﻿ 高斯表面在互连线参数提取中的应用 Application of Gaussian Surface to the Parameter Extraction of Interconnection Line

Applied Physics
Vol.07 No.11(2017), Article ID:22828,7 pages
10.12677/APP.2017.711042

Application of Gaussian Surface to the Parameter Extraction of Interconnection Line

Baojun Chen, Yanjie Ju

School of Electrical and Information Engineering, Dalian Jiaotong University, Dalian Liaoning

Received: Nov. 4th, 2017; accepted: Nov. 17th, 2017; published: Nov. 28th, 2017

ABSTRACT

With the further shrinkage of the feature size of integrated circuit, the reliability problem arising from RC delay of interconnection line becomes the main factor affecting the performance of chip. Affected by the manufacturing technology, the cross section of interconnection line is not regular rectangle, and RC delay is thus intensified. The analysis and computation of the parasitic parameter of such interconnection line using numerical method must firstly describe the rough surface. The surface height of interconnection line in the integrated circuit is not easy to measure. To this end, the paper proposes using Gaussian function to describe the rough surface. The experimental data indicate that the application of such surface to the parameters of interconnection line realizes accurate computation result.

Keywords:Gaussian Function, Interconnection Line, Parameter Extraction

1. 引言

Figure1. The surface of interconnect is rough

2. 生成高斯表面

$\overline{h}={〈h\left(x\right)〉}_{s}={\int }_{-\infty }^{\infty }hp\left(h\right)\text{d}h$ (1)

$〈\text{ }〉$ 表示沿整个粗糙面求高度的平均值，通常我们都选取适当的参考面，使得相对于此参考面的高度 $h\left(x\right)$ 的均值为零，这会给计算带来很大的方便。

$\text{RMS}=\sqrt{\frac{1}{\text{length}}{\int }_{0}^{\text{length}}{h}^{2}\left(x\right)\text{d}x}$ (2)

$C\left(l\right)=〈h\left(x\right)h\left(x+l\right)〉$ (3)

$\rho \left(l\right)=\frac{C\left(l\right)}{{\text{RMS}}^{\text{2}}}=\frac{〈h\left(x\right)h\left(x+l\right)〉}{{\text{RMS}}^{\text{2}}}$ (4)

$U=F\left(x\right)$ ，即可得：

$X={F}^{-1}\left(U\right)$ (5)

$U=X\mathrm{cos}\theta$ , $V=X\mathrm{sin}\theta$ (6)

$F\left(x\right)=\left\{\begin{array}{l}0\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }x<0\\ 1-\mathrm{exp}\left(-{x}^{2}/2{\sigma }^{2}\right)\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ge 0\end{array}$ (7)

$W=F\left(x\right)$ ，即可得

$X={F}^{-1}\left(W\right)=\sqrt{2{\sigma }^{2}\mathrm{ln}\left(\frac{1}{1-W}\right)}$ (8)

3. 粗糙表面参数计算结果

${\sigma }_{\text{discrepancy}}=\frac{\left({\sigma }_{\text{rough}}-{\sigma }_{\text{smooth}}\right)}{{\sigma }_{\text{smooth}}}$ (9)

(a) (b) (c)

Figure 2. Gaussian rough surface (a) RMS = 0.5 um, (b) RMS = 0.2 um, (c) RMS = 0.1 um

Figure 3. Corresponding discrepancy between mean effective conductivity and normal conductivity

Figure 4. Resistance ratio as a function of frequency

Figure 5. Resistance ratio as a function of frequency

4. 结论

Application of Gaussian Surface to the Parameter Extraction of Interconnection Line[J]. 应用物理, 2017, 07(11): 344-350. http://dx.doi.org/10.12677/APP.2017.711042

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