The world as we know it is shaped by electronic states. Be it optical, electrical, or magnetic properties, thermal conductivity, or chemical bonding: almost all macroscopic properties can be traced back to the electronic states on the nanoscale. It is all the more surprising that they remained mostly elusive from an experimental perspective so far.

In this work, we show that the mapping of transitions between electronic states in real space with Ångström resolution is indeed possible using state-of-the-art TEM and EELS [1]. As a model system, we used a 20 nm thick rutile sample oriented in [0 0 1] direction. In this system, the Ti L_2,3 edge splits into contributions from states with e_g and t_2g symmetry, respectively. Fig. 1 shows the experimental L₂-e_g map extracted from the dataset acquired on a double Cs-corrected FEI Titan cubed microscope operated at 80 keV after drift-correction and averaging over 12 unit cells. An asymmetry that is rotated by 90° for nearest neighbors is clearly visible that is caused by the peculiar shape of the e_g states as shown in the charge density distribution. Furthermore, simulations using the multislice [2] and mixed dynamic form factor [3] approaches were performed. As is evident from fig. 1, the simulations are in excellent agreement with the experimental data.

One crucial prerequisite for such asymmetries to appear lies in the local environment of the atom that is being probed [4]. If the atomic site is invariant under a high symmetry point group, many states will be degenerate and their contributions to the scattered intensity will add up to a circularly symmetric map according to Unsöld’s theorem [5]. A prototypical example of this for p-states is shown in fig. 2. Only if the point group symmetry is low enough, the degeneracy is lifted and transitions to individual states can be mapped by selecting a suitable energy window.

This work shows that the mapping of individual electronic states is possible with widely used tools such as TEM and EELS. Thus, it paves the way for exciting new applications such as probing defect states at surfaces and interfaces that could revolutionize material science, as well as our experimental grasp on electronic properties and bonds on the atomic scale.

Acknowledgements: Support by the Austrian Science Fund (FWF), grant nrs. I543-N20, SFB F45 FOXSI, J3732-N27, the ERC under the EU’s 7^th Framework, grant nr. 306447, and the Natural Sciences and Engineering Research Council of Canada (NSERC) is acknowledged.

[1] Löffler et al., submitted
[2] Kirkland, “Electron Energy-Loss Spectroscopy in the Electron Microscope”, Plenum Press 1996
[3] Löffler et al., Ultramicroscopy 131 (2013) 39
[4] Löffler et al., in preparation
[5] Tinkham, “Group Theory and Quantum Mechanics”, Dover Publications 2003

Figures:

Fig. 1: A) Three-dimensional charge density around the central Ti atom in the rutile unit cell. B) Experimental spectrum with individual eg and t2g components. C) Charge density projected along [0 0 1]. D) Exp. data. E) same as D after Gaussian smoothing. F) Sim. image. G) Same as F with added noise. H) Same as G after Gaussian smoothing. Scale bars: 5 Å.

Fig. 2: Examples of calculated intensity distributions for px and py orbitals. If they are non-degenerate, a suitable energy window can be used to distinguish the two components and maps corresponding to each one can be extracted. If they are degenerate, however, only their sum can be recorded, resulting in a circularly symmetric map.

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