Towards full surface Brillouin zone mapping by coherent multi-photon photoemission

We report a novel approach for coherent multi-photon photoemission band mapping of the entire Brillouin zone with infrared light that is readily implemented in a laboratory setting. We excite a solid state material, Ag(110), with intense femtosecond laser pulses to excite higher-order multi-photon photoemission; angle-resolved electron spectroscopic acquisition records photoemission at large in-plane momenta involving optical transitions from the occupied to unoccupied bands of the sample that otherwise might remain hidden by the photoemission horizon. We propose this as a complementary ultrafast method to time- and angle-resolved two-color, e.g. infrared pump and extreme ultraviolet probe, photoemission spectroscopy, with the advantage of being able to measure and control the coherent electron dynamics.


Introduction
Time-and angle-resolved two-photon photoemission spectroscopy (TR-2PP) enables mapping the energy and momentum (k||, k⊥)-resolved electronic structure and dynamics of the occupied and unoccupied electronic bands of solids [1][2][3]. Using excitation frequencies from the infrared (IR) to the ultraviolet (UV) range, TR-2PP has been applied to a wide range of condensed matter systems ranging from pristine metals to complex materials and interfaces [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In TR-2PP spectroscopy, the photon energies ℏ of the pump and the probe laser pulses are chosen such that the pump excites the sample to a real or a virtual intermediate state and the probe induces further upward transition from the excited system to induce photoemission. For photoemission to occur, the combined excitations must impart sufficient energy for the excited electrons to overcome the work function for || near the surface and bulk Brillouin zone center (Γ ̅ -point). To access the entire Brillouin zone, however, photoelectrons must be excited to sufficiently high energies to overcome the photoemission horizon. The photoemission horizon refers to the kinetic energy in the surface parallel motion, || = ℏ 2 || 2 2 ⁄ , which cannot do work against the work function because || of electrons passing through a solidvacuum interface is conserved [18,19]. To induce two-or multi-photon photoemission (mPP), photoelectrons must absorb energy ℏ ≥ || + ( || ) + , where is the photon order, me is electron mass, ħ the reduced Planck's constant, and ( || ) the in-plane momentum dependent initial state binding energy. Therefore, to map out ( || ), i.e. the electronic structure and dynamics of unoccupied bands in full surface Brillouin zone, the experiment must overcome the photoemission horizon either by probing with higher ħ or with higher-order mPP (cf. Fig. 1).
While the unperturbed occupied electronic band structure spanning the full Brillouin zone is routinely measured with extreme ultraviolet (XUV) light available at synchrotron facilities or from gas discharge lamps, band mapping of the structure and dynamics of the unoccupied region requires wavelength tunability and ultrafast time resolution. Table-top high-harmonic generation (HHG) sources pumped by femtosecond laser amplifiers can supply ultrafast XUV light pulses with ħ>20eV energy at high repetition rates [20][21][22][23][24]; such laser systems have been applied to perform time-and angle-resolved photoelectron spectroscopy (TR-ARPES), where an intense infrared or optical pumppulse excites the electronic system, and an XUV-probe pulse interrogates its impact on the electronic band structure of the sample. Remarkable achievements include, for example, probing and switching of exotic properties defined by reduced dimensionality, topological protection, and strong correlation [21,[23][24][25][26][27][28][29][30][31][32].
In this article, we demonstrate a novel approach to probe deep regions of the Brillouin zone that does not require an XUV light source, and gives access to coherent dynamics at a solid surface. Instead of generating high energy photons in a separate non-linear medium, we take advantage of the nonlinear response of the sample to realize a complementary (coherent) approach for band mapping. We

Results and discussion
We elaborate the concept of Brillouin zone mapping by coherent mPP with the example of the pristine Ag(110) surface involving the following scenario [ Fig. 1 [33], scanning tunneling spectroscopy [34,35], surface second harmonic spectroscopy (SSHG) [36], inverse photoemission [37,38], and many-body theory [39]. In Fig. 1 In the following, we confirm the proposed concept experimentally. We measure energy-, and angle-resolved mPP spectra of the Ag(110) surface at room temperature aligned so that its ΓY ̅̅̅̅direction is in the optical plane; details on the ultra-high vacuum system and the optical setup have been reported elsewhere [40,41]. In short, we generate tunable ~20-30 fs laser pulses with a fluence of 1 mJ/cm 2 on the sample and a 1MHz repetition rate by a noncollinear optical parametric amplifier (NOPA), which is pumped by a Clark MXR fiber laser oscillator/amplifier system. p-polarized light excites the Ag(110) sample at a 45° angle of incidence with respect to the detection axis of the hemispherical electron analyzer. Angle-resolved photoemission spectra are acquired with a 2D delay line detector. The angular acceptance angle of the analyzer is 30°; to report mPP spectra over a broader photoemission angle range, the sample is rotated around the normal axis with respect to the optical plane. We report the final state photoelectron energy, Ef, relative to the Fermi level, EF. Figure 2 shows Ef(ky)-resolved mPP data obtained from the Ag(110) surface; the corresponding excitation diagram for excitation with ħ=1.73-eV photons is shown in Fig. 1(b). In green, we plot the  Fig. 1(b)]. The 4PP process is thus enhanced by the Sun ← Soc transition even though the Sun band does not appear as a distinct feature in the 4PP experiment in Fig. 2(a). The resonance between the Shockley surface bands is consistent with its enhancement of SSHG on the Ag(110) surface [36].
Thus, the occupied and unoccupied bands that are otherwise hidden below the photoemission horizon, become accessible in higher-order mPP when exploiting the non-linear response of the sample.
The mPP data presented in Fig. 2(a) thus demonstrates that the non-linear energy conversion process typically applied in separate non-linear media to generate light of higher frequency can similarly be directly excited in a sample and detected through the mPP process. To illustrate the resemblance of both concepts, in the following, we first frequency double the infrared laser pulses in a BBO crystal outside the photoemission apparatus to obtain 3.50-eV photons. Accordingly, || and the maximum reachable in-plane momentum range widen; the Y ̅ -point becomes accessible in a twophoton process [cf. blue circle Fig. 1(a), m=2]. In 2PP [ Fig. 2(b)], the Soc band minimum is detected at Ef 6.9 eV, corresponding to a binding energy of EB 0.1 eV with respect to EF, which is in agreement with the 4PP measurement. Accordingly, a slope of two is obtained for the plot of Ef for Soc in ħdependent measurements [ Fig. 3]. We note, that the additional strongly dispersive bands in Fig. 2(b), which are less prominent though detected in 4PP in Fig. 2(a), are not known and will be discussed in a separate manuscript.
We further point out that the concept of the photoemission horizon and thus the necessity of sufficiently large kinetic energy photoelectrons to probe large k|| is a pure solid-state effect arising from space-periodic arrangement of the lattice atoms. This makes, for example, above-threshold multiphoton photoemission (ATP) [19,40,41,43,44], a phenomenon accompanying mPP driven with IR frequencies, a two-dimensional problem. Conventionally, for mPP excited at the Γ ̅ -point, photoelectrons absorbing more photons than necessary to overcome the work function are treated as ATP. With increasing k||, however, an electron populating a final state may not have sufficient energy in the surface direction to overcome the photoemission horizon. In our mPP data reported for Ag(110), this becomes obvious in Fig. 2(a), where 3PP at the Γ ̅ -point is sufficient to be detected and thus the replica structure in 4PP can be attributed to ATP; by contrast, at the Y ̅ -point, 4PP is not an above threshold process but rather the lowest order of photoemission. These considerations stand in contrast to ATP from nanostructured materials, where band momentum integrated processes obscure the origin of the photoelectrons and thus their in-plane momentum information. Instead, the local field enhancements determine the photoexcitation physics [45]. Similarly, as well the highly non-linear excitation of atoms and molecules, where the concept of above threshold ionization was initially developed [46], is not affected by the considerations discussed here, because k|| is not a conserved quantity.
Finally, we emphasize that the proposed methodology is applicable to probe coherent ultrafast phenomena in the full surface Brillouin zone. For example, interferometrically time-resolved multiphoton photoemission techniques [2,5,6,40,[47][48][49][50] are applicable to those electronic bands. Moreover, it opens a new approach for performing measurements where the excitation field can both drive dipole transitions and interband ballistic electron acceleration in solids [51][52][53]. Similarly, field induced light band structure engineering is especially promising at those long wavelengths and high electric field strength applied in our work [54,55].

Conclusion
In conclusion, we report a coherent approach to perform photoemission spectroscopy, and measure the ultrafast electron dynamics of electronic bands extending deep into the Brillouin zone. Instead of providing large energy photons in the process of HHG, as typically done in optical-pump-XUV-probe TR-ARPES experiments, intense infrared light can be used to induce multi-photon processes leading to photoemission spectral features above the photoemission horizon. Proof of principle data obtained on the Ag(110) surface presented in this article thus provokes the application of mPP to novel materials whose physical and chemical properties are defined by their electronic band structure at the Brillouin zone edges, such as Dirac materials, or transition metal dichalcogenides. Thereby, higher-order mPP yield can be enhanced by resonant driving of selected dipole transitions [41], as well as by excitation at characteristic frequencies of the materials dielectric function, e.g. at the epsilon-near-zero condition, where the materials response is shown to be dominantly non-linear [17,56]. We thus anticipate that time-and angle-resolved photoemission data from such materials will become increasingly accessible in typical laser-based photoemission laboratories without having to resort to more complex XUV pulse generation. Furthermore, the presented methodology can be straightforwardly extended to novel photoelectron detection schemes, opening, for example, the door towards interferometrically timeresolved multi-photon momentum microscopy [57,58].  is only accessible in 4PP, labelled oc (4) . Note that the binding energy axis is only appropriate for the