# Introduction

X-ray reflection (XRR) is a non-destructive analytical technique based on X-ray scattering. It can be used to determine the thickness, roughness, and density of single layer and multi layer thin films and surfaces.

# Working principle

In XRR, an X-ray beam is focused on a sample surface at a low angle and the intensity of the reflected beam is measured at an equal opposing angle. The density profile of the surface can be derived from the detected variation in intensity using the law of Fresnel reflectivity. X-rays focused on the sample at low glancing angles are reflected almost completely. Total external reflection is achieved at angles lower than the critical angle $$a_c$$ as shown in the lower part of Figure 1. Snell's law states that the incident angle  $$\alpha$$ and the refracted/reflected angle  $$\alpha'$$ are related by the equation

$\cos \alpha = n \cos \alpha'$

The Fresnel equations state that

$r = \frac{a_R}{a_I} = \frac{\alpha - \alpha'}{\alpha + \alpha'}; \quad t = \frac{a_T}{a_I} = \frac{2 \alpha}{\alpha + \alpha'}$

where $$r$$ is the amplitude reflectivity, $$t$$ is the amplitude transmittivity, $$a_I, a_R, \text{and } a_T$$ are the amplitudes of the incident, reflected, and refracted beams, respectively .

With angles greater than the critical angle, some of the X-rays propagate into the material, as shown in Figure 2. X-rays are electromagnetic radiation with a wavelength around 1 Å, so they refract in the solid material with a refractive index $$n$$ . The refractive index can be written generally as

$n = 1 - \delta + i \beta$

where $$\delta$$ and $$\beta$$ are refractive constants that depend on the material, the wavelength of the incident beam, and the attenuation coefficient, and $$i$$  is the imaginary unit. The magnitude of $$\delta$$  is usually in the range of 10-5 and the magnitude of $$\beta$$ is usually much smaller .

# Usage

The sample is placed in the sample holder and an appropriate attenuator (usually Cu or Ni), slits, and other devices are set up to control the X-ray beam coming from the source. The setup depends on the manufacturer. An example setup of PANalytical X'Pert Pro is presented in Figure 3. The whole device is presented in Figure 4.

Before the actual measurement, the detector angle and the target placement must be calibrated. After the calibration, the measurement is performed at chosen angles.

Figure 3. Setup of the XRR device PANalytical X'Pert Pro. Figure: Fabian Krahl.

The gathered data yields a plot of the measured intensities at different angles. This data is analyzed by curve fitting using a computer software, such as PANalytical X'Pert Reflectivity. Different curve fitting parameters, such as densities and film thicknesses, are adjusted and optimized to get as precise fit as possible. An example of measurement data is present in Figure 5.

Figure 5. A measured intensity curve of a ZnO thin film deposited on silicon using atomic layer deposition. Figure: Heikki Lappalainen.

# Applications

XRR is a convenient and quick method to analyze thin films and surfaces. For example thin films deposited by atomic layer deposition (ALD) technique can be characterized by XRR. The film thicknesses, as well as densities, and the roughness of the interfaces, can be determined. This also applies for organic/inorganic superlattices deposited by molecular layer deposition (MLD).

XRR is a fast and convenient tool for characterizing thin films and surfaces. It can be used to determine the film thickness, roughness, and density. However, XRR does not provide information about the crystal structure of the material, and multilayered films can be only characterized up to a limited depth.

 1 1 2 V. Holý, J. Kuběna, I. Ohlídal, X-ray reflection from rough layered systems, Physical Review B, 1993, 47, 15896-15903. DOI: 10.1103/PhysRevB.47.15896. 2 1 2 3 4 5 J. Als-Nielsen, D. McMorrow, Elements of Modern X-Ray Physics, Wiley, UK, 2011. 3 1 2 J. Daillant, A. Gibaud, X-Ray and Neutron Reflectivity: Principles and Applications, Springer, Berlin Heidelberg, 2009, DOI: 10.1007/978-3-540-88588-7.

Figure 1. Total reflection is achieved with angles lower than the critical angle. Figure: Heikki Lappalainen.

Figure 2. Reflectivity according to Fresnel reflectivity. Figure: Heikki Lappalainen.

Figure 4. PANalytical X'Pert Pro XRR device. Figure: Heikki Lappalainen.

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