Field-induced collective spin-exciton condensation in a quasi-2D dx2-y2-wave heavy electron superconductor
FField-induced collective spin-exciton condensation ina quasi-2D d x − y -wave heavy electron superconductor Vincent P. Michal
Commissariat `a l’Energie Atomique, INAC/SPSMS, 38054 Grenoble, FranceE-mail: [email protected]
Abstract.
The origin of the spin resonance observed in CeCoIn with Inelastic NeutronScattering is subject to debate. It has been shown recently that in this heavy electroncompound at low temperature an instability to a ground state with coexisting d x − y -wavesuperconductivity and Spin Density Wave (SDW) order in a magnetic field is a corollary ofthe consideration of a collective spin excitation mode in a quasi-2D d x − y -wave Pauli-limitedsuperconductor. This provides a natural scenario for the occurence of the puzzling high-field-low-temperature phase highlighted in CeCoIn . We present perspectives on this ground statetransition and propose directions for future experiment.
1. Background information and model considerations in the spin-fermion approach
Modelling and understanding collective phenomena in correlated electron systems in generaland new superconductors in particular is an interesting and challenging issue. The two-side observation of spin resonances in high-temperature and heavy-fermion superconductorshas raised the question of whether the phenomena in these very different systems have acommon mechanism. Interpreting Inelastic Neutron Scattering data [1] for CeCoIn has recentlygiven rise to interesting debate [2]. We have proposed [3] that the ground state transition inmagnetic field observed [4] in superconducting CeCoIn (superconducting critical temperature T c (cid:39) . explains naturally the confinement of magnetic ordering withinthe superconductor.Historically measurement in CeCoIn phase diagram of a line of discontinuity in the specificheat indicated a second-order transition to a high-field-low-temperature phase (10T < H < . T < . q = (1 / − δ, / − δ, /
2) (in units of the tetragonal crystal Brillouin zone dimensions)with δ (cid:39) .
05 whose appearence coincided with this very transition line gave evidence that theground state had a magnetic character with magnetic coherence length ∼ A [4] extandingwell beyond the vortex cores and ordered magnetic moments of 15% the Bohr magneton [4].Nuclear Magnetic resonance (NMR) experiment [5] supported characterise the magnetic phase.When not superconducting CeCoIn is a semi-metal with non-Fermi liquid resistivityexponent and a Kondo temperature T K ∼
7K while the Kondo lattice coherence temperature Here the term condensation is used in the sense of a transition of a collective excitation mode to static ordering. a r X i v : . [ c ond - m a t . s up r- c on ] J un igure 1. Left: Behaviour of the uniaxial dynamic spin susceptibility (see text for discussion).The effect of the magnetic field is to split the threshold energy Ω into a two-step discontinuity.Right: Cartoon of the predicted resonant mode dispersion and threshold energy at zero field[10], evolution with transverse field in the uniaxial case and criticality at incommensurate wave-vector. T coh ∼
50K [6] yielding low temperature hybridized bands with large effective masses and a largespecific heat discontinuity at the transition to the superconductor. In magnetic field the metalto superconductor transition is observed to be first order at temperature T (cid:46) α M ∼ H orbc20 /H pc20 ( H orbc20 [ H pc20 ] is the orbital [paramagnetic] uppercritical field at zero temperature) which itself follows from a large electron effective mass. Weconsider a constant interaction magnetic instability from the superconducting, non-magneticside. The model starts with a quasi-2D ( α -band in CeCoIn [7]) itinerant heavy electron bandclose to a commensurate SDW ground-state instability [8] with antiferromagnetic hot-spots(points in wave-vector space that belong to the Fermi line and are connected by the diagonalantiferromagnetic ordering wave-vector) which below T c becomes a superconductor with d x − y -wave symmetry gap [9] and include a magnetic field B . In the spin-fermion approach [2, 10] spindensity excitations and fermion excitations couple through a constant g and yield the dynamicspin susceptibility χ − ( q , Ω) = χ − (cid:104) ξ − + | q − Q | − Π xx ( q , Ω) (cid:105) , (1)where ξ is the magnetic correlation length.To account for the magnetic anisotropy of the physicalsystem, spin-density excitations are considered along a single axis (c-axis of the tetragonalcrystal) whose direction is set prependicular to the magnetic field. The dimensionless spinexcitation self-energy due to fermions (sometimes termed spin polarization function) Π( q , Ω)thus writes Π xx ( q , Ω) = − g χ T (cid:88) k ,m,σ (cid:104) G σ ( k , iω m ) G − σ ( k + q , iω m + Ω + i + )+ F σ ( k , iω m ) F + − σ ( k + q , iω m + Ω + i + ) (cid:105) , (2)where G , F and F + are Green’s functions in a superconductor with Zeeman field [3] characterizedby the spectrum of fermion excitations E skσ = s [ (cid:15) k + ∆ k ] / + σB ( s, σ = ± (cid:15) k is the zero-field spectrum in the metal [3], and the d x − y -wave gap in general reads ∆ k = (∆ / (cid:80) n a n [cos( nk a ) − cos( nk b )]). The properties of the Eq. 2 includes magnetic field effect in the Pauli limit. The influence of the electron orbital motions on thespin susceptibility can be evaluated [11] in the linearized Doppler shift approximation and gives correction of theorder B/ ( H pc20 α M ) which is small in a Pauli-limited superconductor as expected. igure 2. Left: Collective mode energy variation with field at incommensurate wavevector q = (0 . , . B ∗ = 0 . [3].2D function Π at one loop level is known for zero field [2] and is considered here in Zeemanmagnetic field (c. f. Fig. 1). Here we consider that the dynamics of the mode is determined bythe superconducting gap such that Eq. 1 yields the condition ξ − + | q − Q | − (cid:60) e Π( q , Ω) = 0(Ω >
0) defining the d x − y -wave superconductor collective spin-excitation mode referred to asspin-exciton. The spectral function (the imaginary part of the dynamic spin susceptibility) ofthe mode looks like a Dirac delta in absence of Landau damping by particle-hole pair creation,which translates into the condition (cid:61) m Π( q , Ω) / ( γ ∆) (cid:28) γ = N g χ / (2 π | v khs × v khs+q | ), N the number of hot-spots and v khs the Fermi velocity at hot-spot.
2. From collective spin-excitation mode in a quasi-2D d x − y -wave superconductorto ground state instability in a transverse magnetic field The fluctuation-dissipation theorem relates the dynamic structure factor measured in InelasticNeutron Scattering experiment [1, 12] with the spectral function of the model (Eq. 1) S ( q , Ω) = 2 (cid:61) m χ ( q , Ω + i + ) / (1 − e − Ω /T ), the latter being the analytic continuation of theimaginary time spin correlation function χ ( q , i Ω n ) = (cid:82) /T dτ (cid:82) d r e − i q · r + i Ω n τ (cid:104) S x ( r , τ ) S x (0 , (cid:105) .The collective spin-excitation mode we discuss is a property of a quasi-2D d x − y -wavesuperconductor on the border of a commensurate SDW continuous ground state transition(an s-wave gap yields a negative Π with no divergence and the logarithmic divergence (Fig.1) associated with the particle-hole excitation threshold in the d-wave case is due to two-dimensionality [2]). This d-wave effect is the one brought about by coherence factors in theBCS formalism.The mode dispersion is schematically represented in the right-hand side of Fig. 1 at zerofield and at the condensation field. The mode spectral weight is maximum at commensuratewave-vector because there the threshold energy is also maximum. Away from this point thedownward dispersion [10] follows from the wave-vector dependence of the d-wave gap in theBrillouin zone. A transverse magnetic field splits the mode into a branch which goes down inenergy and an upper part which becomes damped by the continuum of particle-hole excitations.The consideration of magnetic isotropy gives a splitting between three undamped modes [13].Ongoing Inelastic Neutron Scattering experiment [12] is providing information on the actualmagnetic anisotropy of the system and is showing deviation from strict uniaxiality.The left hand side of Fig. 2 shows the evolution of the resonance energy with field at fixedincommensurate wave-vector q = (0 . , .
45) [3] where we have taken account of the variation ofhe gap magnitude with temperature and magnetic field by considering the superconductor gapequation (we found the zero temperature first order critical field H pc = 0 . correspondingto the band structure considered in [3]). The right-hand side of Fig. 2 shows Fermi pocketsinduced by magnetic field approaching the antiferromagnetic hot-spot locations. Before thepockets reach these points the ground state transition occurs together with reconstruction ofthe Fermi surface. This must be visible with Nuclear Magnetic Resonance.
3. Discussion, perspectives and conclusion
We now turn to some perspectives on this instability of the superconducting ground state. Firstthere is a possibility of a double-q structure which follows from the mode dispersion (Fig. 1)and is consistent with the prediction of Y. Kato et al. [14], there seen as a consequence of theincommensurate ordering wave-vector connecting nested pockets. As was emphasized in [14],this degeneracy is however expected to be lifted by small coupling between the electron orbitalmotion and the magnetic field. We suspect magnetostriction effects also play a part. It hasbeen also conjectured [15] the existence of a phase where SDW lives in a Larkin-Ovchinnikov(LO) superconducting state (where the order parameter spatially varies along the magnetic fielddirection). Because the two orders couple, this should give rise to additional length scale for spacedependence of the SDW order parameter with experimental hallmark as Bragg peaks measurablewith elastic neutron scattering. Such a prediction signalling coexistence between LO and SDWis waiting for experimental verification (this is also true for inhomogeneous superconductivitywith a different spin state such as Pair Density Wave state).We saw that the specificities of the heavy electron superconductor CeCoIn brings allconditions (first order transition [4] resulting from Pauli limiting, tendency towards two-dimensionality [7], pairing symmetry [9], border of antiferromagnetism [8]) for realizing a groundstate instability which is the natural result of the condensation under magnetic field of apreexisting spin collective excitation mode in the d x − y -wave superconductor. As noted in[2], collective phenomena in heavy electron compounds involve a physics that stems from alattice of localized f-electrons coupling with conduction electrons. Understanding the nature ofthe heavy delocalized electrons and their relation with superconductivity remains a significantchallenge.
4. Acknowledgments
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