Modeling of MOS Matching

Abstract

Circuit operation greatly depends on the ability to control and reproduce transistor and process parameters, such as oxide thickness, dielectric constants, doping levels, width and length. Variation in processing was in the past countered by defining process corners: boundaries in parameter variation that accounted for remaining process tolerances. With the improved control over processing, this batch-to-batch variation is largely under control.

However now a new class of phenomena has appeared: statistical variations. In conventional ICs, analog circuits with a differential operation (e.g. analog-to-digital converters) were already affected by this random parameter spread. The remaining variation between otherwise identical components is generally described by “mismatch” parameters. Next to these random phenomena also systematic errors called “offsets” play an increasingly important role Understanding and mitigating these effects requires statistical means and models.

The chapter will focus on the modeling of systematic and random effects that originate from physical, electrical, thermal and lithographical effects in devices causing intra-die variations.

Appendix: Derivation of Spatial Behavior

The Fourier transform is used to analyze the behavior of spatially distributed functions. The spatial Fourier transform and its reverse form in two dimensions are defined as:

Inserting the geometry function as defined in Fig. 15.15:

$$H\left( {{\omega }_{x}},{{\omega }_{y}} \right)=\frac{1}{W\,L}\int_{y=-W/2}^{W/2}{\left( \int_{x=-L/2+D/2}^{L/2+D/2}{{{e}^{-j{{\omega }_{x}}x}}{{e}^{-j{{\omega }_{y}}y}}dx}-\int_{x=-L/2-D/2}^{L/2-D/2}{{{e}^{-j{{\omega }_{x}}x}}{{e}^{-j{{\omega }_{y}}y}}dx} \right)dy.}$$

As the x and y components are independent, the y component can be solved separately:

Using the trigonometric identity: cos (a+b)=cos (a)cos (b)−sin (a)sin (b). After some re-arrangement:

$$\begin{array}{rcl}\displaystyle {\mathcal{G}}(\omega_x,\omega_y)&=&\displaystyle \frac{\sin(\omega_xL/2)}{\omega_xL/2}\frac{\sin(\omega_yW/2)}{\omega_yW/2}[2\sin(\omega_xD_x/2)],\\[16pt]\displaystyle \sigma^2_{\Delta P}&=&\displaystyle \frac{1}{4\pi^2}\int_{\omega_y=-\infty}^{\omega _y=\infty}\int_{\omega_x=-\infty}^{\omega_x=\infty}\left|{\mathcal{G}}(\omega _x,\omega_y)\right|^2\left|{\mathcal{P}}(\omega_x,\omega_y)\right|^2d\omega _xd\omega_y.\end{array}$$

(15.24)

The variance of parameter ΔP between two rectangular devices is then found by substitution of (15.8) and the above-described models for the long and short correlation distance variations in (15.24). Mathematically the white noise model is described in the Fourier domain as a constant: ${\mathcal{P}}(\omega_{x},\omega_{y})={\mathcal{N}}$

Considering that the ω _x and ω _y dimensions can be separated:

Using the standard integrals (CRC handbook 1984, p. 289 form 628 and 630):

(15.25)

$$\int_{x=0}^{\infty }{{{\left[ \frac{\sin (ax)}{x} \right]}^{2}}dx=\frac{ax}{2}}$$

(15.26)

the second integral is easy to solve, the first integral can be re-written into a series of squared sine waves, resulting in a π/2 if (a=L/2) represents the smaller of the two sine coefficients.

$$\sigma^2_{\Delta P}=\frac{{\mathcal{N}}^2}{4\pi^2}\times\frac{\pi/2\times L/2}{(L/2)^2}2^2\times\frac{\pi W/2}{(W/2)^2}=\frac{2{\mathcal{N}}^2}{WL}.$$

This is the first part of the equation. For solving the second part the gradient on the wafer must be modeled. A polar description would describe accurately the circular gradient. As this example only sensitivity parallel to the x axis is considered ω _x =1/2W _D where W _D is the wafer diameter. Now ℘(ω _x )=δ(1/2W _D )

$$\sigma^2_{\Delta P}=\frac{1}{4\pi^2}\int_{\omega_x=-\infty}^{\omega_x=\infty}\delta(1/2W_D)\left[2\sin(\omega_xD_x/2)\frac{\sin(\omega_xL/2)}{\omega_xL/2}\right]^2d\omega_x.$$

It will be clear that the delta function appears at a very low spatial frequency. For that low frequency, the (sin x/x) term can be considered to approach “1”. leaving:

$$\sigma^2_{\Delta P}=\frac{1}{4\pi^2}\int_{\omega_x=-\infty}^{\omega_x=\infty}\delta(1/2W_D)\left[2\sin(\omega_xD_x/2)\right]^2d\omega_x\approx \left(\frac{D_x}{2W_D}\right)^2.$$

Combining both results:

$$\sigma^2_{\Delta P}=\frac{A^2_P}{WL}+S^2_PD_x^2.$$