Multislice - Theory

Theory

The form of multislice algorithm presented here has been adapted from Peng, Dudarev and Whelan 2003. The multislice algorithm is an approach to solving the Schrödinger wave equation:

$\begin{align} -\frac{\hbar^2}{2m} \frac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) &=E\Psi(x,t) \end{align}$

In 1957, Cowley and Moodie showed that the Schrödinger equation can be solved analytically to evaluate the amplitudes of diffracted beam. Subsequently, the effects of dynamical diffraction can be calculated and the resulting simulated image will exhibit good similarities with the actual image taken from a microscope under dynamical conditions. Furthermore, the multislice algorithm does not make any assumption about the periodicity of the structure, as a result this method can be used to simulate HREM images of aperiodic systems as well.

The following section will include a mathematical formulation of the Multislice algorithm. The Schrödinger equation can also be represented in the form of incident and scattered wave as:

$\begin{align} \Psi({\mathbf{r}}) &= \Psi_{0}({\mathbf{r}}) + \int{G({\mathbf{r,r'}})V({\mathbf{r'}})\Psi({\mathbf{r'}})d{\mathbf{r'}}} \end{align}$

where is the Green’s function that represents the amplitude of the electron wave function at a point due to a source at point .

Hence for an incident plane wave of the form the Schrödinger equation can be written as:

$\begin{align} \Psi({\mathbf{r}}) = \exp(i{\mathbf{k\cdot r}}) - \frac{m}{2\pi\hbar^2}\int\frac{\exp(ik\cdot {\mathbf{|r-r'|}})}{{\mathbf{|r-r'|}}} V({\mathbf{r'}})\Psi({\mathbf{r'}})dr' \end{align}$

We then choose the coordinate axis in such a way that the incident beam hits the sample at (0,0,0) in the -direction. Now we consider wave-function with a modulation function for the amplitude of the wave-function. Hence, the modulation function can be represented as:

$\begin{align} \phi({\mathbf{r}}) &= 1 - \frac{m}{2\pi\hbar^2}\int{\frac{\exp}{|{\mathbf{r-r'}}|}V({\mathbf{r'})\phi({\mathbf{r'}})}dr'} \end{align}$

Now we make substitutions with regards to the coordinate system we have adhered.

$\begin{align} {\mathbf{k}} \cdot ({\mathbf{r-r'}}) &= k(z-z') \quad \& \quad |{\mathbf{r-r'}}| \approx (z-z') + ({\mathbf{X-X'}})^2/{2(z-z')} \end{align}$

$\begin{align} \phi({\mathbf{r}}) = 1 -i\frac{\pi}{E\lambda} \int \int \limits_{z'=-\infty}^{z'=z} V({\mathbf{X'}},z') \phi({\mathbf{X'}},z') \frac{1} {i\lambda (z-z')} \exp\left(ik\frac{|{\mathbf{X-X'}}|^2}{2(z-z')}\right)d{\mathbf{X'}}dz' \end{align}$

where, is the wavelength of the electrons with energy

So far we have set up the mathematical formulation of wave mechanics without addressing the scattering in the material. The interaction constant is defined as

$\begin{align} \sigma = \pi/E\lambda \end{align}$

Further we also need to address the transverse spread which is done in terms of Fresnel propagation function

$\begin{align} p({\mathbf{X}},z) = \frac{1}{iz\lambda} \exp\left(ik\frac{{\mathbf{X}}^2}{2z}\right) \end{align}$

In multislice simulation the thickness of each slice over which the iteration is performed is usually small and as a result within a slice the potential field can be approximated to be constant . Subsequently, the modulation function can be represented as:

$\begin{align} \phi({\mathbf{X}},z_{n+1}) = \int p({\mathbf{X}}-{\mathbf{X'}}, z_{n+1}-z_{n}) \phi({\mathbf{X}},z_{n})\exp\left(-i\sigma\int\limits_{z_{n}}^{z_{n+1}}V({\mathbf{X'}},z')dz'\right)dX' \end{align}$

We can therefore represent the modulation function in the next slice

$\begin{align} \phi_{n+1} = \phi({\mathbf{X}},z_{n+1}) = *p_{n} \end{align}$

where, * represents convolution, and defines the transmission function of the slice.

$\begin{align} q_{n}({\mathbf{X}}) = \exp \{-i\sigma \int \limits_{z_{n}}^{z_{n+1}} V({\mathbf{X}},z')dz'\} \end{align}$

Hence, the iterative application of the aforementioned procedure will provide a full interpretation of the sample in context. Further, it should be reiterated that no assumptions have been made on the periodicity of the sample apart from assuming that the potential is uniform within the slice. As a result, it is evident that this method in principle will work for any system. However, for aperiodic systems in which the potential will vary rapidly along the beam direction, the slice thickness has to be significantly small and hence will result in higher computational expense.

Table 1 - Computational efficiency of Dicrete Fourier Transform compared to Fast Fourier Transform

Data Points	N	Discrete FT	Fast FT	Ratio
64	6	4,096	384	10.7
128	7	16,384	896	18.3
256	8	65,536	2,048	32
512	9	262,144	4,608	56.9
1,024	10	1,048,576	10,240	102.4
2,048	11	4,194,304	22,528	186.2

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