Elsevier

Ultramicroscopy

Volume 128, May 2013, Pages 55-67
Ultramicroscopy

A full-scale simulation approach for atom probe tomography

https://doi.org/10.1016/j.ultramic.2013.01.005Get rights and content

Abstract

A versatile approach for simulation of APT measurements is presented. The model is founded on a Voronoi cell partition of 3D space. The partition is used in dual role: First, the atomic structure of the field emitter is depicted in a one to one relationship by single Wigner–Seitz cells. Second, the construction of an adaptive tetrahedral mesh enables solving the Poisson equation on length scales covering seven orders of magnitude. Ion trajectories are computed in full-length comparable to experiments. Contrary to former simulation approaches the sequence of desorbing atoms is determined by field-induced polarization forces.

Both results for cubic lattices in 〈001〉, 〈011〉, and 〈111〉 orientation are presented and the simulation of an APT measurement of a complex crystalline/amorphous layer structure is demonstrated. The example of a grain boundary addresses the new possibility of constructing models with structural defects. In this case, the simulation reveals strong artifacts in the reconstruction even if homogenous evaporation threshold is assumed.

Highlights

► Use of an adaptive mesh enables full-scale solution of the Poisson equation. ► Simulation of emitter structures is controlled by field-induced “polarization forces”. ► Simulated evaporation of a grain boundary resolves strong artefacts.

Introduction

Atom probe tomography (APT) is a unique analysis tool in materials science. Due to its outstanding spatial resolution and its combination with time-of-flight (TOF) mass spectrometry, the method enables a detailed insight into the three dimensional chemical composition at the nano scale [1]. Fundamental to the method is the field-induced evaporation of solid state materials in the presence of extraordinary high electric fields. In APT, ionized atoms are consecutively desorbed from the apex of needle-shaped field emitters. The small curvature radius of the latter allows obtaining a sufficiently high electric field through only moderately high voltages. Also, the tip apex acts as the decisive projective system as trajectories of evaporated species are controlled by the electrostatic field distribution around the emitter. Post processing of the event positions measured by a 2D detector gives access to the desorbed volume.

APT inherits fundamental properties from its predecessor, the field ion microscopy technique (FIM). Already shortly after invention of this technique, people started to develop simulation tools in order to compare with experimental micrographs and to understand contrast formation in FIM. The applied models relied on primarily geometric considerations [2], [3].

The first systematic approach for the simulation of APT measurements was introduced by Vurpillot et al. about a decade ago [4]. For the first time, this pioneering work evaluated the 3D field distribution in the vicinity of a rough field emitter and calculated ion trajectories of field desorbed species. The approach has been based on the solution of the Laplace equation on a regular cubic grid of support points. Also the modelled emitter structure has been composed of cubic shaped atomic cells. Once the potential for the initial input structure has been obtained, the dynamic desorption of atoms is simulated by repetitive execution of three basic steps: First, the algorithm looks for the surface cell which is exposed to the maximum field strength and removes it, i.e. the particular bulk cell becomes a “vacuum” cell. Different thresholds for desorption are taken into account by appropriate scaling of the surface field. Second, the potential is recalculated for the new configuration. And third, the ion-trajectory starting from the origin of the removed atom is computed by integration of Newton´s equation of motion.

Since publication of this general approach for APT simulation, its feasibility has been approved by various studies for the case of precipitates, thin films, and alloys [5], [6], [7]. This way, artefacts in the tomographic reconstruction due to irregular emitter shapes or distinguished desorption properties have been revealed. Inspired by Vurpillot´s work, other groups also started to develop their own simulation codes which essentially follow the same concept. For example, Marquis et al. [8] investigated the tip shape changes during APT measurements of complex multilayer samples in a correlative study with transmission electron microscopy (TEM) and compared experimental APT and TEM results with data derived from simulation. In a theoretical work, Larson et al. [9] showed how APT simulation can be used in order to test the quality of APT reconstruction algorithms.

Despite these successful applications, the need for equidistant grid points puts a severe restriction on possible simulations. Due to the large number of required support points, the size of the simulation system is restricted to about twice the diameter of the evaluated field emitter (≈1003 nm3). For the same reason, the atomic structure of the emitter has to be compatible to a simple cubic lattice of support points.

Recently, Niewieczerzal et al. presented a static multi-scale model in order to predict the image formation for faceted crystals in FIM [10], [11]. In this work the Laplace equation has been already solved on an adaptive mesh based on the finite element method bridging three orders of magnitude in length scale. In a similar way, Haley et al. simulated ion trajectories based on field emitter shapes that were experimentally obtained by TEM tomography [12].

Continuing Vurpillot´s work of a dynamic APT simulation, we present here a new approach which allows the numerical generation of APT raw data including ion trajectories quite comparable to real measurements. Key characteristic is the dual use of a basic mesh of Voronoi cells: On the one hand, this enables an easy representation of the emitter structure because each atom may be directly associated with a Voronoi cell. Therefore the presented framework overcomes any limitation in structure and morphology of possible samples. On the other hand, the three dimensional electrostatic solution, covering seven orders of magnitude in length scale, is computed based on the same mesh of Voronoi cells. Arbitrary local accuracy is enabled due to the adaptive distribution of Voronoi cells in addition to the emitter cells. The only purpose of these cells is to establish the needed support for extending the solution to the full-scale. In a substantially different view, the simulation code considers the field-induced force instead of the field strength to be responsible for atom desorption from the emitter surface.

Section snippets

Geometrical aspects of the Voronoi and Delaunay tessellation

In order to avoid complexity, for the most part the following description will be based on 2D as the extension to 3D remains straight forward.

The Voronoi tessellation of an arbitrary set of points {pi} within a given domain Ω, piΩR2, is constructed by the union of sub-domains V(pi) (Fig. 1a). Point pi is called the generator and Vi is called the Voronoi cell of point pi. Each cell of the Voronoi tessellation is constructed by the intersection of half spacesVi:={pR2:|pi,p|<|pj,p|,ji}

with |.,

Results

Based on the described algorithm, a complete simulation package, “TAPSim”, is offered for free download [26]. The program code can be executed on any modern desktop computer. Typical simulation rates of 300 atoms per minute are reached. In the following section, we present exemplary results obtained with this package.

Discussion

Simulation of field evaporation represents a promising way to improve the understanding of atom probe measurements. However, in order to correlate experimental and simulated data in a reasonable manner, simulated structures should adequately describe all defect features of the samples. In this regard our presented flexible approach is superior to the established scheme introduced by the work of Vurpillot et al. The main weaknesses of the former approach were (i) the description of atomic cells

Conclusion

A flexible approach for simulation of APT measurements is presented. The approach is based on the construction of arbitrary Voronoi cells and offers a well-defined description for real atomic structures of simulated field emitters including arbitrary structural defects.

  • The use of an adaptive support mesh enables a full-scale solution of the Poisson equation. The size of the simulated space is comparable to real measurement conditions. Thus, realistic ion trajectories according to experimental

Acknowledgements

The authors thank F. Vurpillot for his motivating openness in discussing aspects of APT simulation during his stay in Münster, July 2012. Furthermore we thank T. Papenkort for fruitful discussions and the assistance in dealing with some programming problems.

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