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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 value

called the scattering length density (SLD)

•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 value

called the scattering length density (SLD)

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)

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)

NG7 HORIZONTAL NEUTRON REFLECTOMETER (NIST)

•sensitive to the difference of the refractive index (or contrast) across surfaces and interfaces =>

•near surface structure of materials

•sensitive to the difference of the refractive index (or contrast) across surfaces and interfaces =>

•near surface structure of materials

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Specular Reflectivity (θ = θi = θf)

Above the critical angle θc for total reflection, the data show 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) to obtain a real-space profile of the scattering length density as a function of depth.

Above the critical angle θc for total reflection, the data show 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) to obtain a real-space profile of the scattering length density as a function of depth.

Qz = 4 π sin θ/λQz = 4 π sin θ/λ

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Off-Specular Reflectivity information about the length scale of in-plane structural correlations

information about the length scale of in-plane structural correlations

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

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

Typically: a narrow specular peak, evident at Qx=0, can be separated from the underlying diffuse scattering which is broad.

The width of the diffuse peak is indirectly related to the inverse of the coherence length ξ

of the in-plane roughness.

Typically: a narrow specular peak, evident at Qx=0, can be separated from the underlying diffuse scattering which is broad.

The width of the diffuse peak is indirectly related to the inverse of the coherence length ξ

of the in-plane roughness.

For transverse-Qx scans (rocking curve), 2θ is held constant while θi and θf are varied equally in opposite directions (θi + θf = const).

For transverse-Qx scans (rocking curve), 2θ is held constant while θi and θf are varied equally in opposite directions (θi + θf = const).

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Goal of reflectivity measurements: to infer adensity 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

• 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

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Three basic features of reflectivity data (specular)

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.

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.

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)

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)

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.

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.

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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.

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.

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Theory for perfect surfaceThe 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 ρ is very small (~10-6 Å-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

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 ρ is very small (~10-6 Å-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

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Theory for perfect surface

Continuity of ψ and derivative of ψ at z = 0 =>

Continuity of ψ and derivative of ψ at z = 0 =>

and

↓ component perpendicular ↓to the surface

=> reflectance=> reflectance

=> reflectivity=> reflectivity2

41

IzIzTz kkk

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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

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

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)

reflectivityreflectivity

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Thin layerMore interesting and realistic cases involve reflection

from stratified media:thin layer on top of the substrate => interference fringes

More interesting and realistic cases involve reflection from stratified media:

thin layer on top of the substrate => interference fringes

red curve: surface of material with ρ=4.10-6 Å-2

green curve: added thin layer with larger ρ

The fringe spacing at large k0z is ~ /t (a 250 Å film used)

Ability to measure layer thickness with high precision (~3%)

reflectivityreflectivity

nzmz

nzmzmn kk

kkr

20

0

41

z

jzjz k

kk

The critical edge for neutron reflectivity often determined by the

substrate and not the thin film owing to the fact that a neutron

beam is a highly penetrating

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• specular scattering is damped by surface roughness – treat as graded interface. For a single surface with r.m.s roughness :

• specular scattering is damped by surface roughness – treat as graded interface. For a single surface with r.m.s roughness :

• 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

• 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

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Listing the information yielded by the measurements

Measurement feature

Information obtained from a sample of ≈cm2 size

Position of critical edge, Qc

Nuclear (chemical) composition of the neutron-optically thick part of the sample, often the substrate.

Intensity for Q < Qc

Unit reflectivity provides a means of normalization to an absolute scale.

Periodicity of the fringes

Provides measurement of layer thickness. Thickness measurement with uncertainty of 3% is routinely achieved. Thickness measurement to less than 1 nm can be achieved.

Amplitude of the fringes

Nuclear (chemical) contrast across an interface.

Attenuation of the reflectivity

Roughness of an interface(s) or diffusion across an interface(s). Attenuation of the reflectivity provide usually establishes a lower limit (typically of order 1-2 nm) of the sensitivity of reflectometry to detect thin layers.

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Polarized neutron reflectometry

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.

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.

tool to investigate the magnetization profile near the surfaces of crystals, thin films and multilayers.

tool to investigate the magnetization profile near the surfaces of crystals, thin films and multilayers.

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Polarized neutron reflectometry

ApplicationsMultilayersNon colinear magnetismInterface magnetismAllows the study of the magnetic configuration of amultilayer system:access to the magnetisation amplitude and direction ineach layer.Determination of in-depth magnetic profilesAbsolute measurement of the magnetic moment in

μB per f.u. (sum of the spin and orbital moment)But sensitivity only to the in-plane moment.Resolution of the order of 0.1μB (better on simple

systems)No sensitivity to the substrate para/dia-magnetism.No absorption, no phenomenological parameter,

absolutenormalisation.

ApplicationsMultilayersNon colinear magnetismInterface magnetismAllows the study of the magnetic configuration of amultilayer system:access to the magnetisation amplitude and direction ineach layer.Determination of in-depth magnetic profilesAbsolute measurement of the magnetic moment in

μB per f.u. (sum of the spin and orbital moment)But sensitivity only to the in-plane moment.Resolution of the order of 0.1μB (better on simple

systems)No sensitivity to the substrate para/dia-magnetism.No absorption, no phenomenological parameter,

absolutenormalisation.

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Neutron Reflectivity Links

http://www.ncnr.nist.gov/instruments/ng1refl/Fitz.pdfhttp://www.mrl.ucsb.edu/~pynn/Lecture_4_Reflectivity.pdf

http://pathfinder.neutron-eu.net/idb/methods/reflectometryhttp://neutronreflectivity.neutron-eu.net/main/Lectures

http://www.ncnr.nist.gov/programs/reflect/index.htmlhttp://www.ncnr.nist.gov/programs/reflect/NR_article/index.htmlhttp://www.ncnr.nist.gov/programs/reflect/measurements/reflweb1.pdf

http://www.ncnr.nist.gov/instruments/ng1refl/Fitz.pdfhttp://www.mrl.ucsb.edu/~pynn/Lecture_4_Reflectivity.pdf

http://pathfinder.neutron-eu.net/idb/methods/reflectometryhttp://neutronreflectivity.neutron-eu.net/main/Lectures

http://www.ncnr.nist.gov/programs/reflect/index.htmlhttp://www.ncnr.nist.gov/programs/reflect/NR_article/index.htmlhttp://www.ncnr.nist.gov/programs/reflect/measurements/reflweb1.pdf