3.2.3 Lumped Parameter Model

The lumped parameter model uses a simple but very fast photographic model to relate the development time to exposure [53,54]. Based on the assumption of a constant contrast (3.1) can be integrated to give a simple expression for the development rate,

 

$$r(x,z) = r_0 \left(\frac{E(x,z)}{E_0} \right)^\gamma.$$ (3.4)

Here, E(x, z) refers to the exposure energy inside the resist and is related to the normalized aerial image I_i(x) by E(x, z) = E_w I_i(x) I_r(z). E_w denotes the exposure energy needed to generate a feature of width w and I_r(z) is the relative, vertically z-dependent intensity variation inside the resist.^a The integration constants E₀ and r₀ are, respectively, the energy required to exactly clear (w = 0) the photoresist in the allotted development time and the development rate resulting from an exposure of this amount. Another parameter of the lumped parameter model is the effective resist thickness^b

$$D_\mathrm{eff} = \int\limits_0^D \left(\frac{I_r(z)}{I_r(0)} \right)^\gamma\, dz,$$ (3.5)

with D as actual resist thickness. The ``effective'' resist thickness weights the actual height by a term related to the change in the development rate varying from the top to the bottom of the film. In case of absorption the development rate is smaller at the bottom so that the resist film seems to be thicker as it is in reality, i.e., a thinner effective resist is required to be cleared in the same time.

Using a phenomenological two-step development model which is based on the assumption that development occurs first vertically and then horizontally [55], the two segment times can be calculated from (3.4). The sum of these two times equals the total development time and the following expression is derived:^c

 

$$\frac{E_w}{E_0} = \left[ 1+ \frac{1}{D_\mathrm{eff}} \int\limits_... ...)^{-\gamma}\, dx\right]^{\scriptstyle\frac{\scriptstyle1}{\scriptstyle\gamma}}.$$ (3.6)

Note that E_w stands for the required exposure dose to generate a line of width w, and E₀ refers to full resist removal. (3.6) determines the required exposure dose E_w to produce a feature of width w for a given aerial image intensity I_i(x). The two parameters of the model, the effective resist thickness Deff and the contrast $\gamma$, are routinely available in most production and development lithography processes. Hence, linewidth versus exposure curves can be generated extremely fast by means of simulation. This simple model can be used as an initial predictor of results or as the engine of a lithography control scheme.

Footnotes

... thickness,⁰

... resist.^a

Originally, in (3.1) the z-dependence of the development rate was suppressed. However, for the following considerations it is important. Various simple methods to compute I_r(z) are described in Section 5.2.

... thickness^b

The effective resist thickness is defined as Deff = r(x, D) tdev, where D is the resist height, r(x, D) the development rate at the resist bottom, and tdev the development time. Integration of the rate equation dt = r^-1(x, z) dz yields tdev = r₀^-1$\int_{0}^{D}$(E(x, z)/E₀)^{- $\gamma$} dz, whereby (3.4) was inserted. The expression for the effective resist thickness Deff is now found by a combination of its definition with (3.4).

... derived:^c

The vertical development time follows from the definition of the effective resist thickness,⁰ t_z = Deff/r(0, D). The horizontal development time is obtained from integration of the rate equation dt = r^-1(x, z) dx and writes to t_x = r₀^-1$\int_{0}^{w/2}$(E(x, z)/E₀)^{- $\gamma$} dx. Summing up the two times yields tdev = Deff/r(0, D)$\Big[$1 + D^-1eff$\int_{0}^{w/2}$(I_i(x)/I_i(0))^{- $\gamma$} dx$\Big]$ and combines with (E_w/E₀)^$\gamma$ = tdevr(0, D)/Deff to the provided formula.