Non-unitary triplet pairing in the centrosymmetric superconductor LaNiGa{}_{2}

Non-unitary triplet pairing in the centrosymmetric superconductor LaNiGa

Abstract

Muon spin rotation and relaxation experiments on the centrosymmetric intermetallic superconductor LaNiGa are reported. The appearance of spontaneous magnetic fields coincides with the onset of superconductivity, implying that the superconducting state breaks time reversal symmetry, similarly to non-centrosymmetric LaNiC. Only four triplet states are compatible with this observation, all of which are non-unitary triplets. This suggests that LaNiGa is the centrosymmetric analogue of LaNiC. We argue that these materials are representatives of a new family of paramagnetic non-unitary superconductors.

Time-reversal symmetry, non-unitary superconductivity, triplet pairing, muon spin relaxation
pacs:
74.20.Rp, 74.70.Dd

Symmetry breaking is a central concept of physics for which superconductivity provides one of the best understood paradigms. In a conventional superconductor Bardeen et al. (1957) gauge symmetry is broken, while unconventional superfluids and superconductors break other symmetries as well Sigrist and Ueda (1991). Examples include He Lee (1997), cuprate high-temperature superconductors Annett (1990), the ruthenate SrRuO Mackenzie and Maeno (2003) and more recently, non-centrosymmetric LaNiC Hillier et al. (2009). The latter has weak spin-orbit coupling (SOC) Quintanilla et al. (2010), low symmetry and is a non-unitary superconductor. In a non-unitary superconductor the pairing states of the spin-up and spin-down Fermi surfaces are different. At the instability, a spin-up superfluid can coexist with spin-down Fermi liquid. While non-unitary triplet superconductivity is well-established in ferromagnetic superconductors de Visser (2010), its occurrence in parmagnetic LaNiC remains puzzling. Here we provide experimental evidence of this phenomenon in another, compositionally related, but centrosymmetric superconductor: LaNiGa. We also advance an explanation in terms of a coupling between triplet instabilities and paramagnetism that is quite generic and for which these two could provide the first examples of what might be a larger class of materials.

In general unconventional pairing can be difficult to establish in any given material. However evidence for time-reversal symmetry (TRS) breaking in particular can be shown through the detection of spontaneous but very small internal fields Sigrist and Ueda (1991). Muon spin relaxation/rotation (SR) is especially sensitive for detecting small changes in internal fields and can easily measure fields of 0.1 G which corresponds to 0.01 . This makes SR an extremely powerful technique for measuring the effects of TRS breaking in exotic superconductors. Direct observation of TRS breaking states is extremely rare and spontaneous fields have been observed in this way only in a few systems: PrOsSb Aoki et al. (2003), SrRuO Luke et al. (1998) (where TRS breaking was subsequently confirmed by optical measurements Xia et al. (2006)), B-phase of UPt Luke et al. (1993) (although not without controversy de Reotier et al. (1995); Higemoto et al. (2000)), (U,Th)Be Heffner et al. (1990) and more recently LaNiC Hillier et al. (2009), PrPtGe Maisuradze et al. (2010) and (PrLa)(OsRu)Sb Shu et al. (2011). For examples of other systems where the effect is not observed see Refs. Adroja et al. (2005); Anand et al. (2011); Tran et al. (2010); Hillier et al. (2011). Broken TRS in superconductors is especially interesting, because it implies not just unconventional pairing, but the existence of two-fold or higher degeneracy of the superconducting order parameter space Mineev and Samokhin (1999).

The observation of broken TRS in LaNiC was particularly surprising because of the low symmetry of this orthorhombic, non-centrosymmetric, materialHillier et al. (2009). Symmetry analysis has shown that the low dimensionality of this structure, with C point group, gives rise to only 12 possible gap functions. Of these only 4 break TRS and these are all non-unitary triplet pairing states Hillier et al. (2009). These four gap functions are all derived from one dimensional irreducible representations of the point group, implying that the only possible order parameter degeneracy is derived from the triplet Cooper pair spin orientational degree of freedom. A subsequent analysis of the effects of SOC on this system Quintanilla et al. (2010) shows that the SOC always lifts this final degeneracy, leading to a completely non-degenerate order parameter space, which would not be expected to allow spontaneous breaking of TRS at the superconducting transition temperature, Mineev and Samokhin (1999). The only way to reconcile this with the experimental observations of broken TRS in this material is to assume that the effect of SOC is weak on the relevant electron states at the Fermi level in this material. Additional experimental evidence for unconventional pairing in LaNiC has been reported recently Bonalde et al. (2011).

In this letter we report SR results on the centrosymmetric superconductor LaNiGa showing that TRS is broken on entering the superconducting state. This is a centrosymmetric material, which crystallises in the NdNiGa orthorhombic structure, with space group Cmmm (D) Zeng and Lee (2002) (see Fig. 1). Magnetisation and heat capacity measurements have previously shown that LaNiGa is a paramagnetic superconductor, with a T onset of 2.1 KZeng and Lee (2002). Heat capacity measurements have shown a specific heat jump 1.311, which is slightly lower than the expected BCS value of 1.43, and the temperature dependence has the conventional BCS exponential form. Specific heat jumps in TRS-breaking superconductors are sometimes lower and sometimes higher than the BCS value, for example SrRuO Nishizaki et al. (2000), LaNiCPecharsky et al. (1998) and PrPtGeGumeniuk et al. (2008), respectively. Below we analyse the possible order parameter symmetries. Although LaNiGa has a different point group to LaNiC and is centrosymmetric, it is similar to LaNiC in that it has only 12 possible superconducting symmetries Annett (1990). For both materials these are all derived from one-dimensional irreducible representations. In particular, only four of these states break TRS and these are all non-untiary triplet pairing states (see Table 1a).

Figure 1: (color online) The orthorhombic crystal structure of LaNiGa. The red spheres (largest) are La, blue spheres (smallest) are Ni and the black spheres (medium) are Ga.

The sample was prepared by melting together stoichiometric amounts of the constituent elements in a water-cooled argon arc furnace. The SR experiments were carried out using the MuSR spectrometer in longitudinal and transverse geometries. At the ISIS facility, a pulse of muons is produced every 20 ms and has a FWHM of 70 ns. These muons are implanted into the sample and decay with a half-life of 2.2 s into a positron which is emitted preferentially in the direction of the muon spin axis and two neutrinos. These positrons are detected and time stamped in the detectors which are positioned either before, F, or after, B, the sample for longitudinal (relaxation) experiments. The forward and backward detectors are each segmented into 32 detectors. Using these counts the asymmetry in the positron emission can be determined and, therefore, the muon polarisation is measured as a function of time. For the transverse field experiments, the magnetic field was applied perpendicular to the initial muon spin direction and momentum. For a more detailed description of the different instrumental geometry can be found in Ref. Lee et al. (1999); Yaouanc and de Reotier (2011); Schenck (1985).

The sample was powder mounted onto a 99.995+  pure silver plate. Any muons stopped in silver give a time independent background for longitudinal (relaxation) experiments. The sample holder and sample were mounted onto a TBT dilution refrigerator with a temperature range of 0.045-4 K. The stray fields at the sample position are cancelled to within 1 T by a flux-gate magnetometer and an active compensation system controlling the three pairs of correction coils. The transverse field SR (TF-SR) experiment was conducted with applied fields between 5 mT and 60 mT, which ensured the sample was in the mixed state. Each field was either applied above the superconducting transition and the sample was then cooled to base temperature (FC) or the sample was first cooled to base temperature and then the field was applied (ZFC). The sample was cooled to base temperature in zero field and the SR spectra were collected upon warming the sample while still in zero field.

The MuSR spectrometer comprises 64 detectors. In software, the data is mapped to two orthogonal virtual detectors each characterised by a phase offset . The resulting 2 spectra were simultaneously fitted with a sinusoidal oscillating function with Gaussian relaxation:

(1)

where the ith index denotes the sample and background contributions, respectively, is the initial asymmetry, is the Gaussian relaxation rate and is the muon spin precessional frequency. The background term comes from those muons which were implanted into the silver sample holder and therefore this oscillating term has no depolarisation, i.e. =0.0 , as silver has a negligible nuclear moment. Fig. 2 shows a typical spectrum for LaNiGa with an applied field of 40 mT at 50 mK after being FC.

Figure 2: (Color online) The upper graph is a typical muon asymmetry spectra in LaNiGa taken in a transverse field of 40 mT at 0.05 K (shown in the rotating reference frame (RRF) of 6.0 MHz. The line is a fit to the data using Eqn. 1. For clarity, only one of the two virtual detectors have been shown. The lower graph is the zero field SR spectra for LaNiGa. The blue symbols are the data collected at 56 mK and the red symbols are the data collected at 3.0 K. The lines are a least squares fit to the data.

As the muon spin rotation arising from the field distributions associated with the flux line lattice is independent of that arising from the nuclear moments we can write . is assumed to be constant in this temperature region, and is determined from measurements just above . Each data point was collected after field cooling the sample from above . The field dependence of (in s) is related to the superconducting penetration depth (in nm) and coherence length via the relation

(2)

where is the ratio of applied field to upper critical field. From this we have determined and B and hence to be 350(10) nm, 410(3) mT and 28(3) nm respectively (see Fig. 3). This shows that LaNiGa is a type II superconductor, with a superconducting electron density and effective superelectron mass of and , respectively. More details on these calculations can be found in Ref Hillier and Cywinski (1997).

Figure 3: (Color online) The field dependence of at 50mK, after being field cooled. The line is a fit to the data using Eqn. 2.

Now let us consider the longitudinal SR data. The absence of a precessional signal in the SR spectra at all temperatures confirms that there are no spontaneous coherent internal magnetic fields associated with long range magnetic order in LaNiGa at any temperature. In the absence of atomic moments muon spin relaxation is expected to arise entirely from the local fields associated with the nuclear moments. These nuclear spins are static, on the time scale of the muon precession, and are randomly orientated. The depolarisation function, , can be described by the Kubo-Toyabe function Hayano et al. (1979)

(3)

where is the local field distribution width and MHz T is the muon gyromagnetic ratio. The spectra that we observed for LaNiGa are well described by the function

(4)

where is the initial asymmetry, A is the background, and is the electronic relaxation rate (see Fig. 2). It is assumed that the exponential factor involving arises from electronic moments which afford an entirely independent muon spin relaxation channel in real time. The only parameter that shows any temperature dependence is , which increases rapidly with decreasing temperature below T (see Fig. 4). We interpret this increase in as a signature of a coherent internal field with a very low frequency as discussed by Aoki et al. Aoki et al. (2003) for PrOsSb. This increase in has been modelled, assuming that there are uncorrelated, by , where and are the nuclear and electronic contributions respectively. The temperature dependence of agrees with the BCS order parameter (see Fig. 4).

Figure 4: (color online) The left graph shows the temperature dependence of , for LaNiGa in zero-field, which clearly shows the spontaneous fields appearing at T=2.1 K (shown has the vertical line). The line is fit to the data using an approximationCarrington and Manzano (2003) to the BCS order parameter for . The right graph shows the temperature dependence of the electronic relaxation rate, , for LaNiGa in zero-field, which shows no temperature dependence.

Let us now discuss the implications of this result for the pairing symmetry. The group theoretical analysis for the D point group of this system has already been investigated Annett (1990). For the simplest case, where translational symmetry is not broken and SOC does not play a role, this point group has a total of irreducible representations. This leads to possible order parameters, as given in Table 1a. Of these , are unitary and are non-unitary. Only the non-unitary order parameters have a non-trivially complex order parameter that can break TRS. In the case where SOC is large there are only possible states (see Table 1b) and none of them break TRS. Therefore, like LaNiC, LaNiGa must be a non-unitary triplet superconductor with weak SOC. As we have predicted for LaNiCQuintanilla et al. (2010) if the SOC is not zero then a split transition would be expected.

(a) gap function (unitary) gap function (non-unitary) - - - -

(b) gap function with strong SOC

Table 1: The upper table (a) shows the gap function of the homogeneous superconducting states allowed by symmetry, for weak spin-orbit coupling. We have used the standard notation for singlet states and for triplets, where is the vector of Pauli matrices. Here , and are any basis functions in the Brillouin zone of odd parity, such as , and , respectively. Only the four non-unitary triplet states are compatible with our observation of broken TRS. The lower table (b) shows the homogeneous spin-triplet superconducting states allowed by symmetry at , for the case of strong spin-orbit coupling. The spin singlet gap functions are the same as in the weak spin-orbit case. The notation of the possible gap functions is the same as the upper table, except that here , and are (real) constants determined by the microscopic gap equation. Clearly none of these states breaks TRS at .

Until the discovery of non-unitary triplet pairing in LaNiC this state had only been confirmed in ferromagnetic superconductors de Visser (2010). The additional observation of non-unitary triplet pairing in LaNiGa brings up the question of how a triplet superconductor whose normal state is paramagnetic could favour this state. The usual Landau free energy describing a triplet pairing instability in our system is of the form

(5)

where is the order parameter, which relates to the vector through [the possible functional forms of are given in Table 1a]. The triplet instability takes place when , which determines and is independent of whether pairing is unitary or non-unitary. Below , the second of the quartic terms decides which of the two states is most stable. The criterion for non-unitary triplet pairing is Annett (1990)

(6)

On the other hand for a paramagnet there must be an additional term coupling to the magnetization . On symmetry grounds the simplest form of the free energy that takes this into account is

(7)

Here is the normal state susceptibility.

For given the last term on the right hand side of (7) describes an effective magnetic field coupled to . This field vanishes for unitary triplet pairing, but in the non-unitary case it induces a magnetization

(8)

Below , whence . This subdominant order parameter lowers the energy of the non-unitary state compared to the unitary one. Indeed substituting (8) into (7) we recover the simpler expression (5) but with the coefficient replaced with . The condition (6) then becomes

(9)

For a paramagnet the second term on the left hand side is always negative, favouring non-unitary triplet pairing states. This effect would be expected to be strongest in proximity to a Stoner instability. We note that in superconducting ferromagnets de Visser (2010) the same coupling term exists and stabilizes non-unitary triplet pairing states by increasing their relative to unitary states.

In conclusion, zero field and transverse field SR experiments have been carried out on LaNiGa. The zero field measurements show a spontaneous field appearing at the superconducting transition temperature. This provides convincing evidence that time reversal symmetry is broken in the superconducting state of LaNiGa. Symmetry analysis implies non-unitary triplet pairing, in close analogy with the non-centrosymmetric superconductor LaNiC. We propose that these materials could represent a new class of superconductors where a triplet superconducting instability of a paramagnetic state gives rise to non-unitary pairing through a generic coupling to the magnetization.

Acknowledgements.
This works was supported by EPSRC and STFC (U.K.). J.Q. gratefully acknowledges funding from HEFCE and STFC through the South-East Physics network (SEPnet).

Footnotes

  1. The authors of Ref Zeng and Lee (2002) have determined a higher value of by using a 10 % - 90% definition of

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