A metrological atomic force microscope system

In order to perform measurements, the MAFM sensor has to be calibrated against the traceable z-interferometer of the NMM-1. The sampled output signals of the MAFM sensor must be converted in length values using polynomial coefficients, which are derived from the characteristic curves of the output signals of the probing system. The working range and set point (needed for the force control) are determined from the approach and retract curves. From the consecutively repeatedly recorded (two hundred times) characteristic curves the linearity and the repeatability of the polynomial coefficients, i.e. the offset and the slope can be analysed. The residuals r, or rather the deviations of the calibration are calculated according to the equations derived in [7]. The standard deviations of the slope are less than 4·10⁻⁴ nm/digit and the standard deviations of the mean residuals r do not exceed 0.6 nm. The offset and the standard deviations highly depend on temperature. In our system, the temperature dependence of the drift of the bending signals can be estimated as 100 nm/K.

The analog signals of the MAFM are connected to the 16-bit A/D-converter of the digital signal processor (DSP) of the NMM-1. In order to analyse the signal noise, the bending signal d_z and the position data of the z-axis of the NMM-1 l_z were recorded by the A/D-converter of the DSP unit for 10 s. A moving average filter was applied to the raw data in order to remove the drift and calculate the signal noise values. The standard deviation of the signal noise without contact to the measured surface amounts to 0.2 nm (figure 3, red). The standard deviation of the signal noise with contact to the measured surface and without or with control of the NMM-1 is 0.93 nm or 1.41 nm (figure 3, green or blue), respectively. In this case, the MAFM operates as a passive probe system, where the lateral scan motion and vertical scan/control motion are carried out by the NMM-1. The NMM-1 control dynamics is limited by the high masses of the stage and corner mirror of the NMM-1. The signal noise of the bending signals was therefore dominated by the influence of low-frequency disturbances on the stage.

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The measurement uncertainty is defined as the parameter assigned to the measurement result, which characterises the spread of the values that can be reasonably assigned to the measured results [13]. According to the guide to the expression of uncertainty in measurement (GUM), the individual values of each uncertainty component can be evaluated according to type A method, which determines by evaluating several statistically independent measured values from the series of measurements, or type B method, which determines by other means, for example by taking the values from a calibration certificate or personal experience [14].

The measurement uncertainty depends on the entire measurement chain. At the beginning of the measurement chain is the laser source. With the laser interferometric measuring method, the measured path difference is represented by a function of the wavelength of the He-Ne laser. Since the measurements do not take place in a vacuum, the influence of the refractive index n of the medium must be taken into account. Environmental influences such as temperature, air pressure, air humidity and CO₂ content play a role in determining the refractive index and measurement uncertainty. A measurement of the CO₂ content is omitted due to the lower influence and high power loss of the CO₂ sensor [7]. The current refractive index n of the air for a wavelength of 632.991234 nm can be calculated with known data of temperature, air pressure and air humidity and is only carried out after a new sampling of environmental data.

Firstly, the water vapour partial pressure p_v in Pa is calculated from the saturation water vapour partial pressure p_sv in Pa using the relative humidity RH in % and the air temperature T in K derived from the air temperature ϑ in °C (T = ϑ + 273.15 K) according to equation (1) [7, 15, 16].

$$\begin{eqnarray}&&\begin{array}{l}{p}_{{\rm{v}}}=0.01\,{\rm{RH}}{p}_{{\rm{sv}}}\\ =0.01\,{\rm{RH}}\exp \left({A}_{{\rm{Dav}}}{T}^{2}+{B}_{{\rm{Dav}}}T+{C}_{{\rm{Dav}}}+{D}_{{\rm{Dav}}}/T\right)\end{array}\end{eqnarray} \tag{ 1 }$$

with the following coefficients of the Davis formula:

• A_Dav = 1.2378847·10^–5,
• B_Dav = −1.9121316 10^–2,
• C_Dav = 33.93711047,
• D_Dav = −6.3431645·10³.

The correction with the measured values for air pressure p in Pa and air temperature ϑ in °C is carried out according to equation (2) [7, 15].

$$\begin{eqnarray}&&\left(n\left(\vartheta ,p\right)-1\right)=\displaystyle \frac{{c}_{1}p}{{D}_{{\rm{Boe}}}}\displaystyle \frac{1+\left({E}_{{\rm{Boe}}}-{F}_{{\rm{Boe}}}\vartheta \right)p}{1+{G}_{{\rm{Boe}}}\vartheta }\end{eqnarray} \tag{ 2 }$$

Subsequently, the refractive index is corrected with a specified value for the CO₂ content x (default value 300 ppm) and the calculated water vapour partial pressure p_v according to equations (3) and (4) [7, 15].

$$\begin{eqnarray}&&\left(n\left(\vartheta ,p,x\right)-1\right)=\left(n\left(\vartheta ,p\right)-1\right)\left(1+{H}_{{\rm{Boe}}}\left(x-{I}_{{\rm{Boe}}}\right)\right)\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&n\left(\vartheta ,p,{p}_{{\rm{v}}},x\right)=n\left(\vartheta ,p,x\right)-{p}_{{\rm{v}}}{c}_{2}\end{eqnarray} \tag{ 4 }$$

with the following coefficients of the Bönsch equations and substitution constants:

• D_Boe = 93214.6,
• E_Boe = 0.5953·10^–8,
• F_Boe = 0.009876·10^–8,
• G_Boe = 0.0036610,
• H_Boe = 0.5327·10^–6,
• I_Boe = 400,
• Substitution constant c₁ = 2.6822780908·10^–4 for λ = 632.991234 nm,
• Substitution constant c₂ = 3.7061624213·10^–10 for λ = 632.991234 nm.

A simple recursive filtering of the air refractive index minimises the jumps of the length measurement value, which result from the limited resolution of the environmental measurement.

By forming the partial derivatives for each environmental parameter, the relative sensitivities of the refractive index of air per unit can be determined for the working point. The influence of temperature, air pressure and water vapour pressure on the refractive index is quantified with the coefficients of −0.903·10⁻⁶ / K, 0.268·10⁻⁸ / Pa and −3.706·10⁻¹⁰ / Pa, respectively. The products of the coefficients and the uncertainty of the parameter estimation show that a change in the environment temperature has the greatest influence on the refractive index of the air n.

As an incremental measuring method, the evaluation of the measurement signals based on the detection of the number and fraction of the interference fringes starting from a reference point. The fringe counter must be initialised by setting it to zero at a certain position before a measurement, where the length difference between the measuring arm and the reference arm must be known. This length difference is called the dead path length l_t. If the zeroing of the measurement is set at the position of the dead path length (figure 4, position 0), the difference of the beam lengths l_g results from the measured length l_m and the dead path length l_t. For a difference measurement between two measurement positions, the result can be calculated as difference between the measured lengths of the two positions (e.g. l_d = l_m2 − l_m1).

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The uncertainty contribution of the dead path length results from the fluctuation of the vacuum wavelength and of the refractive index and thus from the fluctuation of the wavelength due to change of the environmental conditions. With the same length a of the measuring arm and reference arm, the two arms can compensate each other by the same change of environmental conditions. If the optical paths of the measuring arm and the reference arm are not the same, the change in wavelength causes a change in the fringe counter, although the position of the measured object and the reference mirror do not change.

Taking the influence of the dead path into account, the calculation of the measuring length l_m(t) at time t can be carried out according to the following equation [7]:

$$\begin{eqnarray}&&{l}_{{\rm{m}}}\left(t\right)=\displaystyle \frac{{\lambda }_{{\rm{vac}}}\left(t\right)}{{k}_{{\rm{IF}}}{k}_{{\rm{TF}}}n\left(t\right)}N\left(t\right)+\left(\displaystyle \frac{n\left(0\right){\lambda }_{{\rm{vac}}}\left(t\right)}{n\left(t\right){\lambda }_{{\rm{vac}}}\left(0\right)}-1\right)\cdot {l}_{t}={k}_{1}\left(t\right)N\left(t\right)+{k}_{0}\left(t\right)\end{eqnarray} \tag{ 5 }$$

Here n(0) is the refractive index when the fringe counter is set to zero and n(t) is the current refractive index. Because the beam is folded once at the mirror, an interferometer factor k_IF of 2 is used. The division factor k_TF depends on the electronics used in the NMM-1 for demodulating the interferometer signals and is given as k_TF = 16384 (14-bit arctan register). λ_vac(t) and λ_vac(0) are the current vacuum wavelength and the vacuum wavelength at zeroing, respectively. N(t) is the sum of the counted number of interference fringes multiplied by the division factor and the current arctan value. Based on this calculation for the measuring length and by adding the other uncertainty contributions, a measurement uncertainty for the positioning or individual measuring point can be determined. The uncertainty of the position measurement of the NMM-1 has been discussed in detail in [7, 17]. A positioning uncertainty of 3 nm (k = 2, a level of confidence of 95%) is assumed for the used revision status of the NMM-1 [18].

For the measurement results an overlay of the uncertainty contributions can be assumed. According to GUM, a model of the calculation must be developed in order to assess the uncertainty of the measurement results. For measurements with the MAFM sensor the profile height (z-direction) is calculated as the difference between the calibrated signal and the z-axis position data of the NMM-1. In intermitted contact mode (IM), the profile height (z-direction) is determined as the difference between the calibrated amplitude signal and the z-axis position data of the NMM-1. However, in contact mode (CM) the profile height (z-direction) is given as the difference between the calibrated bending signal and the z-axis position data of the NMM-1. Measurement results using the AFM can be considered at least as the difference between two probing points [7]. For example, in CM the height information H of the measurement result is given as the difference between two measuring points at the times t_i and t_i−1 with i ≥ 1:

$$\begin{eqnarray}&&H\left({t}_{i}\right)={d}_{z}\left({t}_{i}\right)-{d}_{z}\left({t}_{i-1}\right)-\left[{l}_{z}\left({t}_{i}\right)-{l}_{z}\left({t}_{i-1}\right)\right]\end{eqnarray} \tag{ 6 }$$

According to GUM, the term of the standard uncertainty of each uncertainty component u(x_i ) and its sensitivity coefficient c_i are required as uncertainty contribution for the determination of the combined measurement uncertainty of the measurement result. The sensitivity coefficients are determined by the differential quotients and describe the dependence of the measurement result on the input variables [14].

$$\begin{eqnarray}&&\begin{array}{l}{u}^{2}\left(y\right)=\displaystyle \sum _{N}^{1}{u}_{i}^{2}\left(y\right)=\displaystyle \sum _{N}^{1}{\left({c}_{i}{u}_{i}\left({x}_{i}\right)\right)}^{2}\\ \,\,=\displaystyle \sum _{N}^{1}{\left(\displaystyle \frac{\partial f\left({x}_{1},{x}_{2},\cdots {x}_{N}\right)}{\partial {x}_{i}}{u}_{i}\left({x}_{i}\right)\right)}^{2}\end{array}\end{eqnarray} \tag{ 7 }$$

The correlations between the input variables must be taken into account, for example, the correlation between air refractive indexes n at times t_i and t_i−1. In order to achieve the decoupling of the influencing variables for the uncertainty calculation, the environmental data, refractive indexes or wavelengths at times t_i can replaced by the sum of the value at time t_i−1 and the difference value up to the time t_i [7].

$$\begin{eqnarray}&&n\left({t}_{i}\right)=n\left({t}_{i-1}\right)+{\rm{\Delta }}n\left({t}_{i},{t}_{i-1}\right)\end{eqnarray} \tag{ 8 }$$

On one hand this means that only the changes or scattering during the measurement as a result of the scattering of the vacuum wavelength, the scattering of the acquisition of the environmental values as well as the occurring changes of the environmental values affect the difference. On the other hand, the uncertainties of the determination of the values at time t_i−1 are included, for example, as uncertainties of the calibration of the laser light sources and of the environmental sensors as well as the determination of the dead path length.

In order to estimate the drift and the creep of the bending signal, the uncertainty values for the residuals r of the calibration characteristic curves are included. In CM the combined uncertainty for the determination of the step height standards is 5.81 nm in a time period of up to 3 h in the laboratory of the FMT with a temperature stability of ±40 mk, an air pressure fluctuation of ±50 Pa and a humidity fluctuation of ±0.9%. The uncertainty of the residual u(r) and the positioning uncertainty of the NMM-1 u(l_z) are the main contributions.