3.2.2 Point Optimization of the Aerial Image

Full optimization of a lithography process requires thorough and time-consuming calculations of many effects. One simplified approach to this optimization problem is to perform limited calculations at one point in space, e.g., the nominal line edge. The optimization of certain important lithographic parameters is then performed only at this point. In spite of its restriction on a single point the method has often physical significance and the results can be very useful. Additional simplifications of the analysis are achieved by separating the effects of the lithography tool from that of the resist process. This can be done with reasonable accuracy only if the interaction of the imaging tool with the photoresist is known. The lithographic imaging equation [49,50]

 

$$\frac{\partial \ln r(x)}{\partial x} = \gamma \,\frac{\partial \ln I_i(x)}{\partial x}$$ (3.1)

relates the aerial image intensity I_i(x) to the development rate r(x) of the photoresist, whereby the lumped parameter $\gamma$ denotes the resist contrast and x is the horizontal position in the mask/wafer plane relative to the investigated point. The gradient of the logarithm of the aerial image is simply called the log-slope. As can be seen from (3.1) the development rate gradient in lateral direction and thus the resolution is either maximized by higher resist contrast or by a larger log-slope of the aerial image.

Equation (3.1) clearly indicates that the aerial image log-slope is a proper metric to judge the performance of the imaging tool. In particular, the image log-slope, when normalized by multiplication with the feature width w, is directly proportional to exposure latitude expressed as a percent change in exposure to give a percent change in linewidth. The normalized image log-slope NILS [49,50], thus defined

$$\mathit{NILS} = w\, \frac{\partial \ln I_i(x)}{\partial x},$$ (3.2)

has proven to be a powerful metric for optimizing lithography processes by means of simulation [51,52]. The resolution of different features can be conveniently studied by plotting the normalized image log-slope versus defocus. As shown in Figure 3.4 the depth of focus is extremely sensitive to the feature size, a fact that is not evident in the theoretical Rayleigh definition given by (2.2). Hence, resolution and depth of focus cannot be defined independently, but are rather interdependent.

  

Figure 3.4: The normalized image log-slope is directly proportional to exposure latitude and has proven to be a powerful metric in lithography optimization.

Although defocus is a strictly optical phenomenon, the photoresist plays a significant role in determining focus effects. Principally, a better photoresist provides a greater depth of focus, i.e., the minimum acceptable log-slope specification is lower resulting in a larger usable focus range. This relationship between photoresist and the log-slope specification can be determined experimentally by measuring focus-exposure matrices like in Figure 3.2 for many different feature types and sizes. The resulting correlation between normalized image log-slope NILS and exposure latitude  EL is typically expressed by

$$\mathit{EL} = \alpha ( \mathit{NILS} - \beta).$$ (3.3)

Here $\alpha$ is the percent in increase of exposure latitude EL per unit increase of NILS, and $\beta$ is the minimum NILS required to give any image at all in the photoresist. Thus, to a first degree the effect of the photoresist on depth of focus can be characterized by the two parameters $\alpha$ and $\beta$. For an ideal, infinite contrast resist it can be shown that $\alpha$ = 10 and $\beta$ = 0 [53]. The quality of a photoresist with respect to focus and exposure latitude can now be judged by how close $\alpha$ and $\beta$ are to these ideal values.