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Anisotropy of the magnetic susceptibility of UPt3
ELSEVIER
Physica B 230-232 (1997) 9-15
Anisotropy of the magnetic susceptibility of UPt 3 Erwin Schuberth Walther Meissner Institut, Walther Meissner Str. 8, D-85748 Garching, Germany
Abstract Despite great efforts, both experimental and theoretical, the problem of magnetic behavior of UPt3 and its relation to the multiple superconducting phases has not been completely solved. This system continues to show unexpected and often unexplained features. To study the magnetic anisotropy of UPt3 single crystals in a wide range of temperatures and magnetic fields, we used a capacitive torque-meter, where the torque acting on the probe is proportional to (Z~.b - zc)B 2As in the old data of Frings and Franse we find in intermediate fields that Z, and Zb exhibit a maximum around 20 K, while Xc is only slightly enhanced with respect to the room temperature value. The resulting torque maximum around 20 K, however, turns out to be strongly field dependent. On decrease of B, a shift towards lower temperatures sets in below 3 T. This shift becomes nearly horizontal between 0.5 and 0.3 T and the maximum moves down to below 10 K. A second unexpected feature occurred when we followed the temperature dependence of)~a,b - Zc down to 15 mK: it ends suddenly but reversibly at a well-defined border line Li which is also field dependent. For magnetic fields above Be2(0), Li lies between 60 and 100 mK and joins the upper critical field curve at (100 mK, 1.86 T). Below this point it follows Bc2(T) down to the tetracritical point (T *, B*) where it splits off again. These results will be discussed with respect to existing theories of the superconducting order parameter and its relation to the anisotropic magnetism of UPt3. Keywords: UPt3; Heavy electron system; Magnetic susceptibility; Superconductivity
1. Introduction UPt3 is still the most widely studied heavy fermion system because of its unusual superconducting (s.c.) properties. There is a large body of experimental and theoretical work related to this problem. On the experimental side.', the existence of multiple s.c. phases, usually denoted as A, B, and C, is evident from data for the specific heat [1], also under applied field [2] and pressure I-3, 4], for ultrasound attenuation [5, 6-1, flux line dynamics [-7], London penetration depth 1-8], neutron scattering [9-1, tunnelling [10], and for the anisotropy of the lower and upper critical fields [-6, 11, 12]. This is taken as strong evidence for unconventional superconductivity, especially the observation of a double transition in low magnetic fields and the
occurrence of a tetracritical point ( T *, B*) where four phase lines meet, but also the existence of kinks in B c l ( T ) and B c z ( T ). In addition, cs.c.(T) and the London penetration depth 2 ( T ) exhibit power-laws instead of an exponential or (1 - (T/To)4) - 1/2 behavior. For a recent overview see Vorenkamp et al. 1-13]. On the theoretical side, many attempts have been made to analyse the data with respect to an unconventional s.c. order parameter (one or two-dimensional) coupled to a symmetry breaking field for which the most likely candidate considered was the antiferromagnetic phase observed below 5 K in neutron scattering experiments. However, so far, no fully consistent picture has emerged, and no consensus for the dimensionality and symmetry of the order parameter has been established. One of the
0921-4526/97/$17.00 Copyright (C 1997 Elsevier Science B.V. All rights reserved PII S092 1 - 4 5 2 6 ( 9 5 ) 0 0 5 3 3 - 9
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E. Schuberth / Physica B 230-232 (1997) 9-15
most persistent problems is the topology of the phase diagram around the tetracritical point. This topology is the same for all field directions, a fact which is not easily reconcilable with two-dimensional representations of the s.c. order parameter which seem to describe other experimental properties better than one-dimensional representations. The question whether the spin-orbit coupling is strong or weak has also been raised [14]. In the latter case, the spin can freely follow the direction of an external applied field. For recent theoretical reviews see Sauls [-15] and Joynt [16]. In our present work we determined the anisotropy of the static magnetic susceptibility of UPt3 single crystals over a wide range of temperature and magnetic field using a capacitive torque-meter. We find that there are additional magnetic phenomena, not observed so far, which show that the low-field magnetic behavior is different from the high-field one which interrelate superconductivity and magnetism in UPt3 over a wide portion of Be2( T ). Both phenomena are connected to the sizeable anisotropy of the magnetic susceptibility with a maximum of X,,b at 20 K which points to stronger magnetic moments in addition to the weak ones that seem to order antiferromagnetically at 5 K. The latter transition we do not detect in spite of the very high sensitivity of the torque-meter method especially in high fields. We also found no signature of it in a vibrating sample magnetometer around 4 T and in a SQUID magnetometer in fields around 10 G. The capacitive torque-meter was introduced by Brooks et al. [17] and was also used by Schmiedeshoffet al. [18] and Christ et al. [19]. The torque experienced in an external magnetic field (apart from shape effects which are negligible for UPt3 since Za, b,c are small) is given by T = Zaniso
VB2/t~oCOS O
sin O,
where in the case ofUPt3 Zaniso= Z,.b - Zc, O is the angle between the anisotropy axis and the external field, and • is perpendicular to Bext and to c (see Fig. 1). For UPt3, the anisotropy of the susceptibility which is large already at room temperature has been measured 10 years ago by Frings and Franse [20]. We confirmed their data with our first sample
'0
IVIab
~
a
~
~
respel acer
capacitor Fig. 1. Design of the torque meter and orientationof the samples with respect to the external magneticfield. in a vibrating sample magnetometer in a field of 4 T. X, and Zb are identical and show a maximum at 20 K, while Zc increases down to this temperature and then levels off. At the lowest temperatures X, and Zb are about a factor of 2 larger than L,, and we expected a strong torque even at mK-temperatures. These data are reminiscent of a similar case, namely, that of the hexagonal compound PrNi5 where the same magnetic behavior could be quantitatively explained by Pr 3 ÷ states split by the crystal field [21]. However, no specific heat anomaly was found around 20 K for UPt3. But also in PrNi5 only a shoulder in c ( T ) is observed at the temperature of the corresponding maximum in )~1c [22] and for UPt3 the heavy fermion specific heat may mask the magnetic contribution. Arguing from a T 3 In T contribution to the specific heat capacity Frings and Franse and also Stewart [23] interpreted the magnetic anisotropy as a signature of ferromagnetic spin fluctuations. Aeppli et al. [24] found magnetic fluctuations below 20 K in neutron scattering experiments which were attributed to ferromagnetic basal planes coupled antiferromagnetically along the c-direction. A small drop of Za.b below 20 K could be deduced from these data, much smaller however than observed in the direct
E. Schuberth /Physica B 230-232 (1997) 9-15
susceptibility measurements. In analogy to PrNis, we think that the large anisotropy in UPt3 is rather due to the crystal field splitting of the uranium 5f moments leading to a maximum in )~,,b at the temperature at which the first magnetic level is depopulated while in the c-direction only a slightly enhanced Pauli susceptibility is present. This view of an in-plane Van Vleck and an out-of-plane Pauli susceptibility has been recently proposed also by Park and Joynt 1-31].
2. Experiment The design of our torque-meter is shown in Fig. 1. The upper capacitor plate with the narrow cantilever was etched out of O F H C copper. The dimensions of this "tongue" were 0.05 x 2.7 x 0.4 mm 3. The samples, three single crystals with masses between 6.55 and 18.08 rag, came fi'om two different sources and they have been investigated already in our previous heat capacity work [-25, 26] (samples # 1, # 2), and [-27] (samples # i, # 3). They were all cleaved off from larger crystals used in those experiments. The first such crystal was grown by Bucher, Konstanz, with electron beam melting from depleted uranium. Although it has not been annealed so far, we consider it our best sample judging from the high and sharp Tc(520 inK), its high residual resistivity ratio (200), and its sharp X-ray reflexes. The second and third crystals were grown by A.Menovsky, Amsterdam, with the Czochralski method, sample # 3 has been annealed and showed the usual double transitions in the specific heat at T + and T [ . The upper capacitor plate was electrically isolated from the body of the torque-meter (OFHC copper or Ag) by lens paper of 40 gm thickness soaked with GE varnish. The latter was used to give good thermal contact to the plate onto which the samples were glued with DuPont silver epoxy. Estimates showed that the thermal[ contact through the GE-varnish and the silver epoxy was good enough to cool the slightly radioactive samples to below 15 mK with the body temperature below 10mK provided by a dilution refrigerator. The temperatures of the sample holder were determined with a carbon resistor calibrated against NBS fix-
11
point standards (in previous runs) and against a second carbon resistor in the field-compensated region of our cryostat when magnetic fields were applied. The capacitance of the meter was measured with a General Radio 1615-A capacitance bridge excited externally with a frequency of 3-5 kHz. It changed by about 50%, starting from typically 1.1 pF in B = 0, when the field was increased up to 14 T. The torque calculated from the C-traces had the expected B2-dependence in fields above 2 T. Deviations from this behavior both in low and in high fields are discussed below. The B T region between 0 and 7.1 T, 15 mK and 40 K was scanned by either sweeping the field at constant T, or by varying T in a constant field. In the superconducting phase changes of )~aniso were hard to detect in constant field. Due to strong flux pinning, the Meissner effect is very small in the Shubnikov state (we only worked above Be1 here). Only when the field was changed below To, and the sample was warmed up across the upper critical field, the decay of non-equilibrium flux line configurations and shielding currents at Tc made it possible to detect Be2 ( T ) . On the other hand, the upper critical curve could be well distinguished from the adjacent L/dine (see below) because on reversing the field changes below T~ the opposite direction of the shielding currents led to a reversed torque signal at Bc2 , opposing the T-dependence at Li, see Fig. 2.
3. Results The first remarkable result concerns the decrease of the z-anisotropy of UPt3 below 20 K. We find that its T-dependence ends sharply and reversibly, at a well defined temperature T/which depends on B. Below 7,. no T-dependence of Z,.iso = Z~,b -- Z,. within the limit of our sensitivity ( A C / C ~ 5 x 10 -6 ) could be detected, see inset in Fig. 3. Above Ti, Zanisovaried oc T" up to ~ 4 K with v ranging from 1.44 to 1.64. Figs. 3 and 4 show the border line L i ( T , B ) defined by the different Ti(B) which separates the temperature independent region below it from the onset of a detectable T-dependence. The data
E. Schuberth / Physica B 230-232 (1997) 9-15
12
B~ . .
3.5
~.~.. ~__~
e
3.0 ~
1.0 ~ 0.5 ~ Li =
~• T - i n d e p
t
:t
i 0,5
!
i
T-~ndepN°3t'
E 3-
1 0
i
•
0.0
0.1
,
0.2
,, aniso/T-dep ~'~~, T-indep TI.5
i
,
i
0.3
0.4
[KI,5]
a sample #1
• ,.
U Pt
.4.. " t ~ . . ~ v v A• / / r A y
i
,
0
c. ~,.~ 0.2
0.4 0.6 T [K]
0.8
1.0
Fig. 4. Phase diagram of UPt3 and the Lrline as deduced from sample # 2. The symbols have the same meaning as in Fig. 3. The orientation of the c-axis was 41 ° with respect to B~xt.
i i
0.2
,~
2-
,
3
sample #2
"" ~, aniso/T-dep T-indep
0.0 0.0
t
Fig. 2. Two heat-cycle paths used to determine B,2 and L~ and the resulting torque in different fields within the s.c. region. Path 2 was especially suited to distinguish both features on the warming curves.
i
UPt
1.5 ~
Bc2
I~
i
2.0 ~
T
@
i
•
2.5 ~
I, ': ....... t\; .....
i
0.25 0.50 0.75 1.00 1.25
1.5
T [K]
Fig. 3. Phase diagram of UPt3 as deduced from the temperature dependence of the torque in magnetic fields on sample # 1. The c-axis was oriented under 26 ° with respect to the external magnetic field. • and • define the onset and offset of a T-dependence upon warming and cooling, respectively (see inset). The solid lines LdT, B) taken as the detection limit (resolution AC/C = 5xl0 -6) of a T-dependence separate the regions of a T-independent z-anisotropy from the T-dependent area. The open circles define the upper critical field (decay of shielding currents), see text. In the inset interpolated data of the deviation from the low-temperature value in the respective field (here 2.6 T) are plotted versus T 15.
indicated • were taken upon warming and • on cooling through L~. The superconducting transition is indicated by the o-symbols. On some occasions, but not always, torque changes inside the s.c.
region could be detected at the position of the internal phase line B-C (symbols' + ' in Fig. 3). In high fields, Li is nearly independent of B, while at intermediate fields it coincides with the superconducting phase boundary. At lower fields, near (T*, B*), L/ again separates from the s.c. region towards higher temperatures. It is striking that the two phase boundaries L~ and Be2 (T) follow each other over a wide temperature and field range. The possibility that Li extends from higher fields into the s.c. region at 100 mK, being screened by shielding currents and thus detectable only above To, can be ruled out since often we observe a small separation between L~ and Tc (even on cooling) which would not be the case if L~ would lie deep inside the s.c. region. Thus, T~(B) coincides with L~(B) in this field range indicating a simultaneous change in the magnetic state at the superconducting transition. If one extrapolates the low-field Lrline of the first sample from higher temperatures into the s.c. region one directly matches the internal B-C phase boundary at (T*,B*). This behavior is less clear with the second sample, because there we did not find this internal phase line in our torque measurements. Whether the difference in the low-field Lrlines comes from a sample dependence or from the different orientation needs to be studied.
E. S c h u b e r t h / P h y s i c a B 2 3 0 - 2 3 2
The second unexpected result of our torque measurements occurred at high temperatures, when we followed the torque maximum at 20 K to lower fields. This maximum shifted towards lower temperatures below 3 T, first slowly as the field was lowered, but then dramatically below 1 T. It was even suppressed to below 1.5 K in 0.2 T, see Fig. 5. These data imply a field-dependent susceptibility in low magnetic fields. Preliminary magnetization curve measurements indeed reveal an increase of Za,b in fields below 0.5 T, in accord with the Tsweep data. We do not think that these results are due to paramagnetic impurities, since such moments would be isotropic in first order and not felt by the torque meter. They also could not suppress the torque maximum at 20 K. Impurity contents are very low in our crystals anyway.
4. Discussion What is the origin of the anisotropy of the magnetic susceptibility of UPt3 which has been measured long ago? The maximum of Z,.b at 20 K is hard to explain in terms of spin fluctuations. Also,
I
I
I
14-
I
f
•
12 ~
@
•
o
UPt single crystals
o
3
10 ~
oo o
'--'8
m6
sample #1 i2,,j~
#3 ~
d'
®
~""'#2
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.o .
-.
0.2 T, <1.5 K , ~ .
o
_ _
_
._ . . . . . . .
.
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.o ~e
. . -
_ =-=-==.=_ ._o:...::..£.. _.-._._ . . . . . . . . . . .
1'2
1'6
2'o
T [K] Fig. 5. Location of Li and of the torque maximum for single crystals of UPt3. The full circles refer to the annealed probe # 3 oriented below 45 ° between the c-axis arid Bcxt.
(1997) 9 15
13
a paramagnon contribution to Z is very unlikely. DeVisser et al. [28] argue, that a positive curvature N"(EF) > 0 would be necessary to obtain a rising z(T) with rising temperature. But such effects are usually very weak. The comparison to PrNi5 shows that crystal-field-split magnetic ion states are much more likely the cause of this unusual behavior. How the weak itinerant moments which order antiferromagnetically around 5 K (also in the a-b plane) are related to this magnetic behavior and how the Kondo effect and many-body effects influence it needs to be studied. Our low-field Li -line and the suppression of the torque maximum at 20 K give evidence that with increasing field the magnetic character changes between 0.2-0.7 T, depending on the sample and/or the field orientation. Since we find a similar behavior in annealed and as-grown samples, the field-dependent susceptibility in low fields seems not to be related to the structural modulation induced by annealing. There are several effects noticed in the literature which are pointing in the direction of our arguments and which may be related to the phase line Li found here. One is the lattice softening at low temperatures observed by Bullock et al. in ultrasound experiments [-29]. Their onset of decrease of the sound velocity in 3 T coincides with Li(3 T). Then, the point where Li meets Bc2(T) in high fields lies close to the cross-over point for the upper critical field curves for B//a and B//c [-6, 30]. Third, a nearly temperature independent phase line at about 0.1 T has been detected as an anomaly in the magnetostriction by van Dijk et al. [32]. (It was interpreted by the authors as a transition from random orientation of antiferromagnetic domains in low fields to an orientation ± Bext in higher fields.) Fourth, Fraunberger et al. [33] find that contrary to usual expectations the Sommerfeld coefficient ~, is not correlated with the low temperature susceptibility of UPt3 which again demonstrates that there are unknown contributions. Fifth, an anomalous increase of the real part of the ACsusceptibility above 1 T was reported by Signore et al. [34]. The onset of this increase also follows the upper critical curve above B*, and thus Li, although the detection limit of this experiment does not allow the statement that it really coincides with B c 2 ( T ). Finally, in fields above B c 2 ( O ) w e earlier
14
E. Schuberth / Physica B 230-232 (1997) 9-15
observed an upturn of the specific heat capacity below 50 mK 1-25,26] just below Li. What follows for the theory of UPt 3 superconductivity? There is a lot of evidence from neutron scattering, that antiferromagnetic order exists below 5 K and that it is related to the A, B superconducting phases. But why can it not be detected in magnetization and specific heat measurements 1-35]? Recent magnetic X-ray and neutron diffraction experiments in the s.c. state 1-36] also could not be fully explained with the weak antiferromagnetism but demanded a further symmetry breaking mechanism. This could be provided by changes in magnetic moments in low fields as we suggest here. The magnetism of UPt3 which is obviously very complicated seems to consist of more than one magnetic subsystem for the 5f uranium electrons. The first being the weak effective moments of the order of 10-2#B that undergo an AF phase transition at 5-6 K. This system contributes only little to either the magnetic susceptibility and thus to the specific heat but acts on the s.c. order parameter at least in low magnetic fields. The second system consists of larger effective moments originating from crystal field splitting. It is Van Vleck-like in the a, b plane and changes to a temperature independent state below Li. Above (T*, B*) this subsystem is intimately related to the s.c. transition and may lead to the superconducting phases B and C. Of course, more work is needed to further support the scenario outlined here. One crucial point seems to be the question why all the magnetic action in UPt3 seems to take place in the hexagonal basal plane while the c-direction is totally featureless. Many fruitful discussions with Profs. K. Andres, B.S. Chandrasekhar, D. Rainer, Drs. W. Biberacher, D. Einzel, and A. Lerf are gratefully acknowledged.
References [1] R. Fisher, S. Kim, B. Woodfield, N. Phillips, L. Taillefer, K. Hasselbach, J. Flouquet, A. Giorgi and J.L. Smith, Phys. Rev. Lett. 62 (1989) 1411.
[2] K. Hasselbach, L. Taillefer and J. Flouquet, Phys. Rev. Lett. 63 (1989) 93. [3] T. Trappmann, H.v. L6hneysen and L. Taillefer, Phys. Rev. B 43 (1991) 13714. [4] M. Boukhny, G.L. Bullock, B. S. Shivaram and D. Hinks, Phys. Rev. Lett. 73 (1994) 1707. [5] V. Mfiller, Ch. Roth, D. Mauer, E.W. Scheidt, K. Lfiders, E. Bucher and H.E. B6mmel, Phys. Rev. Lett. 58 (1987) 1224. [6] G. Bruls, D. Weber, B. Wolf, P. Thalmeier, B. Liithi, A. deVisser and A. Menovsky, Phys. Rev. Lett. 65 (1990) 2294. [7] R.N. Kleinmann, P.L. Gammel, E. Biicher and D.J. Bishop, Phys. Rev. Lett. 62 (1989) 328. [8] F. Gross, B.S. Chandrasekhar, K. Andres, U. Rauchschwalbe, E. Bucher and B. Liithi B, Physica C 153-155 (1988) 439. [9] G. Aeppli, D. Bishop, C. Broholm, E. Bucher, K. Siemensmeyer, M. Steiner and N. StiJsser, Phys. Rev. Lett. 63 (1989) 676. [10] G. Goll, H.v. L6hneysen, I.K. Yanson, and L. Taillefer, Phys. Rev. Lett. 70 (1993) 2008. [11] S. Adenwalla, S.W. Lin, Q.Z. Ran, Z. Zhao, J.B. Ketterson, J.A. Sauls, L. Taillefer, D.G. Hinks, M. Levy and B.K. Sarma, Phys. Rev. Lett. 65 (1990) 2298. [12] B.S. Shivaram, J.J. Gannon, Jr. and D.G. Hinks, Phys. Rev. Lett. 63 (1989) 1723. [13] T. Vorenkamp, A. deVisser, R. Wester, A.A. Menovsky and J.J. Franse, Phys. Rev. B 48 (1993) 6385. [14] K. Machida, T. Ohmi and M. Ozaki, J. Phys. Soc. Japan 64 (1995) 1067. [15] J.A. Sauls, Adv. in Phys. 43 (1994) 113. [16] R. Joynt, J. Magn. Magn. Mater. 108 (1992) 31. [17] J. Brooks, M.J. Naughton, Y.P. Ma, P.M. Chaikin and R.V. Chamberlin, Rev. Sci. Instr. 58 (1987) 117. [18] G.M. Schmiedeshoff, Z. Fisk and J.L. Smith, Phys. Rev. B 48 (1993) 16417. [19] P. Christ, W. Biberacher, H. Miiller and K. Andres, Solid State Commun. 91 (1994) 451. [20] P.H. Frings and J.J. Franse, Phys. Rev. B 31 (1985) 4355. [21] K. Andres, S. Darack and H.R. Ott, Phys. Rev. B, 19 (1979) 5475. [22] R.S. Craig, S.G. Sankar, N. Marzouk, V.U.S. Rao, W.E. Wallace and E. Segal, J. Chem. Solids, 33 (1972) 2267. [23] G.R. Stewart, J. Appl. Phys. 57 (1985) 3049. [24] G. Aeppli, A. Goldman, G. Shirane, E. Bucher and M.Ch.Lux-Steiner, Phys.Rev.Lett. 58 (1987) 808. [25] E. Schuberth, G. Hofmann, F. Gross, K. Andres and J. Hufnagl, Europhys. Lett. 11 (1990) 249. [26] E.A. Schuberth, K. Kloss, J. Schupp and K. Andres, Z. Phys. B 97 (1995) 55. [27] E.A. Schuberth, B. Strickler and K. Andres, Phys. Rev. Lett. 68 (1992) 117. [28] A. deVisser, A. Menovsky and J.J.M. Franse, Physica B147 (1987) 81.
E. Schuberth / Physica B 230 232 (1997) 9-15 [29] G.L. Bullock, B.S. Shivaram and D.G. Hinks, Europhys. Lett. 21 (1993) 357. [30] N.H. van Dijk, A. deVisser, J.J.M. Franse and L. Taillefer, Journ. Low Temp. Phys. 93 (1993) 101. [31] K.A. Park and R. Joynt, Phys. Rev. B 53 (1996) 12346. [32] N.H. van Dijk, A. deVisser, J.J.M. Franse, S. Holtmeier, L. Taillefer and J. Flouquet, Phys. Rev. B 48 (1993) 1299. [33] G. Fraunberger, M. Baldus and G.R. Stewart, Phys. Rev. B 47 (1993) 3204.
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[34] P.J.C. Signore, B. Andraka, M.W. Meisel, S.E. Brown, Z. Fisk, A.L. Giorgi, J.L. Smith, F. Gross-Alltag, E.A. Schuberth and A.A. Menovsky, Phys. Rev. B 52 (1995) 4446. [35] R.A. Fisher, B.F. Woodfield, S. Kim, N. Phillips, L. Taillefer, A.L. Giorgi and J.L. Smith, Solid State Commun. 80 (1991) 263. [36] E.D. Isaacs, P. Zschack, C.L. Broholm, C. Burns, G. Aeppli, A.P. Ramirez, T.T.M. Palstra, R.W. Erwin, N.Stiicheli and E. Bucher, Phys. Rev. Lett. 75 (1995) 1178.
Physica B 230-232 (1997) 9-15
Anisotropy of the magnetic susceptibility of UPt 3 Erwin Schuberth Walther Meissner Institut, Walther Meissner Str. 8, D-85748 Garching, Germany
Abstract Despite great efforts, both experimental and theoretical, the problem of magnetic behavior of UPt3 and its relation to the multiple superconducting phases has not been completely solved. This system continues to show unexpected and often unexplained features. To study the magnetic anisotropy of UPt3 single crystals in a wide range of temperatures and magnetic fields, we used a capacitive torque-meter, where the torque acting on the probe is proportional to (Z~.b - zc)B 2As in the old data of Frings and Franse we find in intermediate fields that Z, and Zb exhibit a maximum around 20 K, while Xc is only slightly enhanced with respect to the room temperature value. The resulting torque maximum around 20 K, however, turns out to be strongly field dependent. On decrease of B, a shift towards lower temperatures sets in below 3 T. This shift becomes nearly horizontal between 0.5 and 0.3 T and the maximum moves down to below 10 K. A second unexpected feature occurred when we followed the temperature dependence of)~a,b - Zc down to 15 mK: it ends suddenly but reversibly at a well-defined border line Li which is also field dependent. For magnetic fields above Be2(0), Li lies between 60 and 100 mK and joins the upper critical field curve at (100 mK, 1.86 T). Below this point it follows Bc2(T) down to the tetracritical point (T *, B*) where it splits off again. These results will be discussed with respect to existing theories of the superconducting order parameter and its relation to the anisotropic magnetism of UPt3. Keywords: UPt3; Heavy electron system; Magnetic susceptibility; Superconductivity
1. Introduction UPt3 is still the most widely studied heavy fermion system because of its unusual superconducting (s.c.) properties. There is a large body of experimental and theoretical work related to this problem. On the experimental side.', the existence of multiple s.c. phases, usually denoted as A, B, and C, is evident from data for the specific heat [1], also under applied field [2] and pressure I-3, 4], for ultrasound attenuation [5, 6-1, flux line dynamics [-7], London penetration depth 1-8], neutron scattering [9-1, tunnelling [10], and for the anisotropy of the lower and upper critical fields [-6, 11, 12]. This is taken as strong evidence for unconventional superconductivity, especially the observation of a double transition in low magnetic fields and the
occurrence of a tetracritical point ( T *, B*) where four phase lines meet, but also the existence of kinks in B c l ( T ) and B c z ( T ). In addition, cs.c.(T) and the London penetration depth 2 ( T ) exhibit power-laws instead of an exponential or (1 - (T/To)4) - 1/2 behavior. For a recent overview see Vorenkamp et al. 1-13]. On the theoretical side, many attempts have been made to analyse the data with respect to an unconventional s.c. order parameter (one or two-dimensional) coupled to a symmetry breaking field for which the most likely candidate considered was the antiferromagnetic phase observed below 5 K in neutron scattering experiments. However, so far, no fully consistent picture has emerged, and no consensus for the dimensionality and symmetry of the order parameter has been established. One of the
0921-4526/97/$17.00 Copyright (C 1997 Elsevier Science B.V. All rights reserved PII S092 1 - 4 5 2 6 ( 9 5 ) 0 0 5 3 3 - 9
10
E. Schuberth / Physica B 230-232 (1997) 9-15
most persistent problems is the topology of the phase diagram around the tetracritical point. This topology is the same for all field directions, a fact which is not easily reconcilable with two-dimensional representations of the s.c. order parameter which seem to describe other experimental properties better than one-dimensional representations. The question whether the spin-orbit coupling is strong or weak has also been raised [14]. In the latter case, the spin can freely follow the direction of an external applied field. For recent theoretical reviews see Sauls [-15] and Joynt [16]. In our present work we determined the anisotropy of the static magnetic susceptibility of UPt3 single crystals over a wide range of temperature and magnetic field using a capacitive torque-meter. We find that there are additional magnetic phenomena, not observed so far, which show that the low-field magnetic behavior is different from the high-field one which interrelate superconductivity and magnetism in UPt3 over a wide portion of Be2( T ). Both phenomena are connected to the sizeable anisotropy of the magnetic susceptibility with a maximum of X,,b at 20 K which points to stronger magnetic moments in addition to the weak ones that seem to order antiferromagnetically at 5 K. The latter transition we do not detect in spite of the very high sensitivity of the torque-meter method especially in high fields. We also found no signature of it in a vibrating sample magnetometer around 4 T and in a SQUID magnetometer in fields around 10 G. The capacitive torque-meter was introduced by Brooks et al. [17] and was also used by Schmiedeshoffet al. [18] and Christ et al. [19]. The torque experienced in an external magnetic field (apart from shape effects which are negligible for UPt3 since Za, b,c are small) is given by T = Zaniso
VB2/t~oCOS O
sin O,
where in the case ofUPt3 Zaniso= Z,.b - Zc, O is the angle between the anisotropy axis and the external field, and • is perpendicular to Bext and to c (see Fig. 1). For UPt3, the anisotropy of the susceptibility which is large already at room temperature has been measured 10 years ago by Frings and Franse [20]. We confirmed their data with our first sample
'0
IVIab
~
a
~
~
respel acer
capacitor Fig. 1. Design of the torque meter and orientationof the samples with respect to the external magneticfield. in a vibrating sample magnetometer in a field of 4 T. X, and Zb are identical and show a maximum at 20 K, while Zc increases down to this temperature and then levels off. At the lowest temperatures X, and Zb are about a factor of 2 larger than L,, and we expected a strong torque even at mK-temperatures. These data are reminiscent of a similar case, namely, that of the hexagonal compound PrNi5 where the same magnetic behavior could be quantitatively explained by Pr 3 ÷ states split by the crystal field [21]. However, no specific heat anomaly was found around 20 K for UPt3. But also in PrNi5 only a shoulder in c ( T ) is observed at the temperature of the corresponding maximum in )~1c [22] and for UPt3 the heavy fermion specific heat may mask the magnetic contribution. Arguing from a T 3 In T contribution to the specific heat capacity Frings and Franse and also Stewart [23] interpreted the magnetic anisotropy as a signature of ferromagnetic spin fluctuations. Aeppli et al. [24] found magnetic fluctuations below 20 K in neutron scattering experiments which were attributed to ferromagnetic basal planes coupled antiferromagnetically along the c-direction. A small drop of Za.b below 20 K could be deduced from these data, much smaller however than observed in the direct
E. Schuberth /Physica B 230-232 (1997) 9-15
susceptibility measurements. In analogy to PrNis, we think that the large anisotropy in UPt3 is rather due to the crystal field splitting of the uranium 5f moments leading to a maximum in )~,,b at the temperature at which the first magnetic level is depopulated while in the c-direction only a slightly enhanced Pauli susceptibility is present. This view of an in-plane Van Vleck and an out-of-plane Pauli susceptibility has been recently proposed also by Park and Joynt 1-31].
2. Experiment The design of our torque-meter is shown in Fig. 1. The upper capacitor plate with the narrow cantilever was etched out of O F H C copper. The dimensions of this "tongue" were 0.05 x 2.7 x 0.4 mm 3. The samples, three single crystals with masses between 6.55 and 18.08 rag, came fi'om two different sources and they have been investigated already in our previous heat capacity work [-25, 26] (samples # 1, # 2), and [-27] (samples # i, # 3). They were all cleaved off from larger crystals used in those experiments. The first such crystal was grown by Bucher, Konstanz, with electron beam melting from depleted uranium. Although it has not been annealed so far, we consider it our best sample judging from the high and sharp Tc(520 inK), its high residual resistivity ratio (200), and its sharp X-ray reflexes. The second and third crystals were grown by A.Menovsky, Amsterdam, with the Czochralski method, sample # 3 has been annealed and showed the usual double transitions in the specific heat at T + and T [ . The upper capacitor plate was electrically isolated from the body of the torque-meter (OFHC copper or Ag) by lens paper of 40 gm thickness soaked with GE varnish. The latter was used to give good thermal contact to the plate onto which the samples were glued with DuPont silver epoxy. Estimates showed that the thermal[ contact through the GE-varnish and the silver epoxy was good enough to cool the slightly radioactive samples to below 15 mK with the body temperature below 10mK provided by a dilution refrigerator. The temperatures of the sample holder were determined with a carbon resistor calibrated against NBS fix-
11
point standards (in previous runs) and against a second carbon resistor in the field-compensated region of our cryostat when magnetic fields were applied. The capacitance of the meter was measured with a General Radio 1615-A capacitance bridge excited externally with a frequency of 3-5 kHz. It changed by about 50%, starting from typically 1.1 pF in B = 0, when the field was increased up to 14 T. The torque calculated from the C-traces had the expected B2-dependence in fields above 2 T. Deviations from this behavior both in low and in high fields are discussed below. The B T region between 0 and 7.1 T, 15 mK and 40 K was scanned by either sweeping the field at constant T, or by varying T in a constant field. In the superconducting phase changes of )~aniso were hard to detect in constant field. Due to strong flux pinning, the Meissner effect is very small in the Shubnikov state (we only worked above Be1 here). Only when the field was changed below To, and the sample was warmed up across the upper critical field, the decay of non-equilibrium flux line configurations and shielding currents at Tc made it possible to detect Be2 ( T ) . On the other hand, the upper critical curve could be well distinguished from the adjacent L/dine (see below) because on reversing the field changes below T~ the opposite direction of the shielding currents led to a reversed torque signal at Bc2 , opposing the T-dependence at Li, see Fig. 2.
3. Results The first remarkable result concerns the decrease of the z-anisotropy of UPt3 below 20 K. We find that its T-dependence ends sharply and reversibly, at a well defined temperature T/which depends on B. Below 7,. no T-dependence of Z,.iso = Z~,b -- Z,. within the limit of our sensitivity ( A C / C ~ 5 x 10 -6 ) could be detected, see inset in Fig. 3. Above Ti, Zanisovaried oc T" up to ~ 4 K with v ranging from 1.44 to 1.64. Figs. 3 and 4 show the border line L i ( T , B ) defined by the different Ti(B) which separates the temperature independent region below it from the onset of a detectable T-dependence. The data
E. Schuberth / Physica B 230-232 (1997) 9-15
12
B~ . .
3.5
~.~.. ~__~
e
3.0 ~
1.0 ~ 0.5 ~ Li =
~• T - i n d e p
t
:t
i 0,5
!
i
T-~ndepN°3t'
E 3-
1 0
i
•
0.0
0.1
,
0.2
,, aniso/T-dep ~'~~, T-indep TI.5
i
,
i
0.3
0.4
[KI,5]
a sample #1
• ,.
U Pt
.4.. " t ~ . . ~ v v A• / / r A y
i
,
0
c. ~,.~ 0.2
0.4 0.6 T [K]
0.8
1.0
Fig. 4. Phase diagram of UPt3 and the Lrline as deduced from sample # 2. The symbols have the same meaning as in Fig. 3. The orientation of the c-axis was 41 ° with respect to B~xt.
i i
0.2
,~
2-
,
3
sample #2
"" ~, aniso/T-dep T-indep
0.0 0.0
t
Fig. 2. Two heat-cycle paths used to determine B,2 and L~ and the resulting torque in different fields within the s.c. region. Path 2 was especially suited to distinguish both features on the warming curves.
i
UPt
1.5 ~
Bc2
I~
i
2.0 ~
T
@
i
•
2.5 ~
I, ': ....... t\; .....
i
0.25 0.50 0.75 1.00 1.25
1.5
T [K]
Fig. 3. Phase diagram of UPt3 as deduced from the temperature dependence of the torque in magnetic fields on sample # 1. The c-axis was oriented under 26 ° with respect to the external magnetic field. • and • define the onset and offset of a T-dependence upon warming and cooling, respectively (see inset). The solid lines LdT, B) taken as the detection limit (resolution AC/C = 5xl0 -6) of a T-dependence separate the regions of a T-independent z-anisotropy from the T-dependent area. The open circles define the upper critical field (decay of shielding currents), see text. In the inset interpolated data of the deviation from the low-temperature value in the respective field (here 2.6 T) are plotted versus T 15.
indicated • were taken upon warming and • on cooling through L~. The superconducting transition is indicated by the o-symbols. On some occasions, but not always, torque changes inside the s.c.
region could be detected at the position of the internal phase line B-C (symbols' + ' in Fig. 3). In high fields, Li is nearly independent of B, while at intermediate fields it coincides with the superconducting phase boundary. At lower fields, near (T*, B*), L/ again separates from the s.c. region towards higher temperatures. It is striking that the two phase boundaries L~ and Be2 (T) follow each other over a wide temperature and field range. The possibility that Li extends from higher fields into the s.c. region at 100 mK, being screened by shielding currents and thus detectable only above To, can be ruled out since often we observe a small separation between L~ and Tc (even on cooling) which would not be the case if L~ would lie deep inside the s.c. region. Thus, T~(B) coincides with L~(B) in this field range indicating a simultaneous change in the magnetic state at the superconducting transition. If one extrapolates the low-field Lrline of the first sample from higher temperatures into the s.c. region one directly matches the internal B-C phase boundary at (T*,B*). This behavior is less clear with the second sample, because there we did not find this internal phase line in our torque measurements. Whether the difference in the low-field Lrlines comes from a sample dependence or from the different orientation needs to be studied.
E. S c h u b e r t h / P h y s i c a B 2 3 0 - 2 3 2
The second unexpected result of our torque measurements occurred at high temperatures, when we followed the torque maximum at 20 K to lower fields. This maximum shifted towards lower temperatures below 3 T, first slowly as the field was lowered, but then dramatically below 1 T. It was even suppressed to below 1.5 K in 0.2 T, see Fig. 5. These data imply a field-dependent susceptibility in low magnetic fields. Preliminary magnetization curve measurements indeed reveal an increase of Za,b in fields below 0.5 T, in accord with the Tsweep data. We do not think that these results are due to paramagnetic impurities, since such moments would be isotropic in first order and not felt by the torque meter. They also could not suppress the torque maximum at 20 K. Impurity contents are very low in our crystals anyway.
4. Discussion What is the origin of the anisotropy of the magnetic susceptibility of UPt3 which has been measured long ago? The maximum of Z,.b at 20 K is hard to explain in terms of spin fluctuations. Also,
I
I
I
14-
I
f
•
12 ~
@
•
o
UPt single crystals
o
3
10 ~
oo o
'--'8
m6
sample #1 i2,,j~
#3 ~
d'
®
~""'#2
4 2 0
.o .
-.
0.2 T, <1.5 K , ~ .
o
_ _
_
._ . . . . . . .
.
•
.o ~e
. . -
_ =-=-==.=_ ._o:...::..£.. _.-._._ . . . . . . . . . . .
1'2
1'6
2'o
T [K] Fig. 5. Location of Li and of the torque maximum for single crystals of UPt3. The full circles refer to the annealed probe # 3 oriented below 45 ° between the c-axis arid Bcxt.
(1997) 9 15
13
a paramagnon contribution to Z is very unlikely. DeVisser et al. [28] argue, that a positive curvature N"(EF) > 0 would be necessary to obtain a rising z(T) with rising temperature. But such effects are usually very weak. The comparison to PrNi5 shows that crystal-field-split magnetic ion states are much more likely the cause of this unusual behavior. How the weak itinerant moments which order antiferromagnetically around 5 K (also in the a-b plane) are related to this magnetic behavior and how the Kondo effect and many-body effects influence it needs to be studied. Our low-field Li -line and the suppression of the torque maximum at 20 K give evidence that with increasing field the magnetic character changes between 0.2-0.7 T, depending on the sample and/or the field orientation. Since we find a similar behavior in annealed and as-grown samples, the field-dependent susceptibility in low fields seems not to be related to the structural modulation induced by annealing. There are several effects noticed in the literature which are pointing in the direction of our arguments and which may be related to the phase line Li found here. One is the lattice softening at low temperatures observed by Bullock et al. in ultrasound experiments [-29]. Their onset of decrease of the sound velocity in 3 T coincides with Li(3 T). Then, the point where Li meets Bc2(T) in high fields lies close to the cross-over point for the upper critical field curves for B//a and B//c [-6, 30]. Third, a nearly temperature independent phase line at about 0.1 T has been detected as an anomaly in the magnetostriction by van Dijk et al. [32]. (It was interpreted by the authors as a transition from random orientation of antiferromagnetic domains in low fields to an orientation ± Bext in higher fields.) Fourth, Fraunberger et al. [33] find that contrary to usual expectations the Sommerfeld coefficient ~, is not correlated with the low temperature susceptibility of UPt3 which again demonstrates that there are unknown contributions. Fifth, an anomalous increase of the real part of the ACsusceptibility above 1 T was reported by Signore et al. [34]. The onset of this increase also follows the upper critical curve above B*, and thus Li, although the detection limit of this experiment does not allow the statement that it really coincides with B c 2 ( T ). Finally, in fields above B c 2 ( O ) w e earlier
14
E. Schuberth / Physica B 230-232 (1997) 9-15
observed an upturn of the specific heat capacity below 50 mK 1-25,26] just below Li. What follows for the theory of UPt 3 superconductivity? There is a lot of evidence from neutron scattering, that antiferromagnetic order exists below 5 K and that it is related to the A, B superconducting phases. But why can it not be detected in magnetization and specific heat measurements 1-35]? Recent magnetic X-ray and neutron diffraction experiments in the s.c. state 1-36] also could not be fully explained with the weak antiferromagnetism but demanded a further symmetry breaking mechanism. This could be provided by changes in magnetic moments in low fields as we suggest here. The magnetism of UPt3 which is obviously very complicated seems to consist of more than one magnetic subsystem for the 5f uranium electrons. The first being the weak effective moments of the order of 10-2#B that undergo an AF phase transition at 5-6 K. This system contributes only little to either the magnetic susceptibility and thus to the specific heat but acts on the s.c. order parameter at least in low magnetic fields. The second system consists of larger effective moments originating from crystal field splitting. It is Van Vleck-like in the a, b plane and changes to a temperature independent state below Li. Above (T*, B*) this subsystem is intimately related to the s.c. transition and may lead to the superconducting phases B and C. Of course, more work is needed to further support the scenario outlined here. One crucial point seems to be the question why all the magnetic action in UPt3 seems to take place in the hexagonal basal plane while the c-direction is totally featureless. Many fruitful discussions with Profs. K. Andres, B.S. Chandrasekhar, D. Rainer, Drs. W. Biberacher, D. Einzel, and A. Lerf are gratefully acknowledged.
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[34] P.J.C. Signore, B. Andraka, M.W. Meisel, S.E. Brown, Z. Fisk, A.L. Giorgi, J.L. Smith, F. Gross-Alltag, E.A. Schuberth and A.A. Menovsky, Phys. Rev. B 52 (1995) 4446. [35] R.A. Fisher, B.F. Woodfield, S. Kim, N. Phillips, L. Taillefer, A.L. Giorgi and J.L. Smith, Solid State Commun. 80 (1991) 263. [36] E.D. Isaacs, P. Zschack, C.L. Broholm, C. Burns, G. Aeppli, A.P. Ramirez, T.T.M. Palstra, R.W. Erwin, N.Stiicheli and E. Bucher, Phys. Rev. Lett. 75 (1995) 1178.