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A complete small-signal HBT model including AC current crowding effect

本站小编 Free考研考试/2022-01-01



1.
Introduction




Heterojunction bipolar transistors (HBTs) have been extensively used in high-speed analog, digital and mixed-signal integrated circuits (ICs)[1-4]. There are several small-signal equivalent circuit topologies for microwave HBT devices’ modeling and the accuracy of small-signal model plays a crucial role in the design of ICs.



However, most of the reported topologies ignore the AC current crowding effect[5], which is a critical factor for predicting the AC performance of HBT devices, especially in high frequency range. AC current crowding occurs in the high frequency range related to the bypassing effects of base-emitter capacitance. As the small-signal voltage drops along the intrinsic base region, the diffusion capacitance at the edge of the emitter is proportionally higher than that at the center. This phenomenon can cause the small-signal emitter current to crowd near the edge of the base-emitter junction as frequency increases. Therefore, the intrinsic base capacitance is physically connected with the AC current crowding[6].



In recent years, different small-signal equivalent circuits have been proposed with various extraction methods, in which π[7] or T[8, 9] topologies were used to represent the small-signal equivalent circuits. Heung et al.[10] proposed a simple extraction method by using the RC circuit to characterize the AC current crowding effect. However, a simple π-type topology was adopted to extract the intrinsic base capacitance (Cbi) through the Z-parameter equations. In addition, other papers[11-14] have proposed a more complete small-signal model compared with Ref. [10]. Although their methods have shown good accuracy in the extraction process, more cumbersome analytical methods and calculation cost were needed. Zhang et al.[15] proposed a rigorous but concise peeling extraction algorithm. While this method failed to consider the AC current crowding effect, this paper proposes an accurate extraction method with a complete π-type small-signal equivalent circuit considering the AC current crowding effect. This new equivalent-circuit topology is developed based on the small-signal equivalent circuit of an AHBT (agilent heterojunction bipolar transistor) model which has considered the distributed base-collector junction capacitance. Based on the peeling algorithm in Ref. [15] and the T-π transformation presented in Refs. [16–18], a novel method is proposed. In Section 2, an integral small-signal model taking into account the AC current crowding effect is introduced. The parameter extraction procedure is presented in detail in Section 3. Section 4 validates the model performance and a comparison between the proposed model and the conventional model without Cbi. Finally, a conclusion is given in Section 5.




2.
Small-signal model




The complete small-signal equivalent circuit of the HBT device considering the AC current crowding effect is given in Fig. 1.






onerror="this.onerror=null;this.src='http://www.jos.ac.cn/fileBDTXB/journal/article/jos/2021/5/PIC/20080030-1.jpg'"
class="figure_img" id="Figure1"/>



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Figure1.
Complete small-signal equivalent circuit of HBT device including ${C_{{
m{bi}}}} $. $ {{{g}}_{
m{m}}}{
m{ = }}{g_{{
m{m0}}}}{{
m{e}}^{-jomega tau }}$.




In Fig. 1, Rbx, Rcx, Re are the base, collector, and emitter parasitic resistances, respectively. Rbcx and Rbex are the extrinsic base-collector and base-emitter resistances, and Cbcx and Cbex are the extrinsic base-collector and base-emitter depletion capacitances. Rbi and Rci are the intrinsic resistances, and Cbi is the intrinsic base capacitance. The parallel RC circuit composed of Cbi and Rbi characterizes the AC current crowding effect. Rbci, Rbei, Cbci and Cbei are the intrinsic resistances and capacitances, respectively. gm and gm0 are the small-signal and DC transconductances, respectively. τ is the delay time.



Circuit topology demonstrated in Fig. 1 can be simplified to facilitate calculation. We introduce three parameters, Arbe, Arc and Abex with a value range of (0, 1) and stipulate $ {R_{{
m{bex}}}} = {R_{{
m{be}}}}{A_{{
m{rbe}}}}$, ${R_{{
m{bei}}}} = {R_{{
m{be}}}}left( {1-{A_{{
m{rbe}}}}}
ight)$, ${R_{{
m{cx}}}} = {R_{
m{c}}}{A_{{
m{rc}}}}$, ${R_{{
m{ci}}}} = {R_{
m{c}}}left( {1-{A_{{
m{rc}}}}}
ight)$, ${C_{{
m{bex}}}} = {C_{{
m{be}}}}{A_{{
m{bex}}}}$, and ${C_{{
m{bei}}}} = {C_{{
m{be}}}}left( {1-{A_{{
m{bex}}}}}
ight)$.




3.
Extraction procedure





3.1
Parasitic resistances




Since the parasitic resistances Rbx, Rc and Re are independent of the bias condition, we extract them under zero biasing conditions in this work. The calculation equations are given as follows[19]: $ {{R}}_{
m{bx}}= {
m{real}}left({{Z}}_{{11}}{–}{{Z}}_{{12}}
ight) $, $ {{R}}_{
m{c}}= {
m{real}}left({{Z}}_{{22}}{–}{{Z}}_{{12}}
ight) $ and $ {{R}}_{
m{e}}= {
m{real}}left({{Z}}_{{12}}
ight) $. The extracted Rbx, Rc and Re for the 2 × 20 μm2 HBT device are 6.186, 2.561, and 5.526 Ω, respectively. The impedance matrix Zm after de-embedding Rbx, Rc and Re from the total two-port Z-parameters Z can be written as









$$left[ {{Z_{
m{m}}}}
ight] = Z-left[ {begin{array}{*{20}{c}}{{R_{{
m{bx}}}} + {R_{
m{e}}}}&{{R_{
m{e}}}}[2mm]{{R_{
m{e}}}}&{{R_{
m{c}}} + {R_{
m{e}}}}end{array}}
ight].$$

(1)



Through Y–Z transformation, Ym can also be expressed as









$${Y_{{
m{m11}}}} = frac{{{Z_{{
m{be}}}} + {Z_{{
m{bci}}}}}}{N} + {Y_{{
m{bcx}}}},$$

(2)









$${Y_{{
m{m12}}}} = frac{{-{Z_{{
m{be}}}}}}{N}-{Y_{{
m{bcx}}}},$$

(3)









$${Y_{{
m{m21}}}} = frac{{{{
m{g}}_{
m{m}}}{Z_{{
m{be}}}}{Z_{{
m{bci}}}}-{Z_{{
m{be}}}}}}{N}-{Y_{{
m{bcx}}}},$$

(4)









$${Y_{{
m{m22}}}} = frac{{{Z_{{
m{be}}}} + {Z_{{
m{bi}}}}left( {1 + {{
m{g}}_{
m{m}}}{Z_{{
m{be}}}}}
ight)}}{N} + {Y_{{
m{bcx}}}},$$

(5)



where ${Z_{{
m{be}}}} = {R_{{
m{be}}}}/left( {1 + {
m{j}}omega {R_{{
m{be}}}}{C_{{
m{be}}}}}
ight)$, ${Z_{{
m{bci}}}} = {R_{{
m{bci}}}}/left( {1 + jomega {R_{{
m{bci}}}}{C_{{
m{bci}}}}}
ight)$, ${Z_{{
m{bi}}}} = $$ {R_{{
m{bi}}}}/left( {1 + jomega {R_{{
m{bi}}}}{C_{{
m{bi}}}}}
ight)$, ${Y_{{
m{bcx}}}} = 1/{R_{{
m{bcx}}}} + jomega {C_{{
m{bcx}}}}$, $N !=! {Z_{{
m{bi}}}}{Z_{{
m{be}}}} + {Z_{{
m{bi}}}}{Z_{{
m{bci}}}} + {Z_{{
m{be}}}}{Z_{{
m{bci}}}}$.




3.2
Intrinsic part parameters




Once the parasitic resistances are known, the extraction of the intrinsic parameters can be carried out[20]. The circuit is presented in Fig. 2, and the admittance parameter of the intrinsic part Yin can be written as






onerror="this.onerror=null;this.src='http://www.jos.ac.cn/fileBDTXB/journal/article/jos/2021/5/PIC/20080030-2.jpg'"
class="figure_img" id="Figure2"/>



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Figure2.
Small-signal equivalent circuit after de-embedding the extrinsic parameters.










$$left[ {{Y_{{
m{in}}}}}
ight] = left[ {{Y_{
m{m}}}}
ight]-left[ {begin{array}{*{20}{c}}0&00&{{Y_{
m{o}}}}end{array}}
ight].$$

(6)



The circuit in Fig. 2 can be converted to that in Fig. 3 after T-π transformation[21] with $ {{Z}}_{{4}} = {{Z}}_{{2}}{{Z}}_{
m{bcx}}{/}left({{Z}}_{{2}}{{+}{Z}}_{
m{bcx}}
ight) $. The parameters Z1, Z2, and Z3 can be derived from Eq. (6) as follows






onerror="this.onerror=null;this.src='http://www.jos.ac.cn/fileBDTXB/journal/article/jos/2021/5/PIC/20080030-3.jpg'"
class="figure_img" id="Figure3"/>



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Figure3.
Final circuit after T-π transformation.










$${{Z}}_{{1}} = frac{{N}}{{{Z}}_{
m{bci}}}{,}$$

(7)









$${{Z}}_{{2}} = frac{{N}}{{{Z}}_{
m{be}}}{,}$$

(8)









$${{Z}}_{{3}} = frac{{N}}{{{Z}}_{
m{bi}}}{.} $$

(9)



Combining Eqs. (6)–(9), Yin can also be represented by Z1, Z2, Z3, and Z4, which are expressed as[16]









$$left[ {{Y_{{
m{in}}}}}
ight] = left[ {begin{array}{*{20}{c}}{dfrac{1}{{{Z_1}}} + dfrac{1}{{{Z_4}}}}&{-dfrac{1}{{{Z_4}}}}{X dfrac{{{Z_3}}}{{{Z_1}}}-dfrac{1}{{{Z_4}}}}&{dfrac{1}{{{Z_3}}} + dfrac{1}{{{Z_4}}} + X}end{array}}
ight],$$

(10)



in which









$$X = B {{{g}}_{
m{m}}},$$

(11)









$$B = frac{{{Z_1}}}{{{Z_1} + {Z_2} + {Z_3}}}.$$

(12)



Then, we can get that









$${{Z}}_{{1}} = frac{{1}}{{{Y}}_{
m{in11}}+{{Y}}_{
m{in12}}}{,}$$

(13)









$${{Z}}_{{3}} = frac{{{Y}}_{
m{in11}}+{{}{Y}}_{
m{in21}}}{left({{Y}}_{
m{in11}}+{{Y}}_{
m{in12}}
ight)left({{Y}}_{
m{in22}}+{{Y}}_{
m{in12}}
ight)}{,}$$

(14)









$${{Z}}_{{4}} = {-}frac{{1}}{{{Y}}_{
m{in12}}}{.}$$

(15)



Based on the above relationship between Yin and Z1, Z2, Z3, the extraction method reported in Ref. [18] is used to extract the parameters. From Eqs. (13) and (14), we can get









$$frac{{{Z_1}}}{{{Z_3}}} = frac{{{R_{{
m{bi}}}}}}{{{R_{{
m{bci}}}}}}frac{{1 + jomega {R_{{
m{bci}}}}{C_{{
m{bci}}}}}}{{1 + jomega {R_{{
m{bi}}}}{C_{{
m{bi}}}}}},$$

(16)









$${
m{real}}left( {frac{{{Z_1}}}{{{Z_3}}}left( {1 + jomega {T_{{
m{bi}}}}}
ight)}
ight) = frac{{{R_{{
m{bi}}}}}}{{{R_{{
m{bci}}}}}},$$

(17)









$${
m{imag}}left( {frac{{{Z_1}}}{{{Z_3}}}left( {1 + jomega {T_{{
m{bi}}}}}
ight)}
ight) = omega {R_{{
m{bi}}}}{C_{{
m{bci}}}}.$$

(18)



Defining









$$begin{array}{l}{T_{{
m{bi}}}} = {R_{{
m{bi}}}}{C_{{
m{bi}}}},;;;;{F_0} = dfrac{omega }{{{
m{imag}}left( {dfrac{{{Z_1}}}{{{Z_3}}}}
ight)}} = {A_0} + {omega ^2}{B_0},end{array}$$

(19)



where $ {A_0} = 1/{alpha _0} $, ${B_0} = T_{{
m{bi}}}^2/{alpha _0}$, ${alpha _0} = {R_{{
m{bi}}}}left( {{R_{{
m{bci}}}}{C_{{
m{bci}}}} - {T_{{
m{bi}}}}}
ight)/{R_{{
m{bci}}}}$.



Since Z1 and Z3 are given in Eqs. (13) and (14), the ω2 dependence of F0 can be easily plotted in Fig. 4. Then A0 and B0 can be determined. Tbi is defined as






onerror="this.onerror=null;this.src='http://www.jos.ac.cn/fileBDTXB/journal/article/jos/2021/5/PIC/20080030-4.jpg'"
class="figure_img" id="Figure4"/>



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Figure4.
Frequency of F0 versus ω2.










$$ {{T}}_{
m{bi}} = sqrt{frac{{{B}}_{{0}}}{{{A}}_{{0}}}} . $$

(20)



Based on Eq. (7), Z1 can be written as









$$begin{array}{l}{Z_1} = dfrac{{{R_{{
m{bi}}}}}}{{1 + jomega {T_{{
m{bi}}}}}} + dfrac{{{R_{{
m{be}}}}}}{{1 + jomega {T_{{
m{be}}}}}}quadquad + dfrac{{{R_{{
m{bi}}}}}}{{1 + jomega {T_{{
m{bi}}}}}}dfrac{{{R_{{
m{be}}}}}}{{1 + jomega {T_{{
m{be}}}}}}dfrac{{1 + jomega {T_{{
m{bci}}}}}}{{{R_{{
m{bci}}}}}},end{array}$$

(21)









$${Z_1}left( {1 + jomega {T_{{
m{bi}}}}}
ight) = frac{{{R_{
m{x}}}left( {1 + jomega {T_{
m{x}}}}
ight)}}{{1 + jomega {T_{{
m{be}}}}}},$$

(22)



where









$${R_{
m{x}}} = {R_{{
m{bi}}}}{R_{{
m{be}}}}left( {frac{1}{{{R_{{
m{bci}}}}}} + frac{1}{{{R_{{
m{be}}}}}} + frac{1}{{{R_{{
m{bi}}}}}}}
ight),$$

(23)









$${T_{
m{x}}} = frac{{{C_{{
m{bci}}}} + {C_{{
m{be}}}} + {C_{{
m{bi}}}}}}{{dfrac{1}{{{R_{{
m{bci}}}}}} + dfrac{1}{{{R_{{
m{be}}}}}} + dfrac{1}{{{R_{{
m{bi}}}}}}}},$$

(24)









$${T_{{
m{be}}}} = {R_{{
m{be}}}}{C_{{
m{be}}}},$$

(25)









$${T_{{
m{bci}}}} = {R_{{
m{bci}}}}{C_{{
m{bci}}}}.$$

(26)



Defining









$${F_1} = frac{omega }{{{
m{imag}}left( {{Z_1}left( {1 + jomega {T_{{
m{bi}}}}}
ight)}
ight)}} = {A_1} + {omega ^2}{B_1},$$

(27)



where ${A_1} = 1/{alpha _1}$, ${B_1} = T_{{
m{be}}}^2/{alpha _1}$, ${alpha _1} = {R_{
m{x}}}left( {{T_{
m{x}}} - {T_{{
m{be}}}}}
ight)$.



A1 and B1 can be obtained by using the straight-line fitting method. Then, Tbe can be determined as









$${{T}}_{
m{be}} = sqrt{frac{{{B}}_{{1}}}{{{A}}_{{1}}}}{.}$$

(28)



Based on Eqs. (22) and (25), we can get









$${F_2} = {Z_1}left( {1 + jomega {T_{{
m{bi}}}}}
ight)left( {1 + jomega {T_{{
m{be}}}}}
ight) = {R_{
m{x}}}left( {1 + jomega {T_{
m{x}}}}
ight).$$

(29)



By extracting the real and imaginary parts of F2, Rx and Tx can be acquired from Eqs. (30) and (31).









$${
m{real}}left({{F}}_{{2}}
ight) = {{R}}_{
m{x}}{,}$$

(30)









$${
m{imag}}left( {{F_2}}
ight) = omega {R_{
m{x}}}{T_{
m{x}}}.$$

(31)



Based on Eqs. (23) and (24), the expressions of Rx and RxTx can be written as









$${R_{
m{x}}} = left( {1 + frac{{{R_{{
m{bi}}}}}}{{{R_{{
m{bci}}}}}}}
ight){R_{{
m{be}}}} + {R_{{
m{bi}}}},$$

(32)









$${R_{
m{x}}}{T_{
m{x}}} = left( {{T_{{
m{bi}}}} + {R_{{
m{bi}}}}{C_{{
m{bci}}}}}
ight){R_{{
m{be}}}} + {T_{{
m{be}}}}{R_{{
m{bi}}}}.$$

(33)



Rbe, Rbi can be determined from Eqs. (17), (18), (20), (28), (30), and (31). When Rbe and Rbi are obtained, Rbc, Cbc, Cbe and Cbi can be obtained from Eqs. (17), (18), (25), and (26), respectively.



In the end, the remaining parameters are written as ${C_{{
m{bcx}}}} = {
m{imag}}left( {1/{Z_4}-1/{Z_2}}
ight)/omega $, ${R_{{
m{bcx}}}} = {
m{real}}{left( {1/{Z_4}-1/{Z_2}}
ight)^{-1}}$. gm0 and τ are obtained from Eqs. (11) and (12): $g_{
m{m0}}=|X/B| $ and $tau = -{
m{phase}}left( {X/B/{{
m{g}}_{{
m{m0}}}}}
ight)/omega $.




4.
Model verification




An GaAs HBT with 2 μm emitter linewidth was used to validate the accuracy of the equivalent circuit. The adopted device was manufactured in a commercial foundry. The method in Section 3 is applied to extract the parameters of an HBT device with a 2 × 20 μm2 emitter-area under bias points of Bias1 (Vce = 1 V, Ib = 15 μA), Bias2 (Vce = 1 V, Ib = 30 μA) and Bias3 (Vce = 3 V, Ib = 17.5 μA) in the frequency range from 100 MHz to 20 GHz. After extracting all parameters, the Keysight ICCAP software is used to optimize the extracted parameters to further reduce the error between the simulated and measured data. Here, the initial values of Arbe, Arc and Abex are set to 0.5 for optimization. Results of the extraction are compared with the extracted from the small-signal equivalent circuit of an AHBT (agilent heterojunction bipolar transistor) without considering Cbi. Table 1 shows the initial extraction and optimization results of the HBT device under Bias1 and Bias3. The comparisons of the real part and the imaginary part between the simulated and measured S-parameters are plotted in Fig. 5. The accuracy of S-parameters versus frequency shows in Table 2.






ParameterExtractedOptimizedError (%)
Rbx (Ω)Bias16.8977.72311.98
Bias36.8978.6525.42
Rcx (Ω)Bias11.2811.59124.20
Bias31.2811.2710.781
Rci (Ω)Bias11.2810.97024.28
Bias31.2810.96124.98
Re (Ω)Bias16.5265.28918.95
Bias36.5268.41728.98
Rbcx (kΩ)Bias1239.9246.12.584
Bias3571.6569.30.402
Cbcx (fF)Bias134.9234.820.286
Bias330.3228.585.739
Rbex (kΩ)Bias12.0935.500162.8
Bias31.5902.62164.84
Cbex (fF)Bias1208.5127.638.80
Bias3639.5208.667.38
Rbi (Ω)Bias1306.7309.00.750
Bias3627.9541.413.78
Cbi (pF)Bias12.0582.0331.215
Bias31.2711.2035.350
Rbci (kΩ)Bias1140.4139.10.926
Bias3271.5296.89.319
Cbci (fF)Bias11.7641.7730.510
Bias31.0691.0881.777
Rbei (Ω)Bias12.0931.49028.81
Bias31.5900.55365.22
Cbei (fF)Bias1208.5289.939.04
Bias3639.51,04964.03
Ro (kΩ)Bias112.5412.842.390
Bias36.6665.24421.33
Co (fF)Bias123.4820.4312.99
Bias317.6117.630.114
gm0 (mS)Bias179.7780.030.326
Bias3227.5227.90.176
τ (ps)Bias12.3172.82121.75
Bias32.8343.17912.17





Table1.
The initial extraction and optimization results of the HBT under Bias1 (Vce = 1 V, Ib = 15 μA) and Bias3 (Vce = 3 V, Ib = 17.5 μA). Error = |Extracted – Optimized| / Extracted × 100%.



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ParameterExtractedOptimizedError (%)
Rbx (Ω)Bias16.8977.72311.98
Bias36.8978.6525.42
Rcx (Ω)Bias11.2811.59124.20
Bias31.2811.2710.781
Rci (Ω)Bias11.2810.97024.28
Bias31.2810.96124.98
Re (Ω)Bias16.5265.28918.95
Bias36.5268.41728.98
Rbcx (kΩ)Bias1239.9246.12.584
Bias3571.6569.30.402
Cbcx (fF)Bias134.9234.820.286
Bias330.3228.585.739
Rbex (kΩ)Bias12.0935.500162.8
Bias31.5902.62164.84
Cbex (fF)Bias1208.5127.638.80
Bias3639.5208.667.38
Rbi (Ω)Bias1306.7309.00.750
Bias3627.9541.413.78
Cbi (pF)Bias12.0582.0331.215
Bias31.2711.2035.350
Rbci (kΩ)Bias1140.4139.10.926
Bias3271.5296.89.319
Cbci (fF)Bias11.7641.7730.510
Bias31.0691.0881.777
Rbei (Ω)Bias12.0931.49028.81
Bias31.5900.55365.22
Cbei (fF)Bias1208.5289.939.04
Bias3639.51,04964.03
Ro (kΩ)Bias112.5412.842.390
Bias36.6665.24421.33
Co (fF)Bias123.4820.4312.99
Bias317.6117.630.114
gm0 (mS)Bias179.7780.030.326
Bias3227.5227.90.176
τ (ps)Bias12.3172.82121.75
Bias32.8343.17912.17








onerror="this.onerror=null;this.src='http://www.jos.ac.cn/fileBDTXB/journal/article/jos/2021/5/PIC/20080030-5.jpg'"
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Figure5.
S-parameters comparisons in the frequency range from 100 MHz to 20 GHz under the biasing condition: (a) Bias1 (Vce = 1 V, Ib = 15 μA), (b) Bias2 (Vce = 1 V, Ib = 30 μA), (c) Bias3 (Vce = 3 V, Ib = 17.5 μA).






BiasS-parameterWithout Cbi (%)With Cbi (%)
Bias1S1185.1389.31
S1291.7894.65
S2188.3792.63
S2297.0698.39
Bias2S1191.6291.68
S1295.0695.86
S2192.3593.44
S2297.7098.21
Bias3S1191.1692.51
S1293.4996.71
S2193.1793.82
S2296.3899.12





Table2.
The accuracy of S-parameters versus frequency.



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BiasS-parameterWithout Cbi (%)With Cbi (%)
Bias1S1185.1389.31
S1291.7894.65
S2188.3792.63
S2297.0698.39
Bias2S1191.6291.68
S1295.0695.86
S2192.3593.44
S2297.7098.21
Bias3S1191.1692.51
S1293.4996.71
S2193.1793.82
S2296.3899.12





Due to the inaccurate initial value of 0.5 for the partition parameters Arc, Arbe, and Abex defined before extraction, the extracted and optimized values of Rcx, Rci, Rbcx, Rbex, Cbcx and Cbex are slightly larger. From Fig. 6, it can be seen that the proposed model with Cbi shows more accuracy than the one without Cbi, which verifies the effectiveness of the introduced AC current crowding effect.






onerror="this.onerror=null;this.src='http://www.jos.ac.cn/fileBDTXB/journal/article/jos/2021/5/PIC/20080030-6.jpg'"
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Figure6.
(a) Plot of Cbi versus Ib. (b) Plot of Cbi versus Vce.




Fig. 6 shows the decrease of Cbi with Ib and Vce. Results present that Cbi decreases with increasing Ib and Vce, which is consistent with the result shown in Ref. [18]. The experimental results show that the dependence between Cbi and biases accords with the basic capacitance equation.




5.
Conclusion




An improved small-signal equivalent circuit of the HBT device considering the AC current crowding effect is proposed in this paper. This effect is modeled as a parallel RC circuit with Cbi and Rbi. By comparing between the simulated and measured S-parameters under three different biases, the results validate the reliability and availability of the proposed model and the developed extraction method.




Acknowledgements




This work was supported by the National Natural Science Foundation of China (Grant No. 61934006)



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