# X-ray reflectivity

(Redirected from X-ray reflectometry)

X-ray reflectivity (sometimes known as X-ray specular reflectivity, X-ray reflectometry, or XRR) is a surface-sensitive analytical technique used in chemistry, physics, and materials science to characterize surfaces, thin films and multilayers.[1][2][3][4] It is related to the complementary techniques of neutron reflectometry and ellipsometry.

Diagram of x-ray specular reflection

The basic idea behind the technique is to reflect a beam of x-rays from a flat surface and to then measure the intensity of x-rays reflected in the specular direction (reflected angle equal to incident angle). If the interface is not perfectly sharp and smooth then the reflected intensity will deviate from that predicted by the law of Fresnel reflectivity. The deviations can then be analyzed to obtain the density profile of the interface normal to the surface.

The technique appears to have first been applied to x-rays by Lyman G. Parratt in 1954.[5] Parratt's initial work explored the surface of copper-coated glass, but since that time the technique has been extended to a wide range of both solid and liquid interfaces.

The basic mathematical relationship which describes specular reflectivity is fairly straightforward. When an interface is not perfectly sharp, but has an average electron density profile given by ${\displaystyle \rho _{e}(z)}$, then the X-ray reflectivity can be approximated by:[2]:83

${\displaystyle R(Q)/R_{F}(Q)=\left|{\frac {1}{\rho _{\infty }}}{\int \limits _{-\infty }^{\infty }{e^{iQz}\left({\frac {d\rho _{e}}{dz}}\right)dz}}\right|^{2}}$

Here ${\displaystyle R(Q)}$ is the reflectivity, ${\displaystyle Q=4\pi \sin(\theta )/\lambda }$, ${\displaystyle \lambda }$ is the x-ray wavelength (typically copper's K-alpha peak at 0.154056 nm), ${\displaystyle \rho _{\infty }}$ is the density deep within the material and ${\displaystyle \theta }$ is the angle of incidence. Below the critical angle ${\displaystyle Q (derived from Snell's law), 100% of incident radiation is reflected, ${\displaystyle R=1}$. For ${\displaystyle Q\gg Q_{c}}$, ${\displaystyle R\sim Q^{-4}}$. Typically one can then use this formula to compare parameterized models of the average density profile in the z-direction with the measured X-ray reflectivity and then vary the parameters until the theoretical profile matches the measurement.

For films with multiple layers, X-ray reflectivity may show oscillations with wavelength, analogous to the Fabry-Pérot effect. These oscillations can be used to infer layer thicknesses, interlayer roughnesses, electron densities and their contrasts, and complex refractive indices (which depend on atomic number and atomic form factor), for example using the Abeles matrix formalism or the recursive Parratt-formalism as follows:

${\displaystyle X_{j}={\frac {R_{j}}{T_{j}}}={\frac {r_{j,j+1}+X_{j+1}e^{2ik_{j+1,z}d_{j}}}{1+r_{j,j+1}X_{j+1}e^{2ik_{j+1,z}d_{j}}}}e^{-2ik_{j,z}d_{j}}}$

where Xj is the ratio of reflected and transmitted amplitudes between layers j and j+1, dj is the thickness of layer j, and rj,j+1 is the Fresnel coefficient for layers j and j+1

${\displaystyle r_{j,j+1}={\frac {k_{j,z}-k_{j+1,z}}{k_{j,z}+k_{j+1,z}}}}$

where kj,z is the z component of the wavenumber. For specular reflection where the incident and reflected angles are equal, Q used previously is two times kz because ${\displaystyle Q=k_{incident}+k_{reflected}}$. With conditions RN+1 = 0 and T1 = 1 for an N-interface system (i.e. nothing coming back from inside the semi-infinite substrate and unit amplitude incident wave), all Xj can be calculated successively. Roughness can also be accounted for by adding the factor

${\displaystyle r_{j,j+1,rough}=r_{j,j+1,ideal}e^{-2k_{j,z}k_{j+1,z}\sigma _{j}^{2}}}$

where ${\displaystyle \sigma }$ is a standard deviation (aka roughness).

Thin film thickness and critical angle can also be approximated with a linear fit of squared incident angle of the peaks ${\displaystyle \theta ^{2}}$ in rad2 vs unitless squared peak number ${\displaystyle N^{2}}$ as follows:

${\displaystyle \theta ^{2}=({\frac {\lambda }{2d}})^{2}N^{2}+\theta _{c}^{2}}$.

## References

1. ^ V. Holy et al., Phys. Rev. B. 47, 15896 (1993).
2. ^ a b J. Als-Nielsen, D. McMorrow, Elements of Modern X-Ray Physics, Wiley, New York, (2001).
3. ^ J. Daillant, A. Gibaud, X-Ray and Neutron Reflectivity: Principles and Applications. Springer, (1999).
4. ^ M. Tolan, X-Ray Scattering from Soft-Matter Thin Films, Springer, (1999).
5. ^ L. G. Parratt, Phys. Rev. 95, 359 (1954).