Physics:Photoemission orbital tomography
In physics and chemistry, photoemission orbital tomography (POT; sometimes called photoemission tomography) is a combined experimental / theoretical approach which reveals information about the spatial distribution of individual molecular orbitals.[1][2] Experimentally, it uses angle-resolved photoemission spectroscopy (ARPES) to obtain constant binding energy photoemission angular distribution maps, so-called tomograms (also known as momentum maps or [math]\displaystyle{ k }[/math]-maps), to reveal information about the electron probability distribution in molecular orbitals. Theoretically, one rationalizes these tomograms as hemispherical cuts through the molecular orbital in momentum space. This interpretation relies on the assumption of a plane wave final state, i.e., the idea that the outgoing electron can be treated as a free electron, which can be further exploited to reconstruct real-space images of molecular orbitals on a sub-Ångström length scale in two[3][4][5] or three dimensions.[6][7] Presently, POT has been applied to various organic molecules forming well-oriented monolayers on single crystal surfaces or to two-dimensional materials.
Theory
Within the framework of POT, the photo-excitation is treated as a single coherent process from an initial (molecular) orbital [math]\displaystyle{ \Psi_i }[/math] to the final state [math]\displaystyle{ \Psi_f }[/math], which is referred to as the one-step-model of photoemission. The intensity distribution in the tomograms, [math]\displaystyle{ I(k_x,k_y;E_\mathrm{kin}) }[/math], is then given from Fermi's golden rule as[8]
[math]\displaystyle{ I(k_x,k_y;E_\mathrm{kin}) \propto \left| \langle \Psi_f(k_x,k_y;E_\mathrm{kin}) | \vec{A} \cdot \vec{p} | \Psi_i \rangle \right|^2 \times \delta \left(E_i + \Phi + E_\mathrm{kin} - \hbar \omega \right). }[/math]
Here, [math]\displaystyle{ k_x }[/math] and [math]\displaystyle{ k_y }[/math] are the components of the emitted electron's wave vector parallel to the surface, which are related to the polar and azimuthal emission angles [math]\displaystyle{ \theta }[/math] and [math]\displaystyle{ \phi }[/math] defined in the figure as follows,
[math]\displaystyle{ k_x = k \sin \theta \cos \phi }[/math]
[math]\displaystyle{ k_y = k \sin \theta \sin \phi }[/math]
where [math]\displaystyle{ k }[/math] and [math]\displaystyle{ E_\mathrm{kin} = \frac{\hbar^2 k^2}{2m} }[/math] are the wave number and kinetic energy of the emitted electron, respectively, where [math]\displaystyle{ \hbar }[/math] is the reduced Planck constant and [math]\displaystyle{ m }[/math] is the electron mass. The transition matrix element is given in the dipole approximation, where [math]\displaystyle{ \vec{p} }[/math] and [math]\displaystyle{ \vec{A} }[/math], respectively, denote the momentum operator of the electron and the vector potential of the exciting electromagnetic wave. In the independent electron approximation, the spectral function reduces to a delta function and ensures energy conservation, where [math]\displaystyle{ \Phi }[/math] denotes the sample work function, [math]\displaystyle{ E_i }[/math] the binding energy of the initial state, and [math]\displaystyle{ \hbar \omega }[/math] the energy of the exciting photon.
In POT, the evaluation of the transition matrix element is further simplified by approximating the final state by a plane wave. Then, the photocurrent [math]\displaystyle{ I_i }[/math] arising from one particular initial state [math]\displaystyle{ i }[/math] is proportional to the Fourier transform [math]\displaystyle{ \tilde{\Psi}_{i} (\vec{k}) = \mathcal{F}\left\{ \Psi_i(\vec{r}) \right\} }[/math] of the initial state wave function modulated by the weakly angle-dependent polarization factor [math]\displaystyle{ \vec{A} \cdot \vec{k} }[/math]:
[math]\displaystyle{ I_i(k_x,k_y) \propto \left|\vec{A} \cdot \vec{k}\right|^2 \cdot \left| \tilde{\Psi}_{i} (k_x, k_y) \right|^2 \quad \textrm{with} \quad |\vec{k}|^2 = k_x^2 + k_y^2 + k_z^2 = \frac{2m}{\hbar^2} E_\mathrm{kin} }[/math]
As illustrated in the figure, the relationship between the real space orbital and its photoemission distribution can be represented by an Ewald's sphere-like construction. Thus, a one-to-one relation between the photocurrent and the molecular orbital density in reciprocal space can be established. Moreover, a reconstruction of molecular orbital densities in real space via an inverse Fourier transform and applying an iterative phase retrieval algorithm has also been demonstrated.[4]
Experiment
The basic experimental requirements are a reasonably monoenergetic photon source (inert gas discharge lamps, synchrotron radiation or UV laser sources) and an angle-resolved photoelectron spectrometer. Ideally, a large angular distribution ([math]\displaystyle{ k }[/math]-area) should be collected. Much of the development of POT was made using a toroidal analyzer[9] with [math]\displaystyle{ p }[/math]-polarized synchrotron radiation. Here the spectrometer collects the hemicircle of emissions ([math]\displaystyle{ -90^\circ \lt \theta \lt +90^\circ }[/math]) in the plane of incidence and polarization, and the momentum maps are obtained by rotating the sample azimuth ([math]\displaystyle{ \phi }[/math]). A number of commercially available electron spectrometers are now on the market which have been shown to be suited to POT. These include large acceptance angle hemispherical analysers, spectrometers with photoemission electron microscopy (PEEM) lenses and time of flight (TOF) spectrometers.
Applications and future developments
![](/wiki/images/thumb/4/4e/Photoemission_Orbital_Tomography_application.svg/250px-Photoemission_Orbital_Tomography_application.svg.png)
POT has found many interesting applications including the assignment of molecular orbital densities in momentum and real space,[10][11] the deconvolution of spectra into individual orbital contributions beyond the limits of energy resolution,[12] the extraction of detailed geometric information,[13] or the identification of reaction products.[14] Recently, the extension to the time-domain has been demonstrated by combining time-resolved photoemission using high laser harmonics and a momentum microscope to measure the full momentum-space distribution of transiently excited electrons in organic molecules.[15]
The possibility to measure the spatial distribution of electrons in frontier molecular orbitals has stimulated discussions on the interpretation of the concept of orbitals itself.[16] The present understanding is that the information retrieved from photoemission orbital tomography should be interpreted as Dyson orbitals.[17][18]
Approximating the photoelectron's final state by a plane wave have been viewed critically.[19] Indeed there are cases where the plane-wave final state approximation is problematic including a proper description of the photon energy dependence, the circular dichroism in the photoelectron angular distribution[20] or certain experimental geometries.[19] Nevertheless, the usefulness of the plane wave final state approximation has been extended beyond the originally suggested case of [math]\displaystyle{ \pi }[/math]-orbitals of large, planar [math]\displaystyle{ \pi }[/math]-conjugated molecules to three-dimensional molecules,[21] small organic molecules[22] and extended to two-dimensional materials.[23] Theoretical approaches beyond the plane wave final state approximation have also been demonstrated including time-dependent density functional theory calculations[24] or Green's function techniques.[25]
References
- ↑ Woodruff, Phil (2016). Modern Techniques of Surface Science. Cambridge: Cambridge University Press. doi:10.1017/CBO9781139149716. ISBN 9781139149716. http://dx.doi.org/10.1017/CBO9781139149716.
- ↑ Puschnig, P.; Ramsey, M.G. (2018). "Photoemission Tomography: Valence Band Photoemission as a Quantitative Method for Investigating Molecular Films". Encyclopedia of Interfacial Chemistry: 380–391. doi:10.1016/B978-0-12-409547-2.13782-5. ISBN 9780128098943.
- ↑ Puschnig, P.; Berkebile, S.; Fleming, A. J.; Koller, G.; Emtsev, K.; Seyller, T.; Riley, J. D.; Ambrosch-Draxl, C. et al. (30 October 2009). "Reconstruction of Molecular Orbital Densities from Photoemission Data". Science 326 (5953): 702–706. doi:10.1126/science.1176105. PMID 19745118. Bibcode: 2009Sci...326..702P.
- ↑ 4.0 4.1 Lüftner, D.; Ules, T.; Reinisch, E. M.; Koller, G.; Soubatch, S.; Tautz, F. S.; Ramsey, M. G.; Puschnig, P. (14 January 2014). "Imaging the wave functions of adsorbed molecules". Proceedings of the National Academy of Sciences 111 (2): 605–610. doi:10.1073/pnas.1315716110. PMID 24344291. Bibcode: 2014PNAS..111..605L.
- ↑ Kliuiev, P; Latychevskaia, T; Osterwalder, J; Hengsberger, M; Castiglioni, L (22 September 2016). "Application of iterative phase-retrieval algorithms to ARPES orbital tomography". New Journal of Physics 18 (9): 093041. doi:10.1088/1367-2630/18/9/093041. Bibcode: 2016NJPh...18i3041K.
- ↑ Graus, Martin; Metzger, Christian; Grimm, Manuel; Nigge, Pascal; Feyer, Vitaliy; Schöll, Achim; Reinert, Friedrich (April 2019). "Three-dimensional tomographic imaging of molecular orbitals by photoelectron momentum microscopy". The European Physical Journal B 92 (4): 80. doi:10.1140/epjb/e2019-100015-x. Bibcode: 2019EPJB...92...80G.
- ↑ Weiß, S.; Lüftner, D.; Ules, T.; Reinisch, E. M.; Kaser, H.; Gottwald, A.; Richter, M.; Soubatch, S. et al. (November 2015). "Exploring three-dimensional orbital imaging with energy-dependent photoemission tomography". Nature Communications 6 (1): 8287. doi:10.1038/ncomms9287. PMID 26437297. Bibcode: 2015NatCo...6.8287W.
- ↑ Feibelman, Peter J.; Eastman, D. E. (15 December 1974). "Photoemission spectroscopy—Correspondence between quantum theory and experimental phenomenology". Physical Review B 10 (12): 4932–4947. doi:10.1103/PhysRevB.10.4932. Bibcode: 1974PhRvB..10.4932F.
- ↑ Broekman, L.; Tadich, A.; Huwald, E.; Riley, J.; Leckey, R.; Seyller, T.; Emtsev, K.; Ley, L. (June 2005). "First results from a second generation toroidal electron spectrometer". Journal of Electron Spectroscopy and Related Phenomena 144-147: 1001–1004. doi:10.1016/j.elspec.2005.01.022.
- ↑ Dauth, M.; Körzdörfer, T.; Kümmel, S.; Ziroff, J.; Wiessner, M.; Schöll, A.; Reinert, F.; Arita, M. et al. (4 November 2011). "Orbital Density Reconstruction for Molecules". Physical Review Letters 107 (19): 193002. doi:10.1103/PhysRevLett.107.193002. PMID 22181601. Bibcode: 2011PhRvL.107s3002D. http://ir.lib.hiroshima-u.ac.jp/00031970.
- ↑ Zamborlini, Giovanni; Lüftner, Daniel; Feng, Zhijing; Kollmann, Bernd; Puschnig, Peter; Dri, Carlo; Panighel, Mirko; Di Santo, Giovanni et al. (December 2017). "Multi-orbital charge transfer at highly oriented organic/metal interfaces". Nature Communications 8 (1): 335. doi:10.1038/s41467-017-00402-0. PMID 28839127. Bibcode: 2017NatCo...8..335Z.
- ↑ Puschnig, P.; Boese, A. D.; Willenbockel, M.; Meyer, M.; Lüftner, D.; Reinisch, E. M.; Ules, T.; Koller, G. et al. (5 January 2017). "Energy Ordering of Molecular Orbitals". The Journal of Physical Chemistry Letters 8 (1): 208–213. doi:10.1021/acs.jpclett.6b02517. PMID 27935313.
- ↑ Kliuiev, Pavel; Zamborlini, Giovanni; Jugovac, Matteo; Gurdal, Yeliz; Arx, Karin von; Waltar, Kay; Schnidrig, Stephan; Alberto, Roger et al. (December 2019). "Combined orbital tomography study of multi-configurational molecular adsorbate systems". Nature Communications 10 (1): 5255. doi:10.1038/s41467-019-13254-7. PMID 31748503. Bibcode: 2019NatCo..10.5255K.
- ↑ Yang, Xiaosheng; Egger, Larissa; Hurdax, Philipp; Kaser, Hendrik; Lüftner, Daniel; Bocquet, François C.; Koller, Georg; Gottwald, Alexander et al. (December 2019). "Identifying surface reaction intermediates with photoemission tomography". Nature Communications 10 (1): 3189. doi:10.1038/s41467-019-11133-9. PMID 31320632. Bibcode: 2019NatCo..10.3189Y.
- ↑ Wallauer, R.; Raths, M.; Stallberg, K.; Münster, L.; Brandstetter, D.; Yang, X.; Güdde, J.; Puschnig, P. et al. (5 March 2021). "Tracing orbital images on ultrafast time scales". Science 371 (6533): 1056–1059. doi:10.1126/science.abf3286. PMID 33602865. Bibcode: 2021Sci...371.1056W.
- ↑ Truhlar, Donald G.; Hiberty, Philippe C.; Shaik, Sason; Gordon, Mark S.; Danovich, David (2 September 2019). "Orbitals and the Interpretation of Photoelectron Spectroscopy and (e,2e) Ionization Experiments". Angewandte Chemie 131 (36): 12460–12466. doi:10.1002/ange.201904609.
- ↑ Dauth, M; Wiessner, M; Feyer, V; Schöll, A; Puschnig, P; Reinert, F; Kümmel, S (8 October 2014). "Angle resolved photoemission from organic semiconductors: orbital imaging beyond the molecular orbital interpretation". New Journal of Physics 16 (10): 103005. doi:10.1088/1367-2630/16/10/103005. Bibcode: 2014NJPh...16j3005D.
- ↑ Ortiz, J. V. (21 August 2020). "Dyson-orbital concepts for description of electrons in molecules". The Journal of Chemical Physics 153 (7): 070902. doi:10.1063/5.0016472. PMID 32828082.
- ↑ 19.0 19.1 Bradshaw, A M; Woodruff, D P (20 January 2015). "Molecular orbital tomography for adsorbed molecules: is a correct description of the final state really unimportant?". New Journal of Physics 17 (1): 013033. doi:10.1088/1367-2630/17/1/013033. Bibcode: 2015NJPh...17a3033B.
- ↑ Dauth, M.; Graus, M.; Schelter, I.; Wießner, M.; Schöll, A.; Reinert, F.; Kümmel, S. (24 October 2016). "Perpendicular Emission, Dichroism, and Energy Dependence in Angle-Resolved Photoemission: The Importance of The Final State". Physical Review Letters 117 (18): 183001. doi:10.1103/PhysRevLett.117.183001. PMID 27834988. Bibcode: 2016PhRvL.117r3001D.
- ↑ Haag, Norman; Lüftner, Daniel; Haag, Florian; Seidel, Johannes; Kelly, Leah L.; Zamborlini, Giovanni; Jugovac, Matteo; Feyer, Vitaliy et al. (21 April 2020). "Signatures of an atomic crystal in the band structure of a C 60 thin film". Physical Review B 101 (16): 165422. doi:10.1103/PhysRevB.101.165422. Bibcode: 2020PhRvB.101p5422H.
- ↑ Egger, Larissa; Kollmann, Bernd; Hurdax, Philipp; Lüftner, Daniel; Yang, Xiaosheng; Weiss, Simon; Gottwald, Alexander; Richter, Mathias et al. (4 April 2019). "Can photoemission tomography be useful for small, strongly-interacting adsorbate systems?". New Journal of Physics 21 (4): 043003. doi:10.1088/1367-2630/ab0781. Bibcode: 2019NJPh...21d3003E.
- ↑ Schüler, Michael; De Giovannini, Umberto; Hübener, Hannes; Rubio, Angel; Sentef, Michael A.; Werner, Philipp (February 2020). "Local Berry curvature signatures in dichroic angle-resolved photoelectron spectroscopy from two-dimensional materials". Science Advances 6 (9): eaay2730. doi:10.1126/sciadv.aay2730. PMID 32158939. Bibcode: 2020SciA....6.2730S.
- ↑ Wopperer, Philipp; De Giovannini, Umberto; Rubio, Angel (March 2017). "Efficient and accurate modeling of electron photoemission in nanostructures with TDDFT". The European Physical Journal B 90 (3): 51. doi:10.1140/epjb/e2017-70548-3. Bibcode: 2017EPJB...90...51W.
- ↑ Komiya, N.; Hatada, K.; Ota, F.; Krüger, P.; Fujikawa, T.; Niki, K. (October 2017). "Multiple scattering approach to photoemission from the highest occupied molecular orbital of pentacene". Journal of Electron Spectroscopy and Related Phenomena 220: 21–24. doi:10.1016/j.elspec.2017.04.002. https://hal-univ-rennes1.archives-ouvertes.fr/hal-01638858/file/Komiya%20et%20al.%20-%20Multiple%20scattering%20approach%20to%20photoemission%20from.pdf.
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