An improved method assigning three-dimensional atomic potentials to multiple slices in exit-wave simulations of Transmission Electron Microscopy

ToTEM: A software for fast TEM image simulation

“…Unfortunately, parameterised Gaussian functions like in Equation () cannot provide the enough accuracy at the low and high frequency synchronously. Thus, ToTEM contains a $${f_r}( {\vec s} )$$ to amend this error, 24 $$\begin{equation}{f_r}\left( {\vec s} \right) = {f_L}\left( {\vec s} \right) - {f_P}\left( {\vec s} \right)\!,\end{equation}$$where $${f_L}( {\vec s} )$$ and $${f_P}( {\vec s} )$$ are the scattering factors calculated from Lobato's and Peng's coefficients, respectively.…”

“…In the multislice method, as the atomic potential spreads in 3D space, correct assignment of the potential into the slices is crucial to compute the accurate phase change of the exit wave, especially when all atoms vibrate randomly in entire space accounting for the thermal effect. The recently proposed multislice method is implemented in ToTEM to achieve the precise assignment 24 . Figure 1 is the simplification of the new multislice strategy.…”

“…The potential contributing to the k th slice can be integrated from the altitude $$z_k^{i^{\prime}}$$ to $$z_k^{i^{\prime\prime}}$$, given by, 7,24 $$\begin{eqnarray} \def\eqcellsep{&}\begin{array}{l} {V_{z_k^{i^{\prime}} \to z_k^{i^{\prime\prime}}}}\left( {x,y} \right) = \mathop \int \limits_{z_k^{i^{\prime}}}^{z_k^{i^{\prime\prime}}} V\left( {x,y,z} \right)dz\\ = - \pi {a_0}e\mathop \sum \limits_{i = 1}^5 {a_i}\left( {\frac{{4\pi }}{{{b_i}}}} \right)\left| {\rm{erf}}\left( {\frac{{2\pi z_k^{i^{\prime}}}}{{\sqrt {{b_i}} }}} \right)\right.\\ \quad \left.-\; {\rm{erf}}\left( {\frac{{2\pi z_k^{i^{\prime\prime}}}}{{\sqrt {{b_i}} }}} \right) \right|\exp \left[ { - \frac{{4{\pi ^2}({x^2} + {y^2})}}{{{b_i}}}} \right]\!, \end{array} \end{eqnarray}$$where erf is the Gauss error function. Equation () is represented in the real space, a 0 is the first Bohr radius and $${a_i}$$ and $${b_i}$$ are...…”

“…FL and FP have been frequently implemented in most recent simulation programs. Moreover, the slicing strategy is still crucial in the simulation because it determines the scattering contribution of every atom to different slices 24,9,12,25–28 . A general approach is to establish a three‐dimensional (3D) potential matrix for an atom in real space and extract the corresponding scattering potential for those slices enclosing that atom, which has been realised in some simulation tools 3,8,16 …”

“…Moreover, the slicing strategy is still crucial in the simulation because it determines the scattering contribution of every atom to different slices. 24,9,12,[25][26][27][28] A general approach is to establish a three-dimensional (3D) potential matrix for an atom in real space and extract the corresponding scattering potential for those slices enclosing that atom, which has been realised in some simulation tools. 3,8,16 A new software ToTEM with a friendly graphical user interface (GUI) is developed to fulfil the simulation requirements.…”