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Journals >Journal of Semiconductors >Volume 40 >Issue 12 >Page 122901 > Article

Journal of Semiconductors
Vol. 40, Issue 12, 122901 (2019)

A compact two-dimensional analytical model of the electrical characteristics of a triple-material double-gate tunneling FET structure

C. Usha and P. Vimala

Author Affiliations

Department of Electronics and Communication Engineering, Dayananda Sagar College of Engineering, Bangalore-560078, KA, India

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DOI: 10.1088/1674-4926/40/12/122901 Cite this Article

C. Usha, P. Vimala. A compact two-dimensional analytical model of the electrical characteristics of a triple-material double-gate tunneling FET structure[J]. Journal of Semiconductors, 2019, 40(12): 122901 Copy Citation Text

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Abstract

This paper presents a compact two-dimensional analytical device model of surface potential, in addition to electric field of triple-material double-gate (TMDG) tunnel FET. The TMDG TFET device model is developed using a parabolic approximation method in the channel depletion space and a boundary state of affairs across the drain and source. The TMDG TFET device is used to analyze the electrical performance of the TMDG structure in terms of changes in potential voltage, lateral and vertical electric field. Because the TMDG TFET has a simple compact structure, the surface potential is computationally efficient and, therefore, may be utilized to analyze and characterize the gate-controlled devices. Furthermore, using Kane's model, the current across the drain can be modeled. The graph results achieved from this device model are close to the data collected from the technology computer aided design (TCAD) simulation.

$ \frac{{{\partial ^2} \textit{ϕ} (x,y)}}{{\partial {x^2}}} + \frac{{{\partial ^2}\textit{ϕ} (x,y)}}{{\partial {y^2}}} = 0. $

(1)

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$ \textit{ϕ} (x,y) = {C_0}(x) + {C_1}(x)y + {C_2}(x){y^2}. $

(2)

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$ {\textit{ϕ} _1}(x,y) = {C_{10}}(x) + {C_{11}}(x)y + {C_{12}}(x){y^2},\;\;\;{\rm{ 0}} \leqslant x \leqslant {L_1}, $

(3)

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$ {\textit{ϕ} _2}(x,y) = {C_{20}}(x) + {C_{21}}(x)y + {C_{22}}(x){y^2},\;\;\;{{{L}}_{\rm{1}}} \leqslant x \leqslant {L_1} + {L_2}, $

(4)

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$\begin{split} {\textit{ϕ} _3}(x,y) =\, & {C_{30}}(x) + {C_{31}}(x)y + {C_{32}}(x){y^2},\\ & {{{L}}_{\rm{1}}} + {L_2} \leqslant x \leqslant {{{L}}_{\rm{1}}} + {L_2} + {L_3}.\end{split}$

(5)

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$ \frac{{{\rm d}{\textit{ϕ}_1}(x,y)}}{{{\rm d}x}} = \frac{{{\varepsilon _{\rm ox}}}}{{{\varepsilon _{\rm{si}}}}}{\rm{ }}\frac{{{\textit{ϕ}_{{{\rm s}1}}}(x) - {\psi _{{{\rm g}1}}}}}{{{t_{\rm ox}}}},{\rm{under}}\; {\rm{material}} \;{\rm{M}}1\; {\rm{at}}\;y=0, $

(6)

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$ \frac{{{\rm d}{\textit{ϕ} _2}(x,y)}}{{{\rm d}x}} = \frac{{{\varepsilon _{\rm ox}}}}{{{\varepsilon _{\rm{si}}}}}\frac{{{\phi _{{{\rm s}2}}}(x) - {\psi _{{{\rm g}2}}}}}{{_{}{t_{\rm ox}}}},{\rm{under}}\; {\rm{material}} \;{\rm{M}}2\; {\rm{at}}\;y=0, $

(7)

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$ \frac{{{\rm d}{\textit{ϕ}_3}(x,y)}}{{{\rm d}x}} = \frac{{{\varepsilon _{\rm ox}}}}{{{\varepsilon _{\rm{si}}}}}\frac{{{\textit{ϕ} _{\rm s3}}(x) - {\psi _{{{\rm g}_3}}}}}{{_{}{t_{\rm ox}}}},{\rm{unde}}\; {\rm{material}} \;{\rm{M}}3\; {\rm{at}}\;y=0. $

(8)

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$ \frac{{{\rm d}{\textit{ϕ} _1}(x,y)}}{{{\rm d}x}} = \frac{{{\varepsilon _{\rm{ox}}}}}{{{\varepsilon _{\rm{si}}}}}{\rm{ }}\frac{{{\psi _{{{\rm g}1}}} - {\textit{ϕ}_{{{\rm s}1}}}(x)}}{{{t_{\rm{ox}}}}},{\rm{under}}\; {\rm{material}} \;{\rm{M}}1\; {\rm{at}}\;y=t_{\rm{si}}, $

(9)

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$ \frac{{{\rm d}{\textit{ϕ}_2}(x,y)}}{{{\rm d}x}} = \frac{{{\varepsilon _{\rm{ox}}}}}{{{\varepsilon _{\rm{si}}}}}{\rm{ }}\frac{{{\psi _{\rm g2}} - {\textit{ϕ}_{\rm s2}}(x)}}{{{t_{\rm{ox}}}}},{\rm{under}}\; {\rm{material}} \;{\rm{M}}2\; {\rm{at}}\;y=t_{\rm{si}}, $

(10)

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$ \frac{{{\rm d}{\textit{ϕ}_3}(x,y)}}{{{\rm d}x}} = \frac{{{\varepsilon _{\rm{ox}}}}}{{{\varepsilon _{\rm{si}}}}}{\rm{ }}\frac{{{\psi _{\rm g3}} - {\textit{ϕ}_{\rm s3}}(x)}}{{{t_{\rm{ox}}}}},{\rm{under}}\; {\rm{material}} \;{\rm{M}}3\; {\rm{at}}\;y=t_{\rm{si}}. $

(11)

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$ {C_{10}} = \,{\textit{ϕ}_{\rm s1}}(x), $

(12)

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$ {C_{11}} = \frac{{{\varepsilon _{\rm ox}}}}{{{\varepsilon _{\rm{si}}}}}\left[ {\frac{{{\textit{ϕ}_{\rm{si}}} - {\psi _{\rm g1}}}}{{{t_{\rm ox}}}}} \right], $

(13)

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$ {C_{12}} = - \frac{1}{{{t_{\rm si}}}}\frac{{{\varepsilon _{\rm{ox}}}}}{{{\varepsilon _{\rm{si}}}}}\left[ {\frac{{{\textit{ϕ} _{\rm{si}}} - {\psi _{\rm g1}}}}{{{t_{\rm{ox}}}}}} \right], $

(14)

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$ {C_{20}} = \,\,\,\,{\textit{ϕ}_{\rm s2}}(x), $

(15)

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$ {C_{21}} = \frac{{{\varepsilon _{\rm ox}}}}{{{\varepsilon _{\rm{si}}}}}\left[ {\frac{{{\textit{ϕ}_{\rm{si}}} - {\psi _{\rm g2}}}}{{{t_{\rm{ox}}}}}} \right], $

(16)

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$ {C_{22}} = - \frac{1}{{{t_{\rm{si}}}}}\frac{{{\varepsilon _{\rm ox}}}}{{{\varepsilon _{\rm{si}}}}}\left[ {\frac{{{\textit{ϕ} _{\rm{si}}} - {\psi _{\rm g2}}}}{{{t_{\rm{ox}}}}}} \right], $

(17)

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$ {C_{30}} = \,\,\,\,{\textit{ϕ}_{\rm s3}}(x), $

(18)

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$ {C_{31}} = \frac{{{\varepsilon _{\rm{ox}}}}}{{{\varepsilon _{\rm{si}}}}}\left[ {\frac{{{\textit{ϕ}_{\rm{si}}} - {\psi _{\rm g3}}}}{{{t_{\rm{ox}}}}}} \right], $

(19)

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$ {C_{32}} = - \frac{1}{{{t_{\rm{si}}}}}\frac{{{\varepsilon _{\rm{ox}}}}}{{{\varepsilon _{\rm{si}}}}}\left[ {\frac{{{\textit{ϕ} _{\rm{si}}} - {\psi _{\rm g3}}}}{{{t_{\rm{ox}}}}}} \right]. $

(20)

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$ {\textit{ϕ}_{\rm s1}}(0,0)=V_{\rm{bi}}, $

(21)

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$ {\textit{ϕ}_{\rm s1}}({L_1},0)={\textit{ϕ}_{\rm s2}}({L_1},0), $

(22)

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$ \frac{{\partial {\textit{ϕ}_{\rm s1}}}}{{\partial x}} = {\frac{{\partial {\textit{ϕ}_{\rm s2}}}}{{\partial x}}},\;\;{\rm{when}}\;x=L_1, $

(23)

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$ {\textit{ϕ}_{\rm s2}}({L_1} + {L_2},0) = {\textit{ϕ}_{\rm s3}}({L_1} + {L_2},0), $

(24)

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$ \frac{{\partial {\textit{ϕ}_{\rm s2}}}}{{\partial x}} = \frac{{\partial {\textit{ϕ}_{\rm s3}}}}{{\partial x}},\;\;\;\;{\rm{when}}\;x=L_1+L_2, $

(25)

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$ {\textit{ϕ}_{\rm s3}}({L_1} + {L_2} + {L_3},0) = {V_{\rm{bi}}} + {V_{\rm{DS}}}. $

(26)

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$ {\textit{ϕ}_{\rm s}}_1(x) = A{{\rm e}^{\lambda x}} + B{{\rm e}^{ - \lambda x}} + {\psi _{{\rm g1}}},{\rm{ 0}} \leqslant x \leqslant {L_1}, $

(27)

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$ {\textit{ϕ}_{\rm s}}_2(x) = C{{\rm e}^{\lambda (x - {L_1})}} + D{{\rm e}^{ - \lambda (x - {L_1})}} + {\psi _{{\rm g2}}},\;\;\;{L_1} \leqslant x \leqslant {L_1} + {L_2}, $

(28)

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$\begin{split} {\textit{ϕ}_{\rm s}}_3(x) =\, & E{{\rm e}^{\lambda (x - {L_1} - {L_2})}} + F{{\rm e}^{ - \lambda (x - {L_1} - {L_2})}} + {\psi _{{\rm g3}}},\\ & {L_1} + {L_2}\; \leqslant x \leqslant {L_1} + {L_2} + {L_3},\end{split}$

(29)

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$ \lambda = \sqrt {\frac{{2{\varepsilon _{\rm{ox}}}}}{{{\varepsilon _{\rm{si}}}{t_{\rm{ox}}}{t_{\rm{si}}}}}} $

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$ {\psi _{{\rm{g1}}}} = {V_{\rm{gs}}} - {\textit{ϕ}_{\rm m1}} + \chi + {E_{\rm g}}/2 $

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$ {\psi _{\rm g2}} = {V_{\rm{gs}}} - {\textit{ϕ}_{\rm m2}} + \chi + {E_{\rm g}}/2 $

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$ {\psi _{\rm g3}} = {V_{\rm{gs}}} - {\textit{ϕ}_{\rm m3}} + \chi + {E_{\rm g}}/2$

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$ A = \frac{{({V_{\rm{bi}}} - {\psi _{\rm g1}}){{\rm e}^{ - \lambda ({L_1} + {L_2} + {L_3})}} - ({V_{\rm{bi}}} + {V_{\rm{DS}}} - {\psi _{\rm g3}}) + ({\psi _{\rm g1}} - {\psi _{\rm g2}})\cosh \lambda ({L_2} + {L_3}) + ({\psi _{\rm g2}} - {\psi _{\rm g3}})\cosh \lambda {L_3}}}{{{{\rm e}^{ - \lambda ({L_1} + {L_2} + {L_3})}} - {{\rm e}^{\lambda ({L_1} + {L_2} + {L_3})}}}} $

(30)

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$ B = \frac{{({V_{\rm{bi}}} + {V_{\rm{DS}}} - {\psi _{\rm g3}}) - ({V_{\rm{bi}}} - {\psi _{\rm g1}}){{\rm e}^{\lambda ({L_1} + {L_2} + {L_3})}} - ({\psi _{\rm g1}} - {\psi _{\rm g2}})\cosh \lambda ({L_2} + {L_3}) - ({\psi _{\rm g2}} - {\psi _{\rm g3}})\cosh \lambda {L_3}}}{{{{\rm e}^{ - \lambda ({L_1} + {L_2} + {L_3})}} - {{\rm e}^{\lambda ({L_1} + {L_2} + {L_3})}}}} $

(31)

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$ C = A{{\rm e}^{\lambda {L_1}}} + \frac{{{\psi _{\rm g1}} - {\psi _{\rm g2}}}}{2}, $

(32)

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$ D = B{{\rm e}^{ - \lambda {L_1}}} + \frac{{{\psi _{\rm g1}} - {\psi _{\rm g2}}}}{2}, $

(33)

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$ E = C{{\rm e}^{\lambda {L_{21}}}} + \frac{{{\psi _{\rm g2}} - {\psi _{\rm g3}}}}{2}, $

(34)

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$ F = D{{\rm e}^{\lambda {L_2}}} + \frac{{{\psi _{\rm g2}} - {\psi _{\rm g3}}}}{2}, $

(35)

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$ {E_{1x}}(x) = - \frac{{{\rm d}{\textit{ϕ}_{\rm s1}}(x)}}{{{\rm d}x}} = - A\lambda {{\rm e}^{\lambda x}} + B\lambda {e^{ - \lambda x}},\;\;\;{\rm{0}} \leqslant x \leqslant {L_1} $

(36)

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$\begin{split} {E_{2x}}(x) = & - \frac{{{\rm d}{\textit{ϕ}_{\rm s2}}(x)}}{{{\rm d}x}} = - C\lambda {{\rm e}^{\lambda (x - {L_1})}} + D\lambda {{\rm e}^{ - \lambda (x - {L_1})}},\\ & {{{L}}_{\rm{1}}} \leqslant x \leqslant {L_1} + {L_2},\end{split}$

(37)

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$\begin{split} {E_{3x}}(x) \!= & - \frac{{{\rm d}{\textit{ϕ}_{\rm s3}}(x)}}{{{\rm d}x}} \!\!=\! - E\lambda {{\rm e}^{\lambda (\!x - {L_1} - {L_2}\!)}} \!+\! F\lambda {{\rm e}^{ - \lambda (\!x - {L_1} - {L_2}\!)}},\\ & {L_1} + {L_2}\; \leqslant x \leqslant {L_1} + {L_2} + {L_3}.\end{split}$

(38)

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$ {E_{1y}}(x) = - \frac{{{\rm d}{\textit{ϕ}_1}(x,y)}}{{{\rm d}y}} = - {C_{11}}(x) - 2y{C_{12}},\;\;\;{\rm{0}} \leqslant x \leqslant {L_1}, $

(39)

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$ {E_{2y}}(x) \!=\! - \frac{{{\rm d}{\textit{ϕ}_2}(x,y)}}{{{\rm d}y}} \!=\! - {C_{21}}(x) \!-\! 2y{C_{22}},\;\;{{L}_1}\! \leqslant x \leqslant \! {L_1} \!+ {L_2}, $

(40)

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$\begin{split} {E_{3y}}(x) = & - \frac{{{\rm d}{\textit{ϕ}_3}(x,y)}}{{{\rm d}y}} = - {C_{31}}(x) - 2y{C_{32}},\\ & {L_1} + {L_2}\; \leqslant x \leqslant {L_1} + {L_2} + {L_3}.\end{split}$

(41)

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$ {I_{\rm DS}} = q\iint {G{\rm d}x{\rm d}y}, $

(42)

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$ G(E) = {A_1}{E^{{D_1}}}\exp \left( { - \frac{{{B_1}}}{E}} \right), $

(43)

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$E = \sqrt {{E_x} + {E_y}} .$

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C. Usha, P. Vimala. A compact two-dimensional analytical model of the electrical characteristics of a triple-material double-gate tunneling FET structure[J]. Journal of Semiconductors, 2019, 40(12): 122901

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