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.

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.

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

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.

IV. RESULTS AND DISCUSSIONS

In order to validate the presented procedure, an n-p-n emitter-up InP HBT with a 1×15 μm$^{2}$ emitter area was investigated. The S-parameters measured data were realized on-wafer thought employing Keysight 8510C Network Analyzer with Keysight B1500A Semicon-ductor Device Analyzer to supply DC source. With all measured instruments by the control of IC-CAP control software, S-parameters of the range of 0.1-40 GHz were obtained.

Applying the extraction method presented in this study, extracted values of the small-signal model parameters, as well as for the model in ⁽¹⁵⁾ (which does not include $C_{bcx}$ and $C_{bi}$), for the 1×15 μm$^{2}$ InP HBT under two bias points of ($V_{CE}$=2.7 V, $I_{C}$=15.5 mA) and ($V_{CE}$=2.5 V, $I_{C}$=17.5 mA) are summarized in Table 2 and 3, respectively. Fig. 12 shows the comparisons between the measured and modeled S-parameters data for two bias points of ($V_{CE}$=2.7 V, $I_{C}$=15.5 mA) and ($V_{CE}$=2.5 V, $I_{C}$=17.5 mA). From Fig. 12, it can be observed that the presented model was found to improve the prediction of S12 and S22, at the higher frequencies. Additionally, the influence of the base-emitter and collector-emitter metallization capacitances on model accuracy was found to be very limited. The residual error ⁽¹⁵⁾ is adopted to quantify the accuracy of the extraction method and the residual errors between the measured and modeled S-parameters at different bias conditions are less than 2.8%, which is better than the other ones given in previously published literatures ^{(13,15,17,20,22)}. The residual errors between the measured and modeled S-parameters, at different bias conditions, for the presented model and the model in ⁽¹⁵⁾ are listed in Table 4. It is shown that the smaller error values are obtained for the model with $C_{bcx}$ and $C_{bi}$ in this article. This comparison indicates that this model presents a better agreement with the experimental data for different bias points. Therefore, this result shows that the small-signal model and parameter extraction method have high accuracy to model HBTs characteristics.

V. CONCLUSION