Uncertainty evaluation in atomic force microscopy measurement of nanoparticles based on statistical mixed model in a Bayesian framework

The variability of the measured quantity as a function of the main influencing fixed factors and the random effects is described by the mixed model equation as follows:

$$\begin{equation} h_{ijkl} = \beta_0+\sum_{f = 1}^{m_f}\textrm{X}_{fijk}\,\vec{\beta}_{f}+a_{i}+b_{ij}+c_{ijk}+\epsilon_{ijkl,} \end{equation} \tag{ 1 }$$

where h_ijkl stands for the measured height (of a nanoparticle or a step height), X_fijk stands for the effect-type coding matrix for effect f and for which i = 1, ..., 5 refers to the different days, j = 1, ..., 5 to the measurement positions on a given day, k = 1, ..., 3 to the repeated images for a given combination of day and position and l = 1, ..., n_ijk stands for the different measurements on image k taken at position j on day i. n_ijk is the number of nanoparticles (or step heights) measured on the kth repetition of image taken at position j on day i. The realization h_ijkl of h is thus the lth observed value in the kth image taken at position j on the ith day. The total number of nanoparticles n (or step heights) measured is given by

$$\begin{equation*} n = \sum_{i, j, k}n_{ijk}. \end{equation*}$$

The random effects $a_{i}\sim N(0,\sigma_{\textrm{day}}^2)$, $b_{ij}\sim N(0,\sigma_{\textrm{pos}}^2)$, $c_{ijk}\sim N(0,\sigma_{\textrm{im}}^2)$ and $\epsilon_{ijkl}\sim N(0,\sigma^2_{\textrm{res}})$ are mutually independent for all i, j, k and l. The variance $\sigma_{\textrm{day}}^2$ expresses day-to-day variability, $\sigma_{\textrm{pos}}^2$ the variance between positions, $\sigma_{\textrm{im}}^2$ the variance between repeated images and the within image variability $\sigma^2_{\textrm{res}}$ captures the residual variance of the observed quantity within an image, representing the sample polydispersity by the squared standard deviation of its size distribution.

The fixed effects are noted $\vec{\beta}_{f}$, where f = 1, ..., m_f , with a priori m_f  = 5 fixed effects. The intercept β₀ represents the mean response when there are no fixed effects present in the model (1). Fixed effects are categorical variables for which effect-type coding is used (X_fijk matrix) [12]. This coding sets the different categories into contrast while each considered on equal footing. If an effect cancels out, its resulting $\vec{\beta}_f$ is null and induces no shift between β₀ and h_ijkl , which is the second advantage of the effect-type coding over usual dummy coding. Every possible combination of levels of these coding variables provides an opinion $\mathbb{E}[h_{ijkl}|\textrm{X}_{fijk} = \tilde{\textrm{X}}]$ about the measured diameter (or step height). The different opinions are averaged out through a linear opinion pool [16]. The effects are analyzed and interpreted using their corresponding opinions, with effect-type coding in the present work.

Taking the linear interaction of fixed effects into account, equation (1) becomes

$$\begin{equation} h_{ijkl} = \beta_0+\sum_{f = 1}^{m_f}\textrm{X}_{f,ijk}\,\vec{\beta}_{f}+\sum_{f\neq g } \textrm{Y}_{fgijk}\,\vec{\beta}_{fg}+a_{i}+b_{ij}+c_{ijk}+\epsilon_{ijkl,} \end{equation} \tag{ 2 }$$

where Y_fgijk is the combined coding effect matrix expressed as the tensor product of the matrices of single effect coding.

The estimation of the model is made with the restricted maximum likelihood method as it allows to estimate separately the random and the fixed effects [17, 18]. Fixed effects that are not significant at 95% level of confidence are removed from the model.

In the case of nano-gratings, although the four reference samples (listed in table 1) are processed similarly and are made of the same material (Si/SiO₂ layers coated with an uniform layer of platinum), samples S_[1] and S_[3] behave differently than S_[2] and S_[4]: the probe is found to have a significant effect only for the sample S_[1] and S_[3]. We have no clear explanation for this feature. None of the other effects are significant for all the reference samples.

In the case of nanoparticles, probe, tapping force and scan speed do have an effect while the operator and the analyst do not. The number of significant fixed effects is reduced from m_f  = 5 to m_f  = 3. Moreover, it is found that the interaction between the tapping force and the scan speed and the interaction between the tapping force and the probe are significant.

Between n = 140 000 and 280 000 step height measurements on each reference standards are performed according to ISO standard 5436-1 [14] and the Bayesian estimation with the mixed model (2) is carried on the data. The estimated variance of the respective random effects are shown in table 3 for the 4 standards. Clearly, the image repeatability does not bring any variability. For all the standards, the largest contribution comes from the within image variability but contribution from day and position cannot be discarded. Regarding the fixed effects and as already mentioned above, only the probe effect is significant and strangely enough, only for gratings S_[1] and S_[3], but not for gratings S_[2] and S_[4] (see table 4).

Table 3. Variance of the random effects influencing the step height measurements for each grating [nm²].

	σ2[·] for	σ2[·] for	σ2[·] for	σ2[·] for
Effect	for S[1]	for S[2]	for S[3]	for S[4]
Day	0.0075	0.0001	0.5439	1.0923
Position	0.0149	0.0016	0.1623	0.6170
Image repeat.	0.0000	0.0003	0.0002	0.0018
Within image var. ($\epsilon_{\textrm{res}}$)	0.0309	0.0455	0.0105	1.4797

Table 4. Standard uncertainty of the probe fixed effect for each grating.

Grating	$SD\,(\beta_{\textrm{probe}})$ (nm)
S[1]	0.11
S[2]	NS
S[3]	0.25
S[4]	NS

The expected value of the measured mean step height µ_m for the 4 standards $\mathbb{E}\,[\mu_{m}]$ and the corresponding standard uncertainty SD [µ_m ] are extracted from the posterior distribution of the model (see table 5).

Table 5. Expected values and standard uncertainties for the measured mean step height µ_m for each reference standard grating.

Grating	$\mathbb{E}\,[\mu_{m}]$ (nm)	SD [µm ] (nm)
S1	15.91	0.13
S2	42.15	0.01
S3	99.06	0.54
S4	177.04	0.70

The certified mean heights and standard uncertainties of the 4 reference gratings, as given in the calibration certificate, are summarized in table 1. The PDF for the certified step heights µ_c[r] (r = 1, 2, 3, 4) are assumed to follow normal distributions with mean and standard deviation set according to the certificate.

Having the 4 calibration points $(\mathbb{E}\,[\mu_{m}],\mathbb{E}\,[\mu_{c}])_{[r]}$, the calibration curve is obtained by fitting the quadratic regression equation:

$$\begin{equation} \mu_{c[r]} = \alpha + \gamma\, \mu_{m[r]} + \delta\, \mu_{m[r]}^2 + \epsilon. \end{equation} \tag{ 3 }$$

The parameter ε is a normal deviation from the quadratic model. In the Bayesian framework, these parameters are expressed by probabilities and the PDFs for α, γ, δ and ε display the variability present in the calibration points and the model uncertainty. The joint density for (α, γ, δ, ε) is approximated by sampling N times from its distribution through the following procedure:

• take N samples $({\mu_{m}}_{j},{\mu_{c}}_{j})_{[r]}$ (for j = 1, ..., N) from the 4 calibration points (r = 1, 2, 3, 4);
• calculate for j = 1, ..., N the estimated coefficients α_j , γ_j and δ_j , and the mean squared error $s^{2}_j$ by performing N times an ordinary least squares quadratic regression with regression data ${(\mu_{m}}_{j},{\mu_{c}}_{j})_{[r]}$ for r = 1, 2, 3, 4;
• sample a random value ε_j from $\mathcal{N}(0,s^{2}_j)$ (for j = 1, ..., N);
• collect the sample $(\alpha_j,\gamma_j,\delta_j,\epsilon_j)$.

The joint distribution of (α, γ, δ, ε) is approximated by merging the N samples $(\alpha_j,\gamma_j,\delta_j,\epsilon_j)$ for j = 1, ..., N. Table in figure 3 shows the results obtained for the parameters with N = 10⁶. Given the value of δ, the calibration curve can be defined as linear. The 95% confidence interval of the calibration curve and the mean calibration curve are also illustrated in figure 3.

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Similarly to what has been performed for the step heights, an intermediate precision experiment is performed for the nanoparticles, following an appropriate design of experiment. To obtain a sufficient amount of particles on a single image a scan range of 3 µm × 3 µm is chosen with 1024 × 1024 pixels, leading to pixel size of about 3 nm × 3 nm. Three different kinds of tapping probe are used (Olympus AC160TS, Olympus AC240TS, PPP-NCHR) with different spring constants and resonance frequencies. The effect of scan speed in the scanning direction is assessed by considering three different values (i.e. 1.8, 3.6 and 5.4 µm s⁻¹). Also four values of probe oscillation amplitude ratio (i.e. the ratio between the tapping amplitude setpoint and the free tapping amplitude in air), related to the tip tapping force, are tested (i.e. 65%, 70%, 75% and 80%). The electronic feedback controller parameters are chosen by the operator to obtain a good tracking of the topography. These adjustments are subjective choices going into the operator effect. As previously discussed, the operator and the analyst are found to have no significant contribution to the variability of the measurements and is removed from the model.

Executing the experiments, collecting the nanoparticles heights in the different conditions and performing a Bayesian fit with model (2) as described previously, leads to the results shown in tables 6 and 7 for the random and fixed effects on the measurand variability. n = 8259 nanoparticle height measurements were considered for this purpose. Image repeatability is not investigated for nanoparticles because it was shown not significant for step height standards and not considering this effect reduces significantly the measurement time.

Table 6. Estimated variance of the random effects influencing the nanoparticle height measurements.

Effect	$u^2(\cdot)\, (\textrm{nm}^{2}$)
Day	$\mathbb{E}\,[\sigma^2_{\textrm{day}}]$ = 0.16
Position	$\mathbb{E}\,[\sigma^2_{\textrm{pos}}]$ = 0.88
Within image variability ($\epsilon_{\textrm{res}}$)	$\mathbb{E}\,[\sigma^2_{\textrm{res}}]$ = 7.61

Table 7. Estimated standard deviation of the respective fixed effects significantly influencing the nanoparticle height measurements.

Effect	u [nm]
Probe	$SD\,[\beta_{\textrm{probe}}]$ = 0.49
Amplitude ratio	$SD\,[\beta_{\textrm{tapping force}}]$ = 0.52
Scan speed	$SD\,[\beta_{\textrm{speed}}]$ = 0.39

The different levels of the coding variable for the parameters probe, amplitude ratio and scan speed are expressed by posterior PDFs. The average of these level posteriors are considered to calculate the respective standard uncertainties as the standard deviation of these average PDFs, shown in table 7. The main effects of the fixed factors are given in this table, although interaction terms (amplitude ratio · scan speed and probe · scan speed) are also included. The combination of the posterior PDFs of all these parameters in model (2) gives rise to a PDF for the measured height h_m of a single nanoparticle and for the measured mean height µ_m . The expected value $\mathbb{E\,}[h_{m}]$ and the corresponding uncertainty SD [h_m ] for the measured height h_m of a single particle, and $\mathbb{E}\,[\mu_{m}]$ and SD [µ_m ] for the measured mean height µ_m are given in table 8.

Table 8. Expected values and standard uncertainties for the measured particle height h_m and the mean measured height µ_m of the gold nanoparticle sample.

$\mathbb{E}\, [h_{m}]$ (nm)	SD [hm ] (nm)	$\mathbb{E}\,[mu_{m}]$ (nm)	SD [µm ] (nm)
23.39	3.18	23.40	1.19

Results from Bayesian approach have been compared to estimation by the frequentist approach (REML). Fixed effects and random effects are compatible between the two approaches and within their uncertainties. Bayesian approach generally results in smaller uncertainties, which might be explained by correlations better accounted for.

The regression model (3) expresses the relation between the measured and the certified mean step height of the reference standards, so adopting this relation to obtain a calibration curve leads to

$$\begin{equation} q_c = \alpha + \gamma\, q_m + \delta\, q_m^2 + \epsilon, \end{equation} \tag{ 4 }$$

where q_c is the certified quantity of interest and q_m is the measured quantity of interest. The certified PDF of the measurand q_c is obtained by running Monte Carlo through equation (4) with the measured q_m ($q_m = h_m$ and $q_c = h_c$ for the particle height and $q_m = \mu_m$ and $q_c = \mu_c$ for the mean particle height).

The consecutive steps to perform traceable dimensional measurement of nanoparticles are illustrated in figure 4: (a) the particle heights are collected through AFM measurements under given conditions, (b) PDFs are constructed with model (2) and (c), (d) these PDFs are corrected through the application of the calibration curve. The PDFs summary parameters for the special case of the gold nanoparticles under investigation are listed in table 9. Reference value for RM8012 provided in NIST report of investigation is 24.9 nm with an expanded uncertainty of 1.1 nm [21]. The uncertainty detailed in the present work overlaps with value from NIST but is larger. The mean value is slightly smaller than NIST mean value. This might partially be explained by the deformation of the nanoparticles at the contact with substrate as described in other works [9, 23]. Taking this effect into account could be an improvement of the present work, for which the focus was set on investigating instrumental and operation effects.

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Table 9. Expected values and standard uncertainties for the certified particle height h_c and the certified mean height µ_c of the gold nanoparticle sample.

$\mathbb{E}\,[h_{c}]$ (nm)	SD [hc ] (nm)	$\mathbb{E}\,[\mu_{c}]$ (nm)	SD [µc ] (nm)
23.24	3.28	23.25	1.44