Liquid Helium as a reference may provide clarity for some neutron reflectometry experiments 1

1. Introduction

Most neutron scattering experiments involve measuring the intensity of scattering of a nearly plane wave with incident wavevector k → i (and energy E i ) to a scattered wave with wavevector k → f (and energy E f ) [10,32]. Because this paper is focused on specular reflectometry (angles of incidence on to and off the sample surface are equal magnitudes), E i = E f (for cases where E i ≠ E f see [15]). As the incident wave lacks coherence, intensities of the scattered neutron radiation are measured, thus, the phase information of the scattered wave function is lost (neutron interferometry [27,28] is an exception). Armed only with the modulus of the amplitude of the wave function, most neutron scattering experiments involve comparing calculations of the scattering, i.e., the intensity, from a proposed model structure to data. The comparison is not unique and varying levels of ambiguity persist depending upon the specific experiment. At times, incorrect conclusions can be drawn. In the case of neutron reflectometry [13,21], the challenge is to obtain the scattering length density (SLD) averaged over the lateral dimensions of the sample as a function of depth with a measure of confidence. Because the reflectivity varies as ∼ Q − 4 , where Q → = k → f − k → i is the wavevector transfer, the maximum value of Q = Q max (typically < 0.2 A ˚ − 1 ) that can be measured is rather small in comparison to X-ray reflectometry. Consequently, dramatically different SLD profiles can yield remarkably similar reflectivity curves within the measured range of Q.

To address the ambiguity of model fitting in neutron reflectometry, recourse to qualitatively different information is often used. For example, complementary X-ray reflectivity, ellipsometry, microscopy, or scanning probe data may be available. A priori knowledge of the fabrication of the sample may constrain modeling. Sometimes, use of contrast variation of the environment of a sample can improve the confidence of the SLD by generating multiple datasets which are coupled in a known manner. One method to vary contrast is to measure the sample with the beam incident from the front (film/vacuum side) and alternately through the back, i.e., through the substrate [24]. The reflectivities from the two inverted SLD profiles can be complementary (i.e., qualitatively different), thus, providing an additional constraint on model selection. Another method uses a thick underlayer which can improve the model selection through a procedure developed for astronomical speckle holography [31]. For soft-matter systems, measurements are often taken in a fluid reservoir. The fluid composition can be varied to generate a series of coupled data sets [22]. The system of interest in this case should allow the complete exchange of H₂O for D₂O in the reservoir and not the sample. The number of coupled data sets is determined by the accuracy of mixing the fluids. In cases where the film of interest is not magnetic, the film may be grown on a smooth saturated magnetic layer, and polarized neutron beams (with concomitant loss of incident beam intensity) used to generate coupled datasets improving the accuracy of the extracted SLD [30]. In other cases, combinations of contrast matching using a fluid reservoir and a magnetic layer provide additional means to couple data sets and thus highly constrain model selection [11].

In this paper, we demonstrate use of liquid Helium as a reference material. In some cases, complementary data can be produced by measuring a sample in He vapor (essentially vacuum) and liquid He. Helium is advantageous because it is chemically inert and has no significant effect on the neutron spectrum of a beam transiting the ∼ cm length of the liquid (unlike a substrate). The approach does not require specialized samples (unlike a magnetic reference layer sample). For neutron reflectometry experiments that will be performed below 4 K anyway, and given how trivial a sample can be immersed in liquid He, there is possibly no downside to collect data with the sample in and out of liquid He. For cases primarily having layer SLDs of order 1 × 10 − 6 A ˚ − 2 , the change of SLD at the film’s surface by the liquid He reference alters the reflectivity significantly; thus, providing complementary information to test models.

The paper presents data from six samples acquired in the presence of He vapor and alternately liquid He. Model fitting to the data used the dynamical approach of Parratt [25]. On the basis of inspection of fitted SLD profiles and calculation of a simple correlation coefficient, we conclude that for films having layers with small SLD’s, the additional dataset provided by the liquid He reference can be insightful.

2. Sample preparation

Four hard matter and two soft matter samples were prepared for the study. Two of the hard matter samples consisted of Pt deposited on Ta deposited on sapphire (or Si) substrates (adopting the convention of labelling materials from substrate to surface: Al₂O₃/Ta/Pt or Si/Ta/Pt). For the other two hard matter samples, the order of layer deposition was reversed, i.e., (Al₂O₃/Pt/Ta or Si/Pt/Ta). The nominal layer thicknesses were 400 Å for Pt ( SLD = 6.4 × 10 − 6 A ˚ − 2 ) and 200 Å for Ta ( SLD = 3.8 × 10 − 6 A ˚ − 2 ). The factors of ∼ two difference between the layer thicknesses and SLDs can present challenges for analysis of reflectivity data as the data from different combinations of layer order can be similar. The two soft matter samples consisted of the photoresist SU-8 [1] on Si (Si/SU8) and polymethyl methacrylate (PMMA) on Si (Si/PMMA). The nominal layer thicknesses were 700 Å for SU-8 ( SLD = 1.5 × 10 − 6 A ˚ − 2 ) and 400 Å for PMMA ( SLD = 1.1 × 10 − 6 A ˚ − 2 ). Information about the nominal thicknesses and chemical compositions of the layers were provided by the sample growers to the neutron scatterers. The restricted availability of information was intentional. Anticipating future applications using liquid He as a means to change contrast, it is hoped clarity in interpretation of the neutron data may be achieved for some experiments that preclude the availability of data from complementary probes, e.g., use of sample environments that are not compatible with other probes, or samples that are fabricated in situ.

Pt and Ta layers were deposited on Al₂O₃ and Si single-crystalline substrates with a lateral size of 1 cm by 1 cm using DC magnetron sputtering. Before the deposition, the substrates were cleaned by Ar plasma to obtain a smooth surface free of contamination. The base pressure immediately before deposition was < 5 × 10 − 8 mbar . The pressure of the Ar gas (to make the plasma) was 0.013 mbar. The temperature of a substrate during deposition was room temperature. The deposition rates of Pt and Ta were ∼0.5 Å/s and ∼0.7 Å/s, respectively.

Polymer films of two types were deposited on silicon chips by spin-coating solutions of SU-8 (custom dilution in SU-8 2000 thinner, MicroChem) and PMMA (495000 molecular weight, 2% in anisol, MicroChem) resists. The concentration of the solids in the SU-8 solution was adjusted to yield approximately 700 Å film thickness using a spin-coating speed of 5000 rpm. After spin-coating, the films were baked on a hot plate at 180°C for 2 minutes.

3. Experimental methods and results

The reflectivity data were acquired using an unpolarized neutron beam (Beamline-4A, Spallation Neutron Source (SNS) at the Oak Ridge National Laboratory) and a closed-cycle cryostat manufactured by Cryomagnetics [2]. This cryostat is a top-loading system and the sample is placed on the end of a ∼1 m long stick. Alignment of the sample was made by rotating and translating the stick with stepper motors. The sample surface is in the vertical plane and reflection of the neutron beam occurs in the horizontal plane. Neutron wavelength was measured using time-of-flight techniques. Acquisition of reflectivity data involves measurement of the intensity of the specularly reflected beam normalized by the variation of the incident beam spectrum. The incident beam spectra were measured with and without liquid He present in the sample space and were observed to be statistically the same, i.e., liquid He in the cell produced no measurable effect on the spectrum. Samples were somewhat over-illuminated by the neutron beam.

The sample space of the cryostat consists of a ∼1 m-long stainless-steel tube terminated with a single crystal sapphire cylinder (the neutron beam passes transversely through the cylinder). Prior to sample insertion, the sample space was constantly filled with He gas at 1 atm while the temperature in the region of the sapphire cylinder (∼ bottom 20 cm of the sample space) was held at ∼90 K. The sample stick with the sample at its end was inserted into the sample space with the sample on axis and near the center of the sapphire cylinder with respect to its height. The inner space was then pumped to ∼ 10 − 6 mbar in a few seconds and backfilled with He gas to 0.5 atm in a few seconds. The process was repeated five times terminating with a He fill of 0.5 atm. The temperature of the sample was lowered from 90 K to 1.4 K and stable to a few mK in less than one hour.

A reflectivity curve was measured at 1.4 K after condensing ∼ 2 cm 3 of liquid He using additional He gas (introduced for ∼4 minutes at an inlet pressure of 1 atm and temperature of 300 K). The volume of liquid was chosen to ensure the 1 cm by 1 cm sample was completely immersed in liquid. Note the quantity of He to completely immerse the sample in liquid when converted to vapor corresponds to a pressure that is well in excess of the setting of the pressure relief valve. Thus, when the liquid turns to vapor (for instance to remove the sample or to perform a measurement above 4.2 K), the pressure relief valve opens venting excess He and then closes (a reversible and autonomous process). After collecting a measurement with the sample immersed in liquid He, a second reflectivity curve was measured after the temperature was raised to 6 K to ensure that the sample was in Helium vapor only. The same process was repeated three times to measure the six samples. Only three measurements were required, because the specular reflectivities from a pair of samples appear in two well-separated places in the two-dimensional position sensitive (we routinely measure two or three samples at once). The reflectivity data are shown in Figs 1–7.

A Jupyter Notebook framework [3,14] was used to produce the Q-dependent reflectivity, R ( Q ) , for each sample in each environment. Using GenX [4,16], parameters of models of the SLD profiles were optimized to minimize the sum of the absolute value of the difference between the logarithms of the calculated and observed reflectivities (i.e., the log optimizer in GenX was selected). Parameters that were optimized included the SLDs of the layers, layer thickness interface roughness, and detector angle offset error. Optimization of the SLD was constrained to search for solutions within 30% of the literature values for the materials. The data taken for the He vapor and liquid phases were fitted to the same model with the sole exception that for the measurement taken using liquid He that the SLD of liquid He ( ∼ 0.6 × 10 − 6 A ˚ − 2 ) was optimized (the SLD of the He vapor was fixed to be zero). All data and the GenX executables are available to the reader at the Zenodo website [5,6].

4. Discussion

A concern with use of a fluid as a reference material is whether exchange of atoms or molecules between the fluid and the sample significantly affects the chemistry of the sample [17]. For example, the chemical potential difference between a H₂O/D₂O solution and a hydrogenated or deuterated film may drive H or D across the interface. A chemical potential may exist across the He reservoir and film interface; however, at 1.4 K, thermally activated diffusion of He into the film is small. In fact, extrapolation of the diffusion constant, D, of He in Pyrex-brand glass from room temperature [29] to 1.4 K implies the length scale of He diffusion into a sample during the time, t, of a reflectivity measurement (∼12 hours) D t is computationally zero. Kuzmin et al. [20], report the diffusion constant of He in an aerogel is vanishingly small for pressures of <10 mbar. The vapor pressure of He at 1.4 K is 2.7 mbar [7]. Extrapolation of the diffusion constant of helium in Ni, Cu and Pd to 1.4 K is likewise small, such that the diffusion length of He in these materials for a time corresponding to a typical neutron experiment is computationally zero [8]. On the basis of the literature (Refs. [4,5,17]), we conclude thermally activated diffusion of He will not alter our films in a measurable way at 1.4 K. However, there exists the possibility that He enters the films as the films are inserted into the cryostat. During insertion, the films are quenched from 300 K to 90 K in a few seconds in the presence of 1 atm He gas. After insertion, the gas is replaced with 0.5 atm He exchange gas, and the sample cooled to 1.4 K in less than an hour. Some samples may incorporate He while the sample cools from 90 to 1.4 K. We note thermally activated diffusion is suppressed at cryogenic temperatures. Thus, we regard the SLD of the sample in the liquid and vapor to be unchanged at 1.4 K, so our analysis uses the same SLD profile of the film and substrate for both vapor and liquid measurements, and only the SLD of the ambient environment will be different between the two. The reflectivity curves from the models are shown in Figs 1–7 as the solid curves for each sample.

Information in the figures is arranged such that the observed and calculated reflectivities are shown on the left side, and the scattering length density profile(s) used to produce the calculated reflectivity on the right side. Unless otherwise noted, the upper panels of Figs 1–5 show results for the correct ordering of Pt and Ta layers and the bottom panels are for the reversed (incorrect) ordering. The models yielding the results in the bottom panels are known to be inconsistent with the fabrication of the sample.

For the sake of clarity, we define the critical edge of wavevector transfer, Q c , as the value of Q such that R ( Q c ) = 0.5 . Figure 1 illustrates a common feature of all the measurements – namely when the environment of the sample is changed from He vapor to liquid He, the value of Q c shifts to smaller Q consistent with the reduction of SLD contrast between the sample and the ambient environment. The shift of Q c from that using He vapor to liquid He is a measure of the SLD of liquid He – the number density of He atoms in the liquid phase times the coherent scattering length of ⁴He.

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Fig. 1.

Actual sample layer sequence: Pt on Ta on Al₂O₃. Observed reflectivity (circles), calculated reflectivity (curves) and residuals for the (upper) correct and (bottom) inverted (incorrect) models for cases of sample in He vapor (black) and liquid He (red). (inset) SLD profiles of the models.

Refinement of the correct model yields an optimized solution with nearly the expected SLDs for Pt and Ta, and the model mostly lacks systematic errors (residuals are shown above the reflectivity data). Refinement of the incorrect model yields a different optimized solution with a figure-of-merit ∼15% larger than the correct model. The incorrect solution exhibits systematic errors (periodic variation of the residuals) and unphysically large (small) SLD for Pt (Ta). The choice or which model best represents the sample is clear. Aside from the shift of R ( Q ) owing to the difference between Q c , R ( Q ) for the two situations are not qualitatively different. To quantify “not qualitatively different” the Pearson correlation coefficient [26], P, of the two curves normalized to the asymptote of the Fresnel reflectivity, R F = 16 π 2 Q 4 , was evaluated for Q > 0.02 A ˚ − 1 for curves with identical Q-bins. The value is large P = 0.96 ; indicating a significant positive correlation between the two datasets. This result is intended to test whether the reflectivity curves obtained with and without the liquid He reservoir were different beyond the critical edges. For the case of Al₂O₃/Ta/Pt, use of the liquid He reference provided no additional (special) clarity.

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Fig. 2.

Actual sample layer sequence: Pt on Ta on Si. Observed reflectivity (circles), calculated reflectivity (curves) and residuals for the (upper) correct and (bottom) incorrect models for cases of sample in He vapor. Except as noted by the arrow, the dramatically different models produce indistinguishable reflectivities. (inset) SLD profiles of the models.

The Pearson coefficient ranges from −1 to 1, where −1 indicates two anti-correlated distributions, 0 indicates uncorrelated distributions and 1 indicates two correlated distributions. To provide a notion for a value of P that is consistent with an evaluation using two reflectivity curves that are from different samples (one curve each from the Al₂O₃/Ta/Pt and Si/Ta/Pt samples), we found P ∼ 0.6 for Q > 0.02 A ˚ − 1 . In this instance, we selected two samples with nominally the same film architecture but having different substrates. We conclude that P ∼ 0.6 or less suggests complementary, i.e., qualitatively different, data. Note in our evaluation of P, we have normalized the data to R F – this to remove one source of correlation, i.e., the broad ∼ Q − 4 . We note since 2019 there has been growing interest in quantifying information content of reflectivity data [12,23] and applications of Bayesian [33] and machine learning algorithms [9,18] that provide far more sophisticated means to analyze data; one example being whether reflectivity data are the same or different vs. a perturbation to the sample or its environment.

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Fig. 3.

Actual sample layer sequence: Pt on Ta on Si. Observed reflectivity (circles), calculated reflectivity (curves) and residuals for the (upper) correct and (bottom) incorrect models for cases of sample in He vapor (black) and liquid He (red). (inset) SLD profiles of the models. The correct SLD profile (upper) using a model constrained to use the same structural parameters better fits the data than does the incorrect model.

For this case, Fig. 2 shows the fits of the correct (top panel) and incorrect (bottom panel) to the data taken from the Si/Ta/Pt sample in He vapor (only). This example illustrates how dramatically different SLD profiles can yield exceedingly similar reflectivity curves. One difference between the curves is localized to a region of Q ∼ 0.036 A ˚ − 1 (arrow in Fig. 2). Otherwise, both fits are so remarkably good that a researcher without recourse to other information might conclude the incorrect SLD profile was in fact correct. Figure 3 shows the fits to data taken with and without liquid He. For the Si/Ta/Pt sample, the reflectivity of the data taken with the sample immersed in liquid He are like those obtained using He vapor ( P = 0.92 for Q > 0.02 A ˚ − 1 ). For this case, use of the liquid He reference provided no additional (special) clarity.

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Fig. 4.

Actual sample layer sequence: Ta on Pt on Al₂O₃. Observed reflectivity (circles), calculated reflectivity (curves) and residuals for the correct model for cases of sample in He vapor (black) and liquid He (red). (inset) SLD profiles of the model. For these data, no local minimum was found using the incorrect (inverted) layer sequence.

Figure 4 shows only the fits of the correct models to the data. For this sample, when the inverted (incorrect) structure was used as a starting guess, the refinement always evolved to the correct structure. In other words, no local minimum was observed consistent with the incorrect structure. The reflectivities taken with and without liquid He are not qualitatively different ( P = 0.88 for Q > 0.02 A ˚ − 1 ). For this case use of the liquid He reference provided no further clarity.

This case is somewhat like that of Si/Ta/Pt in that the correct structure accurately reproduces a fringe near Q ∼ 0.36 A ˚ − 1 and a fringe near Q ∼ 0.55 A ˚ − 1 when the incorrect structure does not. The reflectivities taken with the sample in He vapor or liquid are not qualitatively different ( P = 0.9 for Q > 0.02 A ˚ − 1 ). For this case use of the liquid He reference provided no further clarity.

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Fig. 5.

Actual sample layer sequence: Ta on Pt on Si. Observed reflectivity (circles), calculated reflectivity (curves) and residuals for the (upper) correct and (bottom) incorrect models for cases of sample in He vapor (black) and liquid He (red). (inset) SLD profiles of the models.

Both models for the soft matter layers on Si required a ∼20 Å thick SiO₂ layer between the Si substrate and soft matter film. For the Si/SU-8 sample, the two curves are qualitatively different ( P = 0.48 for Q > 0.02 A ˚ − 1 ). However, the curves are little more than displaced along Q by a factor corresponding to the SLD of liquid He. Thus, for this case use of the liquid He reference provided no further clarity.

This last case illustrates an opportunity for a liquid-He reference to provide unique information. Because the SLDs of PMMA and liquid He are similar, when the liquid is present, the fringes due to interference of the wave function from either side of the PMMA film are greatly suppressed (Fig. 7 upper). In this case the liquid approximately negates the contrast between vapor (or vacuum) and PMMA. This observation is evidence for the SLD of PMMA being a small positive number like that of liquid He. The observation is not surprising as the density and chemistry of PMMA are well known, so the SLD is readily computed. However, the ability of liquid He to contrast match a film of unknown chemistry (perhaps because of interdiffusion between components due to processing, aging, etc.) may be useful. For the case of Si/PMMA, the data from the sample taken in vapor and liquid phases of He are very different ( P = 0.12 for Q > 0.02 A ˚ − 1 ). In this case, use of a liquid He reference (environment) provided additional clarity to reject a solution attributing a negative SLD to the polymer film).

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Fig. 6.

Sample layer sequence: SU-8 on Si. Observed reflectivity (circles), calculated reflectivity (curves) and residuals for the correct model for cases of sample in He vapor (black) and liquid He (red). (inset) SLD profiles of the models. Note, a thin SiO₂ layer was required to fit the Si/SU-8 data.

We have shown that remarkably similar reflectivity curves can be produced from dramatically different scattering length density depth profiles. In some cases, the difference between the calculated reflectivity and observed reflectivity are subtle. For these cases, if complementary data were not available, then there could be an opportunity for error. For measurements of samples below 4 K, a simple way to acquire complementary data is to make two measurements – measurements of the sample immersed in He vapor and liquid He. If the two measurements are qualitatively similar, then both can be used in model fitting. However, if the two measurements yield qualitatively different data, then potentially unique information can be extracted from a model that accounts for the origin of the difference(s). Of the six samples studied, the approach yielded compelling benefit for one case. We expect the influence of liquid He on the reflectivity to be most pronounced when the SLD of the film (or a layer of the film) and/or substrate is small. Examples of systems that may benefit from the modest change of contrast afforded by He liquid and vapor include films with organic (mostly hydrogenous) materials (the layers are often ones having small scattering length density), so the change about the SLD of an organic film when the He is a vapor or liquid may be significant. Some hard matter systems may also benefit. For example, the scattering length density of a magnetic layer may be small owing to the orientation of one of the two neutron beam polarizations with the magnetism, or the small change of contrast from expulsion of magnetic field by a superconductor (Meissner effect) – potentially of value to studies of superconducting organic films [19].

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Fig. 7.

Sample layer sequence: PMMA on Si. Observed reflectivity (circles), calculated reflectivity (curves) and residuals for a model having a reasonable small positive SLD for PMMA. Note the reduced fringe amplitude for the case of PMMA immersed in liquid He as compared to PMMA in He vapor. Note, a thin SiO₂ layer was required to fit the PMMA/Si data.