Dynamic scaling in sputter grown tungsten thin films

Thin Solid Films 515 (2007) 5541 – 5545 www.elsevier.com/locate/tsf

Dynamic scaling in sputter grown tungsten thin films Luca Peverini a,⁎, Eric Ziegler a , Igor Kozhevnikov b a

European Synchrotron Radiation Facility, BP 220, 38043, Grenoble cedex, France b Institute of Crystallography, Leninsky prospect 59, Moscow 119333, Russia Available online 2 February 2007

Abstract The evolution of a tungsten thin film grown by magnetron sputtering was studied using a dynamic scaling approach. Film growth was followed in-situ and in real-time by monitoring both the specular and the diffuse X-ray scattered intensities as a function of the time of deposition. The analysis of the scattering data allowed us to determine the two Power Spectral Density (PSD) functions, which describe the thin film topography. The time-dependent PSD-function, which describes the dynamic of the external film surface, is found to obey a universal scaling form, which characterizes the thin film growth. The data collapse of these PSDs into a single master curve was achieved using scaling exponents α = 0.18 ± 0.02 and β = 0.06 ± 0.01. In addition, by analyzing the temporal variation of the roughness conformity, it has been demonstrated that the replication factor decreases exponentially with increasing film thickness and spatial frequency. Hence, for a 25 nm thick film the vertical correlation disappears for spatial frequencies p greater than 3.6 μm− 1. © 2007 Published by Elsevier B.V. Keywords: X-ray scattering; Surface roughness; Dynamic scaling

1. Introduction The surface of a film growing under non-equilibrium conditions often develops in agreement with the concept of dynamical scaling [1,2]. In this concept, several attributes, known as scaling exponents, can be used as the signatures in space and time of highly complex film growth processes. By comparing the experimental scaling exponents with the theoretical predictions one should be able to recognize which type of differential equation best describes the film growth and, hence, classify various growth processes within universality classes [2]. Nowadays various experimental techniques are used to determine the scaling exponents (see, e.g. in Ref. [3] and references therein). Among them the X-ray scattering (XRS) technique is a unique method to study the evolution of roughness of a growing film, in-situ, in real-time, and in vacuum environment. Moreover, it is a way of investigating buried interfaces and the correlation between the film and the substrate relief. This paper describes results from in-situ real-time investigations of the roughness evolution during the growth of tungsten

⁎ Corresponding author. Tel.: +33 4 76 88 29 98; fax: +33 4 76 88 23 25. E-mail address: [email protected] (L. Peverini). 0040-6090/$ - see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.tsf.2006.12.035

films using grazing incidence X-ray scattering and a novel experimental apparatus available on the BM5 beamline at the ESRF. We show that, by recording in-situ a single scattering diagram for a grazing angle of a probe beam exceeding the critical angle of total external reflection, it is possible to uniquely determine the two PSD-functions describing the micro-topography of the external surface and the film-substrate roughness conformity. Using this approach, the temporal evolution of the roughness of a sputter-deposited tungsten film has been studied and the thickness-dependent power spectral density (PSD) has been determined to obey a universal scaling form [1]. In accordance with the scaling model [1,2] the 1D PSD-function characterizing the external film topography, may be collapsed into a single master curve provided the scaling exponents are properly chosen. 2. Film growth, scattering measurements and determination of the PSD-functions A super-polished silicon substrate flattened to λ/10 (λ = 632.8 nm) has been processed and a W layer has been progressively grown onto it to reach a final thickness of about 25 nm. The experiment was performed at room temperature and the sputter time was adjusted to observe a significant

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variation of the surface roughness while increasing the processing time. Before process the background pressure in the sputter chamber was below 3.6 × 10− 7 hPa. Then, magnetron sputtering was conducted in pure (99.999%) Ar gas at a working pressure of 13.3 10− 3 hPa. The film growth was realized using a bias voltage of 370 V and a total target current of 30 mA. The pressure set point was achieved by fixing the gas flow rate at a value of 3 sccm. XRS measurements were performed by setting the X-ray energy at 17.5 keV with a double-crystal Si (1 1 1) monochromator and fixing the sample in stationary position. The divergence in the vertical direction was 3 d 10− 6 rad and the spectral purity ΔE/E of the order of 10− 4. The angle of the probe beam with the sample was set to θ0 = 0.5°, i.e., out of the total external reflection. For comparison, the critical angle is 0.25° for bulk tungsten at this wavelength. The X-ray detection was performed using a combination of two detectors operating simultaneously: the first one, consisting of an ionization chamber, monitored the total intensity of the reflected beam and the second one, composed of a cryogenically cooled CCD camera (1024 × 256 pixels, equivalent pixel size: 150 μm), collected the scattered beam. Here the X-ray photons are converted into visible light by a phosphor screen coupled with the entrance window of an image intensifier. Finally, the intensifier output is imaged onto the CCD detector by a lens system. The reflectivity variation as a function of the deposition time, which was recorded at the same time as the XRS diagrams, was analyzed to infer the tungsten deposition rate (12.3 pm/s) and the film density (18 g/cm3). The scattering diagrams were measured while growing the film and at 5 s time intervals. The bi-directional scattering diagram Φ(θ,φ) = (1 / Winc) (dWscatt / dΩ), which describes the power of radiation dWscat scattered from a rough surface within a small solid angle dΩ and normalized to the power of the incident beam Winc, is a function of two scattering angles: the angle θ referencing in the scattering plane (vertical in our case) and the azimuth angle φ referencing in the plane of the sample surface. During our experiment the vertical beam size was set to 0.2 mm and the horizontal size to 5 mm, to favor a high intensity onto the detector. As the scattering diagram is very narrow in the azimuth (horizontal) plane, the scattered intensity was integrated in this direction as Π(θ) = ∫Φ(θ,φ)dφ. An example of scattering measurements processed in this way is shown in Fig. 1 (solid line). The sharp minimum observed on this curve, at the position of the specular peak, is due to the presence of the beamstop placed in front of the CCD detector to prevent it from saturation. In the frame of the first-order scalar perturbation theory the integrated scattering diagram from a rough isotropic film is written in the following form: jðh; hÞ ¼

where Winc and dWscat are, respectively, the radiation power of the incoming beam impinging under an angle θ0 and the radiation power of the beam scattered within an angular interval dθ under an angle θ with respect to the sample surface. The electrodynamic factors Af, As, and Asf have the following expressions: Af ¼ jð1−ef Þ½1 þ rðh0 ; hÞ½1 þ rðh; hÞj2 As ¼ jðef −es Þt ðh0 ; hÞt ðh; hÞj2 Asf ¼ 2Refð1−ef Þðef −es Þ⁎ ½1 þ rðh0 ; hÞ½1 þ rðh; hÞt ⁎ ðh0 ; hÞt ⁎ ðh; hÞg ð2Þ

The dielectric constants of the substrate and the film are noted as εs and εf. The amplitude reflectance and transmittance of a perfectly smooth film of thickness h are noted r(θ,h) and t(θ,h), respectively. The scattering properties of a rough film are thus characterized entirely by three 1D PSD-functions, PSDss( p) and PSDff ( p,h), which describe the roughness of the substrate and the roughness of the external film surface, and PSDsf ( p,h), which determine the statistical correlation (conformity) between film and R substrate roughnesses: PSDij ð pÞ ¼ 4 hfi ðY r Þfj ð0Þicosð pqÞdq with i, j = {s, f }.The stochastic functions fs ðY r Þ and ff ðY r Þ describe the substrate and the film relief, respectively, and Y r is the vector lying in the plane of the surface. The angular brackets denote an ensemble averaging and p is the spatial frequency. In a recent work [4], a new approach for the characterization of growing thin films based on the analysis of two independent sets of scattering data acquired at different grazing angles of incidence has been proposed. Such analysis, using the same physical model of a film, provides convincing argument about the accuracy of the procedure. However, it requires a particularly demanding experimental protocol as two “identical experiments" differing only by the angle of the probe beam must be performed. Practically, all causes of instabilities that may occur during a temporal scan (e.g. X-ray beam and/or sputtering stability) have to be reduced as much as possible. Moreover, it requires a set of samples with statistically identical properties. A careful analysis of formula (1) shows some interesting features which allows us to improve the experimental method

1 dWscat k3 ¼ ½Af ðh0 ; h; hÞd PSDff ð p; hÞ Winc dh 16psinðh0 Þ þAs ðh0 ; h; hÞd PSDss ð pÞ 1 2p þAsf ðh0 ; h; hÞd PSDsf ð p; hÞ; p ¼ jcosh0 −coshj; k ¼ k k ð1Þ

Fig. 1. XRS diagram (circles) measured in-situ and in real-time on a tungsten thin film 24.6 nm thick (solid curve). The red dots denote the points where the electrodynamics factor Asf appearing in formula (1) equal zero.

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used to determine the PSD-functions. The coefficient Asf describes the interference of the waves scattered from both interfaces of a thin film. It is an oscillating function of the scattering angles that equals zero at some angles (see black dots shown in Fig. 1). When Asf is close or equal to zero, it is impossible to determine the PSDsf representing the cross-correlation function of the film. However, since the PSD-function of the substrate roughness can be measured before the film deposition, it is possible to extrapolate the PSD-functions of the external film surface as a function of θ where Asf equals zero, using formula (1). The frequency of oscillation of the coefficient Asf increases with the film thickness and, consequently, the number of zeros observed within the measurable range of scattering angles increases. On the other hand, the number of zeros in Asf decreases quickly when the film thickness is reduced. As an example, for a film thickness of 1.25, there are only three point where Asf = 0. Nevertheless, general physical considerations let us expect PSDff ( p) to be a smooth function of the spatial frequency p. Hensce, the behavior of the function PSDff ( p) is well established even for the thinnest film. To decrease the statistical errors and to interpolate the PSD-function between the measured points, we can represent the function PSDff ( p) in the following way: ! jmax X j PSDff ð pÞ ¼ PSDss ð pÞd 1 þ aj d p ð3Þ j¼1

where typically jmax = 3. The representation in Eq. (3) implies a reasonable physical assumption of the roughness conformity at small spatial frequencies, i.e., PSDff (p) → PSDss (p) at p → 0, allowing the extrapolation of the PSD-functions at low spatial frequencies. Such an analytic continuation is important especially for very thin films, when the number of zeros of the coefficient Asf is small inside the measurable range of spatial frequencies. Once the PSD-function of the external film surface is found, the function PSDsf ( p) can be deduced as well using Eq. (3), the coefficients aj being derived by fitting the calculated scattering diagram to the measured one. A scattering diagram calculated with the found PSD-function is shown superimposed to the experimental data in Fig. 2 (solid curve). The curve is in excellent agreement with the

Fig. 2. The same as in Fig. 1 except that the red solid curve was calculated using the PSD-functions (i.e., PSDss( p), PSDff ( p, h) and PSDsf ( p, h)) extracted from the experimental scattering data. (For interpretation of the reference to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 3. PSDff ( p, h) obtained on the thickest film made in this work (h = 24.6 nm) as compared to its asymptotic at large spatial frequency and showing the development of intrinsic roughness during film deposition.

experimental data, demonstrating the accuracy of this method. By repeating the same procedure for various film thicknesses, it has been possible to determine the two sets of PSDs in the thickness ranging from 1.9 nm to 25 nm. These PSD-functions have a profile characterized by a power-law at small spatial frequency followed by the development of a bump at larger spatial frequency f. This last feature, which has been previously predicted by existing growth models [5], represents the development of intrinsic roughness. An example of experimental PSDff ( p, h) measured in the range of spatial frequency p ∈ [3 d 10− 4, 4.6 d 10− 2 nm− 1] is shown in Fig. 3 together with its asymptotics at large spatial frequencies. 3. Dynamic scaling analysis The PSD-functions provide a rather comprehensive description of the film roughness and allow us to calculate the various statistical parameters characterizing growth processes. In accordance with the Family–Viscek theory [1,2] the 1D PSD-function obeys the scaling relation, which can be written in the following form: PSDff ( p,h) ~ p− 1− 2α g(p1 + 2α h(1 + 2α) / z ) where the scaling function g(u) ~ u for u → 0 and g(u)→constant for u → ∞. Hence, by plotting the “re-normalized” function PSDff (p,h)p1 + 2α versus the “re-normalized” spatial frequency p1 + 2αh(1 + 2α) / z at different film thicknesses h and with a proper choice of the scaling exponents α and z, it is possible to obtain the collapse of all the curves into a single master curve corresponding to the scaling function g(u). Fig. 4 illustrates this effect. The graph was drawn using eight PSD-functions determined from the scattering diagrams measured at 8 different film thicknesses from 1.9 nm to 24.6 nm. The data collapse was clearly observed for a roughness exponent α = 0.18 and a dynamic exponent z = 3. The error on the determination of these values was estimated to be about ± 10%. The straight solid lines indicate the asymptotic behavior of the scaling function at small and large argument values. To observe the data collapse at low film thickness, of the order of 2–4 nm, we had to replace the nominal thickness h by

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Fig. 4. Illustration of the data collapse obtained with eight PSD-functions representing the external tungsten film surface for layer thickness ranging from 1.9 nm to 24.6 nm. Inset: thickness dependence of the rms roughness of the external film surface superimposed to a power law with growth exponent β =α/z = 0.06.

an effective one h′ = h − 1.5 nm. It is conceivable that the scaling exponents we found could only describe the growth dynamic of the film above a certain thickness, e.g. once it gets continuous. The film formed during the initial stage of growth (nominal thickness b1.5 nm), possibly discontinuous, might obey, if at all, to another scaling law. Obviously, a justification of this statement calls for additional experiments. The asymptotic behavior of the PSD-function of the thickest film at high spatial frequency is shown in Fig. 3 (dashed curve). The line drawn corresponds to an inverse power law p− (1 + 2α) with a roughness exponent α = 0.18, the same as the one found from the data collapse. The inset in Fig. 4 shows the dependence of the rms roughness on the film thickness obtained by integration of the PSD-function over the measurable range of spatial frequency p ∈ [3 d 10 − 4 , 4.6 d 10− 2 nm− 1]. As above, we replaced the nominal film thickness h by an effective one h′ = h − 1.5 nm. The solid curve is in accordance with the scaling model, assuming a power-law dependence of the rms roughness with the film thickness σ ~ hβ. The small value of the growth exponent β = α / z = 0.06 expresses the very slow increase of rms roughness with film thickness and is typical in MBE growth system. This last feature will be discussed in more details in another work [6]. We would like to recall here that the same growth exponent was determined from the data collapse. The small raise of the roughness value is often observed in sputter-coated thin films. Just this property explains the extensive application of the sputtering technique to grow very smooth films, e.g., short period multilayer interferential mirrors [7]. However, this slow increase of roughness is only observed for thin W film, when the thickness is less than 25–30 nm. Further film growth results in dramatic development of roughness [8]. A 2.5 nm roughness was measured for a 70 nm thick film, probably due to the progressive crystallization of tungsten. The scaling exponents found from the analysis of the PSDfunction of the external film surface PSDff(p,h) characterize the

Fig. 5. Replication factor versus spatial frequency for different W film thickness: h = 3.83 nm (1), 7.55 nm (2), 15.75 nm (3), and 24.6 nm (4). Inset: replication factor versus film thickness at the fixed spatial frequency p = 0.02 nm− 1.

general law describing the development of intrinsic roughness with film thickness. However, the optical properties of a rough film depend also on the film-substrate conformity. To characterize it quantitatively the replication factor can be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi introduced K ð p; hÞ ¼ PSDsf ð p; hÞ= PSDss ð pÞd PSDff ð p; hÞ [4]. This factor, illustrated in Fig. 5, shows a quick decrease of the vertical correlation with increasing spatial frequency. The inset in Fig. 5 shows an exponential decay of the replication factor with film thickness at fixed spatial frequency. 4. Conclusion In conclusion, we demonstrated that, an XRS diagram measured at a single grazing angle exceeding the critical angle of total external reflection and obtained in-situ during thin film growth is sufficient to determine uniquely the two PSDfunctions, PSDff ( p,h) and PSDsf( p,h), which characterize the statistical properties of the roughness of a film as a function of its thickness. Using these PSDs it is possible to establish the scaling exponents characterizing the thin film growth dynamic and to characterize the roughness conformity evolution. We found that the growth of a tungsten film on a Si substrate for thicknesses ranging from 1.9 nm to 25 nm complies scaling laws, with α = 0.18 ± 0.02 and β = 0.06 ± 0.01. Since the experimental setup does not involve any moving part, it can also be applied to the study of surfaces and/or interfaces in liquid thin films. This approach has a relevant technological impact, since it can be applied to various surface treatments (examples include deposition, ion etching and oxidation). Moreover, the study of the influence of deposition conditions on the scaling exponents can be greatly simplified as compared to the use of the growth models actually available. Acknowledgement One of the authors (I.K.) was supported by the ISTC (project #2297).

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[5] D.G. Stearns, Appl. Phys. Lett. 62 (1993) 1745. [6] L. Peverini, I. Kozhevnikov, E. Ziegler, Phys. Rev. B, (submitted). [7] E. Spiller, Soft X-ray Optics, SPIE Optical Engineering Press, Bellingham, Wa, 1994. [8] L. Peverini, PhD thesis (2004), at http://tel.ccsd.cnrs.fr/tel-00011519/en/.