Inelastic x-ray scattering studies of phonon dispersions in superconductors at high pressures

Although some early studies were done using large volume cells of the piston-cylinder type [9], most IXS experiments under hydrostatic pressure (HP) conditions—and almost exclusively the ones on superconductors—are carried out using the DAC apparatus [10]. The IXS experiments are conventionally performed in transmission geometry through the diamond anvils. Radial scattering geometries using the gasket as an x-ray window [11] are avoided in the energy range of interest due to the overlapping phonon scattering signal from the gasket material (see for instance reference [12] for the beryllium gaskets commonly used in panoramic DACs). In the working geometry, the beam path through the diamonds is ∼50 times longer than the path through the sample. Phonon and Bragg scattering signal from the diamonds inevitably appears and often obstructs the measurements. However, it can be circumvented by rotating the DAC around the compression axis and working on structurally equivalent positions in the reciprocal space of the investigated sample [13]. Parasitic signal can also arise from the pressure transmitting medium (PTM), although this is more rarely encountered. The use of PTM is crucial to ensure the best possible hydrostatic conditions and the preservation of the sample's crystalline quality throughout the measurements. The choice of the PTM in IXS experiments is governed by the standard criteria for chemical inertness to the sample and high hydrostaticity at the working temperature range [14]. In most of the reported HP experiments on superconductors, condensed gases were employed as PTM. Amongst them, helium is considered to be the best choice, particularly for measurements at low temperatures.

Recently, novel piezoelectric-based straining devices have been developed for the transport or thermodynamic studies of uniaxially pressurized samples [15–17].To account for the specific needs of photon scattering experiments (e.g. transmission geometry), dedicated strain cells were developed and used in IXS experiments [18, 19].

They combine a compact design and an extended, unobstructed angular access of the incident (scattered) beam to (from) the strained sample. Their basic working principle relies on controlling the displacement of two moving blocks by varying the voltage across the piezoelectric actuators. The sample under investigation is shaped in the form of a thin bar and firmly mounted with an epoxy across the two moving blocks. Through a suitable variation of the bias voltage, the sample can be compressively or tensily stressed in a continuous and well-controlled manner and with high strain homogeneity at its central part. The applied strain can be obtained by measuring the strained lattice parameters by x-ray diffraction, but also by using a capacitive strain gauge positioned parallel and underneath the mounted sample. The capacitor is kept in a feedback loop, therefore serving as a guarantee of constant strain on the sample during the typically long IXS measurements.

In the study of superconductors high pressures are often necessarily coupled to cryogenic temperatures. While the combination of high pressure and low temperature conditions adds up to the technical complexity, these experiments open the route for an extended (P, T) mapping of the phase diagrams and often enable reaching otherwise inaccessible phases of the studied samples. Dedicated cryostats are typically tailor-made for specific types of DACs. These are developed to meet the requirements for compactness, large window openings, geometrical flexibility, precise motorized translation and rotation, as well as remote pressure control at low temperatures. Optical access is very often also envisaged, allowing the use of the ruby fluorescence lines for the pressure calibration inside the DAC [20].

In addition to the above, in the case of piezoelectric-based uniaxial straining devices another parameter to consider at cryogenic temperatures is the differential thermal contraction between the actuators and the strained sample. The straining cell used for IXS experiments in reference [19] addressed this issue combining a set of compression and tension piezoelectric stacks compensating each other upon cooling.

Sample preparation and handling is a crucial and challenging task for IXS measurements under HP. This is not only because of the small sample size required to fit the DAC sample chamber (typical sample diameter < 100 μm and thickness ∼ 30 μm), but also due to the limited accessible part of the reciprocal space imposed by the DAC design (angular opening of the diamonds ∼ 60°). For this reason, it is essential to prepare and load the sample under investigation in a suitable pre-selected orientation which guarantees that the reciprocal space parts of interest rest accessible. It is worth noting that the sample is not glued or mechanically fixed inside the DAC chamber, remaining therefore free to rotate, for instance across the solidification of the surrounding PTM.

Also in the case of uniaxially straining devices, the needle-like geometry requires a multi-step sample preparation procedure. This includes the wire-saw cutting of the studied crystal in bars, followed by a fine mechanical polishing on all four sides in order to achieve a uniform bar cross-section and to avoid surface cracks. For experiments in transmission geometry, the final bar thickness is selected based on the absorption length of the studied sample in the used incident energy.

Gentle cutting processes, such as femtosecond-laser cutting and focused-ion-beam micromachining, are particularly well-suited for sample preparation both for HP and for uniaxial experiments. Such techniques allow reaching the targeted dimensions in any given orientation, while maintaining the sample quality required for single crystal IXS experiments.

Uranium crystallizes in an orthorhombic structure uniquely encountered for an element under ambient pressure conditions (α-U phase, space group Cmcm). CDW ordering appears below 43 K and induces a structural modulation (α₁ -U) which involves approximately a doubling of the unit cell along the a-axis with respect to the unmodulated α-U structure. The CDW order in uranium was associated to nesting features of its Fermi surface (nesting vector Q_N = (1/2,0,0) in reciprocal lattice units (RLU)). Under pressure application, the CDW disappears, whereas at the same time the superconducting critical temperature T_c is increased. T_c is maximized (∼2 K) when the CDW order is completely supressed at ∼1.5 GPa.

Raymond et al investigated the role of EPI in mediating the competition between superconductivity and CDW ordering in uranium through a combination of HP IXS experiments and ab initio calculations [21]. The authors have measured the softening of the Σ₄ phonon branch around q_N at room temperature and observed the gradual pressure-induced suppression of the dispersion dip until its complete disappearance at 20 GPa (see figure 1). Interestingly, calculations of the electronic structure indicated that the nesting features are rather insensitive to the application of pressure and remain unaltered up to 20 GPa. Nevertheless, calculations of the electron–phonon coupling in the α-U phase revealed a strong suppression under HP. Moreover, the authors showed that—unlike the case of the α-U structure—in the modulated α₁ -U structure (up to 1.5 GPa) the electron-phonon coupling increases under pressure, and related this increase to the rise of the superconducting T_c . All together, these results highlight that the momentum and pressure dependent EPI plays a dominant role in the observed disappearance of the soft mode and in shaping the overall interplay of superconductivity and CDW order in the phase diagram of uranium.

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Following the HP IXS investigation on uranium, Loa and co-authors studied the lattice dynamics in the high pressure α' phase of cerium [22]. The α' phase is isostructural to the α-U phase of uranium and it is stabilized in elemental cerium above 5–6 GPa. Pronounced anomalies were observed in the phonon dispersions of pressurized cerium, strongly resembling the ones reported for the α-U phase. More specifically, a clear dip of the Σ₁ and Σ₄ phonon branches was observed around Q= (1/2,0,0) (figure 2). The similarities between the phonon anomalies in the two systems, which share the same crystal structure but otherwise have very different electronic structures, becomes more intriguing when taking into account that up to date there has been no experimental observation of CDW ordering in cerium. Calculations of the Fermi surface have not revealed particularly strong nesting features at the wavevector where the phonon anomalies are seen. Nevertheless, strong electron-phonon coupling was calculated around Q= (1/2,0,0) due to large electron-phonon matrix elements and was suggested to be the origin of the pronounced dips of the Σ₁ and Σ₄ phonon dispersions. The strong electron-phonon coupling calculated for this phase in comparison to alternative structures stabilized in elemental cerium under pressure, led the authors to the conclusion that the superconducting phase of cerium is the α' phase.

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Many members of the transition metal dichalcogenide family exhibit CDW phases with highly diverse characteristics which compete with superconductivity. The phononic contribution to the stabilization of these electronic phases, as well as to the tuning of their interplay by external parameters is intensively investigated. One of the most studied examples is 2H-NbSe₂, in which incommensurate CDW ordering sets in at T_CDW = 33.5 K and coexists with superconductivity below T_c = 7.2 K. The second order transition to the CDW state is related to the softening of a longitudinal acoustic phonon mode. It has been highlighted that the extended q-width of the observed phonon dip around the CDW ordering vector q_CDW is not in line with the sharply localized dips expected from Fermi surface nesting. Strongly q-dependent electron-phonon coupling has been proposed as the driving force behind the CDW formation and the one determining the ordering wavevector [23].

Leroux et al have investigated the pressure dependence of the soft phonon in 2H-NbSe₂ [24]. Earlier HP experiments have showed the pressure-induced suppression of the CDW order, which disappears above ∼4.5 GPa, while superconductivity remains almost unaffected. In their HP experiment, the authors observed that the strong temperature dependence of the soft phonon survives up to at least 16 GPa, which is much higher than the critical pressure of ∼4.5 GPa above which the CDW is fully suppressed (figure 3). At 16 GPa, the amplitude of the temperature dependent phonon softening is certainly reduced, but remains significant as the phonon energy at q_CDW decreases by almost a factor of 2 (see figure 3).

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The strong temperature dependence of the soft phonon well beyond the stability range of the CDW indicates that an harmonic effects play an important role and should be considered as a contributing mechanism in shaping the phase diagram of this system. Indeed, harmonic ab-initio calculations fail to describe the ground state of 2H-NbSe₂ as a function of pressure, as they predict a CDW instability up to 14 GPa. The authors have carried out a series of calculations which explicitly include the effects of anharmonicity (stochastic self-consistent harmonic approximation). The results of these calculations, which are in very good agreement with the experimentally determined pressure and temperature dependence of the phonon dispersions, succeed in correctly predicting the stability range of the CDW.

In the related system 1T-TiSe₂ CDW order develops at ambient pressure below T_CDW = 200 K. IXS experiments and ab-initio calculations showed that the CDW transition and the accompanying lattice modulation are related to the softening of a transversely polarized phonon mode at the L point of the high temperature trigonal Brillouin zone [25]. Nevertheless, several experimental observations and theoretical investigations are consistent with an excitonic insulator picture, in which exciton condensation plays a major role in driving the CDW modulation in 1T-TiSe₂. It has been argued that as excitons necessarily couple to the underlying phonons, the resulting hybrid phonon-exciton modes are the ones mediating the CDW formation [26]. Superconductivity appears in this system either under pressure or upon Cu intercalation. Unlike the case of Cu intercalation, the superconducting dome under pressure is surrounded by the CDW ordered phase rather than being centered on the critical pressure at which the CDW order disappears. The interplay between the two phases implies that the formation mechanism of the CDW directly relates to the superconducting pairing mechanism. The different effect of chemical intercalation and physical pressure, raised questions on the individual contribution of phonons, excitons and hybrid modes when tuning the interplay of CDW and superconductivity by external parameters.

Maschek et al addressed this issue through a comparative IXS study of Cu_xTiSe₂ and pressurized 1T-TiSe₂ combined with density function perturbation theory (DFPT) calculations [27]. According to their DFPT calculations, the soft mode related to the CDW carries the largest part of the total electron-phonon coupling and should be the most relevant for phonon-mediated superconductivity. The authors investigated experimentally the contribution of this phonon to the superconducting T_c , by following the ratio of the soft mode's linewidth to its energy, Γ/ω. In the case of Cu_xTiSe₂ the experimental results show an increase of the ratio at low temperatures upon approaching the soft L point in the superconducting doped compounds. This is in line with what is expected from DFPT calculations when the electron-phonon coupling of the soft phonon mode is the driving force for the superconducting phase near the critical point of the CDW suppression. Interestingly, in the case of pressurized 1T-TiSe₂ the IXS data reveal no dramatic change of the Γ/ω ratio when the superconducting phase is approached under HP (figure 4). The authors conclude that—unlike the case of Cu-intercalation—the pressure-induced emergence of superconductivity cannot be solely associated to a phononic soft mode and propose a hybrid phonon-exciton model which can explain the experimental observations. More specifically, Cu intercalation induces a shift of the chemical potential of 1T-TiSe₂, deteriorating the exciton formation conditions and subsequently decreasing the hybrid's mode exciton content. Contrarily, HP application has a relatively small effect on the chemical potential. Considering the crossing between the dispersions of the interacting phonon and exciton modes, the authors show that the low energy hybrid mode reaches zero energy at a higher pressure compared to the bare non-interacting phonon. Therefore, the phonon-exciton hybridization provides an explanation for the shift of the superconducting dome with respect to the critical pressure at which the CDW disappears. Moreover, the authors show within the standard BCS (Bardeen-Cooper-Schrieffer) framework that also the superconducting pairing interaction depends strongly on the degree of phonon-exciton hybridization, which in turn is pressure-dependent as it relies upon the energy difference between the bare exciton and phonon modes.

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The existence of charge modulations, in the form of stripes, checkerboard or CDWs and their interplay with high temperature superconductivity in the cuprates has been one of the most debated issues in the literature for more than three decades [28, 29]. Anomalies in the dispersion of high energy optical phonons with a displacement pattern similar to that of the expected charge modulations have been discovered for several families of cuprates (YBa₂Cu₃O_{6 +x} [30], La₂(Sr/Ba)CuO₄ [31], Bi₂Sr_2-xLa_xCu₂O₆ [32]) and interpreted as fingerprints of fluctuating charge stripes [33].

With the more recent discovery of ubiquitous short-range two-dimensional CDW in all the cuprates families [34–39], new investigations of their lattice dynamics have been undertaken and revealed strongly temperature dependent anomalies for the lowest energy phonon branches at the incommensurate CDW ordering wave vector Q_CDW [40–42]. In underdoped YBa₂Cu₃O_{6 +x} (YBCO), where the CDW is best developed with a in-plane correlation lengths up to ∼ 80 Å, the apparition of the CDW at T_CDW > T_c is associated with three distinct fingerprints i) a sharp increase of the quasi-elastic line intensity at Q_CDW down to T_c , which decreases upon further cooling in the superconducting state, ii) a broadening of the lowest energy phonon branches for temperatures down to T_c and iii) a steep sharpening and softening at Q_CDW below T_c (also referred to as a superconductivity-induced giant Kohn anomaly). This behavior contrasts a lot with that, more commonly encountered, of soft-mode driven CDWs in low dimensional metals [23, 43, 44], and indicate an unusual interplay between superconductivity and CDW in these materials.

In the underdoped region of YBCO where the CDW is the strongest, HP results in a steep increase of T_c that almost doubles by 15 GPa [45]. Whereas transport [46] and nuclear magnetic resonance [47] experiments indicate a no or a small suppression of the CDW under pressure, IXS measurements revealed that all the aforementioned low-energy phonon anomalies associated with the CDW at ambient pressure are suppressed by 1 GPa of HP (figure 5) [48]. This suggests a very rapid suppression of the CDW as the superconducting phase is enhanced. The origin of the discrepancy between these experiments, carried out in different types of pressure cells with different PTM remains unclear.

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Another approach to study the interplay between superconductivity and CDW consists in suppressing the former, using e.g. a magnetic field. To date however, the implementation on IXS beamlines of magnets with sufficient strength to obliterate superconductivity in the cuprates (typically > 10 T) have not been realized for technical reasons. As an alternative route, it has been established that the dependence of T_c to uniaxial pressure is highly anisotropic. In YBCO in particular, a-axis compression rather than HP can very effectively suppress superconductivity. Uniaxial compression of the a-axis of underdoped YBCO amounting to ∼1 GPa was shown to decrease T_c from ∼65 K to ∼ 45 K [19], to strengthen the 2D CDW and finally to induce a 3D long-range-ordered CDW (figure 6), that had been only reached so far using magnetic fields larger than ∼ 15 T [49]. This study further revealed that the transition to the 3D CDW is associated with a pronounced, possibly complete, softening of an optical phonon. This illustrates the potential of coupling uniaxial-pressure and spectroscopy to control competing orders and gain insights regarding their formation in the cuprates, and more generally within multi-phase quantum materials.

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