A crib-shaped triplet pairing gap function for an orthogonal pair of quasi-one dimensional Fermi surfaces in Sr$_2$RuO$_4$

5 Pages·0.36 MB·English

by  Kazuhiko Kuroki

A crib-shaped triplet pairing gap function for an orthogonal pair of quasi-one
dimensional Fermi surfaces in Sr RuO
2 4
Kazuhiko Kuroki,∗ Masao Ogata, Ryotaro Arita, and Hideo Aoki
Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan
(February 1, 2008)
The competition between spin-triplet and singlet pairings is studied theoretically for the tight-
1 binding α-β bandsin Sr2RuO4, which arise from two sets of quasi-one dimensional Fermi surfaces.
0 UsingmultibandFLEXapproximation,whereweincorporateananisotropyinthespinfluctuations
0 as suggested from experiments, we show that (i) the triplet can dominate over the singlet (which
2
turns out to be extended s), and (ii) the triplet gap function optimized in the Eliashberg equation

n has an unusual, very non-sinusoidal form, whose time-reversal-broken combination exhibits a crib-
a shaped amplitude with dips.
J

7
PACS numbers: 74.20-z, 74.20Mn

]
l
e
Spin-triplet superconductivity is of great conceptual tion self-consistently for the spin-anisotropy mecha-
-
r importance. It is interesting to ask how the spins in a nism mentioned above. We adopt the fluctuation ex-
st Cooper pair can align, since usually the singlet pairing change (FLEX) approximation14, in which we incorpo-
t. is favoredso that some special mechanism should be en- rate anisotropy in the spin fluctuations. The FLEX re-
a
visaged to account for a triplet superconductivity. The sults arethen fedinto the linearizedEliashbergequation
m
p-wave pairing in superfluid 3He is an outstanding ex- to obtain the gap function. The optimized triplet gap
- ample, where a clear picture of the hard-coreinteraction function turns out to have unexpected, non-sinusoidal
d
favoring the triplet exists. forms, which are in sharp contrast with the k or sink
n x x
o Inthepastseveralyears,Sr RuO 1hasattractedmuch gap functions assumed previously. As a result, the am-
2 4
c attention as a strong candidate for triplet superconduc- plitude of the gap has a shape of a crib along the Fermi
[ tivity. In a seminal paper, Rice and Sigrist suggested a surface,whichmayresolvethecontroversialexperiments.

1 mechanismfortripletpairinginthismaterial,whichthey The origin of the peculiar behavior of the gap function
v callan‘electronicversionofHe’.2Intheirscenariotheor- is traced back to the singular k-dependence in the spin
7 bital degeneracy causes ferromagnetic spin fluctuations, susceptibility due to a nesting of two sets of quasi-one
7 which is considered to favor the triplet pairing. Sub- dimensional Fermi surfaces.
0 sequent experiments indeed suggested triplet pairing.3,4 The ruthenate is essentially a three-band system,
1
0 However,anewpuzzlearosewhenthespinfluctuationin which arises from dxz orbitals aligned linearly along the
1 Sr2RuO4 was found to be antiferromagnetic rather than xaxis,dyz alongthey,anddxy inthexy plane. Thefor-
0 ferromagneticinaneutronscatteringexperiment.5 Usual mertwogiverisetotheα,β bands,whicharewellnested
/ wisdomdictates thatantiferromagneticspin fluctuations duetothequasi-onedimensionalityandcausesantiferro-
t
a lead to singlet d-wave pairing.6 magneticspinfluctuations. Inthispaper,weconcentrate
m Recently, Kuwabara and one of the present authors,7 on the α,β bands, namely, we consider a tight-binding
- and independently Sato and Kohmoto,8 have proposed model,
d
n thatanisotropyinthespin-fluctuation,observedinNMR nn
o experiments for Sr2RuO4,9 may lead to triplet p-wave H =−tX X X(cid:16)cmiσ†cmi′σ+H.c.(cid:17)
c pairing. However, simple functional forms for triplet σ m=xz,yzhii′i
: and singlet gap functions were assumed in ref. 7, i.e.,
v nnn
Xi sdixn2−kxy(2+-wiasvine.kyI)n10reffo.r8,p-twhaeveforamndofcothsekxga−pcwohsekny tfhoer −t′Xσ hXi,ji(cid:16)cxiσz†cyjσz +H.c.(cid:17)+UXi m=Xxz,yznmi↑nmi↓. (1)
r quasi-1D Fermi surfaces are hybridized was discussed
a only qualitatively. The form of the gap function is cru- on a square lattice. Here cm† creates an electron at
iσ
cial in discussing the triplet-singlet competition, so the d (m = xz or yz) orbital. the nearest-neighbor hop-
m
functional form should be optimized. The form of the ping integral t is along the x(y) direction for d (d )
xz yz
gap function is also important in comparing with exper- orbitals. We take t=1 as a unit of energy.
imental results because, e.g., some recent experiments We have also included the next nearest-neighbor hop-
have suggested presence of nodes in the superconduct- pingt′whichcorrespondstoaweakhybridization. When
ing gap,11,12 while a thermal conductivity measurement t′ 6= 0 the two sets of quasi-one-dimensional bands anti-
suggests an isotropic gap.13 cross, and two two-dimensional (rounded-square) bands
The purpose of the present paper is to determine result,whicharetheαandβbands. Theon-siterepulsive
the functional form of the superconducting gap func- interaction,U,isconsideredwithineachorbitals,andin-
1
terorbitalinteractionsareneglectedfor simplicity.15 The V(2) = 1Vzz+V+−− 1V (7)
band filling is n=4/3 electrons per orbital in Sr RuO . s 2 sp sp 2 ch
2 4
In treating the interaction, we employ the FLEX ap-
for singlet pairing,
proximation. This method is a kind of self-consistent
random-phase approximation (RPA) where the dressed 1 1
V(2) =− Vzz− V (8)
Green’s function is used in the RPA diagrams. In the t⊥ 2 sp 2 ch
multiband version of FLEX,16,17 the Green’ function G,
the susceptibilityχ,the self-energyΣ,andthesupercon- for triplet pairing with S =±1 (~d ⊥~z), and
z
ducting gap function φ all become 2×2 matrices, e.g.,
Glm(k,iεn), where l,m denote dyz or dxz orbitals. The V(2) = 1Vzz−V+−− 1V (9)
orbital-indexedmatricesforGreen’sfunctionandthegap tk 2 sp sp 2 ch
functions can be converted into band-indexed ones with
a unitary transformation. As for the spin susceptibility, for triplet pairing with Sz = 0 (~d k ~z). Here ~d is the
we diagonalize the 2×2 matrix χzz and concentrate on d-vector characterizing the triplet pairing gap function.
sp
the larger eigenvalue, denoted as χzz. When the electron-electron repulsion, which causes
fluctuations, is short-ranged (as for the Hubbard U in-
The actual calculation proceeds as follows:
teraction) the spin fluctuations are much stronger than
(i)Dyson’sequationissolvedtoobtainthe renormalized
thechargefluctuations,i.e.,(V ≫V ).15 Whenfluctu-
Green’s function G(k), where k is a shorthand for the sp ch
ations for a certain (‘nesting’) wave vector Q are pro-
wave vector k and the Matsubara frequency, iǫ ,
n
(ii)Thefluctuation-exchangeinteractionV(1)(q)isgiven nounced, the main contribution in the summation in
as18 eq.(6) comes from those satisfying k−k′ ≃Q, which
shouldbethecasewhentheFermisurfaceisnested. The
1 1 present Fermi surface is indeed well nested due to the
V(1)(q)= Vzz(q)+V+−(q)+ V (q). (2)
2 sp sp 2 ch quasi-one-dimensionality.
Now we turn to the results summarized in Fig.1 for
The effective interactions due to longitudinal (zz) and t′ = 0.3 and U = 5. In Fig.1(a), the ridges in |G|2 de-
transverse (+−) spin fluctuations (sp) and that due to lineate α and β Fermi surfaces. These quasi-1D surfaces
charge fluctuations (ch) have the forms Vzz = U2χzz, are strongly nested at q≡(2π/3,q ) and q≡(q ,2π/3)
sp sp y x
V+− = U2χ+−, and V = U2χ , respectively, where (mod 2π), so that the spin susceptibility is ridged in a
sp sp ch ch
the spin and the charge susceptibilities are crib shape as shown in Fig.1(b), with peaks at the cor-
ners, q ≡ (±2π/3,±2π/3). This is consistent with neu-
χzz(q)=χirr(q)[1−Uχirr(q)]−1, (3) tron scattering experiments.5
sp
χ+−(q)=αχirr(q)[1−Uαχirr(q)]−1, (4) Thetripletandsingletgapfunctionsobtainedbysolv-
sp ingtheEliashbergequationareshowninFig.1(c)and(d),
χch(q)=χirr(q)[1+Uχirr(q)]−1, (5) respectively. Remarkably,thetripletgapfunctiontakesa
strangeshape: althoughthesymmetryisconsistentwith
in terms of the irreducible susceptibility χirr(q) = p-wave, its form is far from sink along the rounded-
x
−1 G(k + q)G(k) (N:number of k-point meshes). square Fermi surface. Rather, it has an almost constant
N Pk
Herewehavetakenaccountofthe anisotropyinthespin amplitude on a pair of parallel sides, k ≡ ±2π/3 (mod
x
fluctuation by introducing a phenomelogical parameter 2π), of the square with a vanishing amplitude on the
α `a la Kuwabara-Ogata.7 From the NMR experiments,9 other pair (k ≡ ±2π/3) of parallel sides. This applies
y
we assume α<1. to each of the α and β bands. Of course the symmetry
(iii) V(1) then brings about the self-energy, Σ(k) = dictates that the other solution (p ), rotated by 90 de-
y
1 G(k−q)V(1)(q),whichisfedbacktoDyson’sequa- greesfromwhatisdescribedhere(p ),entersonanequal
N Pq x
tion, and the self-consistent iterations are repeated until footing as we shall discuss below.
convergenceis attained. We take 64×64 k-pointmeshes Why do we have this peculiar behavior for the
andupto16384Matsubarafrequenciesinordertoensure FLEX+Eliashberg optimized gap function ? To begin
convergence at low temperatures. with,superconductivityarisesduetopairscatteringfrom
We determine Tc as the temperature at which the (k,−k) to (k′,−k′) mediated by the pairing interaction
eigenvalue λ of the E´liashberg equation, V(2)(q), where q=k−k′ is the momentum transfer.
From the BCS gap equation we can see that supercon-
T ductivity arises if the quantity
λ φ (k)=−
µ µlm N X X
k′ l′,m′ V(2)(k−k′)φ (k)φ (k′)
×Vµ(l2m)(k−k′)Gll′(k′)Gmm′(−k′)φµl′m′(k′), (6) Vφ =−Pk,k′∈FS µ [φ(k)]2µ µ (10)
Pk∈FS
(2)
reaches unity. Here the pairing interaction V is given is positive and large, where we denote the gap func-
µ
by tion as φ (µ = s for singlet and t for triplet pairing).
µ
As discussed by Kuwabara and Ogata7, and indepen- q ≡ (q ,2π/3). For a given nodal line (vertical for p
x x
dently by Sato and Kohmoto8, the p-wave pairing, with pairing), the pair scatterings across one pair of parallel
φ (k)φ (k+Q)<0,isfavoredwhenspinanisotropyisso sides of the Fermi surface become all favorable as in-
t t
strongastorealizeVzz(Q)>2V+−(Q),i.e.,V (Q)>0. dicated by (cid:13) in Fig.2(a). This gives rise to the near-
sp sp tk
However, the present self-consistent calculation shows constant gap function on that pair of sides of the Fermi
that the situation is a little more involved. A key fac- surface. Bycontrast,thepairscatteringsacrosstheother
tor is the spin fluctuation that is enhanced along a line pair of parallel sides (× in Fig.2(a)) lead to
2π 2π 2π 2π
−V(2)(k −k′,− − )φ (k ,− )φ (k′, )<0
(a) |G|2 t x x 3 3 t x 3 t x 3
p in eq.(10) when kx,kx′ have the same sign. Since −Vt(2)
is negative, the p -wave gap functions on these sides of
x
the Fermi surface is unfavored. This explains the gap
function shown in Fig.1(c).
a b
k 0 Now, let us move on to the competing superconduct-
y
ingstateinthesingletchannel. AlthoughKuwabaraand
Ogata discussed a competition between the triplet p -
x
-p wave and a singlet dx2−y2-(cid:13)wave, we find here that the
-p 0 p real competitor (the most stable singlet state) is unex-
(cid:13) pectedly an extended s-wave rather than dx2−y2-wave.
k
x This is understood as follows. Since Vs(Q) > 0 is re-
pulsive, φ (k)φ (k+Q)<0 has to be satisfied. For the
s s
(b) dx2−y2-wave,thek,k′ =k+QontheFermisurfacethat
10
c zz (a)
5
2p
0 p k'
0
p (cid:13) q
y
q(cid:13) 2p 0
x
k'
f k
(c)
t
k
(b)
k'
k
f
(d)
s(cid:13)
(c)
k'
FIG. 1. The contour plot of the FLEX result for Green’s
k
function(|G|2;a),spinsusceptibility(b),optimizedgapfunc-
tion for the spin triplet (φt; c) or singlet (φs; d) pairing for FIG. 2. Pair scatterings across the Fermi surface which
U = 5, t′ = 0.3, α = 0.8, and T = 0.02. The left(right) favor ((cid:13)) or unfavor (×) px-wave pairing (a), those con-
panelfortheα(β)band,andwhite(black)correspondstopos- tributing to dx2−y2 (b) or to extended s pairings (c). The
itive(negative) amplitudein (c,d). white(grey) areas represent positive(negative) φ.
1.0 along the Fermi surface is displayed in Fig.4, which
has a crib shape with dips at the corners around k ≡
0.8
(±2π/3,±2π/3).20 The dip arises because φ in each of
triplet t
0.6 the p and p channels has already sharp drops at the
x y
corner of the Fermi surface. Thus, in phase-insensitive
l 0.4
experiments, the gap function obtained here may look
singlet
0.2 like a two-dimensional f-wave pairing with nodes along
k = ±k , since a dip and a node are indistinguishable
0 x y
0 0.02 0.04 when the temperature is greater than the dip. Further
T
study on this point is under way.
FIG. 3. Eigenvalues of the Eliashberg equation for triplet
and singlet pairings as a function of temperature for U = 5, DiscussionswithYujiMatsudaaregratefullyacknowl-
t′=0.3, α=0.8. edged. This work is in part funded by the Grant-in-Aid
forScientific Researchfromthe Ministry ofEducationof
Japan.
+ i =
1.0
0.8 ∗
Present address: Department of Applied Physics and
0.6 Chemistry, University of Electro-Communications, 1-5-1
Chofugaoka, Chofu-shi, Tokyo182-8585, Japan
0.4
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s s
(dashed arrows in Fig.2(b)). By contrast, all the k,k′’s 5Y.Sidis,M.Braden,P.Bourges,B.Hennion,S.NishiZaki,
onthe Fermisurfacewithk−k′ ∼(±2π/3,±2π/3)con- Y.Maeno, andY.Mori, Phys.Rev.Lett. 83, 3320 (1999).
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The competition between the triplet and singlet is magneticspinfluctuationsandd-wavepairingbyantiferro-
quantified by the eigenvalues of the Eliashbergequation, magneticfluctuationswasdiscussedbyI.I.MazinandD.J.
shown in Fig.3 as functions of temperature.19 It can be Singh in Phys.Rev.Lett. 82, 4324 (1999).
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seen that for the value of α(= 0.8) adopted here, the
(2000).
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extended-s, although having a greater magnitude of the
(2000).
pairing interaction, has the nodal line running in the
9H.Mukuda,K.Ishida,Y.Kitaoka,K.Asayama,Z.Q.Mao,
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Y.Mori,andY.Maeno,J.Phys.Soc.Jpn.67,3945(1998).
whythispairingisweaker. IfweextrapolateλtoT →0a 10This form of the gap function was proposed in K. Miyake
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is suggested. 11S.NishiZaki, Y.Maeno, and Z.Q. Mao, J. Phys.Soc. Jpn.
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symmetry. A complex linear combination of these two, Y.Maeno, Phys. Rev.Lett. 84, 5387 (2000).
px+ipy,whichbreaksthetimereversalsymmetry,should 13Y.Matsuda, privatecommunications.
bethetruestatebelowTc sincethesuperconductinggap 14N.E. Bickers, D.J. Scalapino, and S.R. White, Phys. Rev.
is maximized for that combination. Lett. 62, 961 (1989); G. Esirgen and N.E. Bickers, Phys.
The absolute value of the gap function for that linear Rev.B 55, 2122 (1997).
combination on the Fermi surface, 15T.Takimoto,Phys.Rev.B62,R14641(2000),hasconsid-
eredinterorbital repulsions, which enhancesorbital fluctu-
|φ(k)|= φt(kx,ky)2+φt(ky,kx)2 1/2, (11) ations to result in a triplet superconductivity.
(cid:2) (cid:3)
16S. Koikegami, S. Fujimoto, and K. Yamada, J. Phys. Soc. SU(2) symmetry in the interaction is broken. However, in
Jpn. 66, 1438 (1997). theTomonaga-Luttingercase,thetripletpairingdominates
17H.KontaniandK.Ueda,Phys.Rev.Lett.80,5619(1998). over the SDW only when the electron-electron interaction
18In eqs.(2),(7) −(9) we have omitted the first- and second- isattractive,insharpcontrastwiththepresentcasewhere
order terms, U and −U2χ0 (in eq.2), which are negligible the interaction is repulsive. The difference comes from a
when spin and/or charge fluctuationsare strong, although different origin of the broken SU(2) here, which is the
we have taken those terms into account in the actual cal- anisotropyintheirreduciblesusceptibility(whicharisesat
culation. asingleelectronlevel)ratherthantheanisotropicelectron-
19The simultaneous enhancement of the triplet pairing and electron interaction.
the spin fluctuations is reminiscent of the Tomonaga- 20Wenotethat,whilethequalitativeresult doesnotdepend
Luttinger theory for purely 1D systems, where one can crucially on t′(=0.3 in thispaper),thequantitativeshape
show that the susceptibilities for the triplet pairing and of thedip does.
the SDW may diverge simultaneously for T →0 when the