Table Of ContentSuperconductivity due to spin fluctuations originating from multiple Fermi surfaces in
a double chain superconductor Pr Ba Cu O
2 4 7 15−δ
Tsuguhito Nakano,1 Kazuhiko Kuroki,1 and Seiichiro Onari2
1Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
2Department of Applied Physics, The University of Nagoya, Nagoya 464-8603, Japan
7
(Dated: February 6, 2008)
0
0 The mechanism of superconductivity in Pr2Ba4Cu7O15−δ is studied using a quasi-one dimen-
2 sional double chain model with appopriate hopping integrals, on-site U, and off-site repulsion V1.
Applying the fluctuation exchange method to this model and solving the Eliashberg equation, we
n
a obtain the doping dependence of superconductivity that is consistent with the experiments. The
J superconducting gap has an extended s-wave-like form, which gives a temperature dependence of
thespin lattice relaxation rate that does not contradict with theexperimental results.
9
]
n It goes without saying that unconventional super-
o conductivity (SC) appears on the CuO planes or lad-
2
c der structures in some cuprates.1,2 Recently, SC in the
-
r cuprateshasbeenfoundinanotherstructure,namely,the
p
CuO double chain. Pr Ba Cu O (Pr247/δ), which
u 2 4 7 15−δ
consists of metallic CuO double-chain and semiconduct-
s
t. ing single-chainbesides the Mottinsulating CuO2 plane,
a showsSCwithTmax ∼15Kinamoderateoxygendefect
c
m concentration range of δ = 0.2−0.63,4,5, which controls
- the band filling of the double chain block. Recent NQR FIG. 1: (a)The effective single band 1D double chain
d study has revealed that the SC occurs in the double- model.(b) Q1D double chain model.
n
chain, and further observed a “charge freezing” just like
o
the one observed in PrBa Cu O (Pr124)6,7,8 at, and
c 2 4 8
[ only at, δ = 0, which implies that the double chain is Therefore,thereasonfortheabsenceofT beforeoxygen
c
near 1/4-filling in Pr247(δ =0) (although there remains reductionisnotclear. Secondly,apreviousestimationof
1 some ambiguity in the band filling estimation).9,10,11
v the spin gapusing density matrix renormalizationgroup
0 Theoretically, Sano et al. proposed a superconducting (DMRG) has shown that the spin gap in a double chain
6 mechanism using a one dimensional (1D) double chain Hubbard model is, if any, very small at 1/4-filling,17 al-
1 model,12 which is essentially based on a superconduct- though the values of the hopping integrals taken there
1 ingmechanismproposedgenerallybyFabrizio13 andalso do not directly correspond to Pr247. Moreover,a recent
0
studied by one of the present authors14 about a decade DMRG analysis shows that the spin gap, if any, is too
7
ago. In a purely 1D double chain system, the band dis- smalltobeestimatednumericallywhentheratiobetween
0
/ persioncanhaveadoublewellstructure,resultinginfour theinterchainandtheintrachainhoppingissmallerthan
at Fermi points for appropriate band fillings, which in turn ∼0.3,18 while this ratio is estimated to be around0.2 in
m results in an opening of the spin gap and dominating Pr247 from first principles calculations.19 Thirdly, it is
- superconducting correlation. Sano et al. determined the of interest whether the relatively high Tc of 15K can be
d tightbinding(TB)parametersforapurely1Dsystemus- explained within a Hubbard-type model with the above
n ing the LDA band dispersionbetweenthe X andS point mentioned ratio as small as ∼ 0.2, because it is obvi-
o
forthedoublechainofYBa O O (Y124)(SeeFig. 2)15. ous that SC does not occur when this ratio is too small,
c 2 4 8
: From this band structure, they have suggested that the namely, when the system is essentially a single chain re-
v appearance of SC upon oxygen reduction is due to the pulsive Hubbard model.20 To resolve these problems, we
i
X increase of the number of Fermi points from two to four propose in this paper a spin fluctuation mediated pair-
as the band filling is increased from ∼1/4-filling. ing mechanism, where the origin of the spin fluctuation
r
a isindeedthe presenceofmultiple FS. We adoptthe fluc-
Although we believe that this theory is correctin that
tuation exchange (FLEX) method to a quasi-one dimen-
theexistenceoffourFermisurfaces(FS)isessentialinthe
sional(Q1D)extendedHubbardmodelfortheCuOdou-
occurrence of SC, there remain several problems, which
ble chain.21,22 A set of TB parameters are determined
have motivated the present study. First, if we look into
from the results of LDA calculation for Y124.16,19 By
thebandstructureintheentiretwodimensionalBrillouin
solving the linearized Eliashberg equation, we obtain a
zone (BZ) shown in Fig.2, the number of the FS along
finite T for an extended s-wave SC, whose band filling
the Γ-Y line remains to be four down to very low band c
fillings (i.e., the inner FS is 2D),16,19 so the variance of dependence is qualitatively similar to the experimental
results.
the FS around 1/4-filling is not simply a change of their
number between two and four as in the pure 1D theory. Note that the original model on the zigzag lattice
2
FIG.2: (Coloronline)(a)TheTBbanddispersionforn=0.7
with the determined parameters along with the LDA results
from ref. 16. (b) The FS (for the noninteracting case) for
FIG.3: TheobtainedTc forsinglet pairing. Theinsetisλas
several band fillings.
functions of temperature. U =2.0 eV, V1 =0 are taken.
(which has two sites in a unit cell and thus results in
a two band system) can be mapped to a single band
model having distant hopping integrals as shown Fig. 1.
Such a mapping is often adopted in the study of zigzag
chains11,13. Note that after this single band mapping,
the BZ will be unfolded in the k direction. Hereafter,
b
ourFLEXresults willbe showninthe unfolded BZ.The
HamiltonianoftheextendedHubbardmodelforthedou-
ble chain is given by
1
H = Xtijc†iσcjσ +UXni↑ni↓+ 2XVijninj, (1)
i,j,σ i i,j
where c† creates an electron of spin σ at site i, t is
iσ ij
a hopping parameter between site i and j, and U and
V isanelectron-electroninteractionbetweensuchsites. FIG. 4: (Color online) (a) The FS and the nodes of the gap
ij
Thevaluesoft aredeterminedbyfittingtheTBdisper- function,and(b)thespinsusceptibilityχs(q)forU =2.0eV
ij
sion to the results of the LDA calculations for the dou- and V1 =0 eV.
ble chains of Y124.19 The fitting result is shown in Fig.
2. With the obtained values of the hopping parameters,
we have found that for the noninteracting case the inner 1V¯ χ¯ V¯ ](k−k′) for triplet pairing. With the obtained
FS around the Γ point remains unless n ≤ 0.1 (n=band G2(kd)candd Γ(k,k′), the linearized Eliashberg equation
filling=number of electrons/number of sites), although λ∆(k) = −T Γ(k−k′)G(k′)G(−k′)∆(k′) is solved.
a topological change does occur near the X point at N Pk′
T isdefinedasT ≡T(λ=1). Inordertoassurethecon-
c c
n ∼ 0.3 (Fig. 2(b)), which corresponds to the change of
vergenceofthecalculation,thesystemsizeandthenum-
the number of Fermi points discussed in ref. 12. In the
ber of Matsubara frequencies are taken as N =512×16
FLEX, the dressed Green’s function is obtained by solv-
sites and 16384, respectively, when only U exists, and
ing the Dyson’s equationG(k)=[G−1(k)−Σ(k)]−1 self-
0 256×16 sites and 16384 are taken when both U and V1
consistently, where G (k) is the undressed Green’s func-
0 are present. As for the band filling range, we have in-
tion and Σ(k) is the self energy written by the spin and
vestigated in the range of n = 0.3 to near half filling.
charge susceptibilities χ¯ (q) = χ¯(q)[I +V¯ (q)χ¯(q)]−1
s,c m,d In the following, we show the irreducible susceptibility
with k ≡ (k,ε = (2n+1)πT) and q ≡ (q,ω = 2lπT).
n l χ(q), spin and charge susceptibilities χ (q) defined as
s,c
χ¯(q)istheirreduciblesusceptibilitywhichiscomposedof
thelargesteigenvaluesofχ¯(q),χ¯ (q). Thesealongwith
s,c
a product of G(k), and V (q) is the magnetic and den-
m,d thegapfunctionareplottedatthelowestMatsubarafre-
sity coupling vertices described with U and V . χ¯(q),
ij quency.
χ¯ (q) and V (q) are the matrices indexed by the ini-
s,c d,m First,wepresenttheresultforU =2.0eVwithoutthe
tial (final) relative displacement of particle-hole pair ∆r
off-siteinteraction. TheobtainedT isshowninFig. 3as
(∆r′). The sizes of these matrices increase as the num- c
functionsofn,andthelargesteigenvalueλvs. T forvar-
ber ofoff-siteinteractionsincreases;namely if U andthe
iousbandfillingsisalsoshownintheinset. Finite values
nearest neighbor (n.n.) repulsion V exist, the matrix
1 ofT areobtainedinawide rangeofbandfilling,andits
sizebecomes3×3. The pairinginteractionsaregivenby c
value graduallyincreaseswith increasingn. The FS (de-
Γs(k,k′) ∝ P∆r,∆r′[32V¯mχ¯sV¯m − 12V¯dχ¯cV¯d](k −k′) for finedasεk−µ+Re(Σ(k))=0)alongwiththenodesofthe
singlet pairing and Γt(k,k′) ∝ P∆r,∆r′[−21V¯mχ¯sV¯m − gapfunction,andthe spinsusceptibility, χs(q), areplot-
3
FIG. 5: λ vs. T for U = 2.0 eV, V1 = 0.7 eV. (Inset) The
estimated values of Tc obtained bylinear extrapolation.
ted in Fig. 4. Three kinds of nesting (hereafter specified
bynestingvectorsQ ,Q andQ )causethepeaksofχ .
1 2 3 s
The most dominant nesting Q arises due to the nesting FIG. 6: (Color online) (a) The FS and the nodes of the gap,
1
betweenthe inner andthe outersets ofFS,andthus is a and(b)thespinandthechargesusceptibilities χs andχc for
direct consequence of the presence of multiple FS, while U =2.0 eV, V1 =0.7 eV.
the other two originate from inner-inner or outer-outer
FS nesting. Q changesfromQ =(3π/4,q )at n=0.5
1 1 a
to (π/2,q ) at n = 0.9 and the values of peak itself in-
a ted, this reduction of χ is caused by the degradation of
s
crease with increasing the band filling because of the FS
the nesting condition due to the renormalization of the
nesting becomes better. There are no nodes intersecting
band for V 6= 0, which results in a stronger warping
1
the FS, but the sign of the gap changes its sign between
of the FS especially around 1/4 filling. The reduction
theinnerandtheouterFS,whichisaconsequenceofthe
of χ is reflected in the reduction of SC especially near
s
repulsivepairinginteractionmediatedbythespinfluctu- n≃0.5. Asforthe directcontributionofthechargefluc-
ationsatthenestingvectorQ1. Consequently,thevalues tuation to the pairing interaction through the 1V¯ χ¯ V¯
of λ and T increases with the increase of the peak value 2 d c d
c term, although χ (Q ) near 1/4 filling is relatively large
ofχ atQ uponincreasingthebandfilling. Wewillcall c 1
s 1 comparedto those forother bandfillings, the peak value
this gap an extended s-wave hereafter.
itselfismuchsmallerthanthatofthe spinsusceptibility,
TheobtainedTc of∼20Kagreeswiththe experimen- so that this contribution should hardly affect SC. Thus
tal result, but the gradual band filling dependence of Tc themaincontributionofthechargefluctuationisthrough
failstoexplainthe“switchon”oftheSCwithoxygenre- the self-energy renormalization.
duction, i.e.,the increaseofnin the realisticbandfilling Although we cannot obtain T within the investigated
c
range. At least within the present approach, the change temperaturerange,judgingtheobtainedtemperaturede-
of the topology of the inner FS (i.e., change of the num- pendence ofλ, it doesnotseemunnaturalto linearlyex-
ber ofFermipointsalongthe X-Sline)doesnotstrongly trapolate log(λ) up to λ = 1 against log(T) in order to
affect SC. estimate the value of T . In the inset of Fig. 5, the esti-
c
This resulthasmotivatedus toinclude the n.n. repul- matedvaluesofT areshown. Ifweassumethattheband
c
sion V . Since the system size and the number of Mat- filling of Pr247/δ=0 is n≃0.5, this result shows quali-
1
subara frequencies are limited, we have calculated λ for tativeagreementwiththeexperimentalresultinthatthe
T > 0.004t in order to assure the convergence. Figure SC appears with moderate electron doping.
2
5 shows the obtained λ as functions of T with V = 0.7 Finally, we discuss the validity of the obtained form
1
eV. The value of λ near n = 0.5 is strongly suppressed of the gap function in the light of the experimental re-
compared to the result without V , and does not seem sults. The NQR measurement has observed 1/T ∝ T2
1 1
toreachunityevenatlowtemperatures,whileforhigher (T is the spin-lattice relaxation rate) without exhibit-
1
filling such a tendency of suppression is smaller. This ing a coherence peak below T .7 This result seems to
c
result can be traced back to the deformation of the FS suggestthat the nodes of the gapintersect the FS. How-
shown in Fig.6, where the nodes of the gapfunction and ever, if we assume that a small amount of oxygen de-
the spin and the charge susceptibilities are shown. Al- fects controlling the band filling work as scatterers, we
though the positions of Q and the gap function do not find that the observed 1/T can be accounted for as fol-
1 1
differsomuchfromtheresultsforV =0,thepeakvalue lows. Taking into account the effect of the impurities
1
of χ , particularly for n = 0.5, turns out to be smaller. and the defects in the unitarity limit, we have calcu-
s
As seen in Fig. 7 (a) and (b), where the obtained FS lated 1/T for the gap functions and the FS obtained by
1
andthevaluesofUχ(Q )forV =0and0.7eVareplot- FLEX.24 As for the temperature dependence of the gap
1 1
4
where T is found to decrease with increasingδ and van-
c
ishesatδ ∼0.6.3ConsideringthatthenestingvectorQ ,
1
Q and Q all become close to (π/2,q ) near half filling
2 3 a
(seeFig. 4andFig. 6),umklappprocessescanbeallowed
to result in a Mott insulating state,13 which cannot be
dealt with in the present method. More study on this
issue will be carried out elsewhere. In a zigzag system
as in the present case, next n.n. interactions may also
playsomeroleassuggestedinexperimentalandtheoret-
FIG. 7: (Color online) (a) The FS and (b) the filling depen- ical studies.8,11 Unfortunately, the calculation including
dence of Uχ(Q1) for V1 = 0 and 0.7 eV. U is fixed at 2.0 such interactions within the present approach has to be
eV.
restricted to small system sizes. Therefore the study on
this effect also remains as future study.
function, ∆(k,T) = ∆0φ(k)tanh(2p(Tc/T)−1) with
∆ /(k T )=4isassumed,whereφ(k)isproportionalto
0 B c
the gap function obtained within the FLEX+Eliashberg
eq. approach. In Fig. 8, the obtained 1/T is shown
1
for U = 2.0 eV, V = 0.7 eV. The values of α/∆ are
1 0
chosen as 0,0.2 and 0.4, where α=c/πN (0) is the pair
0
breaking parameter, c is the impurity concentration and
N (0) is the density of states at the Fermi level in the
0
normal state. When α/∆ = 0, 1/T decays exponen-
0 1
tially since a full gap opens on the FS. For finite α/∆ ,
0
1/T shows a power-law-like decay without exhibiting a
1
coherencepeak,whichresemblestheexperimentalresult. FIG.8: Thecalculated1/T1forseveralvaluesofpairbreaking
Therefore, from these results, we can safely say that the parameter for n=0.7, U =2.0 eV,V1 =0.7 eV.
observed1/T doesnotnecessaryruleoutthe possibility
1
of the fully gapped extended s-wave pairing.
To summarize, we have investigated the superconduc-
tivity of Pr Ba Cu O using an extended Hubbard Present results somewhat resemble that of the recent
2 2 4 15−δ
model on a Q1D double chain with the TB parameters study on NaxCoO2·yH2O in that the extended s-wave
determined from the LDA calculation. By applying the superconductivityoccursdue tothe inner-outerFSnest-
FLEXtothe modelandsolvingthe Eliashbergequation, ing in a disconnected FS system.23 Further studies on
in the absence of V , we have obtained singlet super- such systems may open interesting new physics.
1
conductivity with moderately varying T in a wide band The authors thank N. Yamada for his encouragement
c
filling range. By introducing the n.n interaction V , the throughout this study, T. Oguchi for discussions on the
1
superconductivity is suppressed at a band filling range band structure, and T. Nishida for useful suggestions.
near 1/4-filling, which at least qualitatively explains the We also acknowledge illuminating discussions with Y.
δ dependence ofsuperconductivity. NQRrelaxationrate O¯no, K. Sano, T. Habaguchi, Y. Yamada, and T. Mi-
1/T is also calculated taking into account the effect of zokawa. Numerical calculations were performed at the
1
the impurities and the defects. Power-law-like decaying facilities of Supercomputer Center, ISSP, University of
1/T resemblingthe experimentalresultisobtainedwith Tokyo. This work was supported by Grants-in-Aid for
1
a moderate impurity concentration. Scientific Researchfrom the Ministry of Education,Cul-
We should comment on the higher δ regime (δ >0.5), ture, Sports, Science and Technology of Japan.
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5
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