Traceable thickness measurement of ultra-thin HfO₂ films by medium-energy ion scattering spectroscopy

A series of HfO₂/SiO₂/Si(1 0 0) films were fabricated for certification of thickness by mutual calibration with a length-unit traceable method and a zero-offset method. Six HfO₂ films with nominal thicknesses of 1.0 nm, 1.5 nm, 2.0 nm, 2.5 nm, 3.0 nm, and 4.0 nm were grown on the polished side of Si(1 0 0) substrates by atomic layer deposition. Before the growth of the HfO₂ films, to prevent the diffusion of oxygen atoms from the HfO₂ films to Si(100) substrate, 2 nm SiO₂ layers were grown on the Si(1 0 0) substrates by thermal oxidation, as shown in figure 3. After growth of the HfO₂ films, the wafers were cut into small specimens with sizes of 10 mm  ×  10 mm.

To investigate the traceability of thickness measurement, the determination of well-defined film thicknesses is an important prerequisite. The reference thicknesses of the HfO₂ layers in a series of HfO₂/SiO₂/Si(1 0 0) films were determined by mutual calibration between the average thicknesses by XPS and XRR reported in the pilot study P-190 by 11 national metrology institutes. The linear fitting line showed the good linear relationship between the two average thicknesses of the XPS and XRR. The reference thicknesses were determined from the average values of the XPS thicknesses corrected by the slope, and the XRR thicknesses corrected by the offset value of the linear fitting line. These reference thicknesses can be regarded as traceable values because they are based on the thickness obtained by XRR, where the thickness scale is based on the wavelength of x-rays.

Thicknesses of HfO₂ films were determined from high-resolution (HR) TEM micrographs collected using FEI-F30 microscopes operating at 300 kV. The film thicknesses of the samples were determined from the lattice constant of the Si(1 0 0) substrate. The HfO₂/SiO₂ interface was determined from the point with half of the average contrast of the SiO₂ layer and that of the HfO₂ layer. More than ten TEM images at different locations were obtained.

Thickness of the HfO₂ films was measured using an MEIS system (K-120, K-MAC, Korea), which consists of an ion source, an accelerator, an energy analyser, and a detector, as schematically shown in figure 1. The incident and scattering angles are determined by the geometric arrangement between the ion source, the sample, and the detector. In MEIS analysis, ions generated from an ion source are accelerated to impinge on the sample surface, and the energy of the ions scattered by the nuclei of the constituent atoms is measured by an energy analyser. From the energy distribution of scattered ions, the in-depth locations of the constituent atoms can be determined from kinematic factors, and the quantity of atoms can also be determined from the scattering cross section of the atoms. An electrostatic analyser, magnetic sector analyser, and time of flight (TOF) analyser are widely used for MEIS analysis. Of these, the TOF analyser, because it is the most quantitative, is robust for thickness measurement of ultra-thin oxide films. The TOF analyser acquires scattered particles, both charged and neutralized, and so neutralization correction is not needed in the analysis; therefore, there is no potential error in the charge correction. In addition to this, a full energy spectrum is always obtained, which provides a perfect internal reference.

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Because its scale is based on the lattice constant, HR-TEM is a length-unit traceable thickness measurement method. In particular, in thin films grown on Si(1 0 0) wafers, the crystalline lattice planes of Si can be directly used as an internal standard to measure the absolute film thickness. The lattice distance between Si(1 1 0) planes in the cross-sectional TEM image is 0.543 nm.

An HR-TEM image of an HfO₂/SiO₂/Si(1 0 0) film can be simply converted to an intensity line profile image using the average contrast of the region of interest (ROI), as shown in figure 3. For precise measurement, the aspect ratio of the lattice line should be maximized by aligning the lines parallel to the lattice direction, and as such, parallel to the interface and film surface.

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The locations of the interface and the surface at which the film thickness is measured can be determined from the contrast profile. The location of the SiO₂/HfO₂ interface can be determined from the point of half contrast between the average contrast of the SiO₂ (${{I}_{{\rm Si}{{{\rm O}}_{{\rm 2}}}}}$) and HfO₂ (${{I}_{{\rm Hf}{{{\rm O}}_{{\rm 2}}}}}$) layers, (${{I}_{{\rm Si}{{{\rm O}}_{{\rm 2}}}}}+{{I}_{{\rm Hf}{{{\rm O}}_{{\rm 2}}}}}$)/2. In the same manner, the location of the film's surface can be determined from the average contrast of the HfO₂ (${{I}_{{\rm Hf}{{{\rm O}}_{{\rm 2}}}}}$) and the glue (I_glue) layers using the relation (${{I}_{{\rm glue}}}+{{I}_{{\rm Hf}{{{\rm O}}_{{\rm 2}}}}}$)/2.

The film thickness can be measured from the distance between the SiO₂/HfO₂ interface and the surface of the HfO₂ layer. The thickness of the HfO₂ layer can be simply determined from the ratio of the line width of the HfO₂ layer and the width of 20 lines of Si(1 0 0) planes corresponding to the value of 5.431 nm for ten Si lattice constants [15–17]. The average TEM thicknesses of the HfO₂ films (T_TEM), derived from more than ten TEM images at different locations, are shown in table 1.

The combined standard uncertainty (uc) is calculated from the equation $u_{{\rm c}}^{2}=u_{{\rm m}}^{2}+u_{{\rm d}}^{2}+u_{{\rm r}}^{2}+ u_{{\rm l}}^{2}$. Here, um is the standard uncertainty in the measurement of film thickness, and ud is the standard uncertainty in the definition of the interface and surface, which is related to the offset value of the TEM. It was reported in the range of (0.1–0.2) nm [8–10]. In this study, a small value of 0.1 nm was assigned as the value of ud. Here, ur is the standard uncertainty in the measurement of the line width of the periodic Si(1 0 0) lattice planes. It is the uncertainty in the measurement of the interval of the periodic 20 Si(1 0 0) lattice planes, as shown in figure 3(b). The ur value was measured to be small to be 0.01 nm. Here, ul is the standard uncertainty of the variation of the Si lattice constant; ul is negligibly small to be 0.89  ×  10⁻⁸ nm [15]. The expanded uncertainty (U) was determined from the equation U  =  Ku_c at a 95% confidence level.

In MEIS analysis, the thickness of the HfO₂ films can be determined from the measured number of constituent Hf atoms within a unit area and the number density of the bulk HfO₂. The number of the constituent Hf atoms is measured by simulation of the obtained MEIS spectrum. MEIS is a program developed at K-MAC and used as a simulation tool for MEIS spectra analysis. In the analysis, the Andersen cross section [18] and Chu straggling [19] were used. For the stopping power, fitted values based on the experimental data [20] were used following recent MEIS round robin test results [21]. The simulations took into account the solid angle of the detector, and the kinematic broadening of each element was reliably applied. Multiple scattering effects were included in the calculation.

The thickness of the HfO₂ films (T_MEIS) can be determined by dividing the measured number of Hf atoms in the unit area (atoms cm⁻²) by the number density of the bulk HfO₂ (=8.32  ×  10²² atoms cm⁻³). The number of Hf atoms was measured three times by MEIS. The number of Hf atoms and the thicknesses of HfO₂/SiO₂/Si(1 0 0) films derived from the number of Hf atoms are shown in table 2. The linear fitting results of the measured thickness compared to the reference thicknesses show a slope value of 0.981 and an offset value of 0.017 nm. The slope value of 0.981 means that the measured MEIS thicknesses of the HfO₂ films are close to the reference thicknesses, within 2%.

Mutual calibration using a combination of a zero-offset method and a length-unit traceable method was suggested as a useful method to determine the absolute thickness of ultra-thin oxide films. MEIS can be used as a zero-offset method in the mutual calibration method because the film thickness depends on the number of atoms in the oxide film according to the basis of the number density of c-Si. In this study, the thickness of nanometer HfO₂ films was directly measured by the mutual calibration method from the MEIS spectra and the thicknesses measured by HR-TEM shown in table 1.

In MEIS, the number of scattered ions is related to the scattering cross section of the constituent atoms. In the thickness measurement of an HfO₂ thin film grown on crystalline Si substrate, the signal intensity of the substrate can be a basis for thickness measurement because the number density of crystalline Si is a constant. Figure 4 shows the MEIS spectra of Si(1 0 0) substrate (magenta) without film and HfO₂/SiO₂/Si(1 0 0) film (blue). The fact that the intensities of Si are identical in the substrate without film and in the ultra-thin HfO₂/SiO₂/Si(1 0 0) film is the basis for the thickness measurement.

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For this reason, the relative intensity ratio (R_MEIS  =  I_B/I_A) of film material (B) to substrate (A) can be converted to the thickness (T_MEIS) of the HfO₂ layer. The proportional factor can be determined from the slope, derived from mutual calibration by MEIS, a zero-offset method, and HR-TEM, a length-unit traceable method. The intensities of the crystalline Si substrate (I_A) were determined to be in the energy range from 46 keV to 56 keV from the MEIS spectra shown in figure 4. The intensities of Hf in the six HfO₂ films (I_B) were also determined to be in the energy range from 85 keV to 95 keV. Table 3 shows the MEIS intensities of the substrate (I_A) and of the HfO₂/SiO₂/Si(1 0 0) films (I_B), and their ratios (R  =  I_B/I_A) determined from the three MEIS spectra.

The variations in the values of the signal intensities of the crystalline Si substrate show very small relative standard deviations of 0.46%, 0.43%, and 0.38%. The signal intensities of the Hf peaks show a gradual increase proportional to the reference thicknesses. The variations in the values of the signal intensity ratios (R  =  I_B/I_A) in the three MEIS spectra of the HfO₂/SiO₂/Si(1 0 0) films are also very small at 0.07%, 0.27%, 0.18%, 0.09%, 0.31%, and 0.35%. These reproducible results can be a basis for traceable thickness measurement of ultra-thin oxide films.

In this study, the relative intensity ratios (R_MEIS  =  I_B/I_A) of the film material (B) and substrate (A) were converted to thicknesses. The MEIS thicknesses (T_MEIS) of the HfO₂ films can be determined by mutual calibration from the average relative intensity ratio (R_MEIS  =  I_B/I_A), shown in table 3, and the traceable thicknesses by TEM (T_TEM), shown in table 1. Figure 5 shows the mutual calibration results of the MEIS intensity ratio (R_MEIS) and the average TEM thicknesses (T_TEM). The MEIS intensity ratios are very linearly proportional to the average TEM thicknesses.

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The slope m and offset values c are key parameters for the determination of the film thickness by mutual calibration. The measured value of m is 3.065  ±  0.097; the measured value of c is 0.497 nm  ±  0.066 nm. From the slope and offset values, the certified MEIS thicknesses ($T_{{\rm MEIS}}^{{\rm C}}$) and TEM thicknesses ($T_{{\rm TEM}}^{{\rm C}}$) can be determined as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{{\rm MEIS}}^{{\rm C}}= {{R}_{{\rm MEIS}}}\times 3.065\,{\rm nm}\nonumber \end{align}$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_{{\rm TEM}}^{{\rm C}}={{T}_{{\rm TEM}}}-0.497\,{\rm nm}.\nonumber \end{align}$$

Table 4 shows the certified MEIS thicknesses ($T_{{\rm MEIS}}^{{\rm C}}$) and TEM thicknesses ($T_{{\rm TEM}}^{{\rm C}}$) together with the measured raw thicknesses and the reference thicknesses. The slope and the offset values were derived from linear fitting of the measured and certified thicknesses by MEIS and TEM to the reference thicknesses. The MEIS thicknesses can be used as certified values because the length scale was calibrated from TEM based on the lattice constant of crystalline Si substrate.

Table 4. Reference thicknesses and certified thicknesses of HfO₂ films determined by mutual calibration with MEIS and TEM.

Reference thickness (nm)	Thickness (nm)		Certified thickness (nm)
TRef	RMEIS	TTEM	$T_{{\rm MEIS}}^{{\rm C}}$	$T_{{\rm TEM}}^{{\rm C}}$
0.76	0.24	1.25	0.74	0.75
1.20	0.39	1.60	1.18	1.10
1.64	0.52	2.11	1.60	1.61
2.08	0.66	2.57	2.03	2.07
2.52	0.80	3.03	2.46	2.53
3.36	1.09	3.76	3.33	3.26
slope	3.099	1.008	1.011	1.008
offset	0.015	−0.479	0.015	0.025
R	1.000	0.999	1.000	0.999

Figure 6 shows the linear fitting results of the MEIS thicknesses (T_MEIS) and the reference thicknesses (T_Ref). The slope value (1.011) close to unity means the MEIS thicknesses are identical to the reference thicknesses, within 1.1%. The small offset value of 0.015 nm also means that the MEIS thicknesses are close to the reference thicknesses from P-190 performed by national metrology institutes (NMIs). It means that the role of MEIS as a zero-offset method can be said to be comparable to the role of average XPS thickness in P-190 within the uncertainty level.

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