Observation of Crystalline Nonlocal Volume Plasmon Waves

Abstract

In plasmonics, nonlocal effects arise when the material response to optical excitations is strongly dependent on the spatial correlations of the excitation. It is well known that a classical free electron gas system supports local Drude volume plasmon waves. Whereas a compressible quantum electron gas system sustains hydrodynamic volume plasmons with nonlocal dispersion isotropic across all high-symmetry directions. Here, distinct from Drude and Hydrodynamic plasmon waves, we present the first observation of crystalline nonlocal volume plasmon waves. We use transmission-based momentum-resolved electron energy loss spectroscopy to measure the volume plasmon dispersion of silicon along all the fundamental symmetry axes, up to high momentum values ($q \sim 0.7$ reciprocal lattice units). We show that crystalline nonlocal plasmon waves have a prominent anisotropic dispersion with higher curvature along the light-mass ($ΓK$ \& $ΓL$) axes, compared to the heavy-mass ($ΓX$) axis. We unveil the origin of this phenomenon by experimentally extracting the anisotropic Fermi velocities of silicon. Our work highlights an exquisite nonlocality-induced anisotropy of volume plasmon waves, providing pathways for probing many-body quantum effects at extreme momenta.

Figures (3)

• Figure 1: (a) Drude plasmons are supported by a system of free electron gas in a material, which can move randomly and collide with heavier nuclei. Drude plasmons display a zero-curvature dispersion with momentum $q$. (b) Hydrodynamic Plasmons are supported by a compressible electron gas system, obeying Pauli’s exclusion principle of electrons. Hydrodynamic plasmons exhibit an isotropic parabolic $q$-dependency across all symmetry axes. (c) Here, we introduce a class of crystalline nonlocal volume plasmon waves, which display both nonlocal and highly anisotropic dispersion. Curvature of the crystalline nonlocal plasmons strongly depends on the high-symmetry axes of a material.
• Figure 2: (a) Schematic illustration of a parallel electron-beam incident on the crystalline planes of silicon along (a) $\left[100\right]$ ($\Gamma X$), (b) $\left[111\right]$ ($\Gamma L$), and $\left[110\right]$ ($\Gamma K$) symmetry axes. (d, e, f) The momentum-resolved EELS signal for three different high symmetry directions in silicon. The signal intensity is visualized on a color axis on a logarithmic scale; the extracted maxima for the $q$-EELS signal correspond to the energy of crystalline nonlocal volume plasmon waves, represented by black dots.
• Figure 3: Comparison between theory and experiments for dispersion of crystalline nonlocal volume plasmon waves. (a) Experimental dispersion obtained by fitting the $q$-EELS data with the isotropic electron gas model. (b) Theoretical calculations were performed within the crystalline nonlocal framework implemented in PicoMax software along $\Gamma {\rm X}$, $\Gamma {\rm L}$, and $\Gamma {\rm K}$ high-symmetry directions.