Magnetic Field Penetration Depth in the Heavy-Electron Superconductor UBe13

%e report the observation of a T temperature dependence of the magnetic field penetration depth in UBel3 at low temperatures. %'e show that this behavior is consistent with an anisotropic gap function for an axial p-wave state. Our results further show that the Landau parameter F'l appears to be small.

The heavy-electron superconductor UBe~3 shows anomalous, i.e. , non-BCS, behavior in several proper- ties like the electronic specific heat, ultrasonic at- tenuation, 2 and spin-lattice relaxation.s These proper- ties are directly related to the spectrum of quasiparticle states that can be excited from the superconducting ground state, and it has been argued that they are more consistent with the spectrum expected for a spin-triplet, or odd-parity, paired state.Among experiments yielding more direct information about such an unconventional superconducting ground state are those which investigate the supercurrent properties, such as Josephson effects and magnetic field penetra- tion.The former are particularly sensitive to the sym- metry properties of the ground-state wave function.Partly as a result of material preparation difficulties, however, they have not yet yielded clear-cut answers in the case of UBei3.This Letter describes principal results on the latter experiments in which we have ob- served and calculated the magnetic field penetration depth in UBei3 as a function of temperature.The de- tails of this work will be published elsewhere. 5  By inspecting the basic London equation j'= -(c/4m)X 'A, z'= (m "c'/4me')m/p', (1) which relates the supercurrent density j' to the mag- netic vector potential A and defines a field penetration depth X, one sees that by measuring X(T) one can get information about the following quantities and proper- ties of the superconductor: (i) the value of the effec- tive mass m', (ii) the temperature dependence of the superfluid mass density p', and (iii) (as will be shown below) the possible importance of Fermi-liquid effects   and the value of the Landau parameter F'i.In what follows, we show that our measurements yield information on each of these points.
We obtained internally consistent data on four sam- ples of UBei3, and present here the results for one of them.Its characteristics were as follows: density p = 4.3 glcm, T, = 0.86 K, 10'l0-900/0 transition width = 0.06 K, Meissner effect (in H = 0.3 Oe) =3.5olo.The experiment consisted of observation of the reversible changes in magnetization with tempera- ture of the superconducting sample which was located inside one coil of a sensitive SQUID bridge circuit.
These reversible changes, which must be due to the field penetration effect only (for a more detailed dis- cussion of this point see Ref. 5), were nearly indepen- dent of whether the measuring field (always less than 0.3 Oe) was switched on either above or below T, .
From them the changes in the penetration depth A. (T) -ii.(T;"),with T;, the lowest measuring tem- perature, can be calculated in a straightforward manner.In Fig. 1 we show these observed changes, plotted versus ( T/T, )2 for both the UBe» sample as well as for a reference sample of Sn.
The most striking feature of Fig. 1 is that the tem- perature variation of A. (T) for UBe, 3 is quite unlike that of Sn, a well-known BCS superconductor.In fact, UBe, 3 shows a T" variation of X(T) -X(0), with a temperature exponent K = 2 (see below), inconsistent with the exponential behavior -(l3o/kT) 'i' x exp(b, o/'kT) expected for an isotropic London su- perconductor.
The tin data are, over a significant range of temperature, or nonmagnetic impurities to a p-wave superconductor results in a T behavior of the magnetic penetration depth as well as in a linear low-temperature specific heat and a reduced specific-hcat jump in the so-called gapless regime or dirty limit. 8Gapless superconduc- tivity as an explanation of the observed T law in the penetration depth may be ruled out, however, by the observation of a T' temperature dependence of the specific heat and a large specific-heat discontinuity.
We have therefore reconsidered the theory of the electromagnetic response in a p-wave superconductor with uniaxially anisotropic gap of the form 0 0.~0.2 0.3 (T/Tc) FIG. 1. Incremental magnetic field penetration depth X(T) -X(T;") of UBei3 as a function of (T/T, )' (left scale).The size of the circles shows the experimental uncer- tainty.Also shown, for comparison, are the data for the Sn reference sample (right scale).
ly isotropic energy gap.Fitting the data by this function, we find for tin A. (0) =460 A, in agreement with earlier work.
Before further discussing the UBet3 data, we outline now a calculation of the temperature dependence of the penetration depth.There are two possible physical origins for a power-law behavior of the penetration depth and hence the superfluid density: First, if the gap function of the superconductor under consideration has zeros somewhere on the Fermi surface, for exam- ple point or line nodes, then the number of thermal excitations vanishes at low temperatures according to a power law (kT/Ao)" with same temperature exponent A well-known example for a pair-correlated system with a gap with point nodes is the p-wave superfluid 3He-A with a spin-fluctuation exchange mechanism replacing the usual electron-phonon pairing mechanism.
Second, it has been shawn that the addition of magnet- l a(k T) =S (T)y(i i) (2) with 1 (unit vector) the axis of gap symmetry, 50 the temperature-dependent maximum of the gap, and f some function of k 1 with nodes which will be speci- fied later.
For a superconductor described by the general class of gap functions (2), restricted to a half space z ~0, and exposed to an external field h'"' in the x-y plane, one sees that the London equation (1) has to be gen- eralized in that the superfluid density p' and the penetration depth X have to be replaced by appropriate tensor quantities.
The superfluid density tensor is characterized by the two eigenvalues p'ii and p', according to the principal axes parallel and perpendicular to the vector 1 of gap symmetry.The shielding current j' can then be shown to be purely transverse, if one properly accounts for the order-parameter collective modes within the framework of hydrodynamic theory (for details see Ref. 5 and Millis ).In particular, if we assume the gay orbital degrees of freedom (here represented by 1) to be strongly pinned, only the contribution from the phase gradient of the order parame- ter is important.
For arbitrary directions of 1, the current is not parallel to the vector potential.This is reflected in a penetration depth tensor which has eigenvalues Xt and A. 2 with respect to the directions parallel and perpendicular to the projection L of the vector 1 into the x-y plane; they are related to p'ii and p', as follows: z2t = z'(0) p/p', , )t22= Z'(0) (p/p fi ) [1 -1, '+ (ply/p', ) 1, '], (4) A where 1, is the z component of the vector 1 and A. (0) is the penetration depth at zero temperature.The local mag- netic field h(z) inside the superconductor is finally obtained in terms of the external field h(0) =H'"', the penetration depths Xt and X2, and the directions L and ix L as h(z) = L(L H'"')exp( -z/A.t) +zx L(ix L H'"')cxp( -z/X2).
Lo temperature resuits.--The explicit form of pfi & becomes particularly simple at very low temperatures if one spcctaltzes Eq. ( 2) to thc axial state with f(k 1) = ikxli (two point nodes) and to the polar state with f(k 1) =k 1 (equatorial line of nodes) for which one obtains in the absence of Fermi-liquid effects 20000~' where in the axial state K = 2 (4) and a = 7r (7m /15), and in the polar state K=3 (1) and a =27m((3)/4 (3mln2/2), for the orientations ll (z ).
Gap orientation effects.-It is clear that the predicted penetration depth for a given anisotropic state will depend strongly on the direction of the vector 1, which may be oriented by crystal electric fields, magnetic fields, superflow, and surfaces. 5'o One will therefore generally expect the observed penetration depth to be a mixture of the two eigenvalues of the tensor A. .At very low temperatures, however, the main contribu- tion will originate from the eigenvalue with the lowest temperature exponent ~; i.e. , for the axial state we ex- pect a T2 law from A, 2 (if 1 is not exactly perpendicular to A) and for the polar state the dominating contribu- tion comes from X, , which is linear in T.
Fermi liquid eff-ects can be easily incorporated in our theory.Their importance is reflected in the explicit occurrence (c.f.Ref. 11) of the interaction parameter Fi in the expression for the superfluid density: where pfio i are the superfluid densities in the absence of the Fermi-liquid interaction.At zero temperature O0.2 one is therefore left with a penetration depth renor- (T/T ) malized by a factor [(m'/m )/(1+ Fi /3) ], which is FIG. 2. Comparison of the measured incremental unity if translational invariance may be assumed.It is penetration depth a( T) -a( T;") (circles), plotted vs not entirely clear, however, to what extent the (T/T, )', with theory (full lines, axial state; dashed lines,

Galilean-invariance arguments leading
to the efpolar state; dashed-dotted line, isotropic state).In (a) we fe«ive-mass relation m'/m = I+F't/3 should be even kept &(0)=4200 A fixed and varied Fi as indicated, approximate y va i in a system wit oca ize J e ec- whereas (b) was evaluated for two values of X(0) in the b- 12 h d i tn 1 t sence of Fermi-liquid effects (Fl = 0).The maximum gap trons.Some authors have made use of this relation ~as taken from Eq. i8j. to discuss the origin of the heavy-electron mass.On the other hand, it has been argued'3 that the effective mass shouM scale with Fo rather than with I'&.
The influence of nonmagnetic impurities on the penetration depth of a p-wave superconductor will be discussed in detail in Ref. 5.Here we only note that the low-temperature power laws expected for an axial p-wave superconduc- tor, in contrast to those for the polar state, remain unaltered in the presence of nonmagnetic impurities of not too large concentration.
line, the (local limit) result expected for an isotropic superconductor without Fermi-liquid effects.In Fig.
2(b) the penetration depth for the axial (full lines) and the polar (dashed lines) state was evaluated in the ab- sence of Fermi-liquid corrections (Ff =0) for two values of X(0) as indicated.
It is evident from this figure that the observed T' behavior of the London penetration depth is incon- sistent with the predictions of BCS theory for a pure, isotropic singlet superconductor.
Nor can the temperature dependence be explained by assumption of a gap with a line of nodes (polar state) as the curvature of the dashed lines shows.A gap function with point nodes (axial state), however, allows for a fairly good fit to the experimental data for a certain variety of combinations of the parameters Ft ~20 and X(0) ~8000 A. The parameter combinations A. (0) It should be emphasized that the penetration depth observed in a real experiment need not correspond directly to one of the eigenvalues of the superfluid density tensor which have been used for the fit.Nevertheless, at low temperatures we expect a T or a T2 behavior to dominate depending on whether the state possesses line or point nodes, respectively.The axial state thus represents a possible fit to the experimental data, provided that the 1 vector is not fixed exactly perpendicular to the sample surface over a dis- tance large compared to A. .This will always be the case if the magnetic and/or crystal field orienting effects are sufficiently strong.
The experimental data do not appear to be con- sistent with a large value of the Landau parameter Ff and therefore with approximate validity of the Galilean-invariance effective-mass relation (assuming that the crystal mass m is small).This conclusion fol- lows from the observed pure power-law behavior in X( T) -A.(0) over a large temperature range.In addi- tion, a fit to the data at the very lowest temperatures with Ff large is inconsistent with values of X(0) ex- pected on theoretical5 and experimental3 grounds.If confirmed by absolute penetration-depth measurements, this would lend support to those theories of the heavy-fermion normal state'3 which predict a small Ff and scaling of F$ with m'/m based on an unrenormal- ized compressibility.
%e thank R.
Doll for providing our tin sample.%e