Theory of low-energy electron diffraction for detailed structural determination of nanomaterials: Ordered structures

Abstract

To enable the determination of detailed structures of nanomaterials, we extend the theory of low-energy electron diffraction ͑LEED͒ to become more efficient for complex and disordered systems. Our new cluster approach speeds up the computation to scale as n log n, rather than the current n 3 or n 2 , with n the number of atoms, for example, making nanostructures accessible. Experimental methods to measure LEED data already exist or have been proposed. Potential application to ordered nanoparticles are illustrated here for C 60 molecules adsorbed on a Cu͑111͒ surface, with and without coadsorbed metal atoms, as well as for adsorbed carbon nanotubes. These demonstrate sensitivity to important structural features such as size and deformation of the nanostructures.

Theory of low-energy electron diffraction for nanomaterials—subclusters, automated searches

“…We briefly describe two computationally efficient methods to solve the multiple scattering problem in LEED [1][2][3]: an approximate grid-based method called Sparse-Matrix Canonical Grid or SMCG method; and the approximate "UV" method. Both approaches are iterative and rely in LEED on sufficient inelastic damping of electron wave amplitudes to compensate for the strong elastic scattering.…”

“…SMCG outperforms UV for large numbers of atoms and large interatomic distances. But we also found that a more conventional approach, the Conjugate Gradient (CG) method [14], is more efficient than either SMCG or UV for small numbers of atoms and small interatomic distances [1][2][3]. Our NanoLEED code therefore combines these three methods and selects the most efficient one at each stage: a single structure often includes near, intermediate and distant neighbors, so different methods are applied to different pairs of atoms in the same structure.…”

“…In our applications of NanoLEED, we have found that occasionally the iterations involved in SMCG and UV do not converge well. For example, this happened with SiH 3 clusters which were used to terminate the surface of a silicon nanowire [3]. Removing the hydrogen solved the non-convergence: it may come as a surprise that hydrogen, usually viewed as a weak scatterer, causes poor convergence in multiple scattering, with only a weak dependence on which phase shifts we used for H!…”

“…In recent publications [1][2][3], we have introduced efficient new approaches for calculating intensities of low-energy electron diffraction (LEED) for complex nanostructures, which we collectively call NanoLEED, after the name of our computer code. The accuracy, structural sensitivity and performance of the approximations used were demonstrated for ordered structures -pure C 60 , endohedral C 60 , exohedral C 60 and pure carbon nanotubes adsorbed on Cu(111) -and also for disordered finite-sized structures -silicon nanowires -with various shapes and distortions.…”

“…The accuracy, structural sensitivity and performance of the approximations used were demonstrated for ordered structures -pure C 60 , endohedral C 60 , exohedral C 60 and pure carbon nanotubes adsorbed on Cu(111) -and also for disordered finite-sized structures -silicon nanowires -with various shapes and distortions. The performance gain leads to nlogn scaling of the computational time, instead of the standard n 3 or n 2 scaling, where n is the number of inequivalent atoms, for example. This allows much more complex structures to be analyzed than with conventional LEED methods.…”