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Surface Phase Diagrams Including Anharmonic Effects via a Replica-Exchange Grand-Canonical Method

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Zhou,  Yuanyuan
NOMAD, Fritz Haber Institute, Max Planck Society;

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Zhou, Y. (2020). Surface Phase Diagrams Including Anharmonic Effects via a Replica-Exchange Grand-Canonical Method. PhD Thesis, Technische Universität, Berlin.


Cite as: http://hdl.handle.net/21.11116/0000-0007-EFD5-C
Abstract
Phase diagrams of surfaces in a reactive atmosphere provide detailed information on surface composition and structure at thermodynamic equilibrium at realistic conditions, e.g., temperature (T ) and pressure (p) of the reactive gas. The atomistic structure is a prerequisite to understand and control electronic properties and function of surfaces. For decades, “ab initio atomistic thermodynamic” (aiAT) has been very successful in predicting phase diagrams for surfaces and gas-phase clusters at realistic T, p conditions. The approach was introduced by Scheffler in 1988 and reviewed by Reuter and Scheffler in 2005. However, aiAT phase diagrams usually rely on two approximations: One is that the phase space only consists of a pre-determined list of possible (to be tested) struc- tures compiled by means of informed guess by researchers; the other is that often (but not necessarily) the vibrational contributions from free-energy difference of both sub- strate and adsorbate are neglected. These approximations do not always yield accurate phase diagram, especially at high temperature and/or coverage. In contrast, an unbi- ased sampling of the configurational and compositional space could reveal unexpected (metastable) structures. The work in this thesis paves the way towards calculating sur- face phase diagrams taking accurately into account all anharmonic contributions (e.g., configurational and vibrational entropy), through the unbiased configurational sampling. To this end, we have developed a Replica-Exchange (RE) Grand-Canonical (GC) algorithm that enables the unbiased calculation of complete temperature-pressure phase diagrams of surfaces or clusters in reactive atmospheres including anharmonic effects. Moreover, the multi-canonical sampling within the given model Hamiltonian yields the T -p dependence of all equilibrium observables, e.g., the radial distribution function, when post-processed with the multistate-Bennet-acceptance-ratio (MBAR) approach. MBAR is the Bolzmann-reweighting-based lowest-variance unbiased estimator of both free en- ergies and ensemble average, introduced in 2008 by Shirts and Chodera. If the unbiased configurational sampling is rigorously conducted in the grand-canonical ensemble, all vibrational contributions can be accurately accounted for. Our approach is demonstrated for a model of Lennard-Jones system describing a surface in contact with a gas phase. Furthermore, the algorithm is applied to Si M clusters (M = 2, 4) in contact with an H 2 atmosphere, with all interactions described at the ab initio level, (density-functional theory with generalized gradient corrected exchange-correlation functional). In both cases, we identify the thermodynamically stable phases at T, p conditions. As an example of the insight one can achieve by analyzing configurations sampled via REGC, we inspect the formation of regular vs amorphous structures for Lennard-Jones surface adsorbates and we analyze the order-disorder phase transitions. Moreover, the T -p map for other observables (e.g., number of chemisorbed atoms/molecules and HOMO-LUMO gap) can be evaluated without further sampling. We also analyze at which conditions the aiAT approach yields a good approximation of T -p phase diagrams. Finally, we apply REGC to the study of the phases of Si(100) surface in the H 2 atmosphere. The focus of this study is the characterization of the sur- face and surface+adsorbate structures. The coordination histogram is adopted as the descriptor to distinguish the surface and surface+adsorbate structures with the same composition. This yields a phase diagram populated by several distinct phases includ- ing the H-saturated Si(100)-(3 × 1) phase, which is identified to be thermodynamically stable at around 380 K, in agreement with the reported experimental results. Moreover, we analyze the order-disorder phase transitions and estimate the phase boundary. These results are the first step, though, as performed only on 3 × 3 lateral supercell and needed to confirmed for a larger system. The approach introduced in this thesis can be computationally expensive, espe- cially when interactions are described at the ab initio level, but it is by construction embarrassingly parallel as the replicas do not communicate among each other except for the inexpensive exchange of thermodynamic variables (T and chemical potential of the gas phase) at each swap. Furthermore, in post-production it allows for diverse analy- ses, not necessarily planned before starting the unbiased sampling. Overall, our results demonstrate that the method presented in this thesis is a rigorous, innovative approach for studying the phase stability of surfaces and clusters at reactive atmosphere in an automated fashion.