I. INTRODUCTION
Recently, heterojunction bipolar transistors (HBTs) have been widely applied to design
high-speed analog, digital and mixed-signal integrated circuits because of their high-speed
capability and high-power applications (1-6). Consequently, an accurate small-signal model for HBTs is very important tool used
to characterize transistors and improve fabrication process. There are two usually-used
extraction methods for the determination of elements parameters in small-signal 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 small-signal 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 large-signal model (e.g., VBIC model, Gummel-Poon model) is linearized
(14).
In our previously published article (15), a direct parameter-extraction technique for InP HBT small-signal model was presented,
which is based on peeling algorithm. In the parameter-extraction technique, a π-type
small-signal model was adopted. However, the small-signal model only considered the
principle characteristics of HBTs, ignoring base-collector capacitance distributed
effect and AC current crowding effect. The model ignoring base-collector capacitance
distributed effect, presents a physical limitation to describe the microwave behavior
of the emitter-up HBT structure. If the base-collector capacitance distributed effect
and AC current crowding effect are considered in the small-signal equivalent circuit,
the method for direct parameter extraction would become different.
Most of the other published small-signal 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 base-spreading capacitance in a T-type small-signal model, however,
the small-signal model also does not include the distributed base-collector capacitance
effect, which is valid to describe collector-up structure but appears a physical defect
for the configuration of emitter-up devices. W. B. Tang et al. (21) considered both the AC current crowding effect and base-collector capacitance distributed
effect, however, the small-signal model was developed based on a T-type small-signal
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 emitter-up InP HBT structure.
In this study, to overcome the drawbacks mentioned above, a systematic and rigorous
extraction procedure for a complete InP HBT π-type small-signal 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 small-signal
model, together with its two-port Y-parameters, are derived and shown in Section II.
In order to accurately extract the model elements, the rigorous methodology of deriving
closed-form equations is shown in Section III. In Section IV, the parameter extraction
results and verification for the small-signal model are given and discussed. Finally,
the conclusions are summarized in Section V.
Fig. 2. Complete HBT small-signal equivalent-circuit 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 base-collector parasitic
|
$C_{bcx}$
|
Extrinsic base-collector capacitance
|
|
Intrinsic model
|
$R_{bi}$
|
Base-spreading resistance
|
|
$C_{bi}$
|
Base-spreading capacitance
|
|
$R_{bc}$
|
Dynamic base-collector resistance
|
|
$R_{be}$
|
Dynamic base-emitter resistance
|
|
$C_{bc}$
|
Intrinsic base-collector capacitance
|
|
$C_{be}$
|
Intrinsic base-emitter capacitance
|
|
$g_{m0}$
|
Transconductance
|
|
τ
|
Transit time
|
II. SMALL-SIGNAL MODEL
Fig. 1 displays a 3D representation of an emitter-up InP HBT, associated with small-signal
equivalent circuit. In this work, a π-type small-signal 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 base-collector
parasitic elements, and intrinsic model elements. The parasitic elements is independent
of device operating status. The symbols of all elements in the small-signal model
as shown in Fig. 2 are listed in Table 1.
The Y-matrix of the proposed HBT small-signal model, as shown in Fig. 1, is expressed as (1).
with
where [$Z_{ex}$] is the Z-matrix of the equivalent circuit including the intrinsic
model and extrinsic base-collector parasitic model, [$Z_{in}$] is the Z-matrix of
the intrinsic model, [$Z_{ini}$] and [$Y_{ini}$] are the Z-matrix and Y-matrix of
the model after peeling $C_{bi}$ and $R_{bi}$ off from the intrinsic model, respectively.
And
The extraction steps for the equivalent-circuit elements parameters only depending
on S-parameters 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 S-parameters by peeling peripheral elements from small-signal 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 cold-HBT
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
commonly-used open-collector method.
B. Determination of Extrinsic Base-collector 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 small-signal model as follows which is concluded from (1) and (2).
Combining (3), (4) and (5), we can calculate the Y-matrix of the equivalent circuit including the intrinsic
model and extrinsic base-collector parasitic model, and it is shown as (13).
with D=$R_{bi}$$Z_{be}$+$R_{bi}$$Z_{bc}$+$Z_{bc}$$Z_{be}$. After some calculations,
using the Y-parameters given in (13), $Z_{bi}$ can be extracted which is expressed as:
where $Y_{T}$=$Y_{ex,12}$+$Y_{ex,22}$, ΔY=$Y_{ex,11}$$Y_{ex,22}$-$Y_{ex,12}$$Y_{ex,21}$,
∑Y= $Y_{ex,11}$+$Y_{ex,12}$+$Y_{ex,21}$+$Y_{ex,22}$, and $Y_{ex,ij}$ represent the
Y-parameters of Y-matrix [$Y_{ex}$].
By referring to (6) and (14), adopting some simple analysis, we note that the base-spreading resistance $R_{bi}$
and the base-spreading capacitance $C_{bi}$ can be determined as follows:
The Y-matrix [$Y_{ex}$] is transformed to Z-matrix [$Z_{ex}$]. It can be expressed
as (17).
with A=(1+$g_{m}$$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 base-collector 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 Y-matrix 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 two-port Z-matrix of the INI model is given as (30).
The Z-matrix for the INI model is transformed to Y-matrix [$Y_{ini}$]. It can be written
by (31).
After some calculations, using the Y-matrix 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 Y-parameters 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 Y-parameters of the INI model are calculated using (29)-(31). Fig. 6-11 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.