Optimal aperture sizes and positions for EMCD experiments

ARTICLE IN PRESS Ultramicroscopy 108 (2008) 865– 872

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Optimal aperture sizes and positions for EMCD experiments J. Verbeeck a,c,, C. He´bert b,c, S. Rubino c, P. Nova´k e, J. Rusz d,e, F. Houdellier f, C. Gatel f, P. Schattschneider c a

EMAT, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium CIME & LSME, EPFL, Station 12, 1015 Lausanne, Switzerland ¨t Wien, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria ¨ r Festko ¨rperPhysik, Technische Universita Institut fu d Department of Physics, Uppsala University, Box 530, S-751 21 Uppsala, Sweden e Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-182 21 Prague, Czech Republic f CEMES, 29 rue Jeanne Marvig, BP 94347 31055 Toulouse cedex 4, France b c

a r t i c l e in fo

abstract

Article history: Received 7 December 2007 Received in revised form 21 February 2008 Accepted 26 February 2008

The signal-to-noise ratio (SNR) in energy-loss magnetic chiral dichroism (EMCD)—the equivalent of Xray magnetic circular dichroism (XMCD) in the electron microscope—is optimized with respect to the detector shape, size and position. We show that an important increase in SNR over previous experiments can be obtained when taking much larger detector sizes. We determine the ideal shape of the detector but also show that round apertures are a good compromise if placed in their optimal position. We develop the theory for a simple analytical description of the EMCD experiment and then apply it to dynamical multibeam Bloch wave calculations and to an experimental data set. In all cases it is shown that a significant and welcome improvement of the SNR is possible. & 2008 Elsevier B.V. All rights reserved.

PACS: 82.80.P 61.16.B 32.80.Cy Keywords: Electron energy loss spectrometry Transmission electron microscopy Circular dichroism Coherence

1. Introduction Energy-loss magnetic chiral dichroism (EMCD) is a technique proposed in 2003 [1] to detect circular magnetic dichroism in the transmission electron microscope (TEM). EMCD is the counterpart of X-ray magnetic circular dichroism (XMCD), a nowadays standard technique for investigation of spin and orbital magnetic moments in the synchrotron. The fact that spin-related properties can now be studied with a standard commercial TEM equipped with an energy spectrometer or an energy filter explains the great attraction of this novel technique. The rapidly expanding field of spintronics has seen an increased interest in spin sensitive spectroscopies such as XMCD. Actual questions in spintronics relate to the existence of magnetic dead layers at interfaces posing a barrier for successful spin injection in spin transistors, interlayer coupling in magnetic multilayers [2], the determination and localization of magnetic

 Corresponding author at: EMAT, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium. Tel.: +32 32653249; fax:+32 32653257. E-mail address: [email protected] (J. Verbeeck).

0304-3991/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2008.02.007

moments in dilute magnetic semiconductors or Heusler alloys, to give some examples. As with common microelectronic devices, analytical methods for spin based devices call for spatial resolutions in the range of 10 nm or below. Although there are at least two techniques with high spatial resolution in the range down to 20 nm available on the synchrotron, namely PEEM [3,4] and focusing X-ray optics, their resolution in practice is mostly 450 nm. Within a year after the successful experimental demonstration of EMCD [5] the spatial resolution of this technique could be pushed down to 30 nm, and is approaching the 10 nm level [6]. Since the TEM is the instrument of choice for materials characterization on the nm scale, it is obvious that a reliable procedure to measure EMCD, in combination with the many other analytical capabilities of the TEM, is of great importance for spintronics and other nanomagnetic studies. The chiral signal in the TEM is the difference of two energy loss spectra, taken at ionization edges of typically several 100 eV, and at non-zero scattering angle. These facts make the chiral signal much fainter than the XMCD signal, and questions of how to optimize the signal-to-noise ratio (SNR) by choosing an ideal scattering geometry arise. The problem is not straightforward

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since there are several options to setup the experiment. After the first publication of EMCD spectra [5] two other methods have been devised—taking ðq; EÞ diagrams [6] or using a series of energy filtered diffraction patterns so as to construct dichroic spectra [7]. Furthermore, it was thought originally that a highly parallel incident beam of electrons is needed, but it has been found recently [6] that relatively high convergence angles can be used to advantage, thus increasing the signal strength by an order of magnitude. In this contribution we will investigate the SNR in EMCD experiments via a statistical analysis of the dichroic signal, and use this to optimize the parameters in the experimental setup. We will show that an important increase in SNR can be achieved by using considerably larger detector acceptance angles. This study will lead to recommendations about optimal shape and size of the entrance aperture of the spectrometer. The paper is structured as follows: we start with a summary of the theory behind EMCD and use this to derive formulas for the SNR for the absolute dichroic signal assuming independent Poisson noise in the detector. Next we apply these findings to a simple analytical simulation, a dynamical Bloch wave simulation and finally to an experimentally obtained energy filtered TEM (EFTEM) series. We determine the optimal position, shape and size of the detector aperture (in the diffraction plane) for these cases and discuss the gain in SNR that can be expected.

2. SNR of the dichroic signal When illuminating a sample with two incoming plane waves with wave vector k1 and k2 and components g1 and g2 in the direction perpendicular to the optical axis, we can write the intensity in the diffraction plane after inelastic scattering as [5,8] 2

d s dO dE   4g2 M1 M 2 q:q0 q  q0 :ez ¼ I0 2 þ 0 2 þ 2M 0 cos F 2 0 2  2M00 sin F 2 2 a0 q q q q q2 q0

I ¼ I0

(1)

with q and q0 the wave vector transfer (i.e. distance in the diffraction plane) between a point-like detector and the two diffraction spots caused by the two plane waves as sketched in Fig. 1 and F the phase difference between the two plane waves k1 and k2 . ez is the unit vector in direction of the magnetization, here assumed to be along the optical axis. I0p isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the number of incoming ffi electrons, a0 the Bohr radius, g ¼ 1= 1  v2 =c2 the relativistic 2 factor and d s=dO dE the double differential cross-section. In general, M 1 ; M 2 ; M 0 ; M 00 are functions of the material, the thickness, the incidence angle of plane waves, the excitation energy and elastic scattering. In the most general case, the parameters depend on the wave vector transfer q and numerical simulations are needed.

I+ q

q'

g1

g2

IFig. 1. Sketch of the dichroic setup assuming two incoming beams with wave vector g1 and g2 . The detector position is determined by q and q0 .

The absolute dichroic signal is now defined as the difference between the intensity for two symmetrical detector positions [5]: ID ¼ Iþ  I ¼ DI

(2)

From Eq. (1) we see that an optimal dichroic signal will be obtained on the so-called Thales circle [8] where the vector product has a maximum. To obtain the SNR on the dichroic signal we assume that the noise in the detector is Poisson counting noise and the noise is independent for the two different detector positions. For Poisson noise we have V A ¼ varðAÞ ¼ A¯

(3)

Which gives for the variance on the dichroic signal: varðID Þ ¼ varðIþ Þ þ varðI Þ

(4)

The SNR is then defined as Iþ  I SNRD ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Iþ þ I

(5)

Note that this formula is generally applicable also in the more complicated case of dynamic diffraction and even for experimental data. It is based purely on the method of treating the data and assuming Poisson noise and it is independent on the crosssection formula. For a non-point-like detector we have to integrate the signal over the detector aperture.1 The SNR then becomes R R þ R þ I ds  S I ds ðI  I Þ ds SNRD;S ¼ qSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (6) R  R þ R þ  S I ds þ S I ds S ðI þ I Þ ds With S the area of the detector aperture. Note that the SNR for a fixed aperture S scales is expected as pffiffiffiffi SNR / N (7) With N the number of detected electrons in the aperture. This shows how important it is to optimize the SNR by choosing the right detector S since it is very hard to improve the SNR significantly by e.g. increasing exposure time or beam current.

3. Ideal shape and position of the detector Although the relative strength of the chiral signal in the simple analytic model is strongest on the Thales circle as is evident from Eq. (1), one could argue that in terms of SNR in the final chiral signal it would be advantageous to integrate over a larger detector. In this section we will show for the idealized two-beam case what the effect of non-point-like detectors is on the SNR in the chiral signal. We start with a simple round detector since that is readily available in the microscope as the entrance aperture of the spectrometer. In a second step we show that we can improve the SNR if we optimize the shape of the detector to the signal. For a round detector with radius r d and position qd we determine the SNR of the chiral signal by numerically integrating the signal over the detector area and then applying the formula for the SNR on this signal. We can check the effect of the different positions of the detector by convolving the detector shape with the signal and then applying Eq. (5) on it. This will give an SNR map for a given detector size for all possible positions. The 1 We integrate the intensities assuming that a real detector will actually detect the arrival position of electrons effectively measuring the probability and not the wave function. This is for instance the case if a scintillator is used in combination with a photon detector.

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maximum in this map will point to the ideal detector position for the given size. One could imagine that other shapes of apertures might give rise to a higher SNR. We can test this by a numerical simulation that adds different discrete points (pixels) of the signal together as long as the SNR keeps increasing. A smarter algorithm for finding this optimal shape S can be found when looking at the SNR for an integrated area and requiring that the change in SNR for going from S to S þ dS is zero:

R þ ðI  I Þ dS 2ðIþ  I Þ RS þ ¼ þ   ðI þ I Þ S ðI þ I Þ dS

aperture size and position −8 −6 −4 −2

(8)

(9)

gy [g110]

qSNRD;S ¼0 qS

2ðIþ  I Þ ðIþ þ I Þ

0 2

Unfortunately this still depends on the optimal shape S but we can at least say that the optimal shape must be a contour line of the function: f ¼

867

4 6

(10)

since the right-hand side of Eq. (9) is a constant for a given aperture shape S. An optimized algorithm is to define an aperture A so that ( 1 if f Xa A¼ (11) 0 if f oa And then step through different threshold values a and stop when the total SNR does not improve anymore. This is a much more efficient algorithm compared to adding pixel by pixel to the aperture.

8 −8

−6

−4

−2

0

2

4

6

8

gx [g110] Fig. 2. Optimal position for different round aperture sizes for the simple analytic case. For small apertures the optimal position is on the Thales circle as expected but the optimum shifts up for larger detector sizes. The optimal aperture radius is plotted in dotted line.

SNR vs. aperture radius 1

4. Results

0.9

Three different setups were chosen to test the prediction of the ideal aperture size, shape and position. We start with a simple analytical two beam simulation according to Eq. (1). This can be expanded with a dynamical equation simulating a three beam case. Finally we analyze an experimental data set.

0.8

4.1. Analytical model First of all we make use of Eq. (1) with M1 ¼ M 2 ¼ 1, M 0 ¼ M ¼ 0:1 and F ¼ p=2. This corresponds to equal intensity for the two direct beams (a situation very close to practice), and a maximum dichroic signal of the correct order of magnitude for the 3d ferromagnets (M 00 =M 1 ¼ 0:1). The results for the optimal position for the simple analytical case are shown in Fig. 2 and they show that the larger the aperture, the more we have to shift it up to avoid intersecting the axis connecting the two diffraction spots. The SNR in the final dichroic signal increases significantly for larger apertures as is shown in Fig. 3 and it seems that the upper limit is reached for an aperture radius of around 2:5jg 2  g 1 j. Simulations according to Eq. (11) show that the ideal aperture shape is circular with a radius of approximately 2:5jg 2  g 1 j with a position as is shown in Fig. 2 in dotted line. 00

4.2. Dynamical simulation model As a second step we use first principles simulations. These include elastic scattering events before and after the inelastic collision applying dynamical diffraction theory within the Bloch waves formalism. Inelastic collision is described by means of

SNR [\]

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5 2 2.5 aperture radius [g110]

3

3.5

Fig. 3. SNR for different round aperture radii placed at their optimal position for the simple analytic case. A dramatic increase in SNR is obtainable when choosing larger apertures.

mixed dynamic form factors (MDFFs). Calculations are based on the density functional theory and are implemented as extentions to the WIEN2k code [9]. MDFFs are calculated using the full Coulomb operator without assuming dipole approximation. Details of theory and implementation are summarized in Ref. [10]. We choose as a test case bcc Iron (alpha ferrite). We applied the systematic row approximation using nine beams ð4g; 3g; . . . ; 0; . . . ; 3g; 4gÞ for both incoming and outgoing eigenvalue problem. Simulations are performed as a function of energy and thickness

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−2.5

−2

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−1.5

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gy [g110]

gy [g110]

−2.5

0 0.5

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1

1

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2

2 2.5

2.5 −2

−1

0 gx [g110]

1

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−1

0 gx [g110]

1

2

1

0.45

0.9

0.4

0.8

0.35

0.7

SNR [\]

SNR [\]

Fig. 4. Optimal position for different round aperture sizes for a Bloch wave calculation in a 3-beam case geometry with (a) t ¼ 8 nm and (b) t ¼ 16 nm displayed on top of the integrated L3 signal.

0.6 0.5

0.3 0.25 0.2 0.15

0.4 0.3

0.1

0.2

0.05 0

0.5 1 aperture radius [g110]

1.5

0

0.5 1 aperture radius [g110]

1.5

Fig. 5. Relative SNR for different round aperture radii placed at their optimal position for a Bloch wave calculation in a 3-beam case geometry with (a) t ¼ 8 nm and (b) t ¼ 16 nm.

on a grid of 51  51 pixels in a square area of the diffraction plane spanning from 2:5g to 2:5g in both dimensions, g ¼ ð1 1 0Þ. The Laue circle center for this three beam case is set to (0 0 0) and 200 kV acceleration voltage is taken in agreement with the experiments. Applying the same detector integration techniques as before we see that the result depends sensitively on the thickness of the sample. The ideal position for different round apertures is shown in Fig. 4 for t ¼ 8 and 16 nm. As opposed to the analytical example, ¯ the optimal positions are shifted more towards the 1¯ 10-beam. The SNR increases also considerably with radius much as for the analytical case as shown in Fig. 5. Note that no maximum in the SNR is reached up to aperture radii of 1 reciprocal 110 distance. Most likely, larger apertures would not be practical, ¯ (not included in because weakly excited beams like e.g. 110 systematic row simulations) will influence the signal in a nontrivial way. The ideal shape of the aperture greatly depends on thickness and has quite a complicated contour as shown in Fig. 6. It is clear that the ideal shape is unpractical to use but fortunately Fig. 7 shows that the difference between the SNR obtained with an ideal aperture and the SNR obtained with a circular aperture is not more than 50% for any thickness. The fact that around 50% increase in SNR is possible for the ideal aperture is mainly due to the fact that two quadrants of the diffraction plane can be used simultaneously as opposed to a single circular aperture.

A surface plot of the SNR with respect to thickness (Fig. 8) shows that for all thicknesses the SNR goes up considerably with circular aperture radius, and it also shows that there are optimum thickness ranges around 10 and 30 nm which give the best SNR for this setup. The dependence of the ideal aperture position on the thickness is shown in Fig. 9. For all thicknesses it is advantageous to select the biggest possible aperture radius which was limited in this simulation to one reciprocal vector. The optimal position is plotted for thicknesses between 2 and 40 nm and all positions are in the ¯ spot. The sensitivity of the lower left quadrant, closer to the 1¯ 10 SNR on the exact position is shown in Fig. 10 for a thickness of t ¼ 16 nm. It is shown that the SNR has a plateau around which one can shift the position without affecting the SNR too much. One should, however, avoid that the aperture crosses the ¯ with 110. Therefore a reasonable symmetry line connecting 1¯ 10 rule of thumb could be to put a large aperture with radius r in a position ð1; rÞ expressed in terms of the reciprocal g vector, since one rarely knows the exact thickness and specimen tilt in experiments.

4.3. Experimental data Experimental data for the Fe L2;3 edge was obtained as an EFTEM image series in the image plane using the LACDIF

ARTICLE IN PRESS J. Verbeeck et al. / Ultramicroscopy 108 (2008) 865–872

−2.5

−2.5 −2

−2

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gy [g110]

gy [g110]

869

0 0.5

0 0.5

1

1

1.5

1.5

2

2 2.5

2.5 −2

−1

0

1

2

−1

−2

gx [g110]

0

1

2

gx [g110]

Fig. 6. Numerically determined ideal shape and position of detector for a Bloch wave calculation in a 3-beam case geometry with (a) t ¼ 8 nm and (b) t ¼ 16 nm.

ideal aperture position for selected thicknesses

1.5 −2.5 −2 −1.5

1

−1 −0.5 gy [g110]

relative SNR [/]

optimal circular ideal aperture

0.5

0 0.5 1

0 0

5

10

15 20 25 thickness [nm]

30

35

40

Fig. 7. Relative SNR for an optimal round aperture at optimal position compared to an ‘ideal’ aperture as a function of thickness.

1.5 2 2.5 −2

SNR vs aperture radius for all thicknesses

0

1

2

gx [g110] Fig. 9. Ideal position and size for a circular aperture for different thicknesses. For all thicknesses the SNR keeps increasing with aperture radius so only the position is important here. This figure shows the resulting optimal positions for thicknesses between 2 and 40 nm. The optimal positions are all in the lower left quadrant and ¯ closer to the 1¯ 10-beam.

1 relative SNR [/]

−1

0.8 0.6 0.4 0.2 0 1.5 ap ert

ure

40

1

rad

30

ius

20

0.5

[g

11 ] 0

0

10 0

ess

n thick

[nm]

Fig. 8. Relative SNR for a circular aperture at optimal position vs. radius and vs. sample thickness. Note the strong increase of SNR with increasing aperture radius and the dependence on thickness due to pendello¨sung effects.

technique on the SACTEM TOULOUSE microscope (TECNAI F20 fitted with an image Cs-corrector and GIF tridiem) [7]. An energy slit of 1 eV was chosen and 50 images were taken in a total exposure time of 30 min. The pure bcc Fe sample was oriented in ¯ and 110 spots approximately three beam condition with the 1¯ 10 equally excited. The Laue circle center is on the 000 spot. The approximate thickness of the sample is 16 nm as estimated by comparison with dynamical Bloch wave calculations. LACDIF mode is used [7,11,12] with a convergence angle of 7.8 mrad and an illuminated sample area of around 40 nm diameter. Isochromaticity and drift are measured and corrected directly from the data cubes using scripts based on Gatan Digital Micrograph. Details on the numerical process used will be given in another

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effect of shift of round aperture on relative SNR −2.5 0.4

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r [g

0

110 ]

−1

SNR

gy [g110]

−1.5

0 750

E [eV]

−0.4 2.5 −2

−1

0 gx [g110]

1

Fig. 12. Normalized absolute EMCD signal vs. aperture size obtained from Fig. 11, note the reduction in noise as the aperture increases.

2

Fig. 10. Position sensitivity of the relative SNR using a circular aperture with r ¼ 1g 110 for a thickness of t ¼ 16 nm. Note that the SNR is rather insensitive to a movement of the aperture. This could help to define a standard position which gives a reasonable SNR for all thicknesses and all orientations.

220 200

relative SNR [/]

180 −1

−0.5

160 140

gy [g110]

120 100 0 80 0

0.05

0.1

0.15

0.2

0.25

r [g110]

0.5

Fig. 13. Estimated relative SNR of the absolute dichroic signal vs. aperture size, note the strong increase in SNR as the aperture radius increases.

1

−1

−0.5

0

0.5

1

gx [g110] Fig. 11. Experimental optimal aperture positions drawn on an integrated energy slice around the Fe L3 edge.

paper [13]. Conventional power law background subtraction is performed on each pixel in the x; y; E datacube. This datacube can now be used to extract the EMCD signal making use of different aperture size and positions, which is of special interest in this paper since it allows to compare our theoretical findings with experiments. Different apertures sizes are simulated by convoluting the data with a circular aperture of a given radius. Fig. 11 shows an integrated slice of the datacube around the L3 edge with superimposed the optimal positions of different circular apertures. Note the reasonably good agreement between the predicted optimal positions in Fig. 4b. The obtained dichroic

signal as a function of the aperture size normalized to the maximum in the signal is shown in Fig. 12. It is evident from this figure that the SNR increases significantly for bigger apertures. A rough estimate of the SNR can be made by calculating the root mean square value of the signal in the L2;3 region and compare it to that of the pre-edge region which should only contain noise: qP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 iedge xi S ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P N 2 ipre xi

(12)

with iedge running over the L2;3 region and ipre running over the pre-edge region, and xi the dichroic signal. Fig. 13 shows this estimate and clearly indicates a strong increase in SNR as the aperture radius increases. As the maximum aperture size was limited in this experiment, the expected plateau visible in Fig. 5b is not reached; however, the experiment clearly confirms the significant gain in SNR by taking a larger aperture in an optimal position. The experimental dichroic signal, obtained for the largest aperture in Fig. 11 is shown in Fig. 14.

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multiple scattering effects that mix part of the L3 signal with the L2 signal. Because of the expected symmetry between the L3 and the L2 excitations, this mixing will cancel out part of the expected EMCD signal. The multiple scattering effect can in principle be removed by deconvolution with a low loss spectrum, but the angular dependence of the scattering must be taken into account in the deconvolution. This needs further investigation. It should be noted that the present discussion does not take into account noise arising from the dark count noise of the detector, which can be neglected only as long as it is much smaller than the Poissonian noise assumed here. When recording the signal with an EELS detector using two aperture positions, this is a good approximation since the signal should be considerably larger than the dark count noise. For the EFTEM series method, dark noise can play a role. We use the EFTEM series in this paper mainly to have flexibility in defining the aperture off-line.

4 I+

3.5

I− 5x (I+−I−)

intensity [a.u.]

3 2.5 2 1.5 1 0.5 0 −0.5 700

705

710

715

720

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735

energy loss [eV] Fig. 14. Experimentally obtained EELS spectra in two symmetrical positions I and absolute EMCD signal Iþ  I for the optimal aperture size and position. The EMCD signal is scaled by a factor of 5 to make it better visible.

5. Discussion Analytical and dynamical simulations showed unambiguously that a strong increase in SNR is possible for the absolute EMCD signal. The results depend in a sensitive way on sample orientation, thickness, position and shape of the collection aperture. Comparing the simulated results with a series of energy filtered diffraction patterns pointed out that non-isochromaticity and drift in the image plane must be removed in a first step since these artifacts can strongly distort the EMCD signal [13]. This step is, however, only necessary for energy filtered series of diffraction patterns and is far less a concern when experiments are taken as in [5], making use of two EELS spectra acquisitions in symmetric points in the diffraction pattern or with the E; q method [6]. After the data pretreatment, a good qualitative agreement with the dynamic simulation at t ¼ 16 nm is found. The estimated SNR increases by at least a factor 2 when changing the aperture radius from r ¼ 0:04g 110 to 0:24g 110 (the maximum available in this experiment without hitting the boundaries of the CCD at this camera length). Comparing this to previously published results we see that it was common to use somewhat smaller aperture sizes r  0:1g 200 in [8] and r  0:2g 200 in [6,12]. The ideal aperture radius according to dynamic simulations in Fig. 5b is close to r ¼ 1:5g 110 indicating that a considerable increase in SNR can be expected. In view of the fact that typically the EMCD signal is quite weak, this improvement is of primary importance to obtain the magnetic information from a nanoscale region of the sample. The position of this aperture can in principle be found from analytical simulations, but Fig. 10 shows that the position sensitivity of the SNR is not so critical for large apertures, for small apertures it becomes more critical. The dynamical simulation and the experiment also showed that it is advantageous to shift the aperture slightly away to the Bragg reflection from the symmetry position between the central (0 0 0) and the g spot. This can be understood as the position where an equal inelastic scattering amplitude from 000 and 110 leads to a maximum in the EMCD effect. The final EMCD signal shown in Fig. 14 is relatively strong, although the energy resolution is somewhat limited due to the recording as an energy filtered series with energy selecting slit width of 1 eV. We note that the absolute EMCD effect for the L3 edge is stronger than for the L2 which might be explained by

6. Conclusion As a general trend, we have seen that increasing the circular aperture radius for EMCD experiments leads to a strong increase in SNR. More optimal aperture shapes are possible but impractical since they depend on the exact conditions of the sample and have far too complex shapes. The suboptimal circular aperture, however, still performs quite well and is readily available in any microscope. As a rule of thumb, the aperture can be put near a strongly excited diffraction spot avoiding the crossing of the line connecting that diffraction spot with the 000 spot. Dynamical simulations can help to optimize the position and size further for specific cases. We verified these simulated results for a set of energy filtered diffraction patterns. The trend is followed remarkably well and an increase of SNR of at least a factor 2 was possible. This would require otherwise an increase in current or exposure time by approximately a factor 4. Larger aperture sizes might even give stronger improvements and the dynamic simulations showed that radii of about one reciprocal distance are close to optimal. This size is larger than what was previously used in experiments and shows that significant improvements are possible. The use of EFTEM series to obtain the dichroic signal was necessary in this paper since it allows to define the apertures after recording. Based on the indication given, simple experiments recording two EELS spectra in symmetric positions in the diffraction pattern with a rather large collection angle is easier and faster.

Acknowledgments This work was supported by the European Commission under contract no. 508971 CHIRALTEM. J.V. and F.H. thank the financial support from the European Union under the Framework 6 program under a contract for an Integrated Infrastructure Initiative. Reference 026019 ESTEEM. Thanks to J.P. Morniroli for making the Fe sample available. References [1] C. He´bert, P. Schattschneider, Ultramicroscopy 96 (3–4) (2003) 463. [2] P. Gru¨nberg, R. Schreiber, Y. Pang, M.B. Brodsky, H. Sowers, Phys. Rev. Lett. 57 (19) (1986) 2442. [3] A. Scholl, Thin-film magnetism: PEEM studies, in: K.H.J. Buschow, R.W. Cahn, M.C. Flemings, P. Veyssiere, E.J. Kramer, S. Mahajan (Eds.), Encyclopedia of Materials: Science and Technology, pp. 1–5, in press, ISBN 0-08-043152-6. [4] D.-H. Kim, P. Fischer, W. Chao, E. Anderson, M.-Y. Im, S.-C. Shin, S.-B. Choe, J. Appl. Phys. 99 (2006) 08H303. [5] P. Schattschneider, S. Rubino, C. He´bert, J. Rusz, J. Kunes, P. Nova´k, E. Carlino, M. Fabrizioli, G. Panaccione, G. Rossi, Nature 441 (7092) (2006) 486.

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