Evidence for a nematic component to the hidden-order parameter in URu2Si2 from differential elastoresistance measurements

Charge density waves tuned by biaxial tensile stress

The precise arrangement and nature of atoms drive electronic phase transitions in condensed matter. To explore this tenuous link, we developed a true biaxial mechanical deformation device working at cryogenic temperatures, compatible with x-ray diffraction and transport measurements, well adapted to layered samples. Here we show that a slight deformation of TbTe3 can have a dramatic influence on its Charge Density Wave (CDW), with an orientational transition from c to a driven by the a/c parameter, a tiny coexistence region near a = c, and without space group change. The CDW transition temperature Tc displays a linear dependence witha / c - 1 while the gap saturates out of the coexistence region. This behaviour is well accounted for within a tight-binding model. Our results question the relationship between gap and Tc in RTe3 systems. This method opens a new route towards the study of coexisting or competing electronic orders in condensed matter.

Dominant In-Plane Symmetric Elastoresistance in CsFe_{2}As_{2}

We study the elastoresistance of the highly correlated material CsFe_{2}As_{2} in all symmetry channels. Neutralizing its thermal expansion by means of a piezoelectric-based strain cell is demonstrated to be essential. The elastoresistance response in the in-plane symmetric channel is found to be large, while the response in the symmetry-breaking channels is weaker and provides no evidence for a divergent nematic susceptibility. Rather, our results can be interpreted naturally within the framework of a coherence-incoherence crossover, where the low-temperature coherent state is sensitively tuned by the in-plane atomic distances.

Electronic Nematicity in URu_{2}Si_{2} Revisited

The nature of the hidden-order (HO) state in URu_{2}Si_{2} remains one of the major unsolved issues in heavy-fermion physics. Recently, torque magnetometry, x-ray diffraction, and elastoresistivity data have suggested that the HO phase transition at T_{HO}≈ 17.5  K is driven by electronic nematic effects. Here, we search for thermodynamic signatures of this purported structural instability using anisotropic thermal expansion, Young's modulus, elastoresistivity, and specific-heat measurements. In contrast to the published results, we find no evidence of a rotational symmetry breaking in any of our data. Interestingly, our elastoresistivity measurements, which are in full agreement with published results, exhibit a Curie-Weiss divergence, which we however attribute to a volume and not to a symmetry-breaking effect. Finally, clear evidence for thermal fluctuations is observed in our heat-capacity data, from which we estimate the HO correlation length.

Hidden order and beyond: an experimental-theoretical overview of the multifaceted behavior of URu2Si2

This topical review describes the multitude of unconventional behaviors in the hidden order, heavy fermion, antiferromagnetic and superconducting phases of the intermetallic compound URu₂Si₂ when tuned with pressure, magnetic field, and substitutions for all three elements. Such 'perturbations' result in a variety of new phases beyond the mysterious hidden order that are only now being slowly understood through a series of state-of-the-science experimentation, along with an array of novel theoretical approaches. Despite all these efforts spanning more than 30 years, hidden order (HO) remains puzzling and non-clarified, and the search continues in 2019 into a fourth decade for its final resolution. Here we attempt to update the present situation of URu₂Si₂ importing the latest experimental results and theoretical proposals. First, let us consider the pristine compound as a function of temperature and report the recent measurements and models relating to its heavy Fermi liquid crossover, its HO and superconductivity (SC). Recent experiments and theories are surmized that address four-fold symmetry breaking (or nematicity), Isingness and unconventional excitation modes. Second, we review the pressure dependence of URu₂Si₂ and its transformation to antiferromagnetic long-range order. Next we confront the dramatic high magnetic-field phases requiring fields above 40 T. And finally, we attempt to answer how does random substitutions of other 5f elements for U, and 3d, 4d, and 5d elements for Ru, and even P for Si affect and transform the HO. Commensurately, recent theoretical models are summarized and then related to the intriguing experimental behavior.

One-component order parameter in URu2Si2 uncovered by resonant ultrasound spectroscopy and machine learning

The unusual correlated state that emerges in URu2Si2 below T HO = 17.5 K is known as "hidden order" because even basic characteristics of the order parameter, such as its dimensionality (whether it has one component or two), are "hidden." We use resonant ultrasound spectroscopy to measure the symmetry-resolved elastic anomalies across T HO. We observe no anomalies in the shear elastic moduli, providing strong thermodynamic evidence for a one-component order parameter. We develop a machine learning framework that reaches this conclusion directly from the raw data, even in a crystal that is too small for traditional resonant ultrasound. Our result rules out a broad class of theories of hidden order based on two-component order parameters, and constrains the nature of the fluctuations from which unconventional superconductivity emerges at lower temperature. Our machine learning framework is a powerful new tool for classifying the ubiquitous competing orders in correlated electron systems.

Magnetoelastoresistance in WTe2: Exploring electronic structure and extremely large magnetoresistance under strain

Strain describes the deformation of a material as a result of applied stress. It has been widely employed to probe transport properties of materials, ranging from semiconductors to correlated materials. In order to understand, and eventually control, transport behavior under strain, it is important to quantify the effects of strain on the electronic bandstructure, carrier density, and mobility. Here, we demonstrate that much information can be obtained by exploring magnetoelastoresistance (MER), which refers to magnetic field-driven changes of the elastoresistance. We use this powerful approach to study the combined effect of strain and magnetic fields on the semimetallic transition metal dichalcogenide [Formula: see text] We discover that WTe₂ shows a large and temperature-nonmonotonic elastoresistance, driven by uniaxial stress, that can be tuned by magnetic field. Using first-principle and analytical low-energy model calculations, we provide a semiquantitative understanding of our experimental observations. We show that in [Formula: see text], the strain-induced change of the carrier density dominates the observed elastoresistance. In addition, the change of the mobilities can be directly accessed by using MER. Our analysis also reveals the importance of a heavy-hole band near the Fermi level on the elastoresistance at intermediate temperatures. Systematic understanding of strain effects in single crystals of correlated materials is important for future applications, such as strain tuning of bulk phases and fabrication of devices controlled by strain.

Unveiling hidden multipolar orders with magnetostriction

Broken symmetries in solids involving higher order multipolar degrees of freedom are historically referred to as “hidden orders” due to the formidable task of detecting them with conventional probes. In this work, we theoretically propose that magnetostriction provides a powerful and novel tool to directly detect higher-order multipolar symmetry breaking—such as the elusive octupolar order—by examining scaling behaviour of length change with respect to an applied magnetic field h. Employing a symmetry-based Landau theory, we focus on the family of Pr-based cage compounds with strongly correlated f-electrons, Pr(Ti,V,Ir)₂(Al,Zn)₂₀, whose low energy degrees of freedom are purely higher-order multipoles: quadrupoles \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{O}}_{20,22}$$\end{document}O20,22 and octupole \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal{T}}_{xyz}$$\end{document}Txyz. We demonstrate that a magnetic field along the [111] direction induces a distinct linear-in-h length change below the octupolar ordering temperature. The resulting “magnetostriction coefficient” is directly proportional to the octupolar order parameter, thus providing clear access to such subtle order parameters.

Higher-order multipolar phases are unusual states that can form in correlated materials and are difficult to observe as they do not directly couple to conventional probes. Patri et al. theoretically show that angle-dependent magnetostriction measurements can probe quadrupolar and octupolar ordering.

AC elastocaloric effect as a probe for thermodynamic signatures of continuous phase transitions

Studying the response of materials to strain can elucidate subtle properties of the electronic structure in strongly correlated materials. Here, we focus on the elastocaloric coefficients, forming a second rank tensor quantity describing the relation between entropy and strain. In contrast to the better-known elastoresistivity, the elastocaloric effect is a thermodynamic quantity. Experimentally, elastocaloric effect measurements are demanding since the thermodynamic conditions during the measurement have to be well controlled. In this work, we present a technique to measure the elastocaloric effect under quasiadiabatic conditions. The technique is based on oscillating strain, which allows for increasing the frequency of the elastocaloric effect above the thermal relaxation rate of the sample. We apply the technique to Co-doped iron pnictide superconductors and show that the thermodynamic signatures of second order phase transitions in the elastocaloric effect closely follow those observed in calorimetry experiments. In contrast to heat capacity, elastocaloric effect measurements allow for the electronic signatures to be measured against a small phononic background even at high temperatures and in addition give information on the symmetry of the involved order parameters. This establishes the technique as a powerful complimentary tool for extracting the entropy landscape as a function of strain proximate to a continuous phase transition.

Divergence of the quadrupole-strain susceptibility of the electronic nematic system YbRu2Ge2

A wide range of strongly correlated quantum materials, including some high-temperature superconductors, exhibit “electronic nematic” phases, in which the electronic properties spontaneously break the rotational symmetry of the crystal. However, the role that the corresponding nematic fluctuations play in these complicated systems is unclear, motivating the search for simpler model systems. Here, we identify a particular 4f intermetallic material which exhibits ferroquadrupole order, YbRu2Ge2, as just such a model system. We also provide a robust and accurate method to probe the divergence of an important associated quantity, the quadrupole-strain susceptibility. The temperature dependence of this quantity provides insight into the nature of the interactions that lead to ferroquadrupole order, in this case, magnetoelastic coupling.

Ferroquadrupole order associated with local 4f atomic orbitals of rare-earth ions is a realization of electronic nematic order. However, there are relatively few examples of intermetallic materials which exhibit continuous ferroquadrupole phase transitions, motivating the search for additional materials that fall into this category. Furthermore, it is not clear a priori whether experimental approaches based on transport measurements which have been successfully used to probe the nematic susceptibility in materials such as the Fe-based superconductors will be as effective in the case of 4f intermetallic materials, for which the important electronic degrees of freedom are local rather than itinerant and are consequently less strongly coupled to the charge-carrying quasiparticles near the Fermi energy. In the present work, we demonstrate that the intermetallic compound YbRu2Ge2 exhibits a tetragonal-to-orthorhombic phase transition consistent with ferroquadrupole order of the Yb ions and go on to show that elastoresistivity measurements can indeed provide a clear window on the diverging nematic susceptibility in this system. This material provides an arena in which to study the causes and consequences of electronic nematicity.

Measurement of elastoresistivity at finite frequency by amplitude demodulation

Elastoresistivity, the relation between resistivity and strain, can elucidate the subtle properties of the electronic structure of a material and is an increasingly important tool for the study of strongly correlated materials. To date, elastoresistivity measurements have predominantly been performed with quasi-static (DC) strain. In this work, we demonstrate a method using AC strain in elastoresistivity measurements. A sample experiencing AC strain has a time-dependent resistivity, which modulates the voltage produced by an AC current; this effect produces time-dependent variations in resistivity that are directly proportional to the elastoresistivity, and which can be measured more quickly, with less strain on the sample, and with less stringent requirements for temperature stability than the previous DC technique. Example measurements between 10 Hz and 3 kHz are performed on a material with a large, well-characterized and temperature dependent elastoresistivity: the representative iron-based superconductor Ba(Fe_0.975Co_0.025)₂As₂. These measurements yield a frequency independent elastoresistivity and reproduce results from previous DC elastoresistivity methods to within experimental accuracy. We emphasize that the dynamic (AC) elastoresistivity is a distinct material-specific property that has not previously been considered.

Thermodynamic and electrical transport investigation of URu2Si2-x P x

Magnetic susceptibility, electrical resistivity, and heat capacity results are reported for the chemical substitution series URu₂Si_2-x P _x for [Formula: see text]. This study expands in detail on work recently reported in Gallagher et al (2016 Nat. Commun. 10712), which focused on the small x region of this substitution series. Measurements presented here reveal persistent hybridization between the f- and conduction electrons and strong variation of the low temperature behavior with increasing x. Hidden order and superconductivity are rapidly destroyed for [Formula: see text] and are replaced for [Formula: see text] by a region with Kondo coherence but no ordered state. Antiferromagnetism abruptly appears for [Formula: see text]. This phase diagram differs significantly from those produced by most other tuning strategies in URu₂Si₂, including applied pressure, high magnetic fields, and isoelectronic chemical substitution (i.e. Ru  →  Fe and Os), where hidden order and magnetism share a common phase boundary. Besides revealing an intriguing evolution of the low temperature states, this series provides a setting in which to investigate the influence of electronic tuning, where probes that are sensitive to the Fermi surface and the symmetry of the ordered states will be useful to unravel the anomalous behavior of URu₂Si₂.

Europium-based iron pnictides: a unique laboratory for magnetism, superconductivity and structural effects

Despite decades of intense research, the origin of high-temperature superconductivity in cuprates and iron-based compounds is still a mystery. Magnetism and superconductivity are traditionally antagonistic phenomena; nevertheless, there is basically no doubt left that unconventional superconductivity is closely linked to magnetism. But this is not the whole story; recently, also structural effects related to the so-called nematic phase gained considerable attention. In order to obtain more information about this peculiar interplay, systematic material research is one of the most important attempts, revealing from time to time unexpected effects. Europium-based iron pnictides are the latest example of such a completely paradigmatic material, as they display not only spin-density-wave and superconducting ground states, but also local Eu²⁺ magnetism at a similar temperature scale. Here we review recent experimental progress in determining the complex phase diagrams of europium-based iron pnictides. The conclusions drawn from the observations reach far beyond these model systems. Thus, although europium-based iron pnictides are very peculiar, they provide a unique platform to study the common interplay of structural-nematic, magnetic and electronic effects in high-temperature superconductors.

Direct bulk-sensitive probe of 5f symmetry in URu2Si2

The second-order phase transition into a hidden order phase in URu₂Si₂ goes along with an order parameter that is still a mystery, despite 30 years of research. However, it is understood that the symmetry of the order parameter must be related to the symmetry of the low-lying local electronic [Formula: see text]-states. Here, we present results of a spectroscopic technique, namely core-level nonresonant inelastic X-ray scattering (NIXS). This method allows for the measurement of local high-multipole excitations and is bulk-sensitive. The observed anisotropy of the scattering function unambiguously shows that the 5[Formula: see text] ground-state wave function is composed mainly of the [Formula: see text] with majority [Formula: see text] = [Formula: see text] + [Formula: see text] and/or [Formula: see text] singlet states. The incomplete dichroism indicates the possibility that quantum states of other irreducible representation are mixed into the ground state.

Analogy Between the "Hidden Order" and the Orbital Antiferromagnetism in URu_{2-x}Fe_{x}Si_{2}

We study URu_{2-x}Fe_{x}Si_{2}, in which two types of staggered phases compete at low temperature as the iron concentration x is varied: the nonmagnetic "hidden order" (HO) phase below the critical concentration x_{c}, and unconventional antiferromagnetic (AFM) phase above x_{c}. By using polarization resolved Raman spectroscopy, we detect a collective mode of pseudovectorlike A_{2g} symmetry whose energy continuously evolves with increasing x; it monotonically decreases in the HO phase until it vanishes at x=x_{c}, and then reappears with increasing energy in the AFM phase. The mode's evolution provides direct evidence for a unified order parameter for both nonmagnetic and magnetic phases arising from the orbital degrees-of-freedom of the uranium-5f electrons.

Field-induced spin-density wave beyond hidden order in URu2Si2

URu₂Si₂ is one of the most enigmatic strongly correlated electron systems and offers a fertile testing ground for new concepts in condensed matter science. In spite of >30 years of intense research, no consensus on the order parameter of its low-temperature hidden-order phase exists. A strong magnetic field transforms the hidden order into magnetically ordered phases, whose order parameter has also been defying experimental observation. Here, thanks to neutron diffraction under pulsed magnetic fields up to 40 T, we identify the field-induced phases of URu₂Si₂ as a spin-density-wave state. The transition to the spin-density wave represents a unique touchstone for understanding the hidden-order phase. An intimate relationship between this magnetic structure, the magnetic fluctuations and the Fermi surface is emphasized, calling for dedicated band-structure calculations.

The strongly-correlated electron system URu₂Si₂ possesses a hidden-order phase whose order parameter remains unidentified. Here, the authors demonstrate the development of spin-density-wave phases in URu₂Si₂ under high magnetic fields, providing a potential in-road to understanding this system.

Thermodynamic Measurement of Angular Anisotropy at the Hidden Order Transition of URu_{2}Si_{2}

The heavy fermion compound URu_{2}Si_{2} continues to attract great interest due to the unidentified hidden order it develops below 17.5 K. The unique Ising character of the spin fluctuations and low-temperature quasiparticles is well established. We present detailed measurements of the angular anisotropy of the nonlinear magnetization that reveal a cos^{4}θ Ising anisotropy both at and above the ordering transition. With Landau theory, we show this implies a strongly Ising character of the itinerant hidden order parameter.

Measurement of the B1g and B2g components of the elastoresistivity tensor for tetragonal materials via transverse resistivity configurations

The elastoresistivity tensor mij,kl relates changes in resistivity to strains experienced by a material. As a fourth-rank tensor, it contains considerably more information about the material than the simpler (second-rank) resistivity tensor; in particular, for a tetragonal material, the B1g and B2g components of the elastoresistivity tensor (mxx,xx - mxx,yy and 2mxy,xy, respectively) can be related to its nematic susceptibility. Previous experimental probes of this quantity have focused exclusively on differential longitudinal elastoresistance measurements, which determine the induced resistivity anisotropy arising from anisotropic in-plane strain based on the difference of two longitudinal resistivity measurements. Here we describe a complementary technique based on transverse elastoresistance measurements. This new approach is advantageous because it directly determines the strain-induced resistivity anisotropy from a single transverse measurement. To demonstrate the efficacy of this new experimental protocol, we present transverse elastoresistance measurements of the 2mxy,xy elastoresistivity coefficient of BaFe2As2, a representative iron-pnictide that has previously been characterized via differential longitudinal elastoresistance measurements.

Unfolding the physics of URu2Si2 through silicon to phosphorus substitution

The heavy fermion intermetallic compound URu₂Si₂ exhibits a hidden-order phase below the temperature of 17.5 K, which supports both anomalous metallic behavior and unconventional superconductivity. While these individual phenomena have been investigated in detail, it remains unclear how they are related to each other and to what extent uranium f-electron valence fluctuations influence each one. Here we use ligand site substituted URu₂Si_2-xP_x to establish their evolution under electronic tuning. We find that while hidden order is monotonically suppressed and destroyed for x≤0.035, the superconducting strength evolves non-monotonically with a maximum near x≈0.01 and that superconductivity is destroyed near x≈0.028. This behavior reveals that hidden order depends strongly on tuning outside of the U f-electron shells. It also suggests that while hidden order provides an environment for superconductivity and anomalous metallic behavior, it's fluctuations may not be solely responsible for their progression.

The heavy fermion compound URu₂Si₂ displays a hidden order phase and superconductivity at low temperatures. Here, the authors perform substitution studies—partially replacing silicon with phosphorus—and study the effects on hidden order and superconductivity.