Non-universal Transverse Electron Mean Free Path through Few-layer Graphene
D. Geelen, J. Jobst, E.E. Krasovskii, S.J. van der Molen, R.M. Tromp
NNon-universal Transverse Electron Mean Free Path through Few-layer Graphene
D. Geelen, J. Jobst, E.E. Krasovskii,
2, 3, 4
S.J. van der Molen, and R.M. Tromp
5, 1, ∗ Huygens-Kamerlingh Onnes Laboratorium, Leiden Institute of Physics, Leiden University,Niels Bohrweg 2, P.O. Box 9504, NL-2300 RA Leiden, The Netherlands Departamento de F´ısica de Materiales, Universidad del Pais Vasco UPV/EHU, 20080 San Sebasti´an/Donostia, Spain IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao, Spain Donostia International Physics Center (DIPC), E-20018 San Sebasti´an, Spain IBM T.J.Watson Research Center, 1101 Kitchawan Road,P.O. Box 218, Yorktown Heights, New York, New York 10598, USA (Dated: August 22, 2019)In contrast to the in-plane transport electron mean-free path in graphene, the transverse mean-free path has received little attention and is often assumed to follow the ‘universal’ mean-free path(MFP) curve broadly adopted in surface and interface science. Here we directly measure transverseelectron scattering through graphene from 0 to 25 eV above the vacuum level both in reflection usingLow Energy Electron Microscopy and in transmission using electron-Volt Transmission ElectronMicroscopy. From this data, we obtain quantitative MFPs for both elastic and inelastic scattering.Even at the lowest energies, the total MFP is just a few graphene layers and the elastic MFP oscillateswith graphene layer number, both refuting the ‘universal’ curve. A full theoretical calculationtaking the graphene band structure into consideration agrees well with experiment, while the keyexperimental results are reproduced even by a simple optical toy model.
INTRODUCTION
The mean free path (MFP) of electrons, i.e., the aver-age distance between scattering events, plays a key rolein numerous areas of science and technology. As an elec-tron moves through a medium (gaseous, liquid, solid,or plasma) it will undergo scattering which may be ei-ther elastic, or inelastic due to interaction with phonons,plasmons, nuclei or other electrons. The MFP of elec-trons determines many physical phenomena on all energyscales. At or near the Fermi level in a solid, the MFP isa key ingredient to the transport properties. For exam-ple, ballistic transport is only possible when the MFP islarger than the critical device dimension. At somewhathigher energies (several eV), where electrons can over-come the workfunction of a material and escape into thevacuum, the MFP determines from which depth belowthe surface an electron can escape. Thus, the probingdepth of electrons in a Low Energy Electron Diffraction(or Microscopy) experiment, the electron escape depth inphotoemission experiments, the efficacy of electron emis-sion in electron sources and electron multipliers, and thespatial extent and resolution of electron interactions inScanning Electron Microscopy, all depend on the electronMFP. At higher energies yet (1 keV to 500 keV), the MFPis of key concern in Transmission Electron Microscopy,and in plasmas for the interaction of energetic electronswith other plasma constituents and the solid surfaces incontact with the plasma. Finally, in the few MeV en-ergy range, the relatively short MFP is useful in electronbeam treatment of superficial cancers.For electrons with vacuum energies from just a few eVto tens of keV, the MFP in solids is often assumed tobe described by a ‘universal’ curve, which implies the MFP to depend strongly on energy, but only weakly onmaterial [1]. This ‘universal’ curve shows a minimum inMFP at energies of a few 10’s of eV, increasing at bothlower and higher energies. At the lowest energies notmany excitation mechanisms other than phonons and in-traband transitions are available for scattering, so theMFP is long, presumably up to a 100 nm at 1 eV accord-ing to Ref. [1]. At somewhat higher energies (severaleV to 10’s of eV) surface and bulk plasmon excitationskick in, and the MFP drops to just ∼ a r X i v : . [ c ond - m a t . m t r l - s c i ] A ug (a) (b) E ( e V ) M K Г k || (Å -1 ) . . R . LEEM guneV-TEM gunimaging systemdetector prism sample objective lens (c) (d) eV-TEM nm eV 5.1 eV500 nm FIG. 1. (a) Sketch of the ESCHER setup combining twoelectron guns for reflection (LEEM, red) and transmission(eV-TEM, blue) experiments. (b) Angle-resolved reflected-electron spectroscopy of exfoliated bulk graphite showing elec-tronic bands as minima and band gaps as maxima in elec-tron reflection in agreement with band structure calculations(black lines). Reproduced from Ref. [7]. (c) LEEM image of afree-standing membrane of 1, 2 and 3 layer graphene. (d) Thesame area imaged in eV-TEM. Electron energy (indicated into top right) in both images is chosen for optimal contrast. that electron mean free paths cannot be universal.We describe a set of experiments on thin graphene lay-ers illuminated with electrons with kinetic energies E inthe range from 0 to 25 eV where only the specularly re-flected and the directly transmitted beams are present(first order Low Energy Electron Diffraction can only beexcited above ∼
28 eV, which thus sets an upper limitto a straight-forward, quantitative interpretation). Inaddition to these energy-loss-free coherent beams, elec-trons scatter ‘thermally’ (Debye-Waller scattering) andinelastically. Here, we quantitatively determine the elas-tic scattering fractions in both the specularly reflectedand the transmitted electron beams as a function of en-ergy, using the ESCHER aberration-corrected Low En-ergy Electron Microscope (LEEM) [8–10] equipped withtwo distinct electron sources (see sketch in Fig. 1a). Ina standard LEEM experiment, the sample is illuminatedwith an electron source from the front side of the sam-ple (red in Fig. 1a), and an image is formed from re-flected electrons. However, if the sample is sufficientlythin, one may also use an electron source located behind the sample (blue in Fig. 1a), and utilize electrons trans-mitted through the sample to form an image. Over thelast few years we have developed such a capability en-abling Transmission Electron Microscopy (TEM) exper-iments at electron energies of just a few eV [11], ratherthan 10’s or 100’s of keV as in conventional TEM. Us-ing this ‘electron-Volt Transmission Electron Microscopy’(eV-TEM), we can thus study energy-dependent elastictransmission, in addition to energy-dependent elastic re-flection using standard LEEM, on the same (thin) samplewithin the same instrument.To understand this energy dependence in an idealizedsystem, let us assume that at a particular electron en-ergy, at normal incidence ( k (cid:107) = 0), our sample has abandgap. In the absence of incoherent and inelastic chan-nels, we would then expect a reflectivity of 1 and a trans-missivity of 0 for electrons of that energy [12]. I.e., allelectrons are elastically back-reflected as the sample elec-tronic band structure has no states that would allow theelectrons to propagate within the solid. One can ‘probe’the band structure of the solid above the vacuum levelby measuring electron reflectivity as a function of energyand momentum [13–15]. Figure 1(b) shows the results ofsuch an Angle-Resolved Reflected Electron Spectroscopy(ARRES) experiment on bulk graphite reproduced fromRef. [7]. Reflectivity is plotted as function of energy andin-plane momentum k (cid:107) . The solid lines show theoreticalband structure results from Ref. [16], in good agreementwith the data. Specifically, the high reflectivity (red) re-gion across the Brillouin zone between ∼ E = 5 . E = 10 . n = 1–4). In LEEM, only thespecularly reflected (0,0) LEED beam was used, and ineV-TEM only the directly transmitted beam. Inelasticelectrons were removed from the signal by energy filter-ing, using the magnetic prism array as an efficient in-lineenergy filter [19]. Thus, the image intensities in Fig. 1(c)and (d), normalized to the intensities of the incident elec-tron beams, directly yield the elastic reflectivity R andtransmissivity T . Recording LEEM and eV-TEM im-ages as a function of energy E yields a laterally-resolved,spectroscopic data cube where reflectivity and transmis-sivity spectra can be extracted from every area. Figure r e f l e c t i v i t y R experiment full theory0 5 10 15 20 25 E (eV)0.00.20.40.6 t r an s m i ss i v i t y T experiment 0 5 10 15 20 25 E (eV)full theory (a) (c)(d)(b) FIG. 2. (a) Electron reflectivity R ( E ) as a function of land-ing energy E on 1–4 layer graphene. A general decreasingtrend with strongly layer-dependent oscillations is observed.(b) The electron transmissivity T ( E ) from the same areasalso decreases with energy, but exhibits maxima where R ( E )has minima. (c,d) Theoretical predictions of R ( E ) and T ( E )obtained by ab initio methods [17, 18] reproduce the experi-mental data in all key features. R ( E ) LEEM (a) and T ( E ) eV-TEM (b)spectra for electrons with energies E from 0–25 eV, ob-tained on sample areas with 1–4 layers of graphene.In addition to a decrease with energy, strong mod-ulations that depend not only on energy, but also onthe number of graphene layers are visible for R ( E ) and T ( E ). For 2LG and thicker we find a broad maximumin reflectivity and minimum in transmission between 5and 15 eV, corresponding to the graphite bandgap seenin Fig. 1(b). For 1LG this feature is absent, indicatingthat this gap is a result of interlayer interactions. Infact, between 0 and 5 eV we find n − n is the number of graphene layers that are gen-erally assumed to be caused by inter-layer transmissionresonances that, eventually, merge into the broad mini-mum for many layers [see data on graphite in Fig. 1(b)][16, 20]. Measuring corresponding maxima in transmis-sivity directly [Fig. 2(b)], we here confirm that reflectionminima correlate with transmission maxima. Of course,for 1LG there is no interlayer scattering and thus no min-imum/maximum. The energy dependence of reflectivityand transmissivity for all layer numbers is well repro-duced by ab initio theory in Fig. 2(c) and (d), respec-tively. To obtain these, we calculate the ground-statepotential of the n-layer graphene from first principles inthe local density approximation and use it to obtain thescattering wave functions as described in Refs. [17, 18].To account for inelastic effects, an energy-dependent op-tical potential is used in the scattering calculations. Inaddition, the theoretical reflectivity (obtained for a staticlattice) is scaled down by a factor of 8 to fit the exper- imental reflectivity spectra which accounts for the en-hanced Debye-Waller scattering in free-standing mem-branes. Generally, both reflectivity and transmissivity(elastic signals) shown in Fig. 2(a, b) decrease with in-creasing energy, indicating increasing inelastic scattering.Using the quantitative R ( E ) and T ( E ) data, we can de-rive the inelastic MFP λ inel and elastic MFP λ el , whichcombine to the total MFP λ tot . The total electron MFP λ tot can be obtained from the transmission data alonesince both elastic and inelastic scattering give rise to areduction of the elastically transmitted electron signal I et . The elastic transmissivity T , shown in Fig. 2a, isthus given by T = I et I = e − λ tot /d (1)where I the incident intensity and d the sample thick-ness. Similarly, the inelastic MFP λ inel can be obtainedfrom the sum of reflected and transmitted intensities I er and I et , as the total elastic signal is depleted by inelasticscattering only. T + R = I et I + I er I = e − λ inel /d (2)where R is the elastic reflectivity shown in Fig. 2(a). Fi-nally, the elastic MFP λ el is given by1 λ tot = 1 λ el + 1 λ inel . (3)The experimentally measured λ tot , λ inel and λ el are plot-ted in units of graphene layers as a function of electronenergy in Fig. 3(a), (b) and (c), respectively. Strikingly,the total MFP [Fig. 3(a)] is very short for all layer num-bers, even at energies very close to 0 eV. These valuesfall far short of the large numbers ( ∼
300 layers) sug-gested by the ‘universal’ curve [1], even at energies belowthe graphene π -plasmon energy of ∼ λ inel ≈ λ inel ≈ λ inel at about 6 eV,this is by no means a drastic effect, indicating the strongcontribution of phonon and intra-band excitation lossesat lower energies. The elastic MFP shown in Fig. 3(c)exhibits possibly the most interesting effects. Between0 and 5 eV, where the interlayer resonances occur, λ el is strongly modulated with the graphene layer number:one maximum for one interlayer resonance (2LG), two(three) maxima for 3LG (4LG). The coherent nature ofthis process calls into question the very name ‘mean freepath’ for λ el , but we keep using it for consistency withliterature. Compared to the total and inelastic MFPs, λ el is quite large in this energy range with up to λ el ≈ λ el ≈ λ t o t ( l a y e r s ) λ e l ( l a y e r s ) toy model024 λ i ne l ( l a y e r s ) λ e l ( l a y e r s ) energy-dependenttoy model0 5 10 15 20 25 E (eV)020406080 λ e l ( l a y e r s ) experiment 0 5 10 15 20 25 E (eV)020406080 λ e l ( l a y e r s ) full theory (a)(b)(c) (e)(f)(d) FIG. 3. (a) Energy-dependent total MFP λ tot for electronsimpinging on 1–4 layer graphene obtained from the spectra inFig. 2 using Eq. (1). It decreases with energy, but is gener-ally much lower than the ‘universal’ curve predicts. (b) Theinelastic MFP λ inel [Eq. (2)] shows a similar trend but thelayer-dependent maxima are absent. (c) The elastic MFP λ el [Eq. (3)] exhibits strong, layer-dependent peaks at the ener-gies of high-transmission states where λ el is very long. Thezoomed-in inset shows that the λ el is largest for monolayergraphene in the energy range from 6 eV to 12 eV. (d) An op-tical toy-model based on transfer matrices describes positionand shape of the layer dependent oscillations in λ el well. (e)When an energy-dependent absorption is taken into account,the experimental data in (c) is well described by this sim-ple model over the full energy range. (f) The λ el calculatedfrom the ab initio spectra in Fig. 2(c,d) also reproduces theexperiment well over the full energy range.
15 eV, λ el is largest for 1LG [see inset in Fig. 3(c)] as thebandgap seen in thicker layers is not present for singlelayer, and electron tunneling may play a significant role.In the next band (15–22 eV), an increased λ el is clearlyvisible while layer-number-dependent maxima are broad-ened due to inelastic effects, and no longer resolved. Thissecond high transmission band was predicted by Feenstra et al . [20, 22], but never before observed experimentally.To arrive at a qualitative and intuitive understandingof these results, we turn to a simple toy model. In anal-ogy with an optical multilayer, every graphene layer ismodeled as a semitransparent boundary with reflectiv-ity R = | r | and transmissivity T = | t | where r and t are the reflection and transmission amplitudes, respec-tively. The total reflectivity R and transmissivity T ofa multilayer are then determined by the interference ofall possible wave reflections and transmissions within the multilayer, while gaining a phase φ when propagatingfrom layer to layer. Losses due to absorption, incoher-ent scattering, etc. are taken into account for every layerby setting R + T <
1. The energy-dependent R ( E )and T ( E ) can be calculated using the transfer matrixapproach (see Supplemental Materials [23] for details )frequently used in thin film optics e.g., to describe anti-reflective coatings [28, 29]. Figures 3(d) shows λ el as afunction of electron energy extracted from those R ( E )and T ( El ) using Eqs. (1–3) for the case with moderateabsorption ( R = 0 . T = 0 . R and T , this is a parameter-free toy model(we use literature values of graphite for layer separationand work function, see Supplemental Material [23]). Weobtain better agreement with the experiment at the sec-ond resonance between 15 and 20 eV by taking losses in-creasing with energy into account [Fig. 3(e)]. We opti-mize R ( E ) and T ( E ) by comparing to LEEM specu-lar reflectivity data of 1–8 graphene layers by Hibino etal . [30] (see Supplemental Material [23] for details). Wefind that constant R = 0 . T = 0 . R = 0 . T = 0 . π - and ( σ + π )-plasmon losses above 6 eV leadsto excellent agreement with the data over the full energyrange. This indicates that even this very simple opti-cal toy-model captures the essential physics of electronscattering in multilayer graphene.The full quantum mechanical approach [Fig. 3(f)], cal-culating the elastic MFP from the theoretical R ( E ) and T ( E ) shown in Fig. 2(c,d) again yields good agreementwith the experimental data in Fig. 3(c). In this quantummechanical picture, the maxima in λ el correspond to thetransmission resonances. Comparison with the opticaltoy-model gives us the intuitive understanding that theenhanced transmission is the result of interlayer multi-reflection resonances of the electron waves. The energyrange and scattering-induced broadening of the secondhigh transmission band [15–20 eV in Fig. 3(c)] is well de-scribed by both the toy model [Fig. 3(e)] and the fulltheory [Fig. 3(f)].We have presented the first direct measurements ofelastic electron reflection and transmission data in theenergy range between 0 and 25 eV. While the ‘univer-sal’ MFP curve for electrons in this energy range wouldsuggest very large mean free paths, up to 100 nm at thelowest energies [1], we find that this prediction is far fromtrue. Inelastic MFPs in single and multilayer grapheneare just a few layers, even below the graphene π -plasmonenergy of ∼ λ el and λ inel depend not only on electron energy, and on mate-rial (all carbon here), but also significantly on the de-tails of the electronic structure above the vacuum level.The presence of interlayer resonances gives rise to hightransmission, and very long λ el due to multilayer electroninterference. This can explain the surprising fact that1LG has the shortest total MFP over most of the energyrange. These basic features are reproduced qualitativelyby a simple toy model, and in detailed electronic struc-ture + electron scattering theory. The high transmis-sion/low reflection nLG graphene resonances correspondto electron anti-reflection coatings in our toy-model ana-logue. For other, more complex materials this simpletoy model does not yield valid predictions. Already forother layered crystals such as hexagonal boron nitride[7] or transition metal dichalcogenides [31, 32] only thefull ab initio theory can describe the reflectivity spectracorrectly. This indicates that the MFPs in these ma-terials are also strongly affected by the band structureeffects discussed here. Further measurements on thosematerials will, thus, elucidate our understanding of scat-tering of low-energy electrons with matter more broadly.Moreover, the observed transmission resonances stronglymodify the so-called final state in Angle-Resolved Photo-Emission Spectroscopy (ARPES) for low photon energies[33, 34]. Together with the considerably shorter MFPsfound here compared to the universal curve, this hasbroad implication on the interpretation of ARPES dataat photon energies below 30 eV.The advent of eV-TEM in conjunction with LEEM, ina single instrument, has made it possible to study elasticreflection and transmission of low energy electrons fromthe same sample for the first time. This allows us to de-velop a more complete and detailed understanding of theinteraction of low energy electrons with solids. Theseresults challenge the perceived universality of electronMFP, and demonstrate that such universality cannot ex-ist. The electronic structure of a material depends onits elemental composition, crystal structure, crystal ori-entation, and sample thickness and dominates scatteringat low electron energies. The imaging and spectroscopiccapabilities close to E = 0 demonstrated here enablesother eV-TEM experiments. For instance, using eV-TEM, we have succeeded in imaging single DNA origamimolecules with electron energies below 5 eV, where radia-tion damage appears to be negligible [35, 36], with strongcontrast. Together with the projected spatial resolutionbelow 2 nm in an aberration-corrected instrument, eV-TEM promises new avenues for imaging and spectroscopyin physics, materials science, and life science.We thank Marcel Hesselberth and Douwe Scholma fortheir indispensable technical support, Ruud van Egmond,Arthur Ellis, Raymond K¨ohler and Bert Crama for build-ing the eV-TEM gun assembly and electronics, AniketThete for help with the transfer of graphene membranes and Martin van Exter for discussions on the transfer ma-trix approach. This work was supported by the Nether-lands Organisation for Scientific Research (NWO/OCW)via the VENI grant (680-47-447, J.J.) and the STW-HTSM grant (nr. 12789). It was supported by the Span-ish Ministry of Economy and Competitiveness MINECO,Grant No. FIS2016-76617-P. ∗ [email protected][1] M. P. Seah and W. A. Dench, Surface and Interface Anal-ysis , 2 (1979).[2] R. M. Marsolais, E. A. Cartier, and P. Pfluger, ExcessElectrons in Dielectric Media , edited by J.-P. Jay-Gerinand C. Ferradini (CRC Press, Boca Raton, FL, 1991)p. 43.[3] N. Barrett, E. E. Krasovskii, J.-M. Themlin, and V. N.Strocov, Physical Review B , 035427 (2005).[4] R. Naaman and L. Sanche, Chemical Reviews , 1553(2007).[5] I. M¨ullerov´a, M. Hovorka, and L. Frank, Ultrami-croscopy , 78 (2012).[6] J. Cazaux, Journal of Applied Physics , 064903(2012).[7] J. Jobst, A. J. H. van der Torren, E. E. Krasovskii, J. Bal-gley, C. R. Dean, R. M. Tromp, and S. J. van der Molen,Nature Communications , 13621 (2016).[8] R. Tromp, J. Hannon, A. Ellis, W. Wan, A. Berghaus,and O. Schaff, Ultramicroscopy , 852 (2010).[9] R. Tromp, J. Hannon, W. Wan, A. Berghaus, andO. Schaff, Ultramicroscopy , 25 (2013).[10] S. M. Schramm, J. Kautz, A. Berghaus, O. Schaff, R. M.Tromp, and S. J. van der Molen, IBM Journal of Re-search and Development , 1:1 (2011).[11] D. Geelen, A. Thete, O. Schaff, A. Kaiser, S. J. van derMolen, and R. Tromp, Ultramicroscopy , 482 (2015).[12] J. C. Slater, Physical Review , 846 (1937).[13] V. N. Strocov, H. I. Starnberg, P. O. Nilsson, H. E.Brauer, and L. J. Holleboom, Physical Review Letters , 467 (1997).[14] J. Jobst, J. Kautz, D. Geelen, R. M. Tromp, and S. J.van der Molen, Nature Communications , 8926 (2015).[15] F. Wicki, J.-N. Longchamp, T. Latychevskaia, C. Escher,and H.-W. Fink, Physical Review B , 075424 (2016).[16] H. Hibino, H. Kageshima, F. Maeda, M. Nagase,Y. Kobayashi, and H. Yamaguchi, Physical Review B , 075413 (2008).[17] E. E. Krasovskii, Physical Review B , 245322 (2004).[18] V. U. Nazarov, E. E. Krasovskii, and V. M. Silkin, Phys-ical Review B , 041405(R) (2013).[19] R. M. Tromp, Y. Fujikawa, J. B. Hannon, a. W. Ellis,A. Berghaus, and O. Schaff, Journal of Physics: Con-densed Matter , 314007 (2009).[20] R. M. Feenstra, N. Srivastava, Q. Gao, M. Widom, B. Di-aconescu, T. Ohta, G. L. Kellogg, J. T. Robinson, andI. V. Vlassiouk, Physical Review B , 041406(R) (2013).[21] A. Politano and G. Chiarello, Nanoscale , 10927 (2014).[22] R. Feenstra and M. Widom, Ultramicroscopy , 101(2013).[23] “See Supplemental Material [url] for a description of the transfer-matrix calculations used as toy model, which in-cludes Ref. [25–29],”.[24] L. Rayleigh, Proceedings of the Royal Society A: Math-ematical, Physical and Engineering Sciences , 565(1917).[25] A. Gr¨uneis, C. Attaccalite, T. Pichler, V. Zabolotnyy,H. Shiozawa, S. L. Molodtsov, D. Inosov, A. Koitzsch,M. Knupfer, J. Schiessling, R. Follath, R. Weber,P. Rudolf, L. Wirtz, and A. Rubio, Physical ReviewLetters , 037601 (2008).[26] S. Zhou, G.-H. Gweon, and A. Lanzara, Annals ofPhysics , 1730 (2006).[27] F. Matsui, H. Nishikawa, H. Daimon, M. Muntwiler,M. Takizawa, H. Namba, and T. Greber, Physical Re-view B , 045430 (2018).[28] H. Anders, Thin films in optics (The Focal Press, Londonand New York, 1967).[29] T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, Journal ofPhysics: Condensed Matter , 215301 (2013).[30] H. Hibino, H. Kageshima, F. Maeda, M. Nagase, Y. Kobayashi, Y. Kobayashi, and H. Yamaguchi, e-Journal of Surface Science and Nanotechnology , 107(2008).[31] E. E. Krasovskii, V. N. Strocov, N. Barrett, H. Berger,W. Schattke, and R. Claessen, Physical Review B ,045432 (2007).[32] T. A. de Jong, J. Jobst, H. Yoo, E. E. Krasovskii, P. Kim,and S. J. van der Molen, physica status solidi (b) ,1800191 (2018).[33] V. Strocov, P. Blaha, H. Starnberg, R. Claessen, J.-M.Debever, and J.-M. Themlin, Applied Surface Science , 508 (2000).[34] V. N. Strocov, A. Charrier, J.-M. Themlin, M. Rohlfing,R. Claessen, N. Barrett, J. Avila, J. Sanchez, and M.-C.Asensio, Physical Review B , 075105 (2001).[35] M. Germann, T. Latychevskaia, C. Escher, and H.-W.Fink, Physical Review Letters , 095501 (2010).[36] J.-N. Longchamp, S. Rauschenbach, S. Abb, C. Escher,T. Latychevskaia, K. Kern, and H.-W. Fink, Proceedingsof the National Academy of Sciences114