Atomic resolution
HAADF STEM
tomography using prior physical knowledge and simulated annealing
Authors: Lobato Ivan , De Beenhouwer Jan , Van Dyck Dirk , Van Aert Sandra , Sijbers Jan 
Published: December 2016

Atomic resolution electron tomography using HAADF STEM has become a key tool to get 3D atomic‐scale structural information about the sample under study [1–3]. Different reconstruction algorithms exist including filtered back projection, simultaneous iterative reconstruction (SIRT), discrete tomography [4, 5] and total variation minimization [2]. However, most of these reconstruction techniques do not include prior knowledge concerning the atomistic building blocks of the specimen and the electron specimen interaction. A successful attempt to use atomistic prior knowledge of the specimen in the reconstruction was realized by Goris et al. [3] in which each atom is modeled by a 3D Gaussian function. However, the physical knowledge about the electron specimen interaction was not included. In order to partially overcome these problems, we modelled the specimen as a linear combination of spherical symmetric real functions, which are obtained from HAADF STEM simulations of a single atom. Furthermore, a distance constraint is included which guarantees that the distance between atoms is kept above a physical lower bound. The minimization of the cost function is performed using the simulated annealing technique [6]. The cost function is defined as the sum of the squared differences between the forward model and the projection images plus a Tikhonov regularization term. The advantage of using simulated annealing over other methods is that it statistically guarantees to find (an approximation of) the global optimum and that it allows one to process cost functions with a high degree of nonlinearity, arbitrary boundary conditions, and constraints imposed on the solution [7].

The proposed simulated annealing algorithm was demonstrated on a simulated tomography tilt series consisting of 9 projection images with a limited angular tilt range of 120 degrees of a Au nanoparticle consisting of 6525 atoms. Images were generated using the frozen lattice approach with the MULTEM software [8] with a numerical real space grid of 2048x2048 pixels and the following microscope settings: acceleration voltage (300keV), spherical aberration (0.001mm), defocus (14.03Å) and aperture objective radius (21mrad). The frozen atom simulation is performed by using the Einstein model with 20 configurations, slice thickness of 1Å and the three‐dimensional
rms
displacements of all the atoms are set to 0.085Å. An ideal detector sensitivity is used with 40mrad and 95mrad for the inner and outer circular detector angles, respectively. An area which covers the whole nanoparticle was scanned with a pixel size of 15pm. This image was later convoluted with a Gaussian low pass filter (source broadening) with full width at half maximum of 0.8Å. Poisson noise was generated such that the signal‐to‐noise ratio (SNR) in the resulting images equals 7.  The SNR is defined as the ratio of the standard deviation of the image to the standard deviation of the noise. An example of such a simulated image is shown in Figure 1a.

The result of the simulated annealing based reconstruction method is shown in Figure 1b. The evolution of the cost function during the minimization process is shown in Fig. 2. When comparing position coordinates of all atoms in the reconstructed particle with the input parameters, it has been found that the average distance is less than 8 pm, demonstrating subpixel accuracy.