Optimizing Hybrid Metrology: Rigorous Implementation of Bayesian and Combined Regression

Abstract

Hybrid metrology, e.g., the combination of several measurement techniques to determine critical dimensions, is an increasingly important approach to meet the needs of the semiconductor industry. A proper use of hybrid metrology may yield not only more reliable estimates for the quantitative characterization of 3-D structures but also a more realistic estimation of the corresponding uncertainties. Recent developments at the National Institute of Standards and Technology (NIST) feature the combination of optical critical dimension (OCD) measurements and scanning electron microscope (SEM) results. The hybrid methodology offers the potential to make measurements of essential 3-D attributes that may not be otherwise feasible. However, combining techniques gives rise to essential challenges in error analysis and comparing results from different instrument models, especially the effect of systematic and highly correlated errors in the measurement on the χ² function that is minimized. Both hypothetical examples and measurement data are used to illustrate solutions to these challenges.

1 INTRODUCTION

Hybrid metrology, e.g., the combination of distinct measurement techniques to determine critical dimensions, is an increasingly important approach to meet the needs of the semiconductor industry. A proper use of hybrid metrology may yield not only more reliable estimates for the quantitative characterization of 3-D structures but also a more realistic estimation of the corresponding uncertainties. Ideally it helps to reduce the overall uncertainties by combining the individual strengths of each of the measurement techniques, making sub-nanometer uncertainties a realistic goal as critical dimensions (CDs) approach 10 nm.

Recent developments [1–3] in hybrid metrology at the National Institute of Standards and Technology (NIST) feature the combination of optical critical dimension (OCD) and scanning electron microscope (SEM) measurements. The challenges and possible solutions have been outlined by some of these authors in a previous proceedings paper [3]. Various methods have been presented to combine measurement results from different tool platforms, revealing two related but distinct challenges. There must be an overlapping parameter set for combined regression[3], such that each individual parametric geometry mush share at least one parameter in common (e.g. height). Additionally a priori, each individual method must also yield parametric values that together with their uncertainties are statistically consistent, usually quantified by a Z-test, see Ref. [2] for more details.

This paper can be seen as a continuation of that work and Ref. [4], with an emphasis on the proper treatment of the measurement errors, including highly correlated and systematic errors and their influence on hybrid metrology. Tool-induced errors for OCD and scale errors for SEM are investigated, and it is shown how the parametric uncertainties can be decreased if those issues are addressed accurately in the hybridization.

Since the term hybrid metrology has gained increased significance in the dimensional metrology community outside NIST [5–7] we will start this work with a short overview of two of the most common techniques in Section 2, namely the Bayesian approach and combined regression. The measured targets and the generalized parameter sets that describe them are discussed in Section 3 before we give a detailed description of the performed error analysis both for OCD and SEM in Section 4 and their impact on the hybridization in Section 5. We will close with the conclusions in Section 6.

2 HYBRID METROLOGY

Hybrid metrology has gained significant recognition in recent times as an approach to considerably reduce parametric uncertainties by combining different measurements of the same measurand. We want to use this section to identify the main differences and similarities of two of the most common hybrid approaches, namely the use of a priori information in a Bayesian sense and combined regression. We start with the Bayesian approach and continue with combined regression. In order to keep the notation simple, throughout this section we will assume that only two measurement techniques are combined with each individual method yielding a statistically consistent set of parameters. Note that even if some of the notations are different, the presented approaches are equivalent to those given in Ref. [2]. Additional information for those that are not familiar with all of the terminology of Bayesian data analysis can be found in Refs. [8–10].

2.1 Bayesian Approach

The Bayesian approach treats information provided by each of the measurement tools quite differently. The first tool provides m values of measurement data that can be described as a vector in an m-dimensional real vector space

$$\mathbf{y}={(y_1,\dots ,y_m)}^T\in {\mathbb{R}}^m$$ (1)

that contains only indirect information about the quantity of interest. We therefore need to analyze the data in terms of an inverse problem [11]. A common approach to solve an inverse problem is to set it up as a regression problem. Initially, we need to provide a model function

$$\mathbf{f}:{\mathbb{R}}^n\to {\mathbb{R}}^m,\mathbf{f}\left(\mathbf{p}\right)={\left(f_1\right(\mathbf{p}),\dots ,f_m(\mathbf{p}\left)\right)}^T$$ (2)

that maps the parameters of interest (e.g., the height, the width, etc.), that are identified with a vector in an n-dimensional real vector space p = (p₁, …, p_n)^T, to the simulated quantities. These simulated data are generated in the same m-dimensional space defined by the measurement. The regression problem then amounts to minimizing the difference between the modeled and the measured data. We therefore solve for the parameter vector $\widehat{\mathbf{p}}$ that minimizes the so-called χ² function, measuring the goodness of fit of the simulation to the measurement data given by the weighted norm

$${\chi }^2\left(\mathbf{p}\right)={\Vert \mathbf{f}\left(\mathbf{p}\right)-\mathbf{y}\Vert }^2={\left(\mathbf{f}\right(\mathbf{p})-\mathbf{y})}^T{\mathbf{V}}^{-1}\left(\mathbf{f}\right(\mathbf{p})-\mathbf{y}).$$ (3)

In this formulation we implicitly assume that the measurement data y is a noisy realization of the model, and that we have an additive error model where

$$\mathbf{y}=\mathbf{f}\left(\mathbf{p}\right)+ϵ.$$ (4)

Note that the the errors that are added to each of the model values can also be arranged as an m-dimensional vector. We assume the errors to be normally distributed with zero mean and m × m covariance matrix Σ_ϵ. The matrix $\mathbf{V}\in {\mathbb{R}}^{m\times m}$ in Eq. 3 is usually chosen to be an estimate of Σ_ϵ. We will discuss the importance of a good choice of V in more detail later. If we ignore normalization constants we see that the function in Eq. 3 is proportional to the negative log-likelihood function for the chosen error model [10]. We furthermore assume that we have information about the parameter vector of interest p, such as an estimate μ_p and an uncertainty, or more specifically a covariance matrix Σ_p. This can be based on expert knowledge or an already completed analysis of measurement data from a second measurement tool. If we assume the parameters of interest to be normally distributed we have all the information we need to express this prior information in terms of a probability density function (PDF). This PDF is usually called the prior distribution π_pri. In the case of a normally distributed random vector p it is given by:

$${\pi }_{\text{pri}}:{\mathbb{R}}^n\to \mathbb{R},{\pi }_{\text{pri}}\left(\mathbf{p}\right)={\left({\left(2\pi \right)}^n\text{det}\right({\Sigma }_{\mathit{p}}\left)\right)}^{-1∕2}\text{exp}\left(-\frac{1}{2}{(\mathbf{p}-{\mu }_{\mathit{p}})}^T{{\Sigma }_{\mathit{p}}}^{-1}(\mathbf{p}-{\mu }_{\mathit{p}})\right).$$ (5)

By subtracting the prior distribution, or more precisely its logarithm, from the function in Eq. 3, the negative log-likelihood, we get a function that is proportional to the negative logarithm of the posterior probability distribution. If we again ignore normalization constants we get the function that serves as the modified χ² function

$${\tilde{\chi }}^2\left(\mathbf{p}\right)={\left(\mathbf{f}\right(\mathbf{p})-\mathbf{y})}^T{\mathbf{V}}^{-1}\left(\mathbf{f}\right(\mathbf{p})-\mathbf{y})+{(\mathbf{p}-{\mu }_{\mathit{p}})}^T{{\Sigma }_{\mathit{p}}}^{-1}(\mathbf{p}-{\mu }_{\mathit{p}}).$$ (6)

Note, that the second term that reflects the prior information acts as a penalty or regularization term, penalizing possible solutions to the inverse problem for measurement tool 1 that are not consistent with the prior information. The function in Eq. 6 is finally minimized to find the parameter vector estimate $\widehat{\mathbf{p}}$.

2.2 Combined Regression

In combined regression we start with two distinct sets of measurement data y_A and y_B that come from two different measurement techniques. Their model functions f_A (p_A) and f_B (p_B), depend on parameter vectors p_A and p_B, respectively. The models must have at least one common model parameter in order to perform combined regression. Determining what the common parameters of the two models are can be a challenging task, see Ref. [3] for further details. In combined regression we define the combined χ² function to be the sum of the individual χ² functions for each of the tools. Note, that this is only possible if we assume that the two measurements are independent of each other.

$${\chi }^2\left({\mathbf{p}}_{\mathrm{A}\cup \mathrm{B}}\right)=\chi _{\mathrm{A}}^2\left({\mathbf{p}}_{\mathrm{A}}\right)+\chi _{\mathrm{B}}^2\left({\mathbf{p}}_{\mathrm{B}}\right).$$ (7)

Here p_A⋃B is the vector that consists of the union of the elements in p_A and p_B. The solution ${\widehat{\mathbf{p}}}_{\mathrm{A}\cup \mathrm{B}}$ to the inverse problem in combined regression is then found by minimizing the above combined χ² function.

3 MEASURED TARGETS AND GENERALIZED PARAMETER SETS

The investigated targets and the used geometrical parametrizations have already been described in detail in Ref. [3], so we will only give a brief overview. We investigate finite 30-line arrays of Si on Si with a thin native conformal oxide; see Fig. 1. The nominal widths are 14 nm, 16 nm and 18 nm. A schematic representation of the geometrical parametrizations can be found in Fig. 2. For the OCD analysis the geometry is fully characterized by the height, width, Δtop and Δbot of a single line. The physics by which this geometry interacts with incident light to produce a signal was approximated by the rigorous coupled-wave analysis (RCWA) model that is based on a semi-analytical treatment of Maxwell's equations [12, 13]. The OCD data used in this study has been generated using this RCWA model for 30 lines and a measurement set-up similar to the actual experiment at NIST [14, 15]. The nominal values are height=36.0 nm, width=17.1 nm, Δtop=2.9 nm and Δbot=0.5 nm. The noise that has been added to this simulated data has been generated using the same correlated errors as described in Section 4.1.

Fig 1.

SEM image showing the 30 line arrays. Horizontal field of view is 3.63 μm.

Fig 2.

Initial parameters for (a) optical critical dimension (OCD) and (b) SEM modeling. The parameters that are in bold font are part of the reduced subset of these parameters that has been used in this work.

The initial geometrical parameters for the SEM fit of each line included the line position, its width at half height, and its left and right sidewall edge slopes (represented by the edge widths). Although left and right sidewall slopes of an individual line often differed, this was a manifestation of roughness. On average, left and right edge widths were indistinguishable. Consequently, for hybridization with OCD, which averages over a comparatively large area, it was sufficient to treat sidewall width as a single parameter. Similarly, line position was not important for OCD. Hybridization was limited therefore to two parameters: the width at the middle and Δtop. The geometry is related to the measured signal by JMONSEL, a single-scattering Monte Carlo simulator with a choice among a number of physical models of the scattering phenomena that affect electron transport. We used tabulated Mott cross sections to model scattering of electrons from nuclei and a dielectric function theory approach to secondary electron generation. Electron-phonon interactions and electron refraction (or total internal reflection) at the vacuum/Si interface were included. Details of JMONSELs treatment of these models are available elsewhere[16].

4 ERROR ANALYSIS

Equipped with the proper set of overlapping parameters we will now focus on a more realistic modeling of the present measurement data. We will investigate the influence of tool-induced measurement errors for OCD and the influence of scaling errors for SEM.

4.1 OCD

The specific way in which the OCD measurements are performed, by taking images on the xy plane going through focus, requires increased attention to possible systematic and correlated errors. For example, one can imagine that being slightly off axis in the illumination will affect the symmetry of the entire set of collected images. Determining such correlations from measurement data alone is often not possible and other methods need to be developed in order to give a quantitative description of those effects. We use the Monte Carlo method [17], that is based on the following reasoning. If we denote by ν the vector of k fixed parameters of the measurement set-up we can make use of the following more general model function

$$\mathbf{f}:{\mathbb{R}}^{n+k}\to {\mathbb{R}}^m,\mathbf{f}(\mathbf{p},\nu )={\left(f_1\right(\mathbf{p},\nu ),\dots ,f_m(\mathbf{p},\nu \left)\right)}^T.$$ (8)

The effect that a slight deviation of the parameters ν from the nominal values ν₀ has upon the simulated image can then be estimated from the following five steps:

1. Assume a distribution for ν, based on tool specifications, expert knowledge, etc.

2. Draw a sample {ν_i}_i=1,…,N from the distribution

3. Calculate {f (p, ν_i) = (f₁ (p, ν_i), …, f_m (p, ν_i))^T}_i=1,…,N

4. Define r_i = f (p, ν₀) − f (p, ν_i)

5. Calculate the sample covariance matrix via

$$\tilde{\mathbf{V}}=\frac{1}{N-1}\sum _{i=1}^N({\mathbf{r}}_i-\stackrel{‒}{\mathbf{r}}){({\mathbf{r}}_i-\stackrel{‒}{\mathbf{r}})}^T.$$ (9)

In this paper the ν vector included components for the collection numerical aperture (CNA), illumination numerical aperture (INA), focus heights and phase. This Monte Carlo procedure therefore estimates the effect of errors propagated from variations in those instrument parameters; e.g., we choose the INA to be normally distributed with a mean of 0.13 and a standard deviation of 0.01. Figures 3 and 4 show a graphical representation of the first 752 entries of the sample covariance matrix that correspond to the first four focus heights in X polarization for phase, focus, illumination and collection numerical aperture variations along with the respective concatenated images. One can clearly see both the positive and negative correlation between errors as colored areas off the diagonal. In order to have a reliable model for the measurement errors it is therefore very important to account for this effect in the covariance matrix that is being used in the χ² function; see Eq. 3. This does not only have an influence on the best fit values but also on the estimation of the parametric uncertainties, or more precisely the covariance matrix Σ, diagonal elements of which are the uncertainties in the estimated parameters, given by [2]:

$$\Sigma ={\left({\mathbf{J}}^T{\mathbf{V}}^{-1}\mathbf{J}\right)}^{-1}.$$ (10)

Fig 3.

Estimated covariance matrices for variations in the phase (left) and focus (right). Random phase perturbations normally distributed with a mean of 0 rad and a standard deviation of π/5 rad, Focus offset normally distributed with a mean of 0 nm and a standard deviation of 2 nm. The variance induced by phase variations is three orders of magnitude greater than those from focus variation. Four concatenated OCD intensity profiles are duplicated below the matrices in Figs. 3 and 4 to illustrate the positional dependence of the covariance. In general changes correlate to the position of the finite target.

Fig 4.

Estimated covariance matrices for variations in the illumination numerical aperture (INA) (left) and collection numerical aperture (CNA) (right). INA normally distributed with a mean of 0.13 and a standard deviation of 0.01, CNA normally distributed with a mean of 0.95 and a standard deviation of 0.1. The variances induced by INA and CNA variations are one magnitude less than those from phase errors.

Here J denotes the Jacobian matrix, i.e., the matrix of all first-order partial derivatives of the vector-valued model function f, at the best fit vector $\widehat{\mathbf{p}}$. A comparison of the parametric uncertainties based on using a diagonal V and the full V in Eq. 10 can be found in Tab. 1 below.

Table 1.

Estimates of the parametric uncertainties for simulated OCD data using diagonal V, i.e., only accounting for uncorrelated random noise, and full V, i.e., taking correlations into account.

	width/nm	Δtop/nm	Δbot/nm	height/nm
$\widehat{\mathbf{p}}$	17.09	2.04	0.10	35.37
σ diagonal V	0.05	1.37	0.53	0.97
σ full V	0.14	1.29	0.63	0.92

Note that there is a notable difference between the estimated parametric uncertainties with the most significant change in the uncertainty of the width. It is also very important to note that a given parametric uncertainty might increase or decrease if correlations in the measurement data are taken into account. A general statement about the effect correlated errors have on the estimated parametric uncertainties can thus not be given and it must be investigated for each new problem separately.

4.2 SCALING ERRORS IN SEM

The biggest contribution to the SEM's measurement error in this experiment is due to pixel calibration. Errors in this calibration directly influence the obtained values for the critical dimensions. The usual approach is to simply add the uncertanity due to the calibration in quadrature with the estimated parametric uncertainty after the reconstruction. The estimation of the parameters of interest, i.e., the vector p and their uncertainties in combined regression are based on the combination of the OCD and the SEM data, while the error induced by pixel calibration only affects SEM data; hence it is not possible to simply include the uncertainty due to the calibration afterward. We will therefore model the effect a variation in the scale has on the measurement in a simple way, multiplying the model parameter vector p = (width, Δtop)^T by a scaling parameter κ such that the modified model is given by:

$$\tilde{\mathbf{f}}:{\mathbb{R}}^3\to {\mathbb{R}}^m,\tilde{\mathbf{f}}(\kappa ,\mathbf{p})=\mathbf{f}(\kappa \cdot \mathbf{p})$$ (11)

with f being the SEM's model function. In this description, κ = 1 corresponds to no scale error, κ = 1.01 to 1 % scale error, etc. It is clear that this approach leads to a high parametric correlation in the parameters of the model. We will explain this effect using a simple model with an added scaling factor in the following:

Let

$$\mathbf{f}:\mathbb{R}\to {\mathbb{R}}^m,\mathbf{f}\left(x\right)={\left(f_1\right(x),\dots ,f_m(x\left)\right)}^T,\text{and}\mathbf{J}={\left(J_{i,1}\right)}_{i=1,\dots ,m},J_{i,1}=D{f}_i$$ (12)

be a model function depending on only one parameter x and denote by Df_i the derivative of f_i with respect to this parameter. If we assume the measurement errors to be independent and identically distributed (i.i.d.) with unit variance (hence V = (δ_i,j)_i,j=1,…,m) we have for the estimated covariance matrix

$$\Sigma ={\left({\mathbf{J}}^T{\mathbf{V}}^{-1}\mathbf{J}\right)}^{-1},\text{with}{\mathbf{J}}^T{\mathbf{V}}^{-1}\mathbf{J}=\sum _{i=1}^m{\left[D{f}_i\right]}^2,$$ (13)

which is well defined if det $\left({\mathbf{J}}^T{\mathbf{V}}^{-1}\mathbf{J}\right)={\sum }_{i=1}^m{\left[D{f}_i\right]}^2\ne 0$.

Now assume that we add a scale parameter to the above model by defining a slightly modified function

$$\mathbf{F}:{\mathbb{R}}^2\to {\mathbb{R}}^m,\mathbf{F}(\kappa ,x)=\mathbf{f}(\kappa \cdot x),\text{and}\mathbf{J}={\left(J_{i,j}\right)}_{i=1,\dots ,m;j=1,2},J_{i,1}=x\cdot D{f}_i,J_{i,2}=\kappa \cdot D{f}_i.$$ (14)

The estimated covariance matrix for this two parameter model is then given by

$$\Sigma ={\left({\mathbf{J}}^T{\mathbf{V}}^{-1}\mathbf{J}\right)}^{-1},\text{with}{\mathbf{J}}^T{\mathbf{V}}^{-1}\mathbf{J}=\left[\begin{array}{cc}\hfill {\Sigma }_{i=1}^m{x}^2{\left[D{f}_i\right]}^2\hfill & \hfill {\Sigma }_{i=1}^m\kappa x{\left[D{f}_i\right]}^2\hfill \\ \hfill {\Sigma }_{i=1}^m\kappa x{\left[D{f}_i\right]}^2\hfill & \hfill {\Sigma }_{i=1}^m{\kappa }^2{\left[D{f}_i\right]}^2\hfill \end{array}\right].$$ (15)

However,

$$\det \left({\mathbf{J}}^T{\mathbf{V}}^{-1}\mathbf{J}\right)=\sum _{i=1}^m{x}^2{\left[D{f}_i\right]}^2\cdot \sum _{i=1}^m{\kappa }^2{\left[D{f}_i\right]}^2-{\left\{\sum _{i=1}^{m}kx{\left[D{f}_i\right]}^2\right\}}^2=0.$$ (16)

Since the above term is always equal to zero, the estimated covariance matrix Σ is not defined and we can not assign a parametric uncertainty. But since we have prior information about the scale, the actual error lies between 1–2%, we can use the Bayesian approach as described in [2, 9] under the premise, that the prior information can be expressed in terms of normal distributions. The prior information on the parameter κ is treated as an additional data point, the model function has to account for, such that we have a function that still depends on κ and x but now maps into an (m + 1)-dimensional space, with the (m + 1)-th value simply being κ. This also adds additional terms to the Jacobian, $J_{m+1,1}=\frac{\partial }{\partial \kappa }\kappa =1$ and $J_{m+1,2}=\frac{\partial }{\partial x}\kappa =0$, such that

$$\tilde{\mathbf{F}}:{\mathbb{R}}^2\to {\mathbb{R}}^{m+1},\tilde{\mathbf{F}}(\kappa ,x)={\left(\mathbf{F}\right(\kappa ,x),\kappa )}^T,\text{and}{J}_{m+1,1}=1,J_{m+1,2}=0.$$ (17)

Since we know that $\kappa ~\mathcal{N}(1,{\sigma }_{\kappa }^2)$, we add an additional entry to V, namely $V_{m+1,m+1}={\sigma }_{\kappa }^2$ with σ_k = 0.01 – 0.02, and obtain the estimated covariance matrix, again using Eqs. 10 and 13,

$$\Sigma ={\left({\mathbf{J}}^T{\mathbf{V}}^{-1}\mathbf{J}\right)}^{-1},\text{with}{\mathbf{J}}^T{\mathbf{V}}^{-1}\mathbf{J}=\left[\begin{array}{cc}\hfill {\Sigma }_{i=1}^m{x}^2{\left[D{f}_i\right]}^2+\frac{1}{{\sigma }_{\kappa }^2}\hfill & \hfill {\Sigma }_{i=1}^m\kappa x{\left[D{f}_i\right]}^2\hfill \\ \hfill {\Sigma }_{i=1}^m\kappa x{\left[D{f}_i\right]}^2\hfill & \hfill {\Sigma }_{i=1}^m{\kappa }^2{\left[D{f}_i\right]}^2\hfill \end{array}\right]$$ (18)

and

$$\det \left({\mathbf{J}}^T{\mathbf{V}}^{-1}\mathbf{J}\right)=\left\{\sum _{i=1}^m{x}^2{\left[D{f}_i\right]}^2+\frac{1}{{\sigma }_{\kappa }^2}\right\}\cdot \sum _{i=1}^m{\kappa }^2{\left[D{f}_i\right]}^2-{\left\{\sum _{i=1}^{m}kx{\left[D{f}_i\right]}^2\right\}}^2.$$ (19)

Equation 19 implies that as long as the prior knowledge about κ is not too vague, i.e. σ_κ is not too large, the above term is in general not equal to zero and the estimated covariance matrix is well defined. A graphical representation of the above described phenomenon for the measured SEM data is show in Fig. 5. Note how the χ² surface with the added prior information about κ has a distinct minimum at $\widehat{\mathbf{p}}=(1,17.01\mathrm{nm},2.86\mathrm{nm})$ while it is hard to determine where the minimum is for the χ² surface without prior information. In fact there is not a single distinct minimum, but an infinite set of possible minima. Obviously, $\widehat{\mathbf{p}}=(1,17.01\mathrm{nm},2.86\mathrm{nm})$ is also minimum for the χ² without prior information, but so is any vector $\widehat{\mathbf{p}}=(\kappa ,\frac{17.01}{\kappa }\mathrm{nm},\frac{2.86}{\kappa }\mathrm{nm})$ with κ ≠ 0. Defining a parametric uncertainty is therefore not possible. In contrast, the error estimation for the model with prior information for the scale κ yields a 2 % parametric uncertainty if we assume an error of 2 % in the scale as expected; here this strict linearity only holds since the random errors in the SEM data are much smaller than those attributed to the scale error.

Fig 5.

χ² surface for SEM data without (left) and with (right) Bayesian input for the scale κ.

5 RESULTS

We now combine the results that we found in the previous section in the hybridization of OCD and SEM data by combined regression. As pointed out in Section 2 this is done by minimizing the sum of the respective χ² functions. Note that we use prior information about the scale κ, so that the χ² function for the SEM data is modified as shown in Eq. 6. The individual χ² surfaces in dependence on the width and Δtop are shown in Figures 6 and 7. For these plots the height and Δbot have been fixed for OCD. The plots also show the individual minima and the assigned parametric uncertainties. The χ² surface from the combined regression is shown in Fig. 8, and the results from the combined regression are presented in Tab. 2. The combined minimum is close to the SEM's minimum and the parametric uncertainties for combined regression are lower than the individual ones, even for the parameters that are only present in the OCD model. This is due to the strong parametric correlations in the models.

Fig 6.

χ² surfaces for the SEM data.

Fig 7.

χ² surfaces for the simulated OCD data.

Fig 8.

Combined χ² surface for the hybridization.

Table 2.

Parameter estimates and parametric uncertainties obtained from combined regression.

	width/nm	Δtop/nm	Δbot/nm	height/nm
${\widehat{\mathbf{p}}}_{\mathit{hybrid}}$	17.01	2.94	0.59	36.11
σ hybrid	0.23	0.04	0.58	0.21
σ SEM	0.34	0.06	-	-
σ OCD	0.14	1.29	0.63	0.92

6 DISCUSSION AND CONCLUSION

Following the approach outlined in Ref. [3] we studied the challenges in hybrid metrology due to measurement errors. Those included highly correlated tool-induced errors for the OCD data and systematic errors due to scaling errors in SEM. We have demonstrated how slight variations in the measurement set-up for OCD, e.g., in the focus heights, the phase, INA and CNA lead to highly correlated errors in the measurement data, that manifest themselves as non-zero elements in the sample covariance matrix V. Including those off-diagonal elements in the estimation of a parametric uncertainty can lead to either an increased or a decreased parametric uncertainty compared to the case where only the diagonal of the V matrix is used, depending on the individual nature of that particular full V matrix. Furthermore, we demonstrated the influence of scaling errors on the analysis of SEM data. Attempting to account for such scale errors by including the scale as a fully free parameter would lead to unreasonable results due to strong, or even perfect, correlations. This problem has been solved using prior information about the scale in a Bayesian approach. Finally we demonstrated how the more sophisticated error analyses could be used in the hybridization of OCD and SEM data. With the proper treatment of those errors we could achieve a sub-nanometer parametric uncertainty. It is important to note that the presented framework can be extended to include additional measurement techniques, such as atomic force microscopy (AFM) or critical dimension small angle X-ray scattering (CD-SAXS). In addition it may also be applied across a homogenous multiple-tool platform. However for every added measurement technique it is crucial to perform a careful error analysis in order to use its full capabilities.

Biographies

Mark-Alexander Henn is a guest researcher at the National Institute of Standards and Technology. He received his Diplom in mathematics from the Technical University of Berlin, Germany in 2008, and his PhD degree in theoretical physics from the Technical University of Berlin in 2013. He worked for the Physikalisch-Technische Bundesanstalt in Berlin before. His current research interests include Bayesian data analysis, electro-magnetic modeling, and fourier optics.

Richard Silver received his BA in physics from the University of California at Berkeley and his PhD in physics from the University of Texas at Austin. He is the Surface and Nanostructure Metrology Group Leader, a fellow of SPIE, co-chair of the European Optical Modeling Conference and on the Program Committee of the Metrology, Inspection, and Process Control Conference. He has over 100 archived publications and was a recipient of the 2013 RD 100 Award for Quantitative Hybrid Metrology and the 2013 Intel Outstanding Researcher Award.

John S. Villarrubia received his PhD in physics from Cornell University in 1987. He was a visiting scientist for two years at IBM, where he did scanning tunneling microscopy of silicon surfaces. At the National Institute of Standards and Technology since 1989, he has been working on aspects of nanometer-scale dimensional metrology, particularly modeling the instrument function for Atomic Force and Scanning Electron Microscopes and using these models of the instrument function to correct images for measurement artifacts. Nien Fan Zhang is presently a Mathematical Statistician of the Statistical Engineering Division at the US National Institute of Standards and Technology (NIST). He received MS and PhD in statistics in 1983 and 1985 respectively, at Virginia Polytechnic Institute and State University.

Hui Zhou is a research scientist and a consultant at Dakota Consulting, Inc. His main research area is computational electrodynamics, and his interest spreads to all areas between physics and computing. When he is not at work, you may find him reading a random non-fiction book, playing a game of go, or dreaming of computation and physics in the future.

Bryan M. Barnes is a physicist at the National Institute of Standards and Technology. He received his BA degree in mathematics and physics from Vanderbilt University in 1995, and his MS and PhD degrees in physics from the University of Wisconsin-Madison in 1997 and 2004, respectively. He is the author of more than 40 proceeding and journal papers and holds one patent. His current research interests include optical defect inspection, hybrid metrology, and critical dimension metrology.

Andras E. Vladar, PhD is the leader of the Three-Dimensional Nanometer Metrology Project at the National Institute of Standards and Technology (NIST) USA. He is an expert in scanning electron microscopy and dimensional metrology, one the best-known research scientists and a technical leader of this field. His research interest is in SEM-based sub-10 nm three-dimensional measurements for semiconductor and nanotechnology applications.

Bin Ming: Biography is not available.