Using Cerenkov radiation for measuring the refractive index in thick samples by interferometric cathodoluminescence
Michael Stöger-Pollach, Stefan Löffler, Niklas Maurer, Kristyna Bukvisova
UUsing ˇC erenkov radiation for measuring therefractive index in thick samples byinterferometric cathodoluminescence Michael St¨oger-Pollach a , b 1 Stefan L¨offler a , b Niklas Maurer b Krist´yna Bukviˇsov´a c a University Service Center for Transmission Electron Microscopy (USTEM),Technische Universit¨at Wien, Wiedner Hauptstraße 8-10, 1040 Wien, Austria b Institute of Solid State Physics, Technische Universit¨at Wien, WiednerHauptstraße 8-10, 1040 Wien, Austria c Central European Institute of Technology (CEITEC), Brno University ofTechnology, Purky ˇ n ova 123, Brno 612 00, Czech Republic Abstract
Cathodoluminescence (CL) has evolved into a standard analytical technique in(scanning) transmission electron microscopy. CL utilizes light excited due to theinteractions between the electron-beam and the sample. In the present study wefocus on ˇCerenkov radiation. We make use of the fact that the electron transpar-ent specimen acts as a Fabry-P´erot interferometer for coherently emitted radiation.From the wavelength dependent interference pattern of thickness dependent mea-surements we calculate the refractive index of the studied material. We describethe limits of this approach and compare it with the determination of the refractiveindex by using valence electron energy loss spectrometry (VEELS).
Key words: ˇCerenkov radiation, Cathodoluminescence, VEELS
In recent years cathodoluminescence (CL) in a (scanning) transmission elec-tron microscope (S/TEM) has attracted more and more interest, because itsenergy resolution is independent from the electron source. The only parame-ter influencing the energy resolution is given by the analyzing grating. Thus, corresponding author: e-mail: [email protected], Tel.: +43 (0)1 5880145204, Fax: +43 (0)1 58801 9 45204 Preprint submitted to Elsevier Preprint January 14, 2020 a r X i v : . [ phy s i c s . a pp - ph ] J a n nergy resolutions of µ eV are routinely available. Although such high energyresolution is attractive, CL suffers from some drawbacks. These are the limitedrange of observable energies – which are usually in the range from infra-red(IR) to soft ultra-violet (UV), which is app. 1 - 4 eV (corresponding to wave-lengths in the range of app. 1200 - 300 nm) – and Fabry-P´erot interferencein thin slab-like specimens [1,2,3]. Additionally, the small observed energytransfers are strongly delocalized due to the long-range action of the Coulombforce. Thus, the spatial resolution is also limited. When observing incoherentCL, which is the light being emitted after an electron-hole pair recombina-tion, diffusion of electrons and holes contributes further to a decreased spatialresolution. For many applications, a spatial resolution in the range of a fewnanometers is sufficient, as can be found in mineralogy (see for example: [4])and plasmonics (see for example: [5,6,7,8]).In the present work, we focus on coherent emission of light inside the S/TEM.This means that the electron beam is directly responsible for the creation ofphotons and no detour via the process of electron excitation and de-excitationis required [9]. For this purpose we investigate the light emission of MgO belowthe band gap energy. Due to the fact that the refractive index n for visiblelight is between 1.81 and 1.86 [10], we make use of the ˇCerenkov effect tocoherently create photons inside the sample. 200 keV electrons having a speedof 0 . · c – with c being the vacuum speed of light – are fast enough tofulfill the conditions for ˇCerenkov light emission [11], hence we can use thiseffect as light source. The light is created inside the sample and is partiallyreflected on the lower and upper sample surface. That way, the sample itselfacts as a Fabry-P´erot interferometer.On the other hand, if valence electron energy loss spectrometry (VEELS) shallbe utilized for the determination of the refractive index [12,13], the excitationof ˇCerenkov photons is undesired. Any excitation process causes an energyloss and therefore the emission of ˇCerenkov photons causes a ˇCerenkov loss inthe VEELS spectrum. Consequently, for VEELS experiments beam energiesof ≤
80 keV have to be employed in order to be far below the ˇCerenkov limit[14] of MgO and in order to prevent the VEELS spectrum from being altereddue to ˇCerenkov losses. ˇCerenkov photons have to be emitted [11], as soon as the electron traversesthe specimen with a velocity v e faster than the phase velocity of light insidethe specimen c n = c /n ( λ ), with n ( λ ) as the wavelength dependent refractiveindex of the material and c the vacuum speed of light. In the present CLstudy we use MgO and an electron beam energy of 200 keV. This beam en-2rgy is high enough to generate ˇCerenkov photons. The emission angle of theˇCerenkov photon with respect to the electron trajectory is given bycos ϑ C ( λ ) = c v e n ( λ ) = 1 βn ( λ ) . (1)Even though the electron beam enters the sample with normal incidence,the created light propagates inside the specimen under a certain angle ϑ C ( λ )(being called ϑ C further on). Fig.1 shows a schematic illustration of the ex-perimental geometry. The MgO specimen has a certain dielectric function ε MgO larger than ε = 1. Therefore the angle of total inner reflection for e.g., λ = 504 nm is given assin α = √ ε √ ε MgO = 11 .
83 = 0 . . (2)Consequently, the angle for total inner reflection is 33 ◦ . When following Eq.(1),the corresponding ˇCerenkov emission angle is 38.16 ◦ at the same time. Thus,we are facing at least partial total inner reflection and consequently we haveto treat the specimen as a Fabry-P´erot interferometer (Fig.1). The inner re-flection is only partially, because of the surface roughness. In some earlierstudies the same phenomena were observed, but without making use of it forthe determination of the sample’s refractive index [3,15]. Figure 1. Schematic illustration of the TEM specimen acting as a Fabry-P´erot in-terferometer. The swift electron beam excites ˇCerenkov photons which are emittedunder the ˇCerenkov angle ϑ C . The partial waves A m are interfering with each other, D m are emitted. The difference in the optical path length ∆ l between the optical rays A m and A m +1 for a specimen of given thickness d is∆ l = 2 d (cid:113) n − sin ( ϑ C ) . (3)Because there is reflection at the optically thinner medium, the phase differ-ence ∆ φ is 3 φ = 2 π ∆ lλ = 4 πdλ (cid:113) n − sin ( ϑ C ) . (4)Using sin (arccos( x )) = 1 − x , substituting u = 4 + ( mλ/d ) , and using thecondition for constructive interference ∆ φ = 2 mπ – with m being a positiveinteger – n ( λ ) yields ( β = v e /c ) n ( λ ) = ± √ · (cid:118)(cid:117)(cid:117)(cid:116) u ± (cid:115) u − β (5)The physically meaningful solution is the one having the positive signs. Wesee in Eq.5 that an accurate knowledge of u and thus of the sample thickness d at the position of measurement is of utmost importance. The MgO single crystalline specimen (MaTecK, 99.99% purity) was preparedby mechanical grinding and a final mechanical lapping procedure, in order notto introduce beam damage caused by further ion milling. Beam damage wouldbe responsible for defect states, thus leading to spurious signals in the inco-herent contribution of the CL spectrum. The grinding machine was adjustedto give a wedge angle of 1.3 ◦ , which was verified by optical measurements. Figure 2. Light microscope micrograph of the wedge shaped MgO sample. The wedgeangle is 1.3 ◦ as adjusted and verified by measuring the distance of the interferencefringes. The CL study was performed by employing a TECNAI F20 FEG TEM usingthe scanning mode (STEM) at a beam energy of 200 keV. The high beam4nergy guarantees the excitation of ˇCerenkov radiation acting as light source.A GATAN Vulcan CL detection and analysis system was used.For the EELS experiments, the beam energy was reduced to 80 keV in or-der to prevent the excitation of ˇCerenkov radiation. ˇCerenkov radiation wouldalter the EELS spectrum due to ˇCerenkov losses being present in the bandgap of the low loss part in the spectrum [16]. Albeit ˇCerenkov losses can betreated mathematically [13], avoiding them is the better solution for an accu-rate Kramers-Kronig Analysis (KKA) of the valence EELS (VEELS) spectrum[17].
Figure 3. High-angle annular dark field image of the observed sample. The whitearrow marks the position of the CL line scan acquisition with 300 points of mea-surement over a length of 30 µ m. Determining the sample thickness carefully by using EELS at the position ofthe CL investigations failed because of the large thickness required for the CLexperiments. In this case, multiple scattering becomes dominant, thus forminga Landau-background. In such situation any log-ratio method [18] and multiplescattering deconvolution routines [19] fails. From the optical investigation inFig.2, a linear increase in thickness is justified which is further corroboratedby the fact that no bending of interference maxima in the CL spectra withrespect to sample thickness is observable (see Fig.4).
For the CL experiments the wedge shaped sample was carefully adjusted intothe focal point of the elliptical mirrors of the GATAN Vulcan CL detectionand analysis system. The TEM was operated in scanning mode and the beamwas scanned across the sample at a length of 30 µ m (Fig. 3). Every 100 nm,a CL spectrum was recorded. Due to the wedge shape of the specimen with aslope of 1.3 circ , the sample thickness increased by 680 nm from the start-pointto the endpoint of the scan.Figure 4a shows the acquired spectrum image containing all CL spectra withrespect to the beam position on the sample. There are two prominent featuresalready visible: (i) faint interference fringes and (ii) intensities at 704 nm,5 igure 4. (a) As acquired CL spectra with respect to the beam position. (b) Con-tribution of interference after removal of incoherent CL signals with respect to thebeam position. The bright dots mark the measured interference maxima.
725 nm, 860 nm, 880 nm, 904 nm, and 922 nm which are constant in en-ergy at all thicknesses. The former are caused by interference, the latter arecaused by impurity levels inside the band gap. When removing the incoherentcontribution, which is of equal shape in all spectra, only the spectral frac-tion stemming from interference remains (Fig.4b). As example Fig.5 showsa single measurement at the 1100 nm thick MgO position, its contributionstemming from interference after removal of all incoherent parts, and the cor-responding simulation based on Yamamoto’s theory [1,2,3] using optical data[10]. The CL spectrum was corrected for the system response function. Thissystem response includes the reflectivity of the mirrors, the absorption of thelight guides, the correction for the 500 nm blazed grating, and the wavelengthdependent detection quantum efficiency of the CCD detector.For any interference experiment, the total thickness of the sample is of utmostimportance. Due to the fact that thick samples are needed for this kind ofexperiments, thickness determination by means of EELS fails. Therefore wemake use of the interference pattern in CL and the knowledge about the sam-ple geometry. In the present case the sample was prepared having a wedgeangle of 1.3 ◦ being confirmed with optical methods. Additionally, m has to bea positive integer. Using these two boundary conditions we can make an edu-cated guess for the sample thickness at the starting-point of the line scan. Inthe present experiment, we estimated 580 nm sample thickness. Subsequentlywe varied the sample thickness during the calculation of the reduced thick-ness d red = d/m for all interference maxima (Fig.6). At the correct thickness,all interference maxima should be on a single curve, independently of their6 i n t e n s i t y ( a . u . ) wavelength (nm) raw datainterference (x10)simulation, 1100nm Figure 5. Raw CL spectrum (after correction for the system response function)recorded at a sample thickness of 1100 nm and its contribution from interferenceafter removal of the incoherent spectral fraction. The simulation confirms the inter-pretation of the observed fringes. interference order m (Fig. 6b). Finally, the minimum median of the standarddeviation of the reduced thicknesses with respect to the single curve is givingthe accurate sample thickness at the starting-point. In the present experimentthe thickness at the starting-point is found to be 560 nm. Figure 6. (a) Identified maxima for given m with respect to the sample thickness.(b) The reduced sample thickness d red = d/m which is comparable to the firstinterference maximum m = 1. Inserting the results shown in Fig.6b into Eq.5 and taking u as being u =4 + ( λ/d red ) one can calculate the refractive index for the probed range ofwavelengths. Figure 8 gives the refractive index n in comparison with datafrom literature [10]. In the lower panel, the relative error is given. The error ismainly due to the scratches from the mechanical preparation already visiblein Fig. 2. 7 m e d i a n s t a n d a r d d e v i a t i o n sample thickness (nm) Figure 7. Median standard deviation of d red = d/m for all wavelengths for variousestimated sample thicknesses. r e f r a c t i v e i n d e x Stephens 1952experiment-1.85 0 1.85 400 500 600 700 800 900 r e l . e rr o r ( % ) wavelength (nm) Figure 8. Experimentally determined refractive index in comparison with opticalreference data from [10] and the error of the method with respect to the opticalreference data.
The VEELS experiments were performed at 80 keV at a TECNAI G20 LaB in order to suppress any ˇCerenkov losses. The spectrometer resolution was0.53 eV as full width at half maximum of the zero loss peak (ZLP) at the fullysaturated LaB filament. For data analysis, an 82 nm thin areas of the amplewas selected. The thickness was determined via the log-ratio method [18]. Dataanalysis included multiple scattering deconvolution based on the Fourier-logmethod utilizing a pre-measured vacuum ZLP. That way, the single scatteringdistribution (SSD) was retrieved. The band gap is the onset of the inelasticsignal and is identified to be 7.3 eV [20]. Subsequent Kramers-Kronig Analysis(KKA) was performed using the sample thickness for normalization [13]. Theresulting complex dielectric function ε + iε was further used for calculating8he refractive index n . In Fig.9 the raw experimental spectrum, its SSD andthe real part of ε are shown. Figure 9. 80 keV VEELS spectrum of 82 nm thick MgO and its SSD. By means ofKKA, the dielectric function was determined – here only the real part ε is plotted. Due to the fact that by using CL, we are limited to visible light and thusto energies smaller than the band gap energy of MgO, we have a closer lookto the results of KKA only in the range from 1.2 - 4.0 eV. Within the bandgap, the absorption coefficient κ and thus the imaginary part of ε are zero.Therefore the square root of ε is equal to the refractive index n . r e f r a c t i v e i n d e x wavelength (nm) VEELSCLStephens 1952 Figure 10. Refractive index calculated from VEELS and CL data compared to theoptical reference [10].
In this situation the advantage of VEELS is, that within the band gap ε =2 nκ = 0. Hence ε is a smooth function. Whereas in CL we are restrictedto the quality and smoothness of the sample surface, in VEELS such proper-ties do not dominate the resulting refractive index. Consequently, VEELS ispreferable to interferometric CL as long as thin samples are be investigated.But when investigating optical resonators and light guiding particles of a cer-tain thickness not suitable for EELS, interferometric CL gives the possibility9or the determination of the optical properties. In general ˇCerenkov radiation is seen as a disruptive signal in CL as well as inVEELS. In the present study we make use of the fact, that ˇCerenkov radiationis reflected inside the specimen, thus generating a wavelength and sample-thickness dependent interference spectrum. The sample itself acts as a Fabry-P´erot interferometer, hence the refractive index can be determined with highenergy resolution. The signal-to-noise ratio of the CL spectra determines theaccuracy of this method. Anyhow, VEELS at low beam energies is preferablein the case of MgO.
Acknowledgements
This research was in part supported by the SINNCE project of the EuropeanUnion’s Horizon 2020 programme under the grant agreement No. 810626.
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