Accessing Phonon Polaritons in Hyperbolic Crystals by Angle-Resolved Photoemission Spectroscopy

Recently studied hyperbolic materials host unique phonon-polariton (PP) modes. The ultrashort wavelengths of these modes, as well as their low damping, hold promise for extreme subdiffraction nanophotonics schemes. Polar hyperbolic materials such as hexagonal boron nitride can be used to realize long-range coupling between PP modes and extraneous charge degrees of freedom. The latter, in turn, can be used to control and probe PP modes. Here we analyze coupling between PP modes and plasmons in an adjacent graphene sheet, which opens the door to accessing PP modes by angle-resolved photoemission spectroscopy (ARPES). A rich structure in the graphene ARPES spectrum due to PP modes is predicted, providing a new probe of PP modes and their coupling to graphene plasmons.

Figure 1

Signatures of PP modes in the quasiparticle spectral function $\mathcal{A}(\mathit{k},\omega )$ of a doped graphene sheet placed over a hBN slab, obtained from Eqs. (3) and (6). The Fermi energy is positioned at $\omega =0$. The black linearly dispersing quasiparticle bands display a clear Dirac crossing labeled by (1). The broad spectral feature labeled by (2) is plasmaron excitation related to the emission of the hybrid plasmon mode with highest energy, see Eqs. (4), (5) and accompanying discussion. Its energy is derived from the highest plasmon resonance, labeled by the uppermost green dot in Fig. 2. Emission of Fabry-Pérot PP modes [see Fig. 2] gives rise to four dispersive satellite bands running parallel to the main quasiparticle bands (marked by red arrows). The feature (2) is mainly plasmonic, whereas the satellite bands are mostly due to PP modes. Parameters used: Fermi energy ${ϵ}_F=400\mathrm{meV}$, hBN slab thickness $d=60\mathrm{nm}$, ${\epsilon }_{\mathrm{a}}=1$ (vacuum), ${\epsilon }_b=3.9$ (${\mathrm{SiO}}_2$). The color bar refers to the values of $\hslash \mathcal{A}(\mathit{k},\omega )$ in eV.

Figure 2

(a) Dispersion relation ${\mathrm{\Omega }}_{\mathit{q}}$ of hybrid plasmon modes (solid lines) with parameters as in Fig. 1. The dashed line represents a Dirac plasmon [14]. Horizontal cyan areas denote the hBN reststrahlen bands. The gray-shaded area represents the intraband particle-hole continuum in graphene. Green dots mark the points where the hybrid plasmon group velocity equals $v_F$. (b) The quantity $-\mathrm{Im}[{\mathrm{\Sigma }}_{\lambda }(\mathit{k},\omega )]$ (in units of ${ϵ}_F$ and evaluated at $\mathit{k}=0$) is shown as a function of the rescaled frequency $\hslash \omega /{ϵ}_F$. Green vertical lines denote the values of $\hslash \omega /{ϵ}_F$ at which a hybrid plasmon peak is expected. The inset shows the vertical heterostructure analyzed in this work. (c) Same as in (b) but in the absence of $e\text{−}e$ dynamical screening; obtained by replacing $W_{\mathit{q},\omega }\to V_{\mathit{q},\omega }$ in Eq. (3). Note a polaronic peak at $\hslash \omega \approx -1.42{ϵ}_F$. Results for other values of ${ϵ}_F$ and $d$ can be found in [15].

Figure 3

(a) The quasiparticle spectral function $\mathcal{A}(\mathit{k},\omega )$, evaluated at $|\mathit{k}|={10}^{-3}{k}_F$, for $d=10\mathrm{nm}$. The other parameters are as in Fig. 1. (b) and (c) Dependence on the hBN slab thickness $d$ of the spectral function features highlighted in orange in (a). Different curves correspond to values of $d$ on a uniform mesh from $d=10$ to $d=60\mathrm{nm}$.