Effects of Lifshitz transitions in ferromagnetic superconductors: the case of URhGe

Effects of Lifshitz transitions in ferromagnetic superconductors: the case of URhGe

Abstract

In ferromagnetic superconductors, like URhGe, superconductivity co-exists with magnetism near zero field, but then re-appears again in a finite field range, where the system also displays mass enhancement in the normal state. We present theoretical understanding of this non-monotonic behavior. We explore the multi-band nature of URhGe and associate re-entrant superconductivity and mass enhancement with the finite field Lifshitz transition in one of the bands. We found good agreement between our theory and a number of experimental results for URhGe, such as weakly first order reentrant transition, the dependence of superconducting T on a magnetic field, and the field dependence of the effective mass, the specific heat and the resistivity in the normal state. Our theory can be applied to other ferromagnetic multi-band superconductors.

Ferromagnetic superconductors are exciting systems to study the interplay of magnetism and superconductivity, contrary to the common wisdom that the presence of a ferromagnetic order destroys superconductivity. The coexistence of superconductivity and ferromagnetism has been realised experimentally for uranium-based heavy-fermion compounds, like UGe Saxena et al. (2000), UCoGe Huy et al. (2007) and URhGe Aoki et al. (2001). The materials exhibit a wealth of exotic properties, including, e.g., the appearance of non-Landau damping in magnetic excitations Chubukov, Betouras and Efremov (2014).

Among these systems, URhGe has attracted much attention both experimentally Lévy et al. (2005); Hardy and Huxley (2005); Miyake et al. (2008); Yelland et al. (2011); Hardy et al. (2011); Aoki et al. (2011, 2014); Aoki and Flouquet (2014); Fujimori et al. (2014); Tokunaga et al. (2015); Gourgout et al. (2016); Wilhelm et al. (2017); Braithwaite et al. (2018) and theoretically Hattori and Tsunetsugu (2013); Mineev (2015, 2017). In zero applied magnetic field, it displays ferromagnetism with magnetic moment oriented along the -axis, and spin-triplet superconductivity at a lower temperature Hardy and Huxley (2005). In an external magnetic field along the -axis (), superconductivity disappears at about =2T. This is believed to be caused by the orbital effect of the field Hardy and Huxley (2005). However, at higher magnetic fields, in the range from 8T to 13.5T, it reappears again Lévy et al. (2005) (see Fig.1).

Ferromagnetic spin fluctuations are believed to provide the pairing glue for superconductivity in a ferromagnetic metal Mineev (2015, 2017). Indeed, NMR spin-spin relaxation measurements indicate that uniform longitudinal spin fluctuations (the ones in the direction of a magnetic field) are strongly enhanced in the field range where superconductivity has been observed Tokunaga et al. (2015). Measurements of the specific heat Hardy et al. (2011); Aoki et al. (2011), electric conductivity Miyake et al. (2008); Hardy et al. (2011); Tokunaga et al. (2015); Gourgout et al. (2016), and magnetisation Hardy et al. (2011) indicate that the increase of spin fluctuations is accompanied by the increase of the effective mass of fermions. This is indicative of a critical behavior near a ferromagnetic instability.

In this communication we address the origin of the ferromagnetic instability in a finite field. We argue that it is due to a Lifshitz transition observed  Yelland et al. (2011) in one of the bands which form the electronic structure of URhGe. This Lifshitz transition pushes the system closer to the magnetic instability and enhances the magnetic fluctuations. This in turn leads to re-entrance of superconductivity (RSC) at a finite field.

The minimal model of the electronic structure of URhGe has two bands with non-equal dispersions and band minima shifted by . (see Fig. 1). In a ferromagnetic state at zero field spin-up and spin-down states in both bands are split by an effective exchange field. Both branches of band 1 cross the chemical potential , while both branches of band 2 are above (see the left inset in Fig. 1). At a finite the bands experience additional Zeeman splitting. The exchange field was reported Lévy et al. (2005) to be rather weak (), hence at fields near 10T Zeeman splitting dominates. The dispersions of the two bands with Zeeman splitting are and . As the field increases, the splitting grows, and at some critical field the system experiences a Lifshitz transition, in which spin-up branch of band 2 crosses the chemical potential (the middle inset in Fig. 1). We add Hubbard four-fermion repulsive interaction and analyze the tendency towards magnetic order and magnetically-mediate superconductivity in this field range. The parameters relevant to URhGe are presented in par ()

Figure 1: The experimental and theoretical diagrams of URhGe in an external magnetic field . Triangles are the experimental data  Lévy et al. (2005), solid line is the theoretical result. Superconductivity is present near , absent at intermediate fields, and re-appears at higher fields, with a maximum at around T. In the insets, we show the fermionic dispersion in our model of two electron bands, separated by in the momentum space and exhibiting Zeeman splitting. The Lifshitz transition occurs at , when the spin-up branch of band 2 touches the chemical potential. At higher , this band opens up a new Fermi surface The maximum of is at a field .
Figure 2: (a) Longitudinal (blue/dark grey) and transverse (green/light grey) spin susceptibilities as functions of external magnetic field . (b) The theoretical results for the effective mass of fermions in sub-band , (normalized to its maximum value at ) (solid line) along with extracted from the measurements of magnetisation (triangles). Hardy et al. (2011).

We first compute the longitudinal and transverse susceptibilities at . Within RPA we have Brinkman and Engelsberg (1968):

(1)
(2)

where and are particle-hole susceptibilities of free fermions from the two bands:

(3)

where and .

The results are shown in Fig. 2 a. We see that the uniform longitudinal susceptibility is enhanced in the vicinity of the field , which is somewhat larger than , at which the Lifshitz transition occurs (). The non-monotonic behavior of can be understood by noticing that the denominator in (1) behaves as

(4)

where is the density of states at the Fermi surface of a sub-band with spin projection . At , becomes non-zero, decreases and increases. At higher fields decreases (see right insert in Fig. 1) and decreases. At even higher magnetic field the spin-down sub-band of band 1 undergoes the second Lifshitz transition and becomes unoccupied, in agreement with Yelland et al. (2011). The transverse has much weaker dependence on and does not show a peak around . This agrees with the NMR results Tokunaga et al. (2015).

Figure 3: Feynman diagrams describing the first two orders of the RPA series for (a) longitudinal component of the self-energy, ; (b) transverse component of the self-energy, ; and (c) p-wave pairing vertex.

The effective mass of a conduction electron is given by

(5)

The electron self-energy can be written as a sum of longitudinal and transverse components. The corresponding ring and ladder diagrams Brinkman and Engelsberg (1968); Fay and Appel (1980) are shown in Fig. 3 a and b, respectively:

(6)
(7)

where the effective interactions are

(8)
(9)

Because the susceptibility is enhanced at , the largest contribution to the self-energy comes from intra-band scattering (i.e., the scatterng within band 1 or band 2), while inter-band interactions contribute much less (see Supplemental Material (SM)). Performing frequency integration and following Brinkman and Engelsberg (1968); Fay and Appel (1980) we obtain

(10)

where

(11)
(12)

Here is the Fermi momentum of the sub-band , and the integration limits for the transverse component are and .

The result of the calculation of for the sub-band is shown in Fig. 2 b. As expected, the mass enhancement is peaked at , where the uniform susceptibility is the largest. The effective masses for other sub-bands show similar enhancement (see SM). The theoretical result agrees well with the mass ratio extracted from magnetisation measurements Hardy et al. (2011) (see Fig. 2 b). Using the result for , we computed the specific heat and resistivity. The main contribution to the specific heat comes from the sub-band, and the Sommerfeld coefficient can be estimated as Jacko et al. (2009) . In Fig. 4a we show the calculated and the experimental one from Ref. Hardy et al. (2011). Clearly, both are peaked around and show similar behavior at smaller and higher fields. In Fig. 4 b we show theoretical and experimental results for the prefactor in the expression for the resistivity . Theoretical has been obtained using the Kadowaki - Woods relation Kadowaki and Woods (1986) , the experimental results are from Ref.  Gourgout et al. (2016). Again, the agreement is quite good.

Figure 4: Enhancement of (a) Sommerfeld coefficient measured in Hardy et al. (2011) (triangles) and (b) conductivity coefficient measured in Ref Gourgout et al. (2016) (squares) and calculated with our model (solid line) for the same parameters as in Fig. 1.

We next turn to superconductivity. The reduction and subsequent disappearance of superconductivity at small fields has been argued to be the orbital effect of a field Scharnberg and Klemm (1980); Hardy and Huxley (2005). The reduction of by a vector potential follows Scharnberg and Klemm (1980)

(13)

This form of agrees with the data at small fields Hardy and Huxley (2005). When , vanishes.

To study the re-entrant superconductivity, we do the analysis in two steps. First, we compute within Eliashberg spin-fluctuation formalism, without including the orbital effect of a field. We then use the result for as an effective to estimate from (13). We then use (13) to obtain the actual .

An exchange by ferromagnetic spin fluctuations enhances the pairing vertex in p-wave channel, and below we search for superconducting order with p-wave symmetry. We analyze all fields and keep both Zeeman and exchange splitting. To keep calculations under control, we neglect the feedback from ferromagnetic order on the pairing interaction. This feedback is relevant near a ferromagnetic quantum-critical point Chubukov et al. (2003), but less relevant away from criticality, where our analysis holds.

In Eliashberg theory one needs to solve the set of equations for quasiparticle and the pairing vertex, :

(14)
(15)

where , , , and the interactions are and . We use a standard trick and reduce Eliashberg set to a single equation by introducing . Then is expanded in spherical harmonics and only the p-wave piece, is retained. Integrating over momenta as , where is the solid angle, we obtain an integral equation of in the form

(16)

where

(17)

We introduced and , where is the first spherical harmonic and is the angle between and .

Keeping only the interaction with small momentum transfer, we factorize the pairing between three bands: up and down sub-bands of band 1 and spin-up sub-band of band 2. We recall, however, that effective p-wave pairing interaction between fermions on a given band is the sum of contributions from particle-hole bubbles from all three bands. We solve Eq. (17) for all three bands and find the largest (see SM for details). The result is shown as a solid line in Fig. 5a. At smaller fields , where is slightly above , superconductivity develops on the sub-band. At at it switches to sub-band , and for superconductivity on this band has a maximum at , where the effective mass on this band is also maximal.

Figure 5: (a) Solid line – , obtained within Eliashberg formalism without including the orbital effect of the field. Dashed line – the actual , with both Zeeman/exchange and orbital effects. The actual is always smaller than due to the orbital effect of the external and the exchange fields. (b) The effective field from Eq. 13 as a function of . Orbital effect destroys superconductivity when (blue line) is smaller than (thin black line). The value of changes discontinuously at , when superconductivity switches from sub-band to sub-band , where the effective mass is larger. This gives rise to a jump in the actual in panel (a).

We next include the orbital effect. In Fig. 5b we show from Eq. (13) as a function of external . Orbital effect destroys superconductivity when . We see that this holds at intermediate fields, in the range where without orbital effect superconductivity would develop at sub-band . At higher fields, superconductivity switches to sub-band , where the effective mass and are larger, and is smaller. Each modification increases , which becomes larger than , in which case orbital effects do not destroy superconductivity. We show actual by dashed line in Fig. 5 b and by solid line in Fig. 1. Note that appears discontinuously at , where switches from to sub-band. Inter-band pairing interactions likely smoothen the first-order phase transition. The theoretical profile of vs agrees nicely with the data Lévy et al. (2005) (see Fig. 1).

To summarize, in this communication we argued that the enhancement of the effective mass in URhGe at fields near 10 T and the emergence of RSC around this field are due to Lifshitz transition. We considered the model for URhGe with two electronic bands and analyzed the behavior of the system near a field when the bottom of the spin-up branch of previously unoccupied band 2 sinks below the Fermi level. We first computed transverse and longitudinal spin susceptibilities and argued that the longitudinal susceptibility dramatically enhances in some field range above the Lifshitz transition, while the transverse susceptibility remains flat. This fully agrees with the behavior of longitudinal and transverse susceptibilities, extracted from NMR measurements of the relaxation times, and Tokunaga et al. (2015). We next computed the one-loop self-energy due to magnetically-mediated interaction and obtained the enhancement of the effective mass. The theoretical result for agrees with the experimental data extracted from magnetisation measurements Hardy et al. (2011); Not (). We found good agreement also for the Sommerfeld coefficient and the prefactor for the term in the resistivity  Hardy et al. (2011); Gourgout et al. (2016). We next analyzed superconductivity. We first solved the Eliashberg equation for magnetically-mediated superconductivity without orbital effect of a field and obtained with a maximum at a field where the effective mass is the largest. Superconductivity resides on sub-band at smaller fields and on sub-band at higher fields. We then added addition pair-breaking orbital effect and found that superconductivity exists at small fields, gets destroyed by orbital effect at intermediate fields, and re-appears discontinuously roughly at a field of Lifshitz transition. This behavior fully agrees with the data Lévy et al. (2005) (Fig. 1). The reduction of theoretical at higher fields is somewhat slower than in the data. The reason could be a re-orientational transition, detected at 12T Lévy et al. (2005), in which the magnetic moment rotates towards the field direction, leaving its magnitude unchanged. This spin re-orientation does not increase longitudinal fluctuations but complicates the field dependence of above 12T. Overall, it looks increasingly likely that topological Fermi-surface transitions can account for much of the puzzling physics in nearly magnetic itinerant systems Slizovskiy and Chubukov and Betouras (2015); Slizovskiy and Betouras and Carr and Quintanilla (2015).

Acknowledgements.
We thank R. Fernandes and M. Greven for useful conversations. This work was supported by the EPSRC (YS and JJB) through the grant EP/P002811/1. and by the U.S. Department of Energy through the University of Minnesota Center for Quantum Materials, under award DE-SC-0016371 (A.V.C.).

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