Handbook of Modern Coating Technologies

Neutron reflectivity for the investigation of coatings and functional layers

• Reflection and refraction of neutrons at interfaces

We discuss specular neutron reflectivity in nonpolarized and later in polarized mode including polarization analysis. In the second part, of chapter one, so-called off-specular scattering provid­ing information on the roughness and lateral correlations of interfaces will be introduced. Finally, we will discuss grazing incidence scattering techniques providing lateral information on shorter length scales.

• Reflection from interfaces

In the section we describe how scattering length density profiles are extracted form specular neutron reflection experiments.

• Semiinfinite interfaces

Fig. 4—1 presents the scattering geometry for a typical reflectivity experiment. An incoming neutron beam hits an interface under a comparatively small incident angle 6_; and is fully or partly reflected at an angle 0f. Above the angle of total external reflection part of the intensity is transmitted (refracted) beyond the interface under an angle 0_t. First we discuss the specu­lar reflected intensity for which 6f = 6_; holds. In analogy to classical optics the wave ampli­tude of the reflected and refracted waves can be calculated by introducing an index of refraction n:

k_t    cos6_;

k_; cos6_t

Handbook of Modern Coating Technologies. DOI: https://doi.org/10.1016/B978-0-444-63239-5.00004-4

© 2021 Elsevier B.V. All rights reserved.

Here k, = 2n/A symbolizes the wave vector of the incident neutron beam in air and k_t the wave vector inside the material. A is the wavelength of the neutrons. By introducing the scattering potential for neutrons the index of refraction can be written as

with N being the isotope number density and b being the coherent nuclear scattering length. The scattering length density (SLD) is defined as the product of the number density and the scattering length Nb. For most materials the SLD is positive and as a result n < 1. Due to the relatively weak interaction of neutrons with matter the deviation from n = 1 is small, typically on the order of 10^-5. As an example silicon has a SLD of 207 pm^-2 and a resulting refractive index of n = 0.999998.

Eq. (4.1) does not have a solution as long as cos0 > n and we can define a critical angle of total external reflection cos6_c = n. For incident angles smaller than the critical angle e_c all the intensity is totally externally reflected and only an evanescent wave can penetrate into the first few nanometer beyond the interface. Since typically the interaction potential is very small and as a result the refractive index being very close to 1 the square root in Eq. (4.2) and the cosine in Eq. (4.1) can be Taylor expanded. As a result the critical angle of total external reflection is rewritten as:

Eq. (4.3) contains the incident beam angle and the wavelength and is dependent on the setting of the instrument. To generalize the equation the momentum transfer is introduced as the difference between final and initial neutron wavevectors:

!     !       4П

Qz = \k_f - k,I = — sine

Here we call the momentum transfer Q_z since for specular reflectivity with the incident and exit angle being equal the vector Q is always parallel to the normal of the interface. Combining this equation with the critical angle of total external reflection results in a critical momentum transfer:

4n            /----

Q_c = — sinO_c ^ 4 nNb                                           (4.5)

A

Here we have used the fact that e_c is small and Taylor expanded the sinus. Taking the above example of silicon we calculate a critical momentum transfer for the air—silicon interface of

1. 1 nm^_1, or going back to wavelength and angle, for a 0.5 nm neutron wavelength a critical angle of 0.228 degree. Total external reflection is only observed at the interface with a medium with smaller Therefore a neutron wave in air (n « 1) is totally reflected at a Si surface (n_Si < 1) but not, for example, at an air/vanadium interface (n_V > 1). For momentum transfers larger than the critical momentum transfer Q_z the reflectivity is described by the Fresnel equation:

sind — nsindf sind 1 nsine_t

For larger Q_z the reflectivity drops proportional to Q^—4. Regarding the steep drop off in intensity reflectivity curves are often plotted on a logarithmic scale. Fig. 4—2 depicts the reflectivity for a silicon wafer with no roughness. The intensity is totally externally reflected up to Qc equal to 0.1 nm^—1 and then drops steeply proportional to Q^—4 at larger Q values. For illustration the SLD profile is reproduced in the upper right panel as well.

• Layered structures

Till now we have considered one single interface. For layered structures the Fresnel Eq. (4.6) has to be applied at every single interface to calculate the wave amplitude in the respective

layers. In 1954 Parratt [1] presented a recursion algorithm to solve this problem analytically. To extract quantitative values from reflectivity curves a wide variety of fitting algorithms has been developed basing on the approach proposed by Parratt or the optical transfer matrix method [2].

As an example Fig. 4—3 represents the reflectivity from an oxidized silicon wafer with an oxide layer of 25 nm thickness and no roughness. The corresponding SLD profile is shown in the upper right panel. The incident neutron beam is fully or partially reflected at both the air—silicon oxide (SiO₂) and the SiO₂— silicon interfaces. The reflected waves interfere with each other and as a result a modulation of the reflected intensity is detected. The angles and wavelength for constructive interference can be calculated from Bragg's equation:

n\ = 2dfsindi/f                                                                  (4.7)

with d_f the thickness of the layer. Introducing the momentum transfer the difference between minima and maxima can be calculated from:

.        2n

Д Q = _d                                                                           (4.8)

^df

Due to the refraction effects at each interface this equation holds strictly speaking only for Q values much larger (at least three times) than the critical momentum transfer. For smaller values the optical correction Q‘_z = ^Q2 — Q2, with QZ the momentum transfer inside the film, has to be applied. The oscillations visible in Fig. 4—3 are so-called Kiessig fringes, named after Kiessig [3], who first detected them when doing X-ray reflectivity (XRR) measurements. The second observable, in addition to their distance in Q, regarding the Kiessig fringes is their height in intensity. This height is proportional to the square of the contrast in SLD between the layers, so essentially the larger the contrast the more pronounced the oscillations.

For stacks of layers the reflected waves from each interface interfere and result in addi­tional beating of the oscillations or if a stack of bilayers, of the same thickness is measured result in sharper Bragg reflections similar to crystal Bragg peaks.

• Blurred interfaces

In a next step we consider blurred interfaces resulting in a smearing of the step in the SLD. It is important to note that specular reflectivity measurements cannot distinguish between interface roughness or interdiffusion, since only the SLD profile averaged over the coherence volume of the neutron beam [4] and extracted along the surface normal is probed. Typically, the coherence volume of the neutron beam in the plane of the interface and the scattering plane is relatively large, on the order of 100 pm. For the case of Gaussian roughness or inter­diffusion the SLD profile across an interface can be described by an error function and the reflectivity can be calculated analytically. The final result is an exponential damping:

R(Qz) = Rf (Qz)e^2Q2a2                                     (4.9)

here R_f is the Fresnel reflectivity from a flat interface and a is the roughness parameter or width of the error function. The damping of the reflected intensity in Eq. (4.9) is similar to the Debye—Waller factor in ordinary diffraction.

Fig. 4—4 depicts the reflectivity for a silicon wafer with a Gaussian roughness of 1 nm. For comparison the Fresnel reflectivity for the flat wafer is shown as dotted line and the SLD pro­file is reproduced in the upper right panel. The steeper decay of the specular reflected intensi­ties for larger Q values is well visible. It is important to note that the steeper decay is already significant even though we only assume a roughness of 1 nm. This puts limits to the measure­ments of rough surfaces with NR. Typically, the dynamic range of neutron instruments is on the order of 10^_6 to 10^_8. Regarding this fact the interface roughness should be in the nanome­ter range or even better below. Another important remark with respect to the fitting of data is that due to the nature of the error function the roughness extends relatively far from the inter­face. For this reason one needs to be careful for layers where the roughness is close to the total layer thickness, since then, depending on the fitting routine, unphysical artifacts may appear.

For more details on the technique of reflectivity measurements we refer to literature, for example [5].

• Polarized neutrons

Neutrons have a spin and as a result a magnetic moment. This moment interacts, via dipolar interaction, with the magnetic induction in materials. A powerful tool to investigate the mag­netic properties of materials and in particular of spintronics and functional materials [6,7] is the use of polarized neutrons, which has undergone a rapid development during the past decades [8]. The polarization can be achieved by special devices, such as supermirrors [9], Heusler monochromators [10], or polarized ³He cells [11,12]. The polarization device will field one spin component and result in a loss of at least 50% of the neutrons. The remaining beam is then polarized along one quantization axis which is defined by a magnetic guide field H along the instrument. If needed the polarization can be flipped by Radio-frequqncy (RF) [13] or Mezei [14] spin flippers with the resulting polarization being antiparallel to the guide field. Well- polarized neutron beams have a polarization of better than 99%. The scattering geometry for polarized neutron reflectivity experiments is shown in Fig. 4—5. B symbolizes the magnetic induction in the film Д⁺ and the magnetic moment of the neutron and H the external mag­netic field. p is the angle between H and B. The neutron is insensitive to a magnetic induction parallel to Q. As a result we will only consider in-plane magnetization from now on.

In a first step let us assume that the magnetic induction in the film is parallel to the external magnetic field, p = 0. For this case the nuclear interaction potential of the neutron with the sample becomes modified by the magnetic part, by the Zeemann energy:

where and m denote the magnetic moment and mass of the neutron, respectively, h is Planck's constant over 2n and b_mag the magnetic scattering length, which is b_mag = 2.695 fm per p_B (Bohr magneton) per atom. As seen from Eq. (4.10) the magnetic scattering length does depend on the angle p. However, the total scattering length changes and this has to be taken into account in Eq. (4.2). The resulting refractive indices become:

(4.11)

The " + ” and " — ” means that the neutron spin is parallel and antiparallel to H, since the magnetic moment of the neutron is antiparallel to its spin. As illustration of magnetic scattering Fig. 4—6 depicts simulated reflectivity curves for a flat 50-nm Fe film in saturation on top of a silicon substrate. The reflectivity for neutrons polarized parallel and antiparallel to the magnetic induction in the film is plotted as solid and dashed lines, respectively. The respective SLD profiles are shown in the upper right panel. Iron has a large magnetic moment of almost two Bor magnetons per atom resulting in a magnetic SLD of 6 500 pm^—2. As seen from the figure the two critical wavevectors for total external reflection Q1 and Q^— are clearly separated. In addition, the Kiessig fringes are much more pronounced for the neutron polarization state with the larger SLD contrast as discussed in Section 4.1.1.2. From the normalized difference of the two reflectivities, the spin asymmetry:

the magnetic induction in the film can be extracted in absolute units.

Let us now consider a magnetic induction with a finite angle p with respect to H .In this case the neutrons experience a magnetic induction, which is perpendicular to their quantiza­tion axis. This results in a partial depolarization of the beam. If now after the sample the antiparallel spin state is analyzed so-called spin flip (SF) scattering can be detected. As a result four reflectivities R¹¹, R^——, R^1—, R^—1 can be extracted. From the relative ratio of the four intensities not only the magnitude of the magnetic induction in the sample but also its direction in the plane of the sample surface can be determined. The spin asymmetry mea­sured at a certain Q value only depends on the cosine of the angle between the magnetic
induction in the magnetic layer and the neutron polarization direction SA ~cosip, whereas the spin flip signal is proportional to the square of the sine of that respective angle,

SF ~sin²ip.

• Off-specular scattering

Till now we have discussed specular reflectivity where the incident and exit angles are exactly the same. Fig. 4—7 shows the scattering geometry for off-specular scattering, with a difference in incident and exiting beam angle. As a result from the difference in incident beam angle O, and reflected beam angle Of the momentum transfer has a component parallel to the interface Q_x. The components perpendicular and parallel to the interface can be calculated as follows:

2n

Qx 5 — cosOf - cosOi)                                        (4.13)

2n

Q_z 5     sinOf 1 sinOj)                                                    (4.14)

A

Since now a component in the plane of the interface exists off-specular scattering is sensi­tive to fluctuations in the SLD along the interface. As a result interdiffusion and roughness become distinguishable, since interdiffusion has no lateral fluctuations and such results in no off-specular scattering, whereas roughness has.

Off-specular reflectivity experiments can be analyzed by using the distorted wave Born approximation [15,16]. This approach is a first order distortion theory and requires the SLD profile along the surface normal extracted from specular reflectivity. In the second step in­plane fluctuations are added to the profile and the off-specular scattering is calculated.

Fig. 4—8 depicts off-specular neutron reflectivity data from a NiTi superlattice (SL). Such type of coatings are used as neutron supermirrors. For such mirrors bilayers of different thicknesses are grown on glass substrates. Due to the different thickness of the bilayers a

series of Bragg reflections is generated, which shifts the apparent critical momentum transfer to larger Q values and results in enhanced neutron guiding properties. However, the sample studied here is a NiTi SL with constant bilayer thickness [17]. The intensity in Fig. 4—8 is plotted over ki — kf and ki + kf instead of Q_x and Q_z. The reason is that in the Q_x — Q_z repre­sentation the map becomes contracted close to the critical edge making features in this region difficult to see. The specular reflected intensity is found along the line with ki — kf = 0. On the specular line clearly SL Bragg peaks are visible resulting from the bilayer repeat distance. The horizontal stripe of intensity along the first order peak results from interface roughness between the layers. The refracted beam is visible for k_i — kf > 0 and ki + kf close to zero. At the horizon, where either the incident beam angle or the exiting beam angle reaches the critical angle of total external reflection the Bragg sheet gets dis­torted due to the optical effects. The scattering in this region is called Yoneda scattering and is enhanced due to an constructive interference of the incoming and outgoing wave field [18]. Due to resonant diffuse scattering [19] resulting from the correlated roughness of the layers additional Bragg like reflections are visible along the Bragg sheets in the off-specular regime.

• Grazing incidence small-angle scattering

Finally, we consider a scattering geometry with an outgoing momentum transfer out of the plane defined by the incident wave vector and the surface normal. Scattering along this

direction is generally called grazing incidence small-angle scattering (GISANS). Fig. 4—9 shows the scattering geometry for GISANS. The components perpendicular and parallel to the interface can be calculated as follows:

Qx = у (cosdf - cose і )                                         (4.15)

2n

Qy = — sine_p                                               (4.16)

2n

Q_z =   (sine, 1 sinef)                                         (4.17)

A

Similar to the off-specular scattering the vector of momentum transfer Q has now a com­ponent parallel to the plane of the interface. However, as seen from Eq. (4.15) the magnitude of the momentum transfer along the different directions is different. Typically, along the y- and z-direction Q vectors on the order of 1 nm^-1 are probed, whereas along the x-direction Q typically is around 10^-2 nm^-1. As a result the accessible length scales along the respective directions are on the order of hundreds of nanometers along y and z and tenth of microme­ter along x. To elucidate the interference between SANS, off-specular and NR, Fig. 4—10 shows a comparison between the GISANS scattering pattern (left panel) taken at an incident angle of 0.3 degree, for a crystal formed by polymer micelles and a map where the reflectivity data, including the off-specular scattering, is plotted over Qx and Qz (right panel) [20]. The only difference between the measurements was that the resolution in Q_y-direction was increased for the GISANS measurement. The black line, plotted in both panels of Fig. 4—10, indicates where the two hyper surfaces intersect. For the relaxed resolution the double peak showing up in the GISANS data at Q_y Ф 0 is detected in the scattering plane. A more detailed and pedagogical description of off-specular and grazing incidence scattering was recently published as lecture notes of the French-Swedish Winterschool on Neutron Scattering: Applications to Soft Matter [21].

к

^Гї’

О

оГ

• Instrumentation

To do a reflectometry experiment the reflected intensity from a surface or interface needs to be measured for different Q values. From Eq. (4.4) it can be seen that therefore the incident and exit angle of the neutron beam with respect to the surface and the wavelength has to be defined. This can be achieved in two ways. The first possibility is to keep a constant wavelength and change the incident and exit angle for each Q value. The method is used on monochro­matic neutron reflectometers. Alternatively, for one setting of the angles the wavelength can be varied. Since the neutron carries a mass m_n the wavelength A can be measured from the flight time t_TOF a neutron needs from the source to the detector (distance D_TOf), as long as the scat­tering is elastic, from de Broglie's relationship:

tTOFh

DroFmn

with Planck's constant h. Instruments using this mode of operation are called time-of-flight (TOF) reflectometers and use a pulsed neutron beam with a large band width. Worldwide currently there are at least 59 different neutron reflectometers, 31 of which are operating in TOF mode [22].

Fig. 4—11 shows a schematic of the layout of the neutron reflectometer D17 at the Institute Laue-Langevin (ILL, Grenoble, France) [24]. This instrument has an interesting design since it can be operated in both monochromatic and TOF mode. The respective instrumental configuration is shown in the top and bottom panels of Fig. 4—11.

In monochromatic mode the instrument is rotated by 4 degrees around the monochromator position as shown in Fig. 4—11 (top). Two monochromators are available both made up of SLs leading to a reflected signal similar to the one in Fig. 4—8 with a strong Bragg peak, which is

used to select a monochromatic beam of 5.5 A wavelength at the given reflection angle out of the broad wavelength band coming from the ILL (Grenoble, France) cold source. The nonpolar­izing monochromator is made of a Ni/Ti superlattice as the one in Fig. 4—8, whereas the spin- polarizing monochromator consists of a Si/Fe superlattice embedded in a permanent magnet assembly which leads to a strong asymmetry in the reflectivity of the two neutron spin states. Therefore for the latter monochromator, not only the wavelength is filtered by choosing a well- defined take-off angle but also the spin state parallel to the magnetic field at the monochroma­tor is reflected more than hundred times stronger than the antiparallel spin state leading to a polarization of 98.5% of the monochromatic beam.

As shown in Fig. 4—8 the reflectivity at the critical reflection region is as strong as the Bragg peak reflectivity, which means that the wavelength spectrum coming from the mono­chromator might be contaminated by long wavelengths which are totally reflected from the superlattice and/or substrate in case these are not contrast-matched to the upper medium.

Especially for instruments with a low take-off angle of the monochromator like D17 this is a major issue. Therefor a long-wavelength filter is installed after the monochromator which reflects long wavelengths out of the beam path by using total reflection from stacks of Ni—coated silicon slabs. This filter, however, only works for a limited beam divergence and thus needs an additional collimator, which is installed before the monochromator.

In TOF mode the aforementioned monochromator optics are taken out of the beam and instead a chopper pair is used to pulse the beam [25]. By synchronizing the chopper window opening with the detector acquisition the travel time of the neutrons can be measured and subsequently converted to wavelength [26]. The maximum distance between the choppers and the detector is around 7.7 m on D17. According to Eq. (4.18) this leads to a TOF of around 58 ms for 30 A neutrons, the slowest neutrons used on D17. To avoid an overlap of adjacent pulses the chopper speed is therefore limited to 1000 rpm, corresponding to a pulse frequency of 60 ms. Neutrons with a longer wavelength than 30 A are reflected out of the beam by a Ni—coated silicon wafer, called frame-overlap mirror, working with the same principle as the long-wavelength filter used in monochromatic mode.

In case a spin-polarized beam is needed a polarizer made of a stack of silicon slabs cov­ered by a 4000-nm Si/Fe supermirror coating on top of a 600-nm Gd layer embedded in a permanent magnet assembly is used. Here the different positions in Q of the critical angle of quasi total reflection from the supermirror for the two polarization directions is used to reflect only one spin state, whereas the other spin state is absorbed in the Gd layer. This leads to a polarization of the beam of >99% over a wavelength band of 4—20 A.

Regardless of the mode of operation two collimating slits, 3.6 m apart, define the beam size and horizontal divergence, whereas the full vertical divergence from the neutron guide is focused on the sample by a focusing guide section between the collimating slits as can be seen in Fig. 4—11. If polarization analysis is needed the specularly reflected beam can be analyzed by a supermirror analyzer working by the same principle as the aforementioned polarizer or, in case the full detector image has to be analyzed (off-specular scattering), a polarized ³He—filled cell can be introduced between the sample and the detector (see Fig. 4—11) which absorbs neutrons with their spin antiparallel to the ³He polarization. The manipulation of the neutron spin before and/or after the sample can be realized by using radiofrequency spin flippers [27].

The data is subsequently collected by using a time-resolved two-dimensional detector, which can be placed 1—3.1 m after the sample. As the neutron beam is usually highly colli­mated perpendicular to the surface under investigation and divergent parallel to it the scat­tering pattern is integrated over the parallel direction. Small influences in the reactor power, which are usually below 5%, are corrected by using a low efficiency monitor which is placed before the choppers as indicated in Fig. 4—11.