Lifting of nodes by disorder in extended-$s$ state superconductors: application to ferropnictides

0.4 MB·English

by  V. Mishra

Lifting of nodes by disorder in extended-s state superconductors: application to
ferropnictides
V. Mishra1, G. Boyd1 S. Graser1,2, T. Maier3, P.J. Hirschfeld1, and D.J. Scalapino 4
1University of Florida, Gainesville, FL 32611, USA
2 Center for Electronic Correlations and Magnetism, Institute of Physics,
University of Augsburg, D-86135 Augsburg, Germany
3 Center for Nanophase Materials Sciences and Computer Science and Mathematics Division,
Oak Ridge National Laboratory, Oak Ridge, TN 37831-6494
9 4 Department of Physics, University of California, Santa Barbara, CA 93106-9530 USA
0
(Dated: January 17, 2009)
0
2 We show, using a simple model, how ordinary disorder can gap an extended-s (A1g) symme-
try superconducting state with nodes. The concommitant crossover of thermodynamic properties,
n
particularly the T-dependence of the superfluid density, from pure power law behavior to an acti-
a
J vatedone isexhibited. Wediscuss applications of thisscenario toexperimentson theferropnictide
superconductors.
7
1
PACSnumbers:

]
n
o I. INTRODUCTION ontheelectron-likeFermisurface(“β sheets”),andnoted
c that the next leading channel had dx2−y2 (B1g) symme-
- try. Wang et al.25 studied the pairing problem using the
r When a new class of unconventional superconduc-
p functional renormalization group approach within a 5-
tors is discovered, it is traditional to try to deter-
u orbital framework, also finding that the leading pairing
mine the symmetry class of the order parameter, as
s instability is in the s-wave channel, and that the next
. this may provide clues as to the nature of the pair-
at ing mechanism. Direct measurements of the super- leading channel has dx2−y2 symmetry. For their inter-
m conducting order parameter are not possible, and indi- action parameters, they found however that there were
no nodes on the Fermi surface, but there is a significant
- rect phase-sensitive measurements are sometimes diffi-
d cultbecausetheytypicallyinvolvehigh-qualitysurfaces1. variationinthe magnitudeofthegap. Otherapproaches
n have obtained A gaps which change sign between the
On the other hand, a set of relatively straightforward 1g
o hole and electron Fermi surface sheets but remain ap-
experimental tests are available to determine the ex-
c proximately isotropic on each sheet26,27.
[ istence of low-energy quasiparticle excitations and, in
some cases, their distribution in momentum space. As Wealsorecentlypresentedcalculationsofthespinand
1
in earlier experiences with other candidate unconven- charge fluctuation pairing interaction within a 5-orbital
v
tional systems like the high-T cuprates and heavy RPA framework28, using the DFT bandstructure of Cao
3 c
5 fermion materials, the symmetry class of the newly et al.29 as a starting point. Our results indicated that
6 discovered ferropnictide superconductors2 is in dispute the leading pairing channels were indeed of s (A1g) and
2 at this writing, due in part to differing results on dx2−y2 symmetry, and that one or the other could be
1. superfluid density3,4,5,6,7,8,9, angle-resolved photoemis- the leading eigenvalue, depending on details of interac-
0 sion(ARPES)10,11,12,13,14,15,nuclearmagneticresonance tionparameters. Wealsogaveargumentsastowhythese
9 (NMR)16,17,18,19, Andreev spectroscopy20,21,22,23, and channelsweresonearlydegenerate,andpointedoutsome
0 other probes. In some cases, results have been taken significant differences in the states compared to those
v: to indicate the absence of low-energy excitations, i.e. a found by Kuroki et al.24. Finally, we noted that, within
i fullydevelopedspectralgap. Inothers,low-energypower our treatment and for the interaction parameter space
X
lawshavebeentakenasindicationofthe existenceofor- explored,nodeswerefoundinallstates,generallyonthe
r der parameter nodes. It is not yet clear whether these α sheets for the d-wave case and the β sheets for the s-
a
differences depend on the stoichiometry or doping of the wavecase,butthatinthelattercasetheexcursionofthe
materials, or possibly on sample quality. order parameter of sign opposite to the average sign on
the sheet was small, and might be lifted by disorder28.
In parallel, microscopic theoretical calculations of the
pairinginteractionintheferropnictidematerialshaveat- It is the purpose of this paper to explore the pos-
temptedtopredictthemomentumdependenceoftheor- sibility that the nodes of an extended-s state of the
derparameterassociatedwiththe leadingsuperconduct- type discussed in Ref. 28 are lifted by disorder, and
ing instability. Using a 5-orbitalparameterizationof the consider the consequences for experimental observables.
density functional theory (DFT) bandstructure, Kuroki In the interest of simplicity, we initially neglect many
et al.24 performed an RPA calculation of the spin and complicating aspects of the problem, in particular the
orbitalcontributionstotheinteractiontoconstructalin- multisheet nature of the Fermi surface, and focus pri-
earized gap equation. They found that the leading pair- marily on the sheet found in each case (s or d) which
ing instability had s-wave (A ) symmetry, with nodes has nodes. In this case the problem is similar to one
1g
2
which was studied earlier in the context of potential
extended-s states in a single-band situation for cuprate
superconductors30,31,32. For the pnictides, current inter-
estcentersaroundtheisotropicsign-changingextended-s
Π Β
state proposedby Mazinetal.26, onewhere a gapis mo-
mentumindependentovertwoindependentFermisurface
sheets,buthasa differentsignoneach. Inthis case,it is
knownthatonlyinterbandscatteringispairbreaking,be-
Π(cid:144)2
causeitmixesthetwosignsandsuppressestheoverallor-
der parameter32,33,34,35. This effect has been claimed to
accountforlow-energyexcitationsobservedinNMR37,38
and superfluid density39 experiments. In general, impu- 0 Α Β
rities with screenedCoulombpotentialwill havea larger
intraband component, which is however essentially irrel-
evant for the isotropic case in the sense of Anderson’s
theorem. However, intraband scattering by disorder will
averageanyanisotropyofthe orderparameterpresentin 0 Π(cid:144)2 Π
the conventional s-wave case40. In the case of the pnic-
tides, this intraband component will do the same, with
the effect of lifting the weak nodes found in our micro- FIG. 1: (Color online) Fermi surface and extended-s gap
scopic calculations. We demonstrate this effect, and its eigenfunction from Ref. 28, Fig. 15(d). Red/blue color indi-
consequences,withinasimplemodelwherewetakeweak cates sign of order parameter.
pointlike scatterers, treated in the Born approximation,
as a model for the intraband part of the true disorder
scattering in the pnictides. We believe that such scat-
scattering in the superconducting state is
tering arises primarily from out-of plane dopants like K
in the hole-doped and F in the electron-doped materi-
als; since these dopants sit away from the FeAs plane, G(k,ω)= ω˜τ0+ǫ˜kτ3+∆˜kτ1, (2)
they will certainly produce a significant small-q compo- ω˜2−ǫ˜2 −∆˜2
k k
nentof the scatteringpotential, whichwill mix states on
the same sheet. Our results suggest that an extended-s
wave state with nodes, lifted by disorder in the case of whereω˜ ≡ω−Σ0,ǫ˜k ≡ǫk+Σ3,∆˜k ≡∆k+Σ1,andthe
Σ are the components of the self-energy proportional
the doped superconducting ferropnictides, may explain α
to the Paulimatrices τ in particle-hole (Nambu) space.
the apparentdiscrepanciesamongvarious measurements α
If we assume weak scattering, we may approximate the
in the superconducting states of these materials.
self-energy in the Born approximation as
Σ=n |U(kk′)|2τ G0(k′,ω)τ , (3)
I 3 3
II. MODEL
k′
X
We begin by assuming a superconductor with a sepa- where n is the concentration of impurities. The self-
I
rable pair interaction V(k,k′) = V1Φ1(φ)Φ1(φ′), where energy has Nambu components
φis anangleparameterizingthe electronicmomentumk
on a single circular Fermi surface. To model a situation
ω˜
corresponding to an extended-s state on the β sheet of Σ0(k,ω) = nI |U(k,k′)|2 ω˜2−ǫ˜2 −∆˜2 , (4)
the Fe-pnictide materials,asfound inRef. 28, we choose k′ k′ k′
X
Σ (k,ω) = n |U(k,k′)|2 ǫ˜k′ , (5)
Φ1(φ)=1+rcos2φ, (1) 3 I k′ ω˜2−ǫ˜2k′ −∆˜2k′
X
and
wpaitrhamrete≡r ∆Vk′/V=1 ∆>∼0Φ11,(φen)suwriilnlghatvheatnothdeespnueraeroφrde=r Σ1(k,ω) = −nI |U(k,k′)|2 ω˜2−ǫ˜∆˜2k′−∆˜2 . (6)
π/4,3π/4,.... Note that this function is also suitable k′ k′ k′
X
for modelling the d-wave state on the α Fermi surface
sheets, in the limit r ≫1. Because the effect of disorder Asdiscussedabove,weassumepointlikeimpurityscat-
will renormalize the constant and cos2φ parts of the or- tering U(k,k′)=U , and treat the disorder in Born ap-
0
der parameter differently, we write it in the general case proximation. We further assume particle-hole symmetry
as ∆k =∆0+∆′cos2φ. such that Σ3 = 0, leading to Nambu self-energy compo-
The full matrix Green’s function in the presence of nentsafterintegrationperpendiculartotheFermisurface
3
1
0.9 1
ω˜ r=1.1
Σ0(φ,ω)=Γ (7) 0.8 0.8 r=1.3
* ω˜2−∆˜2k′+φ′ 0.7 00..46 AG
q
Σ1(φ,ω)=−Γ ∆˜k′ , (8) Tc000..56 0.20
* ω˜2−∆˜2k′+φ′ T/c0.4 0 1 2 3
q
wherehi indicatesaveragingoverthecircularFermisur- 0.3
φ
face, and Γ = πnIN0U02 is the normal state scattering 0.2
rate, with N the density of states at the Fermi level.
0 0.1
0
0 10 20 30 40 50 60
Γ/T
III. RESULTS c0
FIG.2: (Coloronline)Suppressionofthecriticaltemperature
The BCS gap equation
Tc/Tc0 vs. normal state scattering rate Γ/Tc0 for anisotropy
1 parameter r = 1.1(dashed),1.3(solid). The inset shows the
∆k = 2TrT Vk,k′τ1G(k′,ωn) (9) two curves at smaller values of the scattering rate. The Tc
Xωn Xk′ suppressioncurveexpectedfrom Abrikosov-Gor’kovtheory41
then reduces to (dotted) is included for comparison.
′ ′ ′ ′ 2.5
∆ = N [V ∆ I +(V ∆ +V ∆)I +V ∆I ]
0 0 1 0 1 0 1 2 3
′ ′ ′ ′ ′ ′ ′
∆ = N [V ∆ I +(V ∆ +rV ∆ )I +rV ∆I ]
0 0 1 0 2 3
2
(10)
with 01.5
Tc
(cos2φ)m−1 φ)/
I = πT , (11) (G
m Ω 1
Xωn * ω˜n2 +∆˜2k′+φ′
q
and where ω = (2n+1)πT is a fermionic Matsubara 0.5
n
frequency.
0
0 90 180 270 360
φ (in degrees)
A. Tc suppression
FIG. 3: (Color online) Normalized spectral gap ΩG(φ)/Tc0
Near Tc I1 = L , I2 = 0 and I3 = L/2 , where L = vs. angleφontheFermisurfaceforanextendeds-wavestate
log 2eγωc , leading to a critical temperature of withr=1.3andΓ/Tc0=0(dashed),0.3(dotted),1.0(solid),
πTc and 3.1 (dashed-dotted).
h i
V
1
T =1.13ω exp − (12)
c c (cid:20) N0(V12+V′2/2)(cid:21) B. Density of states and spectral gap
in the clean limit. In the dirty case, T is suppressed by
c
ordinary disorder until the gap anisotropy is completely In the symmetry broken state, the order parame-
washedout,atvaluesofthenormalstatescatteringrates ter ∆k is renormalized by the off-diagonal self-energy
ΓmanytimeslargerthanTc,asalsofoundbyMarkowitz Σ1(k,ω), but the true spectral gap is determined by
andKadanoff40. InFig. 2,weplotthis behaviorbysolv- bothNambucomponentsΣ1 andΣ0. Onemaydefinean
ing (10) with Φ1 given by Eq. (1). The marginal case angle-dependentspectralgapΩG(φ)byexaminingthe1-
r =1 describes the situation where the nodes just touch particlespectralfunctionA(k,ω)≡−(1/π)ImG11(k,ω),
the Fermi surface without disorder, which has been dis- and plotting either the peak or the energy between the
cussedearlier30. Notethatwhilethescaleoftheplotim- peak and the Fermi level where A falls to one-half its
plies that the critical temperature is rapidly suppressed, peak value. We adopt the latter definition here since it
the scattering rate scale over which this occurs is actu- is similar to one traditionally used in the ARPES com-
allymuchgreaterthanonewouldexpectin,e.g. ad-wave munity, but other measures (e.g. plotting the peak in
superconductor, where a critical concentration Γ ∼ T A(k,ω) or simply ∆˜(ω = 0,φ)) produce very similar re-
c c0
suffices to suppress superconductivity completely. sults. InFig. 3, weshowthe s-wavespectralgapplotted
4
2.5 1
Γ = 0 Γ = 0
Γ = 0.3 T 0.9 Γ = 0.3 T
c0 c0
2 Γ = 1.0 Tc0 0.8 Γ = 1.0 Tc0
Γ = 3.1 T Γ = 3.1 T
c0 0.7 c0
N01.5 =0)0.6
ωN()/ 1 ρρ/ss0(T00..45
0.3
0.5
0.2
0.1
0
0 0.5 1 1.5 2ω/T2.5 3 3.5 4 4.5 00 0.2 0.4 0.6 0.8 1
c0 T/T
c0
FIG.4: (Coloronline)NormalizeddensityofstatesN(ω)/N0 ∗
vs. energy ω for the same parameters and line types as Fig. FIG. 5: (Color online) Superfluid density ρs/m vs. T/Tc0
forr=1.3andsamedisorderparametersandlinetypesasin
3.
Figs. 3-4.
overtheFermisurfaceforvariousvaluesofthescattering
Becausethismaterialisstoichiometric,crystalshavevery
rate Γ. It is clear that the effect of scattering is as de-
longmeanfreepathsofmorethanathousandAngstr¨om
scribed above: the order parameter is averaged over the
and are capable of supporting dHVA oscillations49. It
Fermi surface, which eventually has the effect of lifting
seems very unlikely that anything but order parameter
thenodes,andleadingtoatruegapinthesystem,which
becomes isotropic at sufficiently large scattering rates. nodescanleadtosuchapowerlaw. Thepures+/− state
we consider here with weaknodes willimmediately yield
This can be observed as well in the total density of
ρ (0) − ρ (T) ∼ T, so we take the parameters of the
states, which we exhibit in Fig. 4. In the clean situa- s s
previous section and calculate ρ directly.
tion, there are two coherence-type peaks, corresponding s
Within the current BCS-type model, the xx compo-
to the large and small antinodal order parameter scales
nent of the superfluid density tensor ρ may be written
observedinFig. 3. Theadditionofdisordersmearsthese s
peaksinitiallyandsuppressesthe maximumgapfeature.
This is consistent with the expected behavior of a dirty
ω ∆˜2
nodal superconductor. This behavior continues until the ρ /ρ= cos2φ dω tanh Re k ,
point when the nodes are actually lifted, and the system s,xx * Z 2T (ω˜2−∆˜2k)3/2+φ
acquires a true gap with sharp coherence peaks. When (13)
the gap has become completely isotropic due to disorder where ρ is the full electron density. Note this stan-
averaging, the DOS is identical to the usual BCS DOS, dard expression for the superfluid density of a dirty
as required by Anderson’s theorem. superconductor42 is complete within mean field theory
since vertex corrections to the current response vanish
forsingletsuperconductorsandpointlikescatterers43. In
C. Superfluid density Fig. 5, we show the superfluid density ρ ≡ (ρ +
s s,xx
ρ )/2 calculated from Eq. (13) vs. temperature for
s,yy
The temperature dependence of the superfluid den- various scattering rates44. In the pure system, the su-
sity ρ (T)reflects the distribution of quasiparticlestates perfluid density is linear at the lowest temperatures, re-
s
which contribute to the normal fluid fraction. Early flecting the existence of line nodes. At a temperature
data on powdered samples of R-FeAsOxF1−x (R =Pr3, corresponding to the smaller antinodal gap45, there is a
Sm4,Nd5)nearoptimaldopingshowedexponentialT de- decreaseintherateatwhichthermallyexcitedquasipar-
pendence, asdidmeasurementsonBaxK1−xFe2As2 (Ba- ticles depopulate the condensate,and thereforea change
122)6. More recent experiments on the latter system in slope as visible in the figure. We note that this kink-
doped with Co found a power law temperature depen- likebehaviormaydisappearwhenexcitationsfromother
denceclosetoT2whichevolvedtowardsexponentialwith Fermi surface sheets which contribute at higher temper-
increasing Co concentration8. A T2 dependence is char- atures are included, but also that this type of upward
acteristic of a dirty system with linear nodes46, but the curvature in ρ vs. T was observedby Fletcher et al.9 in
s
authors of Ref. 8 concluded the T2 was more likely due the cleanest LaFePO samples.
tononlocalelectrodynamics47,orpairbreakingduetoin- If disorder is now added without lifting the nodes, the
homogeneity or inelastic scattering. Very recently, how- generic behavior will be quadratic in temperature46, i.e.
ever,alinearT dependencewasmeasuredintheoriginal ρ ≃ ρ (0)(1−aT2/Γ2), where a is a constant of order
s s
ferropnictide superconductor LaFePO,48 with T = 6K. unity. When the nodes are completely lifted, an expo-
c
5
100 1.05 Γ=0.0 T
c0
Tc0 Γ=0.3 Tc0
10−1 αφ()/G 1 Γ=0.6 Tc0
Ω
0)
T= 10−2 0.950 50 100 150 200 250 300 350
ρ(s φ(in degrees)
ρ1−/s10−3 Γ = 0 Tc0 3
Γ = 0.3 Tc0 φ)/ 2
10−4 Γ = 1.0 Tc0 βΩ(G
Γ = 3.1 T 1
c0
10−2 10−1 100 0
T/Tc 0 50 100 φ1(5in0 deg2re0e0s) 250 300 350
FIG. 6: (Color online) log10(1−ρs(T)/ρs(0)) vs. log10T/Tc FIG.7: (Color online)SpectralgapsΩG(φ)/Tc0 vs. φforthe
for r = 1.3 and various scattering rates (Circles: Γ = 0, α (top) and β (bottom) sheets for the 2-band model. Here
triangles: Γ/Tc0 = 0.3, crosses: Γ/Tc0 = 1., and squares: the pairing parameters V11 = 1.0,V12 = −0.6,V22 =1.5 were
Γ/Tc0 =3.1 . Solid line: T; dashed-dottedline: T2. chosen so that reff = 1.3 is the same as in the 1-band case.
ThedensitiesofstatesratiowastakenasN2(0)/N1(0)=1.25.
Scattering rates for intraband scattering correspond to Γ ≡
nential T dependence, ρ ≃ ρ (0)(1−aexp(−∆ /T)) niπN1(0)|U11|2 =0(dashed),0.3(dashed-dotted),0.6(solid).
s s min
mustdominate atthe lowesttemperatures. These power
laws are displayed more precisely in the log-log plot
system, which is not observed. We have also examined
of Fig. 6. The pure system follows a linear-T law
(ρ (0)−ρ (T)∼T)downto the lowesttemperatures,as the above model with smaller values of r >∼ 1, and find
s s
that substantialscattering rates are still required to cre-
expected because of the linear nodes, whereas the dirty
systems follow a T2 law over an intermediate T range, ate a significant spectral gap.
It is possible that our restriction to Born (weak) scat-
also expected because the states are uniformly broad-
teringisinadequate. Unitaryscatterersmodifyprimarily
ened in this range by Born scattering with 1/τ roughly
the states near the nodes, creating a gap without affect-
linear in energy. At lower temperatures, however, the
ing the states at higher energies; smaller normal state
dirty cases no longer exhibit power laws in temperature,
scattering rates can therefore produce a large node lift-
with the curves showing fully developed gaps in Fig. 4
ing effect without suppressing T significantly50. It may
following activated behavior. The Γ/Tc0 = 0.3 case is c
be interesting to explore this effect further, but a priori
marginal in the sense that the effective spectral gap has
thereisnoobviousreasonforthe out-of-planeimpurities
nodes which just touch the Fermi surface, and the figure
which appear to dominate the doped pnictide materials
showsthatwhenthe systemisclosetothisconditionthe
to produce such strong scattering potentials.
low-T behaviorisnota simplepowerlaworexponential.
A final possibility, which we do pursue here, is that in
theferropnictides,thesingleFermisurfacesheetwehave
studied thus far is coupled to other sheets which con-
IV. EFFECT OF ADDITIONAL FERMI trol the T suppression. This appears plausible within
c
SURFACE SHEETS
the context of our microscopic spin-fluctuation pair-
ing calculations28, which show that for extended-s type
While the desired effect of node lifting by disorder states, while the β sheets appear to have weak nodes,
for an extended-s wave state with weak nodes has now the α sheets are more isotropic for the extended-s wave
been exhibited, the amount of disorder required to lift solutions. The pairing on the α and β sheets is strongly
the nodes has also been shown to suppress T substan- coupled by the π,0 (unfolded zone) scattering processes
c
tially. As strong sample-to-sample variations of T have which dominate the spin-fluctuation spectrum. To sup-
c
not been observed in the ferropnictides, this seems in- press the critical temperature, it is clear that not only
consistent with the current overallbody of experimental theβ condensate,butalsotheαcondensatemustbesup-
data. There are several possibilities to explain this dis- pressed. The αcondensatemay thereforebe expectedto
crepancy. Thefirstisthatthenodes,e.g. intheLaFePO act as a reservoir which maintains the critical tempera-
system,maybeaccidentallyextremelyweak. Thisseems ture (provided interband scattering by disorder is rela-
unlikely, since the linear-T behavior extends over a sig- tively weak), while the β condensate is nodal and there-
nificant fraction of T in this material. Were the nodes fore dominates low-temperature and low-energy proper-
c
to be marginal or extremely weak, the phase space from ties.
near-quadraticnodes would actually lead to a T1/2 tem- To test this hypothesis within the philosophy of the
perature variation of the penetration depth in the pure current paper, we take the above model and add to it a
6
single additional sheet with constant pairing amplitude, 2
Γ=0.0 T
andcouplethetwosheetsbyconstantpairingpotentials, c0
Γ=0.3 T
leading to a total pairing interaction. c0
Γ=0.6 T
1.5 c0
V(k,k′) = V1Φ1(k)Φ1(k′)+V2Φ2(k)Φ2(k′) Total
+V12[φ1(k)φ2(k′)+φ2(k)φ1(k′)], (14) ω)/N 1 0.3
N(
where φ (k) = 1 +rcos2φ is understood as before to 0.2
1
describe stateswith konthe β sheet, whereasφ (k)=1 0.5
2 0.1
for k on the α sheet. Note that V is chosen of oppo-
12
site sign to V ,V so as to induce a sign-changing order 0
1 2 0 0.1 0.2 0.3 0.4
parameter between the two sheets. 00 0.5 1 1.5 2 2.5 3
We now considerthe scattering fromnonmagnetic im- ω/Tc0
purities. As discussed by Refs. 33,34,35, it is convenient
to parameterize the scattering potential in terms of an FIG. 8: (Color online) Density of states N(ω)/N vs.
Total
amplitude U22 describing scattering k → k′ on the α ω/Tc0 for 2-band system with reff = 1.3. Same parame-
sheet, similarly U on the β sheet, and U between ters and line types as in Fig. 7.Insert: expanded low-energy
11 12,21
the two, such that the Born self-energy becomes region.
Σ(k∈1,ω) = n |U |2G(k′,ω) 0.5
I 11
"k′∈1 0.45
X
0.4
+ |U12|2G(k′,ω) , (15) 0.35
k′∈2 # Tc0 0.3
X )/G
Ω 0.25
and similarly for k ∈ 2. To compare the 1-band and n(
2-band cases, we now choose pairing parameters such mi 0.2
that the order parameter on the β sheet is nearly iden- 0.15
N/N=1.25
tical to that we had in the 1-band case above with 2 1
0.1 N/N=1.5
r ≡(∆ −∆ )/(∆ +∆ )=1.3. This is il- 2 1
eff max min max min 0.05 1 band
lustratedasthepurespectralgapΩ inFig. 7,wherethe
G
0
correspondingisotropicgapontheαsheetisalsoshown. 0 0.1 0.2 0.3 0.4 0.5 0.6
1−T/T
As disorder is added, the nodes are removed from the β c c0
sheet,asbefore,andafullgapcreated. Atthesametime
theαgapisonlyveryslightlysuppressedinthis process. FIG. 9: (Color online) Normalized spectral gap
These features are evident also in the total DOS shown minΩG(φ)/Tc0 vs. Tc suppression 1 − Tc/Tc0 for vari-
ous impurity concentrations comparing 1-band and 2-band
in Fig. 8 for the same scattering parameters.
models. Parameters are chosen such that r = 1.3 on
eff
Our aim in this section is to see if within a two-band
the 1 (β) sheet for the pure superconducting state in all
picture onecanunderstandqualitativelyunder whatcir-
cases. Diamonds: 1-band model with Tc/Tc0 and ΩG/Tc0
cumstances a substantial spectral gap can be opened on taken from Figs. 2 and 3, respectively. Squares: two-band
the Fermi surface without suppressing Tc substantially model with V11 = 1.0,V12 = −0.6,V22 =1.5, with densities
in the simple 1-bandexample. The naive hypothesis for- of states ratio N2(0)/N1(0) = 1.25 and 1 sheet anisotropy
mulatedabove,thatthe presenceofamoreisotropicgap parameter r = 1.76. Circles: two-band model with same
on a second Fermi surface sheet is sufficient to “protect” V11 = 1.0,V12 = −0.6,V22 =1.5, but with densities of states
T against intraband scattering, is not universally cor- ratio N2(0)/N1(0)=1.5 and r=1.88.
c
rect. This is because the order parameters on the two
sheets are strongly coupled by the interband pair inter-
action. To illustrate which aspects of the problem are (α),isincreased. Anyeffectwhichenhancesthenodeless
importantfor the relativeproportionof gapcreationrel- sheet2 pairing weightmakes Tc lesssusceptible to intra-
ative to T suppression, we compare in Fig. 9 the gap band scattering. Thus we conclude that the amount of
c
created by a certain amount of disorder versus the cor- Tc suppression associated with gap creation depends on
responding suppression of T , for the 1-band case and the details of the situation, and can become quite small.
c
two 2-band cases where the effective 1 sheet anisotropy ThetrueFermisurfacestructureisofcoursemorecom-
r is held fixed, but the ratio of the densities of states plicated than the 2-sheet model considered here, so it is
eff
on the two sheets are varied. It is seen that the critical interesting to ask whether results from the more com-
temperatureisrenderedmorerobustwhenthedensityof plete model show the desired effect. We show in Fig. 10
statesN (0),whichcontrolsthepairingweightonsheet2 calculations for the spectral gap in the extended s-wave
2
7
3 Α1 3 Α2 0.25
2.5 2.5
Tc0 2 Tc0 2 0 0.2
L(cid:144)k1.5 L(cid:144)k1.5 Tc
HWG 1 HWG 1 L(cid:144)WG0.15
0.5 0.5 Hmin 0.1
0 0
0 100 200 300 400 0 100 200 300 400
k k
0.05
3 Β1 3 Β2 0
2.5 2.5 0 0.02 0.04 0.06 0.08 0.1 0.12
Tc0 2 Tc0 2 1-Tc(cid:144)Tc0
L(cid:144)k1.5 L(cid:144)k1.5
HG HG
W 1 W 1 FIG. 11: (Color online) minΩG/Tc0 vs. 1−Tc/Tc0 for same
disorder parameters as Fig. 10.
0.5 0.5
0 0
0 100 200 300 400 0 100 200 300 400
k k ral” due to the proximity of these systems to a situation
wherenearlycircularFermisurfacesheets nestperfectly,
1 in which case spin fluctuation theory predicts a degener-
acy between the extended-s (A ) and d (B ) states.
0.8 1g 1g
Tc00.6 Herewehaveexploredanalternativepossibilitywhich
L(cid:144)k assumes that there are nodes in an extended-s-wave
HWG0.4 (A1g) state for the ideal ferropnictide material, but that
0.2
these nodes are lifted by smallmomentum disorderscat-
0 tering. In this regard, it is interesting that the stoichio-
60 80 100 120 140
k metric ferropnictide LaFePO (with a mean free path of
morethan102nmandcapableofsupportingdHvAoscil-
lations) displays a linear T dependence of the superfluid
FIG. 10: (Color online) Spectral gap vs. Fermi surface arc
density at low temperatures. While many mechanisms
lengthinarbitraryunitsoneachoffourFermisheets(α1,α2,
can give rise to a power law T dependence in the super-
β1 and β2) for realistic spin fluctuation model of Fig. 17 of
Ref. 28. Curves correspond to an arbitrary range of impu- fluid density, we are aware of only one explanation for a
rity concentrations, from red (clean) to blue (dirty). Bottom linear-T law, namely the existence of nodes in the pure
panel: detail of nodal region of β sheets. state. On this basis we speculate that the exponential
behaviors observed in other materials are due to disor-
der. On the other hand, evidence for the opposite trend
(A1g) state obtained from the microscopic calculations has been reported in the Ba1−xKxFe2As2 system, where
of Ref. 28 for various values of intraband disorder scat- the cleanestcrystalsappearto exhibit exponentialT de-
tering. The node lifting phenomenon on the β sheets is pendence,whereasmoredisorderedsamplesexhibitalow
clearlyobserved. InFig. 11,weshowthatrelativelylittle temperature dependence which mimics a power law.
Tc suppressionaccompaniesthisrealisticcase;significant We have therefore explored a scenario in which weak
gaps of order 0.1−0.2Tc are obtained for 1−Tc/Tc0 of nodes in a sign-changing s+/− state on the β sheets of
only ∼10%. the ferropnictides are “lifted” by nonmagnetic disorder.
Qualitative aspects of this phenomenon are obtainable
alreadywithinasimple 1-bandmodelwhereweakpoint-
V. CONCLUSIONS like scatterers are accounted for. With increasing dis-
order, the nodes disappear and are replaced by a fully
There is now evidence that the superfluid density has developedgapandanactivatedtemperaturedependence
a T dependence consistent with a fully developed gap in of ρ . Thus one could imagine that the entire class of
s
some samples, while power laws in T, including linear ferropnictide materials has intrinsic nodes in the ground
T, have been reported in others. It is possible that pa- state order parameter of the analog pure system, which
rameters related to the electronic structure of the pure then disappear in most of the doped, much dirtier ma-
state tune the various materials such that different su- terials. We have shown that in the simplest model the
perconducting ground states are realized. This is “natu- lifting of the nodes corresponds to a significantly larger
8
pairbreaking rate and concomitant T suppression than basedexplanationofthekindgivenhere,wherenodesor
c
observedexperimentally,butthatthisundesirableaspect quasi-nodesarelifted by momentumaveraging,seems to
of the theory is substantially eliminated by the addition us the most likely way to understand the existing data.
of additional bands with more isotropic pair state, as This picture will have implications for other properties,
found in the microscopic theory28. including transport properties, as discussed in Ref. 51
Thejustificationforourneglectofinterbandscattering for the 1-band situation, but there may be interesting
inmostofthis workisbasedonthe empiricalrobustness modifications peculiar to the multiband case. Work on
of T to differences in sample quality, suggesting that this direction is in progress.
c
interband scattering in the sign-changing s state is neg- We note in closing that an alternative approach takes
ligible. In addition, Yukawa-type models of realistic im- the point of view that the intrinsic material is fully
puritypotentialswithfiniterangehavesmalllarge-qam- gapped with a sign-changing s-wave state, and that the
plitudes. Nevertheless, for a quantitative description of inter-Fermisurface scattering leads to the appearance of
thesematerialsbothtypesofscatteringshouldbeconsid- low-temperaturepowerlaw behavior36,37,38. Clearlysys-
ered,andamoremicroscopicunderstandingofthe types tematic electron irradiation or some other source of dis-
ofpotentials introduced,e.g. by a K replacingBa,Srout order which does not change the doping of the system
of the FeAs plane and by a Co substituting directly for would be a very important experimental way to distin-
an Fe would be very useful. guish this proposal from our own.
The simple theory presented in Section II is actually
applicable to a state with arbitrary anisotropy parame-
ter r, allowing us to describe the pure d-wave state for
r ≫ 1, as well as a state with deep minima of the order
parameter (“quasi-nodes”), but no true nodes for r <∼ 1 Acknowledgments
as found, e.g. in Ref. 25. Thus we emphasize that it is
possible that small differences in the electronic structure The authors are grateful for useful communications
of different materials may give rise to slightly different with D.A. Bonn, J. Bobowski, and A. Carrington.
gap structures, and even different symmetry classes28 in Research was partially supported by DOE DE-FG02-
different materials. The implication of current ARPES 05ER46236 (PJH), and the Deutscheforschungsgemein-
experiments,thattheorderparameteranisotropyaround schaft (SG). TAM, DJS, and PJH acknowledgethe Cen-
the various Fermi surface sheets is absent or very weak ter for Nanophase Materials Science, which is sponsored
is, however,difficult to reconcile with experimental data at Oak Ridge National Laboratory by the Division of
which indicate low energy excitations, and a disorder- Scientific User Facilities, U.S. Department of Energy.
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