Chapter II: Experimental techniques and data analysis
C. X-ray Scattering
30 Chapter II
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Fig. 2.9: Schematic diagram of the AFM operation in non contact mode.14
Experimental techniques and data analysis 31
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Powder Diffraction Files covering practically every phase of every known material, crystal phase of the sample is identified from the peak positions of the diffractogram. Homogeneous or uniform elastic strain in the (hkl) direction can also be calculated from the shift in the diffraction peak positions, and the dhkl spacing of the unstrained crystal. X–ray diffraction patters of the nanostructurs were obtained using a commercial XRD set-up (TTRAX III, RIGAKU 2500) using a Cu Kα1 (λ=1.5406Å) radiation with nickel filter. All the data are taken at room temperature in thin film mode of the XRD at a grazing angle of 4°. Here in the Fig. 2.10 we show the commercial photocopy of the XRD instrument.
Fig. 2.10: 18kW X-Ray Diffractometer (TTRAX III, RIGAKU 2500), IIT Guwahati (ii) X ray reflectivity
Almost a century after the discovery of X-ray scattering by materials it has been proved to be one of the most useful techniques to study crystal structure, surfaces and interfaces of thin film. In this thesis specular X-ray reflectivity (XRR) has been used to study the molecular arrangement with the film as well as structural information.18-21 It is very useful in the case of (i) the determination of film thickness with precision of 0.1 nm, (ii) electron density profile throughout the film and (iii) determination of the roughness
32 Chapter II
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of the surface and interface with precision of 0.1 nm. When a electromagnetic wave falls on the interface of two different media then some amount of radiation may be reflected, some amount may be refracted or some amount may be both reflected and refracted depending upon the incident angle and refractive coefficient of the media. X-ray reflectivity is limited to small angle of incidence where it is possible to consider electron density as continuous. The elastic scattering of X-rays with matter can be well described by an index of refraction which characterized the change of direction of the X-ray beam when traversing from the interface of the two materials with different electron densities.
For X-rays the refractive index n is given by 𝑛𝑛 = 1− 𝛿𝛿+𝑖𝑖𝑖𝑖
Where 𝛿𝛿 =2𝜋𝜋𝜆𝜆2𝑟𝑟0𝜌𝜌𝑒𝑒 𝑖𝑖 = 4𝜋𝜋𝜆𝜆 𝜇𝜇𝑙𝑙
Here r0 is the classical electron radius (2.818×10-15Å), ρe is the electron density of the medium, μl is the linear absorption coefficient for the incident photons in the medium.21, 22
For total external reflection to occur, the incident angle, θin must be smaller than the so called critical angle θc, such that 𝜃𝜃𝑖𝑖𝑛𝑛 ≤ 𝜃𝜃𝑐𝑐 = √2𝛿𝛿. This is a very useful phenomenon since under this condition only an evanescent wave propagates below the surface and hence surface sensitivity is considerably enhanced.
The reflection and transmission of X-ray can then be treated as a classical problem of reflection of an electromagnetic wave at an interface.
The Fresnel relationship for a flat surface yields,
Experimental techniques and data analysis 33
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𝑅𝑅𝐹𝐹(𝑞𝑞𝑧𝑧) = ��𝑞𝑞𝑧𝑧 − �𝑞𝑞𝑧𝑧2 − 𝑞𝑞𝑐𝑐2 −32𝑖𝑖𝜋𝜋2𝑖𝑖 𝜆𝜆2 𝑞𝑞𝑧𝑧 +�𝑞𝑞𝑧𝑧2 − 𝑞𝑞𝑐𝑐2 −32𝑖𝑖𝜋𝜋2𝑖𝑖
𝜆𝜆2
��
2
,
with the wave vector transfer q = (0,0,qz= 4πSinθ/λ).
For a medium consisting of slabs of different electron densities and thickness which is shown in Fig. 2.11(a), the reflected intensity can be calculated by applying the boundary condition of the electric and magnetic fields at each interface. This leads to the dynamical theory which takes multiple reflections into account. The dynamical theory is described in detail in the monographs. A recursive formalism was presented the first time by Parratt23, 24 and reflected intensity is given by22
𝑋𝑋𝑗𝑗 = �𝑟𝑟𝑗𝑗�2
�𝑡𝑡𝑗𝑗�2 =𝑒𝑒−2𝑖𝑖𝑘𝑘𝑧𝑧,𝑗𝑗𝑍𝑍𝑗𝑗 𝑟𝑟𝑗𝑗,𝑗𝑗+1 +𝑋𝑋𝑗𝑗+1𝑒𝑒2𝑖𝑖𝑘𝑘𝑧𝑧,𝑗𝑗+1𝑧𝑧𝑗𝑗 1 +𝑟𝑟𝑗𝑗,𝑗𝑗+1𝑋𝑋𝑗𝑗+1𝑒𝑒2𝑖𝑖𝑘𝑘𝑧𝑧,𝑗𝑗+1𝑧𝑧𝑗𝑗 With Fresnel coefficients 𝑟𝑟𝑗𝑗,𝑗𝑗+1 = 𝑘𝑘𝑘𝑘𝑧𝑧,𝑗𝑗−𝑘𝑘𝑧𝑧,𝑗𝑗+1
𝑧𝑧,𝑗𝑗+𝑘𝑘𝑧𝑧,𝑗𝑗+1
However, real surfaces and interfaces have a certain roughness which can be accounted for by the modified Fresnel coefficient given by
𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑙𝑙 = 𝑟𝑟𝑖𝑖𝑑𝑑𝑒𝑒𝑟𝑟𝑙𝑙.𝑒𝑒−2𝑘𝑘𝑖𝑖𝑧𝑧𝑘𝑘𝑓𝑓𝑧𝑧𝜎𝜎2, where σ is the roughness of the interface.
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Fig. 2.11: Schematic representation of (a) the recursive formalism by Parrat and (b) the accessible q-area with specular and rocking scan.
The reflectivity in above equation does present periodic oscillations in reciprocal space with ∆𝑞𝑞𝑧𝑧,𝑗𝑗 = 2𝜋𝜋/𝐷𝐷𝑗𝑗. The oscillations are the result of constructive interferences between the reflected waves at the interfaces and their period gives the thickness 𝐷𝐷𝑗𝑗, of the respective layer.
If a film has a periodic structure in the out-of-plane direction the path difference between the reflected waves from two neighboring scattering planes give rise to Bragg reflections according to Bragg condition, 𝑛𝑛𝜆𝜆 = 2𝑑𝑑𝑓𝑓𝑖𝑖𝑙𝑙𝑓𝑓 sin𝜃𝜃𝐵𝐵𝑟𝑟𝑟𝑟𝐵𝐵𝐵𝐵.
A schematic representation of the specular reflectivity geometry is shown in Fig. 2.12. The incident and reflected beams are in one plane and its normal is perpendicular to the normal of substrate surface.
Experimental techniques and data analysis 35
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Fig. 2.12: Schematic diagram of (a) specular and (b-d) grazing incidence XRD geometry.(a) For specular geometry θi=θf and the normal of the plane spanned by ki and kf. (b) Below the critical angle θc the incoming-ray beam form an evanescent wave which is exponentially damped along the surface normal and which has a component along the interface which give rise to scattering in-plane.
(c) and (d) represent a side view and top view of GIDX geometry.
(iii) Grazing incidence X-ray diffraction
As mentioned already above, for 𝜃𝜃𝑖𝑖𝑛𝑛 < 𝜃𝜃𝑐𝑐 the X-ray reflection is accompanied by an evanescent wave field which propagates along the interface between the two media. The amplitude of this evanescent wave is exponentially damped along the inward surfaces normal with a decay length Λ (typically in the range of few nm)
𝐸𝐸𝑡𝑡~𝑒𝑒𝑖𝑖𝑘𝑘‖𝑟𝑟‖𝑒𝑒(𝑧𝑧⁄ʌ)
Therefore, GIDX is a surface sensitive diffraction method and by varying the angle of incidence the probed depth of the material can be adjusted. A
36 Chapter II
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schematic representation of grazing incidence geometry is shown in Fig.
2.12(b), Fig. (c) and (d) show a side and top view of GIDX geometry respectively. Since the evanescent wave is propagating laterally, scattering planes in the surface plane can give rise to Bragg reflections.25