- Email: [email protected]
Experiments on inelastic electron holography
ARTICLE IN PRESS
Ultramicroscopy 106 (2006) 1012–1018 www.elsevier.com/locate/ultramic
Experiments on inelastic electron holography P.L. Potapova,,1, H. Lichteb, J. Verbeecka, D. van Dycka a
Electron microscopy for Materials Research (EMAT), University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium b Triebenberglabor, Dresden University, D-01062 Dresden, Germany Received 12 July 2005; received in revised form 18 October 2005; accepted 4 May 2006
Abstract Using the combination of an electron biprism and an energy filter, the coherence distribution in an inelastically scattered wave-field is measured. It is found that the degree of coherence decreases rapidly with increasing distance between two superimposed points in the object, and with increasing energy-loss. In a Si sample, coherence of plasmon scattering increases in vacuum with the distance from the edge of the sample. r 2006 Published by Elsevier B.V. Keywords: Inelastic electrons; Holography
1. Introduction Electron holography makes use of coherence properties of incident electrons and provides information about not only the amplitude but also the phase of scattering. The biprism installed above the image plane of the microscope splits the electron image wave in two waves and superimposes them in the image plane. In conventional off-axis holography, one of the two waves is elastically scattered in the object and superposed with the other one passing through vacuum near the object. It should be stressed that only elastically scattered electrons efficiently take part in the formation of such a hologram. Electrons inelastically scattered in the object would not contribute to the interference contrast with the vacuum reference wave, because two waves with an energy difference DE cause rapid beating with the frequency n ¼ DE=h [1]. However, two inelastically scattered waves, which excite the same state in the object and hence suffer also exactly the same energy-loss, hence should show no beat. Therefore, they can build up a stationary hologram, if they are phase-coherent, i.e. have a fixed phase relation. Already the Corresponding author. Tel.: +32 3 218 0472; fax: +32 3 218 0257.
E-mail address: [email protected] (P.L. Potapov). Now with AMD Saxony, Wischdorfer Landstrasse 101, D-01330 Dresden, Germany.
first experiments with a biprism and an energy filter have revealed that the coherence can be partially preserved in an inelastically scattered wave-field [2]; this result was confirmed by Lichte and Freitag in a more elaborate experiment, where the width of coherence was estimated larger than 10 nm in the case of Al-plasmon scattered electrons at an energy transfer of about 15 eV [3]. The coherence width of the electron transmitted through the object under inelastic interaction can be measured from the interference contrast formed by the inelastic waves coming from two different points in the object. This experiment can be performed by means of the electron biprism, which superimposes two waves in the image plane as shown in Fig. 1. Generally, an interference pattern of two waves with equal amplitude a and mutual coherence m ¼ jmj expðixÞ with 0pjmjp1 can be described by IðxÞ ¼ 2a2 ð1 þ jmj cosð2pqc x þ xÞÞ,
(1)
where the biprism is oriented as shown in Fig. 1. The spatial frequency of the fringes qc ¼ b=l is given by the electron wavelength l and by the superposition angle b produced by the biprism. The degree of coherence in the hologram is first of all given by the intensity distribution issued by the electron source. For the usual sources it is given as
1
0304-3991/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.ultramic.2006.05.012
mðd; nÞ ¼ mspat ðdÞmtemp ðnÞ,
(2)
ARTICLE IN PRESS P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
1013
the degree of coherence is generally affected by a factor minel ðE; d; pÞ with 0pjminel ðE; d; pÞjp1, which depends on the energy-loss E, the distance d of superimposed points in the inelastic wave field, and on the nature of the inelastic excitation p. In that case the interference pattern can be expressed as IðxÞ ¼ 2a2 ð1 þ jmspat ðdÞjjminel ðE; d; pÞj cosð2pqc Þ,
(3)
where the phases induced by coherence are still neglected. In order to separate the coherence factor of the inelastic interaction, we have to divide the measured fringe contrast jmspat ðdÞjjminel ðE; d; pÞj by the degree of spatial coherence jmspat ðdÞj of the source. In this way the normalized coherence is a property of the inelastic process independent of characteristics of the instrument actually used. In this paper, we investigate the findings given in Refs. [2,3] more systematically and qualitatively. Coherence of inelastic scattering minel is experimentally studied under different conditions such as varying the biprism shear d and varying energy-loss. The energy of incident electrons was fixed to 300 kV in the present experiments, however, the obtained results might be also relevant for other operating voltages when transforming the mutual coherence function into the mixed dynamic form factor [5]. 2. Experimental method
Fig. 1. Schematic of the experiment for measuring the coherence in an inelastically scattered wavefield. The biprism voltage controls the distance of object points (shear d) subsequently superimposed in the image plane. By means of the energy filter, one can select the energy loss contributing to the interference pattern.
i.e. as the product of spatial coherence mspat ðdÞ and temporal coherence mtemp ðnÞ, where d is a lateral shift between two superimposed waves (Fig. 1) and n is the order of interference of those waves. The later is defined as n ¼ Dl=l where Dl is the maximum shift of two superimposed waves in the longitudinal direction. Temporal coherence is controlled by monocromaticity of the electron source. It can be shown that an electron wave is temporally coherent over the length of l2 =Dl along the direction of propagation, where Dl is the spread of the wavelengths in the source. For a typical field emission gun operating at 300 kV, the coherence length is about 103 nm, which means that the temporal coherence is preserved up to the order of interference n105. Since np103 in usual holograms, jmtemp j 1 and coherence is determined here only by spatial coherence function mspat ðdÞ. At elastic interaction with the object, the degree of coherence is preserved. At inelastic interaction, however,
Energy filtered electron holography was performed in a Philips CM30 FEG electron microscope equipped with a biprism and a post-column Gatan energy filter. The FWHM of the zero-loss peak was about 1.0 eV. All holograms were recorded in a 5 eV energy window selected by a corresponding slit in the energy dispersive plane of the filter. Before performing a measurement, the energy was carefully calibrated using the zero-loss peak. As usual, the mean energy in the window was varied by a small change of the accelerating voltage, which ensures identical focusing conditions for different energies. This way, we selectively examined different specific processes, such as elastic scattering when zero-loss peak selected, or plasmon scattering with plasmon-loss peak selected. For this measurements, special attention has to be paid to ensure that elastic electrons are masked out from the possible contribution to the inelastic interference pattern. Due to the energy spread of the electron source, small tails from the zero-loss peak might be present in the energy region of the, e.g. plasmon-loss. To estimate the influence of these tails, a series of filtered holograms with increasing mean energy-loss were taken in vacuum very far from the specimen. From this, the relative contribution of zero loss tails in a given energy-loss window could be estimated and subtracted. As shown in Fig. 2, the influence of tails decreases rapidly with energy-loss and need to be accounted only at very low losses, or in the situation where the ratio between zero- and plasmon-loss intensities is extremely high. It is interesting to note that interference in zero-loss tails shows a severely degraded fringe contrast
ARTICLE IN PRESS 1014
P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
Fig. 2. (a) Vacuum holograms recorded in the zero-loss and 7.5 eV-loss window. Due to tails in the energy distribution of the electron source, some fringe pattern is detected in energy-loss windows, however, the fringe contrast degrades and (b) its intensity decays rapidly with increasing energy-loss.
compared to that in the zero-loss window (see Fig. 2a). Thus, the influence of tails can be described as adding a uniform background to the inelastic hologram, which reduces rather than enhances the observable fringe contrast. 3. Results and discussion 3.1. Coherence vs. shear Metallic Al shows sharp plasmon peaks of several orders in the EELS spectrum (Fig. 3a). A series of holograms was taken with increasing lateral shear between the two superimposed waves in the homogeneous Al sample. The sample was oriented to the [1 1 0] zone and showed the thickness of 1.4lp (lp is a mean free path for plasmon scattering) in the examined area. Systematic investigation of the influence of orientation and thickness were not yet performed, since no pronounced effects on the measured inelastic coherence were found in the preliminary experiments. At each value of the biprism shear, two holograms were recorded: the first one was recorded in the zero-loss window and, immediately afterwards, the second one with
the window centered at the middle of the first plasmon peak. Fig. 3a shows examples of the observed zero- and plasmon-loss holograms at several d values. In the plasmon-loss holograms, the fringe contrast is much smaller than in the corresponding zero-loss holograms, but still detectable even at the large shear values. As described in Section 1, the contrast of inelastic fringes normalized to that of zero-loss fringes at corresponding shears represents the coherence factor of 15.5 eV plasmon scattering for a given primary energy. The degree of coherence extracted from the fringe contrast is plotted as a function of the shear value in Fig. 3c. Theoretical evaluation using mutual intensity formalism and a free-electron-gas model [4] predicts that the coherence of plasmon scattering should follow the formula: minel ðE; dÞ ¼ K 0 ðkye dÞ= lnðyc =ye Þ, minel ðE; dÞp1,
ð4Þ
where K0 is the pffiffiffi modified Bessel function of order zero (K 0 ðxÞ ex = x at large x); k is the wave number of the primary electron wave; ye ¼ E=E 0 ð1 þ 1=gÞ is the characteristic scattering angle for the given energy-loss E and
ARTICLE IN PRESS P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
1015
Fig. 3. (a) EELS spectrum of Al and (b) energy filtered holograms recorded in the zero- and plasmon-loss (15.5 eV) windows at several shear values d. (c) Fringe contrast in the plasmon-loss window normalized to that in the zero-loss results represents the degree of coherence of plasmon scattering as a function of shear, i.e. the distribution of coherence in the inelastic wave field. (The theoretical curve in Fig. 3c was calculated in Ref. [4] and published there in comparison to experiment.)
primary energy E0 with accounting for the relativistic pffiffiffiffiffiffiffi factor g; yc ¼ 2ye is the critical angle corresponding to the Bethe ridge. Formula (4) is not valid for very small d values, thus, in situations where Eq. (4) leads to minel41, it should be truncated to 1 [4]. We also note that the relativistic corrections are not completely included in Eq. (4) and are only accounted for by taking the relativistic expression for ye. As seen from Fig. 3c, this theoretical
model reproduces the experimental dependence quite reasonably, although the degree of coherence is slightly underestimated by theory for all shears. It is also clear from the experimental data in Fig. 3c and theoretical predictions (Eq. (4)) that the coherence width of plasmon scattering cannot be uniquely defined by a single number. The coherence decays nearly exponentially with the shear d between the two superposed waves, but, strictly
ARTICLE IN PRESS 1016
P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
speaking, never equals zero. Thus, the measured maximal coherence width would depend on the minimal degree of the coherence detectable by the given instrument. For the experimental conditions used in the present work, the coherence of Al plasmon scattering approaches the noise level at shears d30 nm. 3.2. Coherence vs. energy-loss Since both energy and width of plasmon-loss peak vary significantly in different materials, it is possible to study the dependence of coherence of inelastic scattering on energyloss. Fig. 4a shows EELS spectra of several examined materials and the energy windows, in which inelastic holograms were taken. The extracted degree of coherence at a constant shear d of 13.5 nm is plotted as function of
energy-loss in Fig. 4b. Despite random deviations, all data seem to lie on the same curve, which rapidly decays with energy-loss. This finding follows reasonably the theoretical calculations for plasmon energy-losses [4] but here again, the calculated coherence is underestimated especially for low-losses. This discrepancy might be caused by the relativistic effects, which are not completely included in Eq. (4). Note that some peaks in Fig. 4a are associated (at least partially) with scattering processes different from the excitation of bulk plasmons. Namely, there are surface plasmons in Al, single electron excitations in MnO2 and H K-shell ionization in KInPO4(F,H). Nevertheless, the corresponding measurements yield coherence data close to the curve for bulk plasmons. This suggests that the coherence of inelastic scattering is controlled mainly by the value of the energy-loss rather than by the nature of the excitation, which is in agreement with theoretical conclusions in Refs. [4,5].
3.3. Coherence near the edge of sample
Fig. 4. (a) EELS spectra of different materials and (b) the degree of coherence for inelastic scattering measured in the marked energy windows. Most of the energy-losses are associated with excitations of bulk plasmons, although the 7.5 eV window in Al, the 9.5 eV window in MnO2 and the 11 eV window in KInPO4(F,OH) are associated, at least partially, with surface plasmons, single electron transition and H K edge excitation, respectively. Also shown is the theoretical coherence curve for plasmon-losses [4].
Inelastic scattering does not abruptly terminate at the sample edge but extends into vacuum [6,7]. Fig. 5a shows filtered holograms taken in the zero- and plasmon-loss (17 eV) windows near the edge of a silicon sample. As seen from Fig. 5b, the intensity of the inelastic scattering decays with the distance from the edge in quantitative agreement with previous data by Isaakson et al. [6]. Surprisingly, the inelastic interference fringes are visible in vacuum even at a distance of 20–30 nm from the edge (see inset in Fig. 5a). Moreover, the calculated degree of coherence clearly increases with the distance from the edge, at least for the first 15–20 nm (Fig. 5c). Similar results were obtained near the edge of hydrogen bubbles, where inelastic scattering was attributed to ionization of H2 molecules [8]. The intensity of plasmon scattering is quite small at large distances from the edge, thus, the extracted degree of coherence might be subject to artifacts. Therefore, we first tried to exclude the possible influence of tails from the zeroloss peak extending to the plasmon-loss window. These tails could add a continuous background to the inelastic scattering as mentioned in Section 2. As estimated, the influence of tails is only noticeable at distances larger than 25 nm, where the ratio between plasmon- and zero-loss intensities drops below 1:40. Even there, the correction due to the tails does not exceed 1:20 of the calculated degree of coherence. Additionally, the noise in the intensity holograms might result in a false measurement of the degree of coherence. It is intuitively clear that noise will decrease the visibility of the fringes and therefore the measured coherence. Both considered artifacts can result in an underestimated coherence in inelastic holograms of low intensity. Thus, the observed increase of coherence of inelastic scattering in the vacuum should be caused by a physical effect.
ARTICLE IN PRESS P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
1017
Fig. 5. (a) The interference pattern near the specimen edge. The data are collected from filtered holograms recorded in the zero- and plasmon-loss (17 eV) windows near the edge of a Si sample. The biprism shear d was 13.5 nm. Fringe contrast in plasmon scattering is clearly visible even at large distances from the edge (see the inset with enhanced level of intensity). The (b) intensity and (c) degree of coherence for elastic and plasmon scattering were extracted from the corresponding hologram data as a function of the distance from the edge of the sample.
Indeed, such a counterintuitive effect has been recently reproduced by the theoretical simulations using the density-matrix [9] approach and the mutual-intensity [4] formalism. As argued in Ref. [8], the physical reason for this behavior is the increased ambiguity of the which-way information for the electrons passing in the vacuum aside from the sample.
4. Conclusions Electron holograms can be formed by electrons, which experienced inelastic scattering, although the contrast of the interference fringes is much weaker than the one in elastic electron holography. The contrast of the inelastic interference fringes decays rapidly with biprism shear and
ARTICLE IN PRESS 1018
P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
energy-loss. In the vacuum near the edge of a sample, an unexpected behavior of coherence is observed, namely an increase of the contrast in the inelastic fringes.
Acknowledgements The cooperation between the university of Antwerp and TU Dresden was supported by the Francqui Foundation in Belgium. J. Verbeeck would like to acknowledge the financial support of the fund for scientific research Flanders (FWO). Fruitful discussions with P. Schattschneider are acknowledged.
References [1] D. van Dyck, H. Lichte, J.C.H. Spence, Ultramicroscopy 81 (2000) 187. [2] Harscher, H. Lichte, J. Mayer, Ultramicroscopy 69 (1997) 201. [3] H. Lichte, B. Freitag, Ultramicroscopy 81 (2000) 177. [4] J. Verbeeck, D. van Dyck, H. Lichte, P.L. Potapov, P. Schattschneider, Ultramicroscopy 102 (2005) 239. [5] P. Schattschneider, M. Nelhiebel, B. Jouffrey, Phys. Rev. B 59 (1999) 10959. [6] M. Isaakson, J.P. Langmore, H. Rose, Optik 41 (1974) 92. [7] D. Muller, J. Silcox, Ultramicroscopy 59 (1995) 195. [8] P.L. Potapov, H. Lichte, J. Verbeeck, D. van Dyck, P. Schattschneider, in preparation. [9] P. Schattschneider, H. Lichte, Phys. Rev. B 71 (2005) 045130.
Ultramicroscopy 106 (2006) 1012–1018 www.elsevier.com/locate/ultramic
Experiments on inelastic electron holography P.L. Potapova,,1, H. Lichteb, J. Verbeecka, D. van Dycka a
Electron microscopy for Materials Research (EMAT), University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium b Triebenberglabor, Dresden University, D-01062 Dresden, Germany Received 12 July 2005; received in revised form 18 October 2005; accepted 4 May 2006
Abstract Using the combination of an electron biprism and an energy filter, the coherence distribution in an inelastically scattered wave-field is measured. It is found that the degree of coherence decreases rapidly with increasing distance between two superimposed points in the object, and with increasing energy-loss. In a Si sample, coherence of plasmon scattering increases in vacuum with the distance from the edge of the sample. r 2006 Published by Elsevier B.V. Keywords: Inelastic electrons; Holography
1. Introduction Electron holography makes use of coherence properties of incident electrons and provides information about not only the amplitude but also the phase of scattering. The biprism installed above the image plane of the microscope splits the electron image wave in two waves and superimposes them in the image plane. In conventional off-axis holography, one of the two waves is elastically scattered in the object and superposed with the other one passing through vacuum near the object. It should be stressed that only elastically scattered electrons efficiently take part in the formation of such a hologram. Electrons inelastically scattered in the object would not contribute to the interference contrast with the vacuum reference wave, because two waves with an energy difference DE cause rapid beating with the frequency n ¼ DE=h [1]. However, two inelastically scattered waves, which excite the same state in the object and hence suffer also exactly the same energy-loss, hence should show no beat. Therefore, they can build up a stationary hologram, if they are phase-coherent, i.e. have a fixed phase relation. Already the Corresponding author. Tel.: +32 3 218 0472; fax: +32 3 218 0257.
E-mail address: [email protected] (P.L. Potapov). Now with AMD Saxony, Wischdorfer Landstrasse 101, D-01330 Dresden, Germany.
first experiments with a biprism and an energy filter have revealed that the coherence can be partially preserved in an inelastically scattered wave-field [2]; this result was confirmed by Lichte and Freitag in a more elaborate experiment, where the width of coherence was estimated larger than 10 nm in the case of Al-plasmon scattered electrons at an energy transfer of about 15 eV [3]. The coherence width of the electron transmitted through the object under inelastic interaction can be measured from the interference contrast formed by the inelastic waves coming from two different points in the object. This experiment can be performed by means of the electron biprism, which superimposes two waves in the image plane as shown in Fig. 1. Generally, an interference pattern of two waves with equal amplitude a and mutual coherence m ¼ jmj expðixÞ with 0pjmjp1 can be described by IðxÞ ¼ 2a2 ð1 þ jmj cosð2pqc x þ xÞÞ,
(1)
where the biprism is oriented as shown in Fig. 1. The spatial frequency of the fringes qc ¼ b=l is given by the electron wavelength l and by the superposition angle b produced by the biprism. The degree of coherence in the hologram is first of all given by the intensity distribution issued by the electron source. For the usual sources it is given as
1
0304-3991/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.ultramic.2006.05.012
mðd; nÞ ¼ mspat ðdÞmtemp ðnÞ,
(2)
ARTICLE IN PRESS P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
1013
the degree of coherence is generally affected by a factor minel ðE; d; pÞ with 0pjminel ðE; d; pÞjp1, which depends on the energy-loss E, the distance d of superimposed points in the inelastic wave field, and on the nature of the inelastic excitation p. In that case the interference pattern can be expressed as IðxÞ ¼ 2a2 ð1 þ jmspat ðdÞjjminel ðE; d; pÞj cosð2pqc Þ,
(3)
where the phases induced by coherence are still neglected. In order to separate the coherence factor of the inelastic interaction, we have to divide the measured fringe contrast jmspat ðdÞjjminel ðE; d; pÞj by the degree of spatial coherence jmspat ðdÞj of the source. In this way the normalized coherence is a property of the inelastic process independent of characteristics of the instrument actually used. In this paper, we investigate the findings given in Refs. [2,3] more systematically and qualitatively. Coherence of inelastic scattering minel is experimentally studied under different conditions such as varying the biprism shear d and varying energy-loss. The energy of incident electrons was fixed to 300 kV in the present experiments, however, the obtained results might be also relevant for other operating voltages when transforming the mutual coherence function into the mixed dynamic form factor [5]. 2. Experimental method
Fig. 1. Schematic of the experiment for measuring the coherence in an inelastically scattered wavefield. The biprism voltage controls the distance of object points (shear d) subsequently superimposed in the image plane. By means of the energy filter, one can select the energy loss contributing to the interference pattern.
i.e. as the product of spatial coherence mspat ðdÞ and temporal coherence mtemp ðnÞ, where d is a lateral shift between two superimposed waves (Fig. 1) and n is the order of interference of those waves. The later is defined as n ¼ Dl=l where Dl is the maximum shift of two superimposed waves in the longitudinal direction. Temporal coherence is controlled by monocromaticity of the electron source. It can be shown that an electron wave is temporally coherent over the length of l2 =Dl along the direction of propagation, where Dl is the spread of the wavelengths in the source. For a typical field emission gun operating at 300 kV, the coherence length is about 103 nm, which means that the temporal coherence is preserved up to the order of interference n105. Since np103 in usual holograms, jmtemp j 1 and coherence is determined here only by spatial coherence function mspat ðdÞ. At elastic interaction with the object, the degree of coherence is preserved. At inelastic interaction, however,
Energy filtered electron holography was performed in a Philips CM30 FEG electron microscope equipped with a biprism and a post-column Gatan energy filter. The FWHM of the zero-loss peak was about 1.0 eV. All holograms were recorded in a 5 eV energy window selected by a corresponding slit in the energy dispersive plane of the filter. Before performing a measurement, the energy was carefully calibrated using the zero-loss peak. As usual, the mean energy in the window was varied by a small change of the accelerating voltage, which ensures identical focusing conditions for different energies. This way, we selectively examined different specific processes, such as elastic scattering when zero-loss peak selected, or plasmon scattering with plasmon-loss peak selected. For this measurements, special attention has to be paid to ensure that elastic electrons are masked out from the possible contribution to the inelastic interference pattern. Due to the energy spread of the electron source, small tails from the zero-loss peak might be present in the energy region of the, e.g. plasmon-loss. To estimate the influence of these tails, a series of filtered holograms with increasing mean energy-loss were taken in vacuum very far from the specimen. From this, the relative contribution of zero loss tails in a given energy-loss window could be estimated and subtracted. As shown in Fig. 2, the influence of tails decreases rapidly with energy-loss and need to be accounted only at very low losses, or in the situation where the ratio between zero- and plasmon-loss intensities is extremely high. It is interesting to note that interference in zero-loss tails shows a severely degraded fringe contrast
ARTICLE IN PRESS 1014
P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
Fig. 2. (a) Vacuum holograms recorded in the zero-loss and 7.5 eV-loss window. Due to tails in the energy distribution of the electron source, some fringe pattern is detected in energy-loss windows, however, the fringe contrast degrades and (b) its intensity decays rapidly with increasing energy-loss.
compared to that in the zero-loss window (see Fig. 2a). Thus, the influence of tails can be described as adding a uniform background to the inelastic hologram, which reduces rather than enhances the observable fringe contrast. 3. Results and discussion 3.1. Coherence vs. shear Metallic Al shows sharp plasmon peaks of several orders in the EELS spectrum (Fig. 3a). A series of holograms was taken with increasing lateral shear between the two superimposed waves in the homogeneous Al sample. The sample was oriented to the [1 1 0] zone and showed the thickness of 1.4lp (lp is a mean free path for plasmon scattering) in the examined area. Systematic investigation of the influence of orientation and thickness were not yet performed, since no pronounced effects on the measured inelastic coherence were found in the preliminary experiments. At each value of the biprism shear, two holograms were recorded: the first one was recorded in the zero-loss window and, immediately afterwards, the second one with
the window centered at the middle of the first plasmon peak. Fig. 3a shows examples of the observed zero- and plasmon-loss holograms at several d values. In the plasmon-loss holograms, the fringe contrast is much smaller than in the corresponding zero-loss holograms, but still detectable even at the large shear values. As described in Section 1, the contrast of inelastic fringes normalized to that of zero-loss fringes at corresponding shears represents the coherence factor of 15.5 eV plasmon scattering for a given primary energy. The degree of coherence extracted from the fringe contrast is plotted as a function of the shear value in Fig. 3c. Theoretical evaluation using mutual intensity formalism and a free-electron-gas model [4] predicts that the coherence of plasmon scattering should follow the formula: minel ðE; dÞ ¼ K 0 ðkye dÞ= lnðyc =ye Þ, minel ðE; dÞp1,
ð4Þ
where K0 is the pffiffiffi modified Bessel function of order zero (K 0 ðxÞ ex = x at large x); k is the wave number of the primary electron wave; ye ¼ E=E 0 ð1 þ 1=gÞ is the characteristic scattering angle for the given energy-loss E and
ARTICLE IN PRESS P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
1015
Fig. 3. (a) EELS spectrum of Al and (b) energy filtered holograms recorded in the zero- and plasmon-loss (15.5 eV) windows at several shear values d. (c) Fringe contrast in the plasmon-loss window normalized to that in the zero-loss results represents the degree of coherence of plasmon scattering as a function of shear, i.e. the distribution of coherence in the inelastic wave field. (The theoretical curve in Fig. 3c was calculated in Ref. [4] and published there in comparison to experiment.)
primary energy E0 with accounting for the relativistic pffiffiffiffiffiffiffi factor g; yc ¼ 2ye is the critical angle corresponding to the Bethe ridge. Formula (4) is not valid for very small d values, thus, in situations where Eq. (4) leads to minel41, it should be truncated to 1 [4]. We also note that the relativistic corrections are not completely included in Eq. (4) and are only accounted for by taking the relativistic expression for ye. As seen from Fig. 3c, this theoretical
model reproduces the experimental dependence quite reasonably, although the degree of coherence is slightly underestimated by theory for all shears. It is also clear from the experimental data in Fig. 3c and theoretical predictions (Eq. (4)) that the coherence width of plasmon scattering cannot be uniquely defined by a single number. The coherence decays nearly exponentially with the shear d between the two superposed waves, but, strictly
ARTICLE IN PRESS 1016
P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
speaking, never equals zero. Thus, the measured maximal coherence width would depend on the minimal degree of the coherence detectable by the given instrument. For the experimental conditions used in the present work, the coherence of Al plasmon scattering approaches the noise level at shears d30 nm. 3.2. Coherence vs. energy-loss Since both energy and width of plasmon-loss peak vary significantly in different materials, it is possible to study the dependence of coherence of inelastic scattering on energyloss. Fig. 4a shows EELS spectra of several examined materials and the energy windows, in which inelastic holograms were taken. The extracted degree of coherence at a constant shear d of 13.5 nm is plotted as function of
energy-loss in Fig. 4b. Despite random deviations, all data seem to lie on the same curve, which rapidly decays with energy-loss. This finding follows reasonably the theoretical calculations for plasmon energy-losses [4] but here again, the calculated coherence is underestimated especially for low-losses. This discrepancy might be caused by the relativistic effects, which are not completely included in Eq. (4). Note that some peaks in Fig. 4a are associated (at least partially) with scattering processes different from the excitation of bulk plasmons. Namely, there are surface plasmons in Al, single electron excitations in MnO2 and H K-shell ionization in KInPO4(F,H). Nevertheless, the corresponding measurements yield coherence data close to the curve for bulk plasmons. This suggests that the coherence of inelastic scattering is controlled mainly by the value of the energy-loss rather than by the nature of the excitation, which is in agreement with theoretical conclusions in Refs. [4,5].
3.3. Coherence near the edge of sample
Fig. 4. (a) EELS spectra of different materials and (b) the degree of coherence for inelastic scattering measured in the marked energy windows. Most of the energy-losses are associated with excitations of bulk plasmons, although the 7.5 eV window in Al, the 9.5 eV window in MnO2 and the 11 eV window in KInPO4(F,OH) are associated, at least partially, with surface plasmons, single electron transition and H K edge excitation, respectively. Also shown is the theoretical coherence curve for plasmon-losses [4].
Inelastic scattering does not abruptly terminate at the sample edge but extends into vacuum [6,7]. Fig. 5a shows filtered holograms taken in the zero- and plasmon-loss (17 eV) windows near the edge of a silicon sample. As seen from Fig. 5b, the intensity of the inelastic scattering decays with the distance from the edge in quantitative agreement with previous data by Isaakson et al. [6]. Surprisingly, the inelastic interference fringes are visible in vacuum even at a distance of 20–30 nm from the edge (see inset in Fig. 5a). Moreover, the calculated degree of coherence clearly increases with the distance from the edge, at least for the first 15–20 nm (Fig. 5c). Similar results were obtained near the edge of hydrogen bubbles, where inelastic scattering was attributed to ionization of H2 molecules [8]. The intensity of plasmon scattering is quite small at large distances from the edge, thus, the extracted degree of coherence might be subject to artifacts. Therefore, we first tried to exclude the possible influence of tails from the zeroloss peak extending to the plasmon-loss window. These tails could add a continuous background to the inelastic scattering as mentioned in Section 2. As estimated, the influence of tails is only noticeable at distances larger than 25 nm, where the ratio between plasmon- and zero-loss intensities drops below 1:40. Even there, the correction due to the tails does not exceed 1:20 of the calculated degree of coherence. Additionally, the noise in the intensity holograms might result in a false measurement of the degree of coherence. It is intuitively clear that noise will decrease the visibility of the fringes and therefore the measured coherence. Both considered artifacts can result in an underestimated coherence in inelastic holograms of low intensity. Thus, the observed increase of coherence of inelastic scattering in the vacuum should be caused by a physical effect.
ARTICLE IN PRESS P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
1017
Fig. 5. (a) The interference pattern near the specimen edge. The data are collected from filtered holograms recorded in the zero- and plasmon-loss (17 eV) windows near the edge of a Si sample. The biprism shear d was 13.5 nm. Fringe contrast in plasmon scattering is clearly visible even at large distances from the edge (see the inset with enhanced level of intensity). The (b) intensity and (c) degree of coherence for elastic and plasmon scattering were extracted from the corresponding hologram data as a function of the distance from the edge of the sample.
Indeed, such a counterintuitive effect has been recently reproduced by the theoretical simulations using the density-matrix [9] approach and the mutual-intensity [4] formalism. As argued in Ref. [8], the physical reason for this behavior is the increased ambiguity of the which-way information for the electrons passing in the vacuum aside from the sample.
4. Conclusions Electron holograms can be formed by electrons, which experienced inelastic scattering, although the contrast of the interference fringes is much weaker than the one in elastic electron holography. The contrast of the inelastic interference fringes decays rapidly with biprism shear and
ARTICLE IN PRESS 1018
P.L. Potapov et al. / Ultramicroscopy 106 (2006) 1012–1018
energy-loss. In the vacuum near the edge of a sample, an unexpected behavior of coherence is observed, namely an increase of the contrast in the inelastic fringes.
Acknowledgements The cooperation between the university of Antwerp and TU Dresden was supported by the Francqui Foundation in Belgium. J. Verbeeck would like to acknowledge the financial support of the fund for scientific research Flanders (FWO). Fruitful discussions with P. Schattschneider are acknowledged.
References [1] D. van Dyck, H. Lichte, J.C.H. Spence, Ultramicroscopy 81 (2000) 187. [2] Harscher, H. Lichte, J. Mayer, Ultramicroscopy 69 (1997) 201. [3] H. Lichte, B. Freitag, Ultramicroscopy 81 (2000) 177. [4] J. Verbeeck, D. van Dyck, H. Lichte, P.L. Potapov, P. Schattschneider, Ultramicroscopy 102 (2005) 239. [5] P. Schattschneider, M. Nelhiebel, B. Jouffrey, Phys. Rev. B 59 (1999) 10959. [6] M. Isaakson, J.P. Langmore, H. Rose, Optik 41 (1974) 92. [7] D. Muller, J. Silcox, Ultramicroscopy 59 (1995) 195. [8] P.L. Potapov, H. Lichte, J. Verbeeck, D. van Dyck, P. Schattschneider, in preparation. [9] P. Schattschneider, H. Lichte, Phys. Rev. B 71 (2005) 045130.