• Journal of Semiconductors
  • Vol. 42, Issue 5, 052401 (2021)
Jinjing Huang and Jun Liu
Author Affiliations
  • Key Laboratory of RF Circuits and Systems, Ministry of Education, Hangzhou Dianzi University, Hangzhou 310018, China
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    DOI: 10.1088/1674-4926/42/5/052401 Cite this Article
    Jinjing Huang, Jun Liu. A complete small-signal HBT model including AC current crowding effect[J]. Journal of Semiconductors, 2021, 42(5): 052401 Copy Citation Text show less

    Abstract

    An improved small-signal equivalent circuit of HBT concerning the AC current crowding effect is proposed in this paper. AC current crowding effect is modeled as a parallel RC circuit composed of Cbi and Rbi, with distributed base-collector junction capacitance also taken into account. The intrinsic portion is taken as a whole and extracted directly from the measured S-parameters in the whole frequency range of operation without any special test structures. An HBT device with a 2 × 20 μm2 emitter-area under three different biases were used to demonstrate the extraction and verify the accuracy of the equivalent circuit.
    $ \left[ {{Z_{\rm{m}}}} \right] = Z-\left[ {\begin{array}{*{20}{c}} {{R_{{\rm{bx}}}} + {R_{\rm{e}}}} & {{R_{\rm{e}}}}\\[2mm] {{R_{\rm{e}}}} & {{R_{\rm{c}}} + {R_{\rm{e}}}} \end{array}} \right]. $ (1)

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    $ {Y_{{\rm{m11}}}} = \frac{{{Z_{{\rm{be}}}} + {Z_{{\rm{bci}}}}}}{N} + {Y_{{\rm{bcx}}}}, $ (2)

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    $ {Y_{{\rm{m12}}}} = \frac{{-{Z_{{\rm{be}}}}}}{N}-{Y_{{\rm{bcx}}}}, $ (3)

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    $ {Y_{{\rm{m21}}}} = \frac{{{{\rm{g}}_{\rm{m}}}{Z_{{\rm{be}}}}{Z_{{\rm{bci}}}}-{Z_{{\rm{be}}}}}}{N}-{Y_{{\rm{bcx}}}}, $ (4)

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    $ {Y_{{\rm{m22}}}} = \frac{{{Z_{{\rm{be}}}} + {Z_{{\rm{bi}}}}\left( {1 + {{\rm{g}}_{\rm{m}}}{Z_{{\rm{be}}}}} \right)}}{N} + {Y_{{\rm{bcx}}}}, $ (5)

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    $ \left[ {{Y_{{\rm{in}}}}} \right] = \left[ {{Y_{\rm{m}}}} \right]-\left[ {\begin{array}{*{20}{c}} 0 & 0\\ 0 & {{Y_{\rm{o}}}} \end{array}} \right]. $ (6)

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    $ {{Z}}_{{1}} = \frac{{N}}{{{Z}}_{\rm{bci}}}{,} $ (7)

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    $ {{Z}}_{{2}} = \frac{{N}}{{{Z}}_{\rm{be}}}{,} $ (8)

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    $ {{Z}}_{{3}} = \frac{{N}}{{{Z}}_{\rm{bi}}}{.} $ (9)

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    $ \left[ {{Y_{{\rm{in}}}}} \right] = \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}} \right], $ (10)

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    $ X = B {{{g}}_{\rm{m}}}, $ (11)

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    $ B = \frac{{{Z_1}}}{{{Z_1} + {Z_2} + {Z_3}}}. $ (12)

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    $ {{Z}}_{{1}} = \frac{{1}}{{{Y}}_{\rm{in11}}+{{Y}}_{\rm{in12}}}{,} $ (13)

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    $ {{Z}}_{{3}} = \frac{{{Y}}_{\rm{in11}}+{{}{Y}}_{\rm{in21}}}{\left({{Y}}_{\rm{in11}}+{{Y}}_{\rm{in12}}\right)\left({{Y}}_{\rm{in22}}+{{Y}}_{\rm{in12}}\right)}{,} $ (14)

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    $ {{Z}}_{{4}} = {-}\frac{{1}}{{{Y}}_{\rm{in12}}}{.} $ (15)

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    $ \frac{{{Z_1}}}{{{Z_3}}} = \frac{{{R_{{\rm{bi}}}}}}{{{R_{{\rm{bci}}}}}}\frac{{1 + j\omega {R_{{\rm{bci}}}}{C_{{\rm{bci}}}}}}{{1 + j\omega {R_{{\rm{bi}}}}{C_{{\rm{bi}}}}}}, $ (16)

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    $ {\rm{real}}\left( {\frac{{{Z_1}}}{{{Z_3}}}\left( {1 + j\omega {T_{{\rm{bi}}}}} \right)} \right) = \frac{{{R_{{\rm{bi}}}}}}{{{R_{{\rm{bci}}}}}}, $ (17)

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    $ {\rm{imag}}\left( {\frac{{{Z_1}}}{{{Z_3}}}\left( {1 + j\omega {T_{{\rm{bi}}}}} \right)} \right) = \omega {R_{{\rm{bi}}}}{C_{{\rm{bci}}}}. $ (18)

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    $ \begin{array}{l} {T_{{\rm{bi}}}} = {R_{{\rm{bi}}}}{C_{{\rm{bi}}}},\\ \;\;\;\;{F_0} = \dfrac{\omega }{{{\rm{imag}}\left( {\dfrac{{{Z_1}}}{{{Z_3}}}} \right)}} = {A_0} + {\omega ^2}{B_0}, \end{array} $ (19)

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    $ {{T}}_{\rm{bi}} = \sqrt{\frac{{{B}}_{{0}}}{{{A}}_{{0}}}} . $(20)

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    $ \begin{array}{l} {Z_1} = \dfrac{{{R_{{\rm{bi}}}}}}{{1 + j\omega {T_{{\rm{bi}}}}}} + \dfrac{{{R_{{\rm{be}}}}}}{{1 + j\omega {T_{{\rm{be}}}}}}\\ \quad\quad + \dfrac{{{R_{{\rm{bi}}}}}}{{1 + j\omega {T_{{\rm{bi}}}}}}\dfrac{{{R_{{\rm{be}}}}}}{{1 + j\omega {T_{{\rm{be}}}}}}\dfrac{{1 + j\omega {T_{{\rm{bci}}}}}}{{{R_{{\rm{bci}}}}}}, \end{array} $ (21)

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    $ {Z_1}\left( {1 + j\omega {T_{{\rm{bi}}}}} \right) = \frac{{{R_{\rm{x}}}\left( {1 + j\omega {T_{\rm{x}}}} \right)}}{{1 + j\omega {T_{{\rm{be}}}}}}, $ (22)

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    $ {R_{\rm{x}}} = {R_{{\rm{bi}}}}{R_{{\rm{be}}}}\left( {\frac{1}{{{R_{{\rm{bci}}}}}} + \frac{1}{{{R_{{\rm{be}}}}}} + \frac{1}{{{R_{{\rm{bi}}}}}}} \right), $ (23)

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    $ {T_{\rm{x}}} = \frac{{{C_{{\rm{bci}}}} + {C_{{\rm{be}}}} + {C_{{\rm{bi}}}}}}{{\dfrac{1}{{{R_{{\rm{bci}}}}}} + \dfrac{1}{{{R_{{\rm{be}}}}}} + \dfrac{1}{{{R_{{\rm{bi}}}}}}}}, $ (24)

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    $ {T_{{\rm{be}}}} = {R_{{\rm{be}}}}{C_{{\rm{be}}}}, $ (25)

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    $ {T_{{\rm{bci}}}} = {R_{{\rm{bci}}}}{C_{{\rm{bci}}}}. $ (26)

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    $ {F_1} = \frac{\omega }{{{\rm{imag}}\left( {{Z_1}\left( {1 + j\omega {T_{{\rm{bi}}}}} \right)} \right)}} = {A_1} + {\omega ^2}{B_1}, $ (27)

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    $ {{T}}_{\rm{be}} = \sqrt{\frac{{{B}}_{{1}}}{{{A}}_{{1}}}}{.} $ (28)

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    $ {F_2} = {Z_1}\left( {1 + j\omega {T_{{\rm{bi}}}}} \right)\left( {1 + j\omega {T_{{\rm{be}}}}} \right) = {R_{\rm{x}}}\left( {1 + j\omega {T_{\rm{x}}}} \right). $ (29)

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    $ {\rm{real}}\left({{F}}_{{2}}\right) = {{R}}_{\rm{x}}{,} $ (30)

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    $ {\rm{imag}}\left( {{F_2}} \right) = \omega {R_{\rm{x}}}{T_{\rm{x}}}. $ (31)

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    $ {R_{\rm{x}}} = \left( {1 + \frac{{{R_{{\rm{bi}}}}}}{{{R_{{\rm{bci}}}}}}} \right){R_{{\rm{be}}}} + {R_{{\rm{bi}}}}, $ (32)

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    $ {R_{\rm{x}}}{T_{\rm{x}}} = \left( {{T_{{\rm{bi}}}} + {R_{{\rm{bi}}}}{C_{{\rm{bci}}}}} \right){R_{{\rm{be}}}} + {T_{{\rm{be}}}}{R_{{\rm{bi}}}}. $ (33)

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    Jinjing Huang, Jun Liu. A complete small-signal HBT model including AC current crowding effect[J]. Journal of Semiconductors, 2021, 42(5): 052401
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