• Journal of Semiconductors
  • Vol. 41, Issue 6, 062601 (2020)
Shagun Pal and Brijesh Kumar
Author Affiliations
  • Department of Electronics and Communication Engineering, MMMUT Gorakhpur- 273010, India
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    DOI: 10.1088/1674-4926/41/6/062601 Cite this Article
    Shagun Pal, Brijesh Kumar. Mathematical analysis of organic-pass transistor using pseudo-p-OTFTs[J]. Journal of Semiconductors, 2020, 41(6): 062601 Copy Citation Text show less

    Abstract

    Steady state behavior analysis of organic thin film transistor (OTFTs) has been thoroughly researched in the past few decades. Yet, this static logic analysis has drawbacks of high power dissipation and high power consumption, and a large number of prerequisites in the number of transistors for the digital logic circuit application. Hence, to overcome these basic fundamental drawbacks of static logic, the dynamic logic study of organic thin film transistor has been analyzed in this paper. The fundamental basic of dynamic logic is a pass transistor for which logic high and logic low model is designed at an operating voltage of 5 V and frequency of 5 kHz. Additionally, the novel approach of analytical model for organic pass transistor (OPT) circuit is included and verified using MATLAB. The transient individualities of organic pass transistor OPT are examined through Atlas 2-D numerical device simulator. The reduction in the power dissipation along with additional voltage scaling and reduction in the clock frequency such as pipelining may further enable the applications into more complex VLSI ICs.
    $I_{\rm{ds}} = \frac{W}{L}\mu {C_{\rm{i}}}\left[ \left( {V_{\rm{gs}}} - V_{\rm{t}} \right)V_{\rm{ds}} - \frac{1}{2}V_{\rm{ds}}^2 \right].$(1)

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    ${V_{\rm{ds}}} \leqslant {{V_{\rm{gs}}} - {V_{\rm{t}}}} ,$()

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    ${I_{\rm{ds}}} = \frac{W}{L}{\mu _{\rm{l}}}{C_{\rm{i}}}\left[ {\left( {{V_{\rm{gs}}} - {V_{\rm{t}}}} \right){V_{\rm{ds}}}} \right].$(1)

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    ${V_{\rm{ds}}} > {{V_{\rm{gs}}} - {V_{\rm{t}}}} ,$()

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    ${I_{\rm{ds}}} = \frac{W}{{2L}}{\mu _{\rm{s}}}{C_{\rm{i}}}\left[ {{{\left( {{V_{\rm{gs}}} - {V_{\rm{t}}}} \right)}^2}} \right],$(2)

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    $ I_{\rm{ds}}^{\rm{lin}} = - {\mu _{\rm{lin}}}{C_{\rm{ox}}}\frac{W}{L}{\left( {{V_{\rm{gs}}} - {V_{\rm{t}}}} \right)^\gamma }{V_{\rm{ds}}}, \tag{2a}$ ()

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    $ I_{\rm{ds}}^{\rm{sat}} = - {\mu _{\rm{sat}}}{C_{\rm{ox}}}\frac{W}{L}{\left( {{V_{\rm{gs}}} - {V_{\rm{t}}}} \right)^{\gamma + 1}}, \tag{3a}$ ()

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    $ \gamma = 2\frac{{{T_{\rm{o}}}}}{T} - 1. \tag{3b}$ ()

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    $ {\mu _{\rm{l}}} = \frac{{L{g_{\rm{m}}}}}{{W{C_{\rm{i}}}{V_{\rm{ds}}}}},$(4)

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    ${\mu _{\rm{s}}} = \frac{{2L}}{{W{C_{\rm{i}}}}}{\left( {\frac{{\partial \sqrt {{I_{\rm{ds}}}} }}{{\partial {V_{\rm{gs}}}}}} \right)^2}.$(5)

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    $\mu (E) = {\mu _0}\exp \left[ { - \frac{\varDelta }{{kT}} + \left( {\frac{\beta }{{kT}} - \gamma } \right)\sqrt E } \right],$(6)

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    $ {C_{\rm{x}}}\frac{{{\rm{d}}{V_{\rm{x}}}}}{{\rm{d}t}} = {K_{\rm{p}}}{\frac{W}{L}} {\left( {{V_{\rm{dd}}} - {V_{\rm{x}}} - {V_{\rm{tp}}}} \right)^2}. $ (7)

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    $ \int\limits_0^t {{\rm{d}}t} = \frac{{{C_{\rm{x}}}L}}{{W{C_{\rm{ox}}}}}\int\limits_0^{V_{\rm{x}}} {\frac{1}{{ {{V_{\rm{dd}}} - {V_{\rm{x}}} - {V_{\rm{tp}}}} }}} {\rm{d}}{V_{\rm{x}}}. $ (8)

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    $ {t_{\rm{sat}}} = \frac{{{C_{\rm{x}}}L}}{{W{C_{\rm{ox}}}}}\ln \frac{{ {{V_{\rm{dd}}} - {V_{\rm{tp}}}} }}{{ {{V_{\rm{dd}}} - {V_{\rm{tp}}} - {V_{\rm{x}}}} }}, $ (9)

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    $ \frac{{W{C_{\rm{ox}}}{t_{\rm{sat}}}}}{{L{C_{\rm{x}}}}} = \ln \frac{{ {{V_{\rm{dd}}} - {V_{\rm{tp}}}} }}{ {V_{\rm{dd}}} - {V_{\rm{tp}}} - {V_{\rm{x}} }}. $ (10)

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    $ {{\rm e}^{\frac{{W{C_{\rm{ox}}}}}{{L{C_{\rm{x}}}}}t}} = \frac{{ {{V_{\rm{dd}}} - {V_{\rm{tp}}}} }}{{ {{V_{\rm{dd}}} - {V_{\rm{tp}}} - {V_{\rm{x}}}} }}, $ ()

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    $ {{\rm{e}}^{ - {\frac{{W{C_{\rm{ox}}}}}{{L{C_{\rm{x}}}}}t} }} = 1 - \frac{{{V_{\rm{x}}}}}{{ {{V_{\rm{dd}}} - {V_{\rm{tp}}}} }}. $ (11)

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    $ {V_{\rm{x}}} = \left( {{V_{\rm{dd}}} - {V_{\rm{tp}}}} \right)\left( {1 - {{\rm{e}}^{ - {\frac{{W{C_{\rm{ox}}}}}{{L{C_{\rm{x}}}}}t} }}} \right). $ (12)

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    $ {V_{\rm{gs}}} = 0,\;\;{V_{\rm{ds}}} = - {V_{\rm{DD}}}. $ ()

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    $ {C_{\rm{x}}}\frac{{{\rm{d}}{V_{\rm{o1}}}}}{{{\rm{d}}t}} = {K_{\rm{p}}} {\frac{W}{L}} \left[ {\left( {{V_{\rm{gs}}} - {V_{\rm{th}}}} \right){V_{\rm{ds}}} - \frac{{V_{\rm{ds}}^2}}{2}} \right], $ (13)

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    $2{C_{\rm{x}}}\frac{{{\rm{d}}{V_{\rm{o1}}}}}{{{\rm{d}}t}} = 2\left( { - {V_{\rm{tp}}}} \right)\left[\left( {{V_{\rm{out}}} \!-\! {V_{\rm{DD}}}} \right) \!-\! {{\left( {{V_{\rm{DD}}} \!-\! {V_{\rm{out}}}} \right)}^2} \!\right]K_{\rm{p}}\frac{W}{L}.\! $ (14)

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    $ 2C_{\rm{x}}\frac{{\rm{d}}{V_{\rm{o1}}}}{{{\rm{d}}t}} = {2\left( { - {V_{\rm{tp}}}} \right)K - {K^2}} , $ (15)

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    $ 2{C_{\rm{x}}}\frac{{{\rm{d}}{V_{\rm{o1}}}}}{{{\rm{d}}t}} = - K\left[ {2\left( {{V_{\rm{tp}}} + K} \right)} \right]{K_{\rm{p}}}\frac{W}{L}, $ (16)

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    $ \int\limits_{{V_{\rm{DD}}}}^{{V_{\rm{tp}}}} {\frac{{{\rm{d}}{V_{\rm{01}}}}}{{-{{ K(2}}{V_{\rm{tp}}} + K)}} = } \int\limits_0^t \frac{{\rm{d}}t}{{2{C_{\rm{x}}}}} {K_{\rm{p}}}\frac{W}{L}. $ (17)

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    $ \begin{array}{l} \displaystyle\int\limits_{{V_{\rm{DD}}}}^{{V_{\rm{tp}}}} {\frac{{{\rm{d}}{V_{\rm{01}}}}}{{ - {{K}}\left( {{\rm{2}}{{{V}}_{{\rm{tp }}}} + K} \right)}}} = \displaystyle\int\limits_{{V_{\rm{DD}}}}^{{V_{\rm{tp}}}} {\frac{{\dfrac{{ - 1}}{{2{V_{\rm{tp}}}}}}}{{{{ K}}}}} + \displaystyle\int\limits_{{V_{\rm{DD}}}}^{{V_{\rm{tp}}}} {\frac{{\dfrac{1}{{2{V_{\rm{tp}}}}}}}{{{{2}}{V}_{\rm{tp}} + K}}} \\ \quad\quad = \dfrac{{ - 1}}{{2{V_{\rm{tp}}}}}\ln K + \dfrac{1}{{2{V_{\rm{tp}}}}}\ln \left( {2{{{V}}_{{\rm{tp}}}} + K} \right),\\ {\rm{where}}\;K = {{V_{\rm{out}}} - {V_{\rm{DD}}}} , \end{array} $ (18)

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    $ \frac{{{K_{\rm{p}}}\dfrac{W}{L}}}{{2{C_{\rm{x}}}}}t = \frac{1}{{2{V_{\rm{tp}}}}}\ln \left( {1 + \frac{{{V_{\rm{tp}}}}}{{{V_{\rm{tp}}} - {V_{\rm{DD}}}}}} \right). $ (19)

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    $ 1 - {{\rm{e}}^{(K\frac{W}{L}/C_{\rm{x}}){V_{\rm{tp}}}t}} = \frac{{2{V_{\rm{tp}}}}}{{{V_{\rm{out}}} - {V_{\rm{dd}}}}}, $ (20)

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    $ {V_{\rm{out}}} - {V_{\rm{dd}}} = \frac{{2{V_{\rm{tp}}}}}{{1 - {{\rm{e}}^{\frac{{(KW/L){V_{\rm{tp}}}t}}{{C_{\rm{x}}}}}}}}, $ (21)

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    $ {V_{\rm{out}}} = {V_{\rm{dd}}} + 2{V_{\rm{tp}}}{\left( {1 - {{\rm{exp}} } {\frac{{KW/L{V_{\rm{tp}}}t}}{{{C_{\rm{x}}}}}} } \right)^{ - 1}}. $ (22)

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    $ {V_{\rm{out}}} = {V_{\rm{dd}}} + 2{V_{\rm{tp}}}\left(1 + {\exp } {\frac{{KW/L{V_{\rm{tp}}}t}}{{{C_{\rm{x}}}}}} \right) $ (23)

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    $ {C_{\rm{x}}}\frac{{{\rm{d}}{V_{\rm{o1}}}}}{{{\rm{d}}t}} = K_{\rm p} {\frac{W}{L}} \left[ {\left( {{V_{\rm{gs}}} - {V_{\rm{th}}}} \right){V_{\rm{ds}}} - \frac{{V_{\rm{ds}}^2}}{2}} \right], $ (24)

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    $ 2{C_{\rm{x}}}\frac{{\rm{d}}{V_{\rm{o1}}}}{{{\rm{d}}t}} = 2\left( { - {V_{\rm{tp}}}} \right)\left[ {\left( {{V_{\rm{out}}} \!-\! {V_{\rm{DD}}}} \right) \!-\! {{\left( {{V_{\rm{DD}}} \!-\! {V_{\rm{out}}}} \right)}^2}} \right]{K_{\rm{p}}}\frac{W}{L}. $ (25)

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    $ 2{C_{\rm{x}}}\frac{{{\rm{d}}{V_{\rm{o1}}}}}{{{\rm{d}}t}} = {2\left( { - {V_{\rm{tp}}}} \right)K - {K^2}} , $ (26)

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    $ 2{C_{\rm{x}}}\frac{{\rm{d}}{V_{\rm{o1}}}}{{\rm{d}}t} = - K\left[ {2\left( {{V_{\rm{tp}}} + K} \right)} \right]{K_{\rm{p}}}\frac{W}{L}. $ (27)

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    $ \int\limits_{{V_{\rm{DD}}}}^{{V_{\rm{tp}}}} {\frac{{{\rm{d}}{V_{\rm{01}}}}}{{{{ - K(2}}{{{V}}_{{\rm{tp }}}} + K)}}} = \int\limits_0^t {\frac{{{\rm{d}}t}}{{2{C_{\rm{x}}}}}} {K_{\rm{p}}}\frac{W}{L}. $ (28)

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    $\begin{array}{l} \displaystyle\int\limits_{{V_{\rm{DD}}}}^{{V_{\rm{tp}}}}{\dfrac{{{\rm{d}}{V_{\rm{01}}}}}{{{{ - K(2}}{{{V}}_{{\rm{tp }}}} + K)}}} = \displaystyle\int\limits_{{V_{\rm{DD}}}}^{{V_{\rm{tp}}}} {\dfrac{{\dfrac{{ - 1}}{{2{V_{\rm{tp}}}}}}}{{{\rm{ K}}}}}+\displaystyle\int\limits_{{V_{\rm{DD}}}}^{{V_{\rm{tp}}}}{\frac{{\dfrac{1}{{2{V_{\rm{tp}}}}}}}{{{\rm{2}}{{{V}}_{{\rm{tp }}}} + K}}} \\\quad\quad = \dfrac{{ - 1}}{{2{V_{\rm{tp}}}}}\ln K +\dfrac{1}{{2{V_{\rm{tp}}}}}\ln \left( {2{{{V}}_{{\rm{tp}}}} + K} \right),\\{\rm{where}}\;K = {V_{\rm{out}}} - {V_{\rm{DD}}}, \end{array} $ (29)

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    $ \frac{{{K_{\rm{p}}}\dfrac{W}{L}}}{{2{C_{\rm{x}}}}}{t_{\rm{lin}}} = \frac{1}{{2{V_{\rm{tp}}}}}\ln \left( {2 + \frac{{{V_{\rm{tp}}}}}{{{V_{\rm{tp}}} - {V_{\rm{DD}}}}}} \right), $ (30)

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    $ {t_{\rm{lin}}} = \frac{{L{C_{\rm{x}}}}}{{{K_{\rm{p}}}W{V_{\rm{tp}}}}}\ln \left( {2 + \frac{{{V_{\rm{tp}}}}}{{{V_{\rm{tp}}} - {V_{\rm{DD}}}}}} \right). $ (31)

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    $ {t_{\rm{lin}}} = {{S}}\ln \left( {2 + \frac{{{V_{\rm{tp}}}}}{{{V_{\rm{tp}}} - {V_{\rm{DD}}}}}} \right). $ (32)

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    ${I_{\rm{ds}}} = \frac{{{C_{\rm{x}}}}}{{S{V_{\rm{tp}}}}}\left[ {\left( {{V_{\rm{gs}}} - {V_{\rm{t}}}} \right){V_{\rm{ds}}} - \frac{1}{2}{V_{\rm{ds}}}^2} \right].$(33)

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    Shagun Pal, Brijesh Kumar. Mathematical analysis of organic-pass transistor using pseudo-p-OTFTs[J]. Journal of Semiconductors, 2020, 41(6): 062601
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