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Analytical modeling of subthreshold swing in undoped trigate SOI MOSFETs

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Abstract

A new analytical model for the subthreshold swing of nanoscale undoped trigate silicon-on-insulator metal–oxide–semiconductor field-effect transistors (MOSFETs) is proposed, based on the channel potential distribution and physical conduction path concept. An analytical model for the potential distribution is obtained by solving the three-dimensional (3-D) Poisson’s equation, assuming a parabolic potential distribution between the lateral gates. In addition, mobile charges are considered in the model. The proposed analytical model is investigated and compared with results obtained from 3-D simulations using the ATLAS device simulator and experimental data. It is demonstrated that the analytical model predicts the subthreshold swing with good accuracy for different lengthes, thicknesses, and widths of channel.

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Authors and Affiliations

  1. Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

    Hamdam Ghanatian & Seyed Ebrahim Hosseini

Authors
  1. Hamdam Ghanatian
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  2. Seyed Ebrahim Hosseini
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Correspondence to Hamdam Ghanatian.

Appendices

Appendix A

Potential distribution

(a.1) \(C_{2}\) coefficient:

$$\begin{aligned}&C_0 (x)=\varphi _\mathrm{f} (x), \end{aligned}$$
(35)
$$\begin{aligned}&C_1 (x)=\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} }\frac{\varphi _\mathrm{f} (x)-{V}'_\mathrm{gs} }{t_\mathrm{ox} }, \end{aligned}$$
(36)
$$\begin{aligned}&\varphi _\mathrm{sb} (x)=C_0 (x)+C_1 (x)t_\mathrm{Si} +C_2 (x)t_\mathrm{Si}^2, \end{aligned}$$
(37)
$$\begin{aligned}&C_1 (x)+2C_2 (x)t_\mathrm{Si} =\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} }\frac{{V}'_\mathrm{sub} -\varphi _\mathrm{sb} }{t_\mathrm{oxb} }. \end{aligned}$$
(38)

Inserting (35), (36), and (37) into (38), the coefficient \(C_{2}\) is obtained as a function of \(\varphi _\mathrm{f} (x)\) as follows:

$$\begin{aligned}&C_1 (x)+2C_2 (x)t_\mathrm{Si} =\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }({V}'_\mathrm{sub} -\varphi _\mathrm{f} (x)\nonumber \\&\quad +\,\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} }{\varepsilon _\mathrm{Si} t_\mathrm{ox} }(\varphi _\mathrm{f} (x)-{V}'_\mathrm{gs} )+t_\mathrm{Si}^2 C_2 (x)). \end{aligned}$$
(39)

(a.2) \(a_{2}(x, z)\):

Using Eq. 7 in Eq. 15, \(a_{2}\) is obtained as

$$\begin{aligned} a_2 (x,z)=\left( {\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} }\frac{\varphi _\mathrm{f} (x)-{V}'_\mathrm{gs} }{t_\mathrm{ox} }} \right) z+\frac{\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }{V}'_\mathrm{sub} -\varphi _\mathrm{f} (x)\left[ {\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{ox} }\left( {1+\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }} \right) +\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }} \right] +\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{ox} }\left( {1+\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }} \right) {V}'_\mathrm{gs} }{(2t_\mathrm{Si} +\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} ^{2}}{\varepsilon _\mathrm{Si} t_\mathrm{oxb} })}z^{2}. \end{aligned}$$
(40)

(a.3) Coefficients \(k_{1}, k_{2}, \ldots , k_{8}\)

The coefficients \(k_{1}, k_{2}, \ldots , k_{8}\) are as follows:

$$\begin{aligned} k_1= & {} \frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{ox} }, \end{aligned}$$
(41)
$$\begin{aligned} k_2= & {} \frac{\varepsilon _\mathrm{ox} {V}'_\mathrm{gs} }{\varepsilon _\mathrm{Si} t_\mathrm{ox} }, \end{aligned}$$
(42)
$$\begin{aligned} k_3= & {} \frac{\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }{V}'_\mathrm{sub} +\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{ox} }\left( {1+\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }} \right) {V}'_\mathrm{gs} }{(2t_\mathrm{Si} +\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} ^{2}}{\varepsilon _\mathrm{Si} t_\mathrm{oxb} })}, \end{aligned}$$
(43)
$$\begin{aligned} k_4= & {} \frac{\left[ {\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{ox} }\left( {1+\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }} \right) +\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }} \right] }{(2t_\mathrm{Si} +\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} ^{2}}{\varepsilon _\mathrm{Si} t_\mathrm{oxb} })}, \end{aligned}$$
(44)
$$\begin{aligned} k_5= & {} \frac{4\varepsilon _\mathrm{ox} }{W^{2}\varepsilon _\mathrm{Si} t_\mathrm{ox} }, \end{aligned}$$
(45)
$$\begin{aligned} k_6= & {} \frac{4\varepsilon _\mathrm{ox} {V}'_\mathrm{gs} }{W^{2}\varepsilon _\mathrm{Si} t_\mathrm{ox} }, \end{aligned}$$
(46)
$$\begin{aligned} k_7= & {} \frac{4}{W^{2}}\frac{\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }{V}'_\mathrm{sub} +\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{ox} }\left( {1+\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }} \right) {V}'_\mathrm{gs} }{(2t_\mathrm{Si} +\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} ^{2}}{\varepsilon _\mathrm{Si} t_\mathrm{oxb} })}, \end{aligned}$$
(47)
$$\begin{aligned} k_8= & {} \frac{4}{W^{2}}\frac{\left[ {\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{ox} }\left( {1+\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }} \right) +\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }} \right] }{(2t_\mathrm{Si} +\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} ^{2}}{\varepsilon _\mathrm{Si} t_\mathrm{oxb} })}. \end{aligned}$$
(48)

(a.4) Tailor expansion

$$\begin{aligned} \text {e}^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\ldots \end{aligned}$$
(49)

The Tailor expansion of the exponential term in Eq. 20 is as follows:

$$\begin{aligned} \text {e}^{\frac{^{q(C\varphi _\mathrm{f} (x)+D)}}{kT}}=1+\frac{q(C\varphi _\mathrm{f} (x)+D)}{kT}. \end{aligned}$$
(50)

Appendix B

Subthreshold swing

$$\begin{aligned}&\frac{\mathrm{d}\varphi _\mathrm{f} }{\mathrm{d}V_\mathrm{gs} }=\frac{\mathrm{d}M}{\mathrm{d}V_\mathrm{gs} }\exp ((\alpha ^{{\prime }})^{1/2}x)\nonumber \\&\quad +\frac{\mathrm{d}N}{\mathrm{d}V_\mathrm{gs} }\exp (-((\alpha ^{{\prime }})^{1/2}x))-\frac{\mathrm{d}\beta ^{{\prime }}/\mathrm{d}V_\mathrm{gs} }{\alpha ^{{\prime }}}, \end{aligned}$$
(51)

where

$$\begin{aligned}&\frac{\mathrm{d}M}{\mathrm{d}V_\mathrm{gs} }=\frac{-\mathrm{d}D/\mathrm{d}V_\mathrm{gs} }{C}+\frac{\mathrm{d}\beta ^{{\prime }}/\mathrm{d}V_\mathrm{gs} }{\alpha ^{{\prime }}} \nonumber \\&-\frac{\frac{-\mathrm{d}D/\mathrm{d}V_\mathrm{gs} }{C}{+}\frac{\mathrm{d}\beta ^{{\prime }}/\mathrm{d}V_\mathrm{gs} }{\alpha ^{{\prime }}}{-}(\frac{{-}\mathrm{d}D/\mathrm{d}V_\mathrm{gs} }{C}{+}\frac{\mathrm{d}\beta ^{{\prime }}/\mathrm{d}V_\mathrm{gs} }{\alpha ^{{\prime }}})\exp \left( \varGamma \right) }{\exp (-\varGamma ){-}\exp \left( \varGamma \right) },\nonumber \\ \end{aligned}$$
(52)
$$\begin{aligned}&\frac{\mathrm{d}N}{\mathrm{d}V_\mathrm{gs} }{=}\frac{\frac{-\mathrm{d}D/\mathrm{d}V_\mathrm{gs} }{C}{+}\frac{\mathrm{d}\beta ^{{\prime }}/\mathrm{d}V_\mathrm{gs} }{\alpha ^{{\prime }}}-(\frac{-\mathrm{d}D/\mathrm{d}V_\mathrm{gs} }{C}{+}\frac{\mathrm{d}\beta ^{{\prime }}/\mathrm{d}V_\mathrm{gs} }{\alpha ^{{\prime }}})\exp \left( \varGamma \right) }{\exp (-\varGamma ){-}\exp \left( \varGamma \right) },\nonumber \\ \end{aligned}$$
(53)
$$\begin{aligned}&\frac{\mathrm{d}\beta ^{{\prime }}}{\mathrm{d}V_\mathrm{gs} }=\frac{\frac{8}{W^{2}}(-k_1 z+k_9 z^{2})-2k_9 +\frac{8}{W^{2}}y^{2}k_9 }{1+k_1 z-k_4 z^{2}-k_5 zy^{2}+k_8 z^{2}y^{2}}\nonumber \\&+\,\frac{\mathrm{d}D}{\mathrm{d}V_\mathrm{gs} }\frac{q^{2}n_{\text {i}} }{C\varepsilon _\mathrm{Si} kT}, \end{aligned}$$
(54)
$$\begin{aligned}&k_9 =\frac{\frac{\varepsilon _\mathrm{ox} }{\varepsilon _\mathrm{Si} t_\mathrm{ox} }\left( {1+\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} }{\varepsilon _\mathrm{Si} t_\mathrm{oxb} }} \right) }{(2t_\mathrm{Si} +\frac{\varepsilon _\mathrm{ox} t_\mathrm{Si} ^{2}}{\varepsilon _\mathrm{Si} t_\mathrm{oxb} })}, \end{aligned}$$
(55)
$$\begin{aligned}&\frac{\mathrm{d}D}{\mathrm{d}V_\mathrm{gs} }=-k_1 z+k_9 z^{2}+k_5 zy^{2}-\frac{4}{W^{2}}k_9 z^{2}y^{2}. \end{aligned}$$
(56)

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Ghanatian, H., Hosseini, S.E. Analytical modeling of subthreshold swing in undoped trigate SOI MOSFETs. J Comput Electron 15, 508–515 (2016). https://doi.org/10.1007/s10825-016-0817-2

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