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SCIENTIA SINICA Informationis, Volume 49, Issue 3: 342-360(2019) https://doi.org/10.1360/N112017-00200

Threshold voltage and DIBL effect analysis and modeling for FD-SOI MOSFET with high k + SiO$_2$ gate

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  • ReceivedOct 13, 2017
  • AcceptedMar 28, 2018
  • PublishedMar 15, 2019

Abstract

This study aims to propose a gate structure of high k + SiO$_2$ for a fully depleted silicon-on-Insulator (FD-SOI) MOSFET. We developed a two-dimensional model to calculate its subthreshold surface potential of the front gate, threshold voltage, and drain induced barrier lowering (DIBL) effect. Based on the structure and different dielectric permittivity of FD-SOI MOSFET, the MOSFET of the subthreshold state is divided into several distinct rectangular equivalent sources. Furthermore, two-dimensional (2D) boundary value problems of Poisson and Laplace equations are built on the polygon region. Then, we use the method of separation of variables and the eigenfunction expansion to solve the 2D boundary value problems, and obtained their 2D solutions. Computational results show that the high k + SiO$_2$ gate can effectively suppress the degradation of FD-SOI MOSFET threshold voltage, the aggravation of DIBL effect, and the FIBL effect, which are caused by the dielectric permittivity of high k. Since the equations of the model are linear equations, their computational cost is minimal so that the model can be used for not only modeling and simulation of FD-SOI MOSFETs but also as a device model of circuit simulators.


Funded by

国家自然科学基金(61376098)

国家自然科学基金(61076086)


Supplement

Appendix

有限尺寸的栅电容由平行板和边缘电容组成,边缘电容

林志瑗, 杨铨让, 沙玉钧. 电磁场工程基础. 北京: 高等教育出版社, 1984.

\begin{equation}C_{{\rm edge}}=\frac{2\varepsilon}{2\pi}{\rm ln}\frac{2\pi w}{t}=\frac{\varepsilon}{\pi}{\rm ln}\frac{2\pi w}{t}, \tag{33}\end{equation} 式(A1)中$\varepsilon$是介电常数, $w$和$t$分别是栅的宽度和介质材料厚度.因此完整的栅电容是 \begin{equation}C_{{\rm g}}^\prime=\frac{\varepsilon}{\pi}{\rm ln}\frac{2\pi w}{t}+\frac{\varepsilon}{\pi}wl, \tag{34}\end{equation} 式中$l$是栅长度. 单位面积的栅电容是 \begin{equation}C_{{\rm g}}=\frac{C_{{\rm g}}^\prime}{lw}=\varepsilon\bigg[\frac{1}{t}+\frac{1}{lw\pi}{\rm ln}\frac{2\pi w}{t}\bigg], \tag{35}\end{equation} 上式表明单位面积的电容与沟道长度、介电常数、栅宽都有影响. par 下面用式(A3)计算不同介质的单位面积栅长电容. SiO$_2$介电常数是$\varepsilon_{{\rm~ox}}$, 栅介质电容是 \begin{equation}C_{{\rm ox}}=\varepsilon_{{\rm ox}}\bigg[\frac{1}{t_{{\rm ox}}}+\frac{1}{lw\pi}{\rm ln}\frac{2\pi w}{t_{{\rm ox}}}\bigg]=C_0\bigg[1+\frac{t_{{\rm ox}}}{lw\pi}{\rm ln}\frac{2\pi w}{t_{{\rm ox}}}\bigg], \tag{36}\end{equation} 式(A4)中$C_0=\frac{\varepsilon_{{\rm~ox}}}{t_{{\rm~ox}}}$, 是单位面积的平行板栅电容, 称作栅的本征电容. 若用等电容设计, 设高k材料介电常数是$\varepsilon_{{\rm~k}}$, 则有$t_{{\rm~g}}=\frac{\varepsilon_{{\rm~k}}}{\varepsilon_{{\rm~ox}}}t_{{\rm~ox}}$, 高k栅电容是 \begin{equation}C_k=\varepsilon_{{\rm k}}\bigg[\frac{1}{t_{\rm k}}+\frac{1}{\pi lw}{\rm ln}\frac{2\pi w}{t_{\rm k}}\bigg]= C_0\bigg[1+\frac{t_{\rm ox}}{\pi lw}\frac{\varepsilon_{{\rm k}}}{\varepsilon_{{\rm ox}}}\bigg({\rm ln}\frac{2\pi w}{t_{\rm ox}}-{\rm ln}\frac{\varepsilon_{{\rm k}}}{\varepsilon_{{\rm ox}}}\bigg)\bigg]. \tag{37}\end{equation}

高k + SiO$_2$栅的栅电容如图A1所示. $C_{{\rm~k+SiO_2}}$是$C_{{\rm~ox}}^\prime$与$C_{{\rm~k}}^\prime$的串联,等效电容是 \begin{equation}C_{{\rm k+SiO_2}}=\frac{C_{{\rm ox}}^\prime C_{{\rm k}}^\prime}{C_{{\rm ox}}^\prime+C_{{\rm k}}^\prime}. \tag{38}\end{equation}

等效电容设计的MOSFET SiO$_2$层厚度是$t_{{\rm~ox}}$, 令$t_{{\rm~ox}}^\prime=\alpha~t_{{\rm~ox}}~(0\leq\alpha\leq1)$是高k + SiO$_2$栅器件的SiO$_2$层厚度,等效氧化层厚度${\rm~EOT}=(1-\alpha)t_{{\rm~ox}}$, 则有 \begin{align*}& C_{{\rm ox}}^\prime=\varepsilon_{{\rm ox}}\bigg[\frac{1}{\alpha t_{{\rm ox}}}+\frac{1}{\pi lw}\bigg({\rm ln}\frac{2\pi w}{t_{\rm k}}+{\rm ln}\frac{1}{\alpha}\bigg)\bigg], \\ & C_{{\rm k}}^\prime=C_0\bigg[\frac{1}{1-\alpha}+\frac{\varepsilon_{{\rm k}}t_{\rm ox}}{\varepsilon_{{\rm ox}}\pi lw}\bigg({\rm ln}\frac{2\pi w}{t_{\rm ox}}+{\rm ln}\frac{\varepsilon_{{\rm ox}}}{\varepsilon_{{\rm k}}}+\frac{1}{1-\alpha}\bigg)\bigg]. \end{align*} 等效电容计算仅考虑高k材料的影响,故有$C_{{\rm~ox}}^\prime\approx\frac{C_0}{\alpha}$, $C_{{\rm~k}}^\prime\approx~C_0[\frac{1}{1-\alpha}+\frac{\varepsilon_{{\rm~k}}t_{\rm~ox}}{\varepsilon_{{\rm~ox}}\pi~lw}{\rm~ln}\frac{2\pi~w}{t_{\rm~ox}}]$, 式(6)的高k + SiO$_2$栅等效电容为 \begin{equation}C_{{\rm k+SiO_2}}=\frac{C_0\big[\frac{1}{1-\alpha}+\frac{\varepsilon_{{\rm k}}}{\varepsilon_{{\rm ox}}}\frac{t_{\rm ox}}{\pi lw}{\rm ln}\frac{2\pi w}{t_{\rm ox}}\big]}{\big[\frac{1}{1-\alpha}+\alpha\frac{\varepsilon_{{\rm k}}}{\varepsilon_{{\rm ox}}}\frac{t_{\rm ox}}{\pi lw}{\rm ln}\frac{2\pi w}{t_{\rm ox}}\big]} =C_0\left[1+\frac{(1+\alpha)^2\frac{\varepsilon_{{\rm k}}}{\varepsilon_{{\rm ox}}}\frac{t_{\rm ox}}{\pi lw}{\rm ln}\frac{2\pi wlw}{t_{\rm ox}}}{1+\alpha(1+\alpha)\frac{\varepsilon_{{\rm k}}}{\varepsilon_{{\rm ox}}}\frac{t_{\rm ox}}{\pi lw}{\rm ln}\frac{2\pi w}{t_{\rm ox}}}\right]. \tag{39}\end{equation}


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  • Figure 1

    High k + SiO$_2$ gate FD-SOI MOSFET structure diagram

  • Figure 2

    Calculates the coordinate system of the high k + SiO$_2$ gate FD-SOI MOSFET

  • Figure 3

    $\varepsilon_{{\rm~k}}=22$ high k dielectric high k + SiO$_2$ gate FD-SOI MOSFET front gate surface potential with the gate voltage changes

  • Figure 4

    High k + SiO$_2$ gate FD-SOI MOSFET front gate surface potential along the channel length distribution.protect łinebreak (a) $\varepsilon_{{\rm~k}}=7.5$ medium; (b) $\varepsilon_{{\rm~k}}=22$ medium

  • Figure 5

    High k and high k + SiO$_2$ gate FD-SOI MOSFET threshold voltage algorithm flow chart

  • Figure 6

    (a) High k gate, high k + SiO$_2$ gate and SiO$_2$ gate FD-SOI MOSFET threshold voltage and channel length relationship; (b) the relationship between threshold voltage and channel length of high k + SiO$_2$ gate FD-SOI MOSFET with different media

  • Figure 7

    FD-SOI MOSFET threshold voltage and the device's physical structure and material parameters of the relationship. (a) High k gate material unchanged, change the Si film doping concentration; (b) all parameters remain unchanged, only change the thickness of the Si film; (c) high k material unchanged, only change the ratio of high k layer and SiO$_2$ layer; (d) the high k material and device structure remain unchanged, changing only the back gate oxide thickness; (e) high k material and structure unchanged, plus different back gate voltage

  • Figure 8

    High k + SiO$_2$ gate FD-SOI MOSFET threshold voltage and dielectric constant relationship, the abscissa is the high k material dielectric constant. (a) Long channel (60 nm) situation; (b) the channel length is 20 nm short channel condition; (c) different thickness of the high k + SiO$_2$ gate, the channel length is 20 nm situation

  • Figure 9

    (a) The diagrammatic sketch for the leakage electric field intensity of two sides between the gate; (b) the edge effect discussed in the paper

  • Figure 10

    Surface potential distribution of SiO$_2$ gate, high k gate and high k + SiO$_2$ gate FD-SOI MOSFETs. protect łinebreak (a) The channel length is the surface potential $\phi_{\rm~f}(x)$ of 110 nm; (b) the channel length is 20 nm, the surface potential $\phi_{\rm~f}(x)$;protect łinebreak (c) the barrier map of (b), the unit is V

  • Figure 11

    20 nm channel length FD-SOI MOSFET $V_{{\rm~GS}}=0.3$ V, $V_{{\rm~GS}}$ and barrier height difference $\nabla\Phi$ diagram

  • Table 1   Capacitor capacitance per unit area of SiO$_2$ gate, high k + SiO$_2$ gate and high k gate
    Channel length ($l$) 110 nm 20 nm
    Gate capacitance of SiO$_2$ ($C_{\rm~ox}$) (1+0.0664)$C_0$ (1+0.365)$C_0$
    Gate capacitance of high k + SiO$_2$ ($C_{\rm~k+SiO_2}$) (1+0.157)$C_0$ (1+0.637)$C_0$
    Gate capacitance of high k ($C_{\rm~k}$) (1+0.295)$C_0$ (1+1.623)$C_0$
  • Table 2   Figure $V_{\rm~GS}=0.3$ V, 20 nm channel FD-SOI MOSFET surface potential $\phi_{\rm~f}(x)$ data
    $X$ (nm) 50.00 50.8 52.4 55.1 60.00 64.9 67.6 69.2 70.0
    *SiO$_2~\phi_{\rm~f}(x)$ SILVACO 0.5852 0.5174 0.4577 0.4089 0.4042 0.4827 0.5854 0.6859 0.7852
    Model 0.5828 0.5250 0.4771 0.4332 0.4234 0.5041 0.5947 0.7221 0.7807
    *High k + SiO$_2~\phi_{\rm~f}(x)$ SILVACO 0.5852 0.5213 0.4705 0.4330 0.4345 0.5077 0.5979 0.6886 0.7852
    Model 0.5829 0.5288 0.4885 0.4534 0.4516 0.5247 0.6037 0.6818 0.7804
    *High k$~\phi_{\rm~f}(x)$ SILVACO 0.5852 0.5608 0.5305 0.5033 0.5140 0.5949 0.6787 0.7420 0.7852
    Model 0.5824 0.5469 0.5218 0.5007 0.5076 0.5756 0.6415 0.7306 0.7785

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