X-ray Surface Diffraction & Reflectivity Hans-Georg Steinrück Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory XRS 016, 06/1/16
Outline Introduction Surface x-ray diffraction o Surface sensitivity o Technique overview Focus on x-ray reflectivity o Theory o Examples
Introduction Surfaces Outer boundary of any material Dominate interaction with environment Decisive role in numerous natural and technological processes o Nanotechnology / Material science o Catalysis o Energy storage e.g. batteries X-rays: Structure-function relation Ion transport into electrodes Governed by molecular arrangement Molecular thick FETs SAMFETs Transport affected by structure Schmalz, Steinrück et al., Adv. Mat. 5, 4511 4514 (013) Adsorption sites for CO & O Local site configuration and particle size affect binding energy Vatamanu et al., J. Phys. Chem. C 116, 1114 (01) Schauermann & Freund, Acc. Chem. Res. 48, 775 (015)
Surface sensitivity q z = 4π λ sin α Refraction index: n = 1 δ iβ δ = λ r e π ρ e: wavelength dependent scattering ~ 1e 6 β : wavelength dependent absorption ~ 1e 8 neglelible in most cases n 1 > n : Total external reflection Snell s law: n 1 cos α = n cos α in vacuum (n 1 = 1): cos α = n cos α visible light analogous Critical angle α c α = 90 cos α = 1 cos α c = n = 1 δ with cos α c 1 α c / α c δ 1
Surface sensitivity q z = 4π λ sin α Refraction index: n = 1 δ iβ δ = λ r e π ρ e: wavelength dependent scattering ~ 1e 6 β : wavelength dependent absorption ~ 1e 8 neglelible in most cases n 1 > n : Total external reflection Snell s law: n 1 cos α = n cos α in vacuum (n 1 = 1): cos α = n cos α Critical angle α c α = 90 cos α = 1 cos α c = n = 1 δ with cos α c 1 α c / α c δ 1 Angle of refraction n = 1 δ = 1 in Snell s law 1 α = 1 α α = α α c Τ α c 1 α c
Surface sensitivity q z = 4π λ sin α Refraction index: n = 1 δ iβ δ = λ r e π ρ e: wavelength dependent scattering ~ 1e 6 β : wavelength dependent absorption ~ 1e 8 neglelible in most cases n 1 > n : Total external reflection Snell s law: n 1 cos α = n cos α in vacuum (n 1 = 1): cos α = n cos α Critical angle α c α = 90 cos α = 1 cos α c = n = 1 δ with cos α c 1 α c / α c δ 1 Angle of refraction n = 1 δ = 1 in Snell s law Τ α c What happens for α c > α? 1 α = 1 α 1 α c α = α α c
Evanescant wave below the critical angle k z nk 0 sin α k z k 0 α How far do the x-rays penetrate into the material as a function of incoming angle / scattering vector? z-component of amplitude of electromagnetic field inside material E transmitted (z) = E 0 e i wt k z z what is k z? in terms of the incident angle? for small α sin α = α n = 1 α = α α c for α α c purely imaginary α = iα c
Evanescant wave below the critical angle How far do the x-rays penetrate into the material as a function of incoming angle / scattering vector? z-component of amplitude of electromagnetic field inside material E transmitted (z) = E 0 e i wt k z z what is k z? in terms of the incident angle? α = iα c Using k 0 = π/λ and α c = q c λ/4π k z k 0 α = k 0 i α c = i π λ α c = i q c expressed in critical angle α c expressed in critical scattering vector q c
Evanescant wave below the critical angle How far do the x-rays penetrate into the material as a function of incoming angle / scattering vector? z-component of amplitude of electromagnetic field inside material E transmitted (z) = E 0 e i wt k z z what is k z? in terms of the incident angle? k z i q c E evanescant = E 0 e i wt k z z = E 0 e q c z exponentially damped wave with intensity decay length z 1/e : z 1/e 1 q c, typically 100 Å
Evanescant wave below the critical angle The general case for all α: z 1/e = 4π 1 q c q z λ α α c + 4β α α c z 1/e 1 q c What is the intensity just below the interface? Where does the observed scattering come from? The surface enhancement factor Amplitude Intensity Some examples: interference of incident and reflected wave: constructive @ α = α c Up to 16x increased critical angle Below critical angle: Significantly reduced penetration depth: ~ 100 Å Scattering enhanced X-rays are surface sensitive!
Overview X-ray reflectivity XRR Off-specular diffuse scattering Grazing incidence diffraction GISAXS & GID Truncation rods Thin film diffraction Crystallinity, epitaxy, orientation and strain next talk by Apurva Mehta Qiao, Mehta et al., Nano Lett. 15, 4677 4684 (015)
Overview X-ray reflectivity XRR Off-specular diffuse scattering Grazing incidence diffraction GISAXS & GID Truncation rods The scattering geometry 1. X-rays impinge sample under α. Interact with the sample 3. Exit sample according to sample properties under β and θ
X-ray reflectivity - XRR X-ray reflectivity XRR Geometry Off-specular diffuse scattering Grazing incidence diffraction GISAXS & GID Information Truncation rods Scattering vector solely perpendicular to surface Surface normal information Layer thickness Layer density α = β α = varied Surface and interface roughness Surface normal electron density profile θ/θ-scan / butterfly scan q x = 0, q y = 0, q z 0 d Δρ 1/d Δρ
Off-specular diffuse scattering X-ray reflectivity XRR Off-specular diffuse scattering Grazing incidence diffraction GISAXS & GID Truncation rods Geometry Information Example Surface roughness lateral information! Correlation-length ξ Jaggedness h Wetting of a rough surface Tidswell et al., Phys. Rev. Lett 66, 108 (1991) ξ α = fixed, β = varied Detector scan h α + β = fixed, α = varied Rocking scan q x 0, q y = 0, q z 0 ξ Thin film (< 60 Å): Correlation to substrate roughness Thick film: Cappilary waves
GISAXS X-ray reflectivity XRR Off-specular diffuse scattering Grazing incidence diffraction GISAXS & GID Geometry Information Example Truncation rods α = fixed β = varied θ = varied φ = 0 or varied q x 0, q y 0, q z 0 q r = q x + q y Typical θ = low q large in real space Morphology o Surface o Particles Nano- macro-scale density correlations In-situ gold cluster growth Schwartzkopf et al., Nanoscale 5, 5053 (013) size R distance D shape, etc.
GID - GIWAXS X-ray reflectivity XRR Off-specular diffuse scattering Grazing incidence diffraction GISAXS & GID Geometry Information Example α = fixed β = varied θ = varied φ = 0 or varied q x 0, q y 0, q z 0 q r = q x + q y Typical θ = 0 large q small in real space Atomic order Crystallinity unit cell crystal size Molecular orientation of adlayers Truncation rods Checkerboard layering in ionic liquids (free surface) Tamam et al., PRL 106, 197801 (011)
CTR X-ray reflectivity XRR Off-specular diffuse scattering Grazing incidence diffraction GISAXS & GID Truncation rods Geometry Information Example Truncated surface Bragg peak shape change Crystallinity of surface Structure of Graphensupported Ir Nanoparticles Franz et al., PRL 110, 065503 (013) real space reciprocal space α = fixed β = varied θ = varied φ = varied q x 0, q y 0, q z 0 q r = q x + q y Robinson & Tweet, Rep. Prog. Phys. 55, 599 (199) - Crystallographic superlattice - Epitaxy on graphene moire structure - Compressive intraparticle strain
X-ray reflectivity Fresnel reflectivity: Simples case of reflection of x-rays from a single interface Solve Helmholtz equation: propagation of light through medium characterized by refractive index Solution = plane wave: E j = A j e i(ωt k jr) Electro-magnetic field must be continuous at the interface! A i + A r = A t (A i +A r ) sin α = n j n j 1 A t sin α Specularly reflected intensity fraction R α = I(α) I 0 at α = β q z = k r k i = 4π λ sin α R q z = I(q z) I 0 Define reflection & transmission coefficient: r j,j 1 = A r A i & t j,j 1 = A t with k j = n j k 0, equate & solve for r R q z = r j,j 1 = k j,z k j 1,z k j,z + k j 1,z A i = q z q z q c q z + q z q c
X-ray reflectivity R q z = q z q c q c q z + q c q c R F q c is related to electron density ρ for q z > 4q c : R q z q c 4 q z 4 Roughness: Described by Debye-Waller-like factor: r rough = r e q z σ Damping of XRR
Layered systems Qualitativelly: X-ray reflected from different interfaces interfere constructivelly and destructivelly as a function of incoming angle: Kiessig fringes Path length difference changes
Layered systems Some history - Kiessig fringes: First observed by Heinz Kiessig in 1931 for Ni on glass Interferenz von Röntgenstrahlen an dünnen Schichten ANNALEN DER PHYSIK, 5. FOLGE, 1931, BAND 10, HEFT 7 d = 140 Å d = 0 Å
Layered systems Qualitativelly: X-ray reflected from different interfaces interfere constructivelly and destructivelly as a function of incoming angle: Kiessig fringes Path length difference changes Quantitativelly: Calculate reflection and transmission coefficient for each layer Add up iteratively (Parratt) or Matrix formalism Single layer: Phase shift R = r 0,1 + r 1, + r 0,1 r 1, cos k z1 h 1 + r 0,1 r 1, + r 0,1 r 1, cos k z1 h Period of fringes scales inversely with thickness of layers: π/δq
Electron density & several layers Electron density changes: R = r 0,1 + r 1, + r 0,1 r 1, cos k z1 h 1 + r 0,1 r 1, + r 0,1 r 1, cos k z1 h r j,j 1 = k j,z k j 1,z k j,z + k j 1,z, k j = n j k 0 n = 1 δ iβ δ = λ r e ρ e π Electron density contrast determines amplitude of fringes Several layers: Interference of x-rays reflected from different interfaces several phase shifts beating pattern
Master formula Arbitrary density profile: Slice into slabs Calculate via Master formula R(q z ) R F (q z ) 1 ρ න dz ρ z z e iq zz Born approximation: Easily derivable o No multiple scattering o No refraction o No absorption Approximated analytical expression More user friendly
Examples Voltage-Controlled Interfacial Layering in an Ionic Liquid on SrTiO3 T. A. Petach et al., ACS Nano 10, 4565 4569 (016) Novel experimental idea: Minimization of background using tri-block co-polymer as film-stabilizing agent XRR measured at SSRL BL7- & 10- Analysis of XRR with distorted crystal model Double layer formation
Examples Surface Roughness of Water Measured by X-Ray Reflectivity A. Braslau et al., Phys. Rev. Lett. 54, 114 (1985). Synchrotron experiment Lab source experiment Fresnel-XRR Capillary wave roughened XRR Roughness of 3.4 Å Very close to what is expected from thermally excited capillary waves First such measurement of surface roughness of any liquid Synchrotron radiation necessary
Examples OTS on sapphire: Pseudorotational epitaxy Steinrück et al., PRL 113, 156101 (014) The system XRR GID 3 slab model! Molecules vertically aligned Packed in a rotator phase Lattice match with sapphire a OTS = 4.8 Å, a sapphire = 4.76 Å Rotator Crystalline
Examples OTS on sapphire: Pseudorotational epitaxy Steinrück et al., PRL 113, 156101 (014) Geometrical model Critical separation between corresponding lattice sites Loss of coherence length
Examples Si/SiO - nanoscale structure Steinrück et al., ACS Nano 1, 1676 (014) Sapphire: Single interface Silicon: Single interface 1 slab model (Tidswell [1]) Dip model Si and SiO : Bond density mismatch Various oxidation states [] Dangling bonds, hydrogen Theory: Low density region [3] ~ 6 8 missing e - per unit cell area [1] Tidswell et al., PRB 41, 1111 (1990). [] Braun et al., Surf. Sci 180, 79 (1987). [3] Tu et al., PRL 84, 4393 (000).
Take-away messages o Surfaces & interfaces are important & interesting o X-ray scattering can be extremely surface sensitive (by choosing right geometry) o Ideal for in-situ studies buried interfaces o Several techniques are sensitive to different information Off-specular scattering: Lateral roughness GID: Crystallinity of adlayers GISAXS: Morphology of adlayers CRTs: Crystalline surfaces XRR: Surface normal density profile Thickness Density Roughness Thickness Density
Take-away messages Acknowledgements: Dr. Ben Ocko, BNL Prof. Moshe Deutsch Bar-Ilan University, Isreal Thickness Density
Literature General x-rays: J. Als-Nielsen and D. McMorrow, Elements of modern X-ray physics (John Wiley & Sons, New York, USA, 011) D. S. Sivia, Elementary scattering theory: For X-ray and neutron users (Oxford University Press, New York, USA, 011) XRR & GID & off-specular diffuse scattering: M. Deutsch and B. M. Ocko, in Encyclopedia of Applied Physics, edited by G. L. Trigg (VCH, New York, USA, 1998), Vol. 3 J. Daillant and A. Gibaud, X-ray and Neutron Reflectivity: Principles and Applications (Springer, Berlin, Germany, 009) P. S. Pershan and M. Schlossman, Liquid Surfaces and Interfaces: Synchrotron X-ray Methods (Cambridge University Press, Cambridge, UK, 01) M. Tolan, X-ray scattering from soft-matter thin films (Springer, Berlin, Germany, 1999) K. Kjaer, Physica B 198, 100 (1994) J. Als-Nielsen et al., Phys. Rep. 46, 51 (1994) S. K. Sinha et al., Physical Review B 38, 97 (1988) useful webpage: http://www.reflectometry.net/ (by Prof. Adrian R. Rennie, Uppsala, Sweden) GISAXS: G. Renaud et al., Surface Science Reports 64, 55-380 (009) P. Müller-Buschbaum, Anal Bioanal Chem. 376, 3-10, (003) P. Müller-Buschbaum, Materials and Life Sciences Lecture Notes in Physics 776, 61-89 (009) useful webpage: http://gisaxs.com/ (by Dr. Kevin Yager, BNL) Crystal truncation rods: R. Feidenhans l, Surface Science Reports 10, 105-188 (1989) I. K. Robinson and D. J. Tweet, Rep. Prog. Phys. 55, 599 (199)
Links Links to X-ray analysis software: Motofit: https://sourceforge.net/projects/motofit/ by Dr. Andrew Nelson, ANSTO, Australia reference: A. Nelson, J. Appl. Crystallogr. 39, 73 (006) GenX: http://genx.sourceforge.net/ by Dr. Matts Björck, Swedish Nuclear and Fuel Management Company, Sweden reference: M. Björck and G. Andersson, J. Appl. Crystallogr. 40, 1174 (007)