## Abstract

The structures of ultrathin sputtered Cr/C multilayers were determined by high-resolution transmission electron microscopy. The evolution of layer thickness errors, interdiffusion and interfacial roughness were simulated using time series models. The results show that with increasing of interdiffusion and roughness the multilayer thickness ratio changes, thereby influencing the optical performance. All structural parameters show good correlation with and influence adjacent layers. The system errors of the deposition equipment can also be evaluated by the models.

©2011 Optical Society of America

## 1. Introduction

Multilayers, which consist of periodic or aperiodic stacks of two or more alternating materials, are the main near-normal incidence mirrors and grazing incidence focusing components in the EUV, soft X-ray and X-ray regions.

To date most of the crucial problems that prevent further development of multilayers arise from limitations of the fabrication technologies, such as magnetron or ion-beam sputtering. Due to the nanometer-scale requirement, structural imperfections such as layer thickness errors, interfacial roughness and interdiffusion are unavoidable, and often seriously influence the optical performances of multilayers.

For a typical application such as X-ray microscopy, the energy resolution and reflectivity of a narrow-band multilayer depend on the perfection of the periodicity. Simulations [1] have shown that random layer-thickness errors over 5%, or a trend of thickness change of the same size, drastically reduce reflectivity and, especially, energy resolution, decreasing by 50-80%. Some studies of layer thickness errors have included both random and cumulative errors [2], but traditional X-ray reflection methods cannot distinguish between these. Wavelet transforms that analyze the X-ray reflectivity curve can determine the thickness of each layer, but there are large errors with increasing number of layers [3]. For these reasons high-resolution transmission electron microscopy (HRTEM) to observe cross-sectional images of multilayers is the most direct and believable method presently available. Some studies [4] have attempted to derive the thickness of each layer from TEM images, but the results are sample specific and only provide simple statistical analyses which do not help in regulating layer thickness errors or in investigating how to improve this problem.

Interface imperfections including interdiffusion and roughness have always been regarded as the most serious factors in decreasing multilayer reflectivity. Effective methods to characterize them combine X-ray diffuse scattering [5] and reflection [6], but the fitted parameters are always average values and do not reflect the overall trend of interface changes. It is therefore essential to carry out further studies in order to include realistic optical characteristics and manufacturing tolerances in the design process.

In this paper, a new quantitative method using time series techniques to analyze HRTEM images is presented. This provides a more realistic structural model of multilayers. The evolutions of layer thickness error, interdiffusion and interfacial roughness are discussed in turn. The analysis helps to predict the errors for multilayers with increasing numbers of layers, adjusting the deposition time in order to improve the multilayer quality.

## 2. Theory

In order to deposit high precision nanometer-scale multilayers, it is normal to control the deposition time in order to give actual thicknesses close to design values. In the process of deposition, the layer thick- nesses are normally assumed to be directly proportional to the deposition time and so are calculated by integrating the deposition rate with respect to time. However, assuming the deposition rate to be approximately proportional to the sputtering rate is dependent on a complicated deposition environment. Normally, the sputtering rate depends on the plasma voltage and current, the sputtering power, the argon gas pressure, the base vacuum level, and the state of the target (as targets age, the sputtering rate changes). The deposition depends on the reactions of several million, or even billion, target atoms and more many argon atoms and electrons, and is a complex random process. Such processes are inevitably complex functions of many parameters, and also include strong random effects with time due to sudden or continuous changes of many independent or dependent parameters. Thus they cannot readily be described analytically. Hence it is useful to introduce a time series to characterize the deposition process, based on experimental data of multilayer cross sections from HRTEM images, and to evaluate the trends of thickness error, interdiffusion and interfacial roughness with time; the deposition process has very good repeatability in the same deposition environment.

In the actual deposition, the thicknesses of individual layers always deviate from design values. Using *d _{si}* and

*d*to indicate the actual spacer and absorber thicknesses in the

_{ai}*i*th period, the period thickness is

*d*=

_{i}*d*+

_{si}*d*. Over a short time the deposition conditions will change only slightly so that the errors in deposited layers should be similar to those of adjacent periods. Thus the thicknesses and roughenesses of any layer will have better correlation with those of adjacent layers than with those of more distant layers.

_{ai}An ARIMA(*p,I,q*) (auto-regressive integrated moving average) model is often used to describe this kind of process; *p* and *q*, respectively, are the orders of the AR (auto-regressive) and MA (moving average) processes, and *I* is the order of the difference to transfer a non-stationary series to a stationary one. The normal ARIMA(*p*,1,*q*) can be expressed as

*α*} are white noise series. The coefficients

_{t}*φ*are the weights of the layer thickness differences. If the series are stationary, each latent root of the AR part is less than 1 and the coefficient

_{i}*θ*→0 as

_{i}*i*→∞. The values of

*p*and

*q*depend on the autocorrelation and partial correlation of the series, and normally are small positive integers. This model means that in

*p*lags the layer thickness errors are strongly related, while in

*q*lags the systematic random errors are related. The variance of {

*α*}, i.e., systematic random errors can be expressed as

_{t}If the systematic errors become more unstable with time, e.g., the series {*α _{t}*} are dependent on the squares of the previous series but uncorrelated, a simple ARCH(

*m*) (autoregressive conditional heteroskedasticity) model can be used to simulate the residual series to exhibit time-varying volatility clustering – i.e., the errors become larger with time. Based on this model, the systematic errors can be expressed simply by Eq. (3), in which

*σ*is the standard deviation of {

_{t}*α*} and {

_{t}*ε*}~(0,1) is independent of the previous series,

_{t}An important application for time series analysis is to forecast future trends of series. From a forecast origin *h*, the thickness of the layer *h* + *l* can be forecast by using the parameters of previous series. The best forecast satisfies a least squares sum of differences between the simulation and the experimental data.

The layer thicknesses *d _{si}* and

*d*are not easy to identify if there is interdiffusion at the interfaces. If the density profile changes continuously, the positions of interfaces may be defined to be at the maximum of the first-order derivative of the density function.

_{ai}The above discussions, in addition to describing layer thickness errors, may be used for all structural parameters, such as interdiffusion and roughness that satisfy random multilayer deposition processes. In Eqs. (1-3) the thickness *d _{t}* can be replaced by

*V*(interdiffusion) or

_{t}*r*(roughness).

_{t}For non-crystalline ultra-thin multilayers it is normal to use an error function to express the density in the interdiffusion as

*ρ*

_{bulk}is the density of the bulk material and

*Z*is the distance from the centre of the layer. The first derivative of the error function is Gaussian; the standard deviation

*V*is regarded as the interfacial width, equal to the half width at half maximum. In practice, even in a periodic multilayer, the interdiffusion can vary from period to period. In addition, two materials, A and B, which are diffusing form A-on-B and B-on-A interfaces which, in general, are different. Each interface will also have two components, based on the direction of diffusion, either from A to B (in the B layer) and B to A (in the A layer). Thus four functions

*V*

_{1,2,3,4}(

*t*) are needed to express the different interface widths. The first derivative of the density profile function should be continuous, have maxima or minima at the positions of the interfaces, and be zero when there is no interdiffusion. In order to remove the effect of roughness, average density profile functions were calculated from line profiles of 2048 TEM pixels.

## 3. Experiments

Two Cr/C multilayers with 150 layer pairs and different thickness ratios were fabricated using magnetron sputtering. The base vacuum was respectively 5 × 10^{−5}Pa and 8 × 10^{−5}Pa for the two multilayers, and the argon gas pressure was 0.667Pa. The sputtering powers for the Cr and C targets were 20W and 120W respectively. The voltage ranges for sample 1 for the Cr and C targets were 338-340V and 414-415V respectively, and 336-338V and 412-414V for sample 2. The multilayers were deposited on 1 × 2 cm^{2} silicon substrates. The deposition times for sample 1 for Cr and C were 7.0s and 22.8s respectively, and 8.0s and 21.3s for sample 2. The deposition rates of the C layer and Cr layers were about 0.068nm/s and 0.112nm/s respectively.

The samples were measured using high-resolution transmission electron microscopy (Tecnai G2 F20 S-TWIN). The accelerating voltage was 200kV and the image magnifications were in the range 4x10^{5}-10^{6}. The data were read using a FORTRAN programme and analyzed with an EVIEWS programme. The surface roughnesses of the two samples were measured by atomic force microscopy (Bruker AXS Dimension Icon) with line-outs of 512 pixels. The reflectivity curve of sample 1 was measured with a high-resolution hard X-ray diffractometer (BEDE D1), using the Cu Kα line.

## 4. Results

#### 4.1 Evolution of layer thickness errors

In a multilayer, the period thickness determines the wavelength of maximum reflectivity, and the thickness ratio and imperfections influence the reflectivity. For the two multilayer samples, there are no obvious trends of period thickness but there are gradual diverging oscillations for over 80 periods (Fig. 1
). The two multilayers work at almost the same wavelength but have different design thickness ratios *Г* (the ratio of the thickness of the high-Z layer to the period), of 0.34 and 0.38.

Figures 2 and 3 show that the layer thicknesses of the two samples are not constant; the C layers become thinner and the Cr layers thicker with increasing period. If the positions of average density of the two materials are defined as the interfaces, similar trends are observed. Since the period thicknesses are almost constant, the changes of layer thickness must result from interdiffusion between the materials. In fact the actual layer thicknesses are not those of perfect monatomic layers with bulk material density. The density of each layer, including the inter-diffusion regions, changes with depth in the multilayer. This is discussed in more detail in section 4.2. The interdiffusion becomes increasingly signi-ficant as the layer number increases, as can readily be seen from the HRTEM images (Fig. 4 ). As can be seen in Fig. 1, the layer thicknesses oscillate severely near to the substrate and, especially, close to the surface. The near-substrate oscillations are due to the influence of substrate roughness and instability in the initial growth. The near-surface oscillations are caused by significant interdiffusion and oxidation.

Based on the cross-section images of the multilayers, the autocorrelation and partial correlation of the series were calculated. According to the correlograms, *p* = 2 and *q* = 1 were chosen as the parameters of the ARIMA(*p*,1,*q*) model for convenience in comparison and simplification of the models.

The fitted model is shown in Table 1 and the fitted and forecast curves are in Figs. 2 and 3. The results show that over a short time the layer thicknesses have strong correlations especially in the first and second neighboring periods. All the values were determined by setting the confidence level of the model to 95%. Sample 2 has a stronger correlation due to larger interdiffusion. The lag of the ARCH model shows that the Cr (i.e., thinner) layers have more significant cumulative oscillations. Although the nominal thicknesses of the two materials are very different, the systematic errors which depend on the quality of equipment are almost the same. For ultra-thin films thinner than 1nm, the systemic error may larger than 15%.

#### 4.2 Evolution of interdiffusion

The evolution of interdiffusion was also investigated in two ways. The first was to study the evolution of the interface width between the two materials, i.e., for Cr-on-C (*V*1 + *V*2) and C-on-Cr (*V*3 + *V*4). The second was to study the evolution of the degree of interdiffusion in the C layers (*V*2 + *V*3) and Cr layers (*V*1 + *V*4); this is demonstrated in Fig. 5
, where the different patterns show the different types of interdiffusion. For layers near to the substrate, there are stable monatomic zones in the C (i.e., thicker) layers (as seen in Fig. 5) which become narrower as the number of layers increases, eventually disappearing.

Figures 6 and 7 show that the interfacial widths of Cr-on-C and C-on-Cr are similar. The ratio of the interfacial width to half the period thickness changes from 30% near the substrate to 50% near the surface. The first derivative of the density profile shows that the slopes of C-on-Cr zones are smaller; this is because the smaller carbon atoms more easily diffuse into chromium layers (which have an island structure, with gaps) with high kinetic energy during the deposition process.

Figures 6 and 7, and Table 2 , show the results of time series analysis. The interdiffusion in sample 2 has strong correlation because it is the dominant feature of the whole multilayer. In addition, the interfacial widths oscillate more severely near the surface in sample 2 because the thicker Cr layers mean a larger upper limit for interdiffusion.

As can be seen in Fig. 8 , the different trends for the C and Cr layers reveal the important effect of layer thickness on interdiffusion. The interfaces move towards the C layers as interdiffusion increases so that the monatomic zones become narrower and narrower. Hence interdiffusion means that the thicknesses of the Cr layers have the same trend. Because the Cr layers are ultra thin there are no clear monatomic Cr layers even close to the substrate. Over 90% of the Cr layer thicknesses are doped with C atoms and the concentrations of Cr atoms may be very different at different positions even at the same depth in the layer. Such considerations are often not taken into account in multilayer design, which can lead to large differences between target reflectivity and resolution and experimental results. Figure 8 also demonstrates that there is a limit for ultra-thin film deposition, since if the interdiffusion regions are wider than the layer thicknesses, there are no real interfaces to reflect X-rays.

#### 4.3 Evolution of interfacial roughnesses

Characterization of interfacial roughness is very complex, especially for studies of lateral and vertical roughness correlations, and so will not be considered in detail in the present paper. Here the general evolution of root mean square (RMS) roughness is presented in order to understand the overall structure of a multilayer.

Because the Cr and C layers are amorphous and isotropic, it is reasonable to examine one-dimensional roughness from cross-section images.

As can been seen in Figs. 9 and 10 , the evolution of interfacial roughness in the two samples is similar. For the first 10 periods or so the interfacial roughness is influenced by the substrate roughness (about 0.3nm). For higher period numbers, the roughness increases from about 0.15nm to 0.4nm, with obvious oscillations. After about 70 periods the roughness increases more rapidly, in agreement with the interdiffusion trend. Higher roughness loosens the film structure and enlarges the contact surface with the adjacent layer, thus increasing interdiffusion. Larger interdiffusion decreases roughness correlations from layer to layer, but increases uncorrelated roughness. In the ultra-thin multilayers studied here, the two kinds of interfacial roughness (Cr-on-C and C-on-Cr) are very similar, unlike in previous studies [7]. This is because for small layer thicknesses, especially for the metal layer, according to the replication model uncorrelated roughness cannot play a main role and the roughnesses of adjacent layers are closely related. Using a simple linear regression model, the roughnesses (in nanometers) can be expressed as

Equation (5) shows that C layers replicate roughness well, but Cr layers tend to reduce roughness from lower layers but produce more uncorrelated roughness.

AFM can measure surface roughness at the same order of magnitude as TEM, and so was used to verify the TEM measurements. In 100 × 100nm^{2} regions (Fig. 11
), the RMS roughness of sample 1 is 0.267nm and that of sample 2 is 0.417nm, in close agreement with the TEM values.

## 5. Discussion

The traditional method to determine multilayer structural parameters simulates the hard X-ray reflectivity curve by using theoretical modeling. Figure 12 shows a perfect simulation but in practice the results are different from the design targets (Table 3 ). The reasons include: (1) A periodic multilayer model is used; (2) this method cannot distinguish interfacial roughness and interdiffusion; and (3) for the relations between density and geometric layer thickness, this method is only based on mathematical approximations.

The quantitative method presented in the current paper overcomes these disadvantages and shows definite evolution of layer thickness, interdiffusion and roughness, and provides a way of defining more realistic design targets related to the actual structural parameters. In order to improve the quality of multilayers, it is necessary to restrict the evolution of interdiffusion and interfacial roughness. First, better deposition equipment and a cleaner deposition environment can reduce the system errors thereby decreasing random fluctuations of the structural parameters. Second, the thickness ratio and the thinnest film thickness are related to the level of interdiffusion. Further study can help to optimise the thickness ratio to reduce the effect of interdiffusion. Third, materials, such as boron carbide, that readily form dense and stable structures with smooth interfaces should be chosen. Fourth, the deposition process could be halted periodically in order to change deposition parameters, e.g., increasing the power or decreasing the argon pressure to maintain the multilayer density as the number of layers increases. Finally, some appropriate barrier layers can be inserted into various parts of the multilayer structure to improve the roughness.

Due to the expense and difficulty in preparing TEM samples, it was not possible to make sufficient HRTEM measurements on different samples to enable systematic evolutional studies of the structural parameters. However, it is possible make the following statements:

Based on the evolution of layer thickness errors, interdiffusion and interfacial roughness, a new, realistic, multilayer model can be constructed. Ultrathin layers satisfying a given density profile function can be designed, instead of the traditional layer by layer method, and interfacial roughnesses also can be incorporated according to the roughness evolution provided by the present study.

A realistic time series model is useful for studying any structural parameters so that they do not have to be simulated by using an assumed mathematical model. The inclusion of errors in the model can help to estimate the tolerances on reflectivity and resolution. The predictive power of time series analysis is more useful than other mathematical models because time-varying volatility clustering is included and future trends arise from the forecast origin. Multilayers with several hundred layers are often influenced by inherent stresses, and the evolution of structural parameters may vary considerably; time series analysis will be more effective in such cases. In addition, such analyses will be of use in correcting the deposition rate for more complicated aperiodic components such as Laue multilayer lenses.

## 6. Conclusion

The evolution of layer thickness errors, interdiffusion and interfacial roughnesses for ultrathin Cr/C multilayers can be determined from HRTEM images. A quantitative time series analysis was used to simulate these evolutions and provided well fitted models and forecast parameters for subsequent layers. The results directly provide realistic situations for the growth of ultrathin multilayers and new ideas for the design, fabrication and characterization of multilayers.

More measurements for Cr/C multilayer and other material pairs will be carried out in order to acquire more systematic results. The method will also be used to solve further fabrication difficulties in high-resolution and aperiodic multilayers.

## Acknowledgments

This work is supported by the Royal Society of London and the National Science Foundation of China. Hui Jiang is supported by a KC Wong Studentship. We are grateful to Bill Luckhurst (King’s College) for assistance and advice with the atomic force microscope measurements.

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