Plasmon Geometric Phase and Plasmon Hall Shift

The collective plasmonic modes of a metal comprise a simple pattern of oscillating charge density that yields enhanced light-matter interaction. Here we unveil that beneath this familiar facade plasmons possess a hidden internal structure that fundamentally alters its dynamics. In particular, we find that metals with nonzero Hall conductivity host plasmons with an intricate current density configuration that sharply departs from that of ordinary zero Hall conductivity metals. This nontrivial internal structure dramatically enriches the dynamics of plasmon propagation, enabling plasmon wave packets to acquire geometric phases as they scatter. At boundaries, these phases accumulate allowing plasmon waves that reflect off to experience a nonreciprocal parallel shift. This plasmon Hall shift, tunable by Hall conductivity as well as plasmon wavelength, displaces the incident and reflected plasmon trajectories and can be readily probed by near-field photonics techniques. Anomalous plasmon geometric phases dramatically enrich the nanophotonics toolbox, and yield radical new means for directing plasmonic beams.

Popular Summary

Much like photons are quantized oscillations of electromagnetic fields, plasmons are the collective oscillations of electrons and are found in all metals. Plasmons are essential in squeezing and enhancing the interaction between light and matter on the nanoscale, such as in bio-imaging, on-chip communication, and photodetection. There is intense interest in manipulating, exploiting, and probing plasmons in a variety of materials. However, on a fundamental physics level, plasmons in simple metals are typically thought to be devoid of structure, with a vanilla behavior that is well described by their dispersion and lifetime. We show that this naive perspective is far from complete. Our theoretical work demonstrates that plasmons in two-dimensional materials can possess hidden internal structure, which dramatically alters their dynamics and physical properties, such as scattering.

Deep subwavelength plasmons have wavelengths that are greatly compressed compared to the free-space wavelength of light. Conventionally, these plasmons can be described by their charge density and electric fields alone. Our results point to additional internal structure that comprises the local charge current density configuration in a metal, which can take on an intricate, nontrivial spatial pattern when a magnetic field is applied. When these plasmons scatter, their internal structure allows them to pick up nontrivial, tunable geometric phase shifts. This structure can even shift the trajectory by multiple plasmon wavelengths when the plasmon reflects off a boundary.

The internal structure of plasmons reveals uncharted territory for plasmonics, with new ways to manipulate their trajectories and a new playground in which to explore the geometry of quasiparticles.

Figure 1

Plasmon dispersion and internal current density texture. (a),(b) Current density (vector) fields $\mathbf{j}(\mathbf{r})$ configuration for plasmons with ${\sigma }_{xy}=0$ and ${\sigma }_{xy}\ne 0$, respectively. Note that while electric potential ${\varphi }_{\mathbf{E}}(\mathbf{r})$ take on the same longitudinal electric profile, $\mathbf{j}(\mathbf{r})$ adopts a completely different texture. Plasmon dispersion for (c) conventional plasmon (${\omega }_c=0$), and (d) magnetoplasmon modes (${\omega }_c\ne 0$) in a two-dimensional metal [Eq. (5)], both in the nonretarded limit. Current density pseudospinor [Eq. (7)] for ${\omega }_c>0$ exhibits a nontrivial canted texture; when ${\omega }_c=0$, pseudospinor winds only in the $(s_1,s_3)$ plane. Blue band is a complex-conjugate copy of primary plasmon band (see text).

Figure 2

Plasmon geometric phase. Phase shift between incident and reflected plane waves scattered off a boundary at different incident angles and relative magnetic field strength obtained from Eq. (11). While geometric phase shifts appear in both the current density and electric field, here we show the phase shift for (longitudinal) electric field along the directions $-{\mathbf{q}}^i$ (${\mathbf{q}}^r$) away from the interface. Lines from purple (closest to the $x$ axis) to red (farthest away from the $x$ axis) correspond to incident angles ${\varphi }_n=(\pi /2)(n/10)$ with $n$ integers in the range $n\in \{0,\dots ,9\}$. Upper inset: Schematic for the incident and reflected plane wave. Lower inset: Electric field $E^i$ ($E^r$) of the incident (reflected) plane wave exhibits a nontrivial shift $\rho$ tunable by Hall conductivity (magnetic field).

Figure 3

Plasmon Hall shift. (a),(b) Schematic of plasmon Hall shift between the incident and reflected wave packets with incident angle ${\varphi }_0$. The Hall shift $\mathrm{\Delta }{y}_H$ is nonzero when ${\sigma }_{xy}\ne 0$. (c) Plasmon Hall shift for 2D magnetoplasmons (in units of inverse wave vector $q^{-1}$) from Eq. (16) at various incident angles ${\varphi }_0$ can be tuned by the Hall coupling $\eta ={\omega }_c/{\omega }_q$. Lines from purple (closest to the line $\mathrm{\Delta }{y}_H=0$) to red (farthest away from the line $\mathrm{\Delta }{y}_H=0$) correspond to incident angles ${\varphi }_n=(\pi /2)(n/6)$ with $n$ integers in the range $n\in \{0,\dots ,5\}$. The upper inset displays the pseuodospinor orientation on a bulk plasmon isofrequency contour highlighting the different orientations between incident and reflected waves. Lower inset: Contour plot of Hall shift as a function of Hall coupling and ${\varphi }_0$. We have used $\delta =\pi /20$. (d),(e) Numerically computed electric field intensity $[\mathrm{Re}{\mathbf{E}}^i(\mathbf{r})+\mathrm{Re}{\mathbf{E}}^r(\mathbf{r}){]}^2$ from Eq. (13), without and with magnetic field (at ${\eta }_q=1$ limit), for an incident wave with a Gaussian distribution and fixed energy. The width of the wave is $\pi /q$ and the incident angle is $\pi /3$, with a Hall shift $4/q$. Red and blue lines are beam trajectories obtained from Eq. (14).

Figure 4

Nonreciprocal plasmon filter. (a), (b) Plasmon Hall shift displacement at a fixed $\eta ={\omega }_c/{\omega }_q$ are identical for beams with incident angle ${\varphi }_0$ and $-{\varphi }_0$. Gray regions are areas inside which plasmons damp quickly; see text. For plasmon beams incident from the left-hand (right-hand) side, the reflected beam can bypass the damping region when $\eta$ is negative (positive); these reflected beams can be collected at an all-electrical plasmon detector stationed close by ($p\text{−}n$ junction thermoelectric detector [35]). In contrast, plasmon beams incident from the right-hand (left-hand) side pass through the damping region and are damped when $\eta$ is negative (positive). Red (incident) and blue (reflected) lines are beam trajectories obtained from Eq. (14). Parameters used are as follows: the width of the wave is $\pi /q$, the same as Fig. 3, ${\varphi }_0=2\pi /5$, and ${\omega }_c/{\omega }_q=\pm \cot {\varphi }_0$ is taken to maximize the plasmon Hall shift.