Plasmonically assisted channels of photoemission from metals

We analyze recently measured nonlinear photoemission spectra from Ag surfaces that reveal resonances whose energies do not scale with the applied photon energy but stay pinned to multiples of bulk plasmon energy $\hslash {\omega }_p$ above the Fermi level. To elucidate these unexpected and peculiar features we investigate the spectra of plasmons generated in a solid by the optically pumped electronic polarization and their effect on photoemission. By combining quadratic response formalism for calculations of photoemission yield, a nonperturbative approach to inelastic electron scattering, and first-principles calculations for the electronic structure, we demonstrate the dependence of probability amplitude for single- and multiplasmon excitations on the basic parameters characterizing the photon pulse and the system. The resulting multiexcitation spectrum evolves towards a truncated plasmonic coherent state. Analogous concept is extrapolated to interpret plasmon generation by multiphoton excited electronic polarization. Based on this we elaborate a scenario that the thus created real plasmons act as supplementary frequency-locked pump field for non-Einsteinian plasmonically assisted channels of photoemission from metals. The established paradigm enables assignment and assessment of the observed linear $\hslash {\omega }_p$ and nonlinear $2\hslash {\omega }_p$ electron yields from Ag. Such effects may be exploited for selective filtering of optical energy conversion in electronic systems.

Figure 1

(a) Selection of 2PP spectra from three low index surfaces of Ag for excitation field energies $\hslash {\omega }_x$ in the ENZ region where the real part of Ag dielectric function $\varepsilon \left(\omega \right)$ is near zero [5]. The manifolds of all the dots shown above the four representative plasmonic peak maxima make the $2\hslash {\omega }_p$ scaling curves in (b) below. (b) Dependence of the measured final state energies of 2PP electron yields from Ag on the variation of energy $\hslash {\omega }_x$. The data shown by triangles were recorded in the constant inital state mode (CIS) involving the surface state (SS) and the first image potential state (IP1) on Ag(111), the IP1 and the Fermi level $E_F$ on Ag(100), and $E_F$ on Ag(110). They exhibit the standard scaling of 2PP yield energy with $2\hslash {\omega }_x$ in emission from SS and $E_F$, and with $\hslash {\omega }_x$ in emission from IP1. In contrast to this, there also appear yields with energy $\sim 2\hslash {\omega }_p$ above $E_F$ (dots) which do not scale with the multiples of the radiation field energy $p\hslash {\omega }_x$. This $2\hslash {\omega }_p$ emission extends far beyond the ENZ region denoted by vertical shading.

Figure 2

Diagrammatic representation of the sequence of excitation events (9) linearly induced in the electron-plasmon system by pulsed EM field. Full red, dashed blue and purple wiggly lines denote electron, hole and plasmon propagation, respectively. Blue and red circles denote matrix elements $V_{\overline{{\mathbf{k}}_{\mathbf{1}}},{\mathbf{k}}_{\mathbf{1}}}$ and $W_{{\mathbf{k}}_{\mathbf{1}},{\mathbf{k}}_{\mathbf{2}}}$ defined in Eqs. (7) and (8), respectively. The difference of quasiparticle energies of the secondary electron and hole arising from real plasmon absorption is $\mathrm{\Delta }\epsilon \simeq \hslash {\omega }_p$.

Figure 3

Diagrammatic representation of linearly induced plasmon generation processes described by (10) with $\left|\mathrm{\Psi }\right(t)\rangle$ defined in (15) and leading to (18). Meaning of the symbols is same as in Fig. 2. The time evolution after the application of pulsed perturbation (8) is governed by the Hamiltonian (4). Plasmon counting implemented through the operator $\delta (\varepsilon -H_0^{\mathrm{pl}})$ is denoted symbolically by $\mathcal{S}\left(\varepsilon \right)$. Real multiplasmon excitations may occur for excited quasiparticle energies exceeding the multiexcitation threshold $n{\omega }_p$.

Figure 4

Diagrammatic representation of multiplasmon generation driven by electronic polarizarion excited by double action of the laser pulse. All symbols have the same meaning as in Fig. 3. Electron induced plasmon generation may take place either in the intermediate and final electron states (upper panel), or in the final state of the two photon-driven process (lower panel).

Figure 5

Schematic illustration of the temporal behavior of the external EM field amplitude $\mathcal{E}({\omega }_x,t^{\prime },{\sigma }_x)$ (red curve) and $V^{\text{e-pl}}\left(t^{\prime }\right)=V^{\text{e-pl}}{e}^{-\eta |t^{\prime }|}$ (blue curve, $\eta \to 0^{+}$) appropriate to the scattering boundary conditions for preparation of electrons in excited states $|{\mathbf{k}}_{\mathbf{1}}\rangle$ above the Fermi level and their subsequent interaction with plasmons. The horizontal axis illustrates the hierarchy of times established in expression (27).

Figure 6

Diagrammatic illustration of perturbative contributions to plasmon pumped electron yield following the procedures outlined in Refs. [10, 41]. Positive time is running upward. Sketch of Keldysh diagram for plasmonically induced electron yield in (a) quadratic response (analogous to 1PP yield), and in (b) quartic response (analogous to 2PP yield). Full red and dashed blue lines denote propagation of electrons and holes above and below the Fermi level, respectively. Purple wiggly lines indicate plasmons giving rise to electron excitation via electron-plasmon interaction matrix elements $V_{{\overline{\mathbf{k}}}_{\mathbf{1}},{\mathbf{k}}_{\mathbf{1}}}^{\mathbf{q}}$ denoted by blue dots. Vertical full black arrows symbolize time dependent electron yield captured by the electron counter. In the processes depicted in panels (a) and (b), the wiggly lines can be any real plasmon emanating from the primary photoinduced polarization vertices shown in Figs. 3 and 4 or their complex conjugates. (c) Illustration of the mechanism of plasmonic filter for electron yield obtained through temporal juxtaposition of the processes depicted in Fig. 3 and the herein panel (a). This particular yield channel is open for $\hslash {\omega }_x>\hslash {\omega }_p>\varphi$.

Figure 7

(a) First Brillouin zone of fcc Ag lattice. Green dots denote the high symmetry points of the zone and the red rectangle its section in the (110) plane. (b) Band structure of Ag along the high symmetry points in the first Brillouin zone computed within the $GW$ framework. The most prominent interband transitions (ib) in the energy range $\sim 4$ eV are denoted by vertical arrows. Earlier $GW$ results (circles) are taken from Ref. [32]. Experimental points are from Refs. [75] and [76]. (c) Real and imaginary parts of the RPA dielectric function for $\mathbf{q}=\mathbf{0}$ calculated for the band structure displayed in (a). (d) Comparison of the thereof calculated loss function with the experimental data from Ref. [77]. (e) Bulk plasmon dispersion in Ag calculated in RPA for the band structure shown in (b). Circles and diamonds denote experimental points of plasmon dispersion [78, 79].

Figure 8

(a) Electronic band structure along the high symmetry points of the first Brillouin zone in Ag. The magnitude of the band- and momentum-resolved photoexcited quasiparticle decay rate due to one plasmon emission ${\mathrm{\Gamma }}_{\mathbf{k}}$, Eq. (35), is quantified by color bar in the inset. (b) Angle-integrated ${\mathrm{\Gamma }}_{\mathbf{k}}$ as a function of energy. Note the threshold at $\sim E_F+\hslash {\omega }_p$ and very low values below $E_F$ and the $d$-band edge. Yellow line denotes the result of analogous $GW$-based calculation which incorporates the full linear electronic density response of the system.

Figure 9

(a) Band structure of Ag with electronic transitions ib2 and ib1 contributing to CIS resolved rates ${\gamma }_{\mathbf{k}}^{\mathrm{CIS}\left(\hslash {\omega }_p\right)}$ [Eq. (44)] for plasmon-induced hot electron generation processes sketched in Fig. 6. The magnitudes of CIS and CFS transition rates are quantified by color bars in the insets. (b) Angle-integrated ${\gamma }_{\mathbf{k}}^{\mathrm{CIS}\left(\hslash {\omega }_p\right)}$ shown on the energy scale. The CIS amplitudes for transitions from the occupied states closely below $E_F$ to unoccupied bands above $E_F$ are dominant. (c) Electronic transitions ib2 and ib1 contributing to CFS resolved plasmon-induced hot electron generation rates ${\gamma }_{{\mathbf{k}}^{\prime }}^{\mathrm{CFS}\left(\hslash {\omega }_p\right)}$ [Eq. (45)] from all initial to a fixed final $\mathbf{k}$ state at $\sim E_F$ and final ${\mathbf{k}}^{\prime }$ state at $\sim E_F+\hslash {\omega }_p$, respectively. (d) Angle integrated ${\gamma }_{{\mathbf{k}}^{\prime }}^{\mathrm{CFS}\left(\hslash {\omega }_p\right)}$ shown on the energy scale. Here the dominant amplitudes are for transitions to $E_F$ and $\sim E_F+\hslash {\omega }_p$.

Figure 10

(a) Band structure of Ag across the (110)-surface projected bulk Brillouin zone showing the intensities of surface enhanced ${\gamma }_{{\mathbf{k}}^{\prime }}^{\mathrm{CFS}\left(\hslash {\omega }_p\right)}$ quantified by the color bar. (b) Angle integrated surface enhanced ${\gamma }_{{\mathbf{k}}^{\prime }}^{\mathrm{CFS}\left(\hslash {\omega }_p\right)}$ from (a) shown on the energy scale. Shaded area spans the width of $E_F\to E_F+\hslash {\omega }_p$ plasmon-induced electronic transitions. Notable is the surface induced enhancement of all displayed quantities relative to the purely bulk induced values shown in Fig. 9. (c) Isoenergetic distribution of surface enhanced ${\gamma }_{{\mathbf{k}}^{\prime }}^{\mathrm{CFS}\left(\hslash {\omega }_p\right)}$ averaged around ${\epsilon }_{{\mathbf{k}}^{\prime }}\simeq E_F+\hslash {\omega }_p$ [96] and shown across the projected BZ ($a=4.06\AA$). The highest intensities are around $\overline{\mathrm{\Gamma }}, \overline{Y}$, and $\overline{S}$ points.

Figure 11

(a) Band structure of Ag projected onto (110) surface showing the points with maximal intensities of ${\gamma }_{{\mathbf{k}}^{\prime \prime }}^{\mathrm{CFS}\left(2\hslash {\omega }_p\right)}$ (see color bar for computed values). The second plasmon-induced electronic transitions start from resonant intermediate states lying in the orange shaded energy window at $\sim E_F+\hslash {\omega }_p$ that have been occupied in one-plasmon absorption processes illustrated in panels (c) and (d) of Fig. 10. (b) Angle integrated ${\gamma }_{{\mathbf{k}}^{\prime \prime }}^{\mathrm{CFS}\left(2\hslash {\omega }_p\right)}$ from (a) shown on the energy scale. Shaded area straddles the width of plasmonically-induced electronic transitions to $\sim E_F+2\hslash {\omega }_p$. (c) Isoenergetic distribution of ${\gamma }_{{\mathbf{k}}^{\prime \prime }}^{\mathrm{CFS}\left(2\hslash {\omega }_p\right)}$ averaged around ${\epsilon }_{{\mathbf{k}}^{\prime \prime }}\simeq E_F+2\hslash {\omega }_p$ [96] and shown across the projected BZ ($a=4.06\AA$). The accumulation of highest intensities is in the vicinity of $\overline{\mathrm{\Gamma }}$ and $\overline{Y}$ points.

Figure 12

(a) Distribution of the magnitudes of resonant two plasmon-induced electron excitation rates (50) averaged around ${\epsilon }_{{\mathbf{k}}^{\prime \prime }}\simeq E_F+2\hslash {\omega }_p$ [96] and shown across the (110) surface-projected bulk BZ of Ag (see color bar for absolute values). (b) and (c)] Histograms of the same data along the blue and red rectangular paths marked in (a).