Parasitic small-moment-antiferromagnetism and non-linear coupling of hidden order and antiferromagnetism in URu2Si2 observed by Larmor diffraction
P. G. Niklowitz, C. Pfleiderer, T. Keller, M. Vojta, Y.-K. Huang, J. A. Mydosh
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Parasitic small-moment-antiferromagnetism and non-linear coupling of hidden orderand antiferromagnetism in URu Si observed by Larmor diffraction P. G. Niklowitz,
1, 2
C. Pfleiderer, T. Keller,
3, 4
M. Vojta, Y.-K. Huang, and J. A. Mydosh Physik Department E21, Technische Universit¨at M¨unchen, 85748 Garching, Germany Department of Physics, Royal Holloway, University of London, Egham TW20 0EX, UK ZWE FRM II, Technische Universit¨at M¨unchen, 85748 Garching, Germany Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany Institute for Theoretical Physics Universit¨at zu K¨oln, Z¨ulpicher Strasse 77, 50937 K¨oln, Germany Van der Waals-Zeeman Institute, University of Amsterdam, 1018XE Amsterdam, The Netherlands Kamerlingh Onnes Laboratory, Leiden University, 2300RA Leiden, The Netherlands
We report simultaneous measurements of the distribution of lattice constants and the antiferro-magnetic moment in high-purity URu Si , using both Larmor and conventional neutron diffraction,as a function of temperature and pressure up to 18 kbar. We establish that the tiny moment in thehidden order (HO) state is purely parasitic and quantitatively originates from the distribution oflattice constants. Moreover, the HO and large-moment antiferromagnetism (LMAF) at high pres-sure are separated by a line of first-order phase transitions, which ends in a bicritical point. Thusthe HO and LMAF are coupled non-linearly and must have different symmetry, as expected of theHO being, e.g., incommensurate orbital currents, helicity order, or multipolar order. PACS numbers: 61.05.F-,62.50.-p,71.27.+a,75.30.KzKeywords: hidden order, antiferromagnetism, Larmor diffraction, pressure, neutron diffraction
In recent years hydrostatic pressure has become widelyused in the search for new forms of electronic order,because it is believed to represent a controlled andclean tuning technique. Novel states discovered in high-pressure studies include superconducting phases at theborder of magnetism and candidates for genuine non-Fermi liquid metallic states. However, a major un-certainty in these studies concerns the possible role ofpressure inhomogeneities, that originate, for instance, inthe pressure-transmitting medium and inhomogeneitiesof the samples. To settle this issue requires microscopicmeasurements of the distribution of lattice constantsacross the entire sample volume, which, to the best ofour knowledge, has not been available so far.The perhaps most prominent and controversial exam-ple that highlights the importance of sample inhomo-geneities and pressure tuning is the heavy-fermion su-perconductor URu Si . The reduction of entropy at aphase transition at T ≈ . Si over twenty years ago, is still not explained [1, 2, 3]. Theassociated state in turn is known as ’hidden order’ (HO).The discovery of the HO was soon followed by the ob-servation of a small antiferromagnetic moment (SMAF), m s ≈ . − . µ B per U atom [4] then believed to bean intrinsic property of the HO. The emergence of large-moment antiferromagnetism (LMAF) of m s ≈ . µ B per U atom [5] under pressure consequently promptedintense theoretical efforts to connect the LMAF with theSMAF and the HO. In particular, models have been pro-posed that are based on competing order parameters ofthe same symmetry, i.e., linearly coupled order parame-ters, in which the SMAF is intrinsic to the HO [6, 7, 8, 9].This is contrasted by proposals for the HO parameter such as incommensurate orbital currents [10], multipolarorder [11], or helicity order [12], where HO and LMAFbreak different symmetries.The symmetry relationship of HO and LMAF clearlyyields the key to unravelling the nature of the HO state[8, 9]. While some neutron scattering studies of thetemperature–pressure phase diagram suggest that theHO–LMAF phase boundary ends in a critical end point[13], other studies concluded that it meets the boundariesof HO and LMAF in a bicritical point [14, 15, 16, 17].This distinction is crucial, as a critical end point (bicriti-cal point) implies that HO and LMAF have the same (dif-ferent) symmetries, respectively [9]. Moreover, there is asubstantial disagreement w.r.t. the location and shape ofthe HO–LMAF phase boundary (see e.g. Ref. [18]). Thislack of consistency is, finally, accompanied by consider-able variations of the size and pressure dependence of themoment reported for the SMAF [18], where NMR studiesin powder samples suggested the SMAF to be parasitic[19]. Accordingly, to identify the HO in URu Si it is es-sential to clarify unambiguously the nature of the SMAFand the symmetry relationship of HO and LMAF.It was long suspected that the conflicting results aredue to a distribution of lattice distortions due to de-fects. Notably, uniaxial stress studies showed that LMAFis stabilized if the c/a ratio η of the tetragonal crys-tal is reduced by the small amount ∆ η c /η ≈ · − [20]. Hence, the parasitic SMAF may in principle re-sult from a distribution of η values across the sample,with its magnitude depending on sample quality and ex-perimental conditions. In particular, differences of com-pressibility of wires, samples supports or strain gaugesthat are welded, glued or soldered to the samples will (cid:44) (cid:17) (cid:83)(cid:65)(cid:77)(cid:80)(cid:76)(cid:69)(cid:35)(cid:17) (cid:35)(cid:18) (cid:35)(cid:19) (cid:35)(cid:20) (cid:44) (cid:18) (cid:39) (cid:86)(cid:86) (cid:80) (cid:86) (cid:84) (cid:73)(cid:78) (cid:79)(cid:85)(cid:84) (cid:34) (cid:33) (cid:36) (b) (c)(a) FIG. 1: (a) Schematic of Larmor diffraction [22, 23], see textfor details. (b) Typical variation of the polarization P as afunction of the total Larmor phase Φ. (c) Pressure depen-dence of the width of the lattice-constant distribution for thea- and c-axis in URu Si . With increasing pressure the widthof the distribution increases. forcibly generate uncontrolled local strains that stronglyaffect any conclusions about the SMAF signal (see, e.g.,Refs [14, 15, 16, 17]).In this Letter we report simultaneous measurements ofthe distribution and temperature dependence of the lat-tice constants, as well as the antiferromagnetic moment,of a pure single crystal of URu Si utilizing a novel neu-tron scattering technique called Larmor diffraction (LD).This allowed to study samples that are completely free tofloat in the pressure transmitting medium, thereby expe-riencing essentially ideal hydrostatic conditions at highpressures. Our data of the distribution of lattices con-stants f (∆ η/η ) establishes quantitatively that the SMAFis purely parasitic. In addition, we find a rather abrupttransition from HO to LMAF which extends from T = 0up to a bicritical point (preliminary data of T N ( p ) werereported in [21]). We conclude that the HO–LMAF tran-sition is of first order and that HO and LMAF must becoupled non-linearly. This settles the perhaps most im-portant, long-standing experimental issue on the route toidentifying the HO.Larmor diffraction permits high-intensity measure-ments of lattice constants with an unprecedented highresolution of ∆ a/a ≈ − [22, 23]. As shown in Fig. 1 (a)the sample is thereby illuminated by a polarized neutronbeam (arrows indicate the polarization); G is the recipro-cal lattice vector; θ B is the Bragg angle; AD is the polar-ization analyzer and detector. The radio frequency (RF)spin resonance coils (C1-C4) change the polarization di-rection, as if the neutrons undergo a Larmor precessionwith frequency ω L along the distance L = L + L . The total Larmor phase of precession Φ thereby depends lin-early on the lattice constant a : Φ = 2 ω L Lma/ ( π ¯ h ) ( m isthe mass of the neutron [22, 23]).The Larmor diffraction was carried out at the spec-trometer TRISP at the neutron source FRM II. The tem-perature and pressure dependence of the lattice constantswas inferred from the (400) Bragg peak for the a axis andthe (008) Bragg peak for the c axis. The magnetic or-dered moment was monitored with the same setup usingconventional diffraction. For our high-pressure studies aCu:Be clamp cell was used with a Fluorinert mixture [24].The pressure was inferred at low temperatures from the(002) reflection of graphite as well as absolute changesof the lattice constants of URu Si taking into accountpublished values of the compressibility.The single crystal studied was grown by means of anoptical floating-zone technique at the Amsterdam/LeidenCenter. High sample quality was confirmed via X-raydiffraction and detailed electron probe microanalysis.Samples cut-off from the ingot showed good resistanceratios (20 for the c axis and ≈
10 for the a axis) and ahigh superconducting transition temperature T c ≈ . m s ≈ . µ B per U atom, whichmatches the smallest moment reported so far [18].As the Larmor phase Φ is proportional to the latticeconstant a , the polarization P of the scattered neutronbeam reflects the distribution of lattice constants acrossthe entire sample volume [23]. While Larmor diffractionwas recently employed for the first time in a high-pressurestudy [26], measurements of the distribution of latticehave not been exploited to resolve a major scientific issue(for proof of principle studies in Al-alloys see Ref. [22]).In order to explore the origin of the SMAF moment wehave measured the spread of lattice constants, keeping inmind that a distribution of the c/a lattice-constant ratio η may be responsible for AF order in parts of the sam-ple. As shown in Fig. 1 (b) P (Φ) changes only weakly be-tween ambient pressure and 17.3 kbar. Accordingly thedistribution of lattice constants, which is given by theFourier transform of P (Φ), changes only weakly as a func-tion pressure as shown in Fig. 1c). Thus pressure onlyslightly boosts the distribution of lattice constants, butdoes not generate substantial additional inhomogeneities.Moreover, we confirmed that the size of m s remained un-changed tiny after our high-pressure studies.Assuming a Gaussian distribution of both lattice con-stants, we infer the distribution of η . At low pres-sure, we arrive at a full width at half-maximum of f (∆ η/η ) FWHM ≈ . · − . Recalling that the tail of theGaussian distribution beyond ∆ η c /η ≈ · − representsthe sample’s volume fraction in which LMAF forms [20],an average magnetic moment of 0 . µ B is expected in FIG. 2: Temperature dependence of the lattice constants andthermal expansion at various pressures. The HO and LMAFtransitions are indicated by empty and filled arrows, respec-tively. (a) Data for the a -axis; (b) thermal expansion of the a -axis derived from the data shown in (a). (c) Data for the c -axis; (d) thermal expansion of the c -axis derived from thedata shown in (c). our sample. This is in excellent quantitative agreementwith the experimental value and represents the first mainresult we report.We continue with a discussion of the temperature de-pendence of the lattice constants and its change withpressure. At zero pressure in a wide temperature rangebelow room temperature (not shown) the thermal expan-sion is positive for both axes [27]. However, for the c-axis,the lattice constant shows a minimum close to 40 K andturns negative at lower temperatures. This general be-havior is unchanged under pressure.Shown in Fig. 2 are typical low-temperature data forthe a- and c-axis. At ambient pressure T can be barelyresolved. However, when crossing p c ≈ . T . Measurements of the antiferromagnetic moment(see below), identify this anomaly as the onset of theLMAF order below its T N . At the transition the latticeconstant for the a- and c-axis show a pronounced con-traction and expansion, respectively.The pressure and temperature dependence of m s deter-mined at (100) is shown in the inset of Fig. 3 (a). Close to p c ≈ . T N is near base temperature, we ob-serve first evidence of the LMAF magnetic signal, which x0510152025 [17] [16][14] [18]magnetica-axisc-axis T ( K ) LMAFHO (b)p (kbar) D c p / T ( J / m o l K - ) p(kbar) D c p /T D c p /T N m o r d ( m B f. u - ) (a) T (K) pea k he i gh t sc ( a . u . ) FIG. 3: (a) Pressure dependence of the low-temperature mag-netic moment. The inset shows the temperature dependenceof the magnetic peak height at various pressures. (b) Phasediagram based on Larmor diffraction and conventional mag-netic diffraction data. The onset of LMAF is marked by fulland of HO by empty symbols (x marks a transition near basetemperature). For better comparison data of T N (black sym-bols) from Refs. [14, 16, 17, 18] are shown. The T values (redsymbols) of all references are consistent. Inset: background-free specific-heat jump derived from thermal-expansion datavia the Ehrenfest relation (full circles). Heat capacity data istaken from Refs. 1 (square) and 16 (empty circle). rises steeply and already reaches almost its high-pressurelimit at 5 kbar. The pressure dependence was determinedby assuming the widely reported high-pressure value of m s = 0 . µ B and comparing the magnetic (100) and nu-clear (004) peak intensity.The temperature–pressure phase diagram shown inFig. 3 (b) displays T and T N taken from the magneticBragg peak (Fig. 3 (a)) and from the Larmor diffractiondata (Fig. 2), respectively. The different data sets showexcellent agreement (except at p = 0, due to the variableparasitic nature of the SMAF). The main results con-tained in Fig. 3 are (i) the very small value of 0.012 µ B of the average low-temperature ordered moment at zeropressure, (ii) a particularly abrupt increase of the low-temperature moment at p c ≈ . T transitionlines at approximately 9 kbar. Fig. 2 (b) shows that wecan follow this phase boundary from low T up to T .Most importantly, (iv) implies that HO and LMAF arefully separated by a phase boundary which has to be offirst order with a bicritical point, since three second-orderphase transition lines cannot meet in one point. (Note T m a x ( K ) T , T N ( K ) p (kbar) URu Si T max, r T max,M_c AFMHO?
I II
FIG. 4: Extended phase diagram of URu Si based on thedata presented here (diamonds) and signatures in resistivity(brown[28] and grey[16] circles). T max,ρ and T max,M c denotecoherence maxima in the resistivity [28] and magnetisation[25], respectively. The HO might either mask a QCP (I) orreplace LMAF (II) near quantum criticality. that the phase boundaries from the HO and LMAF to thedisordered high-temperature phase are already known tobe of second order from qualitative heat capacity mea-surements [1, 16].) This conclusion is perfectly consistentwith (ii) and (iii), and it excludes a linear coupling be-tween the HO and LMAF order parameters [9].Although our results show that the HO and LMAFmust have different symmetry, they also suggest thatboth types of order may have a common origin. Usingthe Ehrenfest relation, we have converted our thermalexpansion data into a background-free estimate of thespecific-heat jump, ∆ C , at the PM to LMAF transition.The inset in Fig. 3 (b) shows a continuous evolution withpressure of the jumps at the LMAF to PM and corre-sponding HO to PM transitions. The common origin ofboth phases may in fact be related to excitations seen inneutron scattering. Notably, excitations at (1,0,0) onlyappear in the HO phase and may be its salient prop-erty [29, 30]. However, excitations at (1.4,0,0) in the PMstate become gapped in both the HO and LMAF stateand have been quantitatively linked to the specific heatjump at zero pressure [31].Finally, Fig. 4 suggests a new route to the HO. It showsa summary of the pressure evolution of features in the re-sistivity and magnetization, which suggest the existenceof an AF quantum critical point (QCP) at an extrap-olated negative pressure. Remarkably, these features,which are usually denoted as Kondo or coherence temper-ature, all seem to extrapolate to the same critical pres-sure. The HO in URu Si could be understood as emerg-ing from quantum criticality, possibly even masking anAF QCP (case I in Fig. 4). However, considering theremoteness of the proposed QCP, the balance betweenLMAF and HO might rather be tipped by pressure in-duced changes of the properties of URu Si . This hasfor example been suggested in a recent proposal, where HO and LMAF are believed to be variants of the sameunderlying complex order parameter [32]. In such a sce-nario, the HO is more likely to collapse at the proposedQCP as denoted by the dashed line (case II in Fig. 4).The pressure dependence of ∆ C sets constraints on anytheory of the HO involving quantum criticality.In conclusion, using Larmor diffraction to determinethe lattice constants and their distribution at high pres-sures and with very high precision, we were able to showthat the SMAF in URu Si is purely parasitic. Moreover,the HO and LMAF are separated by a line of first-ordertransitions ending in a bicritical point. This is the be-havior expected for the HO being, e.g., incommensurateorbital currents, helicity order, or multipolar order.We are grateful to P. B¨oni, A. Rosch, A. de Visser, K.Buchner, F. M. Grosche and G. G. Lonzarich for supportand stimulating discussions. We thank FRMII for gen-eral support. CP and MV acknowledge support throughDFG FOR 960 (Quantum phase transitions) and MV alsoacknowledges support through DFG SFB 608. [1] T. T. M. 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