Effects of isoelectronic Ru substitution at the Fe site on the energy gaps of optimally F-doped SmFeAsO

1.2 MB·English

by  D. Daghero

Effects of isoelectronic Ru substitution at the Fe site on the energy gaps of optimally
F-doped SmFeAsO
D. Daghero,1 M. Tortello,1 G.A. Ummarino,1 V.A. Stepanov,2
F. Bernardini,3 M. Tropeano,4 M. Putti,5,4 and R.S. Gonnelli1
1Dipartimento di Scienza Applicata e Tecnologia, Politecnico di Torino, 10129 Torino, Italy
2P.N. Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia
3Dipartimento di Fisica, Universita` di Cagliari, 09042 Monserrato (CA), Italy
4Dipartimento di Fisica, Universita` di Genova, via Dodecaneso 33, 16146 Genova, Italy
5CNR-SPIN, Corso Perrone 24, 16152 Genova, Italy
WestudiedtheeffectsofisoelectronicRusubstitutionattheFesiteontheenergygapsofoptimally
F-doped SmFeAsO by means of point-contact Andreev reflection spectroscopy. The results show
3 that the SmFe1−xRuxAsO0.85F0.15 system keeps a multigap character at least up to x = 0.50,
1 and that the gap amplitudes ∆1 and ∆2 scale almost linearly with the local critical temperature
0 TcA. The gap ratios 2∆i/kBTc remain approximately constant only as long as Tc ≥ 30K, but
2 increase dramatically when Tc decreases further. This trend seems to be common to many Fe-
based superconductors, irrespective of their family. Based on first-principle calculations of the
n
bandstructure and of the density of states projected on the different bands, we show that this
a
J trend, as well as the Tc dependence of the gaps and the reduction of Tc upon Ru doping, can be
explained within an effective three-band Eliashberg model as being due to a suppression of the
6
superfluiddensity at finitetemperaturethat,in turns,modifiesthetemperaturedependenceof the
1
characteristic spin-fluctuation energy.

]
n PACSnumbers: 74.50.+r,74.70.Dd,74.45.+c
o
c
- INTRODUCTION louin zone and two electron pockets at the M point (in
r
p the folded Brillouin zone) [2, 3]. In 1111 compounds,
u all the relevant Fermi surface sheets are weakly warped
s The discovery of Fe-based superconductors (FeBS) [1] cylindersparallelto the k axis(as expectedina layered
. z
at withTcashighas55Khasshownthatcupratesnolonger material) while a greater degree of three-dimensionality
m represent the only class of high-Tc compounds. One of isobservedin122compounds. Thesemultiplebandsand
the reasonsof the greatexcitement in the scientific com- their almost perfect nesting in the parent compound ex-
-
d munity and of the impressive amount of work produced plain the AFM instability. The weakeningof the nesting
n up to now is certainly that these materials give the op- inducedbydopinginsteadleadstospinfluctuationsthat
o
c portunitytostudyhigh-Tc superconductivityindifferent wouldactasthegluefortheformationofCooperpairs. A
[ systems, in the hope to enucleate its key elements. The spin-fluctuationmediatedpairingwouldbemainlyinter-

parent stoichiometric compounds of most FeBS are not band and would favour the opening of superconducting
1
superconducting (with few exceptions, like LiFeAs and energy gaps of different sign on different Fermi surface
v
7 LaFePO) but display a metallic behaviour (as opposed sheets, the so-called s± symmetry [4]. Though many
5 to the Mott insulating state of cuprates) and feature a theoretical and experimental results support this theory
7 long-range antiferromagnetic (AFM) spin-density-wave [5, 6] there is not, up to now, a definitive proof of such
3 (SDW) order. Superconductivity appears upon doping a picture. Things are further considerably complicated
.
1 and, in some systems, also by applying pressure; how- bythefactthattheelectronicbanstructureisverysensi-
0 ever, the order of the transition between magnetic and tivetosomefinestructuralparameters,liketheFe-As-Fe
3
superconductingphasesseemsnottobeuniversalthough bond angle and more particularlythe height of the pnic-
1
increasingevidencesarebeingcollectedofaregionofco- togen atom (h ) above the Fe layer. Possibly because
: As
v existenceofsuperconductivityandmagnetism. Contrary of this sensitivity, in many situations the gap structure
i
X to cuprates (where the superconducting region in the of FeBS can vary considerably within the same system,
r phasediagramisdome-shapedandthemaximumTc cor- giving rise to line nodes, point nodes, deep gap minima
a respondstoawell-defined“optimal”doping)inFeBSsu- etc. [7–10]. In 1111 compounds, h has been proposed
As
perconductivity sometimes appears with Tc already very as a switch between high-Tc nodeless superconductivity
close to the maximum and shows a weak doping depen- and low-T nodal superconductivity [11].
c
denceinabroaddopingrange. AcentralfeatureofFeBS
–whichistightlyconnectedtotheoriginofsuperconduc- In the effort to discriminate the effects of different pa-
tivity according to the most widely accepted theories – rameters on the superconducting and magnetic phases
is theirmultibandcharacter. They featureindeedtwoor ofFeBSmany differentchemicalsubstitutions havebeen
three hole pockets around the Γ point of the first Bril- performed. The main effect of aliovalent substitutions is
2
to dope the parent compound with charge, either elec-
4.00 8.6
trons [12, 13] or holes [14] thus allowing to explore the
pwhearesetrdieiadgtroamm.odIsifoyvatlheentlastutibcsetistturtuicotnusre[1,5c–re1a8t]e, i“ncshteemad- om) 3.98 8.5 c-a
ical pressure”, introduce disorder (acting as magnetic or str xis
g (
non-magnetic impurities) etc. A further degree of free- n 3.96 8.4 A
A n
dom is the site of substitution, that can reside either in s ( gs
the spacing layer [14] or in the active one containing Fe -axi 3.94 8.3 trom
[15–18], which is possibly directly involved in the mag- a )
netic pairing via spin fluctuations.
3.92 8.2
Here we report on point-contact Andreev-reflection
0.0 0.1 0.2 0.3 0.4 0.5
spectroscopy (PCARS) measurements performed in
Ru content (x)
SmFe1−xRuxAsO0.85F0.15 withxrangingfrom0to 0.50.
The considerable decrease of T in this series of samples
c
has been attributed to disorder in the Fe sub-lattice [15] FIG. 1: Lattice constants for SmFe1−xRuxAsO0.85F0.15 at
different Ru content x. Lines are only guides tothe eye.
and/or occurrence of a short-range static magnetic or-
der [19]. PCARS results clearly indicate the presence
of a multigap character at all the investigated levels of x Tc (K) ρ0 (mΩ cm)
Ru substitution. The superconducting gapsdecreaseap- 0 52.0 0.33
proximately linearly with the local critical temperature
0.05 42.8 0.87
ofthecontact,TAbut,evenwhenthelatterisreducedby
c 0.10 21.5 1.33
a factor 5 with respect to the optimal value, they show
0.25 28.1 1.20
no sign of nodes, either intrinsic or “accidental”. For
0.30 13.6 1.69
bothgaps,the 2∆/k T ratioisratherconstantdownto
B c
0.50 13.5 0.70
TA >30Kbutthenincreasesconsistentlybelowthiscrit-
c
icaltemperature. Comparisonwithotherresultsinliter-
TABLE I: Resistive critical temperatures and residual resis-
atureindicatesthatmanydifferentFeBSfitinthistrend, tivities (defined as in ref. [15]) for SmFe1−xRuxAsO0.85F0.15
which suggests the possibility to study some properties samples at different Ru contents. Tc and ρ0 for the samples
commontodifferentcompoundsinasinglesamplesseries with x=0.25 and x=0.50 are different from those reported
thatallowsspanningaverywiderangeofcriticaltemper- in ref. [15].
atures. Thanks to ab-initio electronic structure calcula-
tions, the trendof the gaps as a function ofTA has been
c
reproduced within a minimal three-band, s± Eliashberg heatingto450◦ andasecondheatingto1000−1075◦. X-
model. This model also takes into account the so-called ray diffraction analysis showed small amounts of SmOF
“feedback”effect, i.e. the effectofthe condensateonthe (up to 6 %) in the final samples. Figure 1 shows the de-
antiferromagnetic spin fluctuations possibly responsible pendence of the lattice constantsa and c on the Ru con-
for the superconductivity in these compounds. The evo- tentx,indicatingthatRusubstitutionforFeiseffective.
lutionasafunctionofT ofthe temperaturedependence Resistive critical temperatures and residual resistivities
c
ofthe condensatenecessaryto reproducethe experimen- for the samples used in this work arereportedin table I.
tal data looks rather similar to that obtained from Lon- The samples with x= 0.25 and x= 0.50 have higher Tc
donpenetration-depthmeasurementsperformedinother and much improved transport properties (namely, resis-
FeBS, particularly in the region of coexistence of super- tivity, magnetoresistance and Hall mobility) than those
conductivity and magnetism. This fact suggests that, in reported in ref. [15] for the same doping contents, even
agreement with ref. [19], proximity of superconductiv- though they were prepared in the same way. The possi-
ity and magnetism in these samples might be one of the blereasonofthisdifferenceisunderinvestigation. Inany
main reasons for the decrease of T and for the observed case, these samples were particularly suited for PCARS
c
behavior of the energy gaps. measurements, since the longer mean free path makes it
easiertoattainthespectroscopicconditions,asexplained
below.
EXPERIMENTAL DETAILS Point contact spectroscopy is a local, surface-sensitive
technique and it is therefore necessary to avoid any sur-
The polycrystallineSmFe1−xRuxAsO0.85F0.15 samples face degradation or contamination. The samples were
were synthesized as described in Ref. [15]. The starting thus always kept in dry atmosphere, and broken to ex-
mixture of fine powder of SmAs and 99.9% pure Fe O , pose a clean surface prior to point-contact fabrication.
2 3
RuO , FeF , Fe and Ru was pressed in pellets and then Thepointcontactsweremadebyputtingasmalldropof
2 2
put through a two-step reaction process involving a first Agpasteonthatsurface,asdescribedelsewhere[20,21].
3
With respect to the standard “needle-anvil” technique,
)0.023
tmmthhaoieslucncsoottliandnbfigighliufetoaryradtopiofoofnitnhtethenecscouconrrnteytasoacgactetgnstroiceaanbitdneesrhaemrelstroemtchhaeutlalisnocwaiaclsvalyoltiahcdnleiondswgethdhaoenirlnye- -1)0.022 WConductance (1/0000....000012229012 47T.. 20(K)
-100 -50 0 50 100
exposition to air and moisture during the transfer from We (0.021 Voltage (mV) 91.11.1
the glove box (where the point contacts are fabricated) c 13.1
n 14.6
tothecryogenicenvironment. AlthoughtheAgdrophas a 16.0
ct 17.0
a diameter ofatleast50µm,the realelectric contactoc- u 17.6
d0.020
curs only between some of the Ag grains and the sample n 18.0
o 19.0
surface. The true contact is thus the parallel of sev- C 20.0
21.0
eralnanoscopicjunctionsthatcanwellbeinthe ballistic 22.0
0.019 23.0
regime(i.e. havearadiussmallerthantheelectronmean
-40 -30 -20 -10 0 10 20 30 40
free path). In Ref. [15] a rough evaluation of the mean Voltage (mV)
free path in SmFe1−xRuxAsO0.85F0.15 gave ℓ = 3−10
nm without any clear dependence on the Ru content. In
FIG. 2: Temperature dependence of the raw conductance
the cleaner samples with x = 0.25 and x = 0.50, the curveofapointcontacton thex=0.10 sample. Thenormal
same evaluation gives ℓ ≃ 7 nm and ℓ ≃ 20 nm, respec- state resistance is RN =50Ω. The inset shows the curves at
tively. Such small values of the mean free path make 4.2Kandat21.0Kinanextendedvoltagerange,tohighlight
the fulfillment of the ballistic condition a ≪ ℓ (where a theexcess conductance persisting up to about 100 mV.
is the contact radius) be very difficult to achieve. For
instance, with these values of ℓ and the residual resistiv-
ities taken from table I, the Sharvin equation [22] would typical features already observed in SmFeAs(O1−xFx)
requireresistancesofthe orderofseveralkΩ for the con- [23], in LaFeAs(O1−xFx) [20] and in other 1111 com-
tact to be ballistic. The typical experimental resistance pounds. Inparticular,they featureclearmaximarelated
ofthecontactsisinsteadintherange10−100Ω. Indeed, to a presumably nodeless gap, shoulders suggestive of a
many of the contacts were not spectroscopic or showed second larger gap and additional structures that, as re-
heating effects. A large number of measurements was cently shown [24], can be explained as being due to the
then necessary to achieve a relatively small number of strong electron-boson coupling. The excess conductance
successful measurements. All the results reported here, athighvoltage,extendinguptoabout100mV(seeinset)
exceptthoseshowninfig.3,arethusreferredtothesmall is also typical of these systems [24]. The temperature at
fractionof contacts that do not show heating effects and which the Andreev-reflection features disappear and the
gave a clear Andreev-reflection signal. In these cases, conductancebecomesequaltothenormal-stateoneisthe
the existence of many parallel nanojunctions can be in- localcriticaltemperatureofthecontact,orAndreevcrit-
voked to reconcile the actual contact resistance with the icaltemperature TcA. As shownin fig.2 this temperature
requirement of ballistic transport [21]. In some cases, is easy to identify in spectroscopic contacts because it
the Sharvin condition was fulfilled at low temperature also marks the point where conductance curves recorded
but broke down on increasing the temperature because atslightlydifferenttemperaturestarttobesuperimposed
of the decrease in the mean free path. In these cases, to one another (here the curves at 21.0, 22.0 and 23.0 K
the values of the gaps at low temperature can be taken coincide within the experimental noise).
as meaningful anyway, though their temperature depen- In contrast, Figure 3 reports two examples of conduc-
dence and eventually the value of the local critical tem- tance curves that show, together with an Andreev sig-
perature can be slightly affected by the non-ideality of nal, deep and wide dips that, at low temperature, occur
the contact. at energies comparable to those of the large gap. As
shown elsewhere [21, 25] these dips are likely to be due
to the current becoming overcritical in the region of the
RESULTS AND DISCUSSION contact and prevent a proper determination of the gap
amplitudes. On increasing temperature, they move to-
Point-contact Andreev-reflection results ward lower voltage (due to the decrease of the critical
current) causing an apparent shrinkage of the Andreev
Figure 2 reports the temperature dependence of the signal, finally giving rise to a sharp cusp at zero bias.
rawconductance curves(obtainedby numericaldifferen- GoingbacktothecaseofballisticcontactsasinFig.2,
tiationoftheI−V characteristics)ofoneofthecontacts the conductanceat orjust aboveTA canbe usedto nor-
c
that did not show any anomaly. The curves were mea- malize all the curves at T <TA. In principle, a conduc-
c
suredinthex=0.10sample,andthenormal-stateresis- tance curve recorded at a given temperature should be
tanceofthecontactisaround50Ω. Thecurvesshowthe normalizedto the normal-stateconductance at the same
4

1.2 D =3.0 meV D =4.0 meV
1 1
D =17 meV D =17 meV
(cid:3)(cid:7)(cid:9)(cid:6) (cid:12)(cid:24)(cid:14) (cid:10) (cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:11)(cid:10)(cid:10)(cid:9)(cid:5)(cid:2)(cid:10)(cid:6)(cid:12)(cid:15)(cid:15)(cid:13)(cid:15)(cid:6)(cid:3)(cid:3)(cid:14) (cid:12)#(cid:14) (cid:10) (cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:10)(cid:11)(cid:10)(cid:10)(cid:9)(cid:5)(cid:8)(cid:10)(cid:16)(cid:12)(cid:15)(cid:15)(cid:13)(cid:15)(cid:6)(cid:2)(cid:9)(cid:14) (cid:3)(cid:7)(cid:9)(cid:9) 1.1 w21=0.1 xR=N=0 8.0 W5 2 w1=0.1
(cid:10)(cid:9)(cid:16)(cid:15)(cid:9) (cid:10)(cid:6)(cid:3)(cid:15)(cid:9) (cid:3)(cid:7)(cid:9)(cid:3) 1.0
(cid:1)(cid:9)(cid:1)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:19)(cid:22)(cid:25)(cid:10)(cid:12)(cid:14)(cid:3)(cid:3)(cid:3)(cid:3)(cid:7)(cid:7)(cid:7)(cid:7)(cid:3)(cid:3)(cid:9)(cid:9)(cid:2)(cid:8)(cid:3)(cid:9) (cid:30)(cid:31) ! (cid:10)(cid:10)(cid:10)(cid:10)(cid:9)(cid:6)(cid:6)(cid:6)(cid:2)(cid:3)(cid:6)(cid:5)(cid:15)(cid:15)(cid:15)(cid:15)(cid:3)(cid:3)(cid:9)(cid:9) (cid:10) (cid:30)(cid:31) ! (cid:10)(cid:10)(cid:6)(cid:6)(cid:9)(cid:6)(cid:15)(cid:15)(cid:3)(cid:9) (cid:3)(cid:3)(cid:3)(cid:3)(cid:7)(cid:7)(cid:7)(cid:7)(cid:3)(cid:3)(cid:3)(cid:3)(cid:4)"(cid:2)(cid:8) zed conductance 111111......012312 DDwDDw121211======314100.2.1.4.36.6. 0 5mm mmeeVeeVVV RxRx=N=N==0042. .4211 5W0W wwDDDD221111======11330041....9386..8055 mmmmeeeeVVVV
(cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3) (cid:1)(cid:5)(cid:3) (cid:1)(cid:6)(cid:3) (cid:3) (cid:6)(cid:3) (cid:5)(cid:3) (cid:4)(cid:3) (cid:2)(cid:3) (cid:1)(cid:2)(cid:3) (cid:1)(cid:4)(cid:3) (cid:1)(cid:5)(cid:3) (cid:1)(cid:6)(cid:3) (cid:3) (cid:6)(cid:3) (cid:5)(cid:3) (cid:4)(cid:3) (cid:2)(cid:3) mali 11..03 D =3.65 meV D =3.5 meV
(cid:26)(cid:18)(cid:27)(cid:23)(cid:24)(cid:28)(cid:25)(cid:10)(cid:12)(cid:29)(cid:26)(cid:14) (cid:26)(cid:18)(cid:27)(cid:23)(cid:24)(cid:28)(cid:25)(cid:10)(cid:12)(cid:29)(cid:26)(cid:14) Nor 1.2 D12=8.65 meV D12=8.5 meV
w=0.5 w=0.4
1.1 1 R=20 W 1
N
x=0 .30
1.0
FIG. 3: Temperature dependence of two raw conductance 1.2
D = 4.0 meV
curvesincase ofnonballistic contactsinthex=0.1sample. D1= 13.0 meV
1.1 2
Dip features characteristic of non-ideal conduction through w1= 0.5 R=37 W
N
the contact are clearly visible; the conductance minima are 1.0 x=0.50
indicated byarrows. -40 -20 0 20 40
Voltage (mV)
FIG. 4: Some examples of experimental conductance curves,
temperature, but because of the very high upper critical
after normalization (symbols), of point contacts on samples
field of these materials, the latter is not usually acces-
with different Ru content. The curves were all measured at
sible, at least at low T. Using the normal state at TA
c 4.2K.Solidblue(red)linesrepresentthebestfitoftheright
to normalize all the curves is thus a somehow arbitrary (left)sideoftheexperimentalcurvesobtainedwithinthetwo-
choice but, as shown elsewhere [24], is anyway the one band 2D generalized BTK model [24, 26]. The asymmetry
that preserves the weaker structures, i.e. those due to decreases on increasing the Ru vanishing completely at x =
the large gap and, if present, those due to the strong 0.50. The fittingparameters are indicated in the labels.
electron-boson coupling.
Figure 4 shows some examples of low-temperature,
normalized conductance curves in samples with differ- effectmeasurementsperformedinthesesamplesshowin-
ent Ru content. Some important points are immediately deed a considerable decrease of the Seebeck coefficient
clear by looking at these curves. First, none of them with increasing Ru content [28].
displays zero-bias peaks, and the same happens in 100% To extract quantitative information about the ampli-
of the spectroscopic contacts. This points towards the tudeofthegapsfromtheconductancecurves,theymust
absence of line nodes even at the highest Ru contents, be compared with suitable theoretical models. None of
contrary to what has been observed in some other FeBS the models for single-band superconductivity can repro-
away from optimal doping [7–10] [44]. Second, all the duce the shape of the experimental curves of fig.4. In-
curves show more or less marked double-gap features. stead,atwo-bandBlonder-Tinkham-Klapwijkmodel[29]
Third, despite the very large range of doping, the width generalizedto the 2D case [26] and including a broaden-
of the structures does not change very much (note that ing term [30] is the minimal model that can be used in
all the panels have the same horizontal scale). Thus we thiscase. Foreachbandtheparametersofthemodelare
shouldnotexpectmajorvariationsinthegapvaluesupon the energy gap ∆, the broadening parameter Γ and the
Ru doping. Fourth, the asymmetry of the normalized barrier parameter Z. Then, being the total conductance
conductance curves for positive/negative bias – which is the weighed sum of the single-band conductances, the
particularlystronginunsubstituted SmFeAs(O F ) last parameter is the weight w of band 1 (the weight of
0.85 0.15 1
[23] – seems to be reduced by Ru doping. As a matter band2beingconsequentlydeterminedas1−w ).[45]Itis
1
of fact, it is clearly visible even at a first glance in the true that Sm-1111 is not two-dimensionaland thus a 3D
case x = 0.05, becomes discernible only while trying to model should be used; however,as shown elsewhere [24],
fit the data in the cases x = 0.10 and x = 0.25 but al- the latter is much more complicated and for any prac-
most completely disappears for x = 0.30 and x = 0.50. tical purpose one can safely use the 2D one (especially
The real origin of this asymmetry, which is common to when, as it is the case here, the gaps are nodeless). The
mostpoint-contactspectra inFe-basedsuperconductors, lines in Fig.4 represent the best fit of the experimental
is not completely clear yet, though it has been recently data, and the labels indicate the relevant values of the
ascribed to the Seebeck effect [27]. Preliminary Seebeck gaps and of the weight of band 1 in the point-contact
5
conductance,w1. Toaccountfortheresidualasymmetry 15
ofthenormalizedcurves,wechosetofitthepositive-and 1.4 (a) meV)10 D 2
ntievgealyti)v.eA-bsiapsresvidioeusselypasrtaatteeldy,(fobrluxe≥an0d.3r0edthleinaessy,mremspetercy- ctance 11..23 x=0.10 Energy ( 5 D 1
is very small and the difference between the two fits can ondu 1.1 00 5Te1m0pera1t5ure 2(K0) 25
c
be no longer appreciated. d 1.0
e
Figure 5 shows two examples of how the normalized aliz 0.9
m
conductancecurvesevolvewithtemperature,andthere- or 0.8
N
sulting temperature dependence of the gaps extracted 0.7
from the fit. The two cases shown refer to a lightly -40 -20 0 20 40
doped sample (x = 0.10 ) and to a heavily doped one Voltage (mV)
15
sa(xhnod=ullde0se.sr5s0d)ri.seclaeTtrehndeibltleoowitnehsett-htleeamrgopetherreagrtaucpur,erwvcehusircv(hvesebrsethiccooamwllyeclloeeffassr- ce 11..23 (b) Energy (meV)105 DD 21
n x=0.50
set for clarity). In the x = 0.50 case, a dip structure cta 1.1 0 3 6 9 12 15
u Temperature (K)
is also seen to shift to lower energy on increasing the d
n 1.0
o
temperature, possibly giving rise to the small downward c
d 0.9
deviation of the temperature dependence of ∆2 from a alize 0.8
BCS-like ∆(T) curve observed at high temperature. Al- m
or 0.7
though there is no reason to expect the gaps to follow N
a BCS-like curve, the effects of the dip do not allow us 0.6
-40 -20 0 20 40
to discuss whether this deviation is intrinsic or is an ar- Voltage (mV)
tifact due to the small mean free path of the samples.
Incidentally, on the basis of recent calculations within a FIG. 5: Temperature dependence of the normalized conduc-
minimal three-band Eliashberg model [31] one would in- tancecurves(symbols)ofpointcontactsonthesampleswith
stead expect the gaps to be greater than the BCS value x=0.10 (a) and x=0.50 (b) with the relevant two-band fit
(lines). All the curves but the top ones are vertically shifted
in proximity of the critical temperature.
for clarity. The insets show the temperature dependence of
Let us just recall here that the fitting procedure is
the gaps ∆1 and ∆2 as extracted from the fit. The error
generally not univocal, i.e. different sets of parameters
bars indicate the spread of gap values obtained by different
cangivealmostequallygoodfits. Errorbarsintheinsets fits,whentheotherparameters(Γi,Zi andtheweightw1 are
to Fig.5 indicate the spread of gap values resulting from changed as well).
different fits.
Figure 6(a) shows the behavior of the gaps as a func-
tion of the Ru doping x. The data are rather scattered also found in the starting compound SmFeAs(O1−xFx)
but a general trend is anyway discernible. While the [23] (here represented by the vertical error bar on the
small gap ∆ does not vary sensibly on increasing the pointatthemaximumTA)andwasascribedtotheresid-
1 c
Ru content x, the large gap ∆ shows a rapid decrease ual degrees of freedom in the normalization process, to
2
from x=0 to x=0.10 and then remains approximately the asymmetry of the curves and to the fact that the
constant. This behavior is in rough qualitative agree- features related to ∆2 are less sharp than those related
ment with that of the bulk Tc reported in table I. Since to the small gap ∆1. However, in the high-doping range
PCARS is a local probe, the scattering of gap values at (x = 0.3 and x = 0.5), TA depends very little on the
c
the same composition is most probably due to slight in- Ru content and the asymmetry has almost completely
homogeneities in the localdoping content. As long as T disappeared; even large differences in local composition
c
has a strong dependence on the doping content, i.e. up correspond to a small difference in TA. Therefore, the
c
to x=0.25, different point contacts on the same sample spread of gap values accompanied by a small spread in
can thus provide different values of the gaps and of the TA seems to indicate a lack of correlationbetween these
c
local TcA, i.e. the Andreev critical temperature. As a quantities (as observed also in MgB2 with Al and Li co-
matter of fact if one plots the gaps as a function of TA doping [32]).
c
as in fig.6(b), a roughly linear trend of both ∆ and ∆ Since the gaps show an overall linear trend as a func-
1 2
can be appreciated despite the fluctuations in their val- tion of the local TA, it is particularly instructive to plot
c
ues. It is worth reminding that the data reported here the gap ratios 2∆ /k T as a function of TA. This is
i B c c
are already the results of a very careful selection aimed doneinfigure7,whichalsoreportsvariousotherPCARS
at eliminating all the questionable results, so that these data in 1111 and 122 compounds. Only results show-
fluctuationsarenotduetospuriouseffectsthatcanbeas- ing nodeless orderparameters areshown for consistency.
cribed to non-ballistic conduction, heating, or spreading It is clear that the Ru substitution in the optimally F-
resistance. As for the large gap, a large uncertainty was doped Sm-1111 allows spanning a wide range of criti-
6

other different nodeless FeBS of the 1111 and 122 fami-
20 (a) Sm(Fe Ru)As(O F )
1-x x 0.85 0.15 lies.
V)
me 15 Theseresultsappeartobeincontrasttowhatreported
s ( in ref. [33], where an opposite trend is suggested. How-
ap 10 ever, even in the aforementioned paper, several results
g
y reported for FeBS show 2∆/k T ratios which seem to
nerg 5 increasewithdecreasingTc,parBticcularlyforthelargegap.
E
0 Adefinitiveansweronthepossibilityofauniversaltrend
0.0 0.1 0.2 0.3 0.4 0.5
of 2∆ /k T vs. T requires more experimental work,
Ru con tent (x) i B c c
comparing results obtained with different techniques on
20 (b) samples of increasingly better quality.
V)
me 15
aps ( 10 Electronic structure calculations
g
y
erg 5 In order to try to explain the observed PCARS data
n
E withintheEliashbergtheory,wepreliminarilyperformed
0
10 20 30 40 50 electronic structure calculations. In particular, Sm-
Andreev critical temperature (TA) FeAsO electrons and holes density of states (DOS) have
c
been obtained by ab initio calculations performed in
FIG.6: (a)Energygapsasextractedfromthetwo-bandfitof thelocaldensityapproximationtothedensity-functional
theconductancecurves,asafunctionoftheRucontentx. (b) theory (LDA-DFT) [34] as implemented in the all-
EnergygapsasafunctionofthelocalcriticaltemperatureTA
c electron full-potential APW and local orbitals [35, 36]
ofthevariouspointcontacts. Linesaregapscalculatedwithin
code Wien2k [37]. APW has the advantage to treat ex-
thethree-band,s± Eliashberg theory (see text for details).
plicitly Sm 4f electrons within the valence band yield-
ing state-of-the-art band structure dispersion quality.
D DDDP1C , A, D R DDD ,2 ou Lra wFoerAks(O,F) D DDDP1C , A D R DDD,2 other works TfoormsiemduinlataeteRturagsuobnsatlitsuutpieorn-coeullrcocanltcauinlaintigonfosuwreforermpuelra-
15 ,, SSmmFFeeAAss((OO,,FF)) crys. ,, MYaiyteask aewt aa le, tT ablF, eNAdsF(eOA,FsO) units (Pma2 No. 28) where 25% Ru concentration was
,, SSmmFFeeA(Ass(O,P,)FO) 10% ,, SSazambuoe elyt eatl, a(lB, aN,dKF)FeeAsA(sO, Fcr)ys. achieved by the substitution of one Ru out of four Fe,
, Ba(Fe,Co)As 2 2
, Sm(Fe,Ru)2As(2O,F) YCahteens eett aall,, SNmdFFeeAAssO(O,F) retaining the bulk symmetry in the defected cell. The
Tc10 LNua iedty aulk, (eBt aa,lK, S)FmeF2AesA2 sc(rOys,F.) film conservation of symmetry allowed a reliable compari-
kB Naidyuk et al, LaFeAs(O,F) film sonofdoped andundoped band structureswithout fold-
D2/ ARP,ES Ding e t al, (Ba,K)Fe2As2 ing/unfoldingmappingproblems. Muffin tinradiiof2.3,
5 , Terashima et al, Ba(Fe,Co)2As2 1.9, 2.2 and 2.0 Bohr were used for Sm, O, Fe and As,
respectively. Brillouin zone integration was performed
BCS
with tetrahedrons on a 6×6×4 mesh [38]. Since we are
0 interestedinthe superconductingparamagneticphaseof
10 20 30 40 50 60
SmFeAsO, Fe spin polarizationwas not considered. The
Critical Temperature (K)
relevant band structure of this system is given by two
electronandthreeholebandsasitisusualfor1111iron-
FIG. 7: Gap ratios 2∆1/kBTc and 2∆2/kBTc as a function pnictides. Thosearesuperimposedtothe4fSmbandsat
ofTc asdeterminedbypoint-contactAndreev-reflectionspec-
the Fermi energy (E ). To disentangle the contribution
troscopyinSm(Fe1−xRux)AsO0.85F0.15 (blueandredcircles) F
and in various other 1111 and 122 materials. Solid lines are of the Fe 3d bands we used the so-called fat-band rep-
2∆i/kBTc ratioscalculatedwithinthethree-band,s±Eliash- resentation by projecting the wavefunction onto the Fe
berg theory (see text for details). Experimental data from atomic orbitals and thus obtaining the band structures
literature are taken from ref. [24] and references therein. shownbydotsinFig.8. Thesizeofdotsisproportionalto
the d atomic character of the wavefunctionon Fe atoms.
Such a procedure allows to identify a number of
caltemperatures,whichnotonlycoversbutalsoextends parabolic Fe 3d bands around E . Since comparison
F
the range of T values of superconducting Fe-basedcom- withLaFeAsOisrelevant,inFig.8(a)wereporttheband
c
pounds measured so far by PCARS. As already shown structure obtained for the same crystal structure of the
in ref. [24], the 2∆ /k T ratios start to increase be- SmFeAsOcompoundwhereSmhasbeenreplacedbyLa.
i B c
low T ∼30 K. Surprisingly, PCARS data on this single In Fig.8(a) we clearly see the usual set of five parabolic
c
sample series,namelySmFe1−xRuxAsO0.85F0.15,feature Fe d bands, two electrons (e1, e2) and three holes ones
basicallythesamebehaviorasthoseobtainedfrommany (h , h andh ), allofthemcenteredatthe Γ pointsince
1 2 3
7
FIG.8: Color online. SmRuxFe1−xAsOfat-bandstructurein thefourformulaunitsuper-cell. DotsareproportionaltotheFe
3dcharacterofthewavefunctions. ParabolasshowtheresultsofafitoftheFe3dbandswithinthetwo-dimensionalelectrongas
model. Upper and lower horizontal dashed lines are EF with and without F doping, respectively. a) Reference band structure
with Lareplacing Sm. b)Actual SmOFeAsband structureincludingthe4f Sm bands. Thetwo light holes bandshavesimilar
dispersions and have been fitted with a single parabola h1. c) Same as in b) but in the case of SmFe0.75Ru0.25AsO. In b) and
c) the parabolic fit of band e3 is shown only up to an energy close to the one where its hybridization with Sm 4f bands takes
place.
we are using a four formula unit super-cell. This find- calculated effective masses by the free two-dimensional
ingisinagreementwithpreviousLDA-DFT calculations electron gas model N(E) = m∗/π~2. This approxima-
provided we consider that SmFeAsO structure was used tion is evenmore justified by the fact that in the Eliash-
here [3]. As for the band structure of the actual Sm- berg analysis reported below only ratios between DOSs
FeAsO, we find Fe d bands superimposed to Sm 4f ones. (whicharemuchmoreaccurate)enterinthecalculations.
This fact not only makes the interpretation of the band Then, given this assumption, only the band curvature is
structure lesseasybut introducesanhybridizationeffect relevantand thereforeenergy shifts with respect to LDA
between Sm and Fe states. LDA bands in Fig.8(b) show arenotimportantinthe model. The onlyimportantdif-
that Fe electron bands in SmFeAsO are no more simply ference relies on the fact that the number of parabolic
parabolicbuthybridizationintroduceswarpingoutofthe bands taken into consideration is five or four. In this
central part of the Brillouin zone. Hybridization splits regard, the position of E is critical since band e is
F 1
the electron band labeled e in Fig.8(a) in two pieces just above E in the undoped compound. The super-
2 F
named e and e in Fig.8(b). In undoped SmFeAsO e conducting phase is obtained by 15% F doping i.e. in
3 1 3
ande donotcrossE beingtheformertoolowinenergy SmFeAsO F . F substitution adds electrons to the
1 F 0.85 0.15
andhybridizedwithSm4fbands(seedetailsofFig.8(b)) system and in the rigid band approach such effect can
and the upper too high. Therefore, undoped SmFeAsO be coped by the rigid shift of E by 0.15 electrons per
F
will have only one Fe-derivedelectron band e and three formula unit. Given the DOS N(E) of the system, the
2
hole ones, the doubly-degenerate light-hole h and the shift can be simply estimated as 0.15/N(E ). Anyway,
1 F
heavy-hole h . The Fermi surface will be made of four careshouldbetakenindefiningN(E),sincethelocalized
2
nearly cylindrical sheets, one less than LaFeAsO. Such nature of Sm 4f orbitals makes it likely that they do not
a finding seems to be in agreement with ARPES mea- receive the additional doping charge from F. We there-
surements on SmFeAsO where only one electron band is forefilteredoutthis contributionfromN(E)considering
suggested [39]. Since in the Eliashberg approach we are only the contribution of bands e and h . Following
1,2 1,2
interestedintheFedstatesDOS,weneedtodisentangle this approach we get a sort of upper bound for the E
F
that contribution out of the total one that includes the shifttobeabout75meV(90meV)forSmFeAsO F
0.85 0.15
Sm 4f bands. Our choice is to model the band structure (SmFe Ru AsO F ). TheFermilevelsE ofF-
0.75 0.25 0.85 0.15 F
as a superposition of five parabolic bands (including the doped and undoped SmFeAsO and SmFe Ru AsO
0.75 0.25
empty e ) inside the background of 4f states. With the are shown in Fig.8(a)-(c). We see that in the F-doped
1
help of the fat-band representation, we fitted the rele- systems the band e is always partially filled justifying
1
vant bands along the Γ-X direction (Γ-M in the usual our assumption to include its contribution in Eliashberg
two-formulaunitcell)with parabolasshownassolidand calculations. AsfortheeffectofRusubstitution,bycom-
dashed lines in Fig.8. Disregarding the possible warp- paring Fig.8(b) and (c) we see that the effect is modest,
ing of the cylindrical Fermi surfaces, the DOS deriving onlybande isabitdeeperandwithlowereffectivemass
2
fromthe abovementionedbandswasestimatedfromthe as shown in Fig.8(c). Calculated DOS and plasma fre-
8
x=0 meff DOS (st/Ha/Bohr2) ωp (meV) ab-initio
e1 0.739 0.2352 714.04 1.0 tanh[1.76 k sqrt(Tc*/T -1)]
T=52 K, k=0.6192
e2 1.508 0.4800 975.56 c
0.8
h1 0.908 0.2890 1391.91 )
0
h2 2.057 0.6547 844.86 T= 0.6 1.0
x=hee1210.25 000...968568782 000...322018420879 11741823286...104979 rr(T) / (ss00..24 W(T) / W(T=0) 0000....024680 0.2 0k.4 0.6 0.8 1.0 TTTTTccccc=====3542102000KKKKK
h2 1.736 0.5526 864.74 0.0 T / Tc*
0.0 0.2 0.4 0.6 0.8 1.0
TABLEII:Calculated effectivemasses, DOSandplasmafre- T / T*
c
quenciesforSmFe1−xRuxAsO0.85F0.15 at dopinglevelsx=0
and x=0.25. h1 is doubly degenerate.
FIG. 9: Temperature-dependent part of the superfluid den-
sity, ρs(T)/ρs(T = 0) (symbols) calculated by using the
plasmafrequencies obtainedfrom firstprinciplescalculations
quenciesforeachbandandforthetwodopinglevelscon-
sidered are reported in table II. (seetableII)inSmFe1−xRuxAsO0.85F0.15forx=0(Tc=52K).
The line is a fit with the formula tanh(1.76kp(T∗/T −1)),
c
giving k =0.6192. Inset: temperature-dependent part of the
superfluid density (and therefore of the representative boson
Analysis of experimental results within Eliashberg frequency[40])usedintheEliashbergcalculationsatdifferent
theory critical temperatures (see text for details).
Basedonthe resultsofbandstructurecalculationsde-
scribed so far, it is possible to propose an explanation [40],thiscurvewasfittedwithtanh(1.76kp(Tc∗/T −1)),
ofthe experimentaldatabymeansofthe simplestmodel giving k = 0.6192. The values of the calculated super-
thatallowsdescribingtheessentialphysicsofthemateri- fluid density and the relevant fit are shown in Fig. 9 as
als under study. We used the three-band, s± Eliashberg symbols and line, respectively.
theory [31] taking into account the feedback effect [40]. At this point, the only parameters that remain to be
Within this model we have two hole bands (from now determined are two: λ31, λ32. At x = 0 they are ob-
on labeled as 1 and 2) and one equivalent electron band tained by reproducing the two experimental gaps, but
(labeled as 3). The free parameters are N (0), λ , λ , it turns out that they also reproduce the exact experi-
i 31 32
Ω(T) and Γ. N (0) is the DOS at Fermi level, calculated mental critical temperature. Then we assume that the
i
above for x = 0 and x = 0.25 and obtained at all the ratio λ31/λ32 is doping-independent and that k is a lin-
otherdopinglevelsbylinearinterpolationasafunctionof ear function of the experimental critical temperature:
the experimental Tc; λ31 and λ32 are the electron-boson k(Tc) = k(Tc,x=0)Tc/Tc,x=0. This assumption is related
coupling constants between band 3 and band 1 or 2, re- to the temperature dependence of the superfluid density
spectively; Ω(T) is the representative boson energy that andis reasonableto suppose that with increasingx (and
we take as Ω(T) = Ω0tanh(1.76kp(Tc∗/T −1)) where thereforedecreasingofTc)itdecreases[8]. Now,theonly
Ω0 = 2Tc/5 [41]; the electron-boson spectral function freeparameterleftinthe processoffitting the Tc depen-
hasa Lorentzianshape[31] withhalfwidth Γ=Ω0/2[6]; dence of the gaps is λ31 which is fixed by obtaining Tc
T∗ is the feedback critical temperature, determined by coincident with the experimental one. The procedure is
c
∗
solving the Eliashberg equations in the imaginary-axis self-consistent since λ31 is varied until Tc, introduced in
formulation. The electron-bosoncoupling matrix is the formulaforΩ(T),allowsreproducingthe experimen-
tal T . Disorder effects have been neglected as impuri-
c
 0 0 λ31ν1  ties are dominant in the intraband channel and are thus
0 0 λ32ν1 not pair-breaking. Moreover,we also assumed that, as a
 
λ31 λ32 0  firstapproximation,theyarealsoabsentintheinterband
channelsince the two gapsare welldistinct atall doping
where ν = N (0)/N (0) and ν = N (0)/N (0). k was levels.
1 3 1 2 3 2
determined at x = 0 in the following way: first, the Results are shown in Fig. 6(b) as solid lines: the cal-
superfluid density was calculated by using the plasma culated gap values follow rather well the trend of the
frequencies obtained from first principles and reported experimental ones at all temperatures. The same good
in table II, giving also the correct T . Then, since agreementwiththe experimentcanbe seenalsobylook-
c
thetemperature-dependentpartofthesuperfluiddensity ing at the 2∆ /k T ratios as a function of T shown in
i B c c
correspondstothatoftherepresentativebosonfrequency Fig. 7 which also reports many other results from the
9
literature. In this regard, it is also remarkable that it induced disorder mainly enhances intraband scattering
is possible, with a relatively simple model and a small and does not significantly affect the interband one. The
number of free parameters, to reproduce the increase of gap ratios 2∆ /k T strongly increase for T < 30K in
i B c c
the ratio with decreasing T . Since the strength of the a manner which suggests an unexpected increase of the
c
coupling increases with decreasing T , the same does, as electron-boson coupling when T is depressed. Very in-
c c
expected,thetotalelectron-bosoncouplingconstant,λ terestingly, the trend of the gap ratios as a function of
tot
whichisabout3.2atT =52Kandgoesuptoalmost7.3 T inthissinglesystemissuperimposedtotheanalogous
c c
when T =10 K. Another interesting result that comes trend obtained by plotting the data of many Fe-based
c
out from the theoretical analysis is the temperature de- compounds of different families [24]. Needless to say,
pendence of the representative boson frequency (shown this seems to point towards a general, universal prop-
in the inset to Fig. 9 for different critical temperatures) ertyofthisclassofsuperconductors. Byusingthevalues
which, as already stated above, is equivalent to that of of the density of states and of the plasma frequencies
the superfluid density [40]. We can notice that, as T calculated from first principles, we have shown that the
c
decreases,thesuperfluiddensityalsodecreasesasafunc- increase in the gap ratios 2∆ /k T on decreasing T
i B c c
tionofT assuming,belowT =30K,a positivecurvature can be reconciled with a spin-fluctuation-mediated pair-
c
at intermediate temperatures. Similar dependencies (at ingeventhoughthecharacteristicspin-fluctuationenergy
least in the low- and mid-T range) have been obtained has been observed to decrease linearly with T . The key
c c
in penetration depth measurements in Co-doped Ba-122 to solving this puzzle is the feedback effect, i.e. the ef-
samples, as reported in ref. [8]. In that case the effect fect of the condensate on the mediating boson, which is
is less pronounced probably because the results are re- of course only expected when the superconducting pair-
ported only down to about 2T /5 while in our case T ing between electrons is mediated by electronic excita-
c c
drops to T /5 at the highest Ru doping. Moreover, it is tions [40]. An analysis carried out within an effective
c
also interesting to notice that in Co-doped Ba-122 the three-band Eliashberg model shows indeed that the ex-
temperature dependence of the superfluid density looks perimentaldependenceofthegaps(andofthegapratios
slightlymoredepressedinsamples thatbelongto the re- 2∆ /k T ) on T can be explained as being due, in par-
i B c c
gionof coexistence ofsuperconductivity and magnetism. ticular, to a change in the shape of the temperature de-
Thisfactleadsustospeculatethattheobservedbehavior pendence ofthe characteristicbosonenergy,Ω (T),with
0
ofthe superfluiddensity inoursamplesmightbe consid- respect to the optimal-T compound (with no Ru). In-
c
erably influenced by the onset of a short-rangemagnetic deed, a suppression of Ω in the mid-temperature range
0
order which competes with superconductivity and that (which becomes more and more sensible on decreasing
hasbeenobservedbyµSRand75As NQRmeasurements T ) is requiredto obtain the correctcritical temperature
c
in the same samples [19]. Penetration depth measure- and the correct gap values. This finding is in very good
ments, as the ones in ref. [8], would help clarifying this qualitativeagreementwith the experimentalobservation
point as well as the experimental determination of the ofadepressionofthesuperfluiddensityinCo-dopedBa-
temperature dependence of the representativeboson fre- 122 with reduced T [8]. The fact that in the latter case
c
quency, as done in refs. [6, 24, 42]. this reduction is observed in underdoped samples that
fall in the regionof coexistence of magnetismand super-
conductivity further suggests that, also in our samples,
CONCLUSIONS the depressionofthe condensate(that in turns givesrise
toadepressioninthebosonenergy)atfinitetemperature
The isoelectronic substitution of Fe with Ru in opti- may be considerably influenced by the onset of a short-
mally F-doped Sm-1111 is a good way to explore a very range magnetic order competing with superconductivity
wide range of critical temperatures within the same Fe- inducedbyRusubstitution,asrecentlyobservedbyµSR
basedcompound,inprinciplewithoutchangingthe total and75As NQRmeasurementsin the sameset ofsamples
charge of the system [15]. The considerable decrease of [19].
T induced by Ru substitution has been ascribed to dis-
c
order in the Fe sub-lattice [15] and/or to the onset of
a short-range magnetic order [19]. Here we have shown
that, in a wide range of T (from 52 K down to 13.5 K,
c
corresponding to Ru contents ranging from 0 to 50 %)
the Sm(Fe1−xRux)AsO0.85F0.15 system retains its origi- ACKNOWLEDGEMENTS
nal multi-gap character, and also the symmetry of the
gaps remains nodeless. The amplitudes of the two ex-
perimentally detectable gaps, ∆ and ∆ , decrease al- This work was done within the PRIN project No.
1 2
mostlinearlywithT ,buttheyremainwelldistinctdown 2008XWLWF9-005. FB acknowledges support from
c
to the lowest T . This suggests that the substitution- CASPUR under the Standard HPC Grant 2012.
c
10
Gonnelli, Rep.Prog. Phys.74, 124509 (2011).
[25] G.Sheet,S.Mukhopadhyay,andP.Raychaudhuri,Phys.
Rev. B 69, 134507 (2004).
[1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono,
[26] S. Kashiwaya, Y. Tanaka, M. Koyanagi, and K. Ka-
J. Am. Chem. Soc. 130, 3296 (2008).
jimura, Phys. Rev.B 53, 2667 (1996).
[2] I.I.MazinandJ.Schmalian,PhysicaC469,614(2009).
[27] Y. G. Naidyuk, O. E. Kvitnitskaya, I. K. Yanson,
[3] I.I.Mazin,M.D.Johannes,L.Boeri,K.Koepernik,and
G. Fuchs, S. Haindl, M. Kidszun, L. Schultz, and
D.J. Singh,Phys. Rev.B 78, 085104 (2008).
B.Holzapfel,Supercond.Sci.Technol.24,065010(2010).
[4] I.I.Mazin, D.J. Singh,M. D.Johannes, andM.H.Du,
[28] M. Puttiet al., in preparation.
Phys.Rev.Lett. 101, 057003 (2008).
[29] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys.
[5] C.-T.Chen,C.C.Tsuei,M.B.Ketchen,Z.-A.Ren,and
Rev. B 25, 4515 (1982).
Z. X.Zhao, Nature Phys.6, 260 (2010).
[30] A. Plecenik, M. Grajcar, v. Benˇaˇcka, P. Seidel, and
[6] D. S. Inosov, J. T. Park, P. Bourges, D. L. Sun,
A. Pfuch, Phys.Rev. B 49, 10016 (1994).
Y. Sidis, A. Schneidewind, K. Hradil, D. Haug, C. T.
[31] G. A. Ummarino, M. Tortello, D. Daghero, and R. S.
Lin, B. Keimer, et al., NaturePhys. 6, 178 (2010).
Gonnelli, Phys.Rev.B 80, 172503 (2009).
[7] J.-P. Reid, M. A. Tanatar, X. G. Luo, H. Shakeripour,
[32] G. A. Daghero, D. Ummarino, M. Tortello, D. Delaude,
N. Doiron-Leyraud, N. Ni, S. L. Budko, P. C. Canfield,
R.S.Gonnelli,V.A.Stepanov,M.Monni,andA.Palen-
R. Prozorov, and L. Taillefer, Phys. rev. B 82, 064501
zona, Supercond.Sci. Technol. 22, 025012 (2009).
(2010).
[33] D. S. Inosov, J. T. Park, A. Charnukha, Y. Li, A. V.
[8] R. Prozorov and V. G. Kogan, Rep. Prog. Phys. 74,
Boris, B. Keimer, and V. Hinkov, Phys. Rev. B 83,
124505 (2011).
214520 (2011).
[9] J.-P. Reid, M. A. Tanatar, X. G. Luo, H. Shak-
[34] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244
eripour,S.Ren´edeCotret,N.Doiron-Leyraud,J.Chang,
(1992).
B.Shen,H.-H.Wen,H.Kim, et al. (2011), unpublished,
[35] D. Singh, Phys.Rev.B 43, 6388 (1991).
arXiv:1105.2232.
[36] E. Sjo¨stedt, L. Nordstr¨om, and D. J. Singh, Solid State
[10] P.J.Hirschfeld,M.M.Korshunov,andI.I.Mazin,Rep.
Commun. 114, 15 (2000).
Prog. Phys.74, 124508 (2011).
[37] B. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka,
[11] K. Kuroki, H. Usui, S. Onari, R. Arita, and H. Aoki,
andJ.Luitz,WIEN2k: An Augmented Plane Wave Plus
Phys.Rev.B 79, 224511 (2009).
Local Orbitals Program for Calculating Crystal Proper-
[12] S.C. Lee,A.Kawabata, T. Moyoshi, Y.Kobayashi, and
ties, Techn. Universitt Wien, Wien, Austria (2001), see
M. Sato, J. Phys.Soc. Jpn. 78, 043703 (2009).
http://www.wien2k.at.
[13] A.S.Sefat,A.Huq,M.A.McGuire,R.Jin,B.C.Sales,
[38] P.E.Bl¨ochl, O.Jepsen,andO.K.Andersen,Phys.Rev.
D. Mandrus, L. M. D. Cranswick, P. W. Stephens, and
B 49, 16223 (1994).
K.H. Stone, Phys.Rev.B 78, 104505 (2008).
[39] L. Yang, B. Xie, B. Zhou, Y. Zhang, Q. Ge, F. Wu,
[14] H.-H. Wen, G. Mu, L. Fang, H. Yang, and X. Zhu, Eu-
X. Wang, X. Chen, and D. Feng, J. Phys. Chem. Solids
rophys.Lett. 82, 17009 (2008).
72, 460 (2011).
[15] M.Tropeano,M.R.Cimberle,C.Ferdeghini,G.Lamura,
[40] G. A.Ummarino, Phys. Rev.B 83, 092508 (2011).
A. Martinelli, A. Palenzona, I. Pallecchi, A. Sala,
[41] J.PaglioneandR.L.Greene,NaturePhys.6,645(2010).
I.Sheikin,F.Bernardini,etal.,Phys.Rev.B81,184504
[42] M. Tortello, D. Daghero, G. A. Ummarino, V. A.
(2010).
Stepanov, J. Jiang, J. D. Weiss, E. E. Hellstrom, and
[16] N. D. Zhigadlo, S. Katrych, M. Bendele, P. J. W. Moll,
R. S.Gonnelli, Phys.Rev.Lett. 105, 237002 (2010).
M.Tortello,S.Weyeneth,V.Y.Pomjakushin,J.Kanter,
[43] A.A.Golubov,A.Brinkman,Y.Tanaka,I.I.Mazin,and
R.Puzniak,Z.Bukowski,etal.,Phys.Rev.B84,134526
O. V. Dolgov, Phys. Rev.Lett. 103, 077003 (2009).
(2011).
[44] Moreover, none of the curves we measured showed the
[17] M.A.McGuire,D.J.Singh,A.S.Sefat,B.C.Sales,and
finite-energy peaks associated with quasiparticle inter-
D.Mandrus, J. Solid StateChem. 182, 2326 (2009). ferencepredicted,insomeconditions,inthenodelesss±
[18] S.C.Lee,E.Satomi,Y.Kobayashi,andM.Sato,J.Phys.
symmetry [43]. The occurrence and the voltage position
Soc. Jpn. 79, 023702 (2010).
of these peaks is controlled, in the relevant theory, by a
[19] S. Sanna, P. Carretta, P. Bonf`a, G. Prando, G. Allodi,
mixingparameterαwhichhasnotbeenrelatedyettoex-
R. De Renzi, T. Shiroka, G. Lamura, A. Martinelli, and
perimentalparameters.Therefore,eitherthesepeaksare
M. Putti, Phys.Rev. Lett.107, 227003 (2011).
not present because the conditions for their observation
[20] R. S. Gonnelli, D. Daghero, M. Tortello, G. A. Um-
are notfulfilled, ortheyaresmeared out, particularly at
marino, V. A. Stepanov, J. S. Kim, and R. K. Kremer,
the Z values typical of our contacts, by the broadening
Phys.Rev.B 79, 184526 (2009).
effects.
[21] D. Daghero and R. Gonnelli, Supercond. Sci. Technol.
[45] TheBTKmodelprovidesreliableresultsalsointhecase
23, 043001 (2010).
of parallel nano-junctions. Simulations show that the fit
[22] Y.V.Sharvin,Zh.Eksp.Teor.Fiz.48,984(1965),engl.
ofthetotalconductanceinthiscasegives“effective”pa-
Transl. Sov. Phys.-JETP 21, 655 (1965).
rameters. The “effective” gap turns out to be approxi-
[23] D.Daghero,M.Tortello,R.S.Gonnelli,V.A.Stepanov,
mately equaltotheaverageofthegapsoftheindividual
N. D. Zhigadlo, and J. Karpinski, Phys. Rev. B 80,
contacts.
060502(R) (2009).
[24] D. Daghero, M. Tortello, G. Ummarino, and R. S.