X-ray reflectivity and Grazing Incidence Small Angle X-Ray Scattering G. Renaud, CEA-Grenoble, France Département de Recherche Fondamentale sur la Matière Condensée Service de Physique des Matériaux et des Microstructures And ESRF, BM32 beamline grenaud@cea.fr European School on Magnetism New Experimental Aproaches in Magnetism Constantza, Romania, Sept 7-16, 2005
Introduction Nanoparticles , nanowires, thin films and multilayers… have New physical properties (e.g. magnetic, but also electronic, catalytic or photonic) Atomic structure, size, shape & organization Growth conditions & Morphology, temperature … of the substrate surface X-ray complementary to Near Field Microscopy • non destructive - statistical information over mm scale • depth sensitivity, from 20 Å up to several mm • length scale probed : from a few Å to mm • quantitative analysis • following in-time: deposit – annealing – gas adsorption • in situ , in UHV, during growth (and sometimes in real time) • no charge effects : insulating samples (single crystal oxide substrates)
X-Ray Scattering k f k i = − = vector of the reciprocal space q k k G f i hkl Explores Reciprocal Space
Reciprocal Space of nanostructures deposited on a substrate substrate CTR On-site M rod q ┴ α 1/a ┴ diffraction by adsorbate a // A a ┴ A (333) (113) ∆ q // / q // ~ ∆ a // / a // a ┴ S d // a // S (002) (222) ATOMIC STRUCTURE Of NANOPARTICLES (111) (331) Relaxed M (000) q // α ( h 2 + k 2 ) 1/2 α 1/a // Crystal Truncation Rods: (220) interference with adsorbate GISAXS Specular ==> site, d interf . reflectivity z d REGISTRY, i.e. MORPHOLOGY r (z) location of adsorbate at the nm – 100 nm scale D
Grazing Incidence Small Angle Grazing Incidence X-ray X-ray Scattering ( GISAXS) Scattering GIXS (or GID) and X-R Reflectivity (XRR) Structure @ atomic scale Morphology @ nanometer scale • Structure, composition • Shape (facets, equilibrium shape) • Dimensions • Epitaxial relashionships • Size distributions • Relaxation • Organization – Coherent • Growth mode – Incoherent (dislocations) • Density profile • Registry / substrate lattice • Thin film thickness • Intermixing with substrate • Interface roughness • Substrate distortions • Buried layers • ... • ...
Nanostructures (nanoparticles, nanowires, thin films, multilayers …) & x-rays X-RAY M ETHODS AT GRAZING INCIDENCE STRUCTURE OF THIN LAYERS ON SUBSTRATE diffracted beam (GID) Crystalline properties reflected beam density profile depth resolution 10nm-200nm d e p t h Diffuse scattering (GISAXS) < 1 0 n m Morphology of nano islands α i / α c 1 6
X-Ray Reflectivity: Principle n 1 <n 2 Visible Light n 1 Reflectivity: n 2 > 1 n 2 n 1 >n 2 X-Ray n 1 Reflectivity: n 2 n 2 < 1 adapted M. Tolan Univ. Dortmund
Reflection and refraction – Perfect surface λ 2 − − δ = ρ ≈ 4 6 10 .. 10 r E r 0 π E 0 2 λ − − β = µ ≈ 6 9 10 .. 10 E t π 4 α = α cos cos Minus!! n Snell-Descartes law: Dispersion Absorption i t α ≤ α ≥ α cos( ) 1 , i.e. ∃ transmitted wave only if i t c - Incident wave totally externally reflected. α ≤ α , If - Transmitted wave exponentially damped with z. i c r α critical angle for total external α = δ = × λ × ρ ≈ ° 2 0 0 . 1 to 0.5 c π c reflection of X-rays
Reflection and refraction: perfect surface Fresnel equations: • E r E 0 Relationships between the E t amplitudes of incident, transmitted and reflected beam. Intensity Amplitude 2 α − α α − α sin( ) E E = = = ≈ Reflection r r i t i t R r α + α α + α sin( ) E E 0 0 i t i t α α α 2 sin( ) cos( ) 2 2 E = = ≈ E Transmission t i t i t = t T α + α α + α sin( ) E E 0 i t i t 0
Limiting and asymptotic values for Fresnel equations Transmission Reflection α α 2 2 δ 1 2 = ≈ i i t α − α + ( 1 ) δ 1 2 α + α − δ 2 2 α − α − δ 2 α + α + α 2 i i 2 ( 1 ) 2 i i = ≈ α i i i i 2 i 2 r δ 1 2 i α + α − δ 2 2 α + α + ( 1 ) α α 2 i i α 2 i i 2 ≈ ≈ α < δ ...... ...... 2 i i t for δ i Amplitude δ i α + 2 = α << δ 1 .......... ..... .......... 2 r for α i i i δ α 2 = − α >> δ ...... .......... 2 r for ≈ = α >> δ 1 .......... ......... ..... 2 i t for α 2 i 2 α i 2 i i 0 10 4 R F = r² T= t² -1 10 1/Q z 4 4 1/q 3 -2 10 Reflectivity 2 Q c T(q z ) q c Intensity 2 -3 10 π δ 4 2 64 1 Q z 2 = ( ) -4 R F q 10 λ 4 4 q 0,00 0,05 0,10 0,15 0,20 -5 10 Q z = 4π/λ sin Θ 1 q z =4 π / λ sin( α i ) 0,00 0,05 0,10 0,15 0,20 Q z = 4π/λ sin Θ 1 q z =4 π / λ sin( α i )
Exact evaluation of Fresnel reflectivity � Absorption β also play a significant role
Fresnel Reflectivity: R F ( α i ) with absorption Total External Reflection Regime -4 = ( 4π/λ sin α ι ) −4 Q z
Transmission Function with absorption T F = |t| 2
Penetration Depth with absorption − 1 = = λ π Im( ) / 2 L k l , , , i t z i f f [ ] 1 { } 1 / 2 1 / 2 = δ − α + α − δ + β 2 2 2 ( 2 sin ) (sin 2 ) 4 l , , i , i f i f f 2 photoabsorption Λ evanescent regime − ( ) 1 π r ρ 0
The geometry of X-ray reflectivity Transferred momentum :
X-ray reflectivity: main equation Helmholtz equation Formal solution: � Electron density profile adapted from M. Tolan Univ. Dortmund
Reflectivity from multilayers Multiple scattering (dynamical calculation) Matrix formalism: Parrat iterations Parrat, 1954 adapted M. Tolan Univ. Dortmund
Reflectivity from layer on substrate. Ex: PS on Si � Reflectivity used as an everyday laboratory tool to measure the thickness of layers deposited on a substrate adapted from M. Tolan Univ. Dortmund
Rough interfaces: statistics adapted M. Tolan Univ. Dortmund
Reflectivity by a rough surface : which roughness ?
Roughness in multilayers?
Effect of interfacial roughness on reflectivity: single interface 4 1/q 2 ) 1/q 4 exp(-q 2 σ � Reflectivity very efficient to measure (small) (statistically averaged) roughness of surfaces or interfaces.
Roughness at several interfaces
Thin film with surface and interface roughness. Example: PS layer on Si, with roughness � Effects of surface and interface roughness very different � σ 1 , σ 2 and d can be determined independantly
Reflectivity calculation for arbitrary density profiles
Example of fit of reflectivity curve: adapted M. Tolan Univ. Dortmund
Simplier aproach: Kinematical approximation
The « master » formula Example: roughness 2 π σ exp( − z 2 1 ( ) ρ ( z ) = ′ 2 ) ⇒ R ( q ) = R F ( q )exp − q 2 σ 2 2 σ
Kinematical versus dynamical calculation
? ρ (z) R(qz) |T.F.|2 Mat rices Pb: Loss of the phase: 0 10 1.0 -1 10 Different ways to solve the phase problem: 0.8 0.6 -2 10 0.4 0.2 -3 -Inclusion of pre-knowledge 10 0.0 -20 0 20 40 60 -4 Å 10 -Anomalous reflectivity -5 10 -6 10 -7 10 0.0 0.2 0.4 0.6 0.8 1.0
Ex: Multilayers: • Complex index profile A B } xN A B N x 0 A 10 B -1 10 sin θ Β =λ /2d -2 10 -3 10 ∆θ=λ/2/ Nd -4 10 � � � � � � { -5 10 -6 10 -7 10 -8 10 0.0 0.2 0.4 0.6 0.8 1.0 θ, q � X-ray reflectivity used to characterize the thickness, period and roughnesses of multilayers.
Rough surfaces � diffuse scattering qx qz q qz � Lateral features of the roughness – Height-height correlations
Ex.: Roughness correlations in multilayers?
Conclusions on reflectivity Specular reflectivity measures • average density (mass and electron density) • layer thicknesses • interface roughness Off-specular reflectivity probes • Height-height correlations • lateral order at nanometer-micrometer scale • Refraction under grazing incidence � tuneable scattering depth
Why GISAXS ? GISAXS Statistical information Lateral and vertical correlation shape as seen by x-rays: input for diffraction experiments Information about buried objects AFM / STM Local information Detailed shape
Grazing Incidence Small Angle X-ray Scattering (GISAXS) Standard 3D growth ( Volmer-Weber ) Principle Example : 20 Å Ag/MgO(001) 500K q z q y [110] [110] h h ω (001) (001) D (111) (111) α f 1/h 1/h α i d d [100] [100] 2 θ (001) (001) (111) (111) (111) (111) q x h d d 1/h 1/h d 1/d 1/d 2D image around direct beam: Fourier transform of objects 1/D 1/D Morphology • Shape Anisotropic islands: • Sizes truncated square pyramids with (111) facets • Size distributions α i • Particle-particle pair correlation function
Off-specular reflectivity: Probed length scales?
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