Abstract
Strong electron correlations lead to a variety of distinct ground states, such as magnetism, charge order or superconductivity. Understanding the competitive or cooperative interplay between neighbouring phases is an outstanding challenge in physics. CeRhIn5 is a prototypical example of a heavy-fermion superconductor: it orders anti-ferromagnetically below 3.8 K, and moderate hydrostatic pressure suppresses the anti-ferromagnetic order inducing unconventional superconductivity. Here we show evidence for a phase transition to a state akin to a density wave (DW) under high magnetic fields (>27 T) in high-quality single crystal microstructures of CeRhIn5. The DW is signalled by a hysteretic anomaly in the in-plane resistivity accompanied by non-linear electrical transport, yet remarkably thermodynamic measurements suggest that the phase transition involves only small portions of the Fermi surface. Such a subtle order might be a common feature among correlated electron systems, reminiscent of the similarly subtle charge DW state in the cuprates.
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Introduction
Unconventional superconductivity (SC) often appears in strongly correlated materials on the border of magnetism. These same correlations also produce a variety of competing ground states, which can act as new windows to understanding the physics of these correlated materials. Metals containing Ce often show strong electron correlations due to the proximity of the 4f state to the Fermi energy, leading to strong coupling with the conduction electrons and a multitude of competing states. Since these states are typically close in energy, phase transitions can be induced by small environmental changes such as magnetic fields or pressure. Thus, they pose ideal candidates to investigate the interplay of various correlated ground states.
The heavy-fermion superconductor CeRhIn5 received significant attention due to its particularly feature-rich phase diagram1: the local-moment anti-ferromagnetism (AFM, TN~3.8 K) is suppressed by moderate pressure and ultimately quenched by a first-order phase transition at Pc~17 kbar in zero magnetic field. Magnetic fluctuations associated with this quantum critical point are thought to induce unconventional SC with a relatively high transition temperature Tcmax~2.1 K2, similar to the related compounds CeCoIn5 (refs 3, 4, 5, 6) and CeIrIn5 (ref. 7). The main finding of our study is the identification of a previously undetected state akin to a density wave (DW) above a critical field Hc~27 T for fields applied along the c axis (‘DW’, Fig. 1a). The SC and the DW phases share intriguing similarities: both arise in the vicinity of a quantum critical point bounding the AFM phase, and the transition from the AFM metal into both the DW and the SC states5 appears to be first order. As charge DWs (CDWs) and spin DWs (SDWs) are mainly observed in low-dimensional electronic systems, the DW in CeRhIn5 most likely appears due to an instability of a low-dimensional Fermi surface8,9 that coexists with three-dimensional (3D) Fermi surface sheets10.
Our observation of DWs in close proximity to unconventional d-wave SC in CeRhIn5 is reminiscent of the recent developments in the cuprate high-Tc superconductors: their particularly complex phase diagram hosts a variety of ground states of correlated electron matter, some coexisting and others competing with high-temperature SC11,12. One of these states is the spin and charge order, which was initially thought to be a special case for the La-based cuprates such as La2−xBaxCuO4 (refs 12, 13). Yet recently, fluctuating charge order was also observed in YBa2Cu3O7−δ (refs 11, 14) and the electronic structure observed by quantum oscillation measurements agrees with the broken translational symmetry due to charge order15. Moderate magnetic fields have been observed to influence the charge order, as suggested by X-ray diffraction and nuclear magnetic resonance (NMR) experiments16,17. While the lattice distortion and the charge modulation may have a profound effect on the electronic structure of the cuprates, by themselves these DWs are subtle features, which could only very recently be identified experimentally. This naturally leads to the question if similarly subtle charge order in the vicinity of unconventional SC is a common feature among several other classes of materials. Our results thus suggest that the coexistence, or competition, between charger order and unconventional SC might be a more common feature among strongly correlated systems than previously thought.
Results
Resistance anomaly signals first-order transition
The key to detecting this transition in our study is the focused ion beam (FIB)-based fabrication of single crystal microstructures. This technique may become a powerful tool to detect subtle changes in the electronic matter of a wide range of materials (Fig. 1b). Details about device fabrication and an overview of the reproducibility of the results are given in the Methods section and in refs 18, 19.
Figure 2a shows the magnetoresistance ρa(H) and ρc(H) for currents flowing along both the in-plane and the inter-plane direction, respectively, at 380 mK for fields at an angle 20° away from the c axis. These traces immediately highlight the key observation of this study: the in-plane resistivity shows a step-like transition at 34 T to a high-resistive state (red), which more than doubles its value. This jump is followed by a region of negative magnetoresistance up to a minimum in resistivity at 42 T. On decreasing the magnetic field, the resistivity follows an upper hysteresis branch until it drops back onto the lower branch (blue) at 28 T. At the same time, the out-of-plane resistivity ρc(H) remains featureless, indicating that mainly those electrons responsible for the in-plane transport participate in the transition. The magnetic torque τ=M × H was measured simultaneously with the transport samples using a capacitive cantilever technique (Fig. 2b). It shows pronounced de Haas–van Alphen oscillations, again highlighting the crystal quality, but otherwise remains featureless at the fields at which the jumps occur. Consequently, the Ce-4f anti-ferromagnetic order remains unaffected and the transition occurs well within the AFM dome, in agreement with previous specific heat measurements20. Remarkably, the density of states at the Fermi level, therefore, does not change notably as often observed when the Fermi surface is reconstructed by a CDW or SDW. This indicates that the gapped Fermi surface sheet hosts comparatively light electrons, while the heavier sheets remain unaffected by the transition, which may be interpreted as a manifestation of the ‘two fluid’ description of heavy-fermion metals21. The contrasts between the subtle signs for a phase transition in the torque measurements on macroscopic crystals and the sharp features in the charge transport measurements on microstructures is remarkable. As multiple Fermi surface sheets with both two- and three-dimensional character are present in the material, the strong response of ρa to the gapping of comparably few charge carriers suggests a large in-plane electron mobility of the gapped band.
Angular dependence of the critical field
We have studied the angular dependence of the critical field Hc by tilting the magnetic field in the (a,c) plane (Fig. 3). The transition moves to higher field values as the field is rotated towards the a-direction. The upturn in angle does not follow a simple functional dependence; in particular, it does not behave as cos(θ)−1 as one would expect if the out-of-plane field component would be the relevant parameter. Above angles of 60°, the transition becomes unobservable. This is accompanied by a gradual reduction of the relative amplitude of the resistive jump, that is, from its maximum at 20° to zero around 60°, indicating that at higher angles away from the c axis the charge order becomes energetically unfavourable. DWs in 3D metals arise due to a Fermi surface nesting enhancement of the susceptibility. One plausible scenario for a field-induced DW transition would be a progressive enhancement of Fermi surface nesting as the field deforms it in the presence of anisotropic and band-dependent g-factors. Such a mechanism would result in a highly non-trivial angular dependence for the critical magnetic field with respect to the crystallographic axes. The interaction between the f-electrons and the conduction electrons is known to give rise to Lifshitz transitions that change the Fermi surface topology by tuning a parameter such as pressure22 or chemical substitution23. Yet, in contrast to the appearance of the DW, these transitions commonly involve a change in the anti-ferromagnetically ordered and localized 4f states.
Non-linear transport in the DW state
We interpret our observation as evidence for a field-induced DW transition involving in-plane electrons, which does not significantly affect either the c axis resistivity or the magnetic torque. One of the hallmarks of DW phenomena is a pronouncedly non-linear conductivity associated with an additional conduction channel due to the sliding of the DW state in parallel to the remaining, un-gapped quasiparticles8,24. For a DW at small electric fields, Coulomb interactions pin the DW to crystal defects, and the remaining free quasiparticles constitute the only accessible conduction channel. The pinning potential can be overcome by the application of an electric field beyond a certain threshold value Ec, leading to the sliding motion of the DW that contributes to the conductivity. At the critical magnetic field, we observe the onset of this type of non-linearity (Fig. 4). The large hysteresis allows us to make a strong argument: by preparing the system in the lower (blue) and upper (red) branch, we can compare the differential resistance under the exact same conditions of field and temperature. This also allows us to exclude artifacts such as field-dependent contact resistances and self-heating, as these extrinsic sources of error would not depend on the hysteretic state of the sample. The non-linearity was observed in the whole experimentally accessible high-field region up to 45 T, indicating the persistence of the DWs above the hysteretic region.
While the low-resistance branch (blue) remains ohmic up to high current densities, a pronouncedly non-linear differential resistance is observed in the upper branch (red). A well-defined critical current Ic=190 μA separates the regions of pinned and sliding DW. Using the sample length l=34.6 μm and the low-current resistance R=280 mΩ in the upper branch, one finds a depinning-threshold electric field Ec=RIc/l=15.2 mV cm−1. Such Ec of the order of 10 mV cm−1 are commonly observed in incommensurate CDW systems such as NbSe3 and TaS3 (refs 8, 9).
Discussion
The large hysteresis presents us with the rare opportunity of studying a ‘supercooled’ electronic system and to compare the system with and without the DW under the same thermodynamic conditions. At the field-induced transition, the Fermi surface is reconstructed as indicated by a change in the Shubnikov–de Haas frequency spectrum (Fig. 5). Besides a change in amplitude owing to the overall increase of the signal at the transition, two frequencies around 4,680 and 5,400 T appear in the high-field state, indicating the appearance of a new Fermi surface cross-section25. By using a smaller field window for the frequency analysis of the metastable region only (Fig. 5, middle), we can show that these frequencies are absent in the lower and present in the upper branch in the same field region. Since a reconstruction of the Fermi surface indicates the thermodynamic nature of the observed phase transition, the hysteretic appearance of this additional frequency firmly establishes it as a first-order phase transition.
Interestingly, a low-frequency quantum oscillation is found to be strongly enhanced above the DW transition at smaller field angles (Fig. 2). Its amplitude shrinks on increasing the magnetic field, contrary to the usual amplitude growth with field as described by the Lifshitz-Kosevich formalism25. One possible explanation for this unusual behaviour may be the proximity to the field-induced suppression of the AFM order at 50 T and the associated delocalization of the f-electrons. The onset of the AFM order at TN localizes the f-electrons and removes most of their spectral weight from the Fermi surface26. As the magnetic field suppresses the AFM, the f-electrons are expected to gradually delocalize, which leads to a field-dependent increase of the effective mass and thus to a suppression of the oscillatory amplitude at constant temperature. A similar enhancement of effective masses on approaching the AFM quantum critical point by pressure has been previously observed27. This anomalous reduction in amplitude may be related to the previously reported de Haas–van Alphen amplitude anomalies in CeRhIn5 within this field and temperature range28.
An important aspect of the observed DW is to what extent the order includes charge and spin degrees of freedom. Conventional incommensurate CDW and SDW lead to similar non-linear transport characteristics. While the experiment cannot unambiguously distinguish between these states, the experimental evidence seems more consistent with charge order. First of all, SDW-like anti-ferromagnetic transitions often yield pronounced changes in the anisotropic susceptibility χ (ref. 29), which were not detected by our magnetic torque measurements. The sample-size dependence of the hysteretic region presents another indication for a CDW-like state at high magnetic fields. A CDW arises from a coupling of the electronic system to the crystal lattice, and its appearance is accompanied by a lattice distortion. A sample of significantly larger dimensions (Fig. 6, width: 24 μm, thickness: 5.6 μm) was studied under the same conditions. In this larger sample, the hysteretic region is smaller and the transition is broadened compared with the step-like transition in the small sample. One plausible explanation would be a difference in strain relaxation in the two samples. As both microstructures are similarly coupled with the substrate, the elastic force of the mounting glue might impede a structural transition in the small sample while the large sample may more easily relax towards the new equilibrium. This would be in agreement with the observed change in the hysteresis and point towards a coupling of the order parameter to the lattice.
Yet, in the presence of strong magnetic fields, the nesting vector q will be field dependent due to the Zeeman-splitting of the Fermi surface and a pure CDW will become unstable in a similar way as a superconductor transitioning into an Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. It was shown that in this situation an unconventional DW involving both charge and spin degrees of freedom may emerge as the new ground state at high magnetic fields30. Such an unconventional charge order has been observed in high magnetic fields in low-dimensional materials such as (Per)2Pt(mnt)2, which shows remarkable similarities to our findings31. Such DW instabilities are well understood in low-dimensional materials arising from the enhancement of the susceptibility on reducing the dimensionality32. Yet, CeRhIn5 shows nearly isotropic transport properties in zero field (Fig. 7), and further theoretical treatment will be required to understand this phenomenon in the context of low-dimensional Fermi surface sheets coexisting with strongly 3D sheets10 in the presence of sizable magnetic fields. In addition, the driving force of this instability will have to be identified and the observation of a DW emphasizes the need to consider the often neglected role of electron–phonon coupling in heavy-fermion systems. Such coupling, however, is a natural consequence of the sensitivity of the ground state to small changes in bond lengths2.
Given the differences in the electronic structure between CeRhIn5 and YBa2Cu3O7−δ, the signatures of a high-field charge order in both systems share surprising similarities: there is no clear anomaly at the charge order transition in CeRhIn5 observed in the torque measurements, contrary to conventional CDW materials such as NbSe3 (ref. 33). It remains an open question as to why the CDW in YBa2Cu3O7−δ can be well observed by NMR or X-ray diffraction experiments, yet heat capacity, resistivity or magnetic torque studies do not reveal any anomaly at the field-induced transition34. This may result from very small changes in entropy at the transition or be related to the intrinsic time constants of the different measurement techniques, and thus their different sensitivities to fluctuating order. It suggests that other experimental techniques such as high-field NMR should be performed in CeRhIn5, to clarify the so far unknown microscopic structure of the DW. The absence of a magnetization anomaly indicates that the field-induced charge-ordered state in both compounds is a similarly ‘subtle’ order, involving only a relatively small fraction of light carriers. This highlights a different aspect of the similarity between cuprates and Ce-based ‘115’ materials: Apparently, there is an almost undetectably small number of electrons in low-dimensional orbits present in both material classes that can drive a phase transition in moderately high magnetic fields perhaps involving the lattice degrees of freedom, yet without strong influence on the remaining carriers.
Methods
FIB sample fabrication
All presented devices were extracted from macroscopic (>3 mm) single crystals that were characterized by X-ray and checked for signs of indium impurities in a Superconducting Quantum Interference Device (SQUID). The single crystals were then analysed in a scanning electron microscope for surface defects. A clean, scratch-free part of the crystal was selected and the crystal was oriented using X-ray diffraction. In an initial step, a coarse (a,c) lamella with dimensions of ~60 × 60 × (7–8) μm3 was carved out of the crystal bulk with a 2.5 nA, 30 kV Ga2+ FIB (FEI Helios Nanolab 600i). The (a,b) face of the crystal was perpendicular to the ion beam during this process. In the next step, the sample was rotated in situ by 52°, to undercut the slice at 2.5 nA, 30 kV. In the last preparation step, the lamella was polished carefully using a lower current of 430 pA, 30 kV to remove potential damage layers from the high flux beam and to thin the lamella to its final thickness intended for the respective device (2–5 μm).
The lamella was then transferred onto a Si chip with photolithographically prepared Au leads, connecting the final structure to bonding pads. The crystal slice was extracted ex situ and glued into the centre of this chip using a small amount of epoxy glue. In a final procedure, the sample was reintroduced into the FIB, for further carving and electric contacts. The sample was contacted with ion-assisted chemical vapor deposition (ia-CVD) deposited Pt in the FIB, at high ion flux to minimize resistance of the leads (2.5–9 nA, 30 kV). The typical resistance of each lead prepared this way was 20 Ω. In addition, the square platelet was further cut into the shape shown in the main Fig. 1 using low currents (430 pA, 30 kV). Preparing such a sample requires 4–5 full days of FIB work.
This microstructuring procedure is essential for this experiment for four main reasons: The low resistivity of CeRhIn5 leads to very small signals in typically milimetre-sized as-grown crystals at the low currents required to avoid self-heating at low temperatures. After microstructuring the crystal into a long (~30 μm) and thin (~5.5 μm2) bar, the resistance is greatly enhanced by the geometrical factor, facilitating high-precision measurements in the challenging environment of strong magnetic fields. In addition, the FIB-deposited contacts were previously shown to perform reliably in pulsed-field conditions. While the zero-field resistivity at low temperatures is only moderately anisotropic, significant and field-dependent anisotropy develops in magnetic fields. Therefore, it is essential to ensure a homogeneous current profile in the sample to avoid field-dependent current redistribution due to changing anisotropy. The FIB microbars featuring side-contacts on a long bar are ideally suited to reliably distinguish between in- and out-of-plane resistivity. Furthermore, high current densities can be achieved in such devices by applying only moderate currents due to the micron-sized cross-section. While in principle low-resistance contacts could sustain similar current densities with some self-heating, currents of a few Ampere would not be feasible in high magnetic fields due to the extreme Lorentz force acting on the wires. As CeRhIn5 is a good conductor, sustaining the depinning electric field Ec requires a substantial current density that would be impractical in macroscopic crystals. In addition, in micron-sized samples, single domains may be isolated, allowing the study of intrinsic properties without the complications arising from domain dynamics.
Reproducibility
The FIB technique used to microstructure the single crystals of CeRhIn5 studied here involves high-energy ion irradiation, and thus one has to take great care to investigate significant defects or changes to the material that may have been introduced by this preparation. We have found no sign of material degradation in any of the samples and in fact our current experimental evidence points towards a consistently high quality of such devices prepared out of heavy-fermion compounds. We observed consistently large residual resistivity ratios in all prepared devices well exceeding 200. The sample mainly used throughout this study showed residual ratios of ρa(300 K)/ρa(80 mK)~258 and ρc(300 K)/ρc(80 mK)~268. Another evidence for the high quality of our devices is the observation of Shubnikov–de Haas oscillations in fields as low as 4 T. This oscillatory phenomenon originates from the quantization of electronic orbits on the Fermi surface into Landau levels and is known to be exceptionally sensitive to defects (ωcτ~1-criterion)25. Its observation is generally accepted as strong evidence for very clean samples with a long mean-free path. This led us to conclude that the FIB process did not introduce significant scattering. Nine FIB-prepared samples of different shapes and sizes were produced and characterized in fields up to 14 T, and all of them have shown Shubnikov–de Haas oscillations.
Particularly, the possibility of sample artifacts mimicking the reported DW transition was carefully checked. During the FIB process, some peculiar type of defect such as a microcrack could have been introduced into the sample, causing extrinsic physical parameters such as field and temperature to conspire in a way to artificially create the observed resistivity jump. This scenario is highly unlikely, as the jump is accompanied by changes of the Shubnikov–de Haas spectrum, which indicates the thermodynamic nature of the transition. To nonetheless exclude such extrinsic phenomena, a total of nine samples were prepared and four of them studied in high fields, in d.c. fields at the National High Magnetic Field Laboratory in Tallahassee (NHMFL), as well as in pulsed fields in the Pulsed Field Facility at Los Alamos National Laboratory (NHMFL-PFF). All studied samples consistently showed this transition at the same critical fields; however, no hysteresis was observed in pulsed fields, indicating the highly metastable character of the lower branch (Fig. 8). At high ramp rates and high currents, the hysteresis also disappeared in d.c. fields as expected for metastability in first-order transitions.
Additional information
How to cite this article: Moll, P. J. W. et al. Field-induced density wave in the heavy-fermion compound CeRhIn5. Nat. Commun. 6:6663 doi: 10.1038/ncomms7663 (2015).
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Acknowledgements
FIB and scanning electron microscope work were supported by EMEZ and ScopeM at ETH Zurich. L.B. is supported by DOE-BES through award DE-SC0002613. F.B. is supported by DOE-``Science in 100T''. We thank Bertram Batlogg, Brad Ramshaw, Joe Thompson, Suchitra Sebastian, Christoph Geibel and Michael Nicklas for interesting discussions and Philippe Gasser for supporting the FIB work. Work at Los Alamos was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. The NHMFL facility is funded through the US NSF Cooperative Grant No. DMR-1157490, the DOE, and the State of Florida.
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P.J.W.M. and F.R. designed the experiment. P.J.W.M., L.B. and B.Z. performed the d.c. field measurements, and P.J.W.M. and F.B. the pulsed-field experiments. E.D.B. grew and characterized the crystals. S.G. performed the low-field dilution refrigerator resistivity sample characterizations. P.J.W.M., L.B. and F.R. analysed the data and wrote the paper.
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Moll, P., Zeng, B., Balicas, L. et al. Field-induced density wave in the heavy-fermion compound CeRhIn5. Nat Commun 6, 6663 (2015). https://doi.org/10.1038/ncomms7663
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DOI: https://doi.org/10.1038/ncomms7663
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