Momentum-resolved Electron Energy Loss Spectroscopy (qEELS) for Quantum Plasmonics and Metamaterials Prashant Shekhar1, Vaibhav Gaind2, Marek Malac3,4 , Ray Egerton3,4 , Zubin Jacob1* 3 National Institute of Nanotechnology, Edmonton, Alberta, Canada Department of Physics, University of Alberta, Edmonton, Alberta, Canada 1 Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta T6G 2V4, Canada 2 KLA-Tencore, California, USA *
[email protected] 2
Abstract: We report on experimental and theoretical results on EELS from 12nm single-crystal gold films. Our results show that momentum resolution of the electrons gives insight into signatures of non-locality and quantum nature of the excitations. OCIS codes: (160.1190) Anisotropic optical materials, (160.3918) Metamaterials, (160.4236) Nanomaterials
The quantum nature of excitations in plasmonics and the control of non-classical light with metamaterials has opened the doors new applications of nanophotonics [1]. Recent experiments with electron energy loss spectroscopy on plasmonic nanoparticles have shown the effect of electron wavefunction quantization in disagreement with the characteristic predictions of plasmonic spectra using a local dielectric constant model [1]. Effects such as electron tunneling between dimers also require a quantum picture of plasmons [2]. However, there also exists the question whether a non-local model for the dielectric constant can capture these effects without invoking electron quantization in nanoparticles, dimers and thin films [2]. In this paper, we show that along with the energy loss of electrons, the momentum transferred to the plasmons plays a key role in extracting the entire energy-momentum dispersion relation revealing deviations from local dielectric response. We compare experimental results of momentum resolved electron energy loss spectroscopy (q-EELS) to theoretical results obtained by a local dielectric model for thin film gold samples. Our results show deviation from the local dielectric model in the momentum-energy dispersion relation of gold surface plasmons. The developed q-EELS approach can map the non-local as well as quantum nature of collective electronphoton excitations in metallic nanostructures. The schematic shown in figure 1(a) outlines the experimental setup used to acquire the high quality valence EEL spectra for a 12 nm gold film. The single crystal thin film for EELS is obtained from Tedpella. The measurements were conducted with a Hitachi HF 3300 TEM/STEM equipped with a cold field emission gun (CFEG) and a standard Gatan Image Filter (GIF) TridiemTM. The instrument is operated with an incident beam energy of 300 keV in angle-resolved mode to obtain the momentum-resolved EEL spectrum (q-EELS). Many electrons passing through the film experience zero energy loss, resulting in a zero loss peak (ZLP). Diffracted electrons have a characteristic energy loss as well as shifts in their momentum, designated by the momentum vector ( q ). Figure 1(b) shows the EEL spectra for the gold film integrated over a small selected range of angles (or ( q )). The off-axis angular ranges show clear features in the resultant spectra down to very low energies (~1.5eV). Momentum-resolved EELs gives significant insight into the nature of the excitations in a medium by determining the material’s dispersive properties [3] The EEL spectra for the 12 nm gold film was calculated by implementing the method outlined by Bolton and Chen [4]. EEL spectra were calculated for an electron beam with velocity v incident on a thin gold film using a Hertz vector formulation of Maxwell’s equations. Bolton and Chen’s approach allows the calculation of bulk and boundary electron energy loss separately. Our theoretical calculations show an excellent agreement with the 2.5 eV energy loss peak in experiment which occurs at the gold interband transition. It is primarily dominated by losses at the gold-air boundary indicating also the excitation of surface plasmons. Figure 1(d) shows the local photonic density of states (LDOS) close to the surface of a 12 nm gold film. The LDOS is resolved by the in-plane wavevector (kx) as well as the energy, showing the dispersion of the modes present in the film. The gold film used is based on a modified Drude model as seen in [6] which takes into account interband transitions at ~2.6eV and ~3.7eV with adjusted Lorentzian Oscillators. Using this model, one can see the distinct feature arising in the range of ~2.5eV matching the experimental and numerical results seen in figure 1(b) and 1(c) [5]. The interband transitions of gold modify the surface plasmon polariton resonance away from the normal Re(ε)= -1 condition [6] The key effect captured in experiment is the large shift in the energy loss peak at different wavevectors (2.4 ev to 2.6 ev) which cannot be accounted by the local EEL model which predicts shifts of the order of 0.05 ev for similar momenta. This enhanced shift in energy loss peaks with momentum is an indicator of the role of either non-locality or quantum plasmonics. The momentum resolved EEL approach we use can lead to a quantitative comparison of
non-local vs. quantum effects in plasmonics. It can also be adopted to hyperbolic metamaterials to understand the nature of the photonic density of states singularity arising from unique high-k waves.
Fig 1: (a) Schematic of Hitachi HF 3300 TEM/STEM used to obtain momentum-resolved EELS (q-EELS) spectrum for a 12 nm single crystal gold film. An incident energy of 300 keV was used with the beam oriented along the [001] direction of the sample. Electrons are passed through the sample and are diffracted with a characteristic change in momentum and subsequent energy loss. The measured spectra was obtained with a cold field emission gun source and a Gatan Image Filter TridiemTM. (b) Experimental results for angle-integrated spectra at small selected ranges of the momentum vector (q). Selected angular ranges that are off-axis show clear features in the resultant spectra down to 1.5 eV. Features below 1 eV to are superimposed on the strong the zero loss peak (ZLP) and very narrow forward peaked excitations. The intensity is normalized such that the zero loss peak is scaled to 1 at q= 0 nm-1. Appropriate scaling factors are used to show the nature of the features in the spectra when normalized with the ZLP. The experimental data shows that the higher angle (larger momentum) peak occurs at a higher energy. The difference in energy between the peaks is ~0.2 eV. (c) Numerically computed EELS spectra for experimental data from Johnson and Christy [5]. Numerical simulations using the method outlined by Bolton and Chen [4] show a similar feature at ~2.5 eV energy loss as seen in the experimentally obtained values in (b). The numerical data shows much less shift in energy between the higher angle and lower angle data (~0.05eV). The analytical surface loss function ( Im[1/(1+ε(ω)gold)] ) also shown in (c) has similar behavior when using the experimental values of Johnson and Christy . (d) Local Density of States (LDOS) for the 12 nm gold film based on a modified Drude model accounting for interband transitions [5]. The peak in the LDOS using the model matches well with experimental and numerical data seen in (b) and (c). Interband transitions of gold play a significant role in moving the EELS peak away from the Re( )=-1 resonance condition. The large shift in the experimental EELS peak at higher momenta as compared to the theoretical prediction is indicative of non-local or quantum effects warranting further investigation. [1] Z. Jacob, MRS Bulletin, vol. 37, no.8, (2012); J. Scholl et. al. Nature 483, 421-427 (2012) [2] J. Zuloaga et. al. Nano Lett. (2009), 9, 887-891; G. Toscano et. al. Opt. Exp. Vol. 20, (2012) [3] P.A. Midgley, Ultramicroscopy 76, p. 91 (1999) [4] J. P. R. Bolton and M. Chen, J. of Phys.:Condens. Matter, 7, 1995, 3373-3387 [5] P. G. Etchegoin et. al., J. Chem. Phys. 125, 164705 (2006), P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972) [6] A. Pulisciano et. al., Applied Phys. Letters 93, 213109(2008), A. Politano et.al. Gold Bulletin, 42(3), 195–200(2009) [7] We thank Yoshifumi Taniguchi from Hitachi Corporation for help with experiments.
