# Modelling and inverse Problems in Nanometrology

Nanometrology is the science of measurement of distances and displacements of objects at the nanoscale. Mathematical modelling and the numerical simulation in nanometrology supports evaluations of measurements and estimations of uncertainties at the nanoscale level.

# Scatterometry

# Introduction

Scatterometry is the investigation of micro- or nanostructured surfaces regarding their geometry and dimension by measurement and analysis of light diffraction from these surfaces. An example of the experimental setup for scatterometry is shown in Fig. 1a. Scatterometry is an indirect optical measurement, i.e., the sought geometrical parameters of the investigated periodic surface structures have to be reconstructed from the measured light diffraction pattern (inverse problem). In a first step the numerical simulation of the diffraction process for 2D periodic structures is realized by the finite element method (FEM) of Mawell's equations. Then the inverse problem is expressed as a non-linear operator equation where the operator maps the sought mask parameters to the efficiencies of the diffracted plane wave modes and the deviation of the calculated from the measured efficiencies are minimized.

# Mathematical model

Non-imaging metrology methods like scatterometry are in contrast to optical methods not diffraction limited. They give access to the geometrical parameters of periodic structures like structure width (CD), pitch, side-wall angle or line height (cf. Fig. 2). However, scatterometry requires apriori information. Typically, the surface structure needs to be specified as member of a certain class of gratings and is described by a finite number of parameters, which are confined to certain intervals. The inverse diffraction problem has to be solved to determine the structure parameters from a measured diffraction pattern.

The conversion of measurement data into desired geometrical parameters depends crucially on a high precision rigorous solution of Maxwell's equations, which can be reduced to the two-dimensional Helmholtz equation if geometry and material properties are invariant in one direction. The typical transmission conditions of electro-magnetic fields yield continuity and jump conditions for the transverse field components; the radiation conditions at infinity are well established. For the numerical solution, a lot of methods have been developed. We use the finite element method (FEM) and truncate the infinite domain of computation to a finite one by coupling it with boundary elements (cf. Fig. 2).

# Inverse Problem: Least Squares

Apart from the forward computations of the Helmholtz equation, the solution of the inverse problem, i.e. the reconstruction of the grating profiles and interfaces from measured diffraction data, is the essential task in scatterometry. FEM-based optimization procedures, for example included in the DIPOG software of the Weierstrass-Institute for Applied Analysis and Stochastics in Berlin, can be used to reconstruct the profile parameters. The problem is equivalent to the minimization of an objective functional describing the difference between the calculated and the measured efficiency pattern in dependence of the assumed model parameters.

Fig. 3a shows the shape of the objective functional calculated by varying the heights of the Cr- and the CrO-layer of a Chrome on glass mask. For the selected admissible range of the two heights the coordinates of the minimum values of the objective functional are very near to the expected values of 50 nm for hCr and 18 nm for hCrO. However, they were only found if the initial value for hCrO is set to a value smaller than 60 nm where the functional has a ridge parallel to the hCr axis. Otherwise the (second) local minimum of the functional on the other side of this maximum is found. Fig. 3b shows a similar situation for a TaN-EUV mask where line width ($p_2$) and line height ($p_6$) were varied. For gradient-based optimization methods the admissible range of model parameters and their initial values can have a strong influence on the accuracy of the reconstruction results.

# Maximum likelihood Method

The classical method to determine the profile parameters from measured light diffraction patterns by optimization is the least squares (LSQ) method. The norm of the difference between the simulated and the measured data is minimized. The right choice of the weights accounting for the variances in the measured data plays a crucial role. Inappropriate weights of the components in the LSQ sum may result in incorrect reconstructed profile parameters and furthermore an overestimation of the associated uncertainties.

The maximum likelihood estimation (MLE) overcomes this pitfall by addressing the variances of the measurement data $\sigma_j^2=\left(a\cdot f_j\left(\mathbf{p}\right)\right)^2+b^2$ as sought parameters too.

From the assumption that the measurement errors $\sigma_j$ are normally distributed, the likelihood function $\mathcal{L}\left(a,b,\mathbf{p}\right)$ can be written as a function of the error parameters $a$ and $b$ and the geometry parameters $\mathbf{p}$:

\[

\mathcal{L}\left(a,b,\mathbf{p}\right)=\prod_{j=1}^{m}\left(2\pi\left(\left(a\cdot f_j\left(\mathbf{p}\right)\right)^2+b^2\right)\right)^{-1/2}\exp\left[-\frac{(f_j\left(\mathbf{p}\right)-y_j)^2}{2\left(\left(a\cdot f_j\left(\mathbf{p}\right)\right)^2+b^2\right)}\right]

\]

The maximation of the likelihood function $\mathcal{L}\left(a,b,\mathbf{p}\right)$ yields the estimates of the sought parameters $(a,b,\mathbf{p})$.

# Line roughness

To get reliable simulations and reconstructions, line edge roughness (LER) of the line-space structures of the lithographic masks has to be taken into account. The periodicity of the line-space structrues are disturbed due to LER and significant impacts on the measured light diffraction pattern are expected. Investigations with stochastically disturbed edge positions of the absorber lines reveal that the mean efficiencies of the scattering light are damped exponentially in dependence on the standard deviation of the roughness amplitude $\sigma_r$ and the diffraction order $n_j$:

\[ \widetilde{f}_j\left(\sigma_\mathrm{r},\mathbf{p}\right)=\exp(-\sigma_\mathrm{r}^2

k_j^2)\cdot f_j\left(\mathbf{p}\right) \]

where $k_j = 2 \pi n_j/d$, $n_j$ is the order of the diffractive mode and $d$ the period of the line-space structure. $\sigma_\mathrm{r}^2$ denotes the variance of the edge position. In rigorous finite element simulations this expression for the analytical damping factor of the mean scattered efficiencies was confirmed by using computational FEM domains of large periods for the cross section of the lithographic mask containing many line-space pairs with stochastically chose widths, but still with straight edges. Fig. 4 depicts the scheme of the applied 1D LER model.

To verify the proposed systematic impact of LER on the measured efficiencies in terms of the order of the diffractive mode and standard deviation of roughness amplitudes, we go up to investigations on randomly perturbed 2D binary grating. Their edge positions are controlled by an exponentially decaying autocorrelation function allowing a significantly more realistic modelling of line edge roughness. We simulate the diffraction of line-space gratings by that of arrays of strip shaped slits (apertures). In order to create apertures with rough boundary lines, we apply an autocorrelation function to describe the variations along the line edges. Considering lines $\{(x(y),y):\;y\in\mathbb{R}\}$ with random variables $x(y)$, we assume a constant mean value $\langle x(y) \rangle=x_0$ and that the correlation \begin{equation*}

x(y_1,y_2):=\frac{\big\langle [x(y_1)-x_0] [x(y_2)-x_0] \big\rangle }{x_0^2}

\end{equation*}

depends on the distance $r=|y_1-y_2|$ only, i.e. $x(y_1,y_2)=x(r)$. Furthermore, we assume the exponentially decaying autocorrelation function \begin{equation*}

x(r)=\sigma^2 e^{-(r/\xi)^{2\alpha}},\end{equation*} where $\sigma$ is the standard deviation of the edge positions, $\xi$ is the linear correlation length along the line, and $\alpha$ is called roughness exponent. Randomized line edge profiles $x$ are generated by calculating or approximating the associated power spectrum density function $\mathrm{P\!S\!D}(r^{-1})$ that belongs to the autocorrelation function $x(r)$, and then applying to the calculated $\rm P\!S\!D$ a random complex exponential phase term, being uniformly distributed in the range of $[0,2\pi]$. Subsequently, the inverse Fourier transform of that disturbed $\rm P\!S\!D$ provides a rough line edge profile $x$. By this means the rough line edge shown in Fig. 5 was generated for a standard deviation $\sigma$=3 nm, a roughness exponent $\alpha$=0.5 and a correlation length $\xi$=10 nm.

Applying the Fraunhofer approximation the irradiance pattern of illuminated rough apertures far away from the source plane is numerically calculated very efficiently as the 2D-Fourier transform of the light distribution in the aperture plane and then compared to those of the unperturbed, 'non-rough' aperture. Many different ensembles of rough apertures representing variant roughness patterns characterized by different values of $\sigma$, $\xi$, and $\alpha$ were examined. Only a slight increase within a range of maximal 5% were found for the determined $\sigma_r$ compared to the imposed standard deviation $\sigma$ of the associated rough ensemble. Fig. 6 summarizes these findings for many different ensembles up to a correlation length $\xi$ of 150 nm and for two different values of $\alpha$ = 1.0 and $\alpha$ = 0.5.

In summary it can be said, therefore, that the proposed exponential damping factor is quite robust with respect to different values of the correlation length $\xi$ and the roughness exponent $\alpha$.