Neutron Reflectometry

Transcription

1 Neutron Reflectometry Roger Pynn Indiana University and the Spallation Neutron Source

2 Surface Reflection Is Very Different From Most Neutron Scattering Normally, neutrons are very WEAKLY scattered One can ignore doubly scattered neutrons This approximation is called the Born Approximation Below an angle of incidence called the critical angle, neutrons are perfectly reflected from a smooth surface This is NOT wea scattering and the Born Approximation is not applicable to this case Specular reflection is used: In neutron guides In multilayer monochromators and polarizers To probe surface and interface structure in layered systems

3 Why Use Neutron Reflectivity? Neutrons are reflected from most materials at grazing angles If the surface is flat and smooth the reflection is specular Perfect reflection below a critical angle Above the critical angle reflectivity is determined by the variation of scattering length density perpendicular to the surface i.e. we can determine the average density profile normal to the surface of a film on the surface Diffuse scattering gives information on surface & interface roughness Images courtesy of M. Tolan & T. Salditt

4 Various forms of small angle neutron reflection Viewgraph from M. R. Fitzsimmons

5 Randomly Chosen Examples of Neutron Reflectometry Studies Inter-diffusion of polymers Phase separation in bloc copolymers Amphiphilic molecules at air-water interfaces Effect of shear on films of complex fluids Grafting of polymers to surfaces (mushrooms and brushes) Swelling of films exposed to vapor Magnetic structure of multilayers CMR/GMR films Exchange bias and exchange springs Nuclear polarization in spintronic materials

6 Refractive Index for Neutrons r The nucleus - neutron potential is given by : V ( ) So the average potential inside the medium is : V ρ where ρ is called the nuclear Scattering Length Density (SLD) - the same one we used for SANS πh m πh m r bδ ( ) for a single nucleus. ρ volume i b i h The inetic (and total) energy of neutron in vaccuum is E m Inside the medium the total energy is h Conservation of energy gives m h m h m V V h m πh m ρ or 4πρ Since / n refractive index (by definition), and ρ is very small (~ n -λ ρ / π Since generally n <, neutrons are externally reflected from most materials. -6 A - ) we get :

7 Refractive index : The surface cannot change the neutron velocity parallel to the surface so : cosα i.e. Neutrons obey Snell's Law The critical value of critical For quartz (π / λ)sinα critical Note : ( cosα' ).λ( A) for quartz critical z critical ( / n cosα' i.e n cosα/cosα' Since 4πρ (cos α' sin i.e. sin α' sin α 4πρ or α Only Neutrons With Very Low Velocities Perpendicular to a Surface Are Reflected o α o z.5x critical z n for total external reflection is 3 A - ).λ ( A) for nicel α') z z (cos 4πρ α α sin z α) 4πρ 4πρ α

8 Reflection of Neutrons by a Smooth Surface: Fresnel s Law n -λ ρ/π T T R R I I T R I a a a a a a r r r & at z & of continuity ψ ψ ) ( transmitance : ) ( ) ( reflectance : Tz Iz Iz I T Tz Iz Tz Iz I R a a t a a r

9 Reflectivity We do not measure the reflectance, r, but the reflectivity, R given by: R # of neutrons reflected at Qz r.r* # of incident neutrons i.e., ust as in diffraction, we lose phase information Fresnel Reflectivity Measured and Fresnel reflectivities for water difference is due to surface roughness

10 Penetration Depth In the absence of absorption, the penetration depth becomes infinite at large enough angles Because z is imaginary below the critical edge (recall that z z 4πρ), the transmitted wave is evanescent The penetration depth Λ / Im( ) Around the critical edge, one may tune the penetration depth to probe different depths in the sample

11 Surface Roughness Surface roughness causes diffuse (non-specular) scattering and so reduces the magnitude of the specular reflectivity z z θ θ t The way in which the specular reflection is damped depends on the length scale of the roughness in the surface as well as on the magnitude and distribution of roughness x Note that roughness introduces a SLD profile averaged over the sample surface sparling sea model -- specular from many facets each piece of surface scatters indepedently -- Nevot Croce model R R F e σ Iz t z

12 Fresnel s Law for a Thin Film r( z - z )/( z z ) is Fresnel s law z Evaluate with ρ4. -6 A - gives the red curve with critical wavevector given by z (4πρ) / If we add a thin layer on top of the substrate we get interference fringes & the reflectance is given by: Log(r.r*) r r r r r e e i z i and we measure the reflectivity R r.r* t z t Film thicness t substrate If the film has a higher scattering length density than the substrate we get the green curve (if the film scattering is weaer than the substance, the green curve is below the red one) The fringe spacing at large z is ~ π/t (a 5 A film was used for the figure)

13 One can also thin about Neutron Reflection from a Surface as a -d Problem V(z) π ρ(z) h /m n V(z) substrate -4π ρ(z) Where V(z) is the potential seen by the neutron & ρ(z) is the scattering length density z Film Vacuum

14 Multiple Layers Parratt Iteration (954) The same method of matching wavefunctions and derivatives at interfaces can be used to obtain an expression for the reflectivity of multiple layers z z z z i z i z i e X r e X r e T R X,,,,,,,,,, where z z z z r substrate & unit amplitude incident wave) nothing coming bac from inside (i.e. and Start iteration with T X R N N Image from M. Tolan

15 Dealing with Complex Density Profiles Any SLD depth profile can be chopped into slices The Parratt formalism allows the reflectivity to be calculated A thicness resolution of Å is adequate this corresponds to a value of Q z where the reflectivity has dropped below what neutrons can normally measure Computationally intensive!! Image from M. Tolan

16 Neutron Reflectivity unpolarized x -4 x x -4 x -5 Typical Reflectivities - 5 Å Nb on - Si substrate -3 x -4 x -5-6 [3 Å Nb / 5 Å Fe]* on Si substrate q [Å - ] Si substrate res: x -3 Å -

17 The Goal of Reflectivity Measurements Is to Infer a Density Profile Perpendicular to a Flat Interface In general the results are not unique, but independent nowledge of the system often maes 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 Thicness length scale 5 Å Issues include Generally no unique solution for the SLD profile (use prior nowledge) Large samples (~ cm ) with good scattering contrast are needed

18 Analyzing Reflectivity Data We want to find ρ(z) given a measurement of R(Q z ) This inverse problem is not generally solvable Two methods are used:. Modelling Parameterize ρ(z) and use the Parratt method to calculate R(Q z ) Refine the parameters of ρ(z) BUT there is a family of ρ(z) that produce different r(q z ) but exactly the same R(Q z ): many more ρ(z) that produce similar r(q z ). This non-uniqueness can often be satisfactorily overcome by using additional information about the sample (e.g. nown order of layers). Multiple measurements on the same sample Use two different bacings or frontings for the unnown layers Allows r(q z ) to be calculated R(Q z ) can be inverted to give ρ(z) unless ρ(z) has bound states (unusual)

19 Perils of fitting Reflectivity Model Model Scattering length density [Å - ] 5x -6 4x -6 3x -6 x -6 x -6 Model Model Q [Å - ] Depth into sample [Å] Lac of information about the phase of the reflected wave means that profoundly different scattering length density profiles can produce striingly similar reflectivities. Ambiguities may be resolved with additional information and physical intuition. Sample growers Other techniques, e.g., TEM, X-ray Neutron data of very high quality Well-designed experiments (simulation is a ey tool) D. Sivia et al., J. Appl. Phys. 7, 73 (99).

20 Direct Inversion of Reflectivity Data is Possible* Use different fronting or bacing materials for two measurement of the same unnown film E.g. D O and H O bacings for an unnown film deposited on a quartz substrate or Si & Al O 3 as substrates for the same unnown sample Allows Re(R) to be obtained from two simultaneous equations for R and R Re(R) can be inverted to yield a unique SLD profile Another possibility is to use a magnetic bacing and polarized neutrons * Marza et al Biophys Journal, 79,333 () SiO Si or Al O 3 substrate Unnown film HO or DO

21 Planning a Reflectivity Measurement Simulation of reflectivity profiles using e.g. Parratt is essential Can you see the effect you want to see? What is the best substrate? Which materials should be deuterated? If your sample involves free liquid surface you will need to use a reflectometer with a vertical scattering plane Preparing good (i.e. low surface roughness) samples is ey Beware of large islands Layer thicnesses between Å and 5 Å But don t mix extremes of thicness

22 Reflection of Polarized Neutrons Neutrons are also scattered by variations of B (magnetization) Only components of magnetization perpendicular to Q cause scattering If the magnetization fluctuation that causes the scattering is parallel to the magnetic moment (spin) of a polarized neutron, the neutron spin is not changed by the scattering (non-spin-flip scattering) Potential V Fermi pseudo potential: V nuc,mag V nuc V nuc,mag - Depth z V π h N (b n /-b mag )/m N with b nuc : nuclear scattering length [fm] b mag : magnetic scattering length [fm] ( μ B /Atom >.695 fm) N: number density [at/cm 3 ] m N : neutron mass Spin up neutrons see a high potential. Spin down neutrons see a low potential.

23 Typical Non-Spin-Flip Reflectivities zo unpolarized x -4 x -5-6 Neutron Reflectivity R -3 x -4 x -5-6 z 4π(b/ V) q ± z z zo zo 4π[( b/v ) ± cb] Si substrate - 5 Nb on - Si substrate polarized - - μ N -3 x -4 x -5-6 Neutron Reflectivity R - 5 Fe on - Si substrate -3 x -4 x -5-6 Fe surface H - [3 Nb / 5 Fe]* on Si substrate - -3 x -4 x q [ - ] res: x -3 - Courtesy of F. Klose [3 Nb / 5 Fe]* - on Si substrate - -3 x -4 x q [ - ] res: x -3 -

24

25 Summary so Far Neutron reflectometry can be used to determine scattering length density profiles perpendicular to smooth, flat interfaces PNR (polarized neutron reflectometry) allows vector magnetic profiles to be determined Diffuse scattering gives information about surface roughness What new things are we planning?

26 Components of the Scattering Vector in Grazing Incidence Geometry q x is very small Viewgraph courtesy of Gian Felcher

27 Geometry at grazing incidence α f ϕ Resolve with PSD Resolve with Spin Echo α f tight collimation coarse collimation ϕ y z x α i q x (π/λ)( cosα f -cosα i ) q y (π/λ) sinϕ

28 The Classical Picture of Spin Precession spin polarizer st spin-echo arm sample position nd spin-echo arm B O -B O spin analyser P DIFFERENT PATH LENGTHS FOR THE DIFFERENT TRAJECTORIES! Viewgraph sequence by A. Vorobiev

29 spin polarizer Spin Echo Scattering Angle Measurement (SESAME) No Sample in Beam sample position B O -B O spin analyser.5 P DEPOLARIZED i.5 BEAM! ALL SPINS ARE PARALLEL: FULL POLARIZATION P i P f SPIN-ECHO.5 P f ALL SPINS ARE PARALLEL AGAIN: POLARIZATION IS BACK

30 spin polarizer Spin Echo Scattering Angle Measurement (SESAME) Scattering by the Sample B O -B O spin analyser.5 P i FULL POLARIZATION P i > P f.5 P f DEPOLARIZATION!

31 Spin Echo Scattering Angle Measurement (SESAME) Scattering of a Divergent Beam spin polarizer spin analyser B O -B O.5 P i FULL POLARIZATION the same depolarization P i > P f independently of initial traectory.5 P f DEPOLARIZATION!