Ultrafast Second-Harmonic XUV Spectroscopy: A Novel Probe for Symmetry

Abstract

Extreme-ultraviolet second-harmonic generation spectroscopy (XUV-SHG) is a novel spectroscopy that enables probing element-selective symmetry-broken states. This renders XUV-SHG especially useful to study surfaces, interfaces, and symmetry-broken bulk states in otherwise complex chemical environments. In a string of recent works, XUV-SHG was successfully applied to study the role of lithium in various compounds. One of the most striking recent results studied the role of Li symmetry-breaking displacement causing the emergence of polarity in the polar metal LiOsO₃. Furthermore, the directional dependence of the SHG process allows geometry specific measurements. Given the femtosecond nature of the probe pulses, one can readily envision this method to be applied to study interfacial carrier dynamics in complex in-operando environments that are difficult to probe with conventional ultrafast methods.

1 Introduction

Understanding interfacial chemistry, electronic structure as well as symmetry properties at a molecular level is of great importance for investigating the structure-function relationship of matter. A key challenge lies in isolating relevant observables to one or very few atomic layers forming an interface. For decades, surface-sensitive photoemission techniques as well as nonlinear optical and infrared spectroscopy have been successfully applied to study surface chemistry and physical phenomena. Despite the significant progress in understanding surfaces, most of the techniques at hand do not provide the required sensitivities, e.g., at a buried layer or when the interface of interest is emersed in a complex chemical environment. For example, photocatalytic systems typically consist of multi-layered structures consisting of a photoabsorber and catalyst which are immersed in a bath of liquid or gas. In such systems, especially within in operando conditions, elucidating interfacial properties and dynamics is difficult. A similar situation applies to symmetry-broken materials where emerging properties often stem from the lack of symmetry. In complex quantum materials more than one part of the material can contribute to these properties. Disentangling different contributions to symmetry is of great importance to better understand the emergence of novel properties to inform material design and discovery.

There is a lack of methods that allow measuring interfaces and symmetry properties with the sensitivity required to obtain the electronic and molecular structure. Given their high penetration depths, X-rays in general appear suitable due to their sensitivity to core-electrons hence providing element specificity. In standard X-ray absorption spectroscopy (XAS), spectra collected by traversing through a material contain information associated with either side of an interface, i.e., mostly bulk. An emerging method to overcome this limitation is extreme-ultraviolet / soft X-ray second harmonic generation spectroscopy (XUV-SHG) [1]. In this type of spectroscopy, two XUV photons of frequency ω are resonantly absorbed by a core-electron and the resulting SHG photon at frequency 2ω is detected. This process satisfies the relationship I_2ω ~ χ⁽²⁾(ω) I_ω², where χ⁽²⁾(ω) is the second-order nonlinear susceptibility containing material properties pertaining to broken symmetry. The generally low magnitude of the second-order nonlinear response compared to the linear dielectric response of a material requires tens or hundreds of GW peak power, largely requiring free-electron laser (FEL) sources. The second-order nonlinear susceptibility χ⁽²⁾ is a tensor of rank 3. The different tensor components encode information of the nonlinear dielectric response in different planes of the crystal. When a material is inversion-symmetric, all tensor elements are zero making SHG a sensitive probe to broken inversion symmetry. This property allows using SHG to study symmetry within bulk or at any interface. In SHG experiments determining χ⁽²⁾(ω) becomes the central task which can be achieved by fitting a measured I_2ω–I_ω dependence.

The first observation of second harmonic generation in the hard X-ray regime by Shwartz et al. used 7.3 keV photon energy on a diamond crystal [2]. This first observation showed the feasibility of SHG experiments with FELs but found that the response observed appeared not to be sensitive to symmetry-breaking. This was explained by that excited electrons at such high photon energies can be treated as free particles causing the dominant nonlinearity to stem from plasma-like properties. In contrary, Lam et al. demonstrated on graphite films of varying thickness at the carbon K edge [1] in the soft X-ray that indeed at such more moderate photon energies one obtains sensitivity to symmetry. Elemental specificity in XUV-SHG was first demonstrated in GaFeO₃, which has a symmetry-broken bulk state, at the iron M edge [3]. The second harmonic spectra exhibited strong resonances indicative of the large asymmetry of iron within the unit cell of GaFeO₃. A central advantage of XUV-SHG is the capability to measure buried interfaces. This was first demonstrated by Schwartz et al. who studied a perylene-boron junction [4]. Supported by numerical simulations they were able to show that that spectral shifts observed at the boron K edge were dependent on the interfacial bond length. This can be rationalized by the interfacial charge transfer shifting the valence state energies. A series of recent works that specifically studied symmetry-breaking by lithium in different materials will be more closely discussed in the following section.

2 Lithium-Specific Measurement in Symmetry-Broken Solids

Following the initial works in XUV-SHG that showed that the soft X-ray and extreme ultraviolet are suitable regimes for studying elemental resolved symmetry-breaking, a series of works focused on the role of lithium. These works address the inherent difficulty to measure lithium within solids with conventional scattering techniques, such as electron or X-ray scattering. This is due to the low atomic number of lithium having a low scattering cross-section which often results in strong dominance of signals contributed by other heavier elements in presence.

A first study of the role of lithium in quantum materials targeted an interesting physical phenomenon where ferroelectricity and metallicity coexist in a single material which was first theoretically proposed by Anderson and Blount in 1965 [5]. These properties normally contradict each other as free electrons from a metallic state would be expected to shield local dipoles preventing long-range order. However, as predicted in this theoretical work, such polar metals can exist for certain crystals undergoing a second-order phase transition where free electrons cannot entirely shield electric dipoles. It took almost 50 years when Shi et al. successfully synthesized LiOsO₃ and demonstrated its polar metal properties [6]. Since this first discovery large interest has been put forward for such materials as possible building blocks in highly integrated circuitry where polar metals combine metallic conduction with the capability to store information in the long-range dipole order. In the recent years, more materials have been found to belong to this material class[7].

Despite these first discoveries many questions pertaining to the emergence of these properties remained. Especially the exact role of the lithium atoms in LiOsO₃ is difficult to study with conventional scattering techniques. To contribute to the study of this novel material, Berger et al. studied the symmetry-breaking in the polar phase of LiOsO₃ using XUV-SHG specific to lithium (Fig. 1) [8]. In this material, the polar metal phase emerges at temperatures below 140 K where LiOsO₃ undergoes a second-order phase transition with lithium atoms being displaced along a polar axis causing ferroelectricity while free electrons stemming from Os 5d states sustain conductivity of the material rendering it a polar metal. XUV-SHG spectra were measured around the Li K-edge (56–66 eV) by focusing an intense free-electron laser beam onto the material with photon energies in the range of 28–33 eV. It was possible to estimate the displacement of Li from centrosymmetry of about 0.5 Å which is consistent with first neutron diffraction measurements on that material [6]. The experiments in concert with advancements of numerical simulation using time-dependent DFT methods have provided deeper insights into how the XUV-SHG spectrum encodes information about the dielectric environment of a single atomic species in its symmetry-broken state. Further, in this work the importance of considering the directional dependence of the χ⁽²⁾ tensor was studied in [8], which is discussed in detail in Sect. 3.

Fig. 1

XUV-SHG spectra and simulation depending on Li displacement in LiOsO₃ at 60 K. Lines with different shades of green represent different amount of Li displace α along the polar axis from centrosymmetry. The black squares represent measured data. The inset shows pictographically the energy diagram for the experiment conducted in two-photon resonance with the Li 1 s state

Following this first work, the role to symmetry-breaking of the Li atom was further studied in LiNbO₃ where in addition to the second harmonic spectrum for the first time the polarization of the XUV-SHG signal itself was studied [9]. For this, the researchers inserted a polarizer consisting of a multilayer mirror selecting the XUV-SHG signal under 45° angle of incidence [10]. The resulting XUV-SHG polarimetry signal was related to expected symmetries from the crystal structure. Also, this work was the first showing several absorption features of different elements, Li and Nb at the K and N edges respectively, that are spectrally separate but measured in the same experiment indicating contributions to symmetry-breaking by both elements within the unit cell.

3 Interpretation of Second-Order Susceptibility Tensor Components

A crucial aspect for all XUV-SHG measurements is the correct treatment of the experimental geometry and directional dependence of the rank 3 second-order susceptibility tensor. Typically, for given polarizations and crystal orientation one obtains a projection of the weighted tensor components χ⁽²⁾_ijk onto an effective χ⁽²⁾_eff. Figure 2a–d show the calculated independent tensor components for LiOsO₃ at the Li K edge, where the point group results in four independent tensor components. Inspecting the projected charge density (Fig. 2, projection into the z-x-plane) allows to qualitatively interpret the tensor components. For example, χ⁽²⁾_zzz (Fig. 2a) represents a condition where two driving field components and the second-order polarization point into the z direction which is along the polar bond. In this case, one observes a strong dependence of the second-order response at the Li edge which can be rationalized by the strong symmetry-breaking afforded by Li in this direction. In contrary, inspecting χ⁽²⁾_xxy (Fig. 2c) one finds only a comparatively small response since all involved fields and polarization point perpendicular to the polar axis. Taking such geometric effects into account enables one to not only measure element-specific symmetry-breaking but also amplify the response w.r.t. to a specific symmetry-direction of the crystal if that is desired.

Fig. 2

Interpretation of tensor components of LiOsO₃. (a)–(d), four numerically calculated independent tensor components for varying Li atom displacement. (e) Projection of the charge density into the z-x plane with Li atom displaced along the polar axis (dashed gray line). (Panels taken from Ref. 8 with permission from the authors)

4 Outlook

All the works mentioned in Sects. 1, 2, and 3 made use of the high pulse intensities afforded by FEL sources. Using table-top sources, for example driven by high harmonic generation (HHG), can be a route to improve access to XUV-SHG in the coming years. HHG sources generally exhibit pulses in the femtosecond regime with high coherence which renders them in general suitable sources to obtain high peak intensities. Limiting factors are the comparatively low energy per pulse in the pJ to nJ range with only few examples of pulses above 10 nJ energy per pulse. Scaling such sources to higher pulse energy could open routes to table-top XUV-SHG [11]. A first proof-of-principle experiment for table-top XUV-SHG has been performed by Helk et al. who used a post-amplified HHG source that reached ~111 nJ pulse energy and a tight focusing geometry using an off-axis ellipsoidal mirror [12]. In addition, other recent advancements in achieving attosecond FEL pulses [13] provide interesting new opportunities in materials sciences using XUV-SHG spectroscopy in combination with methods that were recently developed to study materials with transient XUV spectroscopies [14,15,16,17,18,19,20,21,22,23,24,25,26,27]. An open topic to study in more detail in the context of strongly driven materials is the interplay between different material responses such as self-induced transparency and two-photon absorption [28] in relation to second-harmonic generation. Despite its novelty XUV-SHG has already shown its importance in studying surfaces, buried interfaces and symmetry properties of relevant materials with emerging properties.