Resonant inelastic soft x-ray scattering on LaPt₂Si₂

In the last two decades, resonant inelastic x-ray scattering (RIXS) has played an essential role in the understanding of many-body physics. By probing the change of energy, momentum, and polarization of the scattered photon, the RIXS technique provides access and information to intrinsic excitations in complex materials [1]. In the latest years, the drastic improvement of the technique has pushed forward the study of systems combining superconductivity (SC) and charge- and/or spin-density waves (CDW and/or SDW). Materials like Fe/Ni-based pnictides [2, 3], transition metal dichalcogenides [4, 5], or cuprate superconductors [6–13] are three examples out of many.

Recently, the quasi-two-dimensional Pt-based rare earth intermetallic material LaPt₂Si₂ has attracted a lot of attention as it exhibits strong interplay between CDW and SC [14]. A first order transition was reported from high-temperature tetragonal to low-temperature orthorhombic phase, accompanied by a CDW transition around T $_\textrm{CDW}$ = 112 K followed by a SC transition at T $_\textrm{c}$ = 1.22 K. LaPt₂Si₂ crystallizes in a CaBe₂Ge₂-type tetragonal structure (space group $P4/nmm$, see figure 1(a)), having a resemblance to the ThCr₂Si₂-type structure found in pnictide and heavy fermion systems. The structure also has close resemblance to the 122 superconducting Fe based systems, where spin fluctuations have been investigated and debated [2, 15, 16]. The striking difference between these two structures is that the former one lacks inversion symmetry in the crystal, resulting in two non-equivalent layers (Si₁–Pt₁–Si₁) and (Si₂–Pt₂–Si₂) arranged in alternating stacking separated by La atoms [17] (as shown in figure 1(a)). This special feature in the crystal structure is reminiscent of noncentrosymmetric SC [14], where the lack of inversion symmetry results in nonuniform lattice potential, creating an asymmetric spin–orbit coupling (SOC).

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To understand the material, we have performed x-ray absorption (XAS) and RIXS experiments on a LaPt₂Si₂ single crystal at the Si 2p ($L_{2,3}$) and La 4d ($N_{4,5}$) edges at room temperature, above the CDW and SC states. To the best of our knowledge, XAS and RIXS measurements have not been reported on this compound so far, and this preliminary study focuses on understanding the features observed above the SC and CDW transitions and how they relate to theoretical models and previous reports of XAS and RIXS from La and Si-based compounds. A review on methods for theoretical modeling on RIXS spectra can be found in [18], and effects of self-interacting interaction beyond standard density functional theory (DFT) are treated in [19].

The XAS spectrum shows sharp peaks at 97.1 eV, 101.47 eV and a dominating broad feature with maximum at 117.4 eV which can be assigned to excitation of atomic-like La $4d^{-1}4f$ states. RIXS spectra excited at these resonances show scattering to La $5p^{-1}4f$ final states, and we also observe x-ray emission (XES) from the dynamically populated $4d^{-1}4f$ $^3D_1$ state to the ground state. The Si XES spectrum is stationary on the emission energy scale, and partial fluorescence yield (PFY) is used to construct the Si 2p XAS spectrum. Using DFT we show that the Si XES spectra can be described in terms of Si s and d local partial density of states (LPDOS). While Pt d-LPDOS is predicted to dominate the occupied states, a sharp localized La f state is found in the unoccupied states, in line with the observed quasi-atomic excitations. XAS and RIXS measurements show that the material's basic interactions can be explained using LPDOS of Si.

The single crystalline sample of LaPt₂Si₂ used in this study was synthesized using the Czochralski pulling method [14]. Crystallographic analysis shows that the LaPt₂Si₂ sample has $P4/nmm$ space group with the crystallographic parameters given in figure 1(a). The VESTA software [20] was used to generate the crystal structure with crystallographic parameters from Gupta et al [14]. The size of the sample used in this study was ≈ 1 mm × 2 mm (figure 1(b)). A diffraction pattern, obtained at room temperature using the Laue backscattering method (figure 1(c)) shows that the sample is a single crystal.

XAS and RIXS spectra were measured at the SPECIES beamline [21, 22] at MAX IV Laboratory in Lund, Sweden. A Si wafer was attached to the sample holder for the beamline monochromator energy calibration (figure 1(b)). Total electron yield (TEY) was measured via the drain current, both on the LaPt₂Si₂ sample, and the Si wafer. Normalization of TEY with respect to beamline flux was achieved via the gold-coated refocusing mirror drain current. The TEY measured from the Si wafer was in good agreement with previous Si $L_{2,3}$ edge XAS measurements [23].

XES spectra were measured using the Scientia model XES-350 spectrometer [24], which is a Rowland spectrometer equipped with three gratings and a micro-channel plate-based delay line detector [25]. For the measurements reported in this article, a grating with 3 m radius and 300 l mm⁻¹ groove density was used. Energy calibration of the spectrometer relative to the monochromator was done using the elastic scattering of the incident photon beam, and the monochromator energy scale was set by the Si $L_{2,3}$ XAS [23]. The overall resolution was estimated from the full width at half maximum (FWHM) of the elastic scattering peak to be 270 meV at 108 eV incident photon energy. All RIXS measurements were carried out in the horizontal plane at 90^∘ scattering angle, and to minimize diffuse elastic scattering the incident radiation was linearly polarized in the horizontal direction.

DFT calculations of the electronic structure of LaPt₂Si₂ have been performed with the Elk code [26] which implements the full-potential augmented plane waves and local orbitals method (FP-APW+lo) [27]. The core electrons are modeled with four component wave functions, and the valence electron are modeled with two-component wave functions, augmented optionally with SOC in (l, s) basis. In tetragonal LaPt₂Si₂ structure, the primitive cell contains ten atoms. Calculations have been performed for this structure, using the lattice parameters reported by Gupta et al [17]. We used the generalized gradient approximation as parameterized in the Perdew–Burke–Ernzerhof (PBE) functional [28] for nonmagnetic, collinear spin-polarized, and noncollinear magnetic including SOC calculations, using a maximum angular momentum l = 8, and a Γ-centered k -point mesh of size $16 \times 16 \times 16$.

In order to consider whether to go beyond DFT to treat the atomic-like localized La 4f electrons, we make the observation that in the PBE-DFT calculations (shown in figure 3) there are no occupied La 4f states, as these reside about 2 eV above the Fermi level. By construction a self-interaction calculation would not change things as the La 4f state are not occupied. Similarly, DFT+U treatment of these states would not change the electronic structure. Therefore we have chosen not to go beyond DFT for calculating the electronic structure. As both highly localized and highly delocalized states contribute to the spectra no attempts were made to explicitly capture the excitations, e.g. by constructing a model Hamiltonian based on the ground-state DFT result.

In the Elk PBE collinear spin-polarized calculation, the electron band structures for majority and minority spins comes out as degenerate, from which we infer that LaPt₂Si₂ is a non-magnetic material. The electronic band structure in Elk PBE SOC calculation is shown in figure 2, with the associated total DOS, and projected partial DOS for different crystallographic sites La, Pt₁, Si₁, Pt₂ and Si₂ are shown in figure 3. At the La site a sharp La f-LPDOS peak dominates the unoccupied states, and two sharp peaks in the La p-LPDOS are found below −15 eV in the occupied states (figure 3(a)). It can be concluded that the La states are localized and do not contribute much to the band formation. The occupied states are dominated by Pt d-LPDOS with a maximum around −4 eV (figures 3(b) and (e)) whereas p-symmetry gives the largest contribution to the electronic structure at the Si sites (figures 3(c) and (f)), apart from a sharp peak in the Si s-LPDOS at around −10 eV. By virtue of the dipole selection rules, x-ray spectra at the Si $L_{2,3}$ edges primarily probe the Si $(s+d)$-LPDOS, and reflect the localized nature of the La f-excitations at the La 4d edges.

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Figure 4(a) shows TEY obtained by the drain current from LaPt₂Si₂ as a function of incident photon energy. There are two sharp peaks at 97.1 eV and 101.47 eV, and a dominating broad feature with maximum at 117.4 eV, which can be assigned to excitation of atomic-like La $4d^{-1}4f$ states, including transitions to the bound LS-forbidden $^3P_1$ and $^3D_1$ states, and the giant $^1P_1$ resonance, respectively. Similar multiplet structures are typically observed in La compounds [29–32], as a consequence of the atomic-like localized nature of the 4f electrons. Notably, the TEY spectrum does not capture the Si 2p edge XAS of LaPt₂Si₂, expected to appear just below 100 eV. This observation demonstrates that the La 4d absorption cross-section totally dominates over Si 2p absorption. Instead, we observe TEY peaks between 104 and 108 eV (figure 4(a)) that match previous XAS spectra of SiO₂ [23]. We attribute these peaks to an oxide layer formed on the surface of LaPt₂Si₂ crystal.

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To address the Si 2p edge absorption spectrum of LaPt₂Si₂ we first turn our attention to the XES map in figure 4(d). A complex pattern is observed, where the main intensity is stationary between 87 eV and 98 eV emission energy, primarily changing its integrated intensity with excitation energy. There is also an elastic peak for which the emission energy equals the excitation energy. A weaker structure disperses at constant 20–24 eV energy loss, and a sharp emission feature is observed at 101.46 eV constant emission energy. The latter is resonantly excited at incident photon energies around 120 eV. These features are denoted in the XES spectrum excited at 129.8 eV, shown in figure 4(c) as:

• (i)  
  Elastic scattering, coinciding with the incident photon energy.
• (ii)  
  A constant energy loss feature which we assign to the scattering to 5p⁻¹4f final states, reached over the 4d⁻¹4f resonances. We tentatively assign the two peaks at 20.8 and 22.5 eV energy loss to final triplet states and the $^1D_2$ state, respectively, following the analysis of Suljoti et al [32] and Miyahara et al [33].
• (iii)  
  A sharp peak at 101.46 eV emission energy assigned to emission from the dynamically excited $4d^{-1}4f$ $^3D_1$ state to the La $4f^{\,\,0}$ ground state.
• (iv)  
  A broad feature at constant emission energy, which we assign to principally Si $L_{2,3}$ emission, i.e. is associated with electrons from the valence band filling Si 2p vacancies.
• (v)  
  Si $L_{2,3}$ emission as in (iv), but as we see, the excitation energy dependence of the two features are different.

The magenta curve in figures 4(a) and (b) is the PFY, constructed to emphasize Si 2p excitations. As Si $L_{2,3}$ emission dominates features (iv) and (v), intensity in the corresponding 85–98 eV emission energy range was integrated to construct the PFY spectrum, while intensity from the dispersing La feature (ii), which contributes in the 110–120 eV excitation-energy range (figure 4(c) inset) was subtracted. The Si $L_{2,3}$ edge shown in the PFY spectrum is found at 99.5 eV, and there is a sharp second intensity increase at 101.4 eV, almost coinciding with the La $^3D_1$ resonance, and another intensity increase around 107 eV.

There is only a faint $({\lt}10\%)$ structure in PFY at the sharp SiO₂ excitations around 104 eV which can be attributed to the surface oxide. For oxidized Si surfaces, a similar observation has been made for oxide layers of around 7 nm thickness [23]. While the fluorescence yield sampling depth in Si is estimated to be 70 nm [23], it is similar in this compound except for the region of the La giant resonance $^1P_1$ where the sampling depth is more than three-fold reduced [34]. We can therefore estimate the oxide layer also for LaPt₂Si₂ to be in the ≈7 nm range.

While the Si $L_{2,3}$ edge PFY spectrum gives information about the bulk material below the surface oxide layer, it does not directly represent the cross-section for Si 2p excitations. The PFY minimum at 117.3 eV coincides with the maximum of the La giant resonance. At these energies, La 4d absorption dominates the XAS spectrum of LaPt₂Si₂. As this process competes with Si 2p absorption, and since La 4d holes do not emit appreciably in the chosen emission energy range, the observed broad dip in the PFY is expected. The mechanism is the same as exploited in the inverse partial yield method [35].

In figure 4(b) we compare the the PFY with the Si s and d LPDOS, separated into contributions from the Si₁ and Si₂ sites. Significant differences between the two sites are predicted, e.g. the absorption close to the edge is primarily of Si₂ s character. The PFY increase around 101.4 eV coincides with increasing Si₁ s-LPDOS, while the 107 eV increase matches the increase in the combined Si d-LPDOS. We assign the main PFY features accordingly.

4.1. Elastic peak (i)

In general, the elastic peak (i) can have contributions from both diffuse reflection and the resonant absorption-emission process. Emission from the giant resonance in La compounds in low-energy electron excited spectra seemingly coincides with the absorption [29], demonstrating that a captured electron is sufficiently localized to recombine with the 4d hole. The giant resonance is also observed in x-ray reflection measurements [30]. Considering the complex index of refraction, n + ik, it has been noted that if $(n-1)^2 \ll k^2 \ll (n+1)^2$ the normal incidence reflectivity varies as the square of the absorption cross section. While this is often the case in this energy range [36], this approximation does not hold at the giant resonance, for which tabulated optical constants [34] imply a high-energy shift of the reflectivity peak relative to the peak in the absorption cross section. This predicted shift is consistent with the observed difference in peak positions when comparing the excitation-energy dependence of the elastic peak intensity and the TEY spectrum (figure 5). While the curves show similarities, there are also significant differences, the TEY spectrum peaks at 117.5 eV, and the maximum in the elastic-peak intensity is around 119 eV. Although we note that the TEY signal may be influenced by disorder in the topmost layers of the crystal, we can conclude that the elastic peak is dominated by diffuse reflection in the region of the giant resonance.

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4.2. Dispersing feature (ii)

4.3. Emission at 101.46 eV (iii)

4.4. Si 2p emission (iv and v)

4.4.1. LPDOS and XES.

The XES spectrum shown in figure 6, excited at 102.87 eV, is shifted to the binding energy scale by subtracting 98.97 eV, as determined by the apparent valence band edge. From the edge, the spectrum shows a plateau down to around −6 eV, followed by a peak with a maximum around −9.5 eV. The experimental results are compared to the Si s and d LPDOS (figure 6) at the two crystallographic sites, Si₁ and Si₂, as denoted in figure 1(a). The low-energy peak can be unambiguously assigned to transitions from states of Si s character as the Si s-LPDOS dominates in this region. In the predicted LPDOS there is a marked difference between the two sites. Whereas the Si₁ $~s$-LPDOS has a rather sharp peak at −9 eV, the Si₂ $~s$-LPDOS is more smeared out. Due to the broadening mechanisms, especially the lifetime broadening of final hole states far from the Fermi level [40] we do not expect this difference to be observable.

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The Si s-LPDOS is not sufficient to capture the plateau in XES spectrum between the valence band edge and −6 eV, as density relative to the −9 eV peak is small in this region. Assuming that the Si d-LPDOS gives a factor of five larger contribution to the XES intensity than the Si s-LPDOS, the intensity in this region can be partly explained (figure 6). Since the Si 2p-d radial overlap typically is larger than the 2p-s overlap, Si d-states are expected to contribute more to Si 2p emission than Si s-states [41], but a factor of five is unusually large. We also note that upscaled Si d-LPDOS introduces spectral structure that is not observed in the experimental spectrum.

4.4.2. Excitation-energy dependence.

The spectral changes as the excitation energy varies across the La $^3D_1$ absorption peak are shown in figure 7. We find that the spectra normalized at 98 eV, show a steep resonant behavior where the intensity of the peak at 90 eV (−9 eV peak in figure 6) increases, and a pronounced shoulder develops around 91 eV, and attains maximum intensity when the incident energy is tuned to the La $^3D_1$ resonance.

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It is tempting to interpret this behavior as due to transitions involving La $4d^{-1}4f$ $^3D_1$ intermediate state. In this state the f electron is localized at the La site, and in principle a valence electron can fill the core hole, to create a final state with a vacancy in the valence band and an excited f electron. The sharp peak in the unoccupied La f-LPDOS of the electronic ground state (figure 3(a)) suggests that the f electron is localized also in the final state. According to dipole selection rules transitions from the valence band to a La 4d hole should be from state with La p or f character. However, the theory does not show appreciable LPDOS of these local symmetries (figure 3(a)), and it lacks steep structures that could directly account for the observations.

Tentatively, we still associate the resonant behavior to transitions to the La 4d level from the valence band. With the excitation energy at 101.47 eV, and the emission energy of the resonant shoulder is 91 eV, the energy loss is 10.5 eV, suggesting that the corresponding hole is in the band that is dominated by Si s states. Especially, the Si₁ site, which is in closest proximity to the La atoms (figure 1(a)) has a sharp peak at the corresponding energy (figure 3(c)). The observations are consistent with a 'cross transition' where the final state has an electron in the band dominated by La f-LPDOS or La p-LPDOS (figure 3(d)) and a vacancy in the band dominated by Si₁ $~s$-LPDOS, and we speculate that excitonic effects, and possibly interference between close lying core hole states may enable pathways to such final states. Finally we note that there is a slight spectral change, when comparing data taken below and above the resonance. They are consistent with selective excitation of the two Si sites, which show large variation at the cross section at these excitation energies (figure 4(b)).

This possible hybridization effects of La with the close Si₁ site might be a key to understanding the predictions put forth in theoretical calculations by Kim et al [42] where the CDW is predicted to be confined to the layer with Pt₁ crystallographic site. We speculate that by understanding these local excitations of La with the Si₁ atoms could indicate how hybridization might aid or prevent the formation of multiple CDW, which are observed in recent experiments [43]. The La atoms are not merely spectators that separate the Si₁–Pt₁–Si₁ and Si₂–Pt₂–Si₂ layers but might play a crucial role in the formation of CDW in LaPt₂Si₂.

XAS and RIXS measurements at Si 2p and La 4d edges of LaPt₂Si₂ are presented, and interpreted in terms of DFT calculations. Atomic-like local La $4d\rightarrow4f$ excitations are found in the absorption spectrum, the LS-forbidden $^3P_1$, and $^3D_1$ excitation as well as the giant $^1P_1$ resonance. Decay to $5p^{-1}4f$ final states are observed, and also the decay of the dynamically populated $^3D_1$ state to the electronic ground state. Observations suggest that resonantly excited $^3D_1$ states also decay via valence band emission. The Si 2p XAS spectra are measured via PFY, and the XES spectra are independent of the excitation energy over most of the energy range. The spectra are assigned using the theoretical LPDOS of Si s and d character at both crystallographic Si sites. We should expect to see a variation in electronic interaction between La and Si atoms as a function of temperature, and an accompanying variation in the Si 2p and La 4d spectra. With site selectivity the question to what extent the CDW primarily is formed in the Si₁–Pt₁–Si₁ or Si₂–Pt₂–Si₂ layers can be addressed. A more detailed study will follow, including a systematic investigation of the temperature dependence across the CDW transition, and the dependence on crystal orientation.

The authors would like to thank Margit Andersson and Jenn-Min Lee for the technical support during the experiments at SPECIES beamline at MAX IV. Y S acknowledges funding from the Area of Advance - Material Sciences from the Chalmers University of Technology and the Swedish Research Council (VR) Starting Grant (Dnr. 2017-05078). M M and E N acknowledge funding from the Swedish Research Council (VR) through a neutron project Grant (Dnr. 2016-06955), the Swedish Foundation for Strategic Research (SSF) within the Swedish national graduate school in neutron scattering (SwedNess), and the Carl Tryggers Foundation for Scientic Research (CTS-18:272). M A acknowledges funding from Stiftelsen for Strategisk Forskning (SSF), through grant RIF14-0064. J-E R acknowledges funding from the Swedish Research Council (VR) and the Carl Tryggers Foundation (CTS). The computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at PDC and NSC, partially funded by the Swedish Research Council through Grant Agreement No. 2018-0597.

All data that support the findings of this study are included within the article (and any supplementary files).