associated with base

$R_{c}$

Series resistance associated with collector

$R_{e}$

Series resistance associated with emitter

Extrinsic basecollector parasitic

$C_{bcx}$

Extrinsic basecollector capacitance

Intrinsic model

$R_{bi}$

Basespreading resistance

$C_{bi}$

Basespreading capacitance

$R_{bc}$

Dynamic basecollector resistance

$R_{be}$

Dynamic baseemitter resistance

$C_{bc}$

Intrinsic basecollector capacitance

$C_{be}$

Intrinsic baseemitter capacitance

$g_{m0}$

Transconductance

τ

Transit time

II. SMALLSIGNAL MODEL
Fig. 1 displays a 3D representation of an emitterup InP HBT, associated with smallsignal
equivalent circuit. In this work, a πtype smallsignal model with pad parasitic elements
applied to HBTs is presented in Fig. 2. According to the physical characteristics of the device, the completed small signal
model is divided into three small modules: parasitic elements, extrinsic basecollector
parasitic elements, and intrinsic model elements. The parasitic elements is independent
of device operating status. The symbols of all elements in the smallsignal model
as shown in Fig. 2 are listed in Table 1.
The Ymatrix of the proposed HBT smallsignal model, as shown in Fig. 1, is expressed as (1).
with
where [$Z_{ex}$] is the Zmatrix of the equivalent circuit including the intrinsic
model and extrinsic basecollector parasitic model, [$Z_{in}$] is the Zmatrix of
the intrinsic model, [$Z_{ini}$] and [$Y_{ini}$] are the Zmatrix and Ymatrix of
the model after peeling $C_{bi}$ and $R_{bi}$ off from the intrinsic model, respectively.
And
The extraction steps for the equivalentcircuit elements parameters only depending
on Sparameters measured data will be depicted in the following section.
III. PARAMETER EXTRACTION PROCEDURE
To accurately and intuitively determine the model elements, the exaction equations
are derived from Sparameters by peeling peripheral elements from smallsignal models
to get reduced ones.
A. Determination of Parasitic Elements
The parasitic elements are independent of the bias conditions. To obtain parasitic
pad parameters, one could employ open and short test structures or cutoff coldHBT
method. However, the cutoff operation method has insufficient accuracy ^{(16)}. Thus, the open and short technique was used to determine parasitic pad parameters
in this study ^{(10)}.
The series resistances ($R_{b}$, $R_{c}$, and $R_{e}$) were determined adopting the
commonlyused opencollector method.
B. Determination of Extrinsic Basecollector Parasitic Elements
The series resistances ($R_{b}$, $R_{c}$, and $R_{e}$), and parasitic pad elements
($C_{pbe}$, $C_{pbc}$, $C_{pce}$, $L_{b}$, $L_{c}$, and $L_{e}$) are removed from
the HBT smallsignal model as follows which is concluded from (1) and (2).
Combining (3), (4) and (5), we can calculate the Ymatrix of the equivalent circuit including the intrinsic
model and extrinsic basecollector parasitic model, and it is shown as (13).
with D=$R_{bi}$$Z_{be}$+$R_{bi}$$Z_{bc}$+$Z_{bc}$$Z_{be}$. After some calculations,
using the Yparameters given in (13), $Z_{bi}$ can be extracted which is expressed as:
where $Y_{T}$=$Y_{ex,12}$+$Y_{ex,22}$, ΔY=$Y_{ex,11}$$Y_{ex,22}$$Y_{ex,12}$$Y_{ex,21}$,
∑Y= $Y_{ex,11}$+$Y_{ex,12}$+$Y_{ex,21}$+$Y_{ex,22}$, and $Y_{ex,ij}$ represent the
Yparameters of Ymatrix [$Y_{ex}$].
By referring to (6) and (14), adopting some simple analysis, we note that the basespreading resistance $R_{bi}$
and the basespreading capacitance $C_{bi}$ can be determined as follows:
The Ymatrix [$Y_{ex}$] is transformed to Zmatrix [$Z_{ex}$]. It can be expressed
as (17).
with A=(1+$g_{m}$$Z_{be}$)[1+$Y_{bcx}$($Z_{bi}$+$Z_{bc}$)], After some calculations
from (17), the following expression can be deduced:
with ZM=$Z_{ex,11}$+$Z_{ex,22}$$Z_{ex,12}$$Z_{ex,21}$, and ZN=$Z_{ex,11}$$Z_{ex,12}$.
After substituting (14) into (18), the $Y_{bcx}$ can be rewritten as:
where
Fig. 3. Extracted $C_{bcx}$ versus frequency.
Finally, through making the real part of (19) equal to zero, the extrinsic basecollector capacitance $C_{bcx}$ can be expressed
as:
The $C_{bcx}$ are determined using (28), and the $C_{bcx}$ characteristics, at $V_{CE}$=2.7 V and $I_{C}$=15.5 mA, is shown
in Fig. 3. The average value over the entire frequency range is assigned to $C_{bcx}$.
C. Determination of Intrinsic Model Elements
After peeling off $C_{bcx}$, the device model is reduced to the intrinsic HBT model.
From (3), the Ymatrix for the intrinsic HBT model can be given by (29).
Next, if the $R_{c}$ circuit consisting of $R_{bi}$ and $C_{bi}$ is also peeled off,
the resultant intrinsic HBT model called INI model is left. The values of $R_{bi}$
and $C_{bi}$ can be evaluated using (15) and (16), respectively. The extracted values of $R_{bi}$ and $C_{bi}$ for the bias condition
($V_{CE}$=2.7 V, $I_{C}$=15.5 mA) are presented in Fig. 4 and 5, respectively.
From (4), the twoport Zmatrix of the INI model is given as (30).
The Zmatrix for the INI model is transformed to Ymatrix [$Y_{ini}$]. It can be written
by (31).
After some calculations, using the Ymatrix given in (31) and referring to (6)(7), the following equations are obtained to determine the intrinsic elements ($R_{bc}$,
$C_{bc}$, $R_{be}$, $C_{be}$, $g_{m0}$ and τ).
where $Y_{ini}$,ij are the Yparameters of the INI equivalent circuit. Eqs. (32)(37) permit to extract the INI model elements ($R_{bc}$, $C_{bc}$, $R_{be}$, $C_{be}$,
$g_{m0}$ and τ) in case the Yparameters of the INI model are calculated using (29)(31). Fig. 611 show the extracted INI model parameters versus frequency for a 1×15 µm$^{2}$ InP
HBT biased at $V_{CE}$=2.7 V and $I_{C}$=15.5 mA.
Fig. 4. Extracted $R_{bi}$ versus frequency.
Fig. 5. Extracted $C_{bi}$ versus frequency.
Fig. 6. Extracted $R_{bc}$ versus frequency.
Fig. 7. Extracted $R_{be}$ versus frequency.
Fig. 8. Extraction of $C_{bc}$.
Fig. 9. Extraction of $C_{be}$.
Fig. 10. Extracted $g_{m0}$ versus frequency.
Fig. 11. Extracted τ versus frequency.