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Abstract
For many years, the Mo/Si multilayer mirror has been the standard choice for normal incidence optics in the EUV/X-ray range below 20 nm. As one goes to longer wavelengths, such as the 46.9 nm Ne-like Ar X-ray laser, the absorption in Mo increases substantially, which is why the Sc/Si mirror was developed to operate in this region. To solve the problem of the metastable interface of Sc/Si mirrors due to diffusion between the two materials, this paper looks at an alternative Ge/Si multilayer mirror that replaces Sc with Ge, which offers potential advantages for fabricating a multilayer structure without the additional diffusion barriers needed with Sc. Ge can also be used to produce a narrow band filter in this wavelength regime.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
For many years the Ne-like Ar X-ray laser [1–3] has been used as XUV/soft X-ray source for experiments done at 46.9 nm. This laser uses an electrically driven capillary discharge and can provide 1 mJ of output in a diffraction limited beam at a repetition rate of 4 Hz making it a very attractive source for doing a large variety of experiments. A compact table-top version of this laser [3] provides 13 µJ of output at 12 Hz for tens of thousands of shots. These lasers have been used in numerous applications, including interferometry of dense plasmas [4], the measurement of optical constants [5], materials ablation [6], the characterization of soft x-ray optics [7], excitation of color centers in crystals [8], and nanopatterning [9].
Since many laser applications require normal incidence high reflectivity X-ray mirrors the Sc/Si multilayer mirrors were developed. These typically have a measured peak reflectivity of 31% at 46.9 nm for normal incidence versus a calculated reflectivity of 67% [10]. A main challenge of these mirrors is that the Sc/Si interface is metastable, and the interfaces diffuse which greatly reduces the theoretical performance of the mirror. A typical Sc/Si mirror designed for 46.9 nm has 10 layers with a period of 25.7 nm where the H(Sc)/H(Si) = 0.786 [10] with H the thickness of a layer. Various efforts have looked at reducing the interdiffusion. Experiments were done looking at adding a boron carbide layer between the Sc and Si to reduce the interdiffusion [11]. Another group [12] experimented with a Cr barrier and observed peak reflectivity of 56% at 44.7 nm in some samples. Recently a three-layer design was demonstrated with a peak reflectivity of 57% at 44.7 nm using SiC as a third layer between Sc and Al to enhance the reflectivity [13].
In this paper we look at an alternative design that uses Ge instead of Sc in combination with the Si. Ge is a better material to coat than Sc and has the potential to be easier to fabricate high quality layers without the need for an additional diffusion barrier and may offer the advantage of a narrower bandwidth with a peak reflectivity similar to the original Sc/Si mirrors.
Multilayer structures of Ge and Si have been studied for use as semiconductors and solar cells. Research measuring these Ge/Si multilayer structures shows very clean interfaces for periods of 3–6 nm with surface roughness of less than 1 nm and small amounts of interdiffusion as compared with Sc/Si [14,15].
In this paper we also show how Ge can be used to produce filters with a narrow bandpass between 44 and 50 nm that could be very useful for reducing background emission in experiments using the Ar X-ray laser at 46.9 nm.
2. Predicting the properties of Ge and Si
To design and model the reflectivity of a multilayer mirror one needs the complex index of refraction for the materials which is defined as n* = n + i k. The Henke tables [16] are the baseline data for these calculations and consist of the non-dimensional constants f1 and f2 where (1 – n) = f1 Nion / (2 Ncrit) gives the real part of the index of refraction and k = f2 Nion / (2 Ncrit) is used to calculate the absorption coefficient. At wavelength λ, Ncrit = π / (r0 λ2) where r0 is the classical electron radius, 2.818 × 10−13 cm [17]. The total absorption coefficient α = Nion σ = (4 π k) / λ = 2 Nion f2 r0 λ where Nion is the ion density of the material. For cold solid materials the ion density Nion is Avogadro’s number NA (6.022 × 1023) times the density ρ (5.32 g/cc for Ge) divided by the atomic weight (AW = 72.64 for Ge). The absorption is measured at discrete energies at LBL and other institutes and then the dimensionless absorption coefficient f2 is fit over the energy range E from 10–30,000 eV. The real part of the index of refraction coefficient f1 is then determined by using the Kramers-Kronig relationship where Znuc is the nuclear charge of the material.
In the Henke Tables f1 is only tabulated from 30–30,000 eV. Since the Ar X-ray laser line is at 46.9 nm or 26.4 eV this presents a challenge in calculating the reflectivity of the multi-layer mirrors.
In this paper we compare the properties of a Sc/Si multilayer mirror with Ge/Si so we require the complex index of refraction for Si, Sc, and Ge. In the case of Si we have more extensive data because Si has been extensively studied and in addition to the Henke data [16] from LBL there is also data from the Handbook of Optical Constants by Palik [18] that extends the optical constants down to energies in the optical regime. For Ge and Sc the original Henke data [16] for f1 begins at 30 eV so we used to the Kramers-Kronig relationship applied to the absorption coefficient f2 in the range 10–30,000 eV to calculate a value of f1 from 10–30,000 eV.
Figure 1 shows a plot of the tabulated Henke data f1 vs energy for Ge from 30–100 eV as well as our calculated value for f1 using the Kramer-Kronig formula in Eq. (1). There is excellent agreement between the calculated value of f1 and the Henke value except for a constant offset of about 0.35 eV for Ge. We subtracted this constant value from our calculated f1 to give a new estimate of f1 for Ge, which is also shown in Fig. 1. Using this new value of f1 together with the Henke value for f2 we then created a table of n,k values that could be used by the IMD code of Windt [19] to calculate the normal incidence reflectivity of a multi-layer mirrors for unpolarized light. Part of this 0.35 eV offset can be explained due to lack of f2 data for energies above 30 keV which means the integration in Eq. (1) is incomplete. Above the Ge K edge at 11.1 keV the absorption coefficient f2 generally falls as the inverse of the square of the energy. If one assumes this dependence for f2 in the integral in Eq. (1) then it is easy to show that for energies much lower than 30 keV where the E2 term in Eq. (1) can be neglected, the integral from 30 keV to infinity is equal to -f2 (at 30 keV) / π. In the case of Ge, f2(at 30 keV) = 0.7034 which means there is a missing integral term of about −0.224. There is also a small positive term missing due to the integral from 0–10 eV that is energy dependent but at an energy of 50 eV where we determine the 0.35 eV offset we estimate that contribution is 0.03 from integrating the Palik data [18] and including that term in the integral would just require a larger offset to normalize the calculations against the Henke tables especially as those low energy values are not included in the Henke tables.
Fig. 1. Optical constant f1 for Ge versus photon energy E from the Henke tables compared with the values calculated from the Kramers-Kronig (K-K) relationship in Eq. (1) with and without an offset of −0.35.
In the case of Sc there is a new compilation of f1 and f2 values in the Henke table from 2006 that come from the research group of Vinogradov and Kondratenko [20] who pioneered the fabrication for Sc/Si mirrors. The new Sc data has better values of the absorption from the 3p electrons that substantially changes the complex index of refraction in the range from 35–60 nm and the new data provides values of f1 from 20–30,000 eV so there is no need to calculate the f1 data below 30 eV. To document the effect of the new Sc data we also extraoplate the original Henke data using the same procudure as for Ge. In the case of Sc there was no offset needed after using the Kramers-Kronig calculation.
Figure 2 shows a plot of the absorption coefficient k vs wavelength for Si, Ge, and Sc. For Sc we include both the historic Henke values from 1993 [16] and the updated values based on Uspenskii’s paper from 2006 [20]. Typically one wants both materials in the multilayer mirror to have small absorption values since this limits the number of multilayers in the mirror. For example, at 46.9 nm the transmission through 0.1 µm of Si is 61.4%, for Ge it is 59.3%, for the historic Sc values from 1993 it is 47%, and for new Sc values from 2006 it is 43.5%. One notices that the big difference between the two values of k for Sc is that the absorption edge has shifted significantly from 35 nm to 44 nm which is more consistent with the 3p photoionization edge at 28.3 eV (43.8 nm) [16].
Fig. 2. Absorption coefficient k vs wavelength for Si, Ge, and Sc. For Sc we include both the historic Henke values from 1993 and the updated values from 2006.
Figure 3 plots the value of 1-n vs wavelength for Si, Ge, and Sc. At 46.9 nm the value of 1-n for Si is 0.196, for Ge is 0.069, for Sc (1993) is 0.033 and for Sc (2006) is −0.059. The reflection from a single interface is [(n1-n2)/(n1+n2)]2. For values of n near one this simplifies to (Δn/2)2. For a multilayer mirror one gets coherent interference between the layers but the reflectivity is larger for larger values of Δn between the two materials. While Δn is larger for the Sc/Si interface there is still good contrast of Δn = 0.127 for the Ge/Si interface. Since the Ge absorption is lower than Sc at 46.9 nm this allows more layers to be effective in the multi-layer and produce a mirror with a smaller bandwidth. Even more important is that the Sc and Si have significant inter-diffusion at the interfaces that causes most of the mirrors that have been built in the past to be less effective than theoretical predictions [10].
Fig. 3. Difference of real part of the index of refraction from unity, 1-n, vs wavelength for Si, Ge, and Sc. For Sc we include both the historic Henke values from 1993 and the updated values from 2006.
3. Simulating the reflectivity of the multi-layer mirrors
To model the reflectivity of the multi-layer mirrors we use the IMD program [19] developed by David Windt that is publicly available. It is based on the calculation methods described in his paper and the original Henke paper [16]. The IMD program uses an input table of n, k values vs wavelength. For Si there is already a detailed table from the optical region to hard X-rays based on the Henke data [16] combined with Palik [18]. For Ge we use the Henke table that we extrapolated to lower energy to cover the range down to 10 eV. For Sc we use both the 1993 Henke data that we extrapolated to 10 eV and the new 2006 Henke data that requires no extrapolation.
For the Ge/Si and Sc/Si multilayer mirrors we used 100 periods in the simulations to explore the peak performance even though calculations show there is virtually no difference between 20 periods and 100 given the absorption coefficients for the materials. Based on the work of Ref. 10 we use a value of 0.4 for gamma, which is the ratio of the thickness of the Ge or Sc to the total thickness of a period. First we did a set of calculations varying the period from 20 nm to 40 nm and looked at the wavelength where the reflectivity peaks as well as the peak value of the reflectivity for a given period. Figure 4 plots the period of the Ge/Si and Sc/Si mirrors vs the wavelength of peak reflectivity. Looking at a wavelength of 46.9 nm in Fig. 4, the Ge/Si mirror has a period of 27.5 nm while the Sc/Si mirror has periods of 27.3 and 26.5 nm for the legacy 1993 data vs the updated 2006 data.
Fig. 4. Period of the Ge/Si and Sc/Si mirrors vs wavelength at peak reflectivity. For Sc we include calculations done with both the historic Henke values from 1993 and the updated values from 2006.
Figure 5 then plots the peak reflectivity vs wavelength as the period was varied from 20–40 nm. The reflectivity of the Ge/Si mirror peaks about 40% near a wavelength of 44 nm. The Sc/Si mirror using the new 2006 data has a very similar shape to the Ge/Si mirror but with a peak reflectivity about 60% at 44 nm while the Sc/Si mirror using the legacy 1993 data has similar, but slightly higher reflectivity than the Ge/Si mirror at 44 nm but continues to increase as one goes to shorter wavelength which is inconsistent with the measured 3p absorption edge at 28.3 eV (43.8 nm).
Fig. 5. Peak reflectivity vs wavelength of peak reflectivity as the period is varied from 20–40 nm for the Ge/Si and Sc/Si multilayer mirrors. For Sc we include calculations done with both the historic Henke values from 1993 and the updated values from 2006.
If we look at the case when the reflectivity peaks at 46.9 nm for each mirror, Fig. 6 plots the reflectivity vs wavelength for each of the multi-layer mirror designs. The Ge/Si mirror has a peak reflectivity of 34.8% with a full-width-half-maximum (FWHM) bandwidth of 3.16 nm. The Ge/Si mirror has a period of 27.5 nm which together with a gamma of 0.4 means that the Ge layer is 11.0 nm thick and the Si layer is 16.5 nm. The Sc/Si mirror (period = 27.3 nm) designed with the legacy 1993 data has very similar performance with slightly higher reflectivity of 37.7% and FWHM bandwidth of 3.91 nm. The Sc/Si mirror (period = 26.5 nm) using the 2006 data has a much higher theoretical reflectivity of 52.7% with a much larger FWHM bandwidth of 5.83 nm. One needs to remember that the measured performance of the Sc/Si mirrors were significantly smaller than predicted by a factor of 2. In 2003 we fabricated several Ge/Si multilayer mirrors with 20 layers and a period of 27.5 nm but were unable to test them at the time due to lack of funding but we are confident these could be built and used successfully today. The smaller bandwidth could be advantageous in some experiments to reduce background and improve the signal to noise in many applications.
Fig. 6. Reflectivity vs wavelength for each of the Ge/Si and Sc/Si multilayer mirror designs optimized for peak reflectivity at 46.9 nm. For Sc we include calculations done with both the historic Henke values from 1993 and the updated values from 2006. Ge/Si has a period of 27.5 nm while the Sc/Si mirrors have periods of 27.3 nm (1993) and 26.5 nm (2006).
Looking at the shape of the Ge absorption curve in Fig. 2 it shows that Ge might also be very useful to create an X-ray filter for use in experiments where one requires a narrow band filter to reduce background emission seen by X-ray CCD detectors. Figure 7 shows the transmission for a 400 nm thick Ge filter versus wavelength. The FWHM bandwidth is 8 nm nicely centered on the 46.9 nm line. In experiments using a bright Ar X-ray laser at 46.9 nm the Ge filter could dramatically reduce the surrounding background emission. Good filters have always been a key component of using X-ray lasers in applications where broadband X-ray CCD cameras are recording the data and the high spectral brightness of the X-ray laser is competing with the continuum emission from the plasma.
4. Conclusions
In this paper we look at the design of a Ge/Si multi-layer X-ray mirror as an alternative for the Sc/Si mirror used for the Ne-like Ar laser at 46.9 nm. While the theoretical peak reflectivity at normal incidence of 35% is lower than predicted for the Sc/Si mirror it is similar to reflectivities that are actually measured for many of the Sc/Si mirrors used in experiments. The Sc based mirrors have fabrication challenges because of the interdiffusion between layers that happens between Sc and Si even though several groups have successfully addressed this issue and pushed the reflectivity of Sc/Si mirrors above 50%. Ge does not suffer from the same diffusion issue as Sc and could offer an interesting alternative for fabricating mirrors in the 44–50 nm range and avoid the need for adding extra barrier layers required for the Sc/Si mirrors. The lower absorption of Ge as compared with Sc enables the Ge/Si mirrors to have a narrower bandwidth. Ideally experiments are needed to confirm the optical constants and the behavior of the interfaces between Ge and Si. Ge can also be used to create a narrow band filter near 46.9 nm to significantly reduce continuum plasma emission and improve the signal to noise of experiments using the Ar X-ray laser.
Funding
Lawrence Livermore National Laboratory (DE-AC52-07NA27344).
Acknowledgments
The author thanks Regina Soufli for valuable advice in preparing this manuscript.
Disclosures
The author declares no conflicts of interest.
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