Abstract
Many correlated electron materials, such as hightemperature superconductors^{1}, geometrically frustrated oxides^{2} and lowdimensional magnets^{3,4}, are still objects of fruitful study because of the unique properties that arise owing to poorly understood manybody effects. Heavyfermion metals^{5}—materials that have high effective electron masses due to those effects—represent a class of materials with exotic properties, ranging from unusual magnetism, unconventional superconductivity and ‘hidden’ order parameters^{6}. The heavyfermion superconductor URu_{2}Si_{2} has held the attention of physicists for the past two decades owing to the presence of a ‘hiddenorder’ phase below 17.5 K. Neutron scattering measurements indicate that the ordered moment is 0.03μ_{B}, much too small to account for the large heatcapacity anomaly at 17.5 K. We present recent neutron scattering experiments that unveil a new piece of this puzzle—the spinexcitation spectrum above 17.5 K exhibits wellcorrelated, itinerantlike spin excitations up to at least 10 meV, emanating from incommensurate wavevectors. The large entropy change associated with the presence of an energy gap in the excitations explains the reduction in the electronic specific heat through the transition.
Main
The central issue in URu_{2}Si_{2} concerns the identification of the order parameter that explains the reduction in the specific heat coefficient, γ=C/T, and thus the change in entropy, through the transition at 17.5 K (ref. 6). Numerous speculations about the ground state have been advanced, from quadrupolar ordering^{7}, to spindensity wave formation^{8}, to ‘orbital currents’^{9} to account for the missing entropy. Here, we present coldneutron timeofflight spectroscopy results that shed some light on the ‘hiddenorder’ (HO) in URu_{2}Si_{2}. We have carried out experiments above and below the ordering temperature to measure how the spin excitations evolve. It is clear from our data that above T_{0} the spectrum is dominated by fast, itinerantlike spin excitations emanating from incommensurate wavevectors at positions located 0.4a^{*} from the antiferromagnetic (AF) points. From the group velocity and temperature dependence of these modes, we surmise that these are heavyquasiparticle excitations that form below the ‘coherence temperature’ and play a crucial role in the formation of the heavyfermion and HO states. The gapping of these excitations, which corresponds to a loss of accessible states, accounts for the reduction in γ through the transition at 17.5 K.
Figure 1 shows the excitation spectrum of URu_{2}Si_{2} at 1.5 K in the H00 plane. The characteristic gaps at ∼2 meV at the AF zone centre (1,0,0) and ∼4 meV at the incommensurate wavevectors (0.6,0,0) and (1.4,0,0) have been known for some time^{10}. The incommensurate wavevector corresponds to a displacement of ∼0.4a^{*} from the AF zone centres (that is, where h+k+l=an odd integer, and is thus forbidden in the bodycentredcubic chemical structure). A scenario for this modesoftening at the incommensurate position was previously described with a model based on oscillatory exchange constants between near neighbours (not uncommon for Ruderman–Kittel–Kasuya–Yoshidatype interactions)^{10,11}.
Figure 2 shows our new neutron results at 20 K in the same H00 plane. Above the phase transition, the sharp spin waves evolve into weak quasielastic spin fluctuations at the zone centre (1, 0, 0), and strong excitations at the incommensurate positions (1±0.4,0,0). We have considered the possibility that these incommensurate excitations may be due to magnetovibrational scattering. This can arise from a shift of the phonon excitations at (2, 0, 0) to (1±0.4,0,0) as allowed through the neutron scattering crosssection for magnetoelastic coupling^{12}. However, with the small moment size of this system, it is improbable that such a scattering process is being observed. It was also originally reported^{10} that the incommensurate excitations were just quasielastic fluctuations; constant Q scans on a tripleaxis spectrometer resolved a peak at a finite energy of 0.6 THz=2.5 meV and a decrease in intensity as a function of energy typical of an overdamped response. What was previously unknown was that the quasielastic fluctuations were only the lower limit of a band of highvelocity spin excitations that extend well above the upper limit of the sharp collective spin excitations of the ordered phase. As the temperature is increased, the (1, 0, 0) fluctuations decrease, and the incommensurate fluctuations remain approximately constant to at least 25 K (ref. 13).
The implications of this discovery are as follows. (1) The incommensurate excitations have a welldefined structure as a function of Q, and thus the electrons are highly correlated above 17.5 K. This is completely unexpected for a system of localized moments in a paramagnetic state, but similar dynamics above the Néel temperature, T_{N}, (with the formation of paramagnons^{14}, for example) has been observed with neutron scattering in other itinerant electron systems such as chromium and V_{2}O_{3} (refs 14, 15). (2) The dispersion is such that the maximum group velocity is at least a factor of two larger than the maximum group velocity of the excitations in the ordered state. A significant restructuring of the Fermi surface must be responsible for the HO state. (3) The gapping of these strong spin fluctuations below 17.5 K must provide a considerable portion of the entropy removal at the transition. These excitations have a structure in reciprocal space such that they occur at several symmetryrelated wavevectors within the first Brillouin zone. The amount of phase space occupied by these excitations is greater than those at the (1, 0, 0) AF zone centre. Figure 3 shows constant energy cuts of the excitations in the H0L plane to emphasize this. Note the weak AF fluctuations at (1, 0, 0) and (2, 0, 1) in comparison with the excitations at the incommensurate positions (see Fig. 4). Also note how a cone of scattering develops at the incommensurate positions, whereas the AF zone centre simply decreases in intensity as energy transfer is increased. For comparison, a cut of the spinwave spectrum at 1.5 K in the same energy window is also shown. The results at 20 K suggest a continuum of excitations within a cone of scattering (as expected for lowlying excitations at the Fermi surface), as opposed to the sharp excitations that are gapped below 17.5 K.
An estimate of the contribution to the electronic specific heat term from the removal of these lowenergy spin fluctuations can be made through an analysis of the spectrum at 20 K. The specific heat of a cone of fast excitations centred on incommensurate wave vectors and extending out by an inverse correlation length κ=ξ^{−1}, is given by
where v_{a} is the cell volume, ρ_{0} is the density of states and f(E)=coth(E/2k_{B}T), where k_{B} is the Boltzmann constant. The upper limit in the energy integral has been taken to be E_{max}=k_{B}T (above this energy, the specific heat contribution is small because df/dT approaches zero exponentially).
If we assume that the density of states is the inverse of a large damping in energy Γ=c ξ^{−1}, then the specific heat becomes
where c is the characteristic spinwave velocity. Previous work has shown that the excitations lie within the HK0 plane around (0.6,0,0) (ref. 16). Note that owing to the dispersion of the spin excitations, the contribution to the heat capacity is linear in temperature, which is exactly the same powerlaw dependence for the electronic specific heat. This term arises owing to these fast itinerant spin excitations in URu_{2}Si_{2} and is the reason for the enhanced linear component of the specific heat above T_{0}=17.5 K. The calculation of a gap opening in the spinexcitation spectrum is equivalent to a calculation of the gap opening in the Fermi surface. This is demonstrated by the similarity of the matrixelementweighted density of spinwave states and infrared spectroscopy measurements, which measure charge excitations^{11,17}. The spin and charge degrees of freedom are strongly coupled.
A characteristic correlation length and spinwave velocity must be extracted from our data to complete this calculation. Our data have been fitted to a standard formula for paramagnetic scattering from a system of correlated spins, also known as the Chou model^{18}. This has been used to extract correlation lengths and spinwave velocities for excitations in the superconducting cuprates, for example^{19,20,21}. With this model, S(q,ω) is represented as
where ℏ is the reduced Planck constant, A is an arbitrary constant, (where δ is the incommensurate wavevector), κ is the inverse of the correlation length ξ, and the frequencies of the damped spin waves are ω_{q}=c q, where c is the spinwave velocity (Γ=κ c). Fitting with these parameters yields a correlation length of ξ=14(2) Å and c=45(10) meV Å (see Fig. 2). This value is similar to the Fermi velocity, reported as ∼8.84×10^{3} m s^{−1} (35 meV Å) (ref. 22), illustrating the itinerant nature of the excitations. It is interesting to note that there is a curious intersection of cuprate and heavyfermion physics with this analysis—the fast excitations seen in the superconductor La_{1.86}Sr_{0.14}CuO_{4} (LSCO), which have an effective mass near an electron mass, seem similar to the excitations in URu_{2}Si_{2}, albeit with a higher velocity because of their lighter effective mass^{19}.
The contribution of the spin excitations to the specific heat with these parameters for URu_{2}Si_{2} is 220±70 mJ mol^{−1} K^{−2} at 20 K. Comparison of this with the data in Fig. 5 shows a good agreement, with a γ of 155(5) mJ mol^{−1} K^{−2}. The gapping of these excitations leads to a dramatic reduction in γ to 38(2) mJ mol^{−1} K^{−2} (refs 23, 24). This is naturally explained through the gaps in the spinexcitation spectrum, which has minima at 2 and 4 meV. The average thermal energy below 17.5 K is less than 2 meV, and thus most of the spin excitations are inaccessible. This is the reason for the reduction of γ through the phase transition. The origin of the size of the specific heat jump at T_{0} is still unresolved at this point, although its lowtemperature limit arises from the removal of the spin contribution to leave a weak remanent ungapped electronic band. A gap opens in the spinexcitation spectrum with a value of ∼110(10) K, but the identity of this ordered state is still unknown^{16}. Through these measurements, we do know that the nature of the transition at T_{0} seems to be dominated by itinerant electron physics rather than localized crystal field effects. This is why URu_{2}Si_{2} shows a transition between itinerant and localized behaviour at T_{0}, as shown through thermal conductivity measurements^{22}. Indeed, we have seen no evidence of crystal field excitations up to at least 10 meV∼110 K. The complicated crystal field schemes that have been used to explain the heat capacity anomaly at 100 K, the HO state and the anomalous thermal expansion at 60 K need to be reexamined within this framework. Our measurements show that the first excited state, if allowed by selection rules, must exist above 200 K.
The role that these excitations play with the formation of the heavyfermion state can be examined through their temperature dependence. Figure 5 shows recent heat capacity measurements, in comparison with inelastic neutron scattering scans at 100 K. Note that the incommensurate features have disappeared at this temperature, often called the ‘coherence temperature’, which marks a crossover from paramagnetic weakly correlated moments to correlated heavyelectron behaviour. They are intimately correlated with the coherence temperature and the formation of the heavyfermion state. This energy scale also describes the temperature dependence of the heatcapacity anomaly at 17.5 K through fits to an activation law^{24}, and the temperature dependence of the incommensurate excitations below 17.5 K (ref. 16). This is further evidence that the gapping of these excitations is directly related to the formation of the HO state.
Our findings place constraints on all new theories of HO in URu_{2}Si_{2}. A minimum theory must account for (1) incommensurate itinerant spin excitations and (2) the lack of crystalline electric fields up to 10 meV. The HO transition seems to be a rearrangement of electrons at the Fermi surface in an itinerant rather than localized electron picture. Excitations out of this state at these incommensurate wavevectors show a modesoftening that is reminiscent of another dynamic ground state that has no translational order but a prominent specificheat anomaly—the superfluid liquid helium transition^{25}. Correlations between the heavy quasiparticles in URu_{2}Si_{2} build up below 100 K, and at 17.5 K, there is a transition to a new condensate that remains unidentified. The true order parameter for this system has yet to be determined.
Methods
Our measurements were made with a single crystal grown at McMaster University by the Czochralski method using elemental U, Ru and Si, followed by annealing in argon at 900 ^{∘}C. The magnetic properties were confirmed through d.c. magnetometry measurements in a superconducting quantum interference device, and heat capacity and resistivity measurements in a physical property measurement system. The neutron scattering measurements were carried out at the Disk Chopper Spectrometer at the NIST Center for Neutron Research in Gaithersburg, Maryland. The 11.5 g URu_{2}Si_{2} crystal was aligned in the H0L plane and placed in a standard Institut LaueLangevin cryostat and measurements were taken using cold neutrons of 2.5 and 5 Å. The identification of the equipment is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the equipment is necessarily the best available for the purpose.
References
Dagotto, E. Correlated electrons in high temperature superconductors. Rev. Mod. Phys. 66, 763–840 (1994).
Greedan, J. E. Geometrically frustrated materials. J. Mater. Chem. 11, 37–53 (2000).
Affleck, I. Quantum spin chains and the Haldane gap. J. Phys. Condens. Matter 1, 3047–3072 (1989).
Dagotto, E. & Rice, T. M. Surprises on the way from one to twodimensional quantum magnets: The ladder materials. Science 271, 618–623 (1996).
Fisk, Z. et al. Heavyelectron metals—new highly correlated states of matter. Science 239, 33–42 (1988).
Tripathi, V., Chandra, P. & Coleman, P. Itinerancy and hidden order in URu2Si2 . J. Phys. Condens. Matter 17, 5285–5311 (2005).
Silhanek, A. V. et al. Quantum critical 5f electrons avoid singularities in U(Ru,Rh)2Si2 . Phys. Rev. Lett. 95, 026403 (2005).
Mineev, V. P. & Samokhin, K. V. de Haas van Alphen effect in metals without an inversion center. Phys. Rev. B 72, 014432 (2005).
Chandra, P., Coleman, P., Mydosh, J. A. & Tripathi, V. Hidden orbital order in the heavy fermion metal URu2Si2 . Nature 417, 831–834 (2002).
Broholm, C. Magnetic excitations and ordering in the heavyelectron superconductor URu2Si2 . Phys. Rev. Lett. 58, 1467–1470 (1987).
Broholm, C. et al. Magnetic excitations in the heavyfermion superconductor URu2Si2 . Phys. Rev. B 43, 12809–12822 (1991).
Squires, G. L. Introduction to the Theory of Neutron Scattering (Dover, New York, 1997).
Buyers, W. J. L. et al. Spin wave collapse and incommensurate fluctuations in URu2Si2 . Physica B 199, 95–97 (1994).
Fawcett, E. Spindensitywave antiferromagnetism in chromium. Rev. Mod. Phys. 60, 209–283 (1988).
Bao, W., Broholm, C., Honig, J. M., Metcalf, P. & Trevino, S. F. Itinerant antiferromagnetism in the Mott compound V1.973O3 . Phys. Rev. B 54, R3726–R3729 (1996).
Wiebe, C. R., Luke, G. M., Yamani, Z., Menovsky, A. A. & Buyers, W. J. L. Search for hidden orbital currents and observation of an activated ring of magnetic scattering in the heavy fermion superconductor URu2Si2 . Phys. Rev. B 69, 132418 (2004).
Bonn, D. A., Garrett, J. D. & Timusk, T. Farinfrared properties of URu2Si2 . Phys. Rev. Lett. 61, 1305–1308 (1988).
Chou, H. et al. Neutronscattering study of spin fluctuations in superconducting YBa2Cu3O6+x (x=0.40,0.45,0.50). Phys. Rev. B 43, 5554–5563 (1991).
Mason, T. E., Aeppli, G. & Mook, H. A. Magnetic dynamics of superconducting La1.86Sr0.14CuO4 . Phys. Rev. Lett. 68, 1414–1417 (1992).
RossatMignod, J. et al. Inelastic neutron scattering study of the spin dynamics in the YBa2Cu3O6+x system. Physica B 192, 109–121 (1993).
Stock, C. et al. Dynamic stripes and resonance in the superconducting and normal phases of YBa2Ca3O6.5 orthoII superconductor. Phys. Rev. B 69, 014502 (2004).
Behnia, K. et al. Thermal transport in the hiddenorder state of URu2Si2 . Phys. Rev. Lett. 94, 156405 (2005).
Maple, M. B. et al. Partially gapped Fermi surface in the heavyelectron superconductor URu2Si2 . Phys. Rev. Lett. 56, 185–188 (1986).
Palstra, T. T. M. et al. Superconducting and magnetic transitions in the heavyfermion system URu2Si2 . Phys. Rev. Lett. 55, 2727–2730 (1985).
Feynman, R. P. in Progress in Low Temperature Physics Vol. 1 (ed. Gorter, C.) (Interscience, New York, 1955).
Acknowledgements
The authors would like to acknowledge helpful discussions with C. Broholm, C. D. Batista, A. Leggett, C. M. Varma, J. S. Gardner, J. S. Brooks, G. S. Boebinger and B. D. Gaulin. This work was made possible by support through the NSF and the state of Florida. L.B. acknowledges support from the NHMFL in house research program and Y.J. acknowledges support from the NHMFL Schuller program. G.M.L., G.J.M. and W.J.L.B. acknowledge support through NSERC and the CIAR. The authors are grateful to the local support staff at the NIST Center for Neutron Research. Data analysis was completed with DAVE, which can be obtained at http://www.ncnr.nist.gov/dave/. The work at NIST is supported in part by the National Science Foundation under Agreement No. DMR0454672. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement No. DMR0084173, by the State of Florida, and by the DOE.
Author information
Authors and Affiliations
Contributions
Scattering experiments were completed by C.R.W., J.A.J., G.J.M., H.D.Z., Y.Q., J.R.D.C., Z.Y. and W.J.L.B. The crystals were grown by J.D.G. and G.M.L. Specific heat measurements were made by Y.J.J. and L.B. Data analysis and writing of the paper was completed by C.R.W., J.A.J., G.J.M., G.M.L., L.B. and W.J.L.B.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
About this article
Cite this article
Wiebe, C., Janik, J., MacDougall, G. et al. Gapped itinerant spin excitations account for missing entropy in the hiddenorder state of URu_{2}Si_{2}. Nature Phys 3, 96–99 (2007). https://doi.org/10.1038/nphys522
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys522
This article is cited by

1D charge density wave in the hidden order state of URu2Si2
Communications Physics (2021)

Intertwined dipolar and multipolar order in the triangularlattice magnet TmMgGaO4
Nature Communications (2019)

Evidence for a nematic component to the hiddenorder parameter in URu2Si2 from differential elastoresistance measurements
Nature Communications (2015)

Direct observation of lattice symmetry breaking at the hiddenorder transition in URu2Si2
Nature Communications (2014)

Momentumresolved hiddenorder gap reveals symmetry breaking and origin of entropy loss in URu2Si2
Nature Communications (2014)