1. Introduction

The Avogadro constant (N _A) is a fundamental physical constant that is closely related to two of the seven SI base units, namely, the unit of mass (kilogram) and the unit of the amount of substance (mole) [1]. Currently, the redefinition of the kilogram is ongoing by metrologists in the world, and N _A is a promising candidate that can be related to the new definition if its relative uncertainty can be reduced from the current 5 × 10⁻⁸ to 2 × 10⁻⁸ or less [2]. The X-Ray Crystal Density (XRCD) is considered as the most precision method to determine N _A [1,3]. According to this method, N _A can be described by the formula$N_A=(M_m/\rho )/(V_0/n)$,where $M_m$is the mole mass, ρ is the macroscopic density of crystal, $V_0$ is the volume of the unit cell, and n is the number of atoms per unit cell. Among these four parameters, the density ρ is one of the biggest obstacles nowadays [3,4]. For density determination, the volume of a perfect single crystal silicon sphere with a diameter 94mm should be measured accurately by optical interferometer. However, the oxide layer, which grew on the sphere’s surface during and after the fabrication, has a great influence on the volume determination [5]. According to Ref. [5], the oxide layer with thickness uncertainty of 1 nm would results in error of nearly 1.9 nm in the diameter of the sphere, so that the corresponding relative uncertainty of N _A will be as large as 6 × 10⁻⁸.

Many attentions have paid to this layer [3–5], and the latest results are given by Kuramoto et al [6] and Becker et al [7]. In Kuramot’s paper, for instance, this layer was measured at 20 different positions by a spectroscopic ellipsometer (SE), then the technique of X-Ray Reflectivity (XRR) and X-ray Photoelectron Spectroscopy (XPS) were applied to check the results of SE as well as the chemical composition of this layer, and the uncertainty of the oxide layer was determined as 0.3 nm. However, they tested the accuracy of SE by comparing the measurement consistence with XRR on just two thermal oxidization disks, which is an improper calibration method since they cannot deduce the calibrated results with specific measured thickness of the oxide layer. Moreover, twenty positions may not be sufficient because it is believed that the thickness of the layer varies with the orientation of the silicon sphere [8]. On the other hand, Becker et al replaced this native oxide layer by a thermal oxide and then measured the thicknesses by XRR directly with 0.2 nm uncertainty. Nevertheless, the determination of the topography of the thermal layer is still necessary. Comparing with SE, XRR is a time-consuming and ineffective method for mapping the surface of the sphere.

In this paper, the thickness of the native oxide layer is determined by SE associated with a sphere-rotating mechanism, and the results of SE are calibrated by XRR. In order to acquire the mean thickness, a novel algorithm of the weighted mean based on equal-area projection theory is applied to the mapped data. We take advantage of the strong points of SE (faster speed) and XRR (independent of optical model). The mapping strategy we proposed does not need rigorous location resolution of the sphere-rotating mechanism, and all the mapping process can be finished within reasonable time.

2. Theories

2.1 SE, XRR, and model of the native oxide layer

SE is an effective instrument for analyzing the thickness and the optical index of thin films. The basic principle of ellipsometry can be described by$\tan \psi \exp (i\mathrm{\Delta })=\rho =R_p/R_s$, where$(\psi ,\mathrm{\Delta })$are elliptical angles measured by the ellipsometer, ρ is the ratio of the reflection coefficients of p-polarization$R_p$and s-polarization$R_s$. The information of the measured film is acquired by fitting the measured elliptical angles to the theoretic ones according to the pre-constructed optical model [9]. Though SE has advantages such as high sensitivity, fast (typically several seconds) and nondestructive measurement, it is an indirect method, which means the thickness measured by SE is dependent on the optical model of the film. Therefore, the results of SE are sometimes in doubt and should be tested carefully.

XRR is an absolute method for determining the film thickness [10]. The principle of XRR is the interference effects between the x-ray reflected from the top and bottom interfaces of the film when the incident angle is close to 90 degree. According to the reflectivity described by Fresnel’s equation, the thickness of the film can be deduced as:

According to the characters of SE and XRR, we take advantage of SE for mapping the sphere by hundreds of points, and use XRR to calibrate SE. Details will be described in Sec.3.

Different from the thermal oxidation having nearly 100% SiO₂, the native oxide layer on the silicon sphere usually contains of sub-oxides. Therefore, the mixture of monoxide silicon and dioxide silicon (SiO)_a(SiO₂)_{1- a} (0<a<1) is applied to represent this layer, and modeled by the Bruggeman Effective Medium Approximation (BEMA) theory. Considering the roughness of the layer, the optical model for SE measurement is shown in Fig. 1 .

 

Fig. 1 Optical model of the native oxide layer on the silicon sphere, c-Si is single crystal silicon.

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The roughness is also modeled using the BEMA theory, assuming 50% native oxide, 50% voids. Optical constants of the crystalline silicon (c-Si), SiO and SiO₂ come from the published data of Aspnes and Edwards [11–13]. All the measurements are carried out in air.

2.2 Mapping strategy

Generally speaking, the sampled points should be distributed as uniformly as possible on the surface of the sphere when estimating the mean thickness of the oxide layer. Several people have discussed the sampling methods in diameter determination [14–16]. In fact, all the sampling theories are based on the consideration that distributing the sample points on the sphere uniformly so that they have the same weight in the mean. The similar application has appeared in geography for many years, which is “equal-area map projection” [17] and can project the earth on the map in equal area. One of the projection methods is cylindrical equal-area projection, whose coordinate transformation formulas are

Figure 2(a) shows an example of the cylindrical equal-area projection. In this case, the distance between points is the same, and equal-weighted-mean can be used for estimating the mean thickness. However, the thing is not as simple as it seems. Usually the position changing of the sphere is by means of controlling the spherical angles but not space distance, therefore, the equal-area sampling requires high angle resolution of the sphere-rotating mechanism because the distributed angles are nonlinear, and the required resolution is much more rigorous along with the increase of the sampling points, which makes this method difficult to be applied to large number sampling.

 

Fig. 2 Two sampling methods. (a) equal-area sampling, (b) equal-spherical angle sampling.

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An alternative mapping strategy based on the equal-spherical angle sampling and the weight mean method is proposed to avoid rigorous angle resolution. For the thicknesses of sampled points ${\rm T}=\left\{t_i|i=1,\dots ,N\right\}$ (N is the total number of sampled points), the estimated mean can be expressed by

Figure 2(b) is an equal-spherical angle sampling, which means the neighboring points have the same spherical angle. It is obviously not an equal-area sampling since the space distance between points is smaller near the poles than that near the equator. However, according to Eq. (2), the weighted mean of the equal-spherical angle sampling can be deduced exactly as

Comparing with the equal-area sampling, the equal-spherical angle sampling is easier to be operated and controllable. It does not need rigorous angle resolution of the sphere-rotating mechanism if the number of sampled points is chose properly. For instance, the step angle of 2-2 phase stepping motor of the sphere-rotating mechanism (See Sec.3.1) is 1.8 degree, then it whose sampled angle is integer multiple of the step angle is appropriate for mapping, and 40 × 20 points with sampled angle of 9 degree were mapped in this paper. In addition, with the help of subdivision technique, the precision of the rotating wheel and the rotating stage in the sphere-rotating mechanism can be better than 0.01 degree.

3. Experiment

3.1 Experimental Set-up

The experimental set-up shown in Fig. 3 consists of a SE, a silicon sphere and a sphere-rotating mechanism.

 

Fig. 3 Experimental set-up. (L) light source, (P) polarizer, (C) compensator, (A) analyzer, (S1), (S2), and (S3) slits, (D) detector, (1) supporting pins, (2) supporting stage, (3) rotating wheels, (4) rotating stage, (5) lifting stage, (6) supporting pillars, (7) adjusting screws, (8) baseboard, (9) adjustable board.

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The utilized SE is of rotating-analyzer type, and the degree of polarization of the reflected light can be measured by inserting a compensator (C). The incidence angle was fixed at 70 degree. Different from the flat sample, the silicon sphere will make the reflected beam divergent and depolarized. To eliminate the depolarization introduced by the spherical surface, a slit (S3) with 1 mm diameter was settled in front of the detector (D), with a distance of 376.1 mm from the top point of the sphere. In this way, only the central area around the top point of the sphere is analyzed by the ellipsometer, and the divergence angle of the reflected beam received by the detector (D) is less than 0.04 degree. Therefore, the uncertainty of the layer thickness introduced by the spherical surface with diameter of 93.6 mm can be reduced to less than 0.05 nm. The light source (L) is deuterium lamps, and the spectral range from 200 nm to 400 nm (about 250 spectral points are sampled) was used since most of the features of the single crystal silicon are in this spectrum. Slits (S1) and (S2) were used for controlling the size of the beam spot.

The silicon sphere is made of single crystal silicon with purity of better than 99.9999%, and the diameter is about 93.6 mm with sphericity of better than 100 nm. The fabrication was carried out by CPEI (China Precision Engineering Institute for Aircraft Industry), and the slurry of the colloidal silicon dioxide was used as the polishing agent for improving the surface condition of the silicon sphere. The roughness of the sphere’s surface was measured to be Ra = 1.33 nm by the 3D-surface profiler.

The sphere-rotating mechanism was putted on the stage of SE and fixed by baseboard (8). During the measurement of SE, the silicon sphere was supported by three pins (1) distributed as an equilateral triangle inside the supporting stage (2). When the measurement position needed be changed, the lifting stage (5) would lift up the sphere by three wheels (3) distributed on the rotating stage (4) symmetrically. Driving the wheels (3) and rotating stage (4) rotating around the x-axis and the z-axis by stepping motors, the whole surface of the sphere could be mapped. When each rotation finished, the lifting stage (5) would lift down to make the sphere is supported by three pins (1) again. The pins and wheels are made of Teflon and polyurethane, which avoid damage to the sphere. The initial position of the sphere can be adjusted by means of changing the adjustable broad (9) by eight screws (7) distributed on the surface and edge of the adjustable stage (9). SE and the stepping motors are controlled by computer software. A single measurement by SE costs about 20 seconds, considering the time for rotating the sphere, it takes about 27 seconds to complete a single measurement. Therefore, the total 1600 points can be measured within 12 hours.

3.2 Calibration by XRR

Since the optical model of the thermal SiO₂ film is widely accepted and routine used, the only fitted parameter in the SE measurement is the thickness of the film. Therefore, the results of SE can be considered as “independent on the model”, and the calibration result is much universal. Five single crystal silicon wafers grown SiO₂ films with nominal thicknesses of 2 nm, 18 nm, 34 nm, 61 nm and 170 nm were prepared by thermal oxidization. SE (incidence angle fixed at 70 degree) and XRR were performed to measure the film thicknesses separately and the results were compared and matched by a straight line of$d_{SE}=(1.013\pm 0.013)d_{XRR}$ with the correlation coefficient of$r=0.999956$, which indicates high consistence between SE and XRR. The calibration curve is shown in Fig. 4 .

 

Fig. 4 Calibration curve of the SE by XRR. d_XRR and d_SE are measurement results of the XRR and SE, and b₁ is the slope of the fitted line.

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All of the thicknesses of the oxide layer measured by SE are calibrated by the fitted line in this paper. The experimental details are described elsewhere [18].

4. Results

4.1 Measurement results

The measured data by SE matched the theatrical model well. The mean and standard deviation of the fitted Mean Square Error (MSE) in all mapped positions of the silicon sphere are 0.82 and 0.21, respectively. Result of SE on one point is shown in Fig. 5 . The mean thickness of the roughness on the sphere measured by SE is 1.41 nm, slightly larger than the result of the 3D surface profiler (Ra = 1.33 nm). The fraction of SiO in the native oxide is also evaluated by SE, which shows a = (3 ± 0.6) %. Then the measured data are fitted again by fixing a = 3.6% as well as a = 2.4%, and the discrepancy 0.03 nm of the mean thickness between these two model is considered as the uncertainty introduced by the optical model.

 

Fig. 5 Measurement result of the layer thickness on one position of the silicon sphere by SE. The MSE of the fitting between measured and theory data is 0.85. PSI and DELTA are the parameters that SE measures directly.

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Eight hundred positions were mapped at first, then the initial position of the sphere was changed randomly by hand and another mapping was repeated. The mapped results are shown in Fig. 6 in$\theta -\varphi$coordinates ($\theta \in \left[0,\pi \right),\varphi \in \left[0,2\pi \right)$). The mean thicknesses and standard deviations of the two measurements are (4.33 ± 0.55) nm and (4.25 ± 0.51) nm.

 

Fig. 6 The mapped results of the oxide layer on 1600 positions of the silicon sphere by SE. Theta and Phi are spherical angles.(a) is the result of 800 positions, and (b) is that of the other 800 positions by changing the initial position of the silicon sphere randomly before the mapping.

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For testing the precision of SE, repeated measurements were performed at the same position of the silicon sphere for about 10 hours. Measurement results were recorded every minute. During the measurement the environmental parameters such as temperature, humidity, and air pressure were monitored. The results of SE measured are shown in Fig. 7 . The standard deviations of Psi and Delta in the full spectral range are better than 0.02 degree and 0.04 degree, and the standard deviation of the thickness is calculated as 0.06 nm, showing an excellent measurement repeatability of the device.

 

Fig. 7 Repeated measurement results by SE versus the temperature in 10 hours. The sampling time interval was one minute.

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The mean thickness of the two measurements is 4.29 nm determined by SE, considering the calibration by XRR in Sec.3.2, the mean thickness of the oxide layer is ‘zoomed out’ to be 4.23 nm.

4.2 Uncertainty budget

The difference of 0.08 nm in mean thickness of the two measurements is considered as the reproducibility of the device. According to the calibration in Sec.3.2, the standard deviation of the calibration for thickness of 4.29 nm is 0.06 nm. The combined uncertainty of the mean thickness is shown in Table 1 .

Table 1. Uncertainty budget of the mean thickness of the oxide layer

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Therefore, the mean thickness of the oxide layer on the silicon sphere is determined to be 4.23 nm with a standard uncertainty of 0.13 nm, indicating that its contribution to the relative uncertainty of N _A being reduced to about 7 × 10⁻⁹ after the error introduce by the oxide layer being corrected.

5. Conclusions

The accurate measurement of the oxide layer on the silicon sphere is considered as one of the primary works to determine N _A with relative uncertainty of 2 × 10⁻⁸ nowadays. According to the principle of the XRCD method, the mean thickness of the oxide layer should be measured with uncertainty of nearly 0.1 nm. Because the thickness distribution of this layer is not uniform on the surface of the sphere, the work presented here measured the mean thickness by means of combining SE and a sphere-rotating mechanism. A weighted mean method based on equal-area projection theory is performed to avoid the rigorous requirement of the angle resolution of the rotating system, and results of SE are calibrated by XRR. The mean thickness of the oxide layer is determined to be 4.23 nm with uncertainty of 0.13 nm, which is in the range of the acceptable level for the current purpose of the Avogadro project.