Superconductivity without inversion symmetry

Finally, it is worth mentioning a different type of unconventional superconductivity. If we refer back to the beginning of this section, we stated that we could write the pair wavefunction as the product of a spin and spatial part. This is not always true, and when it is not, the distinction between singlet and triplet disappears.

To understand how this occurs, we must first consider what various symmetries mean for the lattice and the Fermi surface. We shall consider both inversion and time reversal, as the con- sequences have some similarities, and the combinations of the two are interesting. The following rules are all that are required to understand the changes to the Fermi surface in both cases.

1. The inversion operator_I changesk to₋k

2. Inversion does not change the spin 3. Time reversal_T also changesk to₋k

4. Time reversal does flip the spin

Thus if we have both symmetries, as in a metal with an inversion center at zero field, then the combination IT takes a state to the same k as it started with but flips the spin. If these two states are symmetric, they have the same energy and thus the Fermi surface has a twofold spin degeneracy at all points.

Inversion No Inversion Time Rev ersal No Time Rev ersal I T I T

Figure 1.5: Possible distortions of the spherical Fermi surface with different symmetry constraints. With both inversion and time reversal, degeneracy is enforced. With one but not the other, there will be an surface at₋k if there is one atk, with the symmetry choosing the spin. With neither symmetry, a large variety of shapes are possible, the one drawn is what one might expect if the one above were distorted by a small magnetic field. Note that although the non-inversion surfaces are drawn split into clockwise and anticlockwise sheets, in and out and other more complicated options are also possible. Indeed in and out are perhaps more likely, given that one cannot comb a hairy ball flat.

If time reversal is lost, but inversion is kept, then this degeneracy can be broken. Inversion still requires that there be an equal-energy state at ₋k, with the same spin, but there is no requirement that the opposite spins share the same energy. This is the situation in a metal with a magnetic field applied. A similar state can be obtained by keeping time reversal but breaking inversion. There is still required to be a state at₋kwith the same energy, but now it is required to have the opposite spin. The degeneracy can still be broken, but now the two Fermi surface sheets cannot be classified as spin-up and spin-down, as both spins are present.

Finally, if both symmetries are broken, such as in BiPd with field applied, the Fermi surface is free to take on much more unusual forms. Each of these four cases is sketched for a simple spherical Fermi surface in figure1.5. To begin with, consider the case of a non-centrosymmetric material in the absence of field. This has time reversal but not inversion, so corresponds to the top right panel of the figure. Time reversal relates the spins at k with those at ₋k, as drawn, but does not specify how they change around the Fermi surface. Still, if the spin at one side is required to be opposite to the spin at the other, the spin must change around the surface. This connection of spin and momentum means one cannot write equation1.38as the product of separate spin and orbital parts. Instead, one must use a wavefunction which has spin dependent onkaround the Fermi surface.

At this stage it is worth noting that the loss of inversion symmetry allows, but does not require, a distortion of the Fermi surface. For the Fermi surface to split into two bands with spin varying around the surface one requires an interaction which involves both spin and momentum. The obvious choice is spin-orbit. If the energy scale of spin orbit interactions is larger than the energy scale of the gap, then the two bands are well split, and the current discussion applies. If on the other hand the spin-orbit splitting is small, then one may have inter-band pairing and an ordinary form of superconducting order. A similar argument applies to loss of time reversal — small applied fields break time reversal, but only when they become large enough does one see a significant splitting of the Fermi surface.

This is exemplified in Li2(Pd1−xPtx)3B. Palladium and Platinum are in the same group, the

Pt ion is slightly larger but this only changes the lattice parameter by less than 0.1%. The salient difference is the strength of the spin-orbit splitting. With Pd, the antisymetric band splitting is 30 mV, with Pt it is close to 200 mV. NMR measurements show a complete gap (i.e. s-wave order) in Li2Pd3B, whereas the clear gap is not present in Li2Pt3B, and evidence of line nodes is

seen. Penetration depth measurements also show BCS behaviour in the first case, but non-BCS behaviour in the second. Intermediate dopings show intermediate effects. A good review of this material (and many others) can be found in [21].

When either of both of inversion and time reversal are broken, one degenerate band becomes two distinct bands. Only rarely will either of these move completely above or completely below the Fermi surface, so the material will in general show multi-band physics once the degeneracy

is lifted, even if it did not before. This has various consequences for the superconductivity. The nature of multiband superconductivity varies somewhat depending on the relative interband and intraband interactions. In the absence of interband interactions, superconductivity naturally develops separately on the two bands. The two bands will be different as they are no longer degenerate. In some cases they will show very similar properties to each other, in other cases they may be more different, or superconductivity may not appear at all on one band. As interband interactions appear or strengthen, the properties of the two bands will tend to converge, though two distinct gaps will remain.[22]

Chapter 2

Introduction to URhGe

URhGe was first grown in the 1990s[23,24], as part of a survey of a number of related uranium containing compounds. It crystallises in the orthorhombic structure shown in figure 2.1 and is ferromagnetic with moments partially localised at the U ions and polarised along thec-axis[25,26]. It is unusual in that superconductivity coexists with ferromagnetism at low fields, and in that superconductivity re-enters with fields applied along theb-axis. This re-entrant superconductivity is very robust against magnetic fields, and this complex interplay between different orders is what motivates us to study it. The presence of uranium ions also adds to the interest, providing both the possibility of strong spin-orbit effects and the 5-f valence electrons which show rich heavy fermion physics.

2.1

Ferromagnetism

In the absence of applied field, magnetization measurements[26, 25] indicate a low temperature moment of 0.42µBper formula unit (orµ0M = 0.09 T, ignoring domains), aligned parallel to the

c-axis. The transition temperature is 9.5 K, a value which seems to be independent of sample purity, showing no discernible variation between the Czochralski grown single crystals with single digit RRRs and the quenched melts with RRRs of over 100. Neutron measurements [25] agree on the transition temperature and point to the magnetic moment being located at the uranium sites. They do give a slightly smaller magnetization, but probably still within experimental error. Magnetization measurements at low fields[27] close to and above the Curie temperature indicate the magnetism is slightly anisotropic in thebc-plane, but that thea-axis is much harder. This is borne out by low temperature measurements at higher fields, with fields along theb-axis ultimately causing the moment to align with field, but no such rotation for fields along theaaxis. As the material is metallic at low temperature (i.e. is conductive and has a resistance which usually rises with temperature), the question of localised versus itinerant electron behaviour arises. In a localised ferromagnet, the magnetization arises from electron spins fixed at individual

Figure 2.1: The crystal structure of URhGe, yellow is uranium, green is rhodium, red is ger- manium. The space group is Pnma, and the lattice parameters are a = 682 pm, b = 431 pm andc= 748 pm. Some refs also draw lines connecting the nearby uranium ions along thea-axis, forming zig-zag chains, to indicate the nearest neighbours, however the distance along b is not much longer.

lattice sites, and coupled by some form of interaction. They can be well described by a Heisenberg model, and this model shows such features as a Curie-Weiss susceptibility aboveTCurie, and of course moments localised at some point in the unit cell. It does not explain how the moment can be non-integer multiples ofµB. Nor does it explain any particularly strong interactions with the conductivity — the interaction is limited to the appearance of additional scatterers, as the conduction happens in an unrelated electron population. In an itinerant system on the other hand, the electrons contributing to the magnetism are the conduction electrons. This too is fairly well understood, and can be explained simply by the Stoner model. This is a mean field theory, where alignment of spins creates a magnetic field, which in turn aligns the spins. Provided the energy cost of moving the electrons from the spin-down band to the spin-up band is less than the energy gained by aligning the spins with field, then the lowest energy state is polarised. But one cannot easily recover the Curie-Weiss law from this theory, nor the observation that the moment is centred on the lattice sites.

In reality, the classification of itinerant or localised is a false dichotomy. Most real ferro- magnetic metals show some aspects of both behaviours. One can see how this develops from a localised picture by considering the consequences of involving the electrons of the conduction band in the interactions between lattice sites, such as in RKKY. This necessarily means mixing localised moment with the conduction electrons states. The conduction states will then take on some of the moment’s character. From the itinerant end of the spectrum, one might imagine that the exchange interaction would be strongest in the states which have a large part of the atomic f orbitals mixed in, which will be the ones with a large real space density near the uranium ions. Various theories have been developed to try to unify these two pictures, though a completely

than a critical value (determined by the values of the other coefficients,ax,ay,bx, andby). For

URhGe the various coefficients can be deter- mined from the initial differential susceptibility parallel to thebaxis and Arrott plots of the magnetization for fields parallel to thecaxis and for fieldsH9HRparallel to thebaxis. The

condition for a first order transition for fields close to thebaxis is found to be satisfied, and the computed phase diagram based on the above expression for the free energy is qualitatively compatible with that shown in Fig. 2B.

The theory of superconductivity mediated by the exchange of spin fluctuations is most often considered close to a ferromagnetic- paramagnetic quantum critical point where the longitudinal differential susceptibility diverges at low energy and wave vectors. Only this re- gion of energy–wave vector space then has to be considered (11,12). Under these conditions, a large value of the uniform differential suscep- tibility parallel to the magnetization favors the formation of Cooper pairs with equal spins, whereas a large value of the differential sus- ceptibility perpendicular to the magnetization breaks such pairs. The situation is modified well inside the ferromagnetic state, because the transverse excitations no longer have the same form; they are collective spin waves rather than incoherent overdamped modes. In an isotropic ferromagnet, they can lead to an enhancement of the longitudinal susceptibility due to mode coupling that outweighs their pair-breaking effect (13). For URhGe this same mechanism could be active in a modified form. An im- portant aspect not considered in previous theory is the anisotropy of the spin fluctuation spec- trum for different directions of the wave-vector transfer,qY. For example, a magnetic-field en- ergy is incurred whenqYis not perpendicular to the change in magnetization associated with an excitation (14). For spin rotation excitations in thebcplane, this energy would be absent for wave-vector transfers along theaaxis, and excitations propagating in this direction would consequently have a lower energy than along other directions ofqY. This could favor a polar superconducting order parameter oriented along theaaxis. It is noteworthy that such a state can explain the critical field of the low field super- conductivity (4). Theoretically, the symmetry of such a state would also be consistent with the crystal structure and ferromagnetism with the moments aligned along thebaxis (15).

Over recent years, the application of a magnetic field at very low temperature has been established to be an effective tuning param- eter to drive a number of materials to a quantum critical point or QCEPEexamples are YbRh2Si2

(16), Sr3Ru2O7(17), and URu2Si2(18)^. In the

limit of zero temperature, the divergence of the differential magnetic susceptibility at this point implies a diverging amplitude for zero-point motion (quantum fluctuations) that can desta-

bilize the system relative to other forms of order (19). The behavior in URhGe can be compared with that of the almost-2D material Sr3Ru2O7,

where a new as-yet incompletely identified ground state appears in high quality samples enveloping a QCEP at 7.8 T. For Sr3Ru2O7it

has been argued that superconductivity is not viable because of the large field at which the QCEP occurs (17). For opposite-spin pairing both paramagnetic limitation and orbital limita- tion restrict the maximum field up to which superconductivity can survive. For equal-spin pairing only the second limit applies. This requires that the superconducting coherence length,x0, is small enough to satisfy the re-

lationf0/(2px02)9B(f0is the flux quantum

andBthe magnetic induction); a valuex0G50

)would be compatible with the high field superconducting phase of URhGe.

It appears that the high field supercon- ductivity in URhGe, like the superconductivity at low field in UGe2, is not directly driven by

fluctuations associated with a quantum critical point or QCEP separating ferromagnetism from paramagnetism. In both materials super- conductivity is instead associated with a mag- netic transition between two strongly polarized states, although the transitions differ; in URhGe there is a large change in the transverse moment at the transition, whereas in UGe2only the lon-

Fig. 2.The low tem- perature resistivity and magnetic phase dia- gram for fields in the crystallographicbc plane. (A) The mea- sured resistance for fields in thebcplane. The resistance at 40 mK is represented by the color (top scale). The black areas are regions where the sample has zero resistance and is superconducting. Con- tour lines depict the resistance at 500 mK (bottom scale). The ar- ea where supercon- ductivity occurs at low temperature is seen to correspond to the re- gion over which the resistance is peaked at higher temperature. (B)

A representation of the magnetic phase diagram at low temperature. The thin lines are contours of constant angle,f, of the magnetic moment from thebaxis. The thick line denotes a first order transition across whichfchanges discontinuously. The first order line ends at a QCEP. Beyond this point a sharp crossover behavior still occurs in the field dependence of the moment orientation. The definition offis illustrated in the sketch at the right, with arrows depicting the direction of the magnetization,M, and of the components of the applied field,HbandHc.

2 1 0 µ0 Hc (Tesla) R (arb. units) A

}

_{40 mK}

}

_{500 mK} 9.0 8.5 8.0 7.5 4 2 0 2 1 µ0 Hc (Tesla) 20 15 10 5 0 µ0Hb (Tesla) B 1o 10o 20o 30o 40o 50o 60o 70o = 80o Hc Hb φ M φ

Fig. 3. The field- temperature phase di- agram for applied fields parallel to thebaxis. The color represents the resistivity. Superconduc- tivity occurs throughout the black region where the resistivity is zero. The maximum transi- tion temperature corre- sponds to the field,HR, at which the resistivity has a sharp maximum at higher temperature. The blue solid lines show the position at which the resistance is half its normal-state value (for the data at low field, this was de- termined more precise-

ly in separate measurements). (Inset) The resistance as a function of field at several temperatures corresponding to horizontal cuts through the main figure.

0.8 0.6 0.4 0.2 Temperature (K) 16 12 8 4 0

Applied Magnetic Field, µ₀H_b (Tesla)

10 5 0 R (arb. units) 8 0 R (arb.units) 16 0 µ0Hb(Tesla) 40 mK 200 mK 350 mK 500 mK RE P O R T S

www.sciencemag.org SCIENCE VOL 309 26 AUGUST 2005 1345

LETTERS

–1 0 1 2 0 5 10 0 5 10 15 –2 H_b (T) H_c (T) T (K) Continuous transition line

TCP _QCP First-order plane

Figure 1The schematic field–temperature phase diagram of URhGe for magnetic fields in thebcplane.The magnetization changes discontinuously on crossing the first-order transition surface shaded green that ends at finite temperature along a continuous-transition line. The continuous-transition line bifurcates at a TCP and falls to zero temperature at QCPs shown as blue points. The low-temperature resistivity (of the sample studied) at 50 mK for fields in thebc

plane is shown by the colour shading on the lower face of the axis frame: yellow corresponding to a resistivity of approximately 5_μcm and red to

superconductivity (zero resistivity).

Figure 2 shows our measurements with magnetic fields applied at different angles, γ, from the b axis towards the a-axis direction. The persistence of the sharp peak in the resistivity

In document Electrical transport properties of URhGe and BiPd at very low temperature (Page 41-50)