Structure of Neutron-Scattering Peak in both s++ wave and s+- wave states of an Iron pnictide Superconductor

0.21 MB·English

by  Seiichiro Onari

Structure of Neutron-Scattering Peak in both s -wave and s -wave states
++ ±
of an Iron pnictide Superconductor
Seiichiro Onari1, Hiroshi Kontani2, and Masatoshi Sato2
1 Department of Applied Physics, Nagoya University and JST, TRIP, Furo-cho, Nagoya 464-8602, Japan.
2 Department of Physics, Nagoya University and JST, TRIP, Furo-cho, Nagoya 464-8602, Japan.
(Dated: January 6, 2010)
Westudytheneutron scattering spectrum in iron pnictidesbased on therandom-phaseapproxi-
mationinthefive-orbitalmodel,forfully-gappeds-wavestateswithsign reversal(s±)andwithout
0
1 sign reversal (s++). In the s++-wave state, we find that a prominent hump structure appears just
above the spectral gap, by taking account of the quasiparticle damping γ due to strong electron-
0
electron correlation: As the superconductivity develops, the reduction in γ gives rise to the large
2
overshoot in the spectrum above the gap. The obtained hump structure looks similar to the reso-

n nance peak in the s±-wave state, although the height and weight of the peak in the latter state is
a muchlarger. Inthepresent study,experimentally observed broad spectral peak in iron pnictidesis
J
naturally reproduced byassuming thes++-wave state.
6
PACSnumbers: 74.20.-z,74.20.Rp,78.70.Nx

]
n
o
Since the discovery of superconductivity in iron pnic- tivity ρcr for vanishing T is about 20 µΩcm. However,
c imp c
- tideswithhightransitiontemperature(Tc)nexttohigh- experimental ρcimrp reaches ∼ 750 µΩcm, which corre-
r
p Tc cuprates[1],thestructureofthesuperconducting(SC) sponds to the minimum metallic conductivity 4e2/h per
u gap has been studied very intensively. The SC gap in layer [26]. Since this result supports a conventional s-
s manyironpnictidesisfully-gappedandband-dependent, wave state without sign reversal (s -wave state), we
. ++
t as shown by the penetration depth measurement [2] and haveto resolvethe discrepancybetweenneutronscatter-
a
m theangle-resolvedphotoemissionspectroscopy(ARPES) ing measurements and the impurity effects.
[3, 4], except for P-doped Ba122 [5]. The fully-gapped In this letter, we study the dynamical spin suscepti-
-
d state is also supported by the rapid suppression in 1/T1 bility χs(ω,Q) based on the five-orbital model [9] for
n (∝Tn; n∼4−6) below Tc [6–8]. boths++ ands± wavestates,anddiscussbywhichpair-
o
Inironpnictides,thenestingoftheFermisurface(FS) ing state the experimental results are reproducible. In
c
[ between hole- and electron-pockets is expected to in- the normal state, χs(ω,Q) is strongly suppressed by the
duce the antiferromagnetic (AF) fluctuations in doped quasiparticledamping γ due to strong correlation. How-
2
metal compounds. Since fully-gapped sign-reversing s- ever, this suppression diminishes in the SC state since γ
v
2 wavestate (s±-wavestate)is a naturalcandidate [9,10], isreducedastheSCgapopens. Forthisreason,apromi-
7 it is urgent to clarify the sign reversalin the SC gap via nent hump structure unrelated to the resonance mecha-
4 phase-sensitiveexperiments. Oneofthepromisingmeth- nismappearsinχs(ω,Q)justabove2∆inthe s++-wave
3 ods is the neutron scattering measurement: Existence of state. In the s±-wave state, very high and sharp reso-
.
0 theresonancepeakatanestingwavevectorQisastrong nance peak appears at ωres <2∆. We demonstrate that
1 evidence for AF fluctuation mediated superconductors thebroadspectralpeakobservedinironpnictidesisnat-
9
with sign reversal [11–13]. The resonance condition is urally reproduced based on the s++-wave state, rather
0
: ωres < 2∆, where ωres is the resonance energy and ∆ is than the s±-wave state.
v magnitude of the SC gap at T =0. The resonance peak Now, we study the 10×10 Nambu BCS Hamiltonian
Xi has been observedin many AF fluctuation mediated un- Hˆk composedofthe five-orbitaltight-bindingmodeland
r conventionalsuperconductors,like high-Tc cuprates [14– the band-diagonal SC gap introduced in ref. [25]. The
a 16], CeCoIn [17], and UPd Al [18]. FSs are shown in Fig. 1 (a). Then, the 10×10 Green
5 2 3
function is given by
Neutron scattering measurements for iron pnictides
have been performed [19–22] after the theoretical pre- Gˆ(iω ,k) Fˆ(iω ,k) −1
dictions [23, 24]. Although clear peak structure was ob- Gˆ(iωn,k) ≡ Fˆ†(iωn,k) −Gˆ(−inω ,k)
served in FeSe Te [20] and BaFe Co As [21], (cid:18) n n (cid:19)
0.4 0.6 1.85 0.15 2
its weight is much smaller than that in high-Tc cuprates = (iωnˆ1−Σˆk(iωn)−Hˆk)−1, (1)
and CeCoIn , and the resonance condition ω < 2∆ is
5 res where ωn = πT(2n+1) is the fermion Matsubara fre-
not surely confirmed, as we will discuss later. quency, Gˆ (Fˆ) is the 5×5 normal (anomalous) Green
Nonmagnetic impurity effect also offers us useful function, andΣˆ is the self-energyin the d-orbitalbasis.
k
phase-sensitiveinformation. Theoretically,s -wavestate For a while, we assume that the SC gap for the α-th FS
±
shouldbeveryfragileagainstimpuritiesduetotheinter- is band-independent; |∆ | = ∆. Hereafter, the unit of
α
band scattering [25]; the predicted criticalresidual resis- energy is eV, unless otherwise noted.
2

Here,wehavetoobtainthespinsusceptibilityasfunc-
(a) 8(b) S++ wave Approx. normal
tion of real frequency. For this purpose, it is rather easy Approx. SC
to use the Matsubara frequency method and the numer- V] ∆=0.4 Exact normal
ical analytic continuation (pade approximation). In the FS1 FS3 /e2B 6 U=1.2 Exact SC
µ
present study, however, we perform the analytical con- [
tinuation before numericalcalculationin order to obtain FS2 sχ 4
m
more reliable results. The irreducible spin susceptibility I
2
in the singlet SC state is given by [13] FS4

0
χˆ0l1Rl2,l3l4(ω,q) = N1 d2x (2) 0 1ω [eV] 2
k Z
X
x
tanh GR (x ,k )ρG (x,k) FIG. 1: (Color online) (a) FSs in iron pnictides. (b) ω-
+ thanhx2+TρGl1(l3x ,+k )+GAl4l(2x,k) danepdenthdeenncoermofaIlmsχtast(eω,,wQh)efroertthheeusn+i+t-owfaveenesrtgayteis(∆eV=. T0.h4e)
2T l1l3 + + l4l2 “exact result” is obtained by eq. (2), and the “approximate
x
+ tanh FR (x ,k )ρF† (x,k) result” is obtained by eq. (6).
2T l1l4 + + l3l2
x
+ tanh +ρF (x ,k )F†A(x,k) ,
2T l1l4 + + l3l2 Fermi velocity of the α-th FS, we obtain ρ ≈ (2γ[meV])
i
µΩcm [25]. Since ρ(T) − ρ(0) ∼ (5T[meV]) µΩcm in
where x =x+ω, k =k+q, l =1∼5 representsthe
+ + i
BaFe Co As below 100 K [27], γ(0) due to inelas-
d-orbital, and A (R) represents the advanced (retarded) 1.84 0.16 2
tic scattering is estimated as 2.5T which is comparable
Greenfunction. ρG(x,k)≡(GA(x,k)−GR(x,k))/(2πi)
ll′ ll′ ll′ tothatinover-dopedcuprates. Ifweassumetherelation
and ρF(†)(x,k) ≡ (F(†)A(x,k)− F(†)R(x,k))/(2πi) are
ll′ ll′ ll′ γ(ǫ)∝(πT+ǫ)innearlyAFFermiliquid[28],weobtain
one particle spectral functions. Since ρG,F(x,k) = 0 for
ll′ γ(ǫ)∼2.5(T +ǫ/π).
|x| < ∆, Imχˆ0llR,l′l′(ω,q) = 0 for |ω| < 2∆. That is, the Now, we calculate Imχs(ω,Q) in both normal and
particle-hole excitation gap is 2∆.
s -wave SC states, concentrating on the frequency
++
Then, the spin susceptibility χs(ω,q) is given by the ω ∼2∆. To estimate the renormalizationof Imχs(ω,Q)
multiorbital random-phase-approximation (RPA) with
due to the self-energy,we have to know the value of γ(ǫ)
the intraorbital Coulomb U, the interorbital Coulomb
with |ǫ|∼∆ in both normal and SC states. Considering
U′, the Hund coupling J, and the pair-hopping J′ [9]:
that γ(ǫ) = 2.5(T +ǫ/π) ∼ 2∆ at T = 2.2 meV and
c
ǫ = ∆ ∼ 5 meV in BaFe Co As , in the present
1.85 0.15 2
χs(ω,q)= χˆ0R(ω,q) , (3) study, we simply put γ(ǫ) in the normal state at Tc as
"1−Sˆ0χˆ0R(ω,q)#
Xi,j ii,jj γ(ǫ)=γ (4)
0
where vertex of spin channel Sˆ0 =U, U′, J and J′
l1l2,l3l4 with γ0 & ∆. In the present model, αSt = 0.84 (0.79)
for l = l = l = l , l = l 6=l = l , l = l 6= l = l
1 2 3 4 1 3 2 4 1 2 3 4 for U = 1.3 (1.2) when γ = 0.1 and T = 0.002; the
0
and l = l 6= l = l , respectively. Hereafter, we put
1 4 2 3 T-dependence of α is small when γ is fixed.
J = J′ = 0.15, U′ = U −2J, U = 1 ∼ 1.3, and fix the St 0
In the SC state at T ≪ T , γ(ǫ) = 0 for |ǫ| < 3∆ (=
c
electronnumberas6.1(10%electron-dopedcase). Inthe
particle-hole excitation gap 2∆ plus one-particle gap ∆)
presentmodel,χs(0,q)takesthemaximumvaluewhenq
[12], and its functional form is approximately the same
is the nesting vector Q=(π,π/16). Due to the nesting,
as that in the normal state for |ǫ|&3∆. Then, we put
χs(0,Q)/χ0(0,Q) ≈ 1/(1−α ) is enhanced; α (. 1)
St St
is the maximum eigenvalue of Sˆ0χˆ0R(0,Q) that is called γ(ǫ)=a(ǫ)γ (5)
s
the Stoner factor.
In strongly correlated systems, χs(ω,q) is renormal- where(i)a(ǫ)≪1for|ǫ|<3∆,(ii)a(ǫ)=1for|ǫ|>4∆,
ized by the self-energy correction. In nearly AF met- and(iii)linearextrapolationfor3∆<|ǫ|<4∆. Wehave
als,forexample,thetemperaturedependence ofthe self- confirmedthatthe obtainedresultsareinsensitivetothe
energy induces the Curie-Weiss behavior of χs(0,Q). At boundaryof|ǫ| (4∆inthe presentcase)between(ii)and
themoment,thereisnoexperimentalinformationonthe (iii). Although γ at T ≪ T should be smaller than γ
s c 0
k-, ǫ-, and band-dependences of the self-energy. There- atT =T ,wesimplyputγ =γ hereafter,whichcauses
c s 0
fore, we phenomenologically introduce a band-diagonal underestimation of the peak height of Imχs.
self-energy as ΣˆR(ǫ) = iγ(ǫ)ˆ1. First, we estimate the Figure 1 shows Imχs(ω,Q) obtained by eqs. (2) and
k
value of γ(ǫ) in the normal state. Since the conduc- (3) for U = 1.2, γ = 0.4 and T = 0.01. In the s -
0 ++
tivity is given by σ = e2 N (0)v2/2γ(0), where wave SC state, we put ∆ = γ and a(3∆) = 0.05. In
α α α 0
N (0) and v are the density of states (DOS) and the calculating eq. (2), we use 256 × 256 k-meshes, and
α α
P
3
1000 x-meshes. Although values of ∆ and γ in Fig. 1 16 (a)S++ wave (b)
∆=0.05 U=1.3 8 γ=0.05
araretioveγry/l∆arg∼e t1oiosbctoaninsisetneonutgwhitnhumexepriecrailmaecnctusr.acIyn, tthhee sχ 12 γ0=0.003 6 0
normal0state, Imχs(ω,Q) is suppressed by large quasi- Im 8 4
4 2
particle damping γ0 ∼ ∆. In the SC state, the gap in
Imχs(ω,Q) is 2∆. Since the particle or hole with en- 0 0
8(c) 8(d)
ergy |ǫ| < 3∆ is free from inelastic scattering in the SC
state, the lifetime of particle-hole excitation with energy s 6 γ0=0.075 6 γ0=0.1
χ
|ǫ|<4∆becomeslong. Forthisreason,Imχs(ω,q)shows m 4 4
I
a large hump structure for 2∆ . ω . 4∆ below Tc in 2 2
s++-wave state. 00 0.1 0.2 0.30 0 0.1 0.2 0.3
Unfortunately, we cannot put smaller ∆ and γ in cal- ω ω
culating eq. (2) in the five-orbital model, because of the
computationtime. Tosolvethisproblem,weperformthe FIG. 2: (Color online) Imχs(ω,Q) for s++-wave (solid line)
x-integration in eq. (2) approximately as follows: When and normal (brokenline) states, with γ0 =0∼0.1.
γˆ = γˆ1, the retarded (advanced) 10 × 10 Green func-
tion is expressedas GˆR(A)(x,k)= Um,α(x+(−)iγ−
m,m′ α k
Ekα)−1Ukm′,α∗,whereEkα(α=1∼1P0)istheeigenvalueof einqcuraelasteostfhroamt in(bt)h0e.0n5ortmoa(ld)st0a.t1e,fIomrχωs >in 2th∆e. nAorsmγa0l
Hˆk and Uˆk is the corresponding 10×10 unitary matrix. state decreases gradually, whereas that in the SC state
We promise that Ekα = −Ekα+5 for 1 ≤ α ≤ 5. When depends on γ0 only slightly, since γ(ǫ) ≈ 0 for |ǫ| < 3∆.
γ is sufficiently small, then ρG(F)(x,k) ≈ Ul,αδ(x− Therefore, in the case of γ & ∆, Imχs(ω,Q) in the SC
ll′ α k 0
Eα)Ul′(+5),α∗, and thus eq. (2) becomes state shows a prominent hump structure, and its peak
k k P valueis aboutdoubleofthatinthe normalstate. In(d),
1 f(El)−f(El′ ) experimentalapproximate“sum-rule”atfixedq =Q[21]
χˆ0R (ω,q)≈ k k+q is well satisfied. In Fig. 2 (c) and (d), a relatively large
l1l2,l3l4 N k l,l′ ω+Ekl −Ekl′+q +iΓll′,kq slopefor|ǫ|<2∆isanartifactoftheapproximationdue
XX
Ul1,l′Ul3,l′∗Ul4lUl2l∗+Ul1,l′Ul4+5,l′∗Ul3+5,lUl2l∗ ,(6) to large γ0/∆.
k+q k+q k k k+q k+q k k
h i 8(a) Isotropic S++ wave (b) AnisotropicS++ wave
witWhhΓelnl′,kγqi=s aγsfloarrgγe≪as1∆. , however, we have to check 6 γU0 == 10..31 ∆∆mmainx == 00..00375
to what extent eq. (6) is reliable. Considering that the s
χ
origin of the renormalization of χs is the quasiparticle m4
dampingγ(El)andγ(El′ ),weintroduce thefollowing I
k k+q 2
approximation:
Γll′,kq =b·max{γ(Ekl),γ(Ekl′+q)} (7) 00 0.1 0ω.2 0.3 0.4 0 0.1 0ω.2 0.3 0.4
SwChersetabte≈fo1r |iEskla|,fi|Ettkli′n+gq|p<ara3m∆e,terre.fleΓctlli′n,kgqt≈he0abinsenthcee FanIGd.n3o:rm(Caollo(rbroonkleinne)linIme)χss(tωat,eQs,)wfoirths+∆+m-waxave=(s0o.l0id7lianned)
of quasiparticle damping. In Fig. 1, we show numerical ∆min=0.035.
results given by the present approximationwith b=1.3;
we replace bγ with γ hereafter since b ≈ 1. Since the Next, we study the effect of band-dependent SC gap
0 0
“exact results” given by eq. (2) is quantitatively repro- observed by ARPES measurements [3, 4]. In Fig. 3
duced, we decide to calculate Imχs(ω,Q) using eqs. (6) (a), we put U = 1.3, ∆ = ∆ = 0.07eV for
1,2,4 max
and (7) for more realistic values of ∆ and γ. We veri- FS1,3,4, and ∆ = ∆ = 0.035eV for FS2. In (b),
2 min
fiedthatthepresentapproximationworkswellwhenγ is we introduce the anisotropy of the gap function for only
comparable to or smaller than ∆. FS3,4 with ratio 2; ∆ = ∆ (1−0.5sin2θ ), where
k max k
Figure 2 shows Imχs(ω,Q) obtained by eqs. (6) and θ = tan−1(|k |/(|k |−π)) for FS3(4). Here, we
k y(x) x(y)
(3)forU =1.3andT =0.002. Inthes -waveSCstate, put a(ǫ) in eq. (5) as (i) 0.003/γ for |ǫ| < 3∆ ,
++ 0 min
we put ∆ = 0.05; although it is a few times larger than (ii) 1 for |ǫ| > 4∆ , and (iii) linear extrapolation for
min
the gap for Sm1111 with T = 56K, it is enough smaller 3∆ < |ǫ| < 4∆ . In Fig. 3 (a), Imχs(ω,Q) in-
c min min
than the Fermi energies of electron- and hole-pockets creases rapidly at ω = ∆ + ∆ = 0.105, and it
max min
[9]. In the numerical calculation, we use 1024×1024 k- shows a peak at ω = 0.14. In (b), the peak is located
meshes. Hereafter, we put a(3∆)=0.003/γ . When (a) at ω = 0.125, which is closer to ∆ +∆ = 0.105.
0 max min
γ = 0.003, Imχs(ω,Q) in the SC state approximately In Fig. 3 (a) and (b), the width of the hump peak is
0
4
muchsharperthanthatfortheband-independentSCgap fermion Kondo insulator CeNiSn. As shown in Fig. 1 of
in Fig. 2, since Imχs(ω,Q) is reduced by damping for Ref. [31], neutron scattering spectrum at q =(0,π,0) in
|ω|>4∆ =0.14. WehavealsocalculatedImχs(ω,Q) CeNiSn shows a prominent hump peak structure above
min
for ∆ =∆ and ∆ =∆ , and verified that the the hybridizationgap below the Kondo temperature T ,
3,4 max 1,2 min K
obtained result is similar to Fig. 3. which looks very similar to the spectrum observed in
Here, we make comparison with experiments. The iron pnictides below T [19–22]. This hump structure
c
peak height and the weight in Fig. 3 (b) seems to is well reproduced by the dynamical-mean-field theory
be consistent with the neutron scattering measurements based on the periodic Anderson model [32]. This fact
in iron pnictides [19–22]. In BaFe Co As (T = demonstrates that large hump in Imχs(ω,Q) can ap-
1.85 0.15 2 c
25K), the observed ”resonance energy” is ω = 9.5 pearinstronglycorrelatedsystemswithone-particlegap,
res
meV [21]. According to ref. [3], ∆ /T ≈ 3.5 and without the necessity of the resonance mechanism.
max c
∆min/∆max ≈0.35inmanyironpnictides. (Moresmaller Insummary,we havestudied Imχs(ω,Q)in ironpnic-
∆max,min is reported in ref. [2].) Thus, ∆max+∆min ≈ tides based on the five-orbital model, and revealed that
4.7Tc = 10 meV is comparable to ωres = 9.5 meV a prominent hump structure appears just above 2∆ in
in BaFe1.85Co0.15As2. Moreover, finite Imχs(ω,Q) for the s++-wave state, by taking the strongly correlation
ω & 0.3ωres in ref. [21] may suggest the existence of effect via γ. This hump structure becomes small as αs
SC gap anisotropy. Therefore, the theoretical result in decreasesintheover-dopedregionorq deviatesfromthe
Fig. 3 (b) is well consistent with experimental data. We nesting. At present, experimental data can be explained
haveverifiedthatthe humpstructureofImχs(ω,q)with in terms of the s -wave state very well. Further ex-
++
q =(π,0) is very small for γ0 ∼0.1. perimental efforts are required to determine the height
and width of the ”resonance peak”, and the magnitude
40(a) S+− -wave U=1.2 10(b) S+− -wave U=1.0 relation between ωres and ∆max+∆min.
30 |∆γ|==00..015 8 This study has been supported by Grants-in-Aid for
sχ 0 6 Scientific Research from MEXT of Japan, and by JST,
m 20
I 4 TRIP. Numerical calculations were performed using the
10 facilities of the supercomputer center, ISSP.
2

0 0 0.1 0.2 0.3 0 0 0.1 0.2 0.3
ω ω
FIG. 4: (Color online) Imχs(ω,Q) for s±-wave (solid line) [1] Y.Kamihara,etal.,J.Am.Chem.Soc.130,3296(2008).
and normal (broken line) states, with U =1.2 and 1.0.
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St c
(1998).
width of the resonance peak is wide due to the sample
[14] S. Iikubo,et al., J. Phys. Soc. Jpn. 74, 275 (2005).
inhomogeneity (i.e., nanoscale distribution of T ) [16].
c [15] M. Ito, et al., J. Phys. Soc. Jpn. 71, 265 (2002).
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