Anisotropic chiral d+id superconductivity in NaxCoO2 yH2O

0.9 MB·English

by  Maximilian Kiesel

Anisotropic chiral d+id superconductivity in Na CoO ·yH O
x 2 2
Maximilian L. Kiesel1, Christian Platt1, Werner Hanke1, and Ronny Thomale2
1Institute for Theoretical Physics, University of Wu¨rzburg, Am Hubland, D 97074 Wu¨rzburg and
2 Institut de th´eorie des ph´enom`enes physiques, E´cole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne
(Dated: January 25, 2013)
Since its discovery, the superconducting phase in water-intercalated sodium cobaltates
Na CoO · yH O (x ∼ 0.3, y ∼ 1.3) has posed fundamental challenges in terms of experimental
x 2 2
investigationandtheoreticalunderstanding. Byacombineddynamicalmean-fieldandrenormaliza-
tiongroupapproach,wefindananisotropicchirald+idwavestateasaconsequenceofmulti-orbital
effects,Fermisurfacetopology,andmagneticfluctuations. Itnaturallyexplainsthesingletproperty
3 and close-to-nodal gap features of the superconducting phase as indicated by experiments.
1
0 PACSnumbers: 74.20Rp,74.25.Dw,74.70.-b
2

n
a Introduction. Initiated by the discovery of the teroftheorderparameterhasremainedacontentiousis-
J cuprates, the search for new materials exhibiting uncon- sue. EarlyµSR[6]aswellasmagneticpenetrationdepth

3 ventionalsuperconductivityhasbecomeoneofthemajor measurements [7], have shown evidence against a homo-
2 branches in condensed matter physics [1]. A particularly geneous gap and have been interpreted in favor of line

exciting idea is the concept of chiral superconductivity, nodes. By contrast, the latest specific heat studies [8]
]
n wheretheCooperpaircondensatebreaksparityandtime advocate a two gap scenario for the cobaltates with one
o reversalsymmetryandgivesrisetointerestingedgemode comparably small and another slightly larger gap to fit
c phenomena of the bulk-gapped superconductor. As one the data. This reminds us of similar discussions for the
-
r way to accomplish such a scenario, the lattice can act as iron pnictides, where it is likewise complicated to distin-
p
acustodialsymmetrytoensuretheexact,or,inthepres- guish a nodal from a strongly anisotropic gap [9].
u
ence of disorder and nematic fluctuations, approximate
s In this Letter, we develop a microscopic theory for
. degeneracy of different superconducting instabilities. In
t thenatureofsuperconductivityinthesodiumcobaltates
a such a case, the degeneracy is linked to higher dimen-
m which is consistent with the experimental findings. In-
sional irreducible representations of the lattice symme-
spired by the resemblance to the cuprates, the earliest
- try group, and a chiral superposition of superconducting
d theoreticalproposalsemployedaphenomenologicalRVB
states can be energetically favorable below T . Unfor-
n c theory for the cobaltates [10] supplemented by slave bo-
o tunately, for the square lattice and its C4v group, there son mean-field calculations [11]. A major challenge from
c is no such representation for singlet Cooper pairs, which
[ the beginning has been the choice of an adequate low
in the majority of materials is found to be the generic
energy kinetic theory for the problem: While ARPES
1 sector for superconductivity. This, however, changes for
measurements only observe one Fermi pocket centered
v hexagonal systems, where the E representation of the
2 2 around Γ in the hexagonal Brillouin zone [12–14], band
C lattice symmetry group implies the degeneracy of
6 6v structure calculations indicate the presence of additional
the d and d wave state at the instability level [2],
6 x2−y2 xy e(cid:48) pockets [15]. The absence of the e(cid:48) pockets in the ex-
which can yield a chiral d+id singlet superconductor. g g
5
perimental cobaltate scenario has been assigned to sur-
.
1 Inmanyrespects,thewater-intercalatedsodiumcobal- face effects [16], disorder [17], and electronic correla-
0 tatesNa CoO ·yH O, withasuperconductingdomefor tions [18–20]. It suggests that whatever the microscopic
x 2 2
3
x ∼ 0.3, y ∼ 1.3 at T = 4.5K [3], have been inter- theoryforsuperconductivityinthecobaltatesmaybe, it
1 c
preted as the natural generalization of a square lattice shouldinvolvealow-energykinetictheorywhichexplains
:
v of copper oxide in the high-T cuprates to a triangular the experimental evidence from a single-pocket scenario.
c
i
X lattice of cobalt oxide: The electronic structure can be Such an effective model has been developed by a com-
assumedeffectivelytwo-dimensionalduetotheintercala- bined dynamical mean-field and cluster approximation
r
a tion, andsuperconductivityemergesasafunctionofsuf- approach by Bourgeois et al. [21, 22], which is the start-
ficient Na doping in proximity to magnetic phases. The ing point of our investigations. Employing multi-orbital
experimental evidence, however, remained inconclusive functional renormalization group (fRG) [9, 23–26] to ob-
for a significant amount of time, which did not allow to tain an effective interaction profile for this model, we
drawsubstantiatedconclusionsonthenatureoftheorder finda rich phasediagram forthe sodium cobaltates with
parameter [4]. For example, previous ambiguous indica- ananisotropicd+id-phaseintherelevantdopingregime.
tionsfromKnight-shiftmeasurementsforpolycrystalline Thestronganisotropyofthesuperconductinggapcanex-
samples have only later been clarified by single crystal plaintheexperimentalevidence;itfollowsfromtheinter-
measurements [5], which showed the singlet property of play of multi-orbital hybridization, Fermi surface topol-
the superconducting phase. Similarly, the nodal charac- ogy, and frustrated magnetic fluctuations in the sodium
2
cobaltates.
Cobaltate effective kinetic model. Following the work
byBourgeoisetal.[21,22], allCo andO orbitalsare
3d 2p
taken into account in a finite cluster calculation which is
then mapped to a three-orbital model and fitted against
X-rayabsorptionandARPESdata[22]. Theresultingef-
FIG.1. (Coloronline). (a)Effectivebandstructureresulting
fectivemodelisobtainedfromdynamicalmean-fieldthe-
from (1) with t = 0.1eV, t(cid:48) = −0.02eV and D = 0.105eV.
ory calculations, which adequately take into account the
(b)ThevanHovesingularityisvisibleinthedensityofstates
self-energy effects at a single-particle level. It exhibits
(inset: crystal structure of the CoO layers). The Fermi sur-
2
faceisshownatx=0.1(c),x=0.2(d),andx=0.3(e). The strongly hybridized orbitals formed by an effective t2g
Fermi surface colors indicate the dominant orbital weights. manifold (d˜ ,d˜ ,d˜ ) per site on the triangular Co su-
xy yz zx
(d)sketchesthedivisionoftheBrillouinzoneinto96patches perlattice,afindingwhichisalsoconsistentwithARPES
used in fRG, (e) depicts the nesting vectors Q .
N polarization measurements [27]. The Hamiltonian reads
(cid:88) (cid:16) (cid:17) (cid:88) (cid:88)
H = (t+t(cid:48)δ +Dδ )cˆ† cˆ +h.c. +µ nˆ +U nˆ nˆ
eff αβ ij iασ jβσ iασ 1 iα↑ iα↓
(cid:104)i,j(cid:105),αβ,σ i,α,σ i,α
(cid:32) (cid:33)
1 (cid:88) (cid:88) (cid:88)
+ U nˆ nˆ + J cˆ† cˆ† cˆ cˆ +J cˆ† cˆ† cˆ cˆ , (1)
2 2 iασ iβσν H iασ iβν iαν iβσ P iα↑ iα↓ iβ↑ iβ↓
i,α(cid:54)=β σ,ν σ,ν
where nˆ = cˆ† cˆ , and cˆ† denotes the electron ping J . As the bandwidth in this effective model is
iασ iασ iασ iασ P
creation operator of spin σ = ↑,↓ in orbital α = 1,2,3 smaller than in bare LDA calculations, the interactions
at site i. t represents the hopping mediated by O , t(cid:48) strengths from bare ab-initio calculations likewise have
2pπ
corresponds to a direct Co-Co-hopping, D is the crystal- to be regularized. We set U = 0.37eV, U = 0.25eV,
1 2
field splitting, and µ the chemical potential. We set and J =J =0.07eV.
H P
t = 0.1eV, t(cid:48) = −0.02eV, and D = 0.105eV [21, 22].
WenowproceedbyinvestigatingtheFermisurfacein-
The bandwidth of the effective model is ∼ 0.6eV. This
stabilities of (1) through fRG, where we use the effec-
is a factor 3 smaller than LDA calculations predict
tive multi-orbital band structure as the initial starting
(1.6eV [15] or 2.0eV [28]). This effective three-band
point [9, 23–26]. Through renormalization, we obtain an
model resulting from (1) yields one band intersecting
effectivelow-energytheoryofthescatteringvertexwhich
the Fermi level (Fig.1a). A van Hove singularity oc-
exhibitssuperconductivity. Thepairing2-particlevertex
curs at a doping level of x ≈ 0.09 (Fig.1b). The Fermi
VSC(k,q) is then decomposed into eigenmode contribu-
surface contains one hole pocket around Γ, i.e. the
tions which correspond to the different superconducting
center of the Brillouin zone (Fig.1c). All three hy- form factors VSC(k,q) = (cid:80) cSCfi(k)fi(p). cSC signals
bridized orbitals contribute to the Fermi surface and i i i
the strength of the instability and hence allows to iden-
each has two antipodal dominant regions, indicated by
tify the superconducting phase adopted by the system.
red/green/blue dots in Fig.1c-e, corresponding to the
We employ multi-orbital temperature-flow fRG [25] to
doping x = {0.1,0.2,0.3}. At x ≈ 0.28, the nesting
take into account the interplay between ferromagnetic
of the Fermi surface is optimal, with three nesting wave-
√ (cid:113) (cid:113) and antiferromagnetic fluctuations as well as the multi-
vectors QN ≈ π{(cid:0) 2,0(cid:1),(−√12, 32),(−√12,− 32)} orbitalcharacterofthesodiumcobaltates. Note,thatwe
(Fig. 1e). The interaction part of (1) features intra- mustavoidthedoublecountingofself-energyeffectsthat
orbital Coulomb interaction U , inter-orbital Coulomb havealreadybeenincludedtoobtainthisbandstructure.
1
interaction U , Hund’s rule coupling J , and pair hop- As such, we intentionally do not take into account self-
2 H
3
FIG. 2. (Color online). Phase diagram of the model (1)
as function of doping x and U /U . There are four phases:
1 2
d+id-wave superconductivity (d+id SC, blue), weak ferro-
magnetism (weak FM, green), f-wave superconductivity (f
SC, yellow), and a phase with competitive spin-density wave
andd+id-wavesuperconductivity(SDW/d+idSC,purple
and blue shaded).
FIG. 3. (Color online). Superconducting form factors from
fRG mean-field decoupling of pairing channels (solid lines)
energy effects at the single-particle level which emerge
compared to analytic leading harmonic solutions (dashed
during the RG flow. In total, our procedure is still an
lines): (a,b) f-wave, (c,d) d + id-wave. The representa-
approximation, inthesensethatwefirstrenormalizethe tive parameters are (a,b) U /U = 1.4, x = 0.18 and (c,d)
1 2
single-particle level via a DMFT approach and then sep- U /U = 1.0, x = 0.14. (b) and (d) show a color plot of the
1 2
arately investigate the renormalization of the scattering gap size ∆0 along the Fermi surface (Eq. 2).
vertex through fRG which gives rise to Fermi surface in-
stabilities. This, however, is justified because the scat-
tering vertex evolution under RG is only significant in moniccontributionsinthed-wavesector[30]tobeirrele-
the immediate vicinity of the Fermi level. vant.) Thesystemcouldgenericallyformanylinearcom-
bination d +eiθd of both d-wave solutions which must
Phase diagram. From an itinerant viewpoint of Fermi 1 2
be degenerate at the instability level as protected by lat-
surfaceinstabilities,animportantfeatureof (1)isthedif-
ticesymmetry. Amean-fielddecouplingintheSCpairing
ferent doping location of the van Hove singularity (x =
channel and minimization of the free energy as a func-
0.09) and the optimally nested Fermi surface (x=0.28).
tionofthesuperpositionparameter,canberephrasedby
Notethatbothlocationsarecoincidentandmuchlessre-
satisfying the self-consistent gap equation [2]
vealing for a triangular lattice tight-binding model with
only nearest neighbor hopping [29]. Accordingly, for (cid:18) (cid:19)
small doping, the phases are determined by the large ∆ =−1/N(cid:88)VSC(k,q) ∆k tanh E(k) . (2)
q 2E(k) 2T
densityofstatesattheFermilevelcombinedwithrather k
weaknestingcorrespondingtofinitemomentumtransfer,
We always find d+id to be the energetically preferred
whichintotalpromotesdominantzeromomentumparti-
combination. This is rather generic in a situation of de-
cleholescattering(labeledweakferromagnetism(FM)in
generatenodalSCorderparameters, sincesuchacombi-
Fig.2). Withincreaseddoping,itdependsontheratioof
nation allows the system to avoid nodes in the gap func-
U /U whetherthesystemfavorstripletf-waveorsinglet
1 2
tion(Fig.3d)andmaximizescondensationenergy. Note,
d+id-wave superconductivity. For the former, the sys-
however, that the relative energy gain between a d+id
tem exhibits nodes along the Fermi surface and follows
√ stateandadifferentpossiblesolutionsuchasnodalsingle
the gap function f(k) = sin(ky) − 2cos( 32kx)sin(k2y) d varies significantly depending on the microscopic
(Fig. 3a,b). f-wave is preferred for enhanced U /U , x2−y2
1 2 setup. For example, the condensation energy gain from
as it is seeded by spin alignment stemming from ferro-
d+idforlowerdopinginFig.4cascomparedtoasingle
magnetic fluctuations, which are reduced by U . In the
2 d wave solution will be higher than for larger doping in
case of preferred d-wave superconductivity, we find two
Fig. 4f.
degenerate instabilities associated with the form factors
Anisotropic regime at x ∼ 0.3. The effective model
depicted in Fig. 3c, which relate to the leading harmon-
√ √ in (1) is quantitatively most accurate in the doping
icsd (k)=2cos(k )−cos(kx− 3ky)−cos(kx+ 3ky)
x2−y2 √x 2 √ 2 regime of the superconducting dome of the cobaltates.
and d (k) = cos(kx+ 3ky)−cos(kx− 3ky). (Note that There, we find d + id superconductivity with strong
xy 2 2
throughout parameter space, we always find higher har- SDW fluctuation background (Fig. 2). The enhance-
4
FIG. 5. (Color online). Gap anisotropy η = σ(∆0) in the
∆0
d+id-wave phase (axis annotations as in Fig. 2). Blue (red)
regions indicate a rather homogeneous (anisotropic) d-wave
FIG.4. (Coloronline)Changeofgapanisotropyinthed+id
gap. The inset shows the plot of the gap function on the
superconductingphaseasafunctionofdoping. (a-c)x=0.14.
Fermisurfaceatphysicallysensibledoping:∆ exhibitsclose-
0
The dominant density of states is strongly peaked at the
to-nodal (blue) and larger (red) gap regions along the Fermi
warped edges of the Fermi surface and singles out specific
surface.
particle hole scattering channels which result in a homoge-
neousgap∆ . (d-f)x=0.28. Variousscatteringchannelsare
0
enhancedduetoreducedFermisurfacewarpingandmoreho-
mogenousdistributionofdensityofstates,yieldingastrongly scenariofromspecificheatmeasurements[8],whereboth
anisotropic gap. effective gap scales originate from a single pocket.
Conclusion and perspectives. We have shown that the
experimental evidence of a close-to-nodal singlet super-
ment of magnetic fluctuations is also observed in experi- conducting state in the sodium cobaltates can be de-
ment, anticipating the metal insulator transition regime veloped from a microscopic model taking into account
atx∼0.5. FromtheviewpointofFermisurfacetopology, the multi-orbital nature of the electronic scenario. The
thisisduetoimprovednestingconditionsforalargerpart anisotropic d+id superconductor we find is a combined
of Fermi level density of states (Fig. 1e). As the density effect of magnetic fluctuations, specific Fermi surface
is also more homogeneously distributed along the par- topology at the corresponding Na doping, and multi-
allel sides of the hexagonal Fermi surface, however, this orbital effects. To our conviction, this constitutes the
leads to an increased bandwidth of enhanced particle- sodium cobaltates to be one of the most promising can-
hole channels in the RG flow, which eventually yields an didates for a chiral singlet superconductor, to be fur-
enhancedanisotropyintheseededpairingchannelwhich ther studied experimentally and theoretically. The T
c
gives rise to d + id superconductivity. This trend for might allow for laser-ARPES studies in the supercon-
increasingdopingxisillustratedinFig.4. Atlowerdop- ducting phase, along with more careful investigations of
ing, the more warped Fermi surface clearly singles out time-reversal symmetry breaking than it has been pur-
theparticleholechannelaccordingtoscatteringbetween suedbynow. Likewise, theroleoflatticedistortionsand
theFermisurfaceedges,whichalsopossessthedominant disorder can be interesting to consider, as there might
fractionofdensityofstatesattheFermilevel(Fig.4a-c). be a transition from d+id to d when the custodial C
6v
As doping is increased, the reduced warping allows the symmetry is sufficiently broken. Finally, the desire for
effectively one-dimensional parallel sections of the Fermi anidealchiralsingletsuperconductormightalsowarrant
surfacetodrivemoreparticle-holechannels. Theresultis further work to optimise its two-dimensional character,
an evolution of the seeded superconducting phase from as has recently been reported for intercalated iron-based
a homogeneous to an extremely anisotropic d+id gap superconductors [31]
(Fig. 4d-f). RT thanks G. Baskaran, B. A. Bernevig, S. Blundell,
InFig.5,wehaveplottedthedegreeofgapanisotropy A. Boothroyd, S. A. Kivelson, and S. Raghu for discus-
inthed+idphaseforthesamerangeofparametersasfor sions. MK is supported by DFG-FOR 1162. CP, WH,
the phase diagram in Fig. 2. We use the variance of the and RT acknowledge support by SPP-1458.
gap function, divided by the mean: η = σ(∆0). Notably,
∆0
thephysicallyrelevantregimeforthecobaltatescoincides
with the strongest gap anisotropy, we observe. As seen
in the inset of Fig. 5, this yields a gap structure along a
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