An improved direct extraction method for InP HBT small-signal model

Jun-Jun QI; Hong-Liang LYU; Yu-Ming ZHANG; Yi-Men ZHANG; Jin-Can ZHANG

doi:10.11972/j.issn.1001-9014.2020.03.005

Abstract

In this paper, an improved direct extraction method to extract the model parameters in InP heterojunction bipolar transistor （HBT） small-signal equivalent circuit is presented and successfully applied to small-signal equivalent circuit of InP HBT. The distributed base-collector capacitance effect is taken into consideration in the adopted model. The extracting process of this method, which extracts parameters in turn from the peripheral parasitic elements to the intrinsic internal elements, is clearer than other direct extraction methods. Except for the parasitic parameters, all other parameters are calculated without any simplified approximation. This method relies on S parameters measurement. All of the equivalent circuit parameters are extracted directly from the S parameters without using approximations based on initial values. The direct extraction method is successfully validated on InP HBT in the frequency range of 0.1 ~ 40 GHz, and excellent agreement is achieved between the measured and calculated S parameters over the whole frequency range.

Keywords

direct extraction method InP HBT parameter extraction small-signal model

Introduction

Owing to the characteristics of high-speed and high-frequency, InP HBT has been one of the most promising devices for future applications at microwave and millimeter-wave frequencies [1, 2, 3] . Small-signal model is the footstone of the entire transistor microwave model [4], and therefore accurate InP HBT device model is of great significance for the development of microwave and millimeter-wave integrated circuits.

With scaling down transistors’ size, in general, the structure of C-up HBT devices can ignore the extrinsic base-collector capacitance because the extrinsic area corresponding to the parasitic capacitance can be neglected [5] . Compact with the C-up devices, the E-up devices have a larger extrinsic area and need to consider the distributed base-collector capacitance effect. The effect is represented by extrinsic and intrinsic capacitances. By considering the effect, not only can this more clearly characterize the physical meaning of the base-collector capacitance, but it can further improve the model to make the model more accurate. In Ref.[6], although the authors have considered the effect in small-signal equivalent circuits, the calculation process is complicated and the calculation amount is too large. In Ref.[7,8], the approximations used in the extraction process leads to inaccurate parameter extraction.

In recent years, many methods that extract small signal model parameters have been reported, mainly including direct extraction method [2, 3, 4, 6, 7, 8] and numerical optimization method [9, 10] . The numerical optimization method uses numerical methods to locate the optimal parameter values, and finally obtains simulated results with good fitting characteristics to the measured results. Nevertheless, the method depends on parameters’ initial value and may not even converge. With the direct extraction method, each parameter of the equivalent circuit could be extracted using equations. However, the disorganized extraction process makes the direct extraction method very computationally intensive and complicated. In Ref.[4, 6-7], although the authors used some simplified approximations, they also obtained more complex parameter expressions.

In order to overcome these difficulties, an improved direct extraction method for InP HBT small-signal model is proposed. This method in turn extracts the parameters of small-signal equivalent circuit from the peripheral elements to the internal elements. Compact with the other direct extraction methods, the method has clear extraction process, simple calculation of parameters, and few approximations calculation.

1 Small-signal equivalent circuit model

The adopted hybrid-π equivalent circuit for HBT small-signal modeling is shown in Fig.1 . This equivalent circuit includes two parts, i.e., the inner part contains intrinsic elements, and the outer part contains extrinsic elements [11] . In this model, C pbc 、 C pbe and C pce are pad parasitic capacitances, L c 、 L e and L b are pad parasitic inductances, R c, R b and R e are extrinsic resistances of collector, base, and emitter, respectively. These extrinsic elements are considered to be bias independent.

Figure 1.HBT small-signal equivalent circuit

Intrinsic elements are supposed to be bias dependent, mainly including the dynamic base resistance R bi, the dynamic base-emitter resistance R be, the base-emitter capacitance C be, the base-collector capacitance C bc, the dynamic base-collector resistance R bc, DC transconductance g m0 and delay time τ. Besides, C bcx is extrinsic base-collector capacitance, and it is considered to be bias independent.

2 Parameter extraction procedure

2.1　Extraction of parasitic parameters and the extrinsic resistances

Pad parasitic parameters consist of parasitic capacitances and inductances which are extracted by the Open Test Structure and the Short Test Structure [12], respectively. The extrinsic resistances are extracted by the open-collector method [13], and the equivalent circuit diagram is illustrated in Fig.2 .

Figure 2.open-collector equivalent circuit

The Z-parameters of the open-collector equivalent circuit in Fig. 2 is written as

Z_{11} = R_{b} + R_{b i} + \frac{R_{b e}}{1 + g m \cdot R_{b e}}

Z_{12} = R_{e} + \frac{R_{b e}}{1 + g m \cdot R_{b e}}

Z_{21} = R_{e} + (1 - g m \cdot R_{b e}) \cdot \frac{R_{b e}}{1 + g m \cdot R_{b e}}

Z_{22} = R_{c} + R_{e} + (1 + \frac{R_{b c}}{R_{b e}}) \cdot \frac{R_{b e}}{1 + g m \cdot R_{b e}}

R_{b e} = \frac{η b e K T}{q I b e}

R_{b c} = \frac{η b c K T}{q I b c}

When I b approaches \infty, R be and R bc become very small at approximately 0 because the junction resistance and the junction current are inversely proportional. Moreover, with the increasing of the base current I b, the total resistance of the base gradually approaches the base contact resistance [13], i.e., R b + R bi \approx R b . Therefore, the intrinsic resistances R b 、 R e and R c can be obtained （taking the real part of the Z parameter to indicate the resistance value which makes the extraction result more accurate）:

R_{b} = r e a l (Z_{11} - Z_{12})

R_{e} = r e a l (Z_{12})

R_{c} = r e a l (Z_{22} - Z_{12})

The relationship between real（ Z 11 - Z 12 ）, real（ Z 22 - Z 12 ） and real（ Z 12 ） and 1/ I b is linearly extrapolated to the ordinate to obtain the values of R b, R c and R e, as shown Fig.3 [14] . The extraction values of the bias-independent elements are tabulated in Table 1 .

Figure 3.Plots of the real part of Z₁₁-Z₁₂, Z₂₂-Z₁₂ and Z₁₂ versus 1/I_b

parasitic parameters	V_ce=2.5 V, I_c=12.5 mA
C_pb_c/fF	2.62
C_pb_e/fF	15.60
C_pce/fF	16.20
L_b/pH	52.25
L_c/pH	57.75
L_e/pH	8.88
R_e/Ω	4.27
R_b/Ω	1.77
R_c/Ω	7.31

Table 1. Extraction of extrinsic parameters values

View all Tables

2.2　Extraction of the extrinsic base-collector capacitance

Once all the parasitic elements are de-embedded, only the extrinsic base-collector capacitance C bcx and the intrinsic elements （inside the dashed line） are remained in the equivalent circuit, as shown in Fig.4 . To overcome the problem of unclear parameter extraction process, we need to extract sequentially from the external circuit to the internal circuit, that is, we need to extract C bcx first. However, the intrinsic equivalent circuit of the small signal equivalent circuit needs to be analyzed first before extracting the extrinsic base-collector capacitance.

Figure 4.equivalent circuit after de-embedding off parasitic elements

The Z-parameters of the equivalent circuit after de-embedding in Fig. 4, is written as

[Z] = (\begin{matrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{matrix}) = (\begin{matrix} \frac{g m Z_{b e} R_{b i} + Z_{b e} + R_{b i} + D Y_{b c x}}{A} & \frac{Z_{b e} + D Y_{b c x}}{A} \\ \frac{Z_{b e} + D Y b c x - Z_{b c} Z_{b e} g m}{A} & \frac{Z_{b e} + Z_{b c} + D Y_{b c x}}{A} \end{matrix})

where ,,, , , and

From （10）, the dynamic base resistance R bi can be expressed as

R_{b i} = \frac{(Z_{Δ})}{1 - Y_{b c x} (Σ Z)}

where , and

The extrinsic base-collector capacitance can be expressed by taking the imaginary part of equation （11） equal to 0 as

C_{b c x} = \frac{1}{ω} \frac{- i m a g (Z_{Δ})}{i m a g (Z_{Δ}) i m a g (Σ Z) + r e a l (Z_{Δ}) r e a l (Σ Z)}

Fig.5 shows the extracted result as a function of frequency for ωC bcx at V ce =2.5V, I c =12.5mA.

Figure 5.ωC_bcx versus frequency

2.3　Extraction of the intrinsic elements

Once the parasitic elements and extrinsic base-collector capacitance is de-embedded, the remaining intrinsic elements of the small signal equivalent circuit model can be directly determined. The Z-parameter corresponding to the intrinsic circuit can be written as

[Z_{i n}] = (\begin{matrix} Z_{i n, 11} & Z_{i n, 12} \\ Z_{i n, 21} & Z_{i n, 22} \end{matrix}) = (\begin{matrix} R_{b i} + \frac{Z_{b e}}{1 + g_{m} Z_{b e}} & \frac{Z_{b e}}{1 + g_{m} Z_{b e}} \\ \frac{Z_{b e} - g_{m} Z_{b e} Z_{b c}}{1 + g_{m} Z_{b e}} & \frac{Z_{b e} + Z_{b c}}{1 + g_{m} Z_{b e}} \end{matrix})

From （13）, the intrinsic base resistance R bi can be expressed as

R_{b i} = r e a l (Z_{i n, 11} - Z_{i n, 12})

Fig.6 shows the extracted result as a function of frequency for R bi at V ce =2.5V, I c =12.5mA. Then de-embed off R bi and get a new small signal equivalent circuit, as shown in Fig.7 .

Figure 6.R_biversus frequency

Figure 7.the equivalent circuit of de-embedding R_bi

The Y parameter of the corresponding equivalent circuit （ Fig.7 ） could be expressed as

[Y_{i n t}] = (\begin{matrix} Y_{i n t, 11} & Y_{i n t, 12} \\ Y_{i n t, 21} & Y_{i n t, 22} \end{matrix}) = (\begin{matrix} \frac{1}{Z_{b e}} + \frac{1}{Z_{b c}} & - \frac{1}{Z_{b c}} \\ - \frac{1}{Z_{b c}} + g_{m} & \frac{1}{Z_{b c}} \end{matrix})

From （15）, the intrinsic elements can be expressed as

R_{b c} = - \frac{1}{r e a l (Y_{i n t, 12})}

C_{b c} = - \frac{i m a g (Y_{i n t, 12})}{ω}

R_{b e} = \frac{1}{r e a l (Y_{i n t, 11} + Y_{i n t, 12})}

C_{b e} = \frac{i m a g (Y_{i n t, 11} + Y_{i n t, 12})}{ω}

g_{m 0} = m a g (Y_{i n t, 21} - Y_{i n t, 12})

τ = \frac{- 1}{ω} t a n^{- 1} (\frac{i m a g (Y_{i n t, 21} - Y_{i n t, 12})}{r e a l (Y_{i n t, 21} - Y_{i n t, 12})})

3 Results and discussion

A hybrid-π small-signal equivalent circuit with distributed base-collector capacitance effect was adopted to study the microwave and millimeter-wave behavior of the InP HBT. An improved direct extraction method has been established to accurately extract the small-signal parameters. The improved method extracts parameters from the peripheral circuit to the internal circuit and gives a clearer solution process. Extraction results of small signal equivalent circuit are depicted in table 2, for bias points.

Intrinsic parameters	I_c=2.5mA	I_c=12.5mA	I_c=22.5mA
C_bcx/fF	49.21	35.60	58.31
R_bi/Ω	36.22	8.32	2.45
R_be/Ω	2497.68	121.02	113.50
R_bc/Ω	1.90×10⁴	2.66×10⁴	1.04×10⁴
C_bc/fF	32.03	18.26	16.59
C_be/fF	145.30	718.85	915.30
g_m0/S	0.08	0.60	1.19

Table 2. Extraction of Intrinsic Parameters at V_ce=2.5V

View all Tables

Fig. 8 shows the calculated S -parameters of the small-signal equivalent circuit of the HBT including the distributed base-collector capacitance with the measured data. The comparison shown in Fig. 8 demonstrates a good agreement from 0.1 ~ 40.0 GHz, which also verifies the validity of the model and extraction techniques.

Figure 8.Measured and calculated S-parameters of 1×15 μm² InP HBT between 100 MHz~40 GHz

However, the Smith plots of S parameters do not clearly reflect agreement of fit between measured and calculated data. The residual error between the measured results and the calculated results are quantified using the following equation [16]

E = \frac{1}{4 N} \overset{2}{\sum_{i, j = 1}} \overset{N}{\sum_{k = 1}} \frac{|S_{i j}^{m} (f_{k}) - S_{i j}^{c} (f_{k})|}{m a x i j (|S_{i j}^{m} (f_{k})|)}

where N is the number of frequency points, S ij m （ f k ） and S ij c （ f k ） are the measured and calculated S-parameters at frequency f k, respectively. The residual errors between the measured and modeled S-parameters are around 3.3~3.8%.

4 Conclusion

An improved direct extraction method for the hybrid-π small-signal equivalent circuit with the base-collector capacitance effect has been proposed. The extracting process of this method, which extracts parameters in turn from the peripheral parasitic elements to the intrinsic internal elements, is clearer than other direct extraction methods. Furthermore, this method can extract all intrinsic parameters directly by the equation without approximation and numerical optimization. Good agreement is obtained between calculated and measured results for an InP HBT with 1\times15 μm 2 emitter area over a wide range of bias points up to 40GHz.

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