arXiv:0902.0303v2 [cond-mat.supr-con] 1 Jun 2009
L. Fanfarillo, 1 L. Benfatto, 2,1 S. Caprara, 1 C. Castellani, 1 and M. Grilli 1 1CRS SMC, CNR-INFM and Dipartimento di Fisi a,
Università di Roma La Sapienza, Piazzale A. Moro 2, 00185, Rome, Italy 2
Centro Studi e Ri er he Enri o Fermi, via Panisperna 89/A, 00184, Rome, Italy (Dated: O tober25,2018)
Wederivetheee tivea tionforsuper ondu tingu tuationsinafour-bandmodelforpni tides, dis ussing the emergen e of a single riti al mode out of a dominant interband pairing me ha-nism. We then apply our model to al ulate the para ondu tivity in two-dimensional and lay-ered three-dimensional systems, and ompareour results with re ent resistivity measurements in SmFeAsO
0.8
F0.2
.PACSnumbers:74.40.+k,74.20.De,74.25.Fy
There entdis overyofsuper ondu tivityinpni tides [1℄ hasrenewed the interest in high-temperature super- ondu tivity. Pni tides share many similarities with upratesuper ondu tors,e.g.,thelayeredstru ture, the proximitytoamagneti phase[2℄,therelativelylarge ra-tiobetweenthesuper ondu ting(SC)gapandthe riti- altemperature
T
c
[3,4℄,andthesmallsuperuiddensity [2℄. However, dierently from uprates, the presen e of severalsheets oftheFermisurfa emakesthemultiband hara terofsuper ondu tivityanunavoidableingredient of any theoreti al model for pni tides. Moreover, sin e the al ulatedele tron-phonon oupling annota ount for the high values ofT
c
[5, 6℄, it has been suggested thatthepairingglueisprovidedbyspinu tuations ex- hangedbetweenele tronsindierentbands[68℄. Thus, pni tidesareexpe tedtobesomehowdierentfromother multibandsuper ondu tors(e.g.,MgB2
)wherethemain ouplingme hanismisintraband[9℄.Thiss enarioraisesinterestingquestionsregardingthe appropriatedes riptionofSCu tuationsinamultiband systemdominatedbyinterbandpairing. Theissueis rel-evant, be ause u tuating Cooper pairs above
T
c
on-tributetoseveralobservablequantities,su has,e.g.,the enhan ement of d ondu tivity(para ondu tivity) and ofthediamagneti responseasT
c
isapproa hed[10℄. The natureofSCu tuationsdependsonwhetherthesystem isweaklyorstrongly oupled(andonwhetherpreformed pairsarepresentor not)and awealth ofphysi al infor-mation anbeobtainedfrom para ondu tivityand dia-magneti response,providedatheoreti alba kgroundis establishedto extra tthem.Inthispaper,after introdu ingafour-bandmodel,as appropriate for pni tides, we dis uss the subtleties re-lated to the des ription of SC u tuationsin a system with dominant interband pairing. We then apply our resultsto omputethepara ondu tivityasso iatedwith SCu tuationsabove
T
c
. Weshowthatwheninterband pairing dominates, despite the presen e of four bands, thereareonlytwoindependentu tuatingmodes. Only oneofthemis riti alandyieldsadiverging Aslamazov-Larkin (AL) ontribution to para ondu tivity asT
c
isapproa hed,similarlytothe aseofdominantintraband pairing [11℄. The temperature dependen e of the AL para ondu tivityisthesamederivedforordinary single-band super ondu tors [10, 12℄. Remarkably, within a BCSapproa h, we re overthe AL numeri alprefa tor, whi h is a universal oe ient in two dimensions, and depends instead on the oheren e length perpendi ular tothe planesin thethree-dimensionallayered ase. We alsondthatsubleadingtermswithrespe ttothe lead-ingAL ontribution oulddistinguishbetweendominant inter- and intra-band pairing. Within this theoreti al ba kground, we analyze re ent resistivity data in Sm-FeAsOF[4℄anddis usstheresults.
Atpresent,ARPESmeasurementsinpni tides[3℄ on-rmedthe Fermi-surfa etopologypredi tedbyLDA. It onsists of twohole-like po kets entered around the
Γ
point (labeledα
andβ
, following Ref. [3℄), and two ele tron-like (γ
) po kets entered around the M points of the folded Brillouin zone of the FeAs planes. Moti-vated bythemagneti hara ter of theundoped parent ompound and by the approximate nesting of the hole and the ele tron po kets with respe t to the magneti ordering waveve tor, we assume that pairing mediated by spin u tuations is ee tive only between hole and ele tron bands [8℄. Sin e theβ
band has a Fermi sur-fa e substantially largerthantheα
band, theβ
po ket isexpe tedtobelessnestedtotheγ
po ket,sothatthe interbandβ − γ
ouplingλ
issmallerthantheα − γ
ou-pling,i.e.,η ≡ λ/Λ < 1
. Thetwoele tronpo kets have omparablesizesandforsimpli ityweassumethattheγ
bandsare degenerate. Therefore,the BCSHamiltonian ofourfour-bandmodelis[13℄H =
X
i
H
i
0
+ Λ
X
q
h
Φ
†
γ,q
(Φ
α,q
+ ηΦ
β,q
) + h.c.
i
,
(1) whereH
i
0
=
P
k
ξ
i,k
c
†
i,kσ
c
i,kσ
is theband Hamiltonian,c
(+)
i,kσ
annihilates( reates)afermioninthei = α, β, γ
1
, γ
2
band(withthetwofolddegenerateγ
bandslabeledasγ
1
andγ
2
),ξ
i,k
isthedispersionwithrespe ttothe hemi al potential,Φ
i,q
=
P
inthe
i
-thband,andΦ
γ,q
≡ Φ
γ
1
,q
+ Φ
γ
2
,q
. Sin ewe as-sumedthatpairinga tsbetweenholeandele tronbands only,we anexpressthepairingtermin Eq.1bymeans ofΦ
1
≡ Φ
γ
andΦ
2
≡ Φ
α
+ ηΦ
β
. ThusH
I
= Λ
X
q
(Φ
†
1
Φ
2
+h.c.) = −Λ
X
q
(Φ
†
−
Φ
−
−Φ
†
+
Φ
+
),
(2) whereΦ
±
= (w
1
Φ
1
± w
2
Φ
2
)/
√
2
,w
1
, w
2
are arbitrary numbers satisfyingw
1
w
2
= 1
, and for deniteness wetake
Λ > 0
, whi h isthe ase fora spin-mediatedpair-ingintera tion. FromEq.(2)oneimmediatelyseesthat when interband pairing dominates, the intera tion is a mixtureofattra tion(for
Φ
−
)andrepulsion(forΦ
+
). As a onsequen e,whenweperformthestandard Hubbard-Stratonovi h (HS)de oupling of the quarti intera tion term(2)bymeansoftheHS eldφ
,e
±ΛΦ
†
Φ
=
Z
Dφ e
−|φ|
2
/Λ+
√
±1(Φ
†
φ+h.c.)
,
(3)theHStransformationasso iatedwiththerepulsivepart ontainstheimaginaryunit. Asweshall seebelow,this would requirean imaginaryvalueof the
φ
+
HS eld at thesaddlepoint, that an beavoidedbytherotationof the pairing operatorsΦ
±
dened above, via a suitable hoi eofthew
1,2
oe ients.Inanalogywiththesingle-band ase[14℄,theee tive a tionfortheSC u tuationsreads
S =
X
i
X
k
X
σ
(ξ
i,k
− iε
n
)c
†
i,kσ
c
i,kσ
+
X
q
|φ
+
(q)|
2
Λ
− iφ
∗
+
(q)Φ
+
(q) − iφ
+
(q)Φ
†
+
(q)
+
X
q
|φ
−
(q)|
2
Λ
− φ
∗
−
(q)Φ
−
(q) − φ
−
(q)Φ
†
−
(q)
,
where
k ≡ (k, iε
n
)
,q ≡ (q, iω
m
)
, andε
n
,ω
m
are the Matsubarafermion and boson frequen ies, respe tively. Integratingoutthefermionsweobtainthestandard on-tributiontothea tion,−
P
i
Tr log A
i
kk
′
,wherethetra ea tson momenta, frequen ies, and spins. The elements ofthematri es
A
i
kk
′
areA
i
kk
′
11
= (ξ
i,k
−iε
n
)δ
kk
′
,
A
i
kk
′
22
= −(ξ
i,k
+iε
n
)δ
kk
′
,
withi = α, β, γ
1
, γ
2
,h
A
α
kk
′
i
12
=
w
2
√
2
[φ
−
(k − k
′
) − iφ
+
(k − k
′
)] ,
h
A
β
kk
′
i
12
= η
h
A
α
kk
′
i
12
,
h
A
γ
kk
′
i
12
= −
w
1
√
2
[φ
−
(k − k
′
) + iφ
+
(k − k
′
)] ,
for
γ = γ
1
, γ
2
,andthe[A
i
kk
′
]
21
elements ontainthe om-plex onjugatesof theHS eldsevaluatedat(k
′
− k)
.The
q = k − k
′
= 0
values of theHS elds yield the SC gaps in the various bands, within the saddle-point approximation
φ
±
(q = 0) = ¯
φ
±
. However, due to the presen eoftheimaginaryuniti
asso iatedwith theHS eldφ
+
, in general[A
i
kk
]
∗
21
6= [A
i
kk
]
12
. To re over aHermitian
A
matrixat thesaddlepoint,theintegration ontourofthe fun tionalintegralmust bedeformed to-ward the imaginary axis of theφ
+
eld. This an be avoided if one hoosesw
1
andw
2
in the denition ofΦ
±
in su h awaythatφ
¯
+
= 0
. Inour ase,the hoi e¯
φ
+
= 0
gives∆
α
= −w
2
φ
¯
−
/
√
2
,∆
γ
= w
1
φ
¯
−
/
√
2
and∆
β
= η∆
α
. Hen e,theratioof thegaps inthetwohole bandsissolelydeterminedbytheratioof the ouplings,∆
β
/∆
α
= η
. Usingw
1
w
2
= 1
we haveφ
¯
2
−
= −2∆
α
∆
γ
,andtheequationsfor
∆
α
and∆
γ
read [13℄∆
α
= −Λ(2Π
γ
)∆
γ
,
(4)∆
γ
= −Λ (Π
α
∆
α
+ ηΠ
β
∆
β
) ,
(5)where
Π
i
= N
i
R
ω
0
0
dξ[tanh(E
i
/2T )]/E
i
yields theq = 0
valueofparti le-parti lebubblewhen
T > T
c
,N
i
is den-sityofstatesofthei
-thbandattheFermilevel,ω
0
isthe ut-ofor thepairingintera tion, andE
i
=
pξ
2
+ ∆
2
i
. Hen e,our four-bandmodel for pni tides ee tively re-du es to a two-band model, with one ele tron-like and onehole-likeee tive band. Indeed,dening∆
1
≡ ∆
γ
,∆
2
≡ ∆
α
,andΠ
1
≡ 2Π
γ
,Π
2
≡ Π
α
+ η
2
Π
β
,Eqs.(4)-(5) re overthestandardtwo-bandexpression∆
1
= −ΛΠ
2
∆
2
and
∆
2
= −ΛΠ
1
∆
1
. Sin e∆
1
/∆
2
= −w
1
/w
2
,italsofol-lowsthat
w
2
1
/w
2
2
= Π
2
/Π
1
atT < T
c
.Let us now dis uss the SC u tuations for
T > T
c
, where∆
1,2
= 0
. Toderivetheequivalentofthestandard Gaussian Ginzburg-Landau fun tional [10℄, we expand thea tion uptotermsquadrati in theHS elds,S
G
=
P
ικ
P
q
φ
∗
ι
(q)L
−1
ικ
(q)φ
κ
(q)
,whereι, κ = ±
andL
−1
(q) =
Λ
−1
− Π
ef f
(q)
−i Ξ
ef f
(q)
−i Ξ
ef f
(q)
Λ
−1
+ Π
ef f
(q)
,
(6) withΠ
ef f
(q) ≡
1
2
[w
2
1
Π
1
(q) + w
2
2
Π
2
(q)],
Ξ
ef f
(q) ≡
1
2
[w
2
1
Π
1
(q) − w
2
2
Π
2
(q)].
The riti al temperature is determined by the
ondi-tion
det L
−1
(q = 0) = 0
. In the BCS approximation
Π
i
(0) = N
i
ln(1.13 ω
0
/T
c
)
and, in agreement with theEqs.(4)-(5),weobtain
N
ef f
Λ log(1.13 ω
0
/T
c
) = 1
. Here, the parameterN
ef f
≡
√
N
1
N
2
=
p2N
γ
(N
α
+ η
2
N
β
)
playsthe role of anee tivedensity ofstates. To om-putetheu tuation ontributiontothevariousphysi al quantitiesat
T > T
c
,weevaluateEq.(6)atleadingorder inq
(hydrodynami approximation),using thestandard expansion of the parti le-parti le bubble for a layered system,Π
i
(q) ≈ Π
i
(0) − c
q
2
k
= q
2
x
+ q
2
y
and thez
axisperpendi ulartothelayers. IntheBCSapproximation,e.g.,γ
i
= πN
i
/(8T )
. Weomit thelengthier but standardBCSexpressions forc
i,k
andc
i,z
(see,e.g.,Ref. [10℄),thatwillnotbeexpli itlyusedin thefollowing. BymakingforT > T
c
thesame hoi e pre-viouslyadoptedforT < T
c
,w
2
1
Π
1
(0) = w
2
2
Π
2
(0)
,wesim-plifythestru tureoftheu tuatingmodes. Indeed,with this hoi e, the o-diagonal termsin Eq. (6) yield on-tributionsbeyondhydrodynami sand anbenegle ted. Therefore,theleadingSCu tuationsaredes ribedby
L
−1
(q) ≈
m
−
+ ν(q)
0
m
0
+
− ν(q)
,
(7) wherem
±
= Λ
−1
±
√
Π
1
Π
2
arethemassesofthe olle -tive modes,ν(q) ≡ c
k
q
k
2
+ c
z
q
2
z
+ γ|ω
m
|
,c
k
= (w
1
2
c
1,k
+
w
2
2
c
2,k
)/2
isthestinessalongthelayers,c
z
= (w
2
1
c
1,z
+
w
2
2
c
2,z
)/2
is the stiness in the dire tion perpendi ular to thelayers, andγ = (w
2
1
γ
1
+ w
2
2
γ
2
)/2
is thedamping oe ient. IntheBCS asem
±
= N
ef f
[ln(1.13 ω
0
/T
c
) ±
ln(1.13 ω
0
/T )]
andγ = πN
ef f
/(8T )
.Having dedu ed the hydrodynami a tion of the SC u tuations,we an al ulatethepara ondu tivityalong thelines of Ref. [15℄. Byinspe tion ofEq. (7), one an see that only the
φ
−
mode be omes riti al atT
c
[i.e.,m
−
(T
c
) = 0
,m
+
(T
c
) = 2Λ
−1
℄, thus giving a diverg-ingu tuation ontributiontovariousphysi alquantities
when
T → T
c
. Theleading ontribution totheurrent- urrentresponsefun tion alongthelayersis
δχ(Ω
ℓ
) = 4e
2
T
X
q
c
2
k
q
2
k
L
−−
(q, ω
m
)L
−−
(q, ω
m
+ Ω
ℓ
),
when e the para ondu tivity
δσ
AL
= [Imδχ(Ω)/Ω]
Ω→0
isobtained,after analyti al ontinuationofthe Matsub-ara frequen yΩ
ℓ
to the real frequen yΩ
. Therefore, thesameexpressionsknown forasingle-band super on-du torarefound,althoughwiththeee tiveparametersm
−
, c
z
, γ
. Thepara ondu tivityalongthelayersis inde-pendentofthein-planestinessc
k
,asguaranteedbythe same gauge-invarian e arguments dis ussed in Ref. [15℄ forasingle-bandsuper ondu tor. Theleading ontribu-tions to para ondu tivity along the layersin three and two dimensions(3D and 2D, respe tively) take the AL form[12℄δσ
3D
AL
=
e
2
32~ξ
z
1
√
ǫ
(8)δσ
2D
AL
=
e
2
16~d
1
ǫ
(9)where
ξ
z
=
pc
z
/γ
is the orrelation length in the di-re tionperpendi ulartothelayers,d
isthedistan e be-tweenlayers,andǫ(T ) ≡ πm
−
(T )/(8γT
c
)
isthedimen-sionlessmassofthe riti al olle tivemode. Wepointout thattheaboveexpressionsaregeneralwithina
hydrody-onanyparti ularassumptionaboutthepairingstrength [15℄. WhentheBCSexpressionforthe
Π
i
bubblesholds,m
−
(T ) = N
ef f
ln(T /T
c
)
andthedimensionless massap-pearinginEqs. (8)-(9)issimply
ǫ = log(T /T
c
)
.The al ulated AL para ondu tivity expressions (8)-(9) an be now ompared with the existing results for the pni tides. As it has been widely dis ussed in the ontextof uprates[16℄, forweakly- oupledlayered ma-terials the SC u tuations usually display a 2D-3D di-mensional rossoveras
T
c
is approa hed. However,the interlayer ouplinghasadierentrelevan einthevarious familiesof uprates, with substantial 3D behavior [Eq. (8)℄ in YBa2
Cu3
O6+x
samples, while more anisotropi Bi2
Sr2
CaCu2
O8
orLa2−x
Srx
CuO4
ompounds show2D u tuations [Eq. (9)℄, the 2D-3D rossover being too losetoT
c
tobe learlyobserved. Su hasystemati sur-veyhasnotyetbeenperformedin the aseofpni tides, due also to the limited availability of lean single rys-tal. Indeed, the analysis of the 2D-3D rossovermight be biased in poly rystals by the distribution of riti al temperaturesand by themixing ofthe planarand per-pendi ulardire tions.Havinginmindsu hlimitations,weattemptthe anal-ysis of para ondu tivity in a SmFeAsO
0.8
F0.2
sample [4℄ withT
c
≈ 52
K. To determine the ontribution of SC u tuations to the normal-state ondu tivity,δσ ≡
ρ
−1
−ρ
−1
N
,weneedtoextra tthenormalstateresistivityρ
N
from thedata. Owing tothediverging ondu tivity, thepre ise determination of the normal-state ontribu-tionisimmaterialnearT
c
,butbe omesrelevantforlarger valuesofǫ
. Wettedtheresistivityathightemperature (inthe range between279
K,the highestavailable tem-perature,andabout200
K) he kingthatthequalitative resultswere stable upon small variations of this range. Weusedaquadrati tρ
N
= a + b(T − T
0
) + c(T − T
0
)
2
,with
T
0
= 279
K,and foundthat the resultingpara on-du tivityisroughlytwoordersofmagnitudesmallerthan the2DALresult[Eq. (9)℄. Theslopeinalog-logplotis inagreementwiththe3Dpowerlaw[Eq. (8)℄. Theplot inFig.1 learlyshowsthatthe3Dbehaviorextendsover twode adesof
ǫ
. Thettingparametersa = 1700 µΩ
m,b = 21.9 µΩ
m/K,andc = 0.018 µΩ
m/K2
wereused. The tting with Eq. (8) allows us to determine the pre isevalueof
T
c
andthe oheren elengthξ
z
oftheSC u tuations in the dire tion perpendi ular to the FeAs layers. We ndT
c
= 51.4
K andξ
z
= 19
Å. One an also see that substantial SC u tuations persist up toǫ ≈ (T − T
c
)/T
c
≈ 0.4
, i.e., upto20
KaboveT
c
. Thisu tuatingregimeis thereforemu hlargerthan in on-ventional3Dsuper ondu torsand,despitethe3D behav-iorofthepara ondu tivity, allsforarelevant hara ter of the layered stru ture and for a small planar oher-en elength. Wepointoutthat at even largervaluesof
ǫ
thepara ondu tivitydrasti allydrops,inanalogywith what found in uprates (see, e.g., Ref. [16℄, andrefer-0.01
0.1
1
ε
1e-05
0.0001
0.001
0.01
0.1
∆σ
(
µΩ
cm)
-1
FIG.1:(Coloronline)Comparisonbetweentheexperimental para ondu tivityforaSmFeAsO
0.8
F0.2
samplestudiedinRef. [4 ℄(bla k ir les)andthe2D(dashedline)and3D(solidline) expressionsofthe ALpara ondu tivity[Eqs. (8,9)℄. Forthe 3D para ondu tivity a oheren e lengthξ
z
= 19
Å has been used, while for the 2D ase the stru tural distan e between layersd = 8.4
Åhasbeeninserted.the ALbehaviorin multiband systemsalso depends on the role of the other (non- riti al) olle tive modes. In parti ular,it anbeshown[17℄that,whentheintraband pairingisequallydominantinallbands,the para ondu -tivity mediatedby the non riti al modes may be ome sizable,and theexperimental data should approa h the pure AL ontribution of the riti al mode from above. This is not the ase when the dominant pairing is in-terband andtherefore it isnot surprising that the data forthepni tidesampleanalyzedinthispaperalwayslay belowtheALstraightlinein Fig.1.
In on lusion, we investigated the o urren e of SC u tuationsinamultibandsystemwhereinterband pair-ingdominates, asappropriateforpni tides. In ontrast to the aseof dominantintraband me hanism (as, e.g., in MgB
2
[11℄), in the present situation the HS de ou-plingmustbea ompaniedwithaproperrotationofthe fermioneldswhi hguaranteesaHermitiansaddle-point a tionbelowT
c
. The samerotationleadsto a straight-forwardde ouplingoftheGaussianu tuationsaboveT
c
in the hydrodynami limit. Thus, despite the apparent omplexity of the multiband stru ture in pni tides, we demonstratethattheALexpressionsfor para ondu tiv-ity stay valid notonly as far as the fun tional temper-ature dependen e is on erned, but also regarding the numeri al prefa tors. While in the BCS 2D ase the prefa tor stays universal, in the 3D ase the only dif-feren e is that a suitable redenition of the transverse oheren elengthhasto beintrodu ed. Withthis equip-ment,we onsidered theexperimental resistivitydata of the SmFeAsO0.8
F0.2
sample studied in Ref. [4℄, ndingabove
T
c
. Re ently,anexperimental work[18℄ on u tu-ation ondu tivityin pni tides onrmed thewide u -tuating regime, even though u tuations seem to have 2D hara ter. Thus,further experimentsarein orderto onrm the nature of u tuations in pni tides, and to assesstherelevan eof Cooper-pairu tuations in these newsuper ondu tors.A knowledgments. WewarmlythankD.Dagheroand R. Gonnelli for providing us with the data of Ref. [4℄. This work has been supported by PRIN 2007 (Proje t No. 2007FW3MJX003).
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