Neutron Reflectometry

1. Neutron Reflectometry • measurement of the intensity reflected by a planar surface and/or interfaces • neutrons at thermal energies incident on a surface at a grazing angle of less than 3° • at these small angles, the potential for scattering approximated by a continuous valuecalled the scattering length density (SLD) • sensitive to the difference of the refractive index (or contrast) across surfaces and interfaces => • near surface structure of materials NG7 HORIZONTAL NEUTRON REFLECTOMETER(NIST) The data measured as intensity versus wave-vector transfer, Qz or Q⊥ (difference between the final (kf) and initial (ki) wave-vectors elastic scattering assumed: |kf|=|ki| When θi = θf, specular scattering: used to determine the structure of the material in the z-direction (perpendicular to the surface)

2. Specular Reflectivity (θ = θi=θf) Qz= 4 π sin θ/λ Above the critical angle θcfor total reflection, the datashow finite-size fringes whose separation are inversely related to the film layer thickness After subtraction of the off-specular background, these data can be fit (or inverted) toobtain a real-space profile of the scattering length density as a function of depth.

3. Off-Specular Reflectivity information about the lengthscale of in-plane structural correlations For transverse-Qxscans (rockingcurve), 2θ is held constant while θiand θfare varied equally inopposite directions (θi + θf = const). Typically: a narrow specular peak, evident atQx=0, can be separated from the underlying diffuse scattering which is broad. Thewidth of the diffuse peak is indirectly related to the inverse of the coherence length ξ of the in-plane roughness. Interpretation: difficult => the study of diffuse scattering from rough surfaces has not made much headway. The theory (Distorted Wave Born Approximation) works in some cases only. => Not discussed in this introductory course

4. Goal of reflectivity measurements: to infer a density profile perpendicular to a flat interface • • In general the results are not unique, but independent knowledge of the system often makes them very reliable • • Frequently, layer models are used to fit the data • • Advantages of neutrons include: • – Contrast variation (using H and D, for example) • – Low absorption – probe buried interfaces, solid/liquid interfaces etc • – Non-destructive • – Sensitive to magnetism • – Thickness length scale 10 – 5000 Å

5. Three basic features of reflectivity data (specular) 1) the critical wave-vector transfer. Neutrons are totally reflected below. Given by SLD (for a non-uniform layer: roughly a function of the average SLD). Important: Below the critical angle, neutrons are perfectly reflected from a smooth surface – This is NOT weak scattering and the Born approximation is not applicable to this case (neither near the critical angle) The position of this transition to total reflection yields information about the average SLD of the material. The reflectivity away from the total reflection angle holds information about the change in scattering length density with depth. It can be analyzed to determine a film's total thickness, material composition, periodicity, and even roughness. 2) second feature: the decrease in reflectivity with Qz which, for a smooth sample, becomes proportional to Qz-4. If the surface is not smooth, faster decrease is observed. 3) A thin film can also show oscillations around the continuously decreasing reflectivity; the result of an interference effect between the air/film and film/substrate interfaces. The amplitude proportional to the SLD difference between the film and substrate (SLD contrast). An estimate of the film's thickness given by the oscillation period.

6. Specular reflectivity • The quantity measured in a neutron reflectometry experiment: the intensity reflected from the surface • To calculate the reflectivity of an interface: the time-independent Schrödinger equation • a solution for the wave function, Ψ, representing the neutron wave inside and outside of the reflecting sample.

7. Theory for perfect surface The neutron obeys Schrodinger's equation : The average potential inside the medium is: Then in vacuo: where k0 is neutron wavevector in vacuo and similarly k is the wavevector in a material Since k/k0 = n = refractive index (definition), and since ρis very small (~10-6Å-2 ): Since generally n<1, neutrons are externally reflected from most materials. The surface cannot change the neutron velocity parallel to the surface => neutrons obey Snell's Law: Then =>the critical value of k0z for total external reflection is

8. Theory for perfect surface Continuity of ψ and derivative of ψat z = 0 => and ↓ component perpendicular ↓ to the surface => reflectance => reflectivity

9. perfect surface neutron beam reflecting from a perfectly smooth silicon substrate (surrounded by air). The neutron scattering length density for Si is ρSi = 2.07 x10-6 Å-2 reflectivity solid curve: the calculated reflectivity for the interface (a) dashed curve: a reflectivity curve calculated using the Born approximation The dynamical calculation in the region of Q⊥~ 0.1 Å-1 similar to that obtained by the Born approximation (kinematical case) In the large Q⊥regime, the decay of the curve scales as Q⊥-4 (Fresnel decay)

10. Thin layer More interesting and realistic cases involve reflection from stratified media: thin layer on top of the substrate => interference fringes reflectivity red curve: surface of material with ρ=4.10-6Å-2 green curve: added thin layer with larger ρ The fringe spacing at large k0z is ~ p/t (a 250 Å film used) Ability to measure layer thickness with high precision (~3%) The critical edge for neutron reflectivityoften determined by the substrate andnot the thin film owing to the fact that aneutron beam is a highly penetrating

15. • diffuse scattering is caused by surface roughness or inhomogeneities in the reflecting medium • a smooth surface reflects radiation in a single (specular) direction • a rough surface scatters in various directions • specular scattering is damped by surface roughness – treat as graded interface. For a single surface with r.m.s roughness s:

16. Listing the information yielded by the measurements

17. Polarized neutron reflectometry tool to investigate the magnetization profile near the surfaces of crystals, thin films and multilayers. • applied to important problems such as • the influence of frozen or pinned magnetization on the origin of exchange bias, • the influence of exchange coupling on magnetic domain structures, • the identification of spatially inhomogeneous magnetism in nanostructured systems.

21. Polarized neutron reflectometry Applications n Multilayers n Non colinear magnetism n Interface magnetism Allows the study of the magnetic configuration of a multilayer system: access to the magnetisation amplitude and direction in each layer. n Determination of in-depth magnetic profiles n Absolute measurement of the magnetic moment in μB per f.u. (sum of the spin and orbital moment) n But sensitivity only to the in-plane moment. n Resolution of the order of 0.1μB (better on simple systems) n No sensitivity to the substrate para/dia-magnetism. n No absorption, no phenomenological parameter, absolute normalisation.

22. Neutron Reflectivity Links http://www.ncnr.nist.gov/instruments/ng1refl/Fitz.pdf http://www.mrl.ucsb.edu/~pynn/Lecture_4_Reflectivity.pdf http://pathfinder.neutron-eu.net/idb/methods/reflectometry http://neutronreflectivity.neutron-eu.net/main/Lectures http://www.ncnr.nist.gov/programs/reflect/index.html http://www.ncnr.nist.gov/programs/reflect/NR_article/index.html http://www.ncnr.nist.gov/programs/reflect/measurements/reflweb1.pdf