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Coherence effects in plasmon excitation
Ultramicroscopy28 (1989) 47-48 North-Holland, Amsterdam
47
C O H E R E N C E E F F E C T S IN P L A S M O N E X C I T A T I O N P. S C H A T T S C H N E I D E R and D.-S. SU Institut fiir Angewandte und Technische Physik, Technische Unioersitiit, A-1040 Vienna, Austria
Received at Editorial Office 3 November 1988; presented at Workshop March 1988
The coherent contribution to double-plasmonexcitation in electron energy loss spectra (EELS) of aluminum was obtained by plural scattering deconvolution.Although it was found to be much smaller than reported previously, Kramers-K.ronig analysis shows that the assumption of linear response breaks down in the case of A1.
Coherence in plural excitation of plasmons is an important issue in plasmon spectroscopy. In the last decade, some theoretical [1,2] and experimental [3-6] work has been devoted to the question of coherence in plasmon excitation. In EELS experiments, intensities of some ten to some hundred percent are found at the double plasmon excitation energy, depending on the thickness of the specimen. This is because the fast electron traversing the specimen in an EELS experiment interacts a number of times with the solid state plasma; in each interaction along its trajectory it will loose the plasmon excitation energy Ep with a high probability, hence, the EEL spectrum exhibits a number of peaks at the multiples of the single plasmon excitation energy. This effect masks the coherent excitation of two quanta of plasma oscillation in one event. The incoherent superposition can, in principle, be removed by deconvolution. The reason for separating out the coherent events is primarily to have a test for linear response theory, or to put it another way: How good an approximation is the proportionality of the scattering cross-section to the dynamic structure factor S(q, E ) or to the loss function Im(1/c(q,
E)):
(1)
~3--------~--°cx IS(q, E ) 12 cc Im c(q, E ) OE aq 2
(1)
which is implied by any linear ansatz (e.g. the
golden rule of perturbation theory or the linearity between electric fields and polarization in classical electrodynamics). Secondly, experiments which single out the coherent double-plasmon events might be compared with higher-order scattering theories in order to improve on them. Removal of the incoherent contributions from the double-plasmon loss can be accomplished by any of the available plural-scattering deconvolution methods. We used a method particularly suited for momentum-resolved low-loss spectra [7,8]. Fig. 1 is a measured energy loss spectrum of a 49 nm thick polycrystalline aluminum film, and the same after deconvolution. For the relative excitation probability for coherent excitation of two plasmons in one event (compared to the single plasmon excitation probability) we obtained F 2 = 0.005. This is the lowest value reported thus far. Earlier experiments on F 2 yielded 0.02 [5], 0.07 [3], 0.13 [4]. Theory based upon RPA predicts 0.02-0.17 [1,2]. Recent measurements with thicker films seem to indicate that the coherent contribution increases with film thickness, even for polycrystalline A1. If this is true there is a coherence length in thin films, not necessarily dependent on grain size. An interesting consequence of the existence of coherent double plasmon excitations is the breakdown of linear response theory. This can be
0304-3991/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
P. Schattschneider, D.-S. Su / Coherence effects in plasmon excitation
48
AI 49nm
15
45 ECeV)
30
Fig. 1. Measured spectrum (dashed) of a polycrystalline 49 nm thick AI film and deconvoluted spectrum with plural scattering removed (full line). Note the change of scale.
shown by a calculation of the dielectric function c from the deconvoluted energy loss spectrum. Fig. 2 is Im(c(~)) of a 49 nm thin aluminum film, obtained by Kramers-Kronig analysis of the deconvoluted loss spectrum (fig. 1), drawn as a function of energy loss E = h o~. Any local maximum in Im(c) is caused by a resonant transverse excitation in the medium. The feature at ~ 30 eV would be interpreted as a resonance frequency according to an interband transition in the specimen, whereas we know there is no such thing. Rather, the peak stems from the double-plasmon excitation. The reason for this misinterpretation
must be a wrong assumption, either in KramersKronig analysis or in eq. (1). Since 1/c is analytic, K r a m e r s - K r o n i g analysis holds exactly. The reason for the occurrence of an artifact can only lie in our second assumption, eq. (1). As stated above, this relation between cross-section and Im (c) is based upon linear response theory. In an exact calculation, the right side of eq. (1) would contain infinitely many superlinear terms, all of them considered negligible in linear response. If they were no longer neglected, Kramers-Kronig analysis could not be applied in terms of 1/c. Instead, one would have to extract this function from the deconvoluted loss spectrum prior to KKA, by getting rid of the superlinear terms. The occurrence of an artifact at - 30 eV shows that at the level of accuracy nowadays available in EELS, nonlinear effects in the response of a medium to a fast electron cannot be neglected any longer. We would like to thank M. Adamiker for the K K A computer program. This work was partly sponsored by the Hochschuljubiliiumsstiftung der Stadt Wien.
References
,0
i~
i0
2'5
3'o
~I~vl
Fig. 2. Im(c(~)) obtained from the deconvoluted spectrum fig. 1 by Kramcrs-Kronig analysis. The subsidiary maximum at 30 eV is an artifact caused by assuming linear response.
[1] J.C. Ashley and R.H. Ritchie, Phys. Status Solidi 38 (1970) 425. [2] K.S. Srivastava, S. Singh, P. Gupta and O.K. Harsh, J. Electron Spectrose. Related Phenomena 25 (1982) 211. [3] P.E. Batson and J. Silcox, Phys. Rev. B27 (1983) 5224. [4] J.C. Spencc and A.E. Spargo, Phys. Rev. Letters 26 (1971) 895. [5] D.L. Miscli and A.J. Atkins, J. Phys. C4 (1971) L81. [6] P. Schattschneider, F. F~dermayr and D.-S. Su, Phys. Rev. Letters 59 (1987) 724. [7] P. Schattsclmeider, Phil. Mag. B47 (1983) 555-560. [8] P. Schattsclmeider, M. Zapfl and P. Skalicky, Inverse Problems 1 (1985) 381.
47
C O H E R E N C E E F F E C T S IN P L A S M O N E X C I T A T I O N P. S C H A T T S C H N E I D E R and D.-S. SU Institut fiir Angewandte und Technische Physik, Technische Unioersitiit, A-1040 Vienna, Austria
Received at Editorial Office 3 November 1988; presented at Workshop March 1988
The coherent contribution to double-plasmonexcitation in electron energy loss spectra (EELS) of aluminum was obtained by plural scattering deconvolution.Although it was found to be much smaller than reported previously, Kramers-K.ronig analysis shows that the assumption of linear response breaks down in the case of A1.
Coherence in plural excitation of plasmons is an important issue in plasmon spectroscopy. In the last decade, some theoretical [1,2] and experimental [3-6] work has been devoted to the question of coherence in plasmon excitation. In EELS experiments, intensities of some ten to some hundred percent are found at the double plasmon excitation energy, depending on the thickness of the specimen. This is because the fast electron traversing the specimen in an EELS experiment interacts a number of times with the solid state plasma; in each interaction along its trajectory it will loose the plasmon excitation energy Ep with a high probability, hence, the EEL spectrum exhibits a number of peaks at the multiples of the single plasmon excitation energy. This effect masks the coherent excitation of two quanta of plasma oscillation in one event. The incoherent superposition can, in principle, be removed by deconvolution. The reason for separating out the coherent events is primarily to have a test for linear response theory, or to put it another way: How good an approximation is the proportionality of the scattering cross-section to the dynamic structure factor S(q, E ) or to the loss function Im(1/c(q,
E)):
(1)
~3--------~--°cx IS(q, E ) 12 cc Im c(q, E ) OE aq 2
(1)
which is implied by any linear ansatz (e.g. the
golden rule of perturbation theory or the linearity between electric fields and polarization in classical electrodynamics). Secondly, experiments which single out the coherent double-plasmon events might be compared with higher-order scattering theories in order to improve on them. Removal of the incoherent contributions from the double-plasmon loss can be accomplished by any of the available plural-scattering deconvolution methods. We used a method particularly suited for momentum-resolved low-loss spectra [7,8]. Fig. 1 is a measured energy loss spectrum of a 49 nm thick polycrystalline aluminum film, and the same after deconvolution. For the relative excitation probability for coherent excitation of two plasmons in one event (compared to the single plasmon excitation probability) we obtained F 2 = 0.005. This is the lowest value reported thus far. Earlier experiments on F 2 yielded 0.02 [5], 0.07 [3], 0.13 [4]. Theory based upon RPA predicts 0.02-0.17 [1,2]. Recent measurements with thicker films seem to indicate that the coherent contribution increases with film thickness, even for polycrystalline A1. If this is true there is a coherence length in thin films, not necessarily dependent on grain size. An interesting consequence of the existence of coherent double plasmon excitations is the breakdown of linear response theory. This can be
0304-3991/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
P. Schattschneider, D.-S. Su / Coherence effects in plasmon excitation
48
AI 49nm
15
45 ECeV)
30
Fig. 1. Measured spectrum (dashed) of a polycrystalline 49 nm thick AI film and deconvoluted spectrum with plural scattering removed (full line). Note the change of scale.
shown by a calculation of the dielectric function c from the deconvoluted energy loss spectrum. Fig. 2 is Im(c(~)) of a 49 nm thin aluminum film, obtained by Kramers-Kronig analysis of the deconvoluted loss spectrum (fig. 1), drawn as a function of energy loss E = h o~. Any local maximum in Im(c) is caused by a resonant transverse excitation in the medium. The feature at ~ 30 eV would be interpreted as a resonance frequency according to an interband transition in the specimen, whereas we know there is no such thing. Rather, the peak stems from the double-plasmon excitation. The reason for this misinterpretation
must be a wrong assumption, either in KramersKronig analysis or in eq. (1). Since 1/c is analytic, K r a m e r s - K r o n i g analysis holds exactly. The reason for the occurrence of an artifact can only lie in our second assumption, eq. (1). As stated above, this relation between cross-section and Im (c) is based upon linear response theory. In an exact calculation, the right side of eq. (1) would contain infinitely many superlinear terms, all of them considered negligible in linear response. If they were no longer neglected, Kramers-Kronig analysis could not be applied in terms of 1/c. Instead, one would have to extract this function from the deconvoluted loss spectrum prior to KKA, by getting rid of the superlinear terms. The occurrence of an artifact at - 30 eV shows that at the level of accuracy nowadays available in EELS, nonlinear effects in the response of a medium to a fast electron cannot be neglected any longer. We would like to thank M. Adamiker for the K K A computer program. This work was partly sponsored by the Hochschuljubiliiumsstiftung der Stadt Wien.
References
,0
i~
i0
2'5
3'o
~I~vl
Fig. 2. Im(c(~)) obtained from the deconvoluted spectrum fig. 1 by Kramcrs-Kronig analysis. The subsidiary maximum at 30 eV is an artifact caused by assuming linear response.
[1] J.C. Ashley and R.H. Ritchie, Phys. Status Solidi 38 (1970) 425. [2] K.S. Srivastava, S. Singh, P. Gupta and O.K. Harsh, J. Electron Spectrose. Related Phenomena 25 (1982) 211. [3] P.E. Batson and J. Silcox, Phys. Rev. B27 (1983) 5224. [4] J.C. Spencc and A.E. Spargo, Phys. Rev. Letters 26 (1971) 895. [5] D.L. Miscli and A.J. Atkins, J. Phys. C4 (1971) L81. [6] P. Schattschneider, F. F~dermayr and D.-S. Su, Phys. Rev. Letters 59 (1987) 724. [7] P. Schattsclmeider, Phil. Mag. B47 (1983) 555-560. [8] P. Schattsclmeider, M. Zapfl and P. Skalicky, Inverse Problems 1 (1985) 381.