1. Introduction

Multilayer coatings are nowadays ubiquitous, and their applications cover a broad photon spectrum ranging from visible light [1] to gamma rays [2]. A field that is challenging multilayer technology today deals with x-ray imaging systems for extended and highly diverging x-ray sources. Sources of this kind are typical of high energy density experiments such as those performed at the National Ignition Facility (NIF, LLNL), the OMEGA facility at the Laboratory for Laser Energetics (LLE), the Z-pinch machine at the Sandia National Laboratories (SNLs), and the Megajoule Laser Facility (LMJ). At some of these facilities it has already been demonstrated that robust imaging systems based on x-ray optics can be developed to serve as powerful diagnostic tools for photon energies ≤ 10 keV [3–6]. Here, single metallic coatings or simple multilayer prescriptions can often provide high reflectivity at reasonably steep incidence angles. However, as the range of photon energies of interest moves towards harder x-rays, aperiodic multilayer coatings become necessary in order to achieve high reflectivity combined with large angular acceptance [7], which in turn is necessary to maintain a large field of view (FOV). In this paper we present work we have conducted at LLNL to validate Mo/Si as a viable candidate for this difficult task. We describe the development of a Mo/Si multilayer structure with depth-graded period, designed to maximize average reflectivity and response uniformity over the grazing incidence range of 0.6° ± 0.05° at 16.25 keV. The Mo/Si material combination has been extensively studied over the past three decades [8–15], including non-depth graded applications at hard x-ray energies [7]. Extending the applicability of Mo/Si MLs to highly diverging hard x-ray sources adds new challenges to those already faced by other scientists in the same field. Aquila and co-workers first pointed out the complexity of developing aperiodic Mo/Si MLs for the EUV range, mainly because of the presence of silicide layers at each Mo-Si interface, whose thickness must be carefully controlled in order to reach as-designed ML performance. While the same is true in the hard x-ray regime, additional complications arise as a consequence of the short periods necessary to satisfy Bragg’s law at reasonably steep angles. The latter are necessary to minimize the size of the reflective optics. At hard x-ray energies Mo and Si layers can have thicknesses comparable to, if not shorter than the silicide layers themselves. Our data suggests that in this regime the thickness of the silicide layers varies considerably, making the development of a particular prescription difficult. Furthermore it is well understood that Mo undergoes an amorphous-to-crystalline phase transition around 20 Å, which happens to be well within the range of thicknesses one must consider for hard x-ray wavelengths [12]. Such transition must be avoided due to its detrimental effects on the roughness of the ML and therefore its reflectivity. Despite these complications, our work shows that certain simplifying assumptions can be made when designing and optimizing aperiodic Mo/Si MLs, at least for the range of ML periods we have investigated. We show that even without the use of optimization algorithms, these assumptions yield as-designed performance. As such, we suggest that the Mo/Si material combination can be a viable candidate for hard x-ray multilayers with broad angular acceptance and high average reflectivity.

2. Experimental methods

The multilayer coatings described in this paper were all deposited at LLNL using a custom-built dc magnetron sputtering system. The sputtering was performed using and Ar plasma at a background pressure of 2.0 mTorr. The base pressure was better than 1 × 10⁻⁷ Torr. The Si target power was 75 W and the Mo target power was 50 W. Each sample, whether periodic or aperiodic in structure, consisted of 150 periods of Mo/Si terminated with a 40Å layer of silicon as capping layer. The substrates were 4-inch semiconductor grade silicon (100) wafers, with a micro-roughness measuring less than 2 Å. X-ray reflectivity measurements were performed using a Panalytical Xpert Pro MRD equipped with a Cu K_α anode and a Ge monochromator, and a Rigaku diffractometer equipped with a Mo Kα anode and monochromator. The data was analyzed and fitted using the IMD software package [16], which was also used to design and manually optimize the thicknesses of Mo and Si in the aperiodic structures described later. No numerical algorithm was used to optimize the performance of the aperiodic MLs.

3. Model

Constant d-spacing Mo/Si multilayers have been extensively investigated for more than three decades now [8–12, 14]. Furthermore, aperiodic Mo/Si ML have been investigated by Aquila and co-workers for EUV wavelengths [13]. From the vast amount of data available in literature, it is well understood that intermixing at the Mo-Si interface causes the formation of silicide layers leading to a four-layer model describing each period in the stack, as shown in Fig. 1. The total thickness of the two silicide layers determines the overall contraction of the corresponding period, and must therefore be modeled correctly in order to achieve as-designed performance. For this reason it is important to differentiate between the nominal, pre-silicide formation thickness of Mo and Si (D_Mo and D_Si), and the actual, post-silicide formation thickness of Mo and Si (d_Mo and d_Si). With the assumption that only MoSi₂ makes up the silicide layers across the stack, the conservation of the number of atoms yields [12, 13] the following relations:

 

Fig. 1 (Left) The nominal structure of a single period in the Mo/Si ML. D_Mo and D_Si indicate the thickness of Mo and Si one would have in the absence of silicide formation. (Right) The actual structure of a single period within the ML. Silicide formation at the Mo/Si interfaces requires a description based on a four-layer model. It is our convention to use lower case letters to indicate the thickness of each layer in the period. d_Mo-Si indicates the thickness of the Mo-on-Si silicide, while d_Si-Mo is the thickness of the Si-on-Mo silicide. It is important to note that an overall contraction of the ML period results from the formation of the silicide layers.

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4. Results

A series of periodic and aperiodic ML prescriptions were deposited as part of the development of our target aperiodic structure. In this section we report on their x-ray reflectivity data. In the next section we motivate each deposition, discussing our findings and conclusions.

Fig. 2 shows Cu K_α reflectivity data on two constant d-spacing MLs with (a) d=42.6 Å, Γ = 0.3, and (b) d=33.0 Å, Γ = 0.37. Gamma is defined as D_Mo/((D_Mo + D_Si).

 

Fig. 2 Normalized x-ray reflectivity measured at the Cu Kα energy on Mo/Si constant d-spacing MLs with (a) d=42.6 Å, Γ = 0.3, and (b) d=33.0 Å, Γ = 0.37. Gamma is defined as D_Mo/(D_Mo + D_Si).

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Fig. 3 shows Cu K_α reflectivity data on two aperiodic MLs. Both prescriptions were designed to have (ii) d_Mo constant and equal to 7.9 Å and (ii) d_Si following a power law distribution of the form t_i = a/(i + b)^c with a = 55.56, b =19.83, and c = 0.39. In our notation t_i is the thickness of the i-th layer with i = 1 being the topmost one. However for “aperiodic 1” D_Mo and D_Si were derived using relations (1) and (2) assuming ${\text{d}}_{{\text{MoSi}}_2}^{\text{tot}}=19.8\hspace{0.17em}\mathring{\text{A}}$. On the other hand for “aperiodic 2” the value for ${\text{d}}_{{\text{MoSi}}_2}^{\text{tot}}$ fed into relations (1) and (2) was reduced to 18.4 Å as explained in the next section.

 

Fig. 3 Normalized x-ray reflectivity measured at the Cu Kα energy on two Mo/Si aperiodic MLs obtained assuming (a) ${\text{d}}_{{\text{MoSi}}_2}^{\text{tot}}=19.8\mathring{\text{A}}$, (b) ${\text{d}}_{{\text{MoSi}}_2}^{\text{tot}}=18.4\hspace{0.17em}\mathring{\text{A}}$. Refer to Section 4 for further details.

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Fig. 4 shows Mo Kα reflectivity data taken on sample “aperiodic 2”, with the corresponding IMD fit. Also shown is the reflectivity obtained by extrapolating to 17.5 keV the model resulting from the fit of the Cu Kα “aperiodic 2” data shown in Fig 3.

 

Fig. 4 Normalized x-ray reflectivity of the “aperiodic 2” ML prescription measured with a Mo Kα anode (17.5 keV), and corresponding fit. Also shown is the ML response obtained by extrapolating to 17.5 keV the model resulting from the fit of the Cu Kα “aperiodic 2” data shown in Fig 3.

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5. Discussion

The ML prescription we set out to develop in this work had to maximize its first order average reflectivity at 16.25 keV, at a grazing incidence of 0.6°, with an angular acceptance of ± 0.05°. The latter requirement is defined as (R_max − R_min)/R_ave ≤ 0.5 across the angular range. For this reason an aperiodic prescription with power law behavior was chosen. The precise silicide layer thickness as a function of period number across the stack was a priori unknown, and prevented us from designing the depth-graded prescription right away. Instead, using the four-layer model shown in Fig. 1, we calculated the two ML periods that would satisfy Bragg’s law near the edges of the angular bandwidth for 16.25 keV photons, assuming ${\text{d}}_{{\text{MoSi}}_2}^{\text{tot}}=20\mathring{\text{A}}$ across the stack. Then, two constant d-spacing MLs were deposited each one matching the d-spacing values obtained this way. Their reflectivity data is shown in Fig. 2. Fits to the data suggest that the thickness ratio of the two silicide layers within a period is different for the two samples, as manifested by the different distribution of intensities among Bragg orders. For the d=42.6 Å case, the two silicide layers in each bilayer seem to have similar thicknesses measuring 9.5Å (Mo-on-Si) and 10.5Å (Si-on-Mo). On the other hand, for the d=33.0 Å ML d_Mo-Si is thicker than d_Si-Mo, measuring respectively 12.33 Å and 7.33 Å. We point out that for both samples D_Mo was intentionally kept below 20 Å to avoid an amorphous-to-crystalline transition which greatly increases the interfacial roughness in the ML [12]. As already pointed out D_Mo and D_Si depend on the total silicide layer thickness ( ${\text{d}}_{{\text{MoSi}}_2}^{\text{tot}}$) alone, which is roughly the same for both samples with an average of 19.8 Å.

Based on these results, we proceeded with the design of the ML prescription “aperiodic 1” under the following assumptions: (i) across the graded prescription the thickness of the silicide layers is constant, and (ii) ${\text{d}}_{{\text{MoSi}}_2}^{\text{tot}}=19.8\mathring{\text{A}}$, the average from the constant d-spacing results. The ML structure was designed and optimized manually using IMD. Our numerical simulations indicated that varying the thickness of Si alone was sufficient to get the desired angular bandwidth. For this reason the Mo thickness was kept constant across the stack. This greatly simplified both the deposition and the characterization of the multilayer structure. Cu Kα x-ray reflectivity data was taken on this sample right after deposition, and the result is shown in Fig. 3(a). The dashed line in the figure indicates the as-designed position of the first reflectivity fringe in the first Bragg order. Clearly the distribution of periods across this ML is shifted towards higher values with respect to the target. A fit to the data estimates the shift to be ∼ 1.3Å. The same fit estimates the (constant) silicide layers thickness in each period to be 9.7Å for the Mo-on-Si layer and 8.7Å for the Si-on-Mo layer, for an overall ${\text{d}}_{{\text{MoSi}}_2}^{\text{tot}}=18.4\mathring{\text{A}}$. The fact that the latter is less than what is used in the model to develop this specific ML explains the weaker contraction of the ML period. New values for D_Mo and D_Si were then obtained by feeding the experimentally determined ${\text{d}}_{{\text{MoSi}}_2}^{\text{tot}}$ into relations 1 and 2, while keeping the target (d) thicknesses unchanged.

The ML prescription “aperiodic 2” was deposited based on this adjusted recipe. The data shown in Fig. 3 suggests that the distribution of periods across the stack is now on target. The good agreement between the fit and the data at the first Bragg order demonstrates that the assumptions made during the development phase were reasonable. At higher Bragg orders the disagreement is likely due to minor silicide thickness variations across the stack that are not included in the model. Such disagreement is likely to limit the energy range over which an extrapolation of the model would give a reasonable representation of the ML response. Specifically it is unclear whether extrapolating the model obtained from Cu Kα data would accurately predict ML performance at the target energy of 16.25 keV. In order to quantify the accuracy of the extrapolation, and verify the performance of our ML near-wavelength, we have taken XRR data using a Mo Kα anode. With its 17.5 keV emission line, the latter allows us to test the ML reasonably close to its target wavelength without the need for a tunable source such as a synchrotron. Fig. 4 shows the corresponding data and its fit. Also shown is the ML response estimated from extrapolating the Cu Kα data to 17.5 keV. The intensity of the Cu Kα electric field inside the ML is reduced to 10% within the first 59 periods from the top of the structure. On the contrary the intensity of the Mo Kα never drops below 10% throughout the entire stack. It is no surprise then that the extrapolated Cu Kα model deviates somewhat from the measured response. The fit to the Mo Kα data suggests a ${\text{d}}_{{\text{MoSi}}_2}^{\text{tot}}=17.7\mathring{\text{A}}$, d_Mo =8.3Å, and power law coefficients for the depth graded Si a = 47.3, b =17.5, and c = 0.35.

We conclude by noting that the measured near-wavelength response yields an average reflectivity of 28 % ± 3 % in the angular range (± 0.05°). Additionally we suggest that for a depth graded system characterized by complex intermixing layers such as Mo/Si, numerical extrapolation of a ML model to higher energies may only yield qualitative information on the actual performance of the system.

6. Conclusions

In this work we have developed an aperiodic Mo/Si ML capable of reflecting 16.25 keV photons at 0.6° grazing angle with an average reflectivity of 28%. While aperiodic Mo/Si MLs have already been developed for EUV wavelengths, to the best of our knowledge this is the first time this material combination has been extended to the hard x-ray regime with an aperiodic structure. In this regime we have demonstrated that certain simplifying assumptions can be made regarding the thickness of the silicide layers across the stack. This allows us to achieve as-designed performance without the need for numerical optimization algorithms. At the same time we have shown that Cu Kα characterization may only yield qualitative information regarding the performance of the ML structure at shorter wavelength. Ultimately, we suggest that Mo/Si MLs can be viable candidates for applications in the field of hard x-ray imaging of highly diverging sources.