Magnetic properties of the heavy-fermion superconductors UPt3 and URu2Si2

0038-1098/93 $6.00 + .00 Pergamon Press Ltd

Solid.State Communications, Voi. 85, No. 4, pp. 355-360, 1993. Printed in Great Britain.

MAGNETIC PROPERTIES OF THE HEAVY-FERMION SUPERCONDUCTORS UPt3 AND URu2Si2 S. Wiichner, N. Keller and J.L. Tholence CRTBT-C.N.R.S., BP 166, 38042 Grenoble-C6dex 9, France and J. Flouquet DRFMC-C.E.N.G., BP 85 X, 38041 Grenoble-C6dex, France

(Received 9 September 1992 by P. Burlet) Magnetization measurements have been performed in order to investigate further the temperature dependence of the lower critical field He,, the Meissner effect and the critical currents on single crystals of the anisotropic Heavy-Fermion Superconductors UPt3 and URu2Si 2. For the UPt 3 single crystals, no marked anisotropy in the lower critical field has been observed, but a change in slope could be identified at A T ~ 65 mK below To. Extrapolation of the lower critical field to zero temperature gives a value of 6.5mT in both directions (H I[ c, H _1_c). The lower critical field of URu2Si2 roughly shows a linear temperature dependence in the range from Tc to ,~ 150 mK. At T ~ 150 mK He, seems to pass through a maximum before eventually dropping down. The magnetization measurements have shown that He, of URu2Si 2 is isotropic for H II c and H _L c with a value He, (0) of 3.3 mT. This isotropy of the Hc, contrasts strongly with the anisotropy of upper critical field Hc2(T ). The measured hysteresis of the magnetization of URu2Si2 is more important than in the case of UPt3 and exhibits magnetization jumps at low temperatures. Applying the Bean-model, critical current densities of 3.8 x 103Acm -2 and 2.4 x 104Acm -2 are deduced for UPt3 and URu2Si2 respectively at T/Te ~ 0.2 and IzoHi ,,~ 80mT. 1. INTRODUCTION RECENT intensive studies focus on the unconventional nature of superconductivity in heavy fermion superconductors like UPt3 and URu2Si2 [1]. Special interest was given to UPt3 exhibiting a reduced moment antiferromagnetic (AF) ordering (Tu = 5 K) which is persisting in the superconducting state (Te _ 0 . 5 K ) [2]. The moments of the antiferromagnetic order are oriented parallel to the basal plane of the hexagonal structure. Apparently, they are reducing the symmetry of the lattice. This creates a rich phase diagram with a tetracritical point, established by specific heat [3], thermal expansion [4] and ultrasonic attenuation [5] measurements. Similar to UPt 3, the heavy fermion compound URu2Si2 [6] also shows the coexistence of long-range antiferromagnetism (TN ~ 17 K) and superconductivity (Te ~ 1.2 K). As the reduced moment is aligned parallel to the c-axis of the tetragonal structure, the lattice symmetry is preserved and no splitting of the superconducting transition occurs [7]. This coexistence of

AF and superconductivity make UPt3 and URu2Si 2 important systems to test models of unconventional superconductivity. Magnetic measurements have been performed on different samples of UPt3 and URu2Si2 mainly in the superconducting state. Since, we only applied small magnetic fields (H < 300mT), careful demagnetization corrections had to be made, to account for the influence of the sample-geometry on the local field distribution (Hi). The samples used for these measurements have been shaped as a cube (UPt3 No. 1), as a plate (UPt3 - No. 2) and as a disc (URu2Si2) (cf. Table 1 for details). By assuming the crystallographic axes of the samples to be equal to the principle axes of an ellipsoid, it was possible to calculate the main demagnetization factors (Ni) [8]. Their experimental values (Ni) could be checked by assuming the perfect diamagnetism (B = 0) from 1

N = 1 4 Xa(T ~ 0)' (M = magnetization,

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Ha = applied magnetic field).

356

H E A V Y - F E R M I O N S U P E R C O N D U C T O R S UPt 3 A N D URu2Si 2

Vol. 85, .No. 4

Table 1. The dimensions and demagnetization factors are given along the corresponding principle axes of the samples (UPt3: axes a x b' × e with b' ± a in the basal plane," URu2Si2: axes al × a2 × e) Sample

Mass (mg)

Shape

Dimensions (mm 3)

Demagnetization factors [8]

UPt3 - N o . 1 UPt3 - No. 2 URu2Si2

58.4 25.3 23.9

Cube Plate Disc

1.43 x 1.48 × 1.45 1.64 × 1.38 × 0.57 0.79 × 1.95 × 1.93

0.343 × 0.325 × 0.335 0.604 × 0.172 × 0.218 0.583 × 0.205 x 0.212

They are in good agreement (+5%) with tlie calculated ones. The demagnetization corrections were made with the calculated Ni : Hi = Ha - NiM. Two different experimental set-ups have been used during these measurements. The first one, a non-commercial low field SQUID-magnetometer (Ha < 20mT), was used to perform temperature dependent magnetization measurements in a static magnetic field. The second set-up consists of a classical two-coil extraction magnetometer with a magnetic field strength H a < 0.3 T. This arrangement was used for the isothermal magnetization measurements as a function of applied field. All measurements have been done using a miniaturized dilution refrigerator (q~29mm) [9], which was displaced vertically to perform the sample extraction in both magnetometers. Our aim was the study of the lower critical fields and the critical currents in the superconducting heavy-fermion compounds UPt3 and URu2Si2. The magnetization measurements M ( T ) in the SQUID-magnetometer have mainly been used to explore in more details the temperature dependence of the lower critical field Hc, (T) in the neighborhood of the superconducting transition temperature (To) of the compound UPt 3 [10]. In order to construct the phase diagram Hc,(T ), we applied a deviation criterion to the M ( T ) curves. Assuming a constant local magnetic field Hi, we determined the temperature T* corresponding to the deviation of a certain percentage from the diamagnetic shielding signal. Afterwards this temperature dependence of the corrected field Hi(T*) was identified with the Hc,(T ). One has to keep in mind that the real He, of the bulk sample is approached using the limit when the percentage deviation tends to zero. However, the M ( T ) dependence also contains the contribution of the temperature dependent magnetic field penetration A(T) into the bulk sample. For local fields smaller than the Hc, (T) the change in magnetization M ( T ) can be directly related to the A(T). These results will be published elsewhere [11]. The M(H)-dependence allows to determine the first flux penetration at He, into the superconductor using an extended Bean

model discussed below. The He~(T) can be studied down to very low temperatures without problems. On the other hand, we can deduce the critical currents using the Bean model [12] from the hysteresis curves. Additionally to the standard deviation criteria, an extended Bean model [13, 14] assuming a reversible magnetization in the mixed state was applied to the virgin magnetization curves. In this model the internal fields will correspond to: B = O;

H <

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(1)

(meq represents the reversible magnetization normalized by Hc~, H* is the field characterizing the first complete penetration of flux into the bulk). Using the relation B = # o ( M + H i ) to relate the measured magnetization to the internal fields, the magnetic induction can be expressed by the following analytical expression [14] in a field range of 0 < H <_ H*' B ( H ) = #o[a + bH + O ( H - He, ) x {BI(H-

Hc, ) + B z ( H - Hct)2}],

where a is the possible shift in the zero, b = 1 + Xi is the variation in the initial susceptibility 6x due to a varying superconducting volume fraction and

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1 H > He, 0 H < Hc, "

The values of Xi and a are obtained from a fit of the linear part of the magnetization ( M = x i H + a ) . Therefore, we can describe the induction B with the following parameters: He,, B1, B2. 2. L O W E R C R I T I C A L FIELDS The application of the extended Bean model to UPt3 always gives values for the lower critical field inferior to those obtained by the application of the

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Fig. l. Hq(T) of UPt 3 shown for a 5% deviation criterion and a fit with an extended Bean-model (see text) to the M(H)-curves (open circles and triangles: H II a and black circles and triangles: H ILc). The inset shows Hc, determined from M ( T ) - S Q U I D magnetization measurements. deviation criteria as shown in Fig. 1. Our main result is the isotropy within 5% of He, (T) with the external field parallel and perpendicular to the hexagonal plane. Within logarithmic accuracy the Hc,(T) is given by Hc~ (T) ~ ~0/[47rA2(T)] [15]. Considering a power law behavior for the temperature dependence of the penetration depth ()~2(T) oc [1 - (T/Tc)E]-n), we can describe the Hc~ (T) by the following law:

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The extrapolation of H~,(T) to zero temperature gives a Hc,(O ) = 6.5mT with an exponent n _ 1.3 (sample: UPt3 - No. 1). The extraction method used to measure the magnetization was not sensitive enough to detect any low field anomaly in H~, (T) near T¢. However, SQUID-magnetization measurements on the sample UPt3 - No. 2 reveal the change in slope in the H~, at ,,~ 65 m K below T~ as shown in the insert of Fig. 1 (see also [10]). Specific heat measurements [3, 4] have already shown that a second phase transition takes place at A T ~ 65 mK below T~ in the superconducting phase. The kink in Hc,(T) coincides then with the phase boundary between the A and B phases of the phase diagram [4]. The curves Hc~(T ) obtained from M(H) and M(T) measurements are in good agreement in shape and absolute values. URu2Si2 on the other hand also shows an isotropic behavior of Hc,(T) with the magnetic field parallel to the c-axis or to the tetragonal plane (Fig. 2). Furthermore, Hc~ (T) shows at first sight a very linear temperature dependence from Tc (~- 1.2 K) down to 150inK, where it seems to saturate or present a maximum. A similar linear dependence has already

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Fig. 2. H~, (T) of URu2Si2 as determined from M(H) measurements by deviation criteria (2%, 5%, 15%) and a fit of an extended Bean-model. Isotropic/-/~, (T) for nil a and nil c is shown for the 15%-curve. been observed down to 300mK [16]. However, a linear extrapolation of the Hc,(T ) curves towards zero magnetic field (H ~ 0) would lead in our case to a temperature value of about 0.9 Tc. Therefore, we tried also the same empirical law (2) as in the case of UPt3. As can be seen in Fig. 2, this power law behavior describes quite well the measured dependences. The evaluation of the Hc~ (T) data obtained from the application of the extended Bean model towards T = 0 K using the empirical law (2) gives a value of Hc, (0) = 3.3 mT with an exponent n ~ 1.7. This value for the critical field is smaller than the value (6.5 mT) for UPt3. 3. C R I T I C A L C U R R E N T S Complementary to the initial magnetization after a zero field cooling (ZFC), the complete hysteresis cycles have been studied for magnetic fields up to 200 mT and temperatures down to 70 mK. By Amp~re's law (#0J = rot B), we can calculate directly from the measured hysteresis cycles (corrected for the demagnetization effects) the critical current densities. Technically, this consists in the application of the original Bean model [12] to a slab where the surface S = a × b is perpendicular to the applied field. One finds [17] for the critical current densities:

Je = 6aAM/(3ab- b2); a > b Jc = 12aMJ(3ab - b2)

(3)

with M~ the remanent magnetization. The hysteresis cycles for UPt 3 showed a small (almost temperature independent) paramagnetie contribution of #0M--~ 0.008#0H superimposed. Subtraction of this paramagnetic contribution led to symmetric hysteresis cycles. The critical current densities (Fig. 4) evaluated from the remanent magnetization showed the same isotropic behavior

358

H E A V Y - F E R M I O N S U P E R C O N D U C T O R S UPt3 AND URueSi 2

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The critical current density at T = 0 K can then be evaluated to J~(0) = 3.8 x 103 A c m -2. This value is in good agreement with a J¢(0) of 6.5 x 103Acm -2 recently obtained from magnetic relaxation measurements [18]. The extracted exponent (m-~ 1.15) is slightly different from that of He, (T) in UPt3. A further analysis of the hysteresis cycles gave the isothermal magnetic field dependence of arc showing that the critical current densities are quite small and rise to their maximum values only in very small applied fields H < 100roT. The J~(H) decreases by a factor of approximatively eight with a change in field from zero to 100 roT. This indicates a weak pinning of the vortices in the superconducting phase at low magnetic fields. This absence of strong pinning ~ . . . .

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Fig. 5. Critical current density of UPt 3 deduced from the remanent magnetization, showing isotropic behavior for H II a, H II b and Hi] c (b _L a in the hexagonal plane). centers indicates the high purety of the samples. On the other hand, the existence of a very small Meissner effect (about 2% of the diamagnetic shielding signal in 0.5roT) is commonly interpreted to indicate a strong pinning of the vortices. But there is another possibility to inhibit flux-expulsion. The existence of a Bean-Livingstone surface barrier may limit the expulsion of flux considerably when cooling the sample through the transition in an applied magnetic field [19]. URu2Si 2 exhibits a much bigger hysteresis as shown in Fig. 3. The cycles measured below ,-~ 0.4 K are marked by large magnetization jumps• These magnetization jumps increase in amplitude and decrease in number with increasing temperature and they are reproducible. This effect is common in type II and high Tc superconductors [20]. It can be explained by a sudden depinning of vortices from the pinning centers due to Lorentz forces, liberating a small amount of depinning energy. Considering the small thermal conductivity and low specific heat, this energy leads to a local warming of the sample and a group of vortices can be depinned and penetrate the 3"104

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Fig. 8. Field dependence Je(H) calculated from the hysteresis cycles of URu2Si2 for several temperatures.

bulk sample by an avalanche effect. Afterwards the bulk screening currents will reinstall, giving a magnettization curvature similar to the initial magnetization. Considering these magnetization jumps below 400 inK, we used only the envelop of the curve to determine the lower bound of the critical currents. As in the case of UPt 3, the critical current densities evaluated from the remanent magnetization (Mr) (Fig. 5) showed an isotropic behavior. At first sight a similar linear dependence as for He~ (T) is observed. But Jc(T) can as well be explained by the temperature dependence used for UPt3. The Je(0) then can be evaluated as ,,, 2.4 x 104 Acm -2, which is one order of magnitude bigger than in UPt3. The exponent m _~ 1.7 is the same as n for He, (T). Although the temperature dependences of the He,(T) and the Je(T) are quite well described by the given power laws, the ratio of the exponents (m/n "~ 0.9 for UPt3, m/n _~ 1 for URu2Si2), which should be equal to 3/2 for the relation Je oc He, (T)/A(T), indicates a more complicated relationship between A(T), He,(T) and Je(T). The isothermal critical current density Jc(H) raises continuously but less significantly only below ,~ 100mT [Jc(0)/Je(100mT) ~ 2], indicating also the presence of weak pinning centers at low applied magnetic fields. In conclusion, we determined the lower critical fields and the critical current densities by isothermal magnetization measurements of UPt3 and URu2Si2. Both of them show isotropic behavior of the He, (T) and the Je(T) respectively to the crystallographic orientations. The Je(0) was determined to 3.8 x 103 Acm -3 and 2.4 x 104Acm -3 for UPt 3 and URu2Si2 respectively. In both cases Her (T) and Jc(T) can be fitted with a [1 - (T/Te)2] n general law. The quasi isotropy in H~, by contrast to the anisotropy observed in He2 (factor near 5 for URu2Si2 [7, 21]) cannot be explained in a scheme of conventional non magnetic superconductors where the anisotropy of He~ will be

reversed from that of He2. That confirms the importance of magnetic effects (see [22]). REFERENCES 1.

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7. 8. 9. 10.

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