A possible method to confirm $\pm s$-wave pairing state using the Riedel anomaly in Fe-pnictide superconductors

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by  Daisuke Inotani

A possible method to confirm ±s-wave pairing state using the Riedel anomaly in
Fe-pnictide superconductors
Daisuke Inotani1 and Yoji Ohashi1,21
11Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
2CREST(JST), 4-1-8 Honcho, Saitama 332-0012, Japan
(Dated: January 13, 2009)
We theoretically propose a method to identify s-wave order parameter in recently discovered
9 Fe-pnictidesuperconductors. OurideausestheRie±delanomalyinac-Josephsoncurrentthroughan
0 SI( S)(single-bands-wavesuperconductor/insulator/ s-wavetwo-bandsuperconductor)junction.
0 We±showthattheRiedelpeakeffectleadstovanishinga±c-Josephsoncurrentatsomevaluesofbiased
2
voltage. Thisphenomenondoesnotoccurinthecasewhenthe s-wavesuperconductorisreplaced
±
n byaconventionals-waveone, sothattheobservation ofthisvanishingJosephson currentwould be
a a clear signature of s-wavepairing state in Fe-pnictidesuperconductors.
±
J
3 PACSnumbers: 74.20.Rp,74.50.+r,74.20.-z
1

] In the current stage of research on Fe-pnictide ±s-wave scenario[18, 26]. Since a model calculation in-
n superconductors[1, 2, 3, 4, 5], the symmetry of super- cludingapairinginteractionmediatedbyAFspinfluctu-
o
conducting order parameter is one of the most impor- ations supports ±s-wavesuperconductivity[17], confirm-
c
tant issues. Since the discovery of superconductivity in ing the ±s-wave order parameter in Fe-pnictides is also
-
r LaFeAsO F [1],greatexperimentalandtheoreticalef- crucialfor clarifying the pairing mechanismof these ma-
p 1 x x
forts have−clarified various key properties of these mate- terials.
u
s rials. FeAs-layers form a quasi-two dimensional electron In this paper, we theoretically propose a method
. system, consisting of hole and electron pockets around to confirm the ±s-wave order parameter in Fe-
t
a theΓ-andM-point,respectively[6,7,8,9,10,11,12,13]. pnictide superconductors. In identifying the pairing
m
An antiferromagnetic (AF) phase exists without car- symmetry of unconventional superconductivity, phase-
- rier doping[14], so that the possibility of pairing mech- sensitive experiments are very powerful. For ex-
d
anism associated with AF spin fluctuations has been ample, the so-called π-junction SQUID played cru-
n
o discussed[15,16,17,18]. ThedecreaseofKnightshift[19] cial roles to identify the dx2 y2-wave order parame-
c below the superconducting phase transition tempera- ter in high-Tc cuprates[27]. −Our idea uses the ac-
[ ture T indicates a singlet pairing state. A tunnel- Josephson current I through an SI(±S) (single-band
c J
ing experiment[20], as well as angle-resolved photoemis- s-wave superconductor/insulator/±s-wavesuperconduc-
1
v sionspectroscopy(ARPES)[11, 12], haveshownthat Fe- tor) junction shown in Fig.1. In this case, IJ consists
8 pnictidesaremultigapsuperconductors. TheARPESex- oftwo components associatedwith two bands in the ±s-
1 periment also reports that the order parameter in each wave superconductor. Because of the sign difference of
7 band may have a nodeless s-wave symmetry[11, 12]. two order parameters in the ±s-wave state, these two
1
Whilethisisconsistentwiththeexponentialtemperature current components are found to flow in the opposite
.
1 dependence of the penetration depth far below T [21], direction to each other. In addition, as in the case of
c
0 it seems contradicting with the T3-behavior of NMR- ordinary ac-Josephson current, each current component
09 T1−1[22, 23], implying the existence of nodes. shows the Riedel anomaly[28, 29], where the Josephson
current diverges at a certain value of biased voltage V
:
v As a candidate for the symmetry of order parameter across the junction. These two phenomena are shown
i inFe-pnictidesuperconductors,a±s-wavestatehasbeen to give vanishing total ac-Josephson current I at some
X J
recently proposed[15, 16, 17, 18]. In this pairing state, values of V. This vanishing I does not occur when the
J
r
a nodeless s-wave order parameters in electron and hole order parameters in the two-band superconductor have
bands have opposite sign to each other. This uncon- the same sign. Since the ARPES experiment reports a
ventional superconductivity has been shown to consis- nodelesss-waveorderparameterineachband[11,12],the
tently explain the observed superconducting properties observationof the vanishing ac-Josephsoncurrent would
mention above[11, 12, 19, 20], except for the power-law be a clear signature of ±s-wave state in Fe-pnictides.
behavior of NMR-T1−1[22, 23]. However, some theory To explain details of our idea, we explicitly calculate
groups have pointed out that the NMR result can be theac-Josephsoncurrentthroughthe SI(±S)-junctionin
also explained within the framework of ±s-wave super- Fig.1. The Hamiltonian is given by
conductivity, when one includes impurity scattering[24]
and/or anisotropic Fermi surfaces[25]. It has been also H =H +H +H , (1)
s s T
reportedthat the enhancementof inelastic neutronscat- ±
tering rate at a finite momentum transfer observed in whereH andH ,respectively,describethesingle-band
s s
superconducting Ba K Fe As is consistent with the s-wavesupercond±uctorontheleftofthejunctionand±s-
0.6 0.4 2 2
2
1.8
D
s-wave superconductor – s-wave superconductor =0)] 11..46 D he
s-band: D sIJ(V) heolelect rboann bda: n d : D Dhe D [ (Te h 001...1682
D , h 0.4
D 0.2
0
0 0.2 0.4 0.6 0.8 1
FIG.1: Model SI( S)-junctionwhich weconsiderin thispa-
± T/Tc
per. The left of the junction is a single-band (denoted by
s-band)s-wavesuperconductorwiththeorderparameter∆s.
The s-wavesuperconductorontherightofthejunctionhas FIG.2: Temperaturedependenceoftheorderparameter ∆h
twob±andsdenotedbyh-bandande-band,withtheorderpa- and ∆e obtained from the coupled equations (6) and|(7)|,
| |
rameter∆h and∆e,respectively. IJ istheJosephsoncurrent normalized by the value ∆h(T =0). We set Uhh = Uee =0,
through the junction. Uhe (Nh(0)+Ne(0)) = 1.0, and Nh(0)/Ne(0) = 0.4, where
| |
Nα(0) (α=e,h) is the density of states at the Fermi level in
the normal state of the α-band. ∆h and ∆e have opposite
wave superconductor on the right of the junction. Tun- sign to each other when Uhe > 0, while they have the same
neling effects are described by H . In the BCS approxi- sign when Uhe <0.
T
mation, H is given by
s
Hs = εspasp†σaspσ+ ∆sasp†as†p +h.c. . (2) U Th>e 0±asn-wdaUve st=atUe is=eas0i.lyInobtthaisinceads,e,wEheqns.o(n5e) saentds
Xp,σ Xp h ↑ − ↓ i (6h)egive the couepeled ehqhuations,
Here, as is the creation operator of an electron in the
sF-ebramnidepnw†σeirtghy.th∆esk=inUetsic enhaersgpyaεspsp,imisetahseuroerddefrropmaratmhe- ∆h =−UheXp 2 εep2∆+e |∆e|2 tanh21Tqεep2+|∆e|2,
− ↓ ↑
p q (7)
X
eter inthe s-band, whereU <0 is a pairinginteraction.
s
tdeumFctoiarvsHitay±ms(aiilnnthiEmoqua.glh(m1)bo,adwnedletscoiamldcpeulsylcaratiisbosenusm±[6es,-aw7,taw8v,eo-9sb,uap1n0edr,cs1oy5ns]--, ∆e =−UheXp 2 εhp2∆+h|∆h|2 tanh21Tqεhp2+|∆h|2.
as well as ARPES experiment[11], indicate the existence q (8)
of more than two bands). We also do not discuss the Theseequationshavesolutionsonlywhenthesignof∆h
originofthepairinginteractioninthispaper,butsimply is opposite to the sign of ∆e. We note that ∆h and ∆e
employ the following model Hamiltonian[17, 30] havethe same signwhenUhe <0. We alsonote that the
h-band and e-band have the same T , given by[30]
c
H = εαcα cα
±s p,σX,α=e,h p pσ† pσ Tc = 2γπωce−|Uhe|√N1e(0)Nh(0), (9)
+ α,pαX,p′=′,eq,hUαα′cαp+†q/2↑cα−†p+q/2↓cα−′p′+q/2↓cαp′′+q/2↑, wthheerBeCγS=th1e.o7r8y,.anNdαω(0c)isisththeeodrdeinnsaitryy ocuftsotffateenseargtythine
(3) Fermi level in the normal state of the α-band.
Figure2showsthecalculated∆ and∆ fromEqs. (7)
where cα is the creation operator of an electron in the h e
pσ† and (8). We will use these results in evaluating the ac-
α(= e,h)-band, with the kinetic energy εα, measured
p Josephson current. In this regard, we briefly note that,
from the Fermi level. U is an intraband interaction in
αα although we take U = U = 0 to realize ±s-wave su-
ee hh
the α-band. U (α 6= α) describes a pair tunneling
αα′ ′ perconductivityinasimplymanner,thefollowingdiscus-
between the e-band and h-band. (We take U = U .)
eh he sionsontheac-Josephsoneffectisnotaffectedbydetailed
In the mean-field approximation, Eq. (3) reduces to
values of U , as far as ±s-wave state is realized.
αα′
The tunneling Hamiltonian in Eq. (1) has the form
H s = εαpcαpσ†cαpσ+ ∆αcαp†cα†p +h.c. . (4) H =A+A , where
± ↑ − ↓ T †
pX,σ,α Xp,αh i
Here, the order parameters ∆h and ∆e are given by A= Tpα,kasp†σcαkσ ≡ Aα. (10)
p,kσ α
∆ =U hch ch i+U hce ce i, (5) αX=h,e X
h hh p p he p p
p − ↓ ↑ p − ↓ ↑ Here, Tα is the tunneling matrix element between the
X X p,k
s-bandandα-band,whichsatisfiesthetimereversalsym-
∆ =U hce ce i+U hch ch i. (6) metry, as Tpα,k =Tαp∗, k. Assuming a weak junction, we
e ee −p↓ p↑ he −p↓ p↑ calculatethetunne−ling−currentI ≡−ehN˙ i=ie(A−A )
p p s †
X X
3
within the lowest order in terms of Tα (where N = 6
p,σasp†σaspσ is the total number operpa,ktor of electrsons 5 T/Tcs=0.0 || JJ((hh == +-11Jh))||
on the left of the junction in Fig.1). Then, we find 4 Je
P
3
t
I(t)=e dt′h[A(t)−A†(t),A(t′)+A†(t′)]i0. (11) 2
Z−∞ 0 1
J
HofeHreT,t,haensdtaAt(istt)ic≡aleai(vHesr+aHge±sh)·t·A·ie0−isi(Htask+eHn±isn)tt.heabsence |J|/ 50 T/Tcs=0.9 || JJ((hh == +-11))||
Effects offinite voltageV acrossthe junctionis conve- Jh
4 Je
niently incorporated into Eq. (11) by replacing A(t) by
3
e ieVtA(t). Equation (11) involves both the Josephson
−
current I and quasi-particle current I . Extracting the 2
J q
formercomponent,wefindthatI consistsofthetunnel- 1
J
ingcurrentbetweenthes-andh-band,andthatbetween 0
0 0.5 1 1.5 2
s- and e-band, as
eV / [ |D s(T=0)| + | D h(T=0)| ]
IJ =−2e Im e−2ieVtΠα(ω =eV) , (12)
αX=h,e h i FIG. 3: Calculated ac-Josephson current |J|, normalized by
J0 √GhGe[∆h(T =0) + ∆e(T =0)]/e, as a function of
where bia≡sed voltage|V. (a) T |= 0|. (b) T/Tcs| = 0.9, where Tcs is
t Tc of the superconductor on the left of the junction in Fig.1.
Π (ω)=−i dteiωth[A (t),A (0)]i . (13) The Riedel anomaly can be seen at eV/(∆s + ∆h ) = 1.0
α α α 0 | | | |
and 1.48 in panel (a), and at 0.80 and 1.24 in panel (b). In
Z−∞ addition to these peaks, we also find weak singularities at
Equation (13) can be calculated from the analytic con- eV = ∆α ∆s, for example, eV/(∆s + ∆h ) 0.37 in
tinuation of the corresponding thermal Green’s function panel (|b).|W−e| ta|ke Th = Te , |for|sim|plici|ty.≃For the
h p,ki h p,ki
values of ∆h(T) and ∆e(T), the results in Fig.2 are used.
1/T Values of the interaction Us is chosen as to realize ∆h(T =
Π α(iνn)=− dτeiνnτhTτ{Aα(τ)Aα(0)}i0 0)/∆s(T =0)=1.5.
Z0
= −2T |Tα |2 Gs (p,iω )Gα (k,iω +iν ),
p,k 21 m 12 m n
theresultintoEq. (12),wefindthatthesine-component
Xp,k Xωm of ac-Josephson current (≡ I¯ ) can be written as I¯ =
(14) J J
Jsin(2eV +φ −φ ), where φ and φ are the phases
s h s h
where νn and ωm are the boson and fermion of the order parameter ∆s and ∆h, respectively. The
Matsubara frequencies, respectively, and A(τ) ≡ coefficient J has the form
eτ(Hs+H±s)Ae−τ(Hs+H±s). The off-diagonal Green’s J =J +ηJ , (17)
h e
function is given by
where J and J come from the tunneling current be-
h e
∆
Gλ (p,iω )=− λ (λ=s,α), (15) tween the s-band and e-band and that between s-band
12 m ωm2 +ελp2+|∆λ|2 and h-band, respectively. They are given by[31]
which satisfies Gλ21 =Gλ12∗. J = Gα|∆s||∆α| ∞ dztanh |z|
For simplicity, we approximate the tunneling ma- α e 2 2T
trix element Tpα,k to the value averaged over the θ(|∆s|−|zZ−−∞eV|)θ(|z|−|∆α|)
Fermi surface (≡ hTα i). Executing the momen- ×
p,k |∆ |2−(z−eV)2 z2−|∆ |2
tum summations in Eq. (14), we obtain Π (iν ) = (cid:20) s α
α n
2π2|hTpα,ki|2Ns(0)Nα(0)∆∗s∆αΛ(iνn), where + θp(|z|−|∆s|)θ(|∆α|−p|z+eV|) , (18)
z2−|∆s|2 |∆α|2−(z+eV)2#
1 1
Λ(iν )=T .
n Xωm ωm2 +|∆s|2 (ωm+νn)2+|∆α|2 vwohlevreesGusαef=ul4iπnpefo2rNmsa(0ti)oNnαa(pb0)o|uhTtpαt,hkei|2p.haInseEdqi.ffe(r1e7n),ceηbine--
(16)
Here, N (0) is thepdensity of staptes at the Fermi level in tween ∆h =|∆|iφh and ∆e =|∆e|eiφe, as
s
the normal state of the s-band. As usual, we evaluate
−1 (φ =φ +π),
the ω -summation in Eq. (16) by transforming it into η = e h (19)
m +1 (φ =φ ).
e h
the complex integration. Changing the integration path (cid:26)
so as to be able to carry out the analytic continuation When η =−1 (SI(± S)-junction), the phase difference
in terms of iν , we execute iν → ω+iδ. Substituting between ∆ and ∆ equals π. In this case, the current
n z h e
4
J flows in the opposite direction to J . This leads to again obtain the vanishing ac-Josephson current due to
e h
the suppression of the total Josephson current as J = the same mechanism discussed in this paper.
J −J . Incontrast,whenφ =φ inthecaseofη =+1,
h e e h Multiband superconductivity is affected by even non-
the Josephson current is simply given by the sum of two
magneticimpurities[33],sothatthe Riedelanomalymay
current components, as J =J +J .
h e be weakened by impurity effects. The suppression of
Figure 3(a) shows the magnitude of ac-Josephsoncur-
the Riedel anomaly is also expected when one includes
rent J at T =0, as a function of biased voltageV. Each
anisotropicFermisurfaces. WhentheRiedelpeakinJ is
e
component J and J has a peak at eV = |∆ |+|∆ |
h e s α broadenedandthepeakheightbecomessmallerthanthe
(α = e,h) (Riedel anomaly). The resulting total ac- valueofJ ateV =|∆ |+|∆ |inFig.3(a),thevanishing
h s e
Josephson current J(η = −1) = J −J vanishes when
h e ac-Josephsoncurrent is no longer obtained. In this case,
the voltage V satisfies Jh(eV) = Je(eV). (See the solid however, unless the Riedel peak becomes very broad, J
line in Fig.3(a).) The Riedel peaks at eV = |∆ |+|∆ |
s h would show a dip (peak) structure at eV = |∆ |+|∆ |
s e
and eV = |∆s|+|∆e| guarantee that this condition is when η = −1 (η = +1), which may be still useful
always satisfies at a voltage (≡V ) in the region[32],
0 to confirm ±s-wave superconductivity. Since any real
superconductor more or less has impurities, as well as
|∆ |+Min[|∆ |,|∆ |]<eV <|∆ |+Max[|∆ |,|∆ |].
s h e 0 s h e anisotropic band structure, it is an interesting problem
(20)
how our idea discussed in this paper is modified when
In contrast, when the phase difference between ∆ and
h more realistic situations are taken into account. We will
∆ is absent (η = +1), J does not vanish, but is always
e separately discuss this problem in our future paper.
finite. (SeethedashedlineinFig.3(a).) Thus,theobser-
Toconclude,wehavestudiedapossiblemethodtocon-
vation of the vanishing ac-Josephson current would be a
firmthe±s-wavepairingsymmetryinrecentlydiscovered
clear signature of ±s-wave state in Fe-pnictides.
Fe-pnictide superconductors. Using the Riedel anomaly
The vanishing Josephson current can be also seen at
and the fact that the ac-Josephson current through the
finite temperatures, as shown in Fig.3(b). On the other
SI(±S)-junction consists of two components flowing to-
hand, when |∆ | = |∆ | and J (eV) 6= J (eV) are acci-
h e h e
dentally satisfied, the Riedel peaks in J and J appear ward the opposite direction to each other, we obtained
h e
the vanishing ac-Josephson current at certain values of
atthesamevalueofV,sothatJ(η =−1)=J −J only
h e
has one Riedel peak at eV =|∆ |+|∆ |. In this special biasedvoltage. Thisphenomenonisabsentwhenthe±s-
h s
wave superconductor is replaced by an s-wave supercon-
case,the vanishingJ isnotobtainedeveninthe SI(±S)-
ductor where the order parameters have the same sign.
junction. However, since two different energy gaps have
Since the symmetry of order parameter is deeply related
beenobservedinFe-pnictidesuperconductors[11,12,20],
to the mechanism of superconductivity, our method dis-
we can determine the relative sign of the two order pa-
cussed in this paper would be also helpful in clarifying
rameters corresponding to the observedtwo energy gaps
the mechanism of superconductivity in Fe-pnictides.
by our method.
One canimmediately extend our idea to the case with We would like to thank S. Tsuchiya and R. Watanabe
more than two order parameters. In this case, when for useful discussions. This work was supported by a
all the order parameters do not have the same sign, we GrantinAidfromMext,andtheCTCprogramofJapan.
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