Imaging surface topography with coherent x-ray reflectivity: Theory, kinematics, and simulations

A theoretical formalism is described for understanding coherent x-ray reflectivity (CXR) from the surface of a semi-infinite crystal having a variable surface topography, described by the height profile <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>h</a:mi> </a:math> ( <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>x</b:mi> <b:mo>,</b:mo> <b:mi>y</b:mi> </b:mrow> </b:math> ). The surface topography is imaged as a complex “effective density,” obtained from the phasing and inversion of the coherent x-ray reflectivity data, measured through a rocking scan centered at a vertical momentum transfer <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msubsup> <c:mi>Q</c:mi> <c:mi>z</c:mi> <c:mn>0</c:mn> </c:msubsup> </c:math> and a vertical range <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi mathvariant="normal">Δ</d:mi> <d:msub> <d:mi>Q</d:mi> <d:mi>z</d:mi> </d:msub> </d:mrow> </d:math> . The formalism predicts that the effective density has an amplitude with a maximum located at the surface height for each position within the surface plane. The phase of the effective density has a lateral variation that is controlled by the surface height and a vertical variation that reflects a combination of the interfacial structure and specific choice of measurement conditions. This understanding enables direct observation of nanometer-scale interfacial topography, i.e., <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>h</f:mi> <f:mrow> <f:mo>(</f:mo> <f:mrow> <f:mi>x</f:mi> <f:mo>,</f:mo> <f:mi>y</f:mi> </f:mrow> <f:mo>)</f:mo> </f:mrow> <f:msub> <f:mi>c</f:mi> <f:mi>s</f:mi> </f:msub> </f:mrow> </f:math> (where <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msub> <g:mi>c</g:mi> <g:mi>s</g:mi> </g:msub> </g:math> is the vertical substrate lattice parameter) with <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mspace width="0.16em"/> <h:mi>Å</h:mi> </h:mrow> </h:math> -scale sensitivity to surface height. Numerical simulations illustrate and confirm the theoretical results. These results show how the interpretation of the interfacial density phase obtained by CXR data inversion (i.e., surface topography with respect to a flat surface) is conceptually similar to that previously known for Bragg coherent diffraction imaging (BCDI) measurements of isolated nanoparticles (i.e., lattice displacements with respect to an ideal crystal lattice). This suggests that CXR can be thought of as a form of dark field imaging with respect to the bright field BCDI approach. An implication of these results is that interfacial imaging may bypass some of the significant challenges associated with BCDI imaging of multiple particles having different orientations.