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Fringe contrast in inelastic LACBED holography
ARTICLE IN PRESS
Ultramicroscopy 108 (2008) 407–414 www.elsevier.com/locate/ultramic
Fringe contrast in inelastic LACBED holography Peter Schattschneidera,,1, Jo Verbeeckb,2 a
CEMES CNRS, Toulouse, France Institute for Solid State Physics, Vienna University of Technology, A-1040 Vienna, Austria
b
Received 21 February 2007; received in revised form 15 May 2007; accepted 25 May 2007
Abstract We discuss diffraction holography in a scattering geometry reported by Herring [Ultramicroscopy 104 (2005) 261, Ultramicroscopy 106 (2006) 960] and interpreted in terms of the density matrix of the fast electrons. Whereas the previous description used an approximation replacing the LACBED by a CBED geometry and consequently left some doubts about the conclusions (namely the nondetectability of the MDFF) we now fully include the Fresnel propagator and the biprism operator in order to calculate the density matrix of the inelastically scattered electrons in LACBED geometry. We show that a defocus on the biprism with respect to the sample does not cause a significant effect on the fringe patterns that are formed when the discs are exactly overlapping. An important difference to the CBED geometry is however that the fringe contrast decreases when the shear deviates from a reciprocal lattice vector. This should enable to measure the spatial coherence for smaller shears than is possible in image holography. r 2007 Elsevier B.V. All rights reserved. PACS: 34.80.Pa; 82.80.Pv Keywords: Holography; EELS; Inelastic scattering; Coherence
1. Introduction The original intent of electron holography proposed by Gabor [1] was to overcome the aberrations of the electron lenses for high resolution imaging by measuring the electron wave function instead of intensities. Accurate knowledge of the transfer function of the microscope allows then to estimate the exit wave at the specimen plane. This technique however puts severe demands on the stability of the microscope which makes it still an uncommon solution today. In the meantime, holography at lower resolution has evolved to a powerful tool for the study of electric and magnetic fields that influence the phase of the electron wave in the image [2–8]. Corresponding author.
E-mail address: [email protected] (P. Schattschneider). On leave from: Institute for Solid State Physics, Vienna University of Technology, A-1040 Vienna, Austria. 2 On leave from: EMAT, University of Antwerp, Groenenborgerlaan 171 2020 Antwerpen, Belgium. 1
0304-3991/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2007.05.011
The idea to obtain information on the phase of the electron wave in the diffraction pattern is younger [9,10]. The reasons are manyfold; first it was not so evident what this could bring (Had not the theory of dynamic diffraction been so successful that a verification of the predicted phases in diffraction seemed redundant?). Then it was not obvious to find a port for a biprism in the electron microscope column adapted to shift diffraction patterns. With the development of energy filtering devices and the vision to chemically map single atoms, the question of localisation and coherence of inelastic interactions gained interest [11]. After first claims that single atoms could be seen in energy filtered HR images [12] it became soon clear that coupling between elastic and inelastic scattering plays a dominant role in image formation, hampering a straightforward interpretation. There were a few attempts to demonstrate inelastic coherence in diffraction using a biprism [13,14] or a double crystal interferometer [15]. We can now safely say that the spatial periodicity observed in energy filtered HR images as well as in energy filtered diffraction holograms is mainly an effect of coupling
ARTICLE IN PRESS 408
P. Schattschneider, J. Verbeeck / Ultramicroscopy 108 (2008) 407–414
between inelastic and Bragg scattering and is not a sign of coherence of the inelastic signal. Attempts to measure the delocalisation of the plasma excitation [16,17] were lateron shown to be deemed to failure because of the long range Coulomb interaction between probe and specimen [18]. Similar uncertainty exists for phonon scattering in combination with elastic scattering for high resolution TEM images. Another example we find in STEM-EELS where elastic scattering of the probe inside the specimen occurs in combination with inelastic scattering making it a highly nonlinear process which is a far cry from the intuitive ‘‘single atom column EELS’’ interpretation [19–23]. All these examples point out that on the side of experiments, progress in instrumentation has enabled better measurements using Cs correctors, monochromators, high stability guns or better energy filters. These experiments show better than ever that conventional theories no longer suffice to quantitatively explain some results. Apart from the fundamental interest to measure phase shifts after inelastic scattering, the practical aspects of this technique are considerable: including the mixed dynamic form factor (MDFF) [24] into calculation of the differential cross section would improve chemical microanalysis; there is also suspicion that the Stobbs factor [25–28] is related to some not yet understood details of inelastic interference; and finally inelastic interference in the diffraction pattern seems to be of practical interest in the context of energy loss magnetic chiral dichroism (EMCD, a recently developed technique for the study of atom-specific magnetic moments [29]). The recent publication of a second experiment in this journal claiming that inelastic interference, related in some way to the Stobbs factor, was observed [14] raised doubts that the previous interpretation [30] missed an important point. In fact, that analysis was based on the assumption of a CBED geometry whereas both experiments used a geometry with the specimen lifted with respect to the focussed probe. In Ref. [10] this setup was called CBED þ EBI. Referring to a well-known technique we prefer the term LACBED holography. Since it could not be excluded a priori that the different position of the specimen and notably of the biprism induce some observable degree of inelastic coherence we decided to reanalyse the problem with the correct scattering geometry. This causes considerable complication since it is necessary to invoke Fresnel propagators. In the following we describe the propagation of the fast electron’s density matrix after inelastic scattering in the particular geometry drawn in Fig. 1. The calculation shows that interference fringes in the discs and at their rims appear even under the assumption of angular incoherent inelastic scattering as noticed before. Contrary to the CBED geometry, however, the inelastic fringe contrast in LACBED is predicted to decrease when the shear deviates from a reciprocal lattice vector in a similar way as for inelastic image holography.
A
B
Fig. 1. (A) Ray diagram of the LACBED setup, showing all planes that need to be considered. Note that the specimen is shifted a distance h above the front focal plane (FFP) and the beam is focussed in this FFP. (B) Switching on the biprism which is virtually placed in the FFP to overlap the two diffracted discs.
2. The density matrix of a convergent beam Assume that a pointlike source is projected by an ideal condenser/objective prefield onto the front focal plane (indicated by FFP in Fig. 1) of the microscope. The specimen is lifted by h with respect to the front focal plane. According to Fig. 1 the density matrix of the incident electron is rs ðx; x0 Þ ¼ eiðk=2hÞðx
2
x0 2 Þ
Pðx=dÞPðx0 =dÞ,
(1)
where k is the wave number of the electron and d is the diameter of the illuminated area on the specimen. Eq. (1) is recognized as the product cðxÞc ðx0 Þ of the wave function of the image of the point source at position x0 propagated backwards to the specimen plane by the Fresnel propagator for density matrices k2 iðk=2zÞðx2 x0 2 Þ e (2) 4p2 z2 and by convention z40 for propagation ‘‘downstream’’ the column. Pðx=dÞ is the rectangle function (1 for jxjod=2 and 0 elsewhere) illuminating a disk of diameter d on the specimen. We have neglected here the Fresnel diffraction on the rim of the beam forming aperture. The Fresnel fringes on the border of the disk are thus considered to be negligible, or put differently, geometric optics has been assumed to be valid for the illumination of the specimen. z ¼ ho0 here since the source is propagated back from the front focal plane to the specimen plane. This can be understood by inspection of Fig. 1: it is a patch of a spherical wave converging to the image of the source where z is the distance of the specimen from the focussed source (i.e. the defocus or z-shift in LACBED geometry). All coordinates are two-dimensional vectors in planes perpendicular to the optical axis. Pf ðzÞ ¼
ARTICLE IN PRESS P. Schattschneider, J. Verbeeck / Ultramicroscopy 108 (2008) 407–414
We mention that a real (not pointlike) source would induce additional incoherence into the inelastic signal. This incoherence would act as a multiplicative factor occurring in Eq. (18). It can be taken into account by measuring the coherence in the elastic LACBED disk. 2.1. LACBED geometry A few simplifications are introduced now; we assume that the incident (convergent) beam is Bragg scattered in the specimen before inelastic scattering.1 We also assume a 2-beam case that already shows the principle of coherence; and we take for granted that the crystal is thin enough such that the relative phases between the beams (C 0 and C 1 ) do not depend on the incidence angle. In this case we can treat the propagation of the density matrix as a convolution. With these assumptions we can reconstruct the beam that will scatter inelastically: Right before the inelastic event, the wave function is identical to that we would obtain by propagating back the direct and the Bragg spot from the object to the specimen plane, exactly as we did before in order to construct the wave function of a single convergent beam: 2 2 : cb ðxÞ¼ðC 0 eiðk=2hÞðxx0 Þ þ C 1 eiðk=2hÞðxx1 Þ ÞPðx=dÞ,
rb ðx; x0 Þ ¼ cb ðxÞcb ðx0 Þ.
(4)
Inelastic scattering acts by multiplication with the mutual dynamic object spectrum (or transparency MDOS) Tðx; x0 Þ [32]. The object exit density matrix r0 after inelastic scattering is 0
r0 ¼ rb Tðx; x Þ.
where we convoluted the MDOS for a single atom rE with a function G determining the distribution and the coherence between the different atoms. We assume here that inelastic scattering is incoherent between different atoms2 so we take G to be diagonal and only the distribution g of the atoms remains. S is the mixed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dynamic form factor with variables Q ¼ q2 þ q2e and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q0 ¼ q0 2 þ q2e . The variables q and q0 are coordinates in the diffraction plane. Although rb was a pure state density matrix, the object density matrix r0 describes a mixed state because the MDOS has altered some of the off-diagonal elements. We note for completeness that the Fourier transform used in this paper is defined as Z 0 0 FT x;x0 ½f ðx; x0 Þ:¼ f ðx; x0 Þ eiqxþiq x d2 x d2 x0 . S is the mixed dynamic form factor for scattering on one atom. For an evenly spread distribution of independent scatterers over a patch of diameter d we may use g ¼ Pðjxj=dÞ. We note for later use that [34] FT x;x0 ½Pðx=dÞd2 ðx x0 Þ ¼
(3)
where the spots are located at x0 and x1 . See Fig. 1. Note that z ¼ ho0 as before (backward propagation), and the normalisation is to the area of the rectangle function. The density matrix of this pure state is
¼ FT x;x0 ½r0 ðx; x0 Þ Pf ðz ¼ þhÞ 2
02
Þz=2k
FT x;x0 ½r0 ðx; x0 Þ
2
02
Þh=2k
FT x;x0 ½r0 ðx; x0 Þ
For an arbitrary distribution G of independent scatterers in the object plane we can write Z 0 Tðx; x Þ ¼ Gðx; x0 ÞrE ðx x; x0 x0 Þ d2 x d2 x0 Z ¼ Gðx; x0 Þd2 ðx x0 ÞrE ðx x; x0 x0 Þ d2 x d2 x0 Z ¼ gðxÞrE ðx x; x0 xÞ d2 x ð6Þ
¼ eiðq q
rE ðx; x0 Þ ¼ FT q;q0
0
SðQ; Q ; EÞ 2
Q Q
02
,
(7)
1 Also known as single channeling [21]; a full description would take into account also the dynamic scattering on the lattice after the inelastic event (double channeling approximation). The problem there is that the propagator for the density matrix cannot be written as a convolution integral any longer, and the mathematics becomes considerably harder [19,31].
ð9Þ
where we made use of the fact that the density matrix in the front focal plane and the back focal plane are connected through a Fourier transform. To get the density matrix ri ðx; x0 Þ in the image plane we applied the Fresnel propagator convolution over a distance of z ¼ þh, which turns into a product after Fourier transforming with: FT x;x0 ½Pf ðzÞ ¼ eiðq
#
(8)
rl ðq; q0 Þ ¼ FT x;x0 ½ri ðx; x0 Þ ¼ eiðq q
"
J 1 ðjq q0 jdÞ . jq q0 jd
It is worth commenting here this equation: an uncorrelated arrangement of scatterers in the object leads to partial coherence in the diffraction plane: the coherence function is the well-known Airy disc. In the experiment a LACBED disc is projected onto the screen (or rather onto the GIF entrance aperture). We need therefore the density matrix rl ðq; q0 Þ in the back focal plane of the objective lens. Apart from prefactors this is given by
(5)
with [33]
409
2
q0 2 Þz=2k
.
(10)
Note that the final Eq. (9) is just a generalisation to density matrices of the familiar microscope transfer function for a defocused specimen. Spherical aberration induces an additional phase shift that will be considered in a separate paper. For a specimen 2
Almost; there may be correlated excitations between nearest neighbours but we shall see that the long range Coulomb interaction smooths out these correlations.
ARTICLE IN PRESS P. Schattschneider, J. Verbeeck / Ultramicroscopy 108 (2008) 407–414
410
in focus Eq. (9) recovers the Fourier transform from object to diffraction. After insertion of the quantities into Eq. (9) and repeated use of the convolution theorem we obtain (see Appendix A) rl ðq; q0 Þ ¼ eiðq "
2
ðjC 0 j2 þ jC 1 j2 ÞPðq=kyÞ
q0 2 Þh=2k
rf ðq; q0 Þ
(
)# SðQ; Q0 ; EÞ J 1 ðjq q0 jdÞ 2 jq q0 jd Q2 Q0
ð11Þ
with rf ðq; q0 Þ ¼
For s ¼ g, i.e. exact superposition of the 0 and g discs, the sum of the two direct terms assuming dipole approximation is (see Appendix B, Eq. (B.7))
ð2pzÞ2 iðq2 q0 2 Þh=2k e 2 hk q q 0 0 jC 0 j2 eiðqq Þx0 P ky q q1 0 þ jC 1 j2 eiðqq Þx1 P ky q q0 q q1 iðqx0 q0 x1 Þ þ C0C1 e P P ky ky i q q0 q q1 0 þC 0 C 1 eiðqx1 q x0 Þ P P , ky ky
q2
1 . þ q2e
The phase factors cancel in the direct terms. The third term is an off-diagonal element of the density matrix that becomes visible by the biprism action. It carries the fringes because the phase does not cancel here. We find under the assumption that the diameter d of the illuminated area is much bigger than a lattice constant (see Appendix B, Eq. (B.10)) 2R½rl ðq g=2; q þ g=2Þ 1 iðhqg=kÞ Pðq=kyÞ . ¼ 2R C 0 C 1 e q2 þ q2e
ð12Þ
where y is the convergence half-angle. q0 and q1 are the positions of the zero and Bragg diffracted spots in the diffraction plane as sketched in Fig. 1. As explained above we neglected the Fresnel diffraction on the rim of the convergent discs, therefore Eq. (11) is approximate. The rectangle functions describe the Bragg discs in the LACBED geometry. From experiment we can judge that the used approximation is a good one as long as the Fresnel fringes are faint or we are far away from them (which is the case when measuring contrast in the following).
3. Fringe contrast We calculate the hologram when the biprism is excited as shown on the right in Fig. 1. This operation shifts the half images of the DP by s=2 where s is the shear in the diffraction plane. At position q we see [33,34] rl ðq s=2; q s=2Þ þ rl ðq þ s=2; q þ s=2Þ þ 2R½rl ðq s=2; q þ s=2Þ.
ð13Þ
(14)
ð15Þ
Eq. (15) shows clearly that the contrast in the inelastic hologram is the same as in the elastic disc and the fringes are created by the simple phase factor eiðhqg=kÞ . The only difference between elastic and inelastic+elastic scattering is the fact that the Lorentzian blurrs the diffraction discs in the latter case. This finding is exactly identical to what has been derived for CBED [30]. Although surprising at first glance because the now included Fresnel propagator is known to change the coherence properties, inspection of Eq. (11) shows why this is so: the Fresnel propagator present in rf is exactly compensated by the defocus in rl . One can visualize the situation referring to Fig. 1: the spotlike ‘‘sources’’ are propagated upstream to the specimen, inelastically scattered and propagated downstream to the image plane again. For the case that sag (i.e. not a complete superposition of discs), there is a subtle detail in Eq. (B.6) since then there is a phase factor inside the convolution. This will consequently change the coherence for inelastic scattering. Calculations show that the fringe contrast is reduced. We can estimate the fringe contrast by observing that the convolution in Eq. (B.8) becomes a product after Fourier transforming. We make use of FT of the Lorentz function
1
+hs/k
+hg/k
-hs/k
-hg/k
X
Fig. 2. Sketch of the modulus of the frequency spectrum of the intensity in the BFP with a voltage on the biprism causing a shear s. Note the reduction of the fringe-frequencies if sag.
ARTICLE IN PRESS P. Schattschneider, J. Verbeeck / Ultramicroscopy 108 (2008) 407–414
q/g
|ρl| no bipr.
|ρl| no bipr.
-0.5
-0.5
0
0
0.5
0.5 -0.5
0
0.5
Re (ρl) no bipr. -0.5
0
0
0.5
0.5 0
-0.5
0.5
0.5
0
0.5
|ρl|
|ρl|
-0.4
0
Re (ρl) no bipr.
-0.5
-0.5
For the direct terms we get: ðjC 0 j2 þ jC 1 j2 ÞK 0 ðjxjqe Þ
-0.5
-0.4
-0.2
-0.2
0
0
0.2
0.2
0.4 -0.4 -0.2
0
0.2
0.4 -0.4 -0.2
inelastic
0
411
0.2
J 1 ðjxjkyÞ . jxjky
(17)
Assuming the LACBED discs to be large we get a Fourier spectrum with a zero component from the direct terms and a fringe component from the mixed terms as sketched in Fig. 2. Note that the fringe period is fixed by g and does not change with s which is not so easy to see in Eq. (B.8). The contrast of the fringes can be estimated as the ratio of these frequency components: CðsÞ ¼
2ReðC 0 C 1 ÞK 0 ðjðh=kÞðg sÞjqe Þ . jC 0 j2 þ jC 1 j2
(18)
Describing the angular scattering up to quadrupole terms, changes the K 0 function to a truncated K 0 function as in Ref. [34]. Eq. (18) shows that the contrast of the fringes reduces when sag according to this truncated K 0 function much the same as for plasmon image holography described in Ref. [34] but with a scaling factor depending of the height of the specimen h. This can be understood from the fact that the LACBED discs contain spatial information with a magnification that depends on the height. The big advantage of the LACBED setup over the image holography setup described in Ref. [34] is the fact that it is ideally suited to measure coherence for points that are separated by small distances whereas in image holography this is limited by the diameter of the biprism. In the LACBED setup the diameter of the biprism is less critical as long as the separation jx0 x1 j in the front focal plane is large with respect to the diameter of the virtual biprism in this plane.
elastic
Fig. 3. Numerically calculated density matrices for a 1D setup mimicking the parameters used in Ref. [13]. A two beam case is assumed with g ¼ 6:68 nm1 , d ¼ 5 mm, E 0 ¼ 200 kV, C 0 ¼ C 1 ¼ 1, E ¼ 3 eV. Left column: inelasticþelastic scattering. Right column: only elastic scattering. Top: jrl j with the biprism switched off. Along the diagonal we see two diffraction discs. The off-diagonal elements indicate the presence of coherence between those discs. Note the slightly increased intensity in the centre of the four squares for inelastic scattering which indicates that coherency will fall off if two diffraction discs are not overlapping exactly as opposed to the elastic case where the four squares have an even intensity. This effect stems from the eihðsgÞq=k factor in Eq. (B.8). Middle: Reðrl Þ without biprism. Note the fringes in the off-diagonal elements that will create fringes on the diagonal after the biprism is turned on and moves the off-diagonal elements to the diagonal. Bottom: jrl j with biprism on. Note the fringes along the diagonal which holds the information about the observed intensity in the experiment.
[33] and of the rectangle function:
hs h J 1 ðjxjkyÞ K 0 ðjxjqe Þ d x ðs gÞ C 0 C 1 d x þ k k jxjky
hs J ðjx þ ðh=kÞðs gÞjkyÞ 1 K 0 ðjxjqe Þ ¼ C0C1 d x þ k jx þ ðh=kÞðs gÞjky hs J 1 ðjx ðhg=kÞjkyÞ ¼ C 0 C 1 K 0 x qe ð16Þ . k jx ðhg=kÞjky
4. Numerical simulations To check these analytical results we performed a number of numerical calculations where the Fresnel propagator and a biprism operator were applied to the situation sketched in Fig. 1 assuming a 1D system in order to be able to visually study the 2D density matrices. The results for some intermediate density matrices are plotted in Fig. 3 and show indeed that the main effect of inelastic scattering is a blurring effect of the diffraction discs and that the fringe contrast is equal for elastic and inelastic scattering if s ¼ g. From the image of jrl j without biprism we see that the off-diagonal elements that will create the fringes after applying the biprism have a stronger contrast when s ¼ g (centre diagonal of the off-diagonal square) as compared to the situation sag (away from the centre of the off-diagonal square). This effect is only present for inelastic scattering. We also plot the fringe intensities for different values of the shear in Fig. 4 which clearly shows the reduction of fringe contrast if sag for inelastic+elastic scattering as opposed to only elastic scattering. The 1D simulations qualitatively completely agree with our analytical results but the exact distribution of intensities is different for a 2D case. Therefore, we perform
ARTICLE IN PRESS P. Schattschneider, J. Verbeeck / Ultramicroscopy 108 (2008) 407–414
412
s=g
0.02
inelastic elastic
0.01
0 -0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
s=0.9g 0.02 0.01 0 -0.1
-0.08
-0.06
-0.04
-0.02
0 s=0.8g
0.02
0.04
0.06
0.08
0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.02 0.01 0 -0.1
q/g Fig. 4. Intensity of elastic and inelastic fringes (diagonal of rl ) for different shear values for the same 1D calculation as in Fig. 3. Note the reduction in the fringe contrast for inelastic+elastic scattering as opposed to the pure elastic case.
0.01
also 2D simulations for the situation s ¼ g. Fig. 5 shows traces through perfectly overlapped LACBED discs. The result for s ¼ g is essentially the same as presented in Ref. [30] (the results only differ for sag). The traces can be directly compared with Fig. 6 in Ref. [13] and show that inelastic scattering only influences the distribution but not the fringe contrast in LACBED mode for perfectly overlapping discs.
0.009 0.008 0.007 0.006 0.005 0.004
5. Conclusions
0.003 0.002
inelastic central inelastic off-center
0.001
elastic central
0 -4
-3
-2
-1
0 q
1
2
3
4
[nm-1]
Fig. 5. Traces through a 2D numerical calculation of the fringes formed when overlapping the two diffraction discs in LACBED for a setup equal to Fig. 3 (but 2D now). Linetraces are shown through the centre of the overlapping discs for elastic scattering only and for elastic+inelastic scattering as well as a linetrace outside the overlapping discs a distance q ¼ 2:2 nm1 from the center. Note that also outside the disc, fringes are present with the same contrast as present inside the discs for elastic scattering. These traces are in excellent agreement with the trace presented as Fig. 6 in Ref. [13].
We have derived the contrast of fringes in inelastic LACBED holography. For perfectly overlapping discs (s ¼ g) the contrast is the same as in the previously analysed CBED geometry, the reason being that the newly introduced Fresnel propagator cancels exactly the phase factor from the defocus (z shift). The calculated contrast is in good agreement with the published data, both inside and outside of the LACBED disc. The present calculations are based on the assumption of totally incoherent angular inelastic scattering in the specimen. The observed fringes are therefore probably not a sign of inelastic angular coherence. To verify this conjecture definitely would require a precise quantitative comparison with experiment, having all parameters for the simulation.
ARTICLE IN PRESS P. Schattschneider, J. Verbeeck / Ultramicroscopy 108 (2008) 407–414
Contrary to the CBED case, the fringe contrast in LACBED decreases when the shear deviates from a reciprocal lattice vector, in a similar way as for inelastic image holography. This indicates that in LACBED conditions a mixing of angular and spatial coherence of the inelastic event occurs, thus allowing to qualify the rather pessimistic statement [30] that inelastic diffraction holography does not provide additional information surpassing elastic diffraction holography: this is true for CBED; LACBED may be used to retrieve coherence information. With the dependence of the fringe contrast on the shear—Eq. (18)—which does not occur under CBED conditions one can obtain new information on the spatial coherence of energy loss processes. This is of particular interest for collective excitations: deviations from the predicted K 0 profile would indicate an unexpected high correlation length in the plasma excitation. Another advantage of LACBED over image holography is the possibility to measure the spatial coherence in the density matrix of the fast electrons for small distances. By variation of the shear s or the defocus h one traces the spatial coherence, given by K 0 in Eq. (18). There is no lower limit to the argument of this function whereas in image holography the smallest distance over which electrons emerging from the specimen can be made to interfere is given by the shadow width of the biprism, which in turn is determined by its diameter. This should enable to measure the coherence vs. shear for much smaller values than previously published.
Taking the Fourier transform we get (neglecting prefactors): rf ðq; q0 Þ ¼ FT x;x0 ½rb ¼ FT x;x0 ðri ðx; x0 ÞÞFT x;x0 ðPf ðz; x; x0 ÞÞ 2
02
¼ FT x;x0 ðri ðx; x0 ÞÞ eiðz=2kÞðq q Þ .
ðA:5Þ
Filling in z ¼ h and ri we get: q q q0 q 2 02 0 0 0 rf ðq; q0 Þ ¼ eiðh=2kÞðq q Þ jC 0 j2 eiðqq Þx0 P P ky ky q q q0 q 0 1 1 þ jC 1 j2 eiðqq Þx1 P P ky ky q q q0 q 0 0 1 þ C 0 C 1 eiðqx0 q x1 Þ P P ky ky q q q0 q 0 1 0 þC 0 C 1 eiðqx1 q x0 Þ P P , ðA:6Þ ky ky where we added the effect of a limited convergence halfangle y. The FT of T is constructed from Eqs. (6), the inverse transform of Eqs. (8), (9), as SðQ; Q0 ; EÞ J 1 ðjq q0 jdÞ 2 jq q0 jd Q2 Q0
(A.7)
with Q and Q0 the total momentum transfer as opposed to q and q0 the momentum transfer vector projected on the plane perpendicular to the optical axis. Appendix B For illuminated areas of 20 nm or larger we can safely replace J 1 ðxÞ=x in Eq. (11) by the delta function
Acknowledgements
2
P.S. acknowledges the support of the European Commission, Contract no. 508971 (CHIRALTEM). J.V. acknowledges the FWO-Vlaanderen for financial support under Contract no. G.0147.06 and the financial support from the European Union under the Framework 6 program under a contract for an Integrated Infrastructure Intiative. Reference 026019 ESTEEM.
Appendix A The Fourier transform of r0 , Eq. (5) is a convolution FT x;x0 ½r0 ðx; x0 Þ ¼ FT x;x0 ½rb ðx; x0 Þ FT x;x0 ½Tðx; x0 Þ. ðA:1Þ We can write rb as a backward propagation over z (z ¼ ho0) from the image plane ri where the beam is a focussed spot and note that ri and rb separate in a product of wave functions so that we can do the FTs separately. ri ðx; x0 Þ ¼ CðxÞC ðx0 Þ, 2
413
ðA:2Þ 2
CðxÞ ¼ ½C 0 d ðx x0 Þ½C 1 d ðx x1 Þ, rb ðx; x0 Þ ¼ ri ðx; x0 Þ Pf ðz; x; x0 Þ ¼ cb ðxÞcb ðx0 Þ.
ðA:3Þ ðA:4Þ
02
rl ðq; q0 Þ ¼ eiðq q "
Þh=2k
( 0
rf ðq; q Þ
SðQ; Q0 ; EÞ Q2 Q0
2
)# 2
0
d ðjQ Q jÞ
. ðB:1Þ
Then the q0 integral in the convolution can be performed immediately. We further replace the image coordinates according to the geometry by x0 ¼ hg=2k, and x1 ¼ hg=2k. For non-overlapping discs the P rectangle functions define four quadrants in the ðq; q0 Þ hyperplane: qp0, q0 p0, qX0, q0 X0 . In each quadrant only one of the four terms in Eq. (12) is non-zero. Observing q0 ¼ g=2, q1 ¼ g=2 and abbreviating 0 q g=2 q g=2 0 Pþþ ðq; q Þ:¼P P , ky ky 0 q g=2 q þ g=2 P , Pþ ðq; q0 Þ:¼P ky ky 0 q þ g=2 q g=2 P , Pþ ðq; q0 Þ:¼P ky ky 0 q þ g=2 q þ g=2 P , P ðq; q0 Þ:¼P ky ky q ðB:2Þ P0 ðqÞ:¼P ky
ARTICLE IN PRESS P. Schattschneider, J. Verbeeck / Ultramicroscopy 108 (2008) 407–414
414
the convolution integrals in the density matrix rl can be reduced to Z
SðX; X; EÞ iðh=kÞxðqq0 Þ e , 0 X4 P ðq;q Þ Z SðX; X; EÞ iðh=kÞxðqq0 Þ 2 02 0 d2 x e , jC 1 j2 eiðh=2kÞðq q gðqq ÞÞ X4 Pþþ ðq;q0 Þ Z SðX; X; EÞ iðh=kÞxðqq0 gÞ 2 02 0 C 0 C 1 eiðh=2kÞðq q gðqþq ÞÞ d2 x e , 0 X4 P ðq;q Þ Z þ SðX; X; EÞ iðh=kÞxðqq0 þgÞ 2 02 0 C 1 C 0 eiðh=2kÞðq q þgðqþq ÞÞ d2 x e 0 X4 Pþ ðq;q Þ 2
jC 0 j2 eiðh=2kÞðq q
02
gðqq0 ÞÞ
d2 x
ðB:3Þ with X ¼ ðxx ; xy ; qe Þ. The Fresnel prefactors quadratic in q and q0 cancel with the defocus prefactor in Eq. (11); so the three elements of the density matrix occurring in the hologram, Eq. (13) are s s SðX; X; EÞ rl q ; q P ¼ jC 0 j2 4 2 2 X qs=2 SðQ; Q; EÞ ¼ jC 0 j2 P0 , ðB:4Þ Q4 qs=2þg=2 s s SðX; X; EÞ rl q þ ; q þ P ¼ jC 1 j2 þþ 4 2 2 X qþs=2 SðQ; Q; EÞ ¼ jC 1 j2 P0 , ðB:5Þ Q4 qþs=2g=2 s s rl q ; q þ ¼ C 0 C 1 eiðhg=kÞq 2 2 Z SðX; X; EÞ iðh=kÞxðgsÞ d2 x e . X4 Pþ ðqðs=2Þ;qþðs=2ÞÞ
ðB:6Þ
The first two terms are the diffraction discs convoluted with the dynamic form factor and shifted by s=2, half the shear. For s ¼ g the direct terms are exactly superimposed, and we can write them as s s s s rl q ; q þ rl q þ ; q þ 2 2 2 2 2 2 SðQ; Q; EÞ ¼ ðjC 0 j þ jC 1 j Þ P0 . ðB:7Þ Q4 Multiplying the interference term Eq. (B.6) with eihqs=k eihqs=k , moving the phase factor eihqðsgÞ=k inside the integral and substituting Z ¼ q x results in ihsq=k SðQ; Q; EÞ C0C1 e Q4 h g s g sio eihðsgÞq=k P0 q P0 q þ . ðB:8Þ 2 2 We note that the activation of the biprism does not induce an additional phase ramp (i.e. a different fringe period) when it is located in the upper focal plane. We can also use the dipole approximation for S SðQ; Q0 Þ / q q0 þ q2e ,
(B.9)
which is valid for angles up to the Bethe ridge (many FWHMs).
For s ¼ g the mixed term is 1 iðhg=kÞq P0 . 2R C 0 C 1 e q2 þ q2e
(B.10)
If sag there is a phase term inside the convolution only for the mixed term which will lead to a reduction of the fringe contrast. References [1] D. Gabor, Nature 161 (1948) 777. [2] R.E. Dunin-Borkowski, M.R. McCartney, David J. Smith, S.S.P. Parkin, Ultramicroscopy 74 (1998) 61. [3] B.G. Frost, Ultramicroscopy 75 (1998) 105. [4] E. Snoeck, R.E. Dunin-Borkowski, F. Dumestre, P. Renaud, C. Amiens, B. Chaudret, P. Zurcher, Appl. Phys. Lett. 82 (2003) 88. [5] R.E. Dunin-Borkowski, T. Kasama, A. Wei, S.L. Tripp, M.J. Hytch, E. Snoeck, R.J. Harrison, A. Putnis, Microsc. Res. Technol. 64 (2004) 390. [6] H. Lichte, M. Reibold, K. Brand, M. Lehmann, Ultramicroscopy 93 (3/4) (2002) 199. [7] W.D. Rau, P. Schwander, F.H. Baumann, W. Ho¨ppner, A. Ourmazd, Phys. Rev. Lett. 82 (1999) 2614. [8] J. Li, M.R. McCarthy, D.J. Smith, Ultramicroscopy 94 (2003) 149. [9] R.A. Herring, G. Pozzi, T. Tanji, A. Tonomura, Ultramicroscopy 50 (1) (1993) 94. [10] R.A. Herring, G. Pozzi, T. Tanji, A. Tonomura, Ultramicroscopy 60 (1) (1995) 153. [11] M. Ohtsuki, E. Zeitler, Ultramicroscopy 2 (1977) 1478. [12] H. Hashimoto, A. Kumao, Phys. Status. Solidi. (A) 107 (1988) 611. [13] R.A. Herring, Ultramicroscopy 104 (2005) 261. [14] R.A. Herring, Ultramicroscopy 106 (2006) 960. [15] F. Zhou, E. Plies, Proceedings of the 11th European Congress on Electron Microscopy (EUREM), Dublin, 1996, vol. 2, pp. 555–556. [16] H. Lichte, B. Freitag, Ultramicroscopy 81 (2000) 177. [17] A. Harscher, H. Lichte, J. Mayer, Ultramicroscopy 69 (1997) 201. [18] P. Schattschneider, H. Lichte, Phys. Rev. B 71 (2005) 045130. [19] M. Nelhiebel, Effects of crystal orientation and interferometry in electron energy loss spectroscopy, Ph.D. Thesis, E´cole Centrale Paris, 1999. [20] P. Schorsch, Simulation energiegefilterter elektronenmikroskopischer Bilder unter Beru¨cksichtigung inelastischer Streuprozesse, Ph.D. Thesis, University of Darmstadt, 2001. [21] C.J. Rossouw, L.J. Allen, S.D. Findlay, M.P. Oxley, Ultramicroscopy 96 (3/4) (2003) 299. [22] S.D. Findlay, M.P. Oxley, S.J. Pennycook, L.J. Allen, Ultramicroscopy 104 (2005) 126. [23] L.J. Allen, S.D. Findlay, M.P. Oxley, C. Witte, N.J. Zaluzec, Ultramicroscopy 106 (2006) 1001. [24] H. Kohl, H. Rose, Adv. Electron. Electron. Phys. 65 (1985) 173. [25] M.J. Hy¨tch, W.M. Stobbs, Ultramicroscopy 53 (1994) 191. [26] C.B. Boothroyd, J. Microsc. 190 (1998) 99. [27] A. Howie, Ultramicroscopy 98 (2004) 73. [28] K. Du, K. von Hochmeister, F. Phillipp, Ultramicroscopy 107 (2007) 281. [29] P. Schattschneider, S. Rubino, C. He´bert, J. Rusz, J. Kune, P. Nova´k, E. Carlino, M. Fabrizioli, G. Panaccione, G. Rossi, Nature 441 (2006) 486. [30] J. Verbeeck, Ultramicroscopy 106 (2006) 461. [31] M. Nelhiebel, N. Luchier, P. Schorsch, P. Schattschneider, B. Jouffrey, Philos. Mag. B 79 (6) (1999) 941. [32] H. Rose, Ultramicroscopy 15 (1984) 173. [33] P. Schattschneider, M. Nelhiebel, B. Jouffrey, Phys. Rev. B 59 (16) (1999) 10959. [34] J. Verbeeck, D. van Dyck, H. Lichte, P. Potapov, P. Schattschneider, Ultramicroscopy 102 (2005) 239.
Ultramicroscopy 108 (2008) 407–414 www.elsevier.com/locate/ultramic
Fringe contrast in inelastic LACBED holography Peter Schattschneidera,,1, Jo Verbeeckb,2 a
CEMES CNRS, Toulouse, France Institute for Solid State Physics, Vienna University of Technology, A-1040 Vienna, Austria
b
Received 21 February 2007; received in revised form 15 May 2007; accepted 25 May 2007
Abstract We discuss diffraction holography in a scattering geometry reported by Herring [Ultramicroscopy 104 (2005) 261, Ultramicroscopy 106 (2006) 960] and interpreted in terms of the density matrix of the fast electrons. Whereas the previous description used an approximation replacing the LACBED by a CBED geometry and consequently left some doubts about the conclusions (namely the nondetectability of the MDFF) we now fully include the Fresnel propagator and the biprism operator in order to calculate the density matrix of the inelastically scattered electrons in LACBED geometry. We show that a defocus on the biprism with respect to the sample does not cause a significant effect on the fringe patterns that are formed when the discs are exactly overlapping. An important difference to the CBED geometry is however that the fringe contrast decreases when the shear deviates from a reciprocal lattice vector. This should enable to measure the spatial coherence for smaller shears than is possible in image holography. r 2007 Elsevier B.V. All rights reserved. PACS: 34.80.Pa; 82.80.Pv Keywords: Holography; EELS; Inelastic scattering; Coherence
1. Introduction The original intent of electron holography proposed by Gabor [1] was to overcome the aberrations of the electron lenses for high resolution imaging by measuring the electron wave function instead of intensities. Accurate knowledge of the transfer function of the microscope allows then to estimate the exit wave at the specimen plane. This technique however puts severe demands on the stability of the microscope which makes it still an uncommon solution today. In the meantime, holography at lower resolution has evolved to a powerful tool for the study of electric and magnetic fields that influence the phase of the electron wave in the image [2–8]. Corresponding author.
E-mail address: [email protected] (P. Schattschneider). On leave from: Institute for Solid State Physics, Vienna University of Technology, A-1040 Vienna, Austria. 2 On leave from: EMAT, University of Antwerp, Groenenborgerlaan 171 2020 Antwerpen, Belgium. 1
0304-3991/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2007.05.011
The idea to obtain information on the phase of the electron wave in the diffraction pattern is younger [9,10]. The reasons are manyfold; first it was not so evident what this could bring (Had not the theory of dynamic diffraction been so successful that a verification of the predicted phases in diffraction seemed redundant?). Then it was not obvious to find a port for a biprism in the electron microscope column adapted to shift diffraction patterns. With the development of energy filtering devices and the vision to chemically map single atoms, the question of localisation and coherence of inelastic interactions gained interest [11]. After first claims that single atoms could be seen in energy filtered HR images [12] it became soon clear that coupling between elastic and inelastic scattering plays a dominant role in image formation, hampering a straightforward interpretation. There were a few attempts to demonstrate inelastic coherence in diffraction using a biprism [13,14] or a double crystal interferometer [15]. We can now safely say that the spatial periodicity observed in energy filtered HR images as well as in energy filtered diffraction holograms is mainly an effect of coupling
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between inelastic and Bragg scattering and is not a sign of coherence of the inelastic signal. Attempts to measure the delocalisation of the plasma excitation [16,17] were lateron shown to be deemed to failure because of the long range Coulomb interaction between probe and specimen [18]. Similar uncertainty exists for phonon scattering in combination with elastic scattering for high resolution TEM images. Another example we find in STEM-EELS where elastic scattering of the probe inside the specimen occurs in combination with inelastic scattering making it a highly nonlinear process which is a far cry from the intuitive ‘‘single atom column EELS’’ interpretation [19–23]. All these examples point out that on the side of experiments, progress in instrumentation has enabled better measurements using Cs correctors, monochromators, high stability guns or better energy filters. These experiments show better than ever that conventional theories no longer suffice to quantitatively explain some results. Apart from the fundamental interest to measure phase shifts after inelastic scattering, the practical aspects of this technique are considerable: including the mixed dynamic form factor (MDFF) [24] into calculation of the differential cross section would improve chemical microanalysis; there is also suspicion that the Stobbs factor [25–28] is related to some not yet understood details of inelastic interference; and finally inelastic interference in the diffraction pattern seems to be of practical interest in the context of energy loss magnetic chiral dichroism (EMCD, a recently developed technique for the study of atom-specific magnetic moments [29]). The recent publication of a second experiment in this journal claiming that inelastic interference, related in some way to the Stobbs factor, was observed [14] raised doubts that the previous interpretation [30] missed an important point. In fact, that analysis was based on the assumption of a CBED geometry whereas both experiments used a geometry with the specimen lifted with respect to the focussed probe. In Ref. [10] this setup was called CBED þ EBI. Referring to a well-known technique we prefer the term LACBED holography. Since it could not be excluded a priori that the different position of the specimen and notably of the biprism induce some observable degree of inelastic coherence we decided to reanalyse the problem with the correct scattering geometry. This causes considerable complication since it is necessary to invoke Fresnel propagators. In the following we describe the propagation of the fast electron’s density matrix after inelastic scattering in the particular geometry drawn in Fig. 1. The calculation shows that interference fringes in the discs and at their rims appear even under the assumption of angular incoherent inelastic scattering as noticed before. Contrary to the CBED geometry, however, the inelastic fringe contrast in LACBED is predicted to decrease when the shear deviates from a reciprocal lattice vector in a similar way as for inelastic image holography.
A
B
Fig. 1. (A) Ray diagram of the LACBED setup, showing all planes that need to be considered. Note that the specimen is shifted a distance h above the front focal plane (FFP) and the beam is focussed in this FFP. (B) Switching on the biprism which is virtually placed in the FFP to overlap the two diffracted discs.
2. The density matrix of a convergent beam Assume that a pointlike source is projected by an ideal condenser/objective prefield onto the front focal plane (indicated by FFP in Fig. 1) of the microscope. The specimen is lifted by h with respect to the front focal plane. According to Fig. 1 the density matrix of the incident electron is rs ðx; x0 Þ ¼ eiðk=2hÞðx
2
x0 2 Þ
Pðx=dÞPðx0 =dÞ,
(1)
where k is the wave number of the electron and d is the diameter of the illuminated area on the specimen. Eq. (1) is recognized as the product cðxÞc ðx0 Þ of the wave function of the image of the point source at position x0 propagated backwards to the specimen plane by the Fresnel propagator for density matrices k2 iðk=2zÞðx2 x0 2 Þ e (2) 4p2 z2 and by convention z40 for propagation ‘‘downstream’’ the column. Pðx=dÞ is the rectangle function (1 for jxjod=2 and 0 elsewhere) illuminating a disk of diameter d on the specimen. We have neglected here the Fresnel diffraction on the rim of the beam forming aperture. The Fresnel fringes on the border of the disk are thus considered to be negligible, or put differently, geometric optics has been assumed to be valid for the illumination of the specimen. z ¼ ho0 here since the source is propagated back from the front focal plane to the specimen plane. This can be understood by inspection of Fig. 1: it is a patch of a spherical wave converging to the image of the source where z is the distance of the specimen from the focussed source (i.e. the defocus or z-shift in LACBED geometry). All coordinates are two-dimensional vectors in planes perpendicular to the optical axis. Pf ðzÞ ¼
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We mention that a real (not pointlike) source would induce additional incoherence into the inelastic signal. This incoherence would act as a multiplicative factor occurring in Eq. (18). It can be taken into account by measuring the coherence in the elastic LACBED disk. 2.1. LACBED geometry A few simplifications are introduced now; we assume that the incident (convergent) beam is Bragg scattered in the specimen before inelastic scattering.1 We also assume a 2-beam case that already shows the principle of coherence; and we take for granted that the crystal is thin enough such that the relative phases between the beams (C 0 and C 1 ) do not depend on the incidence angle. In this case we can treat the propagation of the density matrix as a convolution. With these assumptions we can reconstruct the beam that will scatter inelastically: Right before the inelastic event, the wave function is identical to that we would obtain by propagating back the direct and the Bragg spot from the object to the specimen plane, exactly as we did before in order to construct the wave function of a single convergent beam: 2 2 : cb ðxÞ¼ðC 0 eiðk=2hÞðxx0 Þ þ C 1 eiðk=2hÞðxx1 Þ ÞPðx=dÞ,
rb ðx; x0 Þ ¼ cb ðxÞcb ðx0 Þ.
(4)
Inelastic scattering acts by multiplication with the mutual dynamic object spectrum (or transparency MDOS) Tðx; x0 Þ [32]. The object exit density matrix r0 after inelastic scattering is 0
r0 ¼ rb Tðx; x Þ.
where we convoluted the MDOS for a single atom rE with a function G determining the distribution and the coherence between the different atoms. We assume here that inelastic scattering is incoherent between different atoms2 so we take G to be diagonal and only the distribution g of the atoms remains. S is the mixed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dynamic form factor with variables Q ¼ q2 þ q2e and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q0 ¼ q0 2 þ q2e . The variables q and q0 are coordinates in the diffraction plane. Although rb was a pure state density matrix, the object density matrix r0 describes a mixed state because the MDOS has altered some of the off-diagonal elements. We note for completeness that the Fourier transform used in this paper is defined as Z 0 0 FT x;x0 ½f ðx; x0 Þ:¼ f ðx; x0 Þ eiqxþiq x d2 x d2 x0 . S is the mixed dynamic form factor for scattering on one atom. For an evenly spread distribution of independent scatterers over a patch of diameter d we may use g ¼ Pðjxj=dÞ. We note for later use that [34] FT x;x0 ½Pðx=dÞd2 ðx x0 Þ ¼
(3)
where the spots are located at x0 and x1 . See Fig. 1. Note that z ¼ ho0 as before (backward propagation), and the normalisation is to the area of the rectangle function. The density matrix of this pure state is
¼ FT x;x0 ½r0 ðx; x0 Þ Pf ðz ¼ þhÞ 2
02
Þz=2k
FT x;x0 ½r0 ðx; x0 Þ
2
02
Þh=2k
FT x;x0 ½r0 ðx; x0 Þ
For an arbitrary distribution G of independent scatterers in the object plane we can write Z 0 Tðx; x Þ ¼ Gðx; x0 ÞrE ðx x; x0 x0 Þ d2 x d2 x0 Z ¼ Gðx; x0 Þd2 ðx x0 ÞrE ðx x; x0 x0 Þ d2 x d2 x0 Z ¼ gðxÞrE ðx x; x0 xÞ d2 x ð6Þ
¼ eiðq q
rE ðx; x0 Þ ¼ FT q;q0
0
SðQ; Q ; EÞ 2
Q Q
02
,
(7)
1 Also known as single channeling [21]; a full description would take into account also the dynamic scattering on the lattice after the inelastic event (double channeling approximation). The problem there is that the propagator for the density matrix cannot be written as a convolution integral any longer, and the mathematics becomes considerably harder [19,31].
ð9Þ
where we made use of the fact that the density matrix in the front focal plane and the back focal plane are connected through a Fourier transform. To get the density matrix ri ðx; x0 Þ in the image plane we applied the Fresnel propagator convolution over a distance of z ¼ þh, which turns into a product after Fourier transforming with: FT x;x0 ½Pf ðzÞ ¼ eiðq
#
(8)
rl ðq; q0 Þ ¼ FT x;x0 ½ri ðx; x0 Þ ¼ eiðq q
"
J 1 ðjq q0 jdÞ . jq q0 jd
It is worth commenting here this equation: an uncorrelated arrangement of scatterers in the object leads to partial coherence in the diffraction plane: the coherence function is the well-known Airy disc. In the experiment a LACBED disc is projected onto the screen (or rather onto the GIF entrance aperture). We need therefore the density matrix rl ðq; q0 Þ in the back focal plane of the objective lens. Apart from prefactors this is given by
(5)
with [33]
409
2
q0 2 Þz=2k
.
(10)
Note that the final Eq. (9) is just a generalisation to density matrices of the familiar microscope transfer function for a defocused specimen. Spherical aberration induces an additional phase shift that will be considered in a separate paper. For a specimen 2
Almost; there may be correlated excitations between nearest neighbours but we shall see that the long range Coulomb interaction smooths out these correlations.
ARTICLE IN PRESS P. Schattschneider, J. Verbeeck / Ultramicroscopy 108 (2008) 407–414
410
in focus Eq. (9) recovers the Fourier transform from object to diffraction. After insertion of the quantities into Eq. (9) and repeated use of the convolution theorem we obtain (see Appendix A) rl ðq; q0 Þ ¼ eiðq "
2
ðjC 0 j2 þ jC 1 j2 ÞPðq=kyÞ
q0 2 Þh=2k
rf ðq; q0 Þ
(
)# SðQ; Q0 ; EÞ J 1 ðjq q0 jdÞ 2 jq q0 jd Q2 Q0
ð11Þ
with rf ðq; q0 Þ ¼
For s ¼ g, i.e. exact superposition of the 0 and g discs, the sum of the two direct terms assuming dipole approximation is (see Appendix B, Eq. (B.7))
ð2pzÞ2 iðq2 q0 2 Þh=2k e 2 hk q q 0 0 jC 0 j2 eiðqq Þx0 P ky q q1 0 þ jC 1 j2 eiðqq Þx1 P ky q q0 q q1 iðqx0 q0 x1 Þ þ C0C1 e P P ky ky i q q0 q q1 0 þC 0 C 1 eiðqx1 q x0 Þ P P , ky ky
q2
1 . þ q2e
The phase factors cancel in the direct terms. The third term is an off-diagonal element of the density matrix that becomes visible by the biprism action. It carries the fringes because the phase does not cancel here. We find under the assumption that the diameter d of the illuminated area is much bigger than a lattice constant (see Appendix B, Eq. (B.10)) 2R½rl ðq g=2; q þ g=2Þ 1 iðhqg=kÞ Pðq=kyÞ . ¼ 2R C 0 C 1 e q2 þ q2e
ð12Þ
where y is the convergence half-angle. q0 and q1 are the positions of the zero and Bragg diffracted spots in the diffraction plane as sketched in Fig. 1. As explained above we neglected the Fresnel diffraction on the rim of the convergent discs, therefore Eq. (11) is approximate. The rectangle functions describe the Bragg discs in the LACBED geometry. From experiment we can judge that the used approximation is a good one as long as the Fresnel fringes are faint or we are far away from them (which is the case when measuring contrast in the following).
3. Fringe contrast We calculate the hologram when the biprism is excited as shown on the right in Fig. 1. This operation shifts the half images of the DP by s=2 where s is the shear in the diffraction plane. At position q we see [33,34] rl ðq s=2; q s=2Þ þ rl ðq þ s=2; q þ s=2Þ þ 2R½rl ðq s=2; q þ s=2Þ.
ð13Þ
(14)
ð15Þ
Eq. (15) shows clearly that the contrast in the inelastic hologram is the same as in the elastic disc and the fringes are created by the simple phase factor eiðhqg=kÞ . The only difference between elastic and inelastic+elastic scattering is the fact that the Lorentzian blurrs the diffraction discs in the latter case. This finding is exactly identical to what has been derived for CBED [30]. Although surprising at first glance because the now included Fresnel propagator is known to change the coherence properties, inspection of Eq. (11) shows why this is so: the Fresnel propagator present in rf is exactly compensated by the defocus in rl . One can visualize the situation referring to Fig. 1: the spotlike ‘‘sources’’ are propagated upstream to the specimen, inelastically scattered and propagated downstream to the image plane again. For the case that sag (i.e. not a complete superposition of discs), there is a subtle detail in Eq. (B.6) since then there is a phase factor inside the convolution. This will consequently change the coherence for inelastic scattering. Calculations show that the fringe contrast is reduced. We can estimate the fringe contrast by observing that the convolution in Eq. (B.8) becomes a product after Fourier transforming. We make use of FT of the Lorentz function
1
+hs/k
+hg/k
-hs/k
-hg/k
X
Fig. 2. Sketch of the modulus of the frequency spectrum of the intensity in the BFP with a voltage on the biprism causing a shear s. Note the reduction of the fringe-frequencies if sag.
ARTICLE IN PRESS P. Schattschneider, J. Verbeeck / Ultramicroscopy 108 (2008) 407–414
q/g
|ρl| no bipr.
|ρl| no bipr.
-0.5
-0.5
0
0
0.5
0.5 -0.5
0
0.5
Re (ρl) no bipr. -0.5
0
0
0.5
0.5 0
-0.5
0.5
0.5
0
0.5
|ρl|
|ρl|
-0.4
0
Re (ρl) no bipr.
-0.5
-0.5
For the direct terms we get: ðjC 0 j2 þ jC 1 j2 ÞK 0 ðjxjqe Þ
-0.5
-0.4
-0.2
-0.2
0
0
0.2
0.2
0.4 -0.4 -0.2
0
0.2
0.4 -0.4 -0.2
inelastic
0
411
0.2
J 1 ðjxjkyÞ . jxjky
(17)
Assuming the LACBED discs to be large we get a Fourier spectrum with a zero component from the direct terms and a fringe component from the mixed terms as sketched in Fig. 2. Note that the fringe period is fixed by g and does not change with s which is not so easy to see in Eq. (B.8). The contrast of the fringes can be estimated as the ratio of these frequency components: CðsÞ ¼
2ReðC 0 C 1 ÞK 0 ðjðh=kÞðg sÞjqe Þ . jC 0 j2 þ jC 1 j2
(18)
Describing the angular scattering up to quadrupole terms, changes the K 0 function to a truncated K 0 function as in Ref. [34]. Eq. (18) shows that the contrast of the fringes reduces when sag according to this truncated K 0 function much the same as for plasmon image holography described in Ref. [34] but with a scaling factor depending of the height of the specimen h. This can be understood from the fact that the LACBED discs contain spatial information with a magnification that depends on the height. The big advantage of the LACBED setup over the image holography setup described in Ref. [34] is the fact that it is ideally suited to measure coherence for points that are separated by small distances whereas in image holography this is limited by the diameter of the biprism. In the LACBED setup the diameter of the biprism is less critical as long as the separation jx0 x1 j in the front focal plane is large with respect to the diameter of the virtual biprism in this plane.
elastic
Fig. 3. Numerically calculated density matrices for a 1D setup mimicking the parameters used in Ref. [13]. A two beam case is assumed with g ¼ 6:68 nm1 , d ¼ 5 mm, E 0 ¼ 200 kV, C 0 ¼ C 1 ¼ 1, E ¼ 3 eV. Left column: inelasticþelastic scattering. Right column: only elastic scattering. Top: jrl j with the biprism switched off. Along the diagonal we see two diffraction discs. The off-diagonal elements indicate the presence of coherence between those discs. Note the slightly increased intensity in the centre of the four squares for inelastic scattering which indicates that coherency will fall off if two diffraction discs are not overlapping exactly as opposed to the elastic case where the four squares have an even intensity. This effect stems from the eihðsgÞq=k factor in Eq. (B.8). Middle: Reðrl Þ without biprism. Note the fringes in the off-diagonal elements that will create fringes on the diagonal after the biprism is turned on and moves the off-diagonal elements to the diagonal. Bottom: jrl j with biprism on. Note the fringes along the diagonal which holds the information about the observed intensity in the experiment.
[33] and of the rectangle function:
hs h J 1 ðjxjkyÞ K 0 ðjxjqe Þ d x ðs gÞ C 0 C 1 d x þ k k jxjky
hs J ðjx þ ðh=kÞðs gÞjkyÞ 1 K 0 ðjxjqe Þ ¼ C0C1 d x þ k jx þ ðh=kÞðs gÞjky hs J 1 ðjx ðhg=kÞjkyÞ ¼ C 0 C 1 K 0 x qe ð16Þ . k jx ðhg=kÞjky
4. Numerical simulations To check these analytical results we performed a number of numerical calculations where the Fresnel propagator and a biprism operator were applied to the situation sketched in Fig. 1 assuming a 1D system in order to be able to visually study the 2D density matrices. The results for some intermediate density matrices are plotted in Fig. 3 and show indeed that the main effect of inelastic scattering is a blurring effect of the diffraction discs and that the fringe contrast is equal for elastic and inelastic scattering if s ¼ g. From the image of jrl j without biprism we see that the off-diagonal elements that will create the fringes after applying the biprism have a stronger contrast when s ¼ g (centre diagonal of the off-diagonal square) as compared to the situation sag (away from the centre of the off-diagonal square). This effect is only present for inelastic scattering. We also plot the fringe intensities for different values of the shear in Fig. 4 which clearly shows the reduction of fringe contrast if sag for inelastic+elastic scattering as opposed to only elastic scattering. The 1D simulations qualitatively completely agree with our analytical results but the exact distribution of intensities is different for a 2D case. Therefore, we perform
ARTICLE IN PRESS P. Schattschneider, J. Verbeeck / Ultramicroscopy 108 (2008) 407–414
412
s=g
0.02
inelastic elastic
0.01
0 -0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
s=0.9g 0.02 0.01 0 -0.1
-0.08
-0.06
-0.04
-0.02
0 s=0.8g
0.02
0.04
0.06
0.08
0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.02 0.01 0 -0.1
q/g Fig. 4. Intensity of elastic and inelastic fringes (diagonal of rl ) for different shear values for the same 1D calculation as in Fig. 3. Note the reduction in the fringe contrast for inelastic+elastic scattering as opposed to the pure elastic case.
0.01
also 2D simulations for the situation s ¼ g. Fig. 5 shows traces through perfectly overlapped LACBED discs. The result for s ¼ g is essentially the same as presented in Ref. [30] (the results only differ for sag). The traces can be directly compared with Fig. 6 in Ref. [13] and show that inelastic scattering only influences the distribution but not the fringe contrast in LACBED mode for perfectly overlapping discs.
0.009 0.008 0.007 0.006 0.005 0.004
5. Conclusions
0.003 0.002
inelastic central inelastic off-center
0.001
elastic central
0 -4
-3
-2
-1
0 q
1
2
3
4
[nm-1]
Fig. 5. Traces through a 2D numerical calculation of the fringes formed when overlapping the two diffraction discs in LACBED for a setup equal to Fig. 3 (but 2D now). Linetraces are shown through the centre of the overlapping discs for elastic scattering only and for elastic+inelastic scattering as well as a linetrace outside the overlapping discs a distance q ¼ 2:2 nm1 from the center. Note that also outside the disc, fringes are present with the same contrast as present inside the discs for elastic scattering. These traces are in excellent agreement with the trace presented as Fig. 6 in Ref. [13].
We have derived the contrast of fringes in inelastic LACBED holography. For perfectly overlapping discs (s ¼ g) the contrast is the same as in the previously analysed CBED geometry, the reason being that the newly introduced Fresnel propagator cancels exactly the phase factor from the defocus (z shift). The calculated contrast is in good agreement with the published data, both inside and outside of the LACBED disc. The present calculations are based on the assumption of totally incoherent angular inelastic scattering in the specimen. The observed fringes are therefore probably not a sign of inelastic angular coherence. To verify this conjecture definitely would require a precise quantitative comparison with experiment, having all parameters for the simulation.
ARTICLE IN PRESS P. Schattschneider, J. Verbeeck / Ultramicroscopy 108 (2008) 407–414
Contrary to the CBED case, the fringe contrast in LACBED decreases when the shear deviates from a reciprocal lattice vector, in a similar way as for inelastic image holography. This indicates that in LACBED conditions a mixing of angular and spatial coherence of the inelastic event occurs, thus allowing to qualify the rather pessimistic statement [30] that inelastic diffraction holography does not provide additional information surpassing elastic diffraction holography: this is true for CBED; LACBED may be used to retrieve coherence information. With the dependence of the fringe contrast on the shear—Eq. (18)—which does not occur under CBED conditions one can obtain new information on the spatial coherence of energy loss processes. This is of particular interest for collective excitations: deviations from the predicted K 0 profile would indicate an unexpected high correlation length in the plasma excitation. Another advantage of LACBED over image holography is the possibility to measure the spatial coherence in the density matrix of the fast electrons for small distances. By variation of the shear s or the defocus h one traces the spatial coherence, given by K 0 in Eq. (18). There is no lower limit to the argument of this function whereas in image holography the smallest distance over which electrons emerging from the specimen can be made to interfere is given by the shadow width of the biprism, which in turn is determined by its diameter. This should enable to measure the coherence vs. shear for much smaller values than previously published.
Taking the Fourier transform we get (neglecting prefactors): rf ðq; q0 Þ ¼ FT x;x0 ½rb ¼ FT x;x0 ðri ðx; x0 ÞÞFT x;x0 ðPf ðz; x; x0 ÞÞ 2
02
¼ FT x;x0 ðri ðx; x0 ÞÞ eiðz=2kÞðq q Þ .
ðA:5Þ
Filling in z ¼ h and ri we get: q q q0 q 2 02 0 0 0 rf ðq; q0 Þ ¼ eiðh=2kÞðq q Þ jC 0 j2 eiðqq Þx0 P P ky ky q q q0 q 0 1 1 þ jC 1 j2 eiðqq Þx1 P P ky ky q q q0 q 0 0 1 þ C 0 C 1 eiðqx0 q x1 Þ P P ky ky q q q0 q 0 1 0 þC 0 C 1 eiðqx1 q x0 Þ P P , ðA:6Þ ky ky where we added the effect of a limited convergence halfangle y. The FT of T is constructed from Eqs. (6), the inverse transform of Eqs. (8), (9), as SðQ; Q0 ; EÞ J 1 ðjq q0 jdÞ 2 jq q0 jd Q2 Q0
(A.7)
with Q and Q0 the total momentum transfer as opposed to q and q0 the momentum transfer vector projected on the plane perpendicular to the optical axis. Appendix B For illuminated areas of 20 nm or larger we can safely replace J 1 ðxÞ=x in Eq. (11) by the delta function
Acknowledgements
2
P.S. acknowledges the support of the European Commission, Contract no. 508971 (CHIRALTEM). J.V. acknowledges the FWO-Vlaanderen for financial support under Contract no. G.0147.06 and the financial support from the European Union under the Framework 6 program under a contract for an Integrated Infrastructure Intiative. Reference 026019 ESTEEM.
Appendix A The Fourier transform of r0 , Eq. (5) is a convolution FT x;x0 ½r0 ðx; x0 Þ ¼ FT x;x0 ½rb ðx; x0 Þ FT x;x0 ½Tðx; x0 Þ. ðA:1Þ We can write rb as a backward propagation over z (z ¼ ho0) from the image plane ri where the beam is a focussed spot and note that ri and rb separate in a product of wave functions so that we can do the FTs separately. ri ðx; x0 Þ ¼ CðxÞC ðx0 Þ, 2
413
ðA:2Þ 2
CðxÞ ¼ ½C 0 d ðx x0 Þ½C 1 d ðx x1 Þ, rb ðx; x0 Þ ¼ ri ðx; x0 Þ Pf ðz; x; x0 Þ ¼ cb ðxÞcb ðx0 Þ.
ðA:3Þ ðA:4Þ
02
rl ðq; q0 Þ ¼ eiðq q "
Þh=2k
( 0
rf ðq; q Þ
SðQ; Q0 ; EÞ Q2 Q0
2
)# 2
0
d ðjQ Q jÞ
. ðB:1Þ
Then the q0 integral in the convolution can be performed immediately. We further replace the image coordinates according to the geometry by x0 ¼ hg=2k, and x1 ¼ hg=2k. For non-overlapping discs the P rectangle functions define four quadrants in the ðq; q0 Þ hyperplane: qp0, q0 p0, qX0, q0 X0 . In each quadrant only one of the four terms in Eq. (12) is non-zero. Observing q0 ¼ g=2, q1 ¼ g=2 and abbreviating 0 q g=2 q g=2 0 Pþþ ðq; q Þ:¼P P , ky ky 0 q g=2 q þ g=2 P , Pþ ðq; q0 Þ:¼P ky ky 0 q þ g=2 q g=2 P , Pþ ðq; q0 Þ:¼P ky ky 0 q þ g=2 q þ g=2 P , P ðq; q0 Þ:¼P ky ky q ðB:2Þ P0 ðqÞ:¼P ky
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the convolution integrals in the density matrix rl can be reduced to Z
SðX; X; EÞ iðh=kÞxðqq0 Þ e , 0 X4 P ðq;q Þ Z SðX; X; EÞ iðh=kÞxðqq0 Þ 2 02 0 d2 x e , jC 1 j2 eiðh=2kÞðq q gðqq ÞÞ X4 Pþþ ðq;q0 Þ Z SðX; X; EÞ iðh=kÞxðqq0 gÞ 2 02 0 C 0 C 1 eiðh=2kÞðq q gðqþq ÞÞ d2 x e , 0 X4 P ðq;q Þ Z þ SðX; X; EÞ iðh=kÞxðqq0 þgÞ 2 02 0 C 1 C 0 eiðh=2kÞðq q þgðqþq ÞÞ d2 x e 0 X4 Pþ ðq;q Þ 2
jC 0 j2 eiðh=2kÞðq q
02
gðqq0 ÞÞ
d2 x
ðB:3Þ with X ¼ ðxx ; xy ; qe Þ. The Fresnel prefactors quadratic in q and q0 cancel with the defocus prefactor in Eq. (11); so the three elements of the density matrix occurring in the hologram, Eq. (13) are s s SðX; X; EÞ rl q ; q P ¼ jC 0 j2 4 2 2 X qs=2 SðQ; Q; EÞ ¼ jC 0 j2 P0 , ðB:4Þ Q4 qs=2þg=2 s s SðX; X; EÞ rl q þ ; q þ P ¼ jC 1 j2 þþ 4 2 2 X qþs=2 SðQ; Q; EÞ ¼ jC 1 j2 P0 , ðB:5Þ Q4 qþs=2g=2 s s rl q ; q þ ¼ C 0 C 1 eiðhg=kÞq 2 2 Z SðX; X; EÞ iðh=kÞxðgsÞ d2 x e . X4 Pþ ðqðs=2Þ;qþðs=2ÞÞ
ðB:6Þ
The first two terms are the diffraction discs convoluted with the dynamic form factor and shifted by s=2, half the shear. For s ¼ g the direct terms are exactly superimposed, and we can write them as s s s s rl q ; q þ rl q þ ; q þ 2 2 2 2 2 2 SðQ; Q; EÞ ¼ ðjC 0 j þ jC 1 j Þ P0 . ðB:7Þ Q4 Multiplying the interference term Eq. (B.6) with eihqs=k eihqs=k , moving the phase factor eihqðsgÞ=k inside the integral and substituting Z ¼ q x results in ihsq=k SðQ; Q; EÞ C0C1 e Q4 h g s g sio eihðsgÞq=k P0 q P0 q þ . ðB:8Þ 2 2 We note that the activation of the biprism does not induce an additional phase ramp (i.e. a different fringe period) when it is located in the upper focal plane. We can also use the dipole approximation for S SðQ; Q0 Þ / q q0 þ q2e ,
(B.9)
which is valid for angles up to the Bethe ridge (many FWHMs).
For s ¼ g the mixed term is 1 iðhg=kÞq P0 . 2R C 0 C 1 e q2 þ q2e
(B.10)
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