POSITIVE MUON KNIGHT-SHIFT AND SPIN RELAXATION IN HEAVY FERMION SUPERCONDUCTORS UPT3 AND UBE13

Abstract We report muon spin rotation/relaxation (μSR) measurements of the heavy fermion superconductors UBe 13 and UPt 3 . In both materials we find that the muon Knight shift is unchanged in the superconducting state, consistent with odd-parity pairing (such as p-wave). The magnetic field penetration depths in UPt 3 and UBe 13 are extremely long, greater than 10000 A. We find no evidence of a magnetic transition in UBe 13 below 10 K.

Heavy fermion (HF) systems, materials where the conduction electrons have extremely large effective masses m* (as exhibited in such properties as the electronic specific heat y and the cyclotron mass) have been the subject of numerous investigations #I. Measurements of ultrasonic attenuation [ 2 1, magnetic field penetration depth [ 3 1, neutron scattering [4], NMR Knight shift [ $61 and specific heat [7] have provided evidence that the superconducting pairing in UBels and UPt, may be other than s-wave (I=O). The static spin susceptibility and the magnetic field penetration depth are two properties of the host system which reflect the symmetry of the superconducting state. Positive muon spin rotation/ relaxation (uSR), through measurements of the ' Present address: Riken 2-1 Hirosawa, Wako-shi, Saitima 35 l-01, Japan.
" For a review of heavy fermion systems, see ref.
Elsevier Science Publishers B.V. (North-Holland) muon Knight shift and the transverse field muon spin relaxation rate is a sensitive probe of these properties.
In an even-parity (such as s-wave) superconductor, the electrons are paired in states with opposite spin. Thus, the susceptibility of the pair is zero. Only states excited above the superconducting gap contribute to the spin susceptibility which falls from its normal state value to zero at T=O. The spin susceptibility xsc of an s-wave BCS superconductor is given in terms of the Yosida function Y(T) as xsc=xnY( T) where xn is the normal state susceptibility. In real systems, spin-orbit scattering from surfaces or impurities can cause a reduction of this change in the susceptibility [ 8 1. Two examples of triplet-paired states are the ABM and BW states (by analogy with superfluid 3He). As a result of the different pairing symmetry, the susceptibility of these states are XABM =x" and xBw=xn [ $ + f Y(T) 1. In either case, the spin susceptibility differs markedly from that of a spin singlet superconductor; measurement of x allows one to distinguish different pairing symmetries.
In a pSR experiment [9], muons are implanted one at a time and a histogram of muon-decay positrons is obtained as a function of the time after the muon arrival. The number of detected positrons is where B is a time-independent background, rp = 2.2 ps is the muon lifetime, &--0.3 is the initial asymmetry and B (t) is the muon polarization in the direction of the detector. In transverse field experiments, the polarization function is 9 (t) =G,,(t) x cos [ co,, (t) + $1 where w,, = @,,, (r, is the muon gyromagnetic ratio and Bloc is the local field experienced by the muon). The relaxation function G,,(t) is typically approximated by a Gaussian (G,(t)=exp( -a*t*/2)). The flux lattice in the mixed state of a type-II superconductor provides an inhomogeneity AB in the magnetic field throughout the sample which (in high fields) is inversely proportional to the square of the penetration depth [ lo]. Since the relaxation rate 0 is proportional to the field inhomogeneity, measurement of IJ allows one to determine the penetration depth ( (TSC 1 /A'). The temperature dependence of the magnetic field penetration depth A ( T) also provides information about the pairing symmetry. If there are no nodes in the gap, the penetration depth will show little temperature dependence as T-0. If there are zeroes in the gap however, thermal pair-breaking will give a power law temperature dependence in the penetration depth at low temperatures [ 31. The experiments were performed on the M 15 surface muon channel at TRIUMF which provides a nearly 100% spin polarized beam of positive muons. The UPt, sample was a mosaic of pieces 3 mm in diameter and 0.35 mm thick, cut from a single crystal using spark erosion. It was mounted with the b axis along the direction of the muon beam, and along the direction of the main applied field. The polycrystalline UBe,3 sample was approximately 1 cm in diameter and 2 mm thick. Both samples were glued to a pure silver backing attached to the copper sample holder of an Oxford Instruments model 400 toploading dilution refrigerator.
All of the measurements below T, were made following field cooling, except where noted.

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In HF materials, the Knight shift is large because of the large Pauli-like susceptibility of the heavy electrons. The local field B,, is different from the applied field, with the difference being proportional to the applied field (since &XX). Fourier transforms of high transverse field ( -4 kG) data taken in UPt, (and UBe,3 discussed later) exhibit two signals, one from the sample and a smaller amplitude one from the silver backing. Since silver has a small, temperature-independent Knight shift, it provides an internal reference for the sample Knight shift. We find that the frequency shift (comprised of the Knight shift, Lorentz and demagnetizing shifts and a superconducting diamagnetic shift below T,) of the UPt, signal is proportional to the applied field. Measurements at higher temperatures (TX 250 K), show that the Knight shift, measured with the field along the f-axis is proportional to the bulk susceptibility x, also measured with the field along the E-axis (and is not proportional to the susceptibility in the basal plane). This frequency shift at low temperatures (shown in fig. la), is temperature independent and in particular, there is no change below T,. Just above T, the -0.12% ( 1 -n) ) is extremely small, since the demagnetizing factor n is quite close to 1 in our geometry and xd is small in high fields.
We have measured the transverse field relaxation rate (a( 7') ) going from the normal to the superconducting states as a function of external field for several fields Hl(F, up to 3.9 kG. At 3.9 kG (filled circles in fig. 1 ), there is at most an extremely small increase in the relaxation below T,-0.4 K. After subtracting the temperature-independent background, we find b( 0) < 0.06 us-', corresponding to a penetration depth A (0) 3 11000 A.
Broholm et al. have reported transverse field uSR measurements [ 111 of UPt, in external fields somewhat less than 200 G. They extracted a low temperature penetration depth on the order of 8000 A and discussed its temperature/orientation dependence in terms of d-wave superconductivity.
Our results, measured in similarly low fields, are qualitatively consistent with those previously reported. With increasing field however, we find that the magnitude of the low temperature relaxation decreases (illustrated in the upper portion of fig. 1). The relaxation rate has not become field independent even by 3.9 kG as shown in the inset of fig. 1. A key result of the theory for extracting the penetration depth in the mixed state is that the field inhomogeneity AB should be independent of the field in a wide range of fields between H,, and H,, [ lo]. If the measured inhomogeneity is field dependent it generally implies that the measured relaxation does not accurately reflect the penetration depth. In this case, the value of 11000 8, can only serve as a lower bound on the penetration depth which may in fact be much longer.
There are several possible sources of increased relaxation in low fields. One of these is flux pinning, acting to prevent the formation of a uniform flux lattice. This is plausible, in view of the greatly enhanced relaxation observed in zero field cooled measurements (with H,,,=3.9 kG), which is characteristic of strong flux pinning. Shape-dependent inhomogeneities in the demagnetization factor can also provide broadening of the local field, roughly proportional to the sample magnetization ( -47&f), which decreases with increasing field above the lower critical field HCl#2.
Several experiments (see for example ref. [ 141) have detected an anomaly in UPt, around H= 12 kG (for Bile). It has been suggested that such an anomaly may indicate a possible phase boundary between different superconducting states above and below this field. There is a possibility that this feature may be related to our observed field dependence in the relaxation rate. It is, however, not possible to assess the feasibility or magnitude of this effect, due to a lack of theoretical understanding of the superconducting states in UPt,. Since the boundary to the normal state is at much higher field (HC2( 0) -2 T), in general, we do not expect much change in the intrinsic penetration depth with the field He,,<4 kG. UBe13 also possesses a large Knight shift, giving a frequency shift of -0.27% just above T,. Lorentz and demagnetizing fields contribute'-0.09W to this shift. In 3.5 kG, we see that the frequency shift (shown in fig. 2) decreases slightly below T,. This corresponds to a change in the local field at the muon site of about 1 G. This increase in frequency however occurs with the same absolute value B,, ( T-+0) -I?,,( T-T,) x 1 G over the range 50 G<H,,,< 3.5 kG, independent of H,,, (see inset of fig. 2). After removing the fieldindependent frequency shift, we see that the Knight shift itself is essentially the same in the superconducting and normal states.
We note that the positive change in the precession frequency even in fields as small as 50 or 100 G, means that the local field at the muon site is actually significantly larger than in the normal state at these low fields. The source of this small additional field is not yet understood, although it is clearly correlated with the superconductivity of this system (since it appears at T,). Previous high field uSR measurements of Heffner et al. Results of high transverse field (4 kG) relaxation rate measurements above and below the superconducting transition temperature T,= 0.9 K are shown in fig. 3. The absence of any increase below T, indicates that the penetration depth is greater than 0 10000 A m UBe,,. and magnetically ordered ground states. We have performed both zero and transverse field uSR measurements in UBe,, in the temperature range 20 mK< T-c 10 K. The absence of any change in the relaxation rates with temperature implies that the local field from magnetic order below 10 K at the muon site must be smaller than 0.5 G. Except for the unlikely possibility that the muon occupies a site m UBe,3 where the local field is zero by symmetry, this result shows that there is no magnetic order within our sensitivity.

Temperature
(K) Fig. 3. Muon spin relaxation rate in UBe,,, measured in a transverse field of 3.5 kG (circles), and zero tieid (squares). The absence of an increase in the transverse field relaxation below T, indicates that the penetration depth is greater than 10000 A. The temperature independence of the zero field relaxation means that the local field at the muon site due to possible magnetic order must be less than 0.5 G.
quency shift than we report. We do not fully understand the origin of the difference; the different geometry (giving n -0.15 versus n -0.8 in our measurements) might contribute to the difference. We note that zero field measurements (see fig. 3) show no change above and below T,, indicating the extra Both UPt, and UBe,, have muon Knight shifts which are unchanged from the normal to superconducting states. These results are in agreement with '95Pt, 9Be NMR [6,5] measurements of powdered samples as well as induced moment form factor measurements [ 41 using single crystals. Our new results for UBe13 have removed the inconsistency between uSR and these other techniques for the spin susceptibility. The observation of a temperature independent Knight shift is consistent with odd-parity (most likely p-wave) pairing. If the Knight shift was purely of orbital origin, we would not expect any effect below T,. However we expect that there should be a large Pauli spin susceptibility in these materials in view of the large effective masses. In addition, the ratios of the susceptibility to the specific heat coefficient x/y are close to those of simple metals, as would be expected for a Fermi liquid, arguing against a purely orbital susceptibility.
More theoretical and experimental work will be required to reconcile this result with the prevailing picture of d-wave (even parity) pairing in UPtJ.
Although it is possible for spin-orbit scattering to reduce the change in the Knight shift we would argue that this is not the case for these measurements. The mean free path in UPt, is greater than 1000 A [ 2 1 ] which puts this material clearly in the clean limit. The use of powder samples in NMR measurements has led to suggestions of surface scattering; since uSR experiments do not require rf sample penetration, bulk samples are used, which should avoid significant surface scattering.
In both compounds we find that the penetration depths are in excess of 10000 A. Flux confinement measurements [22] of the penetration depth have given a value of 11000 f 2000 8, and 19000 f 2000 8, for the low temperature value J(O) for UBelJ and UPt,, respectively, which are consistent with the limits we can set with uSR. The penetration depth is given in terms of the carrier density and the carrier effective mass ( I2 = m*c2/41cn,e2). Effective masses are large in these systems (for example the cyclotron effective mass m, = 25 +90m, in UPts [ 2 1 ] ) and so we expect that the penetration depths should be long in heavy fermion systems. Indeed, we have obtained shorter values for 1 in systems with lighter effective masses such as URu2Si2 (8600 A) [ 201 and U,Fe (3200 A) [ 201. Since the low field relaxation in UPt, most likely does not directly reflect the penetration depth, we cannot discuss its temperature/orientation dependence as evidence for any particular pairing symmetry.