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\author{A. Hasenkopf and J. Zweck}
\title{Development of components for\\ a ChiralSTEM -- Mode}
\institute{Universität Regensburg\\}
\email{andreas.hasenkopf@physik.uni-regensburg.de}
\leftlogo[1]{UniLogo2}
\rightlogo[1]{TEM-Logo}
\conference{14. European Microscopy Congress, Aachen, September 1--5, 2008}
\begin{document}

\maketitle
\rule{\textwidth}{2mm}

\begin{multicols}{3}

\begin{abstract}
The effect of \textbf{E}nergy-loss \textbf{M}agnetic \textbf{C}hiral \textbf{D}ichroism (EMCD) was proposed by Schattschneider et al. \cite{emcd_proposal} in analogy to XMCD due to similar scattering cross section expressions (wave vector transfer $\vec{q}$ in inelastic electron scattering $\leftrightarrow$ polarization vector $\vec{\epsilon}$ in X-ray absorption). The effect was measured for the first time by Rubino \cite{emcd_nature}.

The state of the art method for recording EMCD-spectra is the ''intrinsic method'' requiring the specimen to fulfill three strict properties (mono crystal, thickness, orientation) due to its use as a beam splitter. In order to avoid these requirements an alternative setup with a twin aperture and Boersch phase plate for generating 2 coherent electron waves with a phase difference of $\pi/2$ is suggested. Chiral contrast in energy-loss spectra is achieved due to a magnetic sample.

To determine whether a twin aperture is suitable for EMCD-measurements -- in particular in STEM-mode -- the wave function and intensity distribution in the sample plane are calculated numerically and the optimum dimensions for the aperture and phase plate are determined. Finally a STEM operation with a twin aperture is realized.
\end{abstract}

\section{Intrinsic method}
\textbf{\scshape Analogon to Circular Polarization}: two scattered coherent electron waves (wave vector transfers: $q_1$, $q_2$, $q_1\perp q_2$ or $q\,'_1$, $q\,'_2$, $q\,'_1\perp q\,'_2$) with a phase difference of $\pi/2$. 

\begin{figure}[h]
\begin{center}
\label{fig_intrinsic}
\includegraphics[width=\textwidth]{poster_aperture_einzel.png}
\caption{setup -- intrinsic method}
\end{center}
\end{figure}

\textbf{\scshape Setup}: see fig. \ref{fig_intrinsic}. 
\begin{itemize}
\item proposed by Nelhiebel \cite{emcd_intrinsic} and used in prior EMCD-measurements\cite{emcd_nature,pd_rubino,pd_hurm},
\item mono crystalline sample as beam splitter, 
\item two Bragg reflexes with phase difference $\pi/2$ in the diffraction plane according to dynamic diffraction theory (see \cite{pd_rubino,pd_hurm}).
\end{itemize}

\textbf{\scshape Deficiencies}:
\begin{itemize}
\item signal low,
\item resolution low (improved by convergent beam technique \cite{mcd_in_eels_2}),
\item long duration of measurement.
\end{itemize}

\textbf{\scshape Needed Improvements}:
\begin{itemize}
\item higher intensity within
\item small specimen area
\item[$\Rightarrow$] focussed beams $\to$ STEM?!
\end{itemize}

\section{Twin Aperture}
\textbf{\scshape Suggestion}: see fig. \ref{fig_twin}
\begin{itemize}
\item twin aperture $\to$ two coherent partial electron waves,
\item condenser lens $\to$ focussing,
\item electrostatic Boersch phase plate (annular electrode) \cite{phaseplate} $\to$ application of phase shift to one partial wave
\end{itemize}

\begin{figure}[h]
\begin{center}
\label{fig_twin}
\includegraphics[width=\textwidth]{poster_aperture_twin.png}
\end{center}
\caption{setup -- twin aperture}
\end{figure}
% one electron wave is shifted in phase such that the phase difference of the two waves in the sample plane is $\pi/2$ and such that a similar diffraction pattern is observed in the diffraction plane as for the intrinsic method. 

%Therefor the possibility of a STEM operation with such a twin aperture was investigated due to the wish to perform EMCD measurement in a STEM mode in the future.

\section{Numerical Calculations}

\textbf{\scshape Intensity Distribution}: $|\psi_{BFP}|^2$, see fig. \ref{fig_linescan}
\begin{eqnarray}
\psi_{BFP}(X,\,Y) &=& \mathrm{J}_1\left(\frac{2\pi R\sqrt{X^2+Y^2}}{\lambda f}\right)\frac{\lambda f}{\pi R\sqrt{X^2+Y^2}} \nonumber \\
&& \cdot\left( e^{-\frac{id_1X}{\lambda f}}+e^{-\frac{id_2X}{\lambda f}-i\varphi}\right),\nonumber \\
\Delta\phi(X) &=& \left((d_2-d_1)\frac{X}{\lambda f}+\varphi+\pi\right)\mathrm{mod}(2\pi)-\pi \nonumber
\end{eqnarray}

\begin{itemize}
\item in sample plane $\equiv$ back focal plane (BFP)
\item influence of phase shift on intensity in BFP
\item what are optimum parameters for $R$, $d_1$, $d_2$ and $\varphi$?
\end{itemize}

\begin{figure}[h]
\includegraphics[width=\textwidth]{poster_linescan.png}
\caption{intensity distribution and phase difference}
\label{fig_linescan}
\end{figure}

%As one can see the maximum of the intensity distribution and the $\pi/2$-polygon coincide. Thus it is possible to adjust a phase shift to one electron wave such that the phase difference in the BFP reaches it's optimum value of $\pi/2$ at the maximum of the intensity distribution in the BFP.

\textbf{\scshape Definition}: quality parameter $U$ and weighting function $\Upsilon$:
\begin{eqnarray}
\label{eqn_usignal}
U &=& \frac{\int\limits_{-\infty}^\infty\Upsilon(X)|\psi_{BFP}(X,\,0)|^2\,\mathrm{d}X}{\int\limits_{-\infty}^\infty|\psi_{BFP}(X,\,0)|^2\,\mathrm{d}X} \nonumber\\
\label{eqn_upsilon_1}
\Upsilon(X) &=& 2\,\mathrm{sgn}\left(\Delta\phi(X)\right)\left|\left(\frac{\Delta\phi(X)}{\pi}+\frac{1}{2}\right)\mathrm{mod}\,1-\frac{1}{2}\right| \nonumber\\
\label{eqn_sn}
\mathrm{S/N} &=& \frac{\int\limits_a^b|\psi_{BFP}(X,\,0)|^2\,\mathrm{d}X}{\int\limits_{-\infty}^\infty|\psi_{BFP}(X,\,0)|^2\,\mathrm{d}X},\;[a,\,b]: \mathrm{FWHM} \nonumber
\end{eqnarray}

\begin{itemize}
\item $U$: coarse measure for percentage of intensity with  phase difference $= \pi/2$ ,
\item S/N: ratio between intensity within FWHM and total intensity,
\item $\Upsilon$: zigzag weighting function (see fig. \ref{fig_upsilon}) defined to fulfill:

\begin{equation}
\Upsilon(X) = \left\{\begin{array}{rcl}
\pm1 &\mathrm{for}& \Delta\phi(X) = \pm\frac{\pi}{2} \\
0 &\mathrm{for}& \Delta\phi(X) = i\pi,\;i=0,\,\pm1 \\
\end{array}\right. .\nonumber
\end{equation}
\end{itemize}

\textbf{\scshape Optimum Parameter Set}: determined by brute-force method
\begin{itemize}
\item numerical calculation: $U$, FWHM and S/N for each parameter set,
\item saving results in MySQL database $\to$ graphical evaluation: fig. \ref{fig_sql_1} and \ref{fig_sql_2}
\item[$\Rightarrow$] \textbf{aperture}: maximum radius and minimum spacing 
%(see fig. \ref{fig_linescan})
\item[$\Rightarrow$] chosen parameters: Radius = $28.5\mu$m, Spacing = $60.7\mu$m, Phase shift = $0.57\pi$
\end{itemize}

\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{poster_upsilon.png} 
\end{center}
\caption{weighting function $\Upsilon$ (for legend see fig. \ref{fig_linescan})}
\label{fig_upsilon}
\end{figure}

%whereby $J_1$ is the Bessel function, $\Delta\phi$ is the phase difference between two electron waves in the diffraction plane, $X$ is an coordinate in that plane, $d_{1,2}$ are the spacings of the apertures' centers from the optical axis, $R$ is the aperture radius, $\lambda$ is the electron wave length, $f$ is the focal length of the lens and $\varphi$ is the phase shift due to a phase plate.

\begin{figure}[hbt]
\label{fig_sql_1}
%\includegraphics[width=.8\textwidth]{poster_sql_u.png}
\includegraphics[width=\textwidth]{poster_usignal_2.png}
\caption{evaluation -- $U$}
\end{figure}

\begin{figure}[hbt]
%\includegraphics[width=.8\textwidth]{poster_sql_sn.png}
\includegraphics[width=\textwidth]{poster_sn_2.png}
\caption{evaluation -- Signal/Noise (for legend see fig. \ref{fig_sql_1})}
\label{fig_sql_2}
\end{figure}

%The data sets generated in this way are stored in a MySQL database and are plotted in dependence of the spacing between the apertures in figures \ref{fig_sql_1} and \ref{fig_sql_2}. Figure \ref{fig_sql_1} shows the dependence of the quantity $\mathfrak{U}$ and figure \ref{fig_sql_2} the dependence of the S/N ratio for different aperture radii (7, 11, 17, 21 and 27 $\mu$m) on the aperture spacing.

%These results suggest the use of large aperture radii but small aperture spacings. According to that the parameters given in the legend of figure \ref{fig_linescan} (Radius: 28$\mu$m, Spacing: 62$\mu$m) were chosen further regarding the available area on the used SiN membrane for production of a twin aperture.

\section{Experiments}
\textbf{\scshape Production}: electron lithography 
\begin{itemize}
\item holes in SiN membrane (100$\mu$m$\times$100$\mu$m)
\item evaporation of gold
\end{itemize}

\textbf{\scshape STEM}: operation with condenser twin aperture
\begin{itemize}
\item sample: gold particles on a carbon foil.
\item comparison: conventional aperture $\leftrightarrow$ twin aperture
\item[$\Rightarrow$] no influence on image quality
\end{itemize}

\begin{figure}[hbt]
\label{fig_goldstem}
\subfigure[twin aperture]{\label{fig_goldstem_a}
%\includegraphics[width=.475\textwidth]{gold_einzel_50k.png}
\includegraphics[width=.45\textwidth]{8061-5png.png}}
\subfigure[STEM image]{\label{fig_goldstem_b}\includegraphics[width=.45\textwidth]{gold_twin_50k.png}}
\caption{twin aperture (w/o phase plate) \& STEM image of gold particles using twin aperture.}
\end{figure}

%Figure \ref{fig_goldstem} shows images observed in STEM-mode with the CM30 at a magnification of $5\cdot10^4$. By means of red arrows two particles are marked in both figures. Figure \ref{fig_goldstem}(a) shows the image obtained with an einzel aperture and \ref{fig_goldstem}(b) the image obtained using the twin aperture.

%As one can see the use of a twin aperture has no apparent negative influence on image quality.

\section{Outlook}
\begin{multicols}{2}
\textbf{\scshape Electron Lithography}: phase plate
\begin{itemize}
\item negative resist PMMA\cite{pmma}
\item[$\Rightarrow$] deposition of gold $\to$ annular electrode
\item deposition of insulating layers and gold overlays
\end{itemize}
\textbf{\scshape EMCD}: record spectra in
\begin{itemize}
\item diffraction mode
\item STEM-mode
\end{itemize}
\end{multicols}

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