Hidden order in URu2Si2

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Physica B 359–361 (2005) 986–993 www.elsevier.com/locate/physb

Hidden order in URu2Si2 F. Bourdarota, A. Bombardib, P. Burleta, M. Enderlec, J. Flouqueta, P. Lejayd, N. Kernavanoisc, V.P. Mineeva, L. Paolasinib, M.E. Zhitomirskya, B. Fa˚ka, a

De´partement de Recherche Fondamentale sur la Matie`re Condense´e, SPSMS, CEA Grenoble, 38054 Grenoble Cedex 9, France b European Synchrotron Radiation Facility, 6 r J. Horowitz, BP 220, 38043 Grenoble Cedex 9, France c Institut Laue-Langevin, BP 156, 38042 Grenoble Cedex 9, France d Centre de Recherches sur les Tre`s Basses Tempe´ratures, CNRS, BP 166, 38042 Grenoble Cedex 9, France

Abstract Neutron-scattering and specific-heat measurements of the heavy-fermion superconductor URu2 Si2 under hydrostatic pressure and with Rh-doping, UðRu0:98 Rh0:02 Þ2 Si2 ; show the existence of two magnetic phase transitions. At T m  17:5 K; a second-order phase transition with strong anomalies in the specific heat and other macroscopic and transport properties are accompanied by an antiferromagnetically ordered dipolar moment of only 0:03mB : At T M oT m under pressure, pX5 kbar; or small Rh doping, a first-order phase transition gives rise to small anomalies in the specific heat and a large ordered moment of  0:3mB ; but with the same magnetic structure. The results can be understood in terms of a hidden order parameter c; which is linearly coupled to the ordered moment m. It follows that m and c have the same symmetry and hence both break time-reversal symmetry. r 2005 Elsevier B.V. All rights reserved. PACS: 74.70.Tx; 75.25.þz; 75.30.Kz Keywords: Heavy-fermion; Hidden order; Uranium compound; Neutron scattering

1. Introduction URu2 Si2 is a moderate heavy fermion compound ðCe =T  1180 mJK2 mole1 ) where superconductivity coexists with antiferromagnetic (AFM) order below the superconducting transition at T sc  1:5 K [1]. The AFM order is characterized by a propagaCorresponding author. Fax: +33 438 78 5109.

E-mail address: [email protected] (B. Fa˚k).

tion vector k ¼ ð001Þ with the ordered dipolar moments pointing along the c-axis of the bodycentered tetragonal structure (space group I4/mmm) [2]. The magnitude of the ordered moment is very weak, only 0:03mB at low temperatures. The ordered moment develops at T m ¼ 17:5 K; where clear anomalies are observed in bulk properties. However, it is difficult to reconcile the small size of the ordered moment with e.g. the large jump of DC=T m ¼ 0:3 JK2 mol1 in the specific heat at T m [1]. Recent

0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.318

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elastic neutron-scattering experiments in a high magnetic field along the c-axis show that the ordered moment mðHÞ at low temperatures persists up to applied fields of 17 T [3]. The observation of an inflection point in mðHÞ at H  7 T is in agreement with the predictions of a model [4] where the ordered moment m is linearly coupled to a second (hidden) order parameter c; which gives rise to the large specific-heat anomaly. To elucidate the origin of the hidden order, we have performed specific heat and neutron scattering measurements on URu2 Si2 under pressure [5,6] and doped with 2% Rh [6,7].

doped compound, U(Ru0:98 Rh0:02 )2 Si2 ; also shows two transitions: a magnetic first-order transition at T M ¼ 8:3 K with an entropy jump of DSðT M Þ ¼ 23 mJK1 mol1 and a second-order transition at T m ¼ 13:2 K; which resembles very strongly the lambda-type anomaly in the pure compound. No superconducting transition is observed in U(Ru0:98 Rh0:02 )2 Si2 : Neutron elastic scattering measurements were performed on the larger pieces of the U(Ru0:98 Rh0:02 )2 Si2 and URu2 Si2 crystals, using the thermal triple-axis spectrometer IN22 at the high-flux reactor of Institut Laue-Langevin (ILL) with a ( and the cold triple-axis wavelength l of 2:36 A ( spectrometer IN14 at the ILL with l ¼ 4:24 A:

2. Rhodium doping

Higher-order contamination from the monochromator was carefully filtered out using pyrolytic graphite and Be filters. Fig. 2 shows the temperature dependence of the magnetic Bragg peak intensity at Q ¼ ð100Þ after background subtraction, I M / m2 ; for URu2 Si2 and U(Ru0:98 Rh0:02 )2 Si2 : In U(Ru0:98 Rh0:02 )2 Si2 ; a large ordered moment develops in the LMAF phase below T M ¼ 7:9ð2Þ K with m ¼ 0:25ð2ÞmB in the zerotemperature limit. The rapid drop of the ordered moment at T M is characteristic of a first-order transition. There is no sign of hysteresis. A much smaller moment is observed in the SMAF phase at higher temperatures, m  0:013mB at T ¼ 11:4 K:

Cylindrical single-crystals of diameter 5 mm and length 20 mm of both URu2 Si2 and U(Ru0:98 Rh0:02 )2 Si2 were grown by the Czochralski technique using a triarc furnace. The crystals were annealed in ultra-high vacuum after growth. Small discs ð 150 mgÞ were cut for the specific-heat measurements. The electronic contribution to the specific heat, C e ; was estimated by subtracting the specific heat from the non-magnetic isomorphic compound ThRu2 Si2 [8]. Fig. 1 shows C e =T as a function of temperature T for the two compounds. The pure compound, URu2 Si2 ; shows two transitions: a superconducting transition at T sc  1:5 K and a lambda-type anomaly at T m ¼ 17:5 K: The

U(Ru0.98 Rh0.02 )2Si 2

0.06

1 10-3

0.6 Tm (0%)

m 2 (µ B2)

Ce /T (J mol-1 K-2)

URu2 Si 2

URu 2 Si 2

0.5

U(Ru 0.98 Rh0.02 ) Si 2 2

0.4

0.04

0.02

0.3

5 10-4

Tm (2%) TM (2%) Tm (0%)

Tm (2%) 0.2 TM (2%)

TSC (0%)

0.00 0

0.1 0.0

0

5

10

15

20

Temperature (K)

Fig. 1. Electronic specific heat (after subtraction of the phonon contribution obtained from ThRu2 Si2 Þ divided by temperature for URu2 Si2 (blue circles) and for U(Ru0:98 Rh0:02 )2 Si2 (red squares).

0

0 10 5

10 15 Temperature (K)

20

Fig. 2. Neutron-scattering measurements of the temperature dependence of the magnetic Bragg peak intensity m2 at Q ¼ ð100Þ for U(Ru0:98 Rh0:02 )2 Si2 (open red circles, left scale) and URu2 Si2 (closed blue circles, right scale). The transition temperatures from the specific-heat measurements are shown by arrows. The line is a fit to Eq. (2). Error bars are smaller than the symbol size.

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The less sharp intensity variation below T m is characteristic of a second-order phase transition. We note a curious decrease of the magnetic intensity at the lowest temperatures below T M (see Fig. 2), also observed in magnetic X-ray scattering measurements (see below). The existence of a weak ordered moment is more clearly seen in a log-plot of the magnetic intensity, in Fig. 3. All the transition temperatures observed by neutron scattering clearly correlate with those of the specific-heat measurements. Also, the observed saturation value of the moment of 0:25mB in the LMAF phase of U(Ru0:98 Rh0:02 )2 Si2 is compatible with that of 0:16mB estimated from the size of the specific heat jump, which can be calculated from the AFM energy in molecular field theory,
 Z TN m0 2 2 ¼ C mag ðTÞ dT, (1) 3kB T N 0 msat with msat ¼ 3:62mB : Resonant magnetic X-ray scattering measurements at the M 4;5 absorption edge of uranium were performed on the doped compound U(Ru0:98 Rh0:02 )2 Si2 at the ID20 beam-line of the European Synchrotron Radiation Facility (ESRF). Longitudinal (parallel to l) and transverse (parallel to h) scans were performed on the Q ¼ ð005Þ magnetic Bragg reflection in the p2s polarization channel. The integrated intensity of the magnetic scattering is shown in Fig. 4 for a longitudinal scan. The nearly constant intensity 10 -1

TM (2%)

URu2Si2

10 -2 m2 (µB2 )

U(Ru0.98Rh0.02 )2Si2

10 -2 U(Ru 0.98 Rh 0.02 ) 2 Si 2 10 -3

Q=( 0 0 5 )

10 -4

Tm (2%)

10 -5

TM (2%)

10 -6 10 -7

0

5

10

15

Temperature (K)

Fig. 4. Resonant magnetic X-ray scattering measurements of the temperature dependence of the integrated intensity of the Q ¼ ð005Þ magnetic Bragg peak in U(Ru0:98 Rh0:02 )2 Si2 : The transition temperatures from the specific-heat measurements are shown by arrows. 10 -2 U(Ru0.98 10

κ (Å-1)

988

Rh0.02) Si 2 2

TM (2%)

-3

Q = (005)

10 -4 Tm (2%) 10 -5 ∆Q//[001]

10 -6 10

∆Q//[100]

-7

0

5

10

15

Temperature (K)

Fig. 5. Temperature dependence of the inverse correlation length k of the Q ¼ ð005Þ magnetic Bragg peak for U(Ru0:98 Rh0:02 )2 Si2 measured by X-rays. Blue closed circles are from longitudinal scans and open red circles from transverse scans.

Tm (2%) Tm (0%)

10 -3

10 -4

10 -5

0

5

10

15

20

Temperature (K)

Fig. 3. Neutron-scattering measurements of the temperature dependence of the magnetic Bragg peak intensity m2 at Q ¼ ð100Þ on a logarithmic intensity scale. Symbols as in Fig. 2. The dashed line corresponds to the sensibility of the measurements.

below T M  8:3 K and the rapid drop above T M corroborate the first-order character of the phase transition. The intensity does not drop to zero above T M but has a long tail extending up to temperatures in the vicinity of T m : This is very similar to the neutron scattering data (cf. Fig. 3). The intrinsic width k (taken as the half width at half maximum) of the longitudinal and transverse Q-scans of the (005) magnetic Bragg peak after deconvolution with the instrumental resolution is shown as a function of temperature in Fig. 5. The

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m2 ¼ m20 ½1  ðT=T N Þa 

(2)

to the data, with N ¼ M or m. In the SMAF phase (N ¼ m), ao2:5; while in the LMAF phase (N ¼ M), a  8:5: We also note that the large moment M ¼ 0:33mB for p46:4 kbar is independent of pressure within the precision of the measurements. The absence of magnetic scattering

0.15 11.8 kbar 8.2 kbar 6.4 kbar 4.5 kbar 0 kbar

URu 2 Si2

0.10 2

B

Neutron-scattering measurements under pressure on the pure compound show strong similarities with the Rh-doped compound. At hydrostatic pressures above 5 kbar, a large-moment phase appears with a transition temperature T M ðpÞ that is smaller than that of the smallmoment phase, T m ðpÞ: The variation of the ordered moment with temperature is steep at T M ðpÞ; which suggests that the transition is first order, as in U(Ru0:98 Rh0:02 )2 Si2 : Since the slope dT M =dp has to be infinite at T ¼ 0 for a first-order transition, it is delicate to fine-tune the pressure to determine the transition temperature T M ðpÞ with precision at low pressures. In addition, it is technically difficult to obtain perfect hydrostatic pressure conditions for neutron scattering at pressures above 5 kbar. Hydrostatic pressure conditions can be obtained using helium gas-pressure cells, since the helium also acts as pressure-transmitting medium. Our neutron scattering measurements using a helium gas-pressure cell show the absence of the LMAF phase up to pressures of 5.0 kbar. However, the LMAF phase is observed already at 4.5 kbar using a standard clamp cell (of steel or CuBe), where the use of fluorinert as pressure transmitter induces pressure variations and deviations from hydrostatic conditions throughout the sample volume. This effect is clearly evidenced by a much smoother intensity variation of the magnetic Bragg peaks at the transition. At higher pressures, the

2

3. Hydrostatic pressure

transition sharpens up since the smaller slope dT M ðpÞ=dp reduces the effect of pressure variations Dp on T M : The temperature dependence of the magnetic intensity, M 2 ðTÞ; after background subtraction and normalization to nuclear Bragg peaks, is shown in Fig. 6 for selected pressures. The measurements were performed at the ILL on the ( and on IN22 D15 diffractometer with l ¼ 0:85 A ( with l ¼ 2:36 A on a cylindrical crystal (4.8 mm diameter and 4 mm length) cut from the same batch as used in the high-field measurements of Ref. [3]. Standard ILL CuBe or steel clamp pressure cells of inner diameter of 5 or 6 mm were used. The pressure was determined from the lattice spacing of a NaCl single crystal mounted next to the sample inside the pressure cell. The temperature variation of the magnetic intensity is much faster near T M than near T m : This can be quantified by fitting the expression

m (µ )

width is resolution limited for longitudinal scans below T M and only weakly broadened for transverse scans 1 (the resolution is 8:9 104 4 ( and 5:9 10 A ; respectively). At T M ; k increases rapidly but nearly saturates again well into the SMAF phase. Here, the correlation length x ¼ 1=k is of the order of 1000 A˚. Towards T m ; the width increases rapidly again. In the pure compound, neutron scattering measurements also show a finite correlation length of the order of 300–500 A˚ in the SMAF phase, but this correlation length appears to be temperature independent [9].

989

0.05

0.00 0

5

10

15

20

Temperature (K)

Fig. 6. Neutron-scattering measurements of the temperature dependence of the (100) magnetic Bragg peak intensity M 2 in URu2 Si2 for different applied pressures. The lines are fits to Eq. (2).

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at the ð0; 0; 2l þ 1Þ reflections shows that the ordered moments are along the c-axis at all pressures, i.e. the SMAF and LMAF phases have the same AFM structure. From data of M 2 ðTÞ; such as those shown in Fig. 6, we determine the transition temperature T M ðpÞ as the onset of the magnetic order, defined in Eq. (2). This allows us to construct the p–T phase diagram for the LMAF phase. The small moment being difficult to observe in pressure cells due to the high background, we use the results from resistivity measurements [10–13] to determine T m : The phase diagram is shown in Fig. 7. This phase diagram is different from a preliminary one, where the midpoint rather than the onset of the intensity variation was taken as criteria for the transition temperature [6]. As said, the SMAF phase is characterized by a small moment,  0:03mB ; and a finite correlation length. The transition between the paramagnetic phase and the SMAF phase at T m is second order and is accompanied by large anomalies in bulk properties. The transition at T M between the LMAF and the SMAF phases, on the other hand, is first order in character. The LMAF phase is true long-range order and the ordered moment is relatively large, M ¼ 0:33mB :

25

Transition temperature (K)

URu2 Si2 20

Tm

15

TM

4. Landau free energy The p–T phase diagram shown in Fig. 7 can be understood in the framework of a hidden order parameter that breaks time-reversal symmetry. For a system with two coupled order parameters, the Landau free energy can be written as F ¼ ac c2 þ am m2 þ 2gcm þ bc c4 þ bm m4 þ 2bi c2 m2 ,

ð3Þ

where c is a hidden primary order parameter of unknown origin that gives rise to the huge specificheat anomaly and m is a secondary order parameter corresponding to the tiny ordered moment. Shah et al. [4] calculated the magneticfield dependence of the ordered moment mðHÞ using Eq. (3) and predicted an inflection point for mðHÞ only in the case ga0: Subsequent high-field neutron scattering experiments showed indeed such an inflection point [3], which gives strong support for the scenario ga0; in which case there is a linear coupling between the two order parameters m and c: Because the coupling term cm is allowed only if m and c transform according to the same irreducible representation and since m breaks time-reversal symmetry, it follows that c also must break time-reversal symmetry. The order parameters m and c develop below the transition temperature T m : A first-order transition T M occurs below the second-order transition T m if pffiffiffiffiffiffiffiffiffiffiffi bi 4 bc bm : Eq. (3) then gives two possible phase diagrams, shown schematically in Fig. 8. In the case g ¼ 0 (left panel of Fig. 8), there is no small moment to the left of the T M ðpÞ transition

10 SMAF

T

LMAF

5

γ=0

γ=0

Tm

Tm

0 0

5 Pressure (kbar)

10

Fig. 7. Pressure–temperature phase diagram of URu2 Si2 showing the SMAF and LMAF phases determined from resistivity data (open circles, Refs. [10–13]) and neutron scattering data (closed circles for pressure clamps, closed squares for hydrostatic helium cells). The lines are guides to the eye.

TM

m=0 ψ=0

ψ > m> 0 m=0

TM

ψ=0 pM

pM

m>ψ>0 pc

p

Fig. 8. Schematic phase diagram of the functional Eq. (3) for g ¼ 0 (left panel) and ga0 (right panel).

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line and no hidden order c to the right. The firstorder transition line T M ðpÞ joins the second-order line T m ðpÞ in a bi-critical point. In the case ga0 (right panel), the two linearly coupled order parameters m and c coexist below T m at all pressures, the two transition lines T M ðpÞ and T m ðpÞ are split, and T M ðpÞ terminates in a critical end point at pc : Our experimental data clearly support the scenario with ga0: (i) The ordered moment mðHÞ has an inflection point as a function of magnetic field [3]. (ii) A small moment is observed to the left of the T M ðpÞ line and a large moment to the right. (iii) The first-order transition line T M ðpÞ is split from the second-order line T m ðpÞ:

5. Discussion 5.1. Large moment and inhomogeneities Our neutron scattering measurements on URu2 Si2 show the existence of a large pressureindependent moment, M  0:33mB ; above a pressure of 6.4 kbar. NMR measurements by Matsuda et al. [14] also show a large pressure independent moment in the LMAF phase. At pressures between 4.5 and 6.4 kbar, the magnetic intensity in our measurements increases with pressure. This could be interpreted as a smooth increase in the magnitude of the ordered moment, as originally done by Amitsuka et al. [15] in their neutronscattering measurements, or in terms of an increasing volume fraction of a large-moment phase LMAF, as suggested by Matsuda et al. [14]. The absence of a large-moment phase at 5 kbar in a helium-gas pressure cell (hydrostatic conditions) and the presence of a large moment at 4.5 kbar in a clamp cell (non-hydrostatic conditions), using the same crystal, suggests that URu2 Si2 is very sensitive to deviations from hydrostaticity or pressure variations. As a consequence, for pressures near the transition line between the SMAF and the LMAF phases, the sample is inhomogeneous and the intensity varia-

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tion is most likely due to a changing volume fraction of the LMAF phase, as suggested by Matsuda et al. [14]. Inhomogeneities can arise not only from pressure variations, but also from impurities and other defects in the crystalline structure. Heavy-fermion materials are in general quite sensitive to such imperfections, and URu2 Si2 is not an exception. Unusual temperature dependence of the ordered moment has, e.g., been found in some URu2 Si2 samples [9]. We believe that the large pressure region in which Matsuda et al. [14] observed changes in the volume fraction of the LMAF phase is a signature of sample inhomogeneity, possibly related to the use of an uncharacterized polycrystal crushed into powder. This sample sensitivity is also evidenced in the zero-field mSR measurements of Amitsuka et al. [16], performed on a different (better?) crystal. There, no oscillatory signal (evidence for the large moment) is observed below 6 kbar, while measurements above 7 kbar show a nearly constant volume fraction. This narrow inhomogeneity region of only 1 kbar contrasts sharply with the large inhomogeneity region of several kbars for the sample of Matsuda et al. [14] and also for the sample used in early high-pressure neutron work by Amitsuka et al. [15]. 5.2. Small moment The small ordered moment, characteristic of the zero-pressure SMAF phase, has only been directly observed by neutron and X-ray scattering. Other techniques such as NMR and mSR; which are sensitive to weak local magnetic fields arising from small ordered moments, have given conflicting results. E.g., the weak oscillatory component in the zero-field mSR measurements reported by Luke et al. [17] in URu2 Si2 at ambient pressure was not observed by other groups [18–21]. However, the absence of oscillations below 6 kbar and the presence of an oscillatory component above 7 kbar in the zero-field mSR measurements by Amitsuka et al. [16] on an apparently high-quality URu2 Si2 single crystal is an important result, and in agreement with our results. This observation leads us to speculate that the measurements by Luke et al. might have been made on a crystal

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where strain, impurities, or dislocations could have induced an LMAF phase in part (10%) of the sample. The large moment reported at ambient pressure in NMR measurements [22] was observed on a sample with a particularly large pressure region where a mixture of LMAF and SMAF phases was observed.

Table 1 Lattice parameters a and c and the unit cell volume V ¼ a2 c at T ¼ 10 K of U(Ru1x Rhx )2 Si2

a c V

x¼0

x ¼ 0:02

Ratio

4.12473 9.55896 162.6304

4.12356 9.57057 162.7355

0.99972 1.00121 1.00065

5.3. First-order transition Our neutron-scattering measurements suggest that the transition between the SMAF and the LMAF phases is first order. This is corroborated by the rapid change in the mSR oscillating amplitude (‘‘volume fraction’’) as a function of temperature and the nearly constant frequency of the spontaneous muon spin rotation [16]. A firstorder transition line between the SMAF and the LMAF phases is also found in the thermal expansion measurements by Motoyama et al. [23] under pressure along the a and the c axis. Their p–T phase diagram is similar, but not identical, to ours. The position of the line T M ðpÞ is quite sensitive to the sample quality, and the onset pressure pM for the LMAF phase (T M ðpM Þ ¼ 0) varies from 4 to 8 kbar. They conclude that T M joins T m at a critical point. This is in contrast to our neutron-scattering measurements, where the first-order character is preserved all the way up to 11.8 kbar.

URu2 Si2 : They found a quite strong increase of the magnetic peak intensity with stress in the basal plane (s parallel to [100] or to [110]) and hardly any changes with sk½001: However, it is unclear whether uniaxial stress in the basal plane actually increases the small moment of the SMAF phase or whether a phase transition to the LMAF phase occurs. In fact, the correlation length has no significant pressure dependence in the work of Yokoyama et al. [24] and is of the order of 300–500 A˚ at all s; i.e. characteristic of the SMAF phase. Also, the observed temperature dependence of the magnetic intensity shows no sign of a firstorder transition. The importance of the lattice parameter a is also indicated by the thermal expansion measurements by Motoyama et al. [23], who find that a shrinks and c expands as the system enters the LMAF phase, i.e. similar to the effect of Rh doping.

5.4. Rh doping

6. Conclusion

We have seen that URu2 Si2 under pressure behaves similarly to the weakly Rh-doped compound, U(Ru0:98 Rh0:02 )2 Si2 : This observation is non-trivial, since Rh doping is quite different from hydrostatic pressure, for two reasons: Rh doping (i) adds electrons to the system and (ii) decreases the lattice parameter a while c increases, in such a way that the volume increases. This is illustrated in Table 1, where results of high-resolution X-ray powder diffraction measurements at T ¼ 10 K performed on the ID31 beam line at the ESRF are shown. It seems that the lattice parameter a is the crucial one for URu2 Si2 : This is confirmed by the neutron scattering measurements under uniaxial pressure by Yokoyama et al. [24] on

Clamp cells not using helium as pressure transmitter are known to produce non-hydrostatic pressure conditions. When such a technique is applied to heavy-fermion materials, which in general are sensitive to the sample preparation, there is always a risk to obtain inhomogeneous samples. In the case of URu2 Si2 under pressure, the pressure region over which the first-order transition to the LMAF phase occurs varies widely between different samples, suggesting that this region is the effect of strain and non-hydrostatic pressure, rather than URu2 Si2 being intrinsically inhomogeneous. Combining our specific-heat, neutron, and X-ray scattering measurements on U(Ru0:98

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Rh0:02 )2 Si2 ; we find that the SMAF and LMAF phases are distinctly different, and that the small and the large moment have different characteristics, in particular concerning the correlation length. The p–T phase diagram of the pure compound, URu2 Si2 ; shows the existence of a first-order transition line between the SMAF and the LMAF phases. This line does not join the second-order transition line between the SMAF and the paramagnetic phases for pressures up to 11.8 kbar, which suggests a linear coupling between the magnetic moment and a hidden order parameter, which breaks time-reversal symmetry. This result is in agreement with high-field neutron scattering measurements. References [1] T.T.M. Palstra, et al., Phys. Rev. Lett. 55 (1985) 2727. [2] C. Broholm, et al., Phys. Rev. B 43 (1991) 12809.

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