Charge density wave with anomalous temperature dependence in UPt2Si2

TL;DR

This study identifies a charge density wave in UPt2Si2 below 320 K, revealing its unusual temperature-dependent behavior and its interplay with electronic coherence and magnetic order.

Contribution

It reports the discovery of a charge density wave in UPt2Si2 with an anomalous temperature dependence, linking it to electronic and magnetic phenomena.

Findings

Charge density wave discovered below 320 K.

Unusual shift from commensurate to incommensurate wavevector.

Interplay between CDW, electronic coherence, and magnetic order.

Abstract

Using single crystal neutron and x-ray diffraction, we discovered a charge density wave (CDW) below 320 K, which accounts for the long-sought origin of the heat capacity and resistivity anomalies in UPt2Si2. The modulation wavevector, Qmod, is intriguingly similar to the Fermi surface nesting wavevector of URu2Si2. Qmod shows an unusual temperature dependence, shifting from commensurate to incommensurate position upon cooling and becoming locked at ~ (0.42 0 0) near 180 K. Bulk measurements indicate a cross-over toward a correlated coherent state around the same temperature, suggesting an interplay between the CDW and Kondo-lattice-like coherence before coexisting antiferromagnetic order sets in at TN = 35 K.

Equations

$$I(\bm{q})\approx\Big{|}\sum_{\nu}(\bm{q}\cdot\bm{\epsilon}_{\nu})f_{\nu}(\bm{q}\pm\bm{Q}_{\rm{mod}})\Big{|}^{2}\delta(\bm{q}-\bm{G}\pm\bm{Q}_{\rm{mod}}),$$

Full text

In strongly correlated 5$f$-electron systems, interactions with comparable strength compete due to the extended nature of the wave-functions. This competition leads to a rich variety of exotic states, which can hardly be understood with conventional models from $d$- or 4$f$- electrons physics StewartRMP1 . In metallic U-based heavy fermion compounds with strong hybridization effects of the surrounding ligands, an exceptional coexistence of unusual phases occurs as, for example, in the hidden order superconductor URu2Si2. The nature of the $``$hidden order$"$ parameter responsible for the heat capacity anomaly is still debated, more than 30 years after its discovery MydoshRMP .

UPt2Si2 is a closely related intermetallic compound of the U$T_{2}M_{2}$ ($T$: transition metal, $M$: Si or Ge) family whose Pt-5$d$ electrons hybridize with the U-5$f$ states. UPt2Si2 adopts the CaBe2Ge2 crystal structure and orders magnetically at $T_{\rm{N}}$ = 35 K with wavevector $\bm{Q}_{\rm{M}}$ = (1 0 0), where ferromagnetic (FM) $ab$-planes are stacked antiferromagnetically (AFM) along $c$-axis, with a large ordered magnetic moment of $\approx$ 2 $\mu_{B}$ SteemanJPCM ; NieuwenhuysPRB ; ProkesArxive . Consequently, UPt2Si2 has long been considered a rare example of a uranium intermetallic compound with localized 5$f$-electrons where magnetism can be explained within a simple crystal-field level scheme NieuwenhuysPRB . Several recent studies Grachtrup2012 ; ElgazzarPRB ; Grachtrup2017 ; LeePRL , however, questioned the degree of electron localization in this system. High field measurements suggest that phase transitions under applied magnetic field should be understood in terms of Fermi surface effects Grachtrup2012 . This approach has been further supported by a density functional theory (DFT) calculation which favors a scenario where the 5$f$-electrons are mostly itinerant ElgazzarPRB . A recent inelastic neutron scattering study reveals a dual nature, both itinerant and local, in the magnetic excitation spectrum of UPt2Si2 at low temperatures LeePRL .

The crystal structure of UPt2Si2 and its relationship with electronic properties have been another important longstanding controversy Otop2004 ; SullowJPSJ ; SullowPRL . The x-ray diffraction measurements found large anisotropic thermal factors on one of the two Pt and Si sites, which were interpreted as an indication of strong crystallographic disorder SullowJPSJ . This purported disorder was thought to be responsible for the highly anisotropic resistivity with Anderson localization along the $\it{c}$-axis.

The origin and the nature of the disorder in this moderately heavy fermion system, however, has long been open to questions. Most U$T_{2}M_{2}$ compounds crystallize into the ThCr2Si2 type structure (space group $I$4$mmm$) and only a few, including UPt2Si2, have been reported to adopt the primitive CaBe2Ge2 type structure. Therefore, when a specific heat anomaly and a concomitant cusp in the electrical resistivity around room temperature were found BleckmannJMMM , it was anticipated that there could be a structural transition between the two closely related crystal structures as in UCo2Ge2 Endstra91 , leading to disorder. However, x-ray diffraction measurements on a polycrystalline sample did not find any indication of a structural change. Moreover, a comparative electrical transport study of as-grown and annealed samples showed that the effect of the presumed disorder does not depend on synthesis conditions or impurity level BleckmannJMMM .

Here, we report single crystal neutron and x-ray diffraction measurements, which reveal the discovery of an periodic lattice modulation that most likely results from a charge density wave (CDW) which develops in UPt2Si2 below $T_{s}$ $\approx$ 320 K. The appearance of the CDW is responsible for the heat capacity and electrical resistivity anomalies and resolves the controversy surrounding the putative “disorder”. The CDW wavevector ($\tau$ 0 0) shows an interesting temperature dependence, shifting up on cooling from a commensurate value of $\tau$ = 0.40 just below $T_{s}$ to lock-in value at $\tau$ $\approx$ 0.42 around 180 K. Around the same temperature, electrical transport data indicates a cross-over from a local to a coherent electronic state on cooling, suggesting an interplay between the CDW and Kondo-lattice physics in this system. This unusual type of CDW that involves strongly correlated U-5$f$ states appears at relatively high temperatures and coexists at low temperatures with an antiferromagnetic order. Intriguingly, the incommensurate CDW modulation wavevector, $\bm{Q}_{\rm{mod}}$ = ($\tau$ 0 0) with $\tau$ $\approx$ 0.4, resembles the incommensurate wavevector of URu2Si2 where the softening of magnetic excitation is observed  BareilleNatComm ; ElgazzarNatMat , implying that similar underlying physics may be at play.

Neutron diffraction measurements were carried out at the Wide Angle Neutron Diffractometer (WAND) and HB1 Polarized Triple-Axis Spectrometer (PTAX) at the High Flux Isotope Reactor (HFIR), ORNL and E4 two-axis diffractometer E4 at the Helmholtz-Zentrum Berlin. For the PTAX measurement, the neutron spin was aligned parallel to wavevector transfer, $\bm{q}$, so that nuclear and magnetic scattering were measured in Non-Spin-Flip (NSF) and Spin-Flip (SF) channel, respectively. The flipping ratio of PTAX was approximately 15. Synchrotron x-ray experiments were performed at the A1 beamline at Cornell High Energy Synchrotron Source (CHESS). We carried out density-functional theory (DFT) calculations via the Generalized Gradient Approximation (GGA) using Perdew-Burke-Ernzerhof (PBE) functional DFT1 , as implemented in the VASP package DFT2 ; SuppleExp .

Figure 1 (a) presents single crystal neutron diffraction data obtained at the WAND neutron diffractometer at T = 50 K, which is above TN = 35 K but well below the reported heat capacity anomaly temperature of $\approx$ 320 K BleckmannJMMM . We observe a pattern of satellite peaks offset from the original P4/nmm Bragg reflections by a wavevector $\bm{Q}_{\rm{mod}}$ = ($\tau$ 0 0) or (0 $\tau$ 0), where $\tau$ $\approx$ 0.42. We investigated the origin of these superlattice reflections by performing neutron polarization analysis on the (2 0.42 0) and (2 1.58 0) satellite peaks at 50 K at HB1 triple-axis spectrometer as shown in Fig. 1 (e) and (f), respectively. NSF scattering dominates the intensity of both reflections, confirming their non-magnetic, charge/lattice origin. Our findings thus correct the previous reports on the atomic disorder SullowJPSJ , which in fact resulted from the mis-assignment of the observed changes of the lattice principal Bragg reflection intensities to disorder rather than lattice modulation induced by CDW.

By virtue of the WAND diffraction patterns covering wide reciprocal space area, Fig. 1 (a), we can clearly identify the polarization of the observed atomic displacements. As shown in Figs. 1 (b) and (c), the intensities of longitudinal superlattice peaks, (H $\pm\tau$ 0 0), or, equivalently, (0 K $\pm\tau$ 0), are absent, or very weak compared to those of transverse superlattice peaks, ($\pm\tau$ K 0) and (H $\pm\tau$ 0). For example, the integrated intensity of a longitudinal (4.42 0 0) peak is about 240 times smaller compared to the corresponding transverse (4 0.42 0) peak. For lattice Bragg peaks near the diagonal (H H 0) direction we find all 4 satellite reflections with comparable intensity, Fig. 1 (d). There exist weak reflections at (H K 0) with H + K = odd inconsistent with P4/nmm symmetry, which, however, turns out to be due to an internal strain ProkesArxive .

The intensity of a satellite peak at a momentum transfer $\bm{q}$ can be expressed as KBLEE ; IgorPRB ,

[TABLE]

e.g. using Taylor expansion of Jacobi-Anger identity. Here, $\bm{\epsilon}_{\nu}$ is the displacement of atom ${\nu}$ in the unit cell, $f_{\nu}$ is the partial structure factor of atom ${\nu}$, and $\bm{G}$ is the reciprocal lattice vector. The $(\bm{q}\cdot\bm{\epsilon}_{\nu})$ prefactor in the satellite reflection intensity is responsible for the observed polarization dependence of the diffraction pattern, indicating that the atomic displacements, $\bm{\epsilon}_{\nu}$, in UPt2Si2 are mostly transverse, either along the $\it{a}$-axis direction for $\bm{Q}_{\rm{mod}}$ $\approx$ (0 0.4 0), or along the $\it{b}$-axis direction for $\bm{Q}_{\rm{mod}}$ $\approx$ (0.4 0 0). While the observed pattern of superlattice reflections can be regarded as overlaid patterns from two $\bm{Q}$-domains, within our resolution we did not detect any clear sign of tetragonal symmetry breaking in the main Bragg reflections.

To understand the atomic modulation pattern in real space, we performed DFT calculation by relaxing the atomic positions in a 4$\times$1$\times$1, 5$\times$1$\times$1, and 6$\times$1$\times$1 supercell. For the 5$\times$1$\times$1, case, we find a reduction of the total energy by 8.6 meV per unit cell with the presence of the CDW. This confirms the CDW to be unidirectional, that is, each Q-domain is described by a single wave vector. As shown in Fig. 2 (a), there are two kinds of Pt-Si layers in UPt2Si2. In the Si(2)-Pt(2)-Si(2) layer, a square lattice of Pt atoms is sandwiched between Si atoms, while in the Pt(1)-Si(1)-Pt(1) layer the situation is inverted. Our calculation shows that about 90% of displacement is in the $ab$-plane of the Si(2)-Pt(2)-Si(2) layer, in agreement with previously reported putative disorder SullowJPSJ . It is interesting that the lattice modulation of sandwiched metallic layers is quite ubiquitous TMDC1 as in the Pt-based layered superconductor LaPt2Si2 CDWSooranKim ; CDWUeda ; CDWFalkowski . It should be, however, noted that the period of the LaPt2Si2 superstructure is close to 3 unit cells compared to the $\approx$ 5 unit cell periodicity in UPt2Si2. Here, we expect the strongly correlated 5$f$ electron states to participate in the bonding due to hybridization between U-5$f$ and Pt-5$d$ states.

The calculated real space displacement pattern of Pt(2) layer is shown in Fig. 2 (b), where the dashed red rectangle represents the calculated supercell. When the CDW modulation wavevector is given along the $\it{a}$-axis, the atomic displacement is almost along the $\it{b}$-axis. Figure 2 (c) shows the Fourier transform of the Pt atomic positions, which can well explain the polarization of the observed reflections. Black, red, and blue dots correspond to principal Bragg peaks, satellites peaks from Fig. 2 (b) structure and its 90*∘*-rotated domain, respectively.

Figure 3 (a) presents the evolution of the superlattice Bragg peak at $\bm{q}$ = (2 $\tau$ 0) with temperature. The intensity, Fig. 3 (c), keeps increasing as the sample is cooled down, indicating gradual development of the superlattice modulation. Here, we find a second-order-like transition with a transition temperature Ts = 319(8) K, consistent with the previously reported heat capacity and electrical resistivity anomaly temperature BleckmannJMMM . Fitting the temperature dependence of the order parameter to expression $I\propto(T_{c}-T)^{2\beta}$, we obtain the critical exponent $\beta$ = 0.39(10) consistent with the mean-field behavior. Below $T_{\rm{N}}$ = 35 K, the CDW coexists with the antiferromagnetic order.

Surprisingly, as shown in Fig. 3 (b), the satellite peak moves away from the nearly commensurate position of $\tau$ = 0.400(5) observed around 300 K and becomes locked-in at an incommensurate value 0.418(2) below $\approx$ 180 K suggesting an unusual thermal evolution of the CDW. As the energy to lock-in to the commensurate lattice is a function of the CDW amplitude, which increases at low temperature, an opposite, commensurate to incommensurate lock-in of CDW upon cooling observed here is extremely rare TokuraPRL ; DeanPRX . Such a CDW wavevector shift suggests that the electronic structure of Uranium plays an important role at low temperatures.

Previously reported bulk measurements indicate that around the same temperature the $\it{a}$-axis resistivity shows a broad maximum AmitsukaPhysB ; SuppleBulk and the magnetic susceptibility begins to deviate from the Curie-Weiss law NieuwenhuysPRB . These observations suggest an onset of a “Kondo-lattice type” coherence that marks a crossover from local moments and weakly correlated electrons to correlated, heavy-electron behavior. Therefore, the unusual gradual change of $\bm{Q}_{\rm{mod}}$ from commensurate to incommensurate is likely related to a transition from local-moments towards a system with a dual character of the 5$f$ electrons LeePRL facilitated by the 5$f$-5$d$ hybridization and indicates an interplay between CDW and Kondo-lattice like coherence in UPt2Si2. While there are a few systems that show the mere coexistence of CDW and Kondo states StewartRMP1 ; KondoPRB05 ; KondoSciRep17 , we suggest UPt2Si2 as the first example of coupling between CDW and Kondo-lattice coherence effects.

The nature of the CDW has been investigated in more detail with a synchrotron x-ray diffraction measurement. A careful inspection of the diffraction pattern reveals that besides the first harmonic superlattice reflections with $\bm{Q}_{\rm{mod}}$, there are also weak superlattice reflections indexable as the second and the third harmonics located around $q$ = (4 3-$\delta$K 2), Fig. 4 (a). At high temperature, weak second and third harmonics nearly overlap at $q$ = (4 3.8 2) as higher harmonics are found at (4 3+2$\bm{Q}_{\rm{mod}}$ 2) and (4 5-3$\bm{Q}_{\rm{mod}}$ 2), while $\bm{Q}_{\rm{mod}}$ is commensurate, with $\tau$ = 0.4. As the temperature decreases, the second and third harmonic peaks get stronger in intensity and split apart as the incommensurability increases. Below Tcoh, their positions lock-in at the corresponding incommensurate values, 2$\tau$ and 3$\tau$, confirming their higher harmonics origin.

In Fig. 4 (b), we show the temperature dependence of the integrated intensities of the first three harmonics. The onset of the principal harmonics at Ts precedes a long tail of critical fluctuation precursor to the CDW at T $>$ Ts, which is emphasized in an energy-integrating measurement without an analyzer. There is also a hint of a second phase transition around 270 K, where the intensities of higher harmonics begin to increase, suggesting that the CDW order parameter becomes two-component, with two displacement polarizations. The existence of higher harmonics indicates that the atomic CDW modulation is not purely sinusoidal, as is also noted in DFT calculation.

It is provocative to compare UPt2Si2 with URu2Si2. While these systems seem to be very different at the first glance, one with mainly itinerant and the other with mainly local 5$f$ electron states, they indeed share many common features. They have closely related crystal structures and the same magnetic structures although the magnetic order in URu2Si2 seems to hover in the background, ready to appear under pressure, dilution, or magnetic field AmitsukaPRL ; KnafoNatComm . Thermoelectric measurements JohannsenPRB suggest that both systems show Fermi surface reconstruction with a partial gap opening in hidden order or antiferromagnetic states, respectively SuppleCompare .

Here, we find that both systems have two similar important wavevectors at play SuppleWavevectors . In URu2Si2, strong and coherent magnetic excitations are observed around the antiferromagnetic wavevector, $\bm{Q}_{\rm{0}}$ = (1 0 0), and the incommensurate wavevector, $\bm{Q}_{\rm{1}}$ $\approx$ (1$\pm$0.4 0 0) BroholmPRL ; WiebeNatPhy , whose origin is commonly believed to be Fermi surface nesting. UPt2Si2, as we find here, develops a CDW with $\bm{Q}_{\rm{mod}}$ = (0.4 0 0) and antiferromagnetic order with $\bm{Q}_{\rm{M}}$ = (1 0 0). Very recent STM measurement shows Moiré pattern in URu2Si2 possibly from a CDW related to $\bm{Q}_{\rm{1}}$ arXiv . It is tempting to suggest that similar wavevectors characterizing the respective ground states in URu2Si2 and UPt2Si2 might be more than just a pure coincidence, demanding further attention.

In conclusion, our diffraction experiments demonstrate that a CDW appears near room temperature in a strongly correlated U-based intermetallic compound, UPt2Si2. In the ground state, this CDW coexists with an AFM order, which appears at a much lower temperature. Higher-harmonic CDW reflections, which arise on cooling below $\approx$ 270 K indicate non-sinusoidal lattice modulation that is also supported by DFT calculations. Furthermore, the CDW wavevector exhibits unique temperature dependence. The commensurate-incommensurate transition on cooling and lock-in near the coherence temperature challenges our current understanding of CDW phenomena and requires taking into account the 5$f$-ligand hybridization. This suggests Kondo-lattice type physics and the dual nature of U 5$f$-electrons at play in UPt2Si2. The similarity of wavevectors characterizing electronic order in the two seemingly different systems, URu2Si2 and UPt2Si2, requires further studies, which might shed light on our understanding of the hidden order phase and unconventional superconductivity.