X-ray Reflectivity: Theory, Application And Sample Preparation

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Surface Specific X-ray ScatteringX-ray Reflectivity:Theory, application and sample preparationHans-Georg SteinrückStanford Synchrotron Radiation Lightsource,SLAC National Accelerator LaboratoryXRS 2018, 07/16/18

Outline Introduction Surface x-ray diffractiono Surface sensitivityo Technique overview Focus on x-ray reflectivityo Theoryo Applicationo Sample preparation

IntroductionSurfaces Outer boundary of any material Dominate interaction with environment Decisive role in numerous natural andtechnological processeso Nanotechnology / Material scienceo Catalysiso Energy storage – e.g. batteriesX-rays: Structure-function relationIon transport into electrodesMolecular thick FETs – SAMFETsTransport affected by structureSchmalz, Steinrück et al., Adv. Mat. 25, 4511–4514 (2013)Adsorption sites for CO & O2Local site configuration and particle sizeaffect binding energyGoverned by molecular arrangementVatamanu et al., J. Phys. Chem. C 116, 1114 (2012)Schauermann & Freund, Acc. Chem. Res. 48, 2775 (2015)

Surface sensitivity𝑞𝑧 4𝜋sin 𝛼𝜆Refraction index:𝒏 𝟏 𝜹 𝒊𝜷𝛿 𝜆2 𝑟𝑒𝜌 :2𝜋 𝑒wavelength dependent scattering 1𝑒 6𝛽 : wavelength dependent absorption 1𝑒 8 – neglelible in most cases 𝒏𝟏 𝒏𝟐 : Total external reflectionSnell’s law:𝑛1 cos 𝛼 𝑛2 cos 𝛼 ′in vacuum (𝑛1 1):cos 𝛼 𝑛2 cos 𝛼 ′visible light analogous

Surface sensitivity𝑞𝑧 4𝜋sin 𝛼𝜆Refraction index:𝒏 𝟏 𝜹 𝒊𝜷𝛿 𝜆2 𝑟𝑒𝜌 :2𝜋 𝑒wavelength dependent scattering 1𝑒 6𝛽 : wavelength dependent absorption 1𝑒 8 – neglelible in most cases 𝒏𝟏 𝒏𝟐 : Total external reflectionSnell’s law:Critical angle 𝛼𝑐Angle of refraction𝑛1 cos 𝛼 𝑛2 cos 𝛼 ′𝛼 ′ 90 cos 𝛼 ′ 1𝑛 1 𝛿 1 𝛼𝑐2 2in vacuum (𝑛1 1):cos 𝛼𝑐 𝑛2 1 𝛿2in Snell‘s lawcos 𝛼 𝑛2 cos 𝛼 ′with cos 𝛼𝑐 1 𝛼𝑐2 /2𝛼2𝛼 ′21 1 22𝛼𝑐 2𝛿2 1 𝜶′ 𝜶𝟐 𝜶𝟐𝒄𝛼𝑐21 2

Surface sensitivity𝑞𝑧 4𝜋sin 𝛼𝜆Refraction index:𝒏 𝟏 𝜹 𝒊𝜷𝛿 𝜆2 𝑟𝑒𝜌 :2𝜋 𝑒wavelength dependent scattering 1𝑒 6𝛽 : wavelength dependent absorption 1𝑒 8 – neglelible in most cases 𝒏𝟏 𝒏𝟐 : Total external reflectionSnell’s law:Critical angle 𝛼𝑐Angle of refraction𝑛1 cos 𝛼 𝑛2 cos 𝛼 ′𝛼 ′ 90 cos 𝛼 ′ 1𝑛 1 𝛿 1 𝛼𝑐2 2in vacuum (𝑛1 1):cos 𝛼𝑐 𝑛2 1 𝛿2in Snell‘s lawcos 𝛼 𝑛2 cos 𝛼 ′with cos 𝛼𝑐 1 𝛼𝑐2 /2𝛼𝑐 2𝛿2 1 What happens for 𝜶𝒄 𝜶 ?𝛼2𝛼 ′2𝛼𝑐21 1 1 222𝜶′ 𝜶𝟐 𝜶𝟐𝒄

Evanescant wave – below the critical angleHow far do the x-rays penetrate into the materialas a function of incoming angle / scattering vector?𝒛-component of amplitude of electromagnetic field inside material𝐸transmitted (𝑧) 𝐸0 𝑒 𝑖what is 𝑘𝑧′ ?𝑤𝑡 𝑘𝑧′ 𝒛in terms of the incident angle ?𝑘′𝑧 𝑛𝑘0 sin 𝛼 ′ for small 𝛼 ′ 𝑠𝑖𝑛 𝛼 ′ 𝛼 ′ 𝑛 1𝑘′𝑧 𝑘0 𝛼′𝛼′ 𝛼 2 𝛼𝑐2for 𝛼 𝛼𝑐 purely imaginary𝛼 ′ 𝑖𝛼𝑐

Evanescant wave – below the critical angleHow far do the x-rays penetrate into the materialas a function of incoming angle / scattering vector?𝒛-component of amplitude of electromagnetic field inside material𝐸transmitted (𝑧) 𝐸0 𝑒 𝑖what is 𝑘𝑧′ ?𝑤𝑡 𝑘𝑧′ 𝒛in terms of the incident angle ?𝛼 ′ 𝑖𝛼𝑐Using 𝑘0 2𝜋/𝜆 and 𝛼𝑐 𝑞𝑐 𝜆/4𝜋𝑘′𝑧 𝑘0𝛼′2𝜋 𝑘0 𝑖 𝛼𝑐 𝑖 𝛼𝑐𝜆𝑞𝑐 𝑖 2expressed in critical angle 𝛼𝑐expressed in critical scattering vector 𝑞𝑐

Evanescant wave – below the critical angleHow far do the x-rays penetrate into the materialas a function of incoming angle / scattering vector?𝒛-component of amplitude of electromagnetic field inside material𝐸transmitted (𝑧) 𝐸0 𝑒 𝑖what is 𝑘𝑧′ ?𝑘′𝑧 𝑖 𝑤𝑡 𝑘𝑧′ 𝒛in terms of the incident angle ?𝑞𝑐2𝐸evanescant 𝐸0′𝑒 𝑖 𝑤𝑡 𝑘𝑧𝑧 𝒒𝒄𝑬𝟎 𝒆 𝟐 𝒛exponentially damped wave withintensity decay length 𝒛𝟏/𝒆 :𝟏𝒛𝟏/𝒆 𝒒 , typically 100 Å𝒄

Evanescant wave – below the critical angleThe general case for all 𝛂:𝑧1/𝑒 𝜆4𝜋2𝛼 2 𝛼𝑐21𝑞𝑐2 𝑞𝑧22Where does the observed scattering come from? 4𝛽 2 𝛼 2 𝛼𝑐2𝑧1/𝑒What is the intensity just below the interface?𝟏 𝒒𝒄The surface enhancement factorAmplitudeIntensitySome examples:interference of incident and reflected wave: constructive @ 𝛼 𝛼𝑐Up to 16x increased critical angleBelow critical angle: Significantly reduced penetration depth: 100 Å Scattering enhanced X-rays are surface sensitive!

OverviewX-ray reflectivityXRROff-speculardiffuse scatteringGrazing incidence diffractionGISAXS & GIDTruncationrodsThe scattering geometry1. X-rays impinge sample under αDirection of q important, not only magnitude𝒒 𝒌𝑓 𝒌𝑖2. Interact with the sample3. Exit sample according to sampleproperties under β and 2θ𝑞 𝑞𝑥cos 𝛽 cos 2𝜃 cos 𝛽 cos 2𝜃2𝜋𝑞𝑦 cos 𝛽 sin 2𝜃 𝜆𝑞𝑧sin 𝛼 sin 𝛽

X-ray reflectivity - XRRX-ray reflectivityXRROff-speculardiffuse scatteringGeometryGrazing incidence diffractionGISAXS & GIDTruncationrodsInformationScattering vector solely perpendicular to surface Surface normal information Layer thickness Layer density Surface and interface roughnessα βSurface normal electron density profileα varied𝜃/2𝜃-scan / butterfly scanqx 0, qy 0, qz 0𝒅 𝚫𝝆𝚫𝝆 𝟏/𝒅

Off-specular diffuse scatteringX-ray reflectivityXRROff-speculardiffuse scatteringGeometryGrazing incidence diffractionGISAXS & GIDTruncationrodsInformationExample Surface roughnesslateral information! Correlation-length ξ Jaggedness hWetting of a rough surfaceTidswell et al., Phys. Rev. Lett 66, 2108 (1991)ξα fixed, β variedDetector scanα β fixed, α variedRocking scanqx 0, qy 0, qz 0hξThin film ( 60 Å):Correlation to substrateroughnessThick film:Cappilary waves

GISAXSX-ray reflectivityXRROff-speculardiffuse scatteringGeometryGrazing incidence diffractionGISAXS & GIDInformationTruncationrodsExampleIn-situ gold cluster growthTypical 2θ 2 Schwartzkopf et al., Nanoscale 5, 5053 (2013)low q large in real space Morphologyα fixedβ varied2θ variedφ 0 or variedqx 0, qy 0, qz 0𝑞𝑟 𝑞𝑥2 𝑞𝑦2o Surfaceo Particles Nano- macro-scaledensity correlations size R distance D shape, etc.

GID - GIWAXSX-ray reflectivityXRROff-speculardiffuse scatteringGeometryGrazing incidence diffractionGISAXS & GIDInformationTypical 2θ 20 large q small in real space Atomic orderα fixedβ varied2θ variedφ 0 or variedqx 0, qy 0, qz 0𝑞𝑟 𝑞𝑥2 𝑞𝑦2 Crystallinity unit cell crystal size Molecular orientationof adlayersTruncationrodsExampleCheckerboard layering inionic liquids (free surface)Tamam et al., PRL 106, 197801 (2011)

CTRX-ray reflectivityXRROff-speculardiffuse scatteringGeometryGrazing incidence diffractionGISAXS & GIDInformationTruncated surface Bragg peak shape change Crystallinity of surfacerealspaceTruncationrodsExampleStructure of Graphensupported Ir NanoparticlesFranz et al., PRL 110, 065503 (2013)reciprocalspaceα fixedβ varied2θ variedφ variedqx 0, qy 0, qz 0𝑞𝑟 𝑞𝑥2 𝑞𝑦2Robinson & Tweet, Rep. Prog. Phys. 55, 599 (1992)- Crystallographic superlattice- Epitaxy on graphene moire structure- Compressive intraparticle strain

X-ray reflectivitySpecularly reflected intensity fraction𝐼(𝛼)𝑅 𝛼 𝐼0at 𝛼 𝛽4𝜋𝑞𝑧 𝑘𝑟 𝑘𝑖 sin 𝛼𝜆𝑅 𝑞𝑧𝐼(𝑞𝑧 ) 𝐼0Fresnel reflectivity: Simples case of reflection of x-rays from a single interface Solve Helmholtz equation:propagation of light through medium characterized by refractive indexSolution plane wave: 𝐸𝑗 𝐴𝑗 𝑒 𝑖(𝜔𝑡 𝐤𝑗𝐫)Electro-magnetic field must be continuous at the interface!𝑛𝐴𝑖 𝐴𝑟 𝐴𝑡 (𝐴𝑖 𝐴𝑟 ) sin 𝛼 𝑗 𝐴𝑡 sin 𝛼′𝑛𝑗 1Define reflection & transmission coefficient: 𝒓𝒋,𝒋 𝟏 with 𝑘𝑗 𝑛j 𝑘0 , equate & solve for 𝒓 𝑨𝒓𝑨𝒊𝑅 𝑞𝑧 𝑟𝑗,𝑗 1& 𝒕𝒋,𝒋 𝟏 2𝑨𝒕𝑨𝒊𝑘𝑗,𝑧 𝑘𝑗 1,𝑧 𝑘𝑗,𝑧 𝑘𝑗 1,𝑧2 𝑞𝑧 𝑞𝑧2 𝑞𝑐2𝑞𝑧 𝑞𝑧2 𝑞𝑐22

X-ray reflectivity𝑅 𝑞𝑧 𝑞𝑧 𝑞𝑐2 𝑞𝑐2𝑞𝑧 𝑞𝑐2 𝑞𝑐22 𝑅𝐹𝑞𝑐 is related to electron density 𝜌for 𝑞𝑧 4𝑞𝑐 : 𝑅 𝑞𝑧 𝑞𝑐4𝑞𝑧42Roughness:Described by Debye-Waller-like factor: 𝑟rough 𝑟 𝑒 Damping of XRR𝑞 𝜎2 𝑧2

Layered systemsQualitativelly:X-ray reflected from different interfaces interfereconstructivelly and destructivelly as a function ofincoming angle: Kiessig fringesPath length difference changes

Layered systemsSome 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 7d 1420 Åd 220 Å

Layered systemsQualitativelly:X-ray reflected from different interfaces interfereconstructivelly and destructivelly as a function ofincoming angle: Kiessig fringesPath length difference changesQuantitativelly: Calculate reflection and transmissioncoefficient for each layer Add up iteratively (Parratt) or Matrix formalismSingle layer:Phase shift22𝑟0,1 𝑟1,2 2𝑟0,1 𝑟1,2 𝐜𝐨𝐬 𝟐𝒌𝒛𝟏 𝒉𝑅 2 21 𝑟0,1𝑟1,2 2𝑟0,1 𝑟1,2 𝐜𝐨𝐬 𝟐𝒌𝒛𝟏 𝒉 Period of fringes scales inversely withthickness of layers: 2π/Δq

Electron density & several layersElectron density changes:22𝑟0,1 𝑟1,2 2𝑟0,1 𝑟1,2 𝐜𝐨𝐬 𝟐𝒌𝒛𝟏 𝒉𝑅 2 21 𝑟0,1𝑟1,2 2𝑟0,1 𝑟1,2 𝐜𝐨𝐬 𝟐𝒌𝒛𝟏 𝒉𝑟𝑗,𝑗 1 𝑘𝑗,𝑧 𝑘𝑗 1,𝑧, 𝑘 𝑛j 𝑘 0𝑘𝑗,𝑧 𝑘𝑗 1,𝑧 𝑗𝑛 1 𝛿 𝑖𝛽𝜆2 𝑟𝑒 𝜌𝑒𝛿 2𝜋 Electron density contrast determinesamplitude of fringesSeveral layers:Interference of x-rays reflectedfrom different interfaces several phase shifts beating pattern

Master formulaArbitrary density profile: Slice into slabs Calculate via Master formula𝑅(𝑞𝑧 )1 𝑅F (𝑞𝑧 )𝜌 2 𝜌 𝑧𝑑𝑧𝑒 𝑖𝑞𝑧 𝑧 𝑧Born approximation: Easily derivableo No multiple scatteringo No refractiono No absorption Approximated analytical expression More user friendly

ApplicationsIn-situ Study of Si Electrode Lithiation with X-ray ReflectivityCao, Steinrück et al., Nano Lett. 16, 7394-7401 (2016) & Adv. Mater. Interfaces 4, 1700771 (2017).Silicon: A promising high capacity anode for Li-ion batteriesLi in Sio Theoretical capacity 3580 mAh/g - 10 times higher than graphiteo BUT: Volume expansion & other issues limit commercialization Fundamental understanding of Si lithiation process and SEIIn-situ XRR measuredat 12 keV, SSRL BL 2-1reaction limited, layer-by-layer three-stage lithiation mechanism

ApplicationsThe nanoscale structure of the electrolyte–metal oxide interfaceSteinrück et al., Energy Environ. Sci., 11, 594-602 & Nano Lett. 18, 2105-2111 (2018).XRR measured atSSRL BL7-2Analysis of XRR withdistorted crystal model Double layer formation

ApplicationsSurface Roughness of Water Measured by X-Ray ReflectivityA. Braslau et al., Phys. Rev. Lett. 54, 114 (1985). SynchrotronexperimentFresnel-XRR Lab sourceexperimentCapillary wave roughened XRR Roughness of 3.24 Å Very close to what is expected from thermally excited capillary waves First such measurement of surface roughness of any liquid Synchrotron radiation necessary

ApplicationsOTS on sapphire: Pseudorotational epitaxySteinrück et al., PRL 113, 156101 (2014)The system 3 slab model! Molecules vertically aligned Packed in a rotator phase Lattice match with sapphireaOTS 4.82 Å, asapphire 4.76 ÅXRRGIDRotatorCrystalline

ApplicationsOTS on sapphire: Pseudorotational epitaxySteinrück et al., PRL 113, 156101 (2014)Geometrical modelCritical separation betweencorresponding lattice sites Loss of coherence length

ApplicationsSi/SiO2 - nanoscale structureSteinrück et al., ACS Nano 12, 12676 (2014)Sapphire: Single interfaceSilicon: Si and SiO2: Bond density mismatch Various oxidation states [2] Dangling bonds, hydrogen Single interface 1 slab model (Tidswell [1]) Theory: Low density region [3] “Dip“ model 6 – 8 missing e- per unit cell area[1] Tidswell et al., PRB 41, 1111 (1990). [2] Braun et al., Surf. Sci 180, 279 (1987). [3] Tu et al., PRL 84, 4393 (2000).

Sample preparationCleaning Essential: Any surface contamination will effect XRR Very sample specific: Cleaning methods may also effect sampleExamples:o Ultrasonication in organic solvents: EAT (Ethanol, Acetone, Toluene (polar and non-polar)) Removes organic contaminationo Piranha acid (H2SO4 and 30% hydrogen peroxide H2O2, dangerous, use SOP) Strongly acidic and a strong oxidizer Removes organic contaminationo UV-ozone (dry, simple to use) UV irradiation, creates ozone Decomposes organic mattero Rinsing with ultra-pure watero RCA clean (silicon wafer technology, dangerous, use SOP)o Oxygen plasma cleaning

Sample preparationSurface roughness Check surface roughness via laboratory XRR, AFM, etc. Typically, root-mean-square roughness 10 Å required for XRR 3 Å required for molecular resolution Calculate expected XRR to test feasibility2Roughness:Described by Debye-Waller-like factor: 𝑟rough 𝑟 𝑒 Damping of XRRBackground𝑞 𝜎2 𝑧2

Sample preparationExperimental setupo Sample size: Ideally larger than footprint at critical angle, otherwise corrections arenecessaryo Vertical beam size: Optimize with respect to sample size (footprint) Optimize with respect to diffuse scattering (divergence)o Horizontal beam size: Optimize with respect to possible sample translation (beam damage)o X-ray energy: Higher energy: Less radiation damage (absorption cross-section 1/E3, valid up to 30 keV, where Compton dominates), spec. organics Lower energy: Larger angles, mostly important for footprint effects Consider absorption edges of compounds in sample Consider transmission through e.g. liquid

Sample preparationSample environmento Solid-vapor interfaces: Use Helium or Nitrogen environment Reduces beam damage (oxidation) Reduces background (air scattering)o Solid-liquid interfaces: Significant background/absorption from bulk liquid scattering (signal to noise) Minimize transmission length (convoluted with footprint problem) Use thin-film cells to minimize liquid scattering (Fenter et al., Prog. Surf. Sci. 77, 171–258 (2004)) Use film-stabilizing agent (e.g. polymer, Petach et al., ACS Nano 10, 4565 4569 (2016))

Take-away messageso Surfaces & interfaces are important & interestingo X-ray scattering can be extremely surface sensitive (by choosing right geometry)o Ideal for in-situ studies – buried interfaceso 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 RoughnessThicknessDensity

Take-away messagesAcknowledgements: XRS 2018 organizing chairs, support from the DOE-BES Toney group, Chuntian Cao, Mike Toney SSRL engineers, beamline engineers and scientistsThicknessDr. Ben OckoBrookhavenNational LaboratoryProf. Moshe DeutschBar-Ilan University,IsrealDensity

LiteratureGeneral x-rays: J. Als-Nielsen and D. McMorrow, Elements of modern X-ray physics (John Wiley & Sons, New York, USA, 2011) D. S. Sivia, Elementary scattering theory: For X-ray and neutron users (Oxford University Press, New York, USA, 2011)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. 23 J. Daillant and A. Gibaud, X-ray and Neutron Reflectivity: Principles and Applications (Springer, Berlin, Germany, 2009) P. S. Pershan and M. Schlossman, Liquid Surfaces and Interfaces: Synchrotron X-ray Methods (Cambridge University Press,Cambridge, UK, 2012) 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. 246, 251 (1994) S. K. Sinha et al., Physical Review B 38, 2297 (1988)useful webpage: http://www.reflectometry.net/ (by Prof. Adrian R. Rennie, Uppsala, Sweden)GISAXS: G. Renaud et al., Surface Science Reports 64, 255-380 (2009) P. Müller-Buschbaum, Anal Bioanal Chem. 376, 3-10, (2003) P. Müller-Buschbaum, Materials and Life Sciences Lecture Notes in Physics 776, 61-89 (2009)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 (1992)

LinksLinks to X-ray analysis motofit/by Dr. Andrew Nelson, ANSTO, Australiareference: A. Nelson, J. Appl. Crystallogr. 39, 273 (2006)GenX:http://genx.sourceforge.net/by Dr. Matts Björck, Swedish Nuclear and Fuel Management Company, Swedenreference: M. Björck and G. Andersson, J. Appl. Crystallogr. 40, 1174 (2007)

X-ray Reflectivity: Theory, application and sample preparation Surface Specific X-ray Scattering . Outline Introduction Surface x-ray diffraction o Surface sensitivity o Technique overview

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