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Stimulated excitation electron microscopy and spectroscopy A. Howie Department of Physics, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 27 July 2014 Received in revised form 15 September 2014 Accepted 18 September 2014
Recent advances in instrumentation for electron optics and spectroscopy have prompted exploration of ultra-low excitations such as phonons, bond vibrations and Johnson noise. These can be excited not just with fast electrons but also thermally or by other external sources of radiation. The near-field theory of electron energy loss and gain provides a convenient platform for analysing these processes. Possibilities for selected phonon mapping and imaging are discussed. Effects should certainly be observable in atomic resolution structure imaging but diffraction contrast imaging could perhaps be more informative. Additional exciting prospects to be explored include the transition from phonon excitation to single atom recoil and the boosting of energy loss and gain signals with tuned laser illumination. & 2014 Elsevier B.V. All rights reserved.
Keywords: Near-field excitation Phonon imaging Atomic displacement Laser-boosted interactions
1. Introduction Over the past decade, Harald Rose's celebrated design for a hexapole aberration corrector [1] has not only fathered outstanding achievements in high resolution structure imaging with the conventional transmission electron microscope (CTEM) [2,3] but has also been accorded tremendous commercial success as a key component in world-wide sales of both the CTEM and the scanning transmission electron microscope (STEM). In the dedicated STEM, great technical success has also been enjoyed by Ondrej Krivanek's independently developed quadrupole–octopole system [4]. Encouraged by these advances in electron optics and other developments with monochromators and spectrometers, electron microscopists are keenly looking for new fields to explore. Very recently the importance of excitations with very low energy ħωokT has emerged as a fresh challenge, firstly through the identification of Johnson noise as a significant cause of e-beam decoherence [5] and secondly with the prospect of spatially localised spectroscopy of phonons and atomic bond vibrations [6–8]. Despite their different microscopic natures, these two phenomena share the common feature that the interaction with the fast electron takes place with pre-existing thermal excitation. Still more dramatic examples of the same effect arise when the excitations concerned are driven to high levels by an external periodic field [9]. As described in Section 2, the recently introduced near-field interaction approach [10–12] to energy loss and gain processes appears to provide a better basis for handling these problems than the usual classical theory of electron energy loss spectroscopy (EELS). How best to combine useful imaging with the improved spectroscopic capability now becoming available is an interesting
issue raised by these developments. For electrons interacting with phonons in the sample, we can with confidence dismiss concern that, the spatial resolution attainable might be limited by the characteristic long range Coulomb impact parameter v/ω that applies to purely electronic excitations of angular frequency ω by a fast electron of velocity v. Anomalous absorption theory in transmission imaging as well as the experimentally observed role of phonon scattering in STEM HAADF imaging provide ample evidence, even in ionic crystals, that phonon scattering operates with atomic scale impact parameters. Any residual long range component may often be negligible since v/ω, measuring the optimum interaction wavelength along the beam as well as the characteristic impact parameter for a free electron, can exceed the typical sample thickness. With the near-field theory described in Section 2, the energy loss and gain processes involved in phonon scattering can be handled more naturally using the familiar phonon potential allowing various options for mapping or imaging to be analysed in Section 3. Tuned laser excitation offers further interesting prospects for boosting the strength of electron energy losses and gains in the infrared and optical regions which are briefly explored in Section 4. Before embarking on these topics however it is important to recall that that the ability to drive selected extremely low frequency modes to high excitation levels by externally applied periodic fields has already been exploited for low resolution imaging. Fig. 1 provides an update to a recent summary [13] of this work and perhaps shows that there may be other ways besides improved electron spectroscopy to enter this new territory including scanning electron acoustic microscopy (SEAM), time resolved scanning optical microscopy (TRSOM) and radio frequency TEM in addition to tip-enhanced Raman spectroscopy (TER), cathodolumenescence
http://dx.doi.org/10.1016/j.ultramic.2014.09.006 0304-3991/& 2014 Elsevier B.V. All rights reserved.
Please cite this article as: A. Howie, Stimulated excitation electron microscopy and spectroscopy, Ultramicroscopy (2014), http://dx.doi. org/10.1016/j.ultramic.2014.09.006i
A. Howie / Ultramicroscopy ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
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108
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STM emap EELS
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Pump – probe TEM
RF TEM
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I
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E O
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TRSOM
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Fig. 1. Time/frequency range and spatial resolution of various microscopy techniques including those (hatched) using low frequency driving fields and identified in the text.
(CDL), photoemission electron microscopy (PEEM) and scanning tunnelling microscopy (STM) techniques. Some of the complexities of vibration modes in thin slabs and cantilevers (involving Lamb waves, flexural modes and Rayleigh modes) which the EM community may soon need to address, have already been explored in pulsed e-beam experiments, where the different dispersion relations can be directly measured [14]. The stroboscopic method using sinusoidal driving fields and phase sensitive detection would probably provide the most powerful entry to this exciting new field if it could be massively extended to phonon frequencies and atomic resolution imaging. In the alternative pump–probe technique using periodic MHz pulses from a laser, many phonon modes are simultaneously excited but in favourable cases the different frequencies can still be resolved by following the time evolution of the response [15].
2. EELS theory in the presence of additional external stimulation 2.1. Fluctuation dissipation theorem – energy gains and losses For very low excitation energies where thermal excitation must be considered, the fluctuation dissipation theorem [16,17] shows that the familiar dielectric excitation expressions for penetrating beams or for aloof beams should be multiplied by a thermal correction factor F(T). Since we can separately observe energy losses as well as energy gains of the fast electron (corresponding respectively to the creation and annihilation of an excitation in the object), it is useful to express F(T) as the sum of these two contributions. F ðT Þ ¼ F loss ðT Þþ F gain ðT Þ ¼
exp ðħω=kTÞ 1 ħω þ ¼ coth 2kT 1 exp ðħω=kTÞ 1 exp ðħω=kTÞ
ð1Þ Longstanding experimental data [18] for energy gains and losses for fast electron interaction with optical phonons have been shown [19] to be reasonably consistent with Eq. (1) by considering the effect of the fast electron in driving the upwards and downwards transitions between the energy levels of a simple harmonic oscillator. The fluctuation dissipation theorem and Eq. (1) are more general however and would apply to other excitations such as Johnson noise. Using photon-induced near-field electron microscopy (PINEM), Zewail et al. reported a much more extreme example of energy gains
and losses [9]. Here the laser excitation level was so great (corresponding to a very high temperature in Eq. (1)) that the multiple gains and losses observed were effectively equal.
2.2. Near-field theory for spatially resolved EELS In the classical picture for excitation in a dielectric medium, widely used in electron microscopy, it is assumed that the field from the passing electron causes a polarisation response in the medium generating an induced field which acts back on the fast electron to slow it down. More consistent with quantum electrodynamics however is the idea that it is only necessary to consider the electromagnetic field at the position of the fast electron no matter how it arises. Since energy-momentum conservation between photons and electrons is impossible with free photons, we have to deal with the evanescent or near-field electromagnetic waves arising near a dielectric or radiation source. Even when no other external stimulation is involved, these fields will still arise at characteristic frequencies corresponding to the zero-point motion of the excitations in the medium. Calculations of the near-field interaction near a dielectric cylinder [10] were able to simulate the multiple losses and gains observed [9] near a carbon nanotube exposed to high intensity laser pulses. Using the standard eikonal theory, it also proved possible [11] to compute the dependence of the loss and gain probabilities on the fast electron impact parameter. In these calculations the absolute strength of the evanescent field, connected of course to the laser pulse intensity and the dielectric properties of the carbon nanotube, had to remain as a fitting parameter. However when there is no thermal or other external stimulation and only the zero-point modes arise the strength of the near-fields can in many cases easily be calculated. Although we may expect that the energy loss probabilities which are then obtained (with no energy gains) are consistent with the results of the EELS theory used previously this has so far been checked in only the simple case of non-relativistic dipole excitation. The near-field excitation theory does give significantly different results from the classical theory when more structured fast electron illumination is used. In the case of aloof beam holography for instance, with the coordinate system shown in Fig. 2, the evanescent waves induced on the classical theory will all have the same symmetry as the illumination with respect to the y coordinate. Thus when the wave amplitudes on the two paths are equal the evanescent wave
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et al. [25], ascribed tail features on their zero loss peak (with FWHM¼ 120 meV) as due to phonons and showed that the changes in these features observed between different columns of a SrTiO3 crystal correlated with their atomic composition.
Evanescent x wave qy y ... 2D
3
3.2. Acoustic phonon excitations
z X0
Lens
f
Image
L Fig. 2. Showing small-angle scattering and fringe shifts in aloof beam inelastic holography.
for qz ¼ ω/v and a given qy component will have the form s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "
# h i h ω 2 ωzi þ q2y ϕðx; y; zÞ ¼ A exp 7 i cos qy y exp x v v
For momentum transfer though not of course for energy transfer in fast electron scattering, a phonon can be regarded as frozen. Much can therefore be learnt from the established theory of diffraction contrast imaging or structure imaging in a crystal subjected to a static atomic displacement function RðrÞ ¼ AðqÞexp ½iq:r: In the diffraction contrast theory, the generally used deformable ion scattering potential [26] then takes the form for small amplitude A, V ðrÞ-V ðr þ Rðr ÞÞ ¼ ∑ g V g exp ½ig ðr þ RðrÞÞ ¼ i∑g AðqÞ gV g exp ½iðg þ qÞ r
ð3Þ ð2Þ
In addition to changing the z component of the fast electron momentum (corresponding to the energy loss or gain process) there will be correlated lateral momentum transfers of 7ħqy occurring equally at the two paths y¼ 7D. As shown in Fig. 2, the symmetry of the fast electron wave will thus be preserved after the inelastic scattering though, because of the small angle of scattering in the y direction, there will be a shift of any fringes formed for example after passing through a lens. In the near-field theory the symmetrical cosine wave of Eq. (2) can be supplemented with equal probability by a similar wave with sin[qyy] dependence. This acts with opposite amplitude at the two paths y¼ 7D becoming the dominant factor when qyD4π/4. The symmetry of the fast electron wave is thus immediately reversed. In this respect the near-field theory provides a proper foundation for the rather ad-hoc assumption previously made [20] about the significance of the quantity qyD in explaining decoherence effects in aloof beam electron holography. Clearly the sin[qyD] term has no effect in the usual illumination situation where there is only a single electron path (D¼0). The near-field theory might also make a difference when two electrons (possibly-phase correlated) follow very closely on the same path. In the limit when they coincide, the classical theory gives a loss probability four times higher than for a single electron.
3. Phonons 3.1. Phonon effects in EM imaging In traditional unfiltered imaging, the absorption effect in CTEM and the high-angle annular dark field (HAADF) signal in the STEM indicate that phonon scattering can be highly localised. Following theoretical consideration of the phonon scattering potential, a similar conclusion has been reached by Rez [21]. Through a comparison of HAADF images taken at 300 K and 1100 K in a decagonal quasicrystal, an intriguing demonstration of the influence of phonon scattering at the atomic level was given by Abe et al. [22]. The scattering from a given atomic column depends not just on the atomic number of the atoms but also on their vibration amplitudes. Going beyond the simple absorption effect in CTEM, the flexural vibration amplitude as a function of position in a carbon nanotube was directly observed in a high resolution image [23]. The contribution of phonon scattering within the CTEM objective aperture is less clear but has been observed in some cases. In a pioneering electron holography experiment, Boothroyd and Dunin-Borkowski [24] found interesting differences between the centre band (normal CTEM image) and the side-band (phonon-free) image. Very recently Egoavil
In structure imaging, the rigid ion approximation is generally used V ðr Þ-∑n vðr n Rðr n Þ ¼ i∑ g AðqÞ ðq þ gÞV g exp½iðg þ qÞ r
ð4Þ
Here for simplicity we have considered an acoustic phonon in a monatomic crystal where the atomic potential v(r) has a Fourier transform related in the usual way to the Fourier component Vg of the crystal potential. In Eq. (3), the scattering is entirely linked to Bragg reflection and the small-angle scattering k-kþq does not occur. Such scattering does take place in the rigid ion model but is generally small in comparison with the Bragg scattering contributions and is certainly further reduced in the case of a metal due to electronic screening. Screening effects could then be included by dividing the g¼0 term in Eq. (3) by the Fermi Thomas dielectric 2 function εðqÞ ¼ 1 þ q0 =q [27]. The importance of the Bragg scattering contribution due to the movement of ions has also been exploited in surface EELS work with low energy electrons by Ibach [28] and is known as impact scattering. Of course for microscopy we are interested in phonons which might be localised at interfaces or other defects. The above analysis should however still apply for penetrating beam investigations either by making the component of q normal to the interface imaginary or perhaps better by making a space and time Fourier decomposition of the localised phonon strain field. The phonon amplitude coefficients should then be written as A(q,ω) and could be frequency-selected as appropriate either by spectroscopy (when it becomes adequate) or by excitation with a tuned driving force. In many cases where the wave properties are determined by the specimen boundary conditions with the complications noted in the introduction, standing wave combinations would be more appropriate than the travelling wave used in Eqs. (3) and (4). We may also note that the quasi-static phonon potential used here which neglects the radiation field and other electromagnetic refinements, is already in the near-field form required by the theory described in Section 2. For transmission microscopy work it may be preferable to resort to diffraction contrast methods, using an aperture to form an image from the scattering near a particular Bragg spot g. Provided strong many-beam dynamical situations can be avoided, the dependence of the scattering amplitude on the quantity g A(q,ω)Vg would then yield the mode polarisation. Although the maximum signal occurs at larger values of g, these should still be attainable if necessary in tilting experiments and the spectrometer required to select particular phonon frequencies would not need to use a large angular aperture. These advantages could be lost in the nowadays more fashionable structure imaging approach where many strong Bragg reflections are involved. The phonon scattering can then best be considered in terms of transitions between Bloch wave states in the dispersion surface picture of Fig. 3. Because the energy losses and gains are so
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A
be supplemented by a more delocalised interaction from the g ¼0 contribution where dielectric screening has to be considered. Here the established theory [27] uses a dielectric response function of the form ε1 ε0 ε1 2 ε1 þ2 εðωÞ ¼ ε1 þ ; ω2T ¼ ð6Þ ωL ¼ ω2r 2 ε0 þ 2 ε0 ðω=ωT Þ 1
C
B
D
z Fig. 3. Dispersion surface representation of intraband (broken arrows) g ¼0 transitions and intraband (solid arrows) g a 0 transitions. For the very small energy changes in phonon scattering, the elastic and inelastic surfaces virtually superimpose. Transitions from wave points A and B near the axis would be relevant for structure imaging. Diffraction contrast imaging could use points near the Brillouin zone boundary such as C and weak beam transitions C-D.
small, the energy surfaces are all effectively indistinguishable on the scale of the diagram from the zero loss case. The g¼0 term in Eq. (3) can drive only intraband transitions corresponding to rather small momentum changes qz in the z direction. As already argued however these transitions may be too weak to detect. The other terms for ga0 all cause interband transitions with changes of Bloch wave symmetry and much larger momentum changes in the z direction. These transitions could complicate the interpretation of structure images for which any atomic displacements ideally do not vary with distance along the beam and preserve the 2D projection. Such complications are familiar in diffraction contrast imaging, including weak beam imaging which exploits still higher momentum transfers (Fig. 3) to improve the spatial resolution both laterally and in depth but does not reach the atomic level. Effects of phonons in atomic resolution images may indeed be apparent [22,25] but for selection of specific frequencies a much greater spectrometer aperture would be necessary unless a suitable tuneable excitation source can be found. For an aloof beam travelling at velocity v in the z direction and at distance x outside a free surface, the potential associated with Eq. (4) would take the (non-relativistic) form V ðρ; xÞ ¼ i∑0g AðqÞ ðq0 þ gÞV g exp ½iðg þ q0 Þ ρexp ½ x√ ðg þ q0 Þ 2
ð5Þ
∑0g now
Here the summation includes only those g vectors with no x component normal to the surface and q0 is the projection of q in the y–z plane. The very weak g¼ 0 term extending far from the surface is poorly localised. Coupling to the ga 0 impact scattering effect would then need very small atomic scale values of x that would probably be more attainable in glancing angle (REM) mode than in the usual parallel aloof beam mode.
Here ε0 and ε1 are respectively the measured dielectric constants at zero frequency and at a frequency high compared to optical phonon frequencies but below the frequencies of electronic excitations; ωr is the resonance frequency (k/Mr)1/2 where k is the bond stretching constant and Mr the reduced mass; ωT and ωL are the frequencies of the transverse and longitudinal optical phonons. All of this applies in the optical interaction region at small values of q and could certainly be relevant for aloof beam electron spectroscopy since the usual condition for surface modes ε(ω)¼ 1 can be satisfied for a frequency ω lying in the range ωT oωoωL provided the damping is small. The long-wave optical phonon modes at the two surfaces of a thin slab can be strongly coupled and this was indeed observed in the early work with electron spectroscopy [18]. At very small aloof beam impact parameters or with penetrating beams it could once again be possible to achieve much higher spatial resolution from the scattering in the vicinity of Bragg reflections. The effect of screening in this less well explored territory is not so clear but failures of the rigid ion potential detected in optical phonon spectroscopy with neutrons have been successfully described using shell models [19]. 3.4. Individual atomic recoil effects The phenomenon of large atomic displacement and knock-on damage is well known for high-angle (4π/2) collisions of high energy electrons. However the transition from phonon excitation to single atom recoil which must take place for medium momentum transfers does not seem to have been investigated. Knock-on would seem to require firstly that all the momentum delivered by the fast electron goes to just one atom and secondly that this is not immediately dissipated in phonon excitation. The maximum energy of an acoustic phonon is about ħg1vs/2 where vs is the velocity of sound and g1 is a first order reciprocal lattice vector. On scattering through a large angle θ, a highly focused STEM probe will deliver its momentum change p¼ 2ħk sin(θ/2)ENħg1 at least to a single atomic column and perhaps even to just one or two atoms in the column. A single atom of mass M may then acquire an energy p2/2 M¼pvatom/2 exceeding the maximum phonon energy when vatom 4vs/N. This condition may herald a breakdown of the single phonon excitation model. A complete breakdown would be signalled by the more extreme situation vatom 4vs when the struck atom would crash into its neighbours before they receive any intimation that it is on the move. Detailed study of the transition from phonon excitation to single atom recoil would seem to be an attractive new topic for electron microscopy. For the excitation of molecular bond vibrations a similar transition will occur leading to dissociation when the energy transfer exceeds the bond energy but here dissociation can also arise from electronic excitation processes.
3.3. Optical phonons An analysis similar to that described in Eqs. (3) and (4) can be made for a crystal containing more than one atom or ion per unit cell and thus supporting both acoustic and optic phonon modes. For mapping of the excitation of specific optical phonon modes similar conclusions about the relevance of diffraction contrast experience would apply. Now however it would be best to select scattering in the vicinity of Bragg reflections where the different atoms in the unit cell contribute with different phases to the structure factor. Once again these well-localised interactions will
4. Tuned laser boosting of spatial and energy resolution in EELS Although there are obvious attractive possibilities in the whole of the optical region for combining the spatial resolution of EELS with the spectral resolution of photons, progress has so far been limited to some cathodoluminescence studies [29,30]. As already noted, much more striking boosting of the energy loss and gain probabilities at ħω¼ 2.3 eV were obtained for an aloof beam
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E x
z
Fig. 4. Schematic diagram showing the excitation of an x-dipole mode on a nanosphere. The associated near-field wave can interact with a passing electron giving energy losses and gains.
y
E
z
e Fig. 5. Laser excitation of the antisymmetrical “hot spot” plasmon between two spheres.
passing near a carbon nanotube under pulsed laser illumination in a pump–probe timing experiment [9]. With sacrifice of the time resolution of pulsed operation, more systematic exploration of the dielectric resonances of nanostructures could be made by combining EELS with continuous tuned laser irradiation. The situation for laser excitation of the x-dipole normal to the e-beam direction on a nano-sphere is depicted in Fig. 4 but z-dipole excitation parallel to the e-beam could also be used. Significant boosting of loss probabilities could obviously be very useful for weak losses as well as in probing situations such as Raman or catalytic hot spots in configurations inaccessible to the usual EELs geometry. As shown in Fig. 5, a suitably polarised and directed laser beam could be used to stimulate the asymmetric coupled mode between two spheres which interacts with the e-beam only when this is displaced from the mid-point between the spheres. Laser stimulated e-beam excitation processes could present a fascinating test bed for the near-field theory of EELS outlined in Section 2. A preliminary theory has already been presented [31] but treats both the laser illumination and the e-beam interaction as perturbations. For the nano-sphere of Fig. 4, it would be more relevant to have a theory describing the steady state equilibrium between the dipole near-field excitation taking full account of its coupling to the surface plasmon on the sphere as well as to the external laser beam. In the case of a small sphere of radius r ¼a, the coupling of the lQ1 dipole mode to an incident optical plane wave will be significantly reduced by the usual relation exp ikr cos θ ¼ Σ l jl ðkrÞP l ð cos θÞ
ð7Þ
5
Even so the laser may be able to drive the dipole mode and the surface plasmons coupled to it to quite high excitation levels n limited eventually either by plasmon decay processes (including radiation into other outgoing directions in the far-field) or by plasmon frequency shifts due to anharmonic effects. It seems more reasonable then to treat the fast electron interaction with the dipole near-field as a perturbation causing energy loss transitions n-n þ1 and energy gain transitions n-n-1 with relative probabilities of nþ 1 and n respectively. It would be an attractive challenge to measure these relative loss and gain probabilities as a function of laser intensity, laser detuning and electron impact parameter near an isolated nano-sphere. According to the nearfield picture of the process, this should be the whole story for the interaction of the fast electron. To add to the near-field interaction a further interaction with the plasmon modes in the manner of the conventional energy loss theory as has recently been suggested [32] would seem to risk double counting.
5. Conclusions Clearly the main contribution of electron microscopy to the study of phonons and other low frequency excitations must lie in achieving imaging or spatially resolved spectroscopy and these are topics that have not been significantly explored in the vast volume of earlier works using broad beam spectroscopy. The scattering transitions in the phonon potential discussed here involve both normal (small q) and umklapp (q þg) momentum transfers with the latter being probably more intense and providing much higher spatial resolution. Diffraction contrast imaging with use of specific Bragg reflections may be very relevant and more informative than structure imaging in studying these effects. The transition from phonon excitation to single atom recoil which takes place with increasing momentum transfer from the fast electron would also seem to be a promising subject for study. Although the inelastic scattering from molecular bond vibrations has not been specifically considered here, they probably constitute the best known case of localised phonons. In many cases they have energies ħω between 50 meV and 500 meV with scattering cross sections in the range 10 5–10 4 Å2 and also show both low and high momentum transfer characteristics [8]. More definite conclusions about the degree of image localisation achievable also emerge from analysis of the scattering as a function of impact parameter [7]. Atomic resolution of bond vibrations in individual surface molecules has already been observed by inelastic tunnelling spectroscopy in STM [33]. The possibility of using tuned laser illumination of nanostructures to boost the electron energy loss and gain probabilities in the infrared and optical regions is attractive, offering opportunities for useful applications as well as for systematic testing of the nearfield excitation theory.
Acknowledgements I am grateful to Javier Garcia de Abajo, Ray Egerton, Ondrej Krivanek and Peter Rez for several useful discussions and for showing me some of their work in advance of publication. References [1] H. Rose, Outline of a spherically corrected semiaplanatic medium-voltage transmission electron microscope, Optik 85 (1990) 19–24. [2] M. Haider, S. Uhlemann, E. Schwan, H. Rose, B. Kabius, K. Urban, Electron microscopy image enhanced, Nature 392 (1998) 768–769. [3] C.L. Jia, M. Lentzen, K. Urban, Atomic-resolution of oxygen in perovskite ceramics, Science 299 (2003) 870–873.
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Stimulated excitation electron microscopy and spectroscopy A. Howie Department of Physics, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, UK
art ic l e i nf o
a b s t r a c t
Article history: Received 27 July 2014 Received in revised form 15 September 2014 Accepted 18 September 2014
Recent advances in instrumentation for electron optics and spectroscopy have prompted exploration of ultra-low excitations such as phonons, bond vibrations and Johnson noise. These can be excited not just with fast electrons but also thermally or by other external sources of radiation. The near-field theory of electron energy loss and gain provides a convenient platform for analysing these processes. Possibilities for selected phonon mapping and imaging are discussed. Effects should certainly be observable in atomic resolution structure imaging but diffraction contrast imaging could perhaps be more informative. Additional exciting prospects to be explored include the transition from phonon excitation to single atom recoil and the boosting of energy loss and gain signals with tuned laser illumination. & 2014 Elsevier B.V. All rights reserved.
Keywords: Near-field excitation Phonon imaging Atomic displacement Laser-boosted interactions
1. Introduction Over the past decade, Harald Rose's celebrated design for a hexapole aberration corrector [1] has not only fathered outstanding achievements in high resolution structure imaging with the conventional transmission electron microscope (CTEM) [2,3] but has also been accorded tremendous commercial success as a key component in world-wide sales of both the CTEM and the scanning transmission electron microscope (STEM). In the dedicated STEM, great technical success has also been enjoyed by Ondrej Krivanek's independently developed quadrupole–octopole system [4]. Encouraged by these advances in electron optics and other developments with monochromators and spectrometers, electron microscopists are keenly looking for new fields to explore. Very recently the importance of excitations with very low energy ħωokT has emerged as a fresh challenge, firstly through the identification of Johnson noise as a significant cause of e-beam decoherence [5] and secondly with the prospect of spatially localised spectroscopy of phonons and atomic bond vibrations [6–8]. Despite their different microscopic natures, these two phenomena share the common feature that the interaction with the fast electron takes place with pre-existing thermal excitation. Still more dramatic examples of the same effect arise when the excitations concerned are driven to high levels by an external periodic field [9]. As described in Section 2, the recently introduced near-field interaction approach [10–12] to energy loss and gain processes appears to provide a better basis for handling these problems than the usual classical theory of electron energy loss spectroscopy (EELS). How best to combine useful imaging with the improved spectroscopic capability now becoming available is an interesting
issue raised by these developments. For electrons interacting with phonons in the sample, we can with confidence dismiss concern that, the spatial resolution attainable might be limited by the characteristic long range Coulomb impact parameter v/ω that applies to purely electronic excitations of angular frequency ω by a fast electron of velocity v. Anomalous absorption theory in transmission imaging as well as the experimentally observed role of phonon scattering in STEM HAADF imaging provide ample evidence, even in ionic crystals, that phonon scattering operates with atomic scale impact parameters. Any residual long range component may often be negligible since v/ω, measuring the optimum interaction wavelength along the beam as well as the characteristic impact parameter for a free electron, can exceed the typical sample thickness. With the near-field theory described in Section 2, the energy loss and gain processes involved in phonon scattering can be handled more naturally using the familiar phonon potential allowing various options for mapping or imaging to be analysed in Section 3. Tuned laser excitation offers further interesting prospects for boosting the strength of electron energy losses and gains in the infrared and optical regions which are briefly explored in Section 4. Before embarking on these topics however it is important to recall that that the ability to drive selected extremely low frequency modes to high excitation levels by externally applied periodic fields has already been exploited for low resolution imaging. Fig. 1 provides an update to a recent summary [13] of this work and perhaps shows that there may be other ways besides improved electron spectroscopy to enter this new territory including scanning electron acoustic microscopy (SEAM), time resolved scanning optical microscopy (TRSOM) and radio frequency TEM in addition to tip-enhanced Raman spectroscopy (TER), cathodolumenescence
http://dx.doi.org/10.1016/j.ultramic.2014.09.006 0304-3991/& 2014 Elsevier B.V. All rights reserved.
Please cite this article as: A. Howie, Stimulated excitation electron microscopy and spectroscopy, Ultramicroscopy (2014), http://dx.doi. org/10.1016/j.ultramic.2014.09.006i
A. Howie / Ultramicroscopy ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Fig. 1. Time/frequency range and spatial resolution of various microscopy techniques including those (hatched) using low frequency driving fields and identified in the text.
(CDL), photoemission electron microscopy (PEEM) and scanning tunnelling microscopy (STM) techniques. Some of the complexities of vibration modes in thin slabs and cantilevers (involving Lamb waves, flexural modes and Rayleigh modes) which the EM community may soon need to address, have already been explored in pulsed e-beam experiments, where the different dispersion relations can be directly measured [14]. The stroboscopic method using sinusoidal driving fields and phase sensitive detection would probably provide the most powerful entry to this exciting new field if it could be massively extended to phonon frequencies and atomic resolution imaging. In the alternative pump–probe technique using periodic MHz pulses from a laser, many phonon modes are simultaneously excited but in favourable cases the different frequencies can still be resolved by following the time evolution of the response [15].
2. EELS theory in the presence of additional external stimulation 2.1. Fluctuation dissipation theorem – energy gains and losses For very low excitation energies where thermal excitation must be considered, the fluctuation dissipation theorem [16,17] shows that the familiar dielectric excitation expressions for penetrating beams or for aloof beams should be multiplied by a thermal correction factor F(T). Since we can separately observe energy losses as well as energy gains of the fast electron (corresponding respectively to the creation and annihilation of an excitation in the object), it is useful to express F(T) as the sum of these two contributions. F ðT Þ ¼ F loss ðT Þþ F gain ðT Þ ¼
exp ðħω=kTÞ 1 ħω þ ¼ coth 2kT 1 exp ðħω=kTÞ 1 exp ðħω=kTÞ
ð1Þ Longstanding experimental data [18] for energy gains and losses for fast electron interaction with optical phonons have been shown [19] to be reasonably consistent with Eq. (1) by considering the effect of the fast electron in driving the upwards and downwards transitions between the energy levels of a simple harmonic oscillator. The fluctuation dissipation theorem and Eq. (1) are more general however and would apply to other excitations such as Johnson noise. Using photon-induced near-field electron microscopy (PINEM), Zewail et al. reported a much more extreme example of energy gains
and losses [9]. Here the laser excitation level was so great (corresponding to a very high temperature in Eq. (1)) that the multiple gains and losses observed were effectively equal.
2.2. Near-field theory for spatially resolved EELS In the classical picture for excitation in a dielectric medium, widely used in electron microscopy, it is assumed that the field from the passing electron causes a polarisation response in the medium generating an induced field which acts back on the fast electron to slow it down. More consistent with quantum electrodynamics however is the idea that it is only necessary to consider the electromagnetic field at the position of the fast electron no matter how it arises. Since energy-momentum conservation between photons and electrons is impossible with free photons, we have to deal with the evanescent or near-field electromagnetic waves arising near a dielectric or radiation source. Even when no other external stimulation is involved, these fields will still arise at characteristic frequencies corresponding to the zero-point motion of the excitations in the medium. Calculations of the near-field interaction near a dielectric cylinder [10] were able to simulate the multiple losses and gains observed [9] near a carbon nanotube exposed to high intensity laser pulses. Using the standard eikonal theory, it also proved possible [11] to compute the dependence of the loss and gain probabilities on the fast electron impact parameter. In these calculations the absolute strength of the evanescent field, connected of course to the laser pulse intensity and the dielectric properties of the carbon nanotube, had to remain as a fitting parameter. However when there is no thermal or other external stimulation and only the zero-point modes arise the strength of the near-fields can in many cases easily be calculated. Although we may expect that the energy loss probabilities which are then obtained (with no energy gains) are consistent with the results of the EELS theory used previously this has so far been checked in only the simple case of non-relativistic dipole excitation. The near-field excitation theory does give significantly different results from the classical theory when more structured fast electron illumination is used. In the case of aloof beam holography for instance, with the coordinate system shown in Fig. 2, the evanescent waves induced on the classical theory will all have the same symmetry as the illumination with respect to the y coordinate. Thus when the wave amplitudes on the two paths are equal the evanescent wave
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et al. [25], ascribed tail features on their zero loss peak (with FWHM¼ 120 meV) as due to phonons and showed that the changes in these features observed between different columns of a SrTiO3 crystal correlated with their atomic composition.
Evanescent x wave qy y ... 2D
3
3.2. Acoustic phonon excitations
z X0
Lens
f
Image
L Fig. 2. Showing small-angle scattering and fringe shifts in aloof beam inelastic holography.
for qz ¼ ω/v and a given qy component will have the form s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "
# h i h ω 2 ωzi þ q2y ϕðx; y; zÞ ¼ A exp 7 i cos qy y exp x v v
For momentum transfer though not of course for energy transfer in fast electron scattering, a phonon can be regarded as frozen. Much can therefore be learnt from the established theory of diffraction contrast imaging or structure imaging in a crystal subjected to a static atomic displacement function RðrÞ ¼ AðqÞexp ½iq:r: In the diffraction contrast theory, the generally used deformable ion scattering potential [26] then takes the form for small amplitude A, V ðrÞ-V ðr þ Rðr ÞÞ ¼ ∑ g V g exp ½ig ðr þ RðrÞÞ ¼ i∑g AðqÞ gV g exp ½iðg þ qÞ r
ð3Þ ð2Þ
In addition to changing the z component of the fast electron momentum (corresponding to the energy loss or gain process) there will be correlated lateral momentum transfers of 7ħqy occurring equally at the two paths y¼ 7D. As shown in Fig. 2, the symmetry of the fast electron wave will thus be preserved after the inelastic scattering though, because of the small angle of scattering in the y direction, there will be a shift of any fringes formed for example after passing through a lens. In the near-field theory the symmetrical cosine wave of Eq. (2) can be supplemented with equal probability by a similar wave with sin[qyy] dependence. This acts with opposite amplitude at the two paths y¼ 7D becoming the dominant factor when qyD4π/4. The symmetry of the fast electron wave is thus immediately reversed. In this respect the near-field theory provides a proper foundation for the rather ad-hoc assumption previously made [20] about the significance of the quantity qyD in explaining decoherence effects in aloof beam electron holography. Clearly the sin[qyD] term has no effect in the usual illumination situation where there is only a single electron path (D¼0). The near-field theory might also make a difference when two electrons (possibly-phase correlated) follow very closely on the same path. In the limit when they coincide, the classical theory gives a loss probability four times higher than for a single electron.
3. Phonons 3.1. Phonon effects in EM imaging In traditional unfiltered imaging, the absorption effect in CTEM and the high-angle annular dark field (HAADF) signal in the STEM indicate that phonon scattering can be highly localised. Following theoretical consideration of the phonon scattering potential, a similar conclusion has been reached by Rez [21]. Through a comparison of HAADF images taken at 300 K and 1100 K in a decagonal quasicrystal, an intriguing demonstration of the influence of phonon scattering at the atomic level was given by Abe et al. [22]. The scattering from a given atomic column depends not just on the atomic number of the atoms but also on their vibration amplitudes. Going beyond the simple absorption effect in CTEM, the flexural vibration amplitude as a function of position in a carbon nanotube was directly observed in a high resolution image [23]. The contribution of phonon scattering within the CTEM objective aperture is less clear but has been observed in some cases. In a pioneering electron holography experiment, Boothroyd and Dunin-Borkowski [24] found interesting differences between the centre band (normal CTEM image) and the side-band (phonon-free) image. Very recently Egoavil
In structure imaging, the rigid ion approximation is generally used V ðr Þ-∑n vðr n Rðr n Þ ¼ i∑ g AðqÞ ðq þ gÞV g exp½iðg þ qÞ r
ð4Þ
Here for simplicity we have considered an acoustic phonon in a monatomic crystal where the atomic potential v(r) has a Fourier transform related in the usual way to the Fourier component Vg of the crystal potential. In Eq. (3), the scattering is entirely linked to Bragg reflection and the small-angle scattering k-kþq does not occur. Such scattering does take place in the rigid ion model but is generally small in comparison with the Bragg scattering contributions and is certainly further reduced in the case of a metal due to electronic screening. Screening effects could then be included by dividing the g¼0 term in Eq. (3) by the Fermi Thomas dielectric 2 function εðqÞ ¼ 1 þ q0 =q [27]. The importance of the Bragg scattering contribution due to the movement of ions has also been exploited in surface EELS work with low energy electrons by Ibach [28] and is known as impact scattering. Of course for microscopy we are interested in phonons which might be localised at interfaces or other defects. The above analysis should however still apply for penetrating beam investigations either by making the component of q normal to the interface imaginary or perhaps better by making a space and time Fourier decomposition of the localised phonon strain field. The phonon amplitude coefficients should then be written as A(q,ω) and could be frequency-selected as appropriate either by spectroscopy (when it becomes adequate) or by excitation with a tuned driving force. In many cases where the wave properties are determined by the specimen boundary conditions with the complications noted in the introduction, standing wave combinations would be more appropriate than the travelling wave used in Eqs. (3) and (4). We may also note that the quasi-static phonon potential used here which neglects the radiation field and other electromagnetic refinements, is already in the near-field form required by the theory described in Section 2. For transmission microscopy work it may be preferable to resort to diffraction contrast methods, using an aperture to form an image from the scattering near a particular Bragg spot g. Provided strong many-beam dynamical situations can be avoided, the dependence of the scattering amplitude on the quantity g A(q,ω)Vg would then yield the mode polarisation. Although the maximum signal occurs at larger values of g, these should still be attainable if necessary in tilting experiments and the spectrometer required to select particular phonon frequencies would not need to use a large angular aperture. These advantages could be lost in the nowadays more fashionable structure imaging approach where many strong Bragg reflections are involved. The phonon scattering can then best be considered in terms of transitions between Bloch wave states in the dispersion surface picture of Fig. 3. Because the energy losses and gains are so
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A
be supplemented by a more delocalised interaction from the g ¼0 contribution where dielectric screening has to be considered. Here the established theory [27] uses a dielectric response function of the form ε1 ε0 ε1 2 ε1 þ2 εðωÞ ¼ ε1 þ ; ω2T ¼ ð6Þ ωL ¼ ω2r 2 ε0 þ 2 ε0 ðω=ωT Þ 1
C
B
D
z Fig. 3. Dispersion surface representation of intraband (broken arrows) g ¼0 transitions and intraband (solid arrows) g a 0 transitions. For the very small energy changes in phonon scattering, the elastic and inelastic surfaces virtually superimpose. Transitions from wave points A and B near the axis would be relevant for structure imaging. Diffraction contrast imaging could use points near the Brillouin zone boundary such as C and weak beam transitions C-D.
small, the energy surfaces are all effectively indistinguishable on the scale of the diagram from the zero loss case. The g¼0 term in Eq. (3) can drive only intraband transitions corresponding to rather small momentum changes qz in the z direction. As already argued however these transitions may be too weak to detect. The other terms for ga0 all cause interband transitions with changes of Bloch wave symmetry and much larger momentum changes in the z direction. These transitions could complicate the interpretation of structure images for which any atomic displacements ideally do not vary with distance along the beam and preserve the 2D projection. Such complications are familiar in diffraction contrast imaging, including weak beam imaging which exploits still higher momentum transfers (Fig. 3) to improve the spatial resolution both laterally and in depth but does not reach the atomic level. Effects of phonons in atomic resolution images may indeed be apparent [22,25] but for selection of specific frequencies a much greater spectrometer aperture would be necessary unless a suitable tuneable excitation source can be found. For an aloof beam travelling at velocity v in the z direction and at distance x outside a free surface, the potential associated with Eq. (4) would take the (non-relativistic) form V ðρ; xÞ ¼ i∑0g AðqÞ ðq0 þ gÞV g exp ½iðg þ q0 Þ ρexp ½ x√ ðg þ q0 Þ 2
ð5Þ
∑0g now
Here the summation includes only those g vectors with no x component normal to the surface and q0 is the projection of q in the y–z plane. The very weak g¼ 0 term extending far from the surface is poorly localised. Coupling to the ga 0 impact scattering effect would then need very small atomic scale values of x that would probably be more attainable in glancing angle (REM) mode than in the usual parallel aloof beam mode.
Here ε0 and ε1 are respectively the measured dielectric constants at zero frequency and at a frequency high compared to optical phonon frequencies but below the frequencies of electronic excitations; ωr is the resonance frequency (k/Mr)1/2 where k is the bond stretching constant and Mr the reduced mass; ωT and ωL are the frequencies of the transverse and longitudinal optical phonons. All of this applies in the optical interaction region at small values of q and could certainly be relevant for aloof beam electron spectroscopy since the usual condition for surface modes ε(ω)¼ 1 can be satisfied for a frequency ω lying in the range ωT oωoωL provided the damping is small. The long-wave optical phonon modes at the two surfaces of a thin slab can be strongly coupled and this was indeed observed in the early work with electron spectroscopy [18]. At very small aloof beam impact parameters or with penetrating beams it could once again be possible to achieve much higher spatial resolution from the scattering in the vicinity of Bragg reflections. The effect of screening in this less well explored territory is not so clear but failures of the rigid ion potential detected in optical phonon spectroscopy with neutrons have been successfully described using shell models [19]. 3.4. Individual atomic recoil effects The phenomenon of large atomic displacement and knock-on damage is well known for high-angle (4π/2) collisions of high energy electrons. However the transition from phonon excitation to single atom recoil which must take place for medium momentum transfers does not seem to have been investigated. Knock-on would seem to require firstly that all the momentum delivered by the fast electron goes to just one atom and secondly that this is not immediately dissipated in phonon excitation. The maximum energy of an acoustic phonon is about ħg1vs/2 where vs is the velocity of sound and g1 is a first order reciprocal lattice vector. On scattering through a large angle θ, a highly focused STEM probe will deliver its momentum change p¼ 2ħk sin(θ/2)ENħg1 at least to a single atomic column and perhaps even to just one or two atoms in the column. A single atom of mass M may then acquire an energy p2/2 M¼pvatom/2 exceeding the maximum phonon energy when vatom 4vs/N. This condition may herald a breakdown of the single phonon excitation model. A complete breakdown would be signalled by the more extreme situation vatom 4vs when the struck atom would crash into its neighbours before they receive any intimation that it is on the move. Detailed study of the transition from phonon excitation to single atom recoil would seem to be an attractive new topic for electron microscopy. For the excitation of molecular bond vibrations a similar transition will occur leading to dissociation when the energy transfer exceeds the bond energy but here dissociation can also arise from electronic excitation processes.
3.3. Optical phonons An analysis similar to that described in Eqs. (3) and (4) can be made for a crystal containing more than one atom or ion per unit cell and thus supporting both acoustic and optic phonon modes. For mapping of the excitation of specific optical phonon modes similar conclusions about the relevance of diffraction contrast experience would apply. Now however it would be best to select scattering in the vicinity of Bragg reflections where the different atoms in the unit cell contribute with different phases to the structure factor. Once again these well-localised interactions will
4. Tuned laser boosting of spatial and energy resolution in EELS Although there are obvious attractive possibilities in the whole of the optical region for combining the spatial resolution of EELS with the spectral resolution of photons, progress has so far been limited to some cathodoluminescence studies [29,30]. As already noted, much more striking boosting of the energy loss and gain probabilities at ħω¼ 2.3 eV were obtained for an aloof beam
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E x
z
Fig. 4. Schematic diagram showing the excitation of an x-dipole mode on a nanosphere. The associated near-field wave can interact with a passing electron giving energy losses and gains.
y
E
z
e Fig. 5. Laser excitation of the antisymmetrical “hot spot” plasmon between two spheres.
passing near a carbon nanotube under pulsed laser illumination in a pump–probe timing experiment [9]. With sacrifice of the time resolution of pulsed operation, more systematic exploration of the dielectric resonances of nanostructures could be made by combining EELS with continuous tuned laser irradiation. The situation for laser excitation of the x-dipole normal to the e-beam direction on a nano-sphere is depicted in Fig. 4 but z-dipole excitation parallel to the e-beam could also be used. Significant boosting of loss probabilities could obviously be very useful for weak losses as well as in probing situations such as Raman or catalytic hot spots in configurations inaccessible to the usual EELs geometry. As shown in Fig. 5, a suitably polarised and directed laser beam could be used to stimulate the asymmetric coupled mode between two spheres which interacts with the e-beam only when this is displaced from the mid-point between the spheres. Laser stimulated e-beam excitation processes could present a fascinating test bed for the near-field theory of EELS outlined in Section 2. A preliminary theory has already been presented [31] but treats both the laser illumination and the e-beam interaction as perturbations. For the nano-sphere of Fig. 4, it would be more relevant to have a theory describing the steady state equilibrium between the dipole near-field excitation taking full account of its coupling to the surface plasmon on the sphere as well as to the external laser beam. In the case of a small sphere of radius r ¼a, the coupling of the lQ1 dipole mode to an incident optical plane wave will be significantly reduced by the usual relation exp ikr cos θ ¼ Σ l jl ðkrÞP l ð cos θÞ
ð7Þ
5
Even so the laser may be able to drive the dipole mode and the surface plasmons coupled to it to quite high excitation levels n limited eventually either by plasmon decay processes (including radiation into other outgoing directions in the far-field) or by plasmon frequency shifts due to anharmonic effects. It seems more reasonable then to treat the fast electron interaction with the dipole near-field as a perturbation causing energy loss transitions n-n þ1 and energy gain transitions n-n-1 with relative probabilities of nþ 1 and n respectively. It would be an attractive challenge to measure these relative loss and gain probabilities as a function of laser intensity, laser detuning and electron impact parameter near an isolated nano-sphere. According to the nearfield picture of the process, this should be the whole story for the interaction of the fast electron. To add to the near-field interaction a further interaction with the plasmon modes in the manner of the conventional energy loss theory as has recently been suggested [32] would seem to risk double counting.
5. Conclusions Clearly the main contribution of electron microscopy to the study of phonons and other low frequency excitations must lie in achieving imaging or spatially resolved spectroscopy and these are topics that have not been significantly explored in the vast volume of earlier works using broad beam spectroscopy. The scattering transitions in the phonon potential discussed here involve both normal (small q) and umklapp (q þg) momentum transfers with the latter being probably more intense and providing much higher spatial resolution. Diffraction contrast imaging with use of specific Bragg reflections may be very relevant and more informative than structure imaging in studying these effects. The transition from phonon excitation to single atom recoil which takes place with increasing momentum transfer from the fast electron would also seem to be a promising subject for study. Although the inelastic scattering from molecular bond vibrations has not been specifically considered here, they probably constitute the best known case of localised phonons. In many cases they have energies ħω between 50 meV and 500 meV with scattering cross sections in the range 10 5–10 4 Å2 and also show both low and high momentum transfer characteristics [8]. More definite conclusions about the degree of image localisation achievable also emerge from analysis of the scattering as a function of impact parameter [7]. Atomic resolution of bond vibrations in individual surface molecules has already been observed by inelastic tunnelling spectroscopy in STM [33]. The possibility of using tuned laser illumination of nanostructures to boost the electron energy loss and gain probabilities in the infrared and optical regions is attractive, offering opportunities for useful applications as well as for systematic testing of the nearfield excitation theory.
Acknowledgements I am grateful to Javier Garcia de Abajo, Ray Egerton, Ondrej Krivanek and Peter Rez for several useful discussions and for showing me some of their work in advance of publication. References [1] H. Rose, Outline of a spherically corrected semiaplanatic medium-voltage transmission electron microscope, Optik 85 (1990) 19–24. [2] M. Haider, S. Uhlemann, E. Schwan, H. Rose, B. Kabius, K. Urban, Electron microscopy image enhanced, Nature 392 (1998) 768–769. [3] C.L. Jia, M. Lentzen, K. Urban, Atomic-resolution of oxygen in perovskite ceramics, Science 299 (2003) 870–873.
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