θ-2θ and ω-2θ X-ray Diffraction Profiles

X-ray diffraction is one of the most powerful and widely used techniques for determining the phase, crystal structure, crystallinity, thickness and orientation of thin films, etc. Combined with refinement methods, x-ray diffraction can provide detailed crystal structure with remarkable accuracy. Among the many configurations that x-ray diffraction can used in, θ-2θ and ω scans are the most popular. In case of powdered materials, θ-2θ scan provides a number of diffraction peaks which, along with peak intensities, help in identifying crystal phases and their relative concentration. In case of thin films, however, θ-2θ scans need to be replaced by ω-2θ scans. This has to do with the way substrate surface, on which thin film is grown, is prepared.

Properties of a material depend on its constituent elements and relative placement of atoms of these elements within the material. Different crystal phases of the same material exhibit quite different physical properties. However, naturally occurring stable phases of a material are quite limited in number, thus imposing a severe restriction on their use in device applications. Thin film technology is a way to circumvent this problem. Phases, which are otherwise unstable in bulk, can be crystallized by making use of epitaxial constraints from substrate surface. Here, symmetry of atomic arrangement on the surface of the substrate, and lattice mismatch between the substrate surface and deposited material play a crucial role, along with other parameters such as temperature, pressure, etc.

Epitaxial constraints, however, are often not sufficient to ensure a good quality single crystal thin film. To exercise a better control over growth dynamics, substrates surfaces are prepared at a small angle from the out-of-plane crystal planes (fig. 1). This angle is known as 'miscut angle' and the surface is known as 'vicinal surface' with high Miller indices. Vicinal surfaces, in general, consist of well defined and uniformly distributed atomic step-terrace features. Step-walls, terraces and step-edges have disparate free energies. This disparity in surface free energies combined with epitaxial constraints (and other growth parameters) facilitate a uniform crystal growth throughout the substrate surface.

Fig. 1: Preparation of a vicinal surface and its effect on thin film. Here, c represents crystal plane direction and n represents surface normal.

A film grown on such a surface is likely to have a similar surface morphology and hence, similar disparity between the crystal planes and film surface directions. Difference between θ-2θ and ω-2θ stems from this disparity.

Relevant definitions: 

1. θ - defined as angle between the incident x-ray and sample surface
2. ω - defined as angle between the incident x-ray and out-of-plane crystal axis
3. 2θ - defined as angle between the incident and scattered x-rays.


A diffraction spot is observed when incident x-ray vector k, scattered x-ray vector k' and the reciprocal lattice vector K for a specific crystal planes satisfy the following condition:

 k - k' = K

which is Bragg's law in terms of reciprocal vectors. Under the assumption of elastic scattering, the above equation leads to Bragg's diffraction law:

                                                                         nλ = 2d sin θB                                     

where θB is the Bragg angle at which the above condition is satisfied and a diffraction peak is observed. The angle between the crystal planes and incident x-rays is θB and between scattered and incident x-rays is 2θB. Thus, Bragg's law implies θ-2θ geometry, similar to the way a mirror behaves. Consequently, the x-ray source and the detector are always maintained in θ-2θ geometry and scanned over a range of θ values to detect diffraction peaks (lattice planes).


Case 1: When out-of-surface crystal planes and substrate surface are parallel

In this case, the angle between crystal planes and incident x-rays (θ) is same as that between substrate surface and incident x-rays (ω). Therefore, θ = ω (always) and hence, θ-2θ scan is same as ω-2θ scan as shown in fig. 2. (In ω-2θ scan, angle between the substrate surface surface and incident x-rays is measured and scanned over a range.)  

Fig. 2: Diffraction with substrate surface (red) and crystal planes (black) parallel.

Case 2: When out-of-surface crystal planes and substrate surface are NOT parallel

In case there is a non-zero miscut angle, the angle between crystal planes and incident x-rays (θ) that between substrate surface and incident x-rays (ω) are no longer equal. Bragg's law demands x-ray source, detector and crystal planes to be in θ-2θ geometry in order for diffraction peaks to be observed. However, now the angle (ω) between surface and incident x-rays does not equal θ (fig.2)! There is a constant 'offset' (δ) between ω and θ such that |θ - ω| = δ. As a result, if we now scan over ω, the diffraction peaks are observed at ω ≠ θ but at ω = θ ± δ. Thus, in ω-2θ scan ω ≠ (2θB)/2 implying reflection geometry (with respect to surface) does not apply. It is for this reason, we usually refer to ω-2θ scan in case of thin films whereas θ-2θ scan in scan of powdered samples.

Fig. 3: Diffraction with finite angle between substrate surface (red) and crystal planes (black)

Thus, the only difference between ω-2θ and θ-2θ profiles is the reference surface/plane used to measure the angle of incident x-rays. This, however, has no bearing on the actual 2θB which depends only on the crystal planes and incident and scattered x-ray vectors, and not the surface. For this reason, the position of peaks on both ω-2θ and θ-2θ profiles are identical.

Why bother about ω?

Before we can begin quantitative measurements on any XRD instrument, the source, the detector and the sample need to be aligned to make sure that the measured angles are correct. Initial goniometer alignment often involves measurement of angles with respect to the sample surface as it is the only observable and measurable surface (before alignment); the location of crystal planes is unknown. Once the angular positions of the detector and the source is established relative to the surface, the hunt for out-of-plane crystal planes begins (this can take hours). When the out-of-plane crystal planes are found, the offset between the surface and the crystal planes can be easily obtained. This offset is known as 'ω-offset' (denoted above by δ). If this offset is not accounted for, a shift of δ in peak position is observed in ω-scan and θ-scan (these scans are also known as 'rocking curves').

Initial alignment results in a θ-2θ reflection geometry between the source, the detector and substrate surface. By accommodating the ω-offset (i.e. δ), we establish a θ-2θ reflection geometry between the source, the detector and crystal planes.

Fig. 4: Thin film alignment.