## Section snippets

## Rationale for plasmonically assisted electron emission model

Since the advent of modern spectroscopic techniques and the development of quantum many-body theory a great deal of attention and work have been focused on the various aspects of mutual interactions of electromagnetic waves, plasmonic fields and electrons in inhomogeneously structured dielectric media. The plasmonic phenomena arising from the interplay of these interactions have aroused considerable and unabated interest, partly because of their potential roles in communication and computing

## Model description of excitation of electron–plasmon system by EM field

Symmetry selection rules do not permit direct excitation of longitudinal bulk plasmons in metals by purely transversely polarized EM fields. Therefore, the intriguing “non-Einsteinian” photoemission modes described in the Introduction call for interpretations going beyond the sole effect of electron excitation by dielectrically screened EM fields [5], [57], [58], [59] which oscillate with the frequencies of the applied external fields. We introduce a novel paradigm which involves real plasmon

## Perturbative treatment of bulk plasmon-assisted electron emission from metals

Formation and structure of plasmonic coherent states (9) which can drive plasmoemission are controlled by the saturated mode occupation amplitudes $\left\{{\alpha}_{{\mathbf{q}}^{\prime}}\right\}$. As they stand, these amplitudes are ${\mathbf{q}}^{\prime}$-resolved and electronic band ($\mathbf{k}$-band) integrated quantities retrievable from expressions (23) and (28) of Ref. [50]. As such they do not provide much insight into the dynamics of intra- and interband electronic transitions giving rise to plasmon excitations. Hence, in the following we shall prefer to

## Model description of surface plasmon-induced electron excitations at metal surfaces

Low index surfaces of some metals may exhibit surface projected band gaps. This means that there are no electrons with energies within the gap that can propagate normal to the surface. The effective one-electron potential at such surfaces can support the set of quasi-two-dimensional (Q2D) surface state (SS) and image potential state (IP) bands. Localization of SS and IP electrons in the direction perpendicular to the surface is only few atomic radii over the image potential well whereas their

## Quantum formulation in the scattering boundary conditions

The use of the velocity gauge in the calculations of transition amplitudes usually provides faster converging results. Hence, we rewrite (43), (44) starting from integral representation of the evolution operator to obtain the SBC limit of the transition amplitude in either picture $\underset{SBC}{lim}{T}_{f,i}^{vel}(t,{t}_{0})=-\frac{i}{\u0127}{\int}_{{t}_{0}}^{t}d\tau \u3008\u3008{\text{coh}}_{\mathrm{pu}},{\varphi}_{f},\left(\tau \right)||{V}_{S}^{\prime}{U}_{S}({H}^{\prime},\tau ,{t}_{0})||{\varphi}_{i},{\text{coh}}_{\mathrm{pu}},\left({t}_{0}\right)\u3009\u3009.$$=-\frac{i}{\u0127}{\int}_{{t}_{0}}^{t}d\tau \u3008\u3008{\text{coh}}_{\mathrm{pu}},{\varphi}_{f}||{V}_{I}^{\prime}\left(\tau \right){U}_{I}({H}^{\prime},\tau ,{t}_{0})||{\varphi}_{i},{\text{coh}}_{\mathrm{pu}},\u3009\u3009.$ Here ${V}_{S}^{\prime}={V}^{\prime}$ and ${V}_{I}^{\prime}\left(\tau \right)={e}^{i{H}_{0}\tau}{V}_{S}^{\prime}{e}^{-i{H}_{0}\tau}$ consistent with (42) vanishes in the remote past

## Theoretical framework

Depending on the boundary conditions specific to a particular problem, the Volkov ansatz-derived electron states may participate in emission or scattering processes as either initial, intermediate or final states [77], [83], [84], [85], [89], [93], [101]. The initial field-dressed band states are conventionally termed Floquet or Bloch-Floquet states whereas the final outgoing field-dressed electron states are designated Volkov states [89], [93], [101]. In the present problem of electron

## Discussion of plasmoemission from surface Floquet bands on $\mathbf{Ag}$(111)

Signatures of Floquet sidebands in plasmoemission rates predicted by the RHS of expression (84) largely depend on two factors. The first is related to the values of generalized Bessel functions ${J}_{n}(x,y)$ with parametric variables ${U}_{p}$, ${\beta}_{s}$ and ${Z}_{s}$ in relation to $\u0127{\omega}_{s}$ of the interacting electron-SP system. The second one pertains to the overall magnitude of (90) which determines the weight of each $\delta $-function in the sum over the Floquet band index $n$ in (84). Here we shall separately inspect the high

## Pump–probe picture of perturbative and nonperturbative plasmoemission from surface bands

The paradigmatic one-electron pump–probe picture of multiphoton photoemission ($m$PP) is intrinsically perturbative in that it describes amplitudes of $m$th order electron excitations induced by the repeated action of one or more external EM fields [76]. In this scenario absorption of photons can pump an electron over the ladder of real or virtual intermediate states until the absorption of probe photon(s) brings it to the real final outgoing wave state that supports the emission current obeying

## Summary and conclusions

In this work we have studied plasmonically induced electron excitation processes that can generate stationary electron currents emanating from metal surfaces. We have exploited a full analogy of these phenomena with standard photoemission to formulate perturbative and nonperturbative pictures of plasmonically induced electron yields. The relevant emission probabilities appear even order in the applied plasmon field, as it should be in the description of externally driven stationary quantities

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

The authors are grateful to M. Reutzel, A. Li and H. Petek for sharing the experimental data on multiphoton photoemission from Ag surfaces. D.N. acknowledges the financial support from the Croatian Science Foundation (Grant No. UIP-2019-04-6869).

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