Potential antiferromagnetic fluctuations in hole-doped iron-pnictide superconductor Ba_{1-x}K_{x}Fe_{2}As_{2} studied by ^{75}As nuclear magnetic

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by  Masanori Hirano

Typeset with jpsj2.cls <ver.1.2> Full Paper
Potential antiferromagnetic fluctuations in hole-doped iron-pnictide superconductor
Ba K Fe As studied by 75As nuclear magnetic resonance
1−x x 2 2
Masanori Hirano1 ∗, Yuji Yamada1, Taku Saito1, Ryo Nagashima1, Takehisa Konishi2,
Tatsuya Toriyama1, Yukinori Ohta1,5, Hideto Fukazawa1,5, Yoh Kohori1,5, Yuji Furukawa3,
Kunihiro Kihou4,5, Chul-Ho Lee4,5, Akira Iyo4,5, and Hiroshi Eisaki4,5
2
1Department of Physics, Chiba University, Chiba 263-8522, Japan
1
0 2Department of Chemistry, Chiba University, Chiba 263-8522, Japan
2 3Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA
4National Institute of Advanced Industrial Science and Technology, Tsukuba 305-8568, Japan
n
5JST, Transformative Research-Project on Iron Pnictides (TRIP), Chiyoda, Tokyo 102-0075, Japan
a
J

2 We have performed 75As nuclear magnetic resonance (NMR) and nuclear quadrupole res-
1 onance (NQR) on single crystalline Ba1−xKxFe2As2 for x = 0.27-1. 75As nuclear quadruple
resonance frequency (νQ) increases linearly with increasing x. The Knight shift K in normal
]
state shows Pauli paramagnetic behavior with slight temperature T dependence. The valueof
n
o K increases gradually with increasing x.By contrast, nuclearspin-lattice relaxation rate 1/T1
c innormalstatehasastrongT dependence,whichindicatesexistenceoflargeantiferomagnetic
- (AF) spin fluctuations for all x. The T dependence of 1/T1 shows a gap-like behavior below
r
p approximately 100 K for 0.6 < x < 0.9. These behaviors are well explained by the change of
u bandstructurewithexpansionofholeFermisurfacesandshrinkanddisappearanceofelectron
s FermisurfacesatBrillouinzone(BZ)withincreasingx.Theanisotropyof1/T1,representedby
.
t aratioof1/T1ab to1/T1c,isalwayslargerthan1forallx,whichindicatesthatthestripe-type
a
AF fluctuations is dominant in this system. The K in superconducting (SC) state decreases,
m
which corresponds toappearanceof spin-singlet superconductivity.TheT dependenceof1/T1
- inSCstateindicatesmultiple-SC-gapfeature.Asimpletwogapmodelanalysisshowsthatthe
d
largersuperconductinggapgraduallydecreaseswithincreasingxfrom0.27to1andsmallergap
n
decreases rapidly and nearly vanishes for x>0.6 where theelectron pockets in BZ disappear.
o
c
[ KEYWORDS: iron pnictide superconductor, nuclear magnetic resonance, nuclear quadrupole resonance,
spin fluctuations,superconducting gap
2
v
1
1. Introduction ity, magnetic penetration-depth measurements
8
0 Soon after the discovery of LaFeAsO1−xFx with su- in Ba(Fe1−xCox)2As2 have shown existence of
6 perconducting (SC) transition temperature T = 26 K nodes in underdoped and overdoped region.10) For
c
0. byKamiharaetal.in2008,1)ironbasedsuperconductors BaFe2(As1−xPx)2, angle resolved thermal conductivity
1 have been studied extensively all over the world. High- suggestedtheclosednodalloopslocatedattheflatparts
1 est superconducting transition temperature T of 56 K of the electron Fermi surface,11) while three dimensional
1 was soon observed in RFeAsO2,3) (“1111”, Rcis a rare- nodal structure on hole Fermi surface was proposed
:
v earth element), which is a high T discovered next to by theoretical calculation using fluctuation exchange
c
Xi the value in cuprate system. Many iron based super- (FLEX) approximation.12)
conductors were observed one after another in AFeAs In Ba1−xKxFe2As2 (BKFA), many experiments re-
r
a (“111” A is an alkaline element), Fe Se (“11”), and vealedappearanceofmultiplefullSCgaparoundx=0.4
1+δ
AEFe2As2 (“122”, AE is an alkaline-earth element). whereTchasamaximumvalueof38K.13–15)Signchang-
Among them, AEFe As (AE = Ba, Ca, Sr) occupies ing s±-wave which is mediated by spin fluctuations well
2 2
a singular position, since high quality and large single explainsalotofexperimentalresults.16–19)T dependence
crystals were grown. A parent compound BaFe2As2 has ofspin-latticerelaxationrate1/T1 canalsobe explained
ThCr Si type crystal structure and has an antiferro- bys±-wave.20,21)However,itispointedouttheoretically
2 2
magnetic (AF) and tetragonal to orthorhombic crystal that s±-waveis veryfragile to nonmagnetic impurity,22)
structure phase transition at 140 K. The orthorhombic while iron-pnictide superconductors are experimentally
andAFphasewassuppressedandsuperconductivityap- robust against nonmagnetic impurity. No sign changing
peared in hole doped Ba1−xKxFe2As2,4) electron doped s++-wave which is mediated by orbital fluctuations and
Ba(Fe1−xCox)2As2,5)andBaFe2(As1−xPx)26,7)inwhich is robustagainstnonmagneticimpurity isanotherpossi-
isovalentPsubstitutionforAsactsaschemicalpressure. blecandidatefortheCooperpairinginthissystem.23–26)
Indeed, the direct application of pressure for BaFe As Hence, the SC pairing symmetry in optimum region is
2 2
also induces superconductivity.8,9) still an open question. However, there is consensus that
As for SC gap structure, thermal conductiv- the SC gap structure itself is fully gapped one.
By contrast, NQR and specific heat, thermal conduc-
tivity, and magnetic penetration-depth measurements
∗[email protected]
2 J.Phys.Soc.Jpn. FullPaper AuthorName
for x = 1 (KFe As ) revealed appearance of nodal SC reports the single crystal growth of an end member of
2 2
gap.27–29) Smallangleneutronscattering(SANS) exper- the system, KFe As . The ratio of Ba and K was de-
2 2
iment has pointed out that stable isotropic hexagonal termined by the energy dispersive X-ray spectroscopy.
vortex lattice exists when magnetic fields applies paral- Experimental error of the evaluated values was within
lel to crystal c axis, which supports nodes in horizontal 5%. The c-axis parameter of the single crystals was de-
direction.30) Recentmuonspin relaxationmeasurements termined by X-ray diffraction analysis and the relation
of KFe As also supports the SANS results.31) How- between the composition x and lattice parameter c is
2 2
ever, recent angle resolved photoemission spectroscopy consistentwiththe formerreportedrelationbetweenthe
(ARPES)andspecificheatmeasurementsindicatenodes nominal composition x and lattice parameter by Rotter
inverticaldirection.32,33) Thehorizontalnodalstructure et al.4)
withs±-wave19) andnodal-lineSCgapstructurewithd- We determined the Tc and superconducting vol-
wave34)aretheoreticallyproposedtothisendcompound. ume fraction of the samples with a commercial
ItisnoteworthythatSCgapstructuresinBKFAaredif- superconducting-quantum-interference-device magne-
ferent in optimally doped x = 0.4 and in heavily over- tometer. The T ’s of the samples are 38.5 K (x = 0.27),
c
doped x= 1. 36.5 K (x = 0.31), 38 K (x = 0.39), 30.5 K (x = 0.58),
Important feature of BKFA is a wide range real- 26.5 K (x = 0.64), 20.5 K (x = 0.69), 4.5 K (x = 0.94),
ization of SC phase for 0.2 < x ≤ 1.4) In BKFA, 3.5 K (x = 1) and the volume fraction for all the x’s is
the band structure changes associated with hole dop- approximately 100% except for x = 0.27. For x = 0.27,
ing. For x = 0.4, cylindrical hole Fermi surfaces exist the fraction is approximately 60% because the crystal
around Γ point and cylindrical electron surfaces around contains the phase-separated antiferromagnetically
XpointsoftheBrillouinzone(BZ).Goodnestingcondi- orderedphase40) as we reportthe experimentalevidence
tion between disconnected hole and electron Fermi sur- of the phase separation below. We also determined
faces enhances AF spin fluctuations and is favorable the T in the NMR-measurement magnetic field by
c
for superconductivity mediated by AF spin fluctuations. the change of inductance of NMR detecting coil with
Within this picture, s±-wave superconductivity is ex- changing T.
pected.16–19) Interestingly, the inelastic neutron scatter- The NMR/NQRexperimentonthe 75As nucleus (I =
ing revealed that there is incommensurate spin fluctua- 3/2,γ/2π=7.292MHz/T)wascarriedoutusingphase-
tionQ=((1±2δ)π,(1±2δ)π,0)withδ =0.16forx=1at coherent pulsed NMR/NQR spectrometers and a super-
excitation energies above 3 eV35) where electron Fermi conducting magnet between approximately 3 and 7 T.
surfaces completely disappear and apparent good nest- The measurement was performed using a 4He cryostat.
ingconditionbetweeninterbandsatFermileveldoesnot The NMR spectra were measured by sweeping the ap-
exit.Ontheotherhand,thereisanothertheoreticalpro- plied fields at a constant resonance frequency. Magnetic
posal that SC is mediated by orbital fluctuations, which field was applied parallelto the crystal ab plane and the
leads to nearly orbital independent SC gap s .23–26) c axis.Field alignmentwas performedwith the eye.The
++
Hence, there remains much work to clarify the SC gap originof the Knightshift K =0 of the 75As nucleus was
symmetry in ironpnictide superconductors.Band struc- determined by the 75As NMR measurement of GaAs.41)
ture of BKFA changes associated with hole doping. The The NQR spectra were measured by sweeping the fre-
hole Fermi surfaces expand with increasing x, whereas quencyinzeromagneticfield.T wasmeasuredbyasat-
1
electron Fermi surfaces shrink gradually and disappear uration recovery method at center of As-spectrum. We
for x > 0.6.36–38) In BKFA, T changes smoothly with obtained T at a fixed frequency of 37.15 or 43.75 MHz
c 1
increasingx.Itisveryimportanttoclarifythechangeof with an externalfield of 5.06-5.09or 5.9-5.96T in the T
SCgapstructuresinBKFAwithinterpolatingoptimally range of 2-300 K, respectively.
doped and overdoped regions. The relation between AF
3. Results and discussion
spinfluctuationsandT shouldbealsostudied.Recently,
c
we succeeded in synthesizing high quality large single 3.1 NQR frequency
crystals of BKFA for 0.31 ≤ x ≤ 1, which provides us a InFig.1,weshowtheNQRfrequencyν againstsub-
Q
unique opportunity to solve above problems. stitution concentration x. The experimentally obtained
NMR and/or NQR is suitable for study of static and ν was evaluated from the 75As NQR spectra of BKFA
Q
dynamicalmagneticproperties,andprovidesvaluablein- as shown in the inset of Fig 1. Bars for each data point
formation on SC gap symmetry and the gap structures correspondtothefullwidthathalfmaximum(FWHM)
throughKnightshift(K)andspinlatticerelaxationrate of the spectra. It is expected that the spectral broaden-
(1/T ) measurements. In this paper, we report a 75As- ing was brought out because of the random distribution
1
NMR/NQR study of BKFA. of Ba2+ and K+ in the crystal. Here, z parameter in-
As
dicates z-coordinate of As site in the unit cell.
2. Experimental
The ν generally depends on temperature and these
Q
Single crystals of BKFA (x = 0.27, 0.31, 0.39, 0.58, spectra were obtained at the different measurement
0.64, 0.69, 0.94, 1) were grown by the BaAs and KAs temperatures. However, note that such temperature
self-flux method. X-raydiffractionshowedthat the crys- dependence of ν in each compound is within the bar
Q
tals had a tetragonal ThCr Si type structure with no in Fig. 1. Hence, it is clear that the x dependence of
2 2
impurity phase. The detailed procedure of single crystal the experimental ν is nearly linear. The principal axis
Q
growth is basically the same described in ref. 39 which of the electric field gradient is along the crystal c-axis
J.Phys.Soc.Jpn. FullPaper AuthorName 3
tal approximation for alloying effect, where we made
calculations for hypothetical materials Ba1−xCsxFe2As2
and Ca1−xKxFe2As2 to see the electronic structure of
Ba1−xKxFe2As2. We used experimental lattice parame-
ters a and c reported in ref. 4. The obtained Fermi sur-
faces using the experimental z parameter of As in the
ThCr Si tetragonalstructure are consistentwith Fermi
2 2
surfaces reported by de Haas-van Alphen effect37) and
ARPES.15,36,38) The Fermi surfaces obtained using the
optimizedvalueofthezparameterwasnotconsistentfor
the end member of the system: electron Fermi surfaces
donotdisappearatthecornerXpointoftheBZevenfor
x = 1. This tendency is already reported by Singh with
thesameprocedureusinglocaldensityapproximation.43)
In this system changes in the lattice parameters, Fe-
Fe interatomic distance, and the position of the As ions
upon doping are closely interconnected with the elec-
tronicstructure.Inallthecalculatedresults,ν exhibits
Q
monotonic increase with x and has a same order of the
value compared with the experimental ν . Of all the re-
Q
sults,itissurprisingthattheexperimentalν hasagood
Q
agreementwiththe νQ calculatedforCa1−xKxFe2As2 in
the virtual crystal approximation since such agreement
can rarely be seen in other strongly electron-correlated
Fig. 1. (color online) Nuclear quadrupole resonance frequency compounds. Such good agreement between experiments
νQ against concentration x. BCFA, CKFA, and BKFA and theory suggests that the electronic structure calcu-
are abbreviation for Ba1−xCsxFe2As2, Ca1−xKxFe2As2, and
lated assuming the experimental z parametersof As can
Ba1−xKxFe2As2,respectively. Theexperimental data of νQ for
x=0 werequoted fromref.44. Closedcircles areexperimental explain the experimentalelectronic state ofBKFA fairly
results ofνQ obtained from75As NQRmeasurements asshown well.
intheinset.Opensquaresanddiamondsaretheresultsofband-
structure calculation using the experimentally obtained z pa-
3.2 NMR spectra
rameter of As inThCr2Si2 tetragonal structure. z parameter is
In Fig. 2(a) and (c), we show the NMR spectra of
z-coordinate of As siteinthe unitcell. Closedsquares anddia-
monds arethecalculated resultsbyoptimizingthez parameter BKFA with magnetic field paralleland perpendicular to
with GGA. The NQR spectra in the inset were obtained at 40 the ab plane above T or T . In Fig. 2(b), we show the
c N
(x=0.39and0.58),35(x=0.64),30(x=0.69),10(x=0.94), temperature dependent NMR spectra of x = 0.27. We
and4.2K(x=1).
usedthe NMR dataforx=0 fromref.44.Because75As
nucleus has quantum number I = 3/2, the nuclear-spin
HamiltonianwithanexternalmagneticfieldH isgiven
since the As site has a local fourfold symmetry around ext
by
the c-axis. Hence, asymmetry parameter η at As site
ν
ibsilibtaysiocfalfilyniηte=η 0in. Hthoewfeovlelor,wwinegwsuilblsdeicstciuonss§th3e.2p.oTsshie- H=−γHextIz−γH~ext·K¯·I~+ 6Q 3Iz2′ −I(I +1) , (2)
(cid:0) (cid:1)
definitionofν atAssiteinthesematerialsisasfollows:
Q (H~ ·K¯ ·I~)/H =−(K sinθ)(I cosθ−I sinθ)
ext ext ab x z
3e2qQ
νQ = , (1) +(Kccosθ)(Ixsinθ+Izcosθ),
2hI(2I−1)
(3)
where h, eq, eQ represent the Planck constant, the elec-
Iz′ =Ixsinθ+Izcosθ, (4)
tric field gradient (EFG), and the nuclear quadrupole
moment, respectively. The main contribution of change where K¯ (Kab and Kc) and θ represent the Knight shift
in ν can be simply interpreted as a linear increase of tensor (Knight shifts) and the angle between quantiza-
Q
effective ligand valency by substitution of K for Ba in tion axis and the crystal c axis, respectively. Here, we
addition to the change in lattice parameter. These two took the directionof the externalmagnetic field as prin-
changes directly bring out the change in the EFG. cipal axis of quantization axis and we also assumed the
This tendency is also confirmed by the systemat- diagonalizedK¯ andtheabsenceofin-planeanisotropyof
ics obtained in the calculated electronic structures for Kab. The detailed procedure to obtain Kab andKc from
BKFA. We performed the electronic structure calcula- the NMR spectra is described elsewhere.45)
tionusingWIEN2Kcode,42) wherethefull-potentiallin- The FWHM of the center peak for x=1 is about 48
earized augmented-plane-wave method (FLAPW) with ± 2 kHz. Hence, the analysis using the above expression
the generalized gradient approximation (GGA) for elec- works well for x = 1 and it turned out that the actual
tron correlations is used. We also used the virtual crys- field misalignment was within approximately 5% accu-
4 J.Phys.Soc.Jpn. FullPaper AuthorName
center line is affected only by the K and the line width
c
ofthecenterlinewasrathersharper:theFWHM ofthe
center peak for these composition is approximately 2-3
times broader than that for x = 1. The satellite lines
of x = 0.58, 0.64, and 0.69 seem to have small different
structures. It probably arises from appearance of local
asymmetries by K doping. Because η does not affect the
central transition, we assume η = 0 for the following
discussion.
The NMR spectra of x = 0.27 are symmetric at 100
and 160 K. However, the overall feature of the spectra
slightly changes below about 60 K. In addition to the
symmetriccenterandsatellitelines,broadpeakappears:
thepositionofthispeakisjustbelowthecenterposition.
Below T this feature becomes obvious. This is because
c
the paramagnetic region and the antiferromagnetically
ordered region are phase-separated in this compound.
This phenomenon was previously reported for polycrys-
talline samples.40) However, recent µSR study of under-
dopedBKFAshowsamicroscopiccoexistenceofSCand
AF phases without phase separation.46)
3.3 Knight shifts
In Fig. 3(a) and (b), we show the temperature depen-
denceofKnightshiftK(T)ofBKFAwithmagneticfield
parallel and perpendicular to the ab plane. For all the
data, the K(T) exhibits weak temperature dependence.
Thistendencyissimilartotheisovalentsubstitutionsys-
temBaFe2As2−xPx.47)Althoughwecouldnotobtainthe
K forx=0.58,0.64,and0.69,systematicdevelopment
ab
of the K indicates that the K for x=0.58, 0.64, and
c ab
0.69 exists between x = 0.27 and 1 and that it also ex-
hibits weaktemperaturedependence.With increasingx,
K(T)atsametemperaturebasicallyincreases.Thisisat-
Fig. 2. (coloronline)TheNMRspectraofBKFAwithmagnetic
tributabletotheincreaseofdensityofstatesattheFermi
field(a)paralleland(c)perpendiculartotheabplaneaboveTc
orTN.Thedataforx=0wasquotedfromref.44.Themeasure- levelwithholedoping.Thus,staticmagneticpropertyis
menttemperaturesare141,100,60,50,30,25and4.2Kforx= nearly temperature independent and shows the slight x
0, 0.27, 0.39, 0.58, 0.64, 0.69 and 1, respectively. Solidlines de- dependence: spin fluctuation at around ~q =~0 is not sig-
note the fitted parameter obtained from the analysis described
nificant in this system.
in the text. (b) The temperature dependent NMR spectra of
ThehyperfinecouplingconstantA parallelorperpen-
x=0.27. i
dicular to the ab-plane for x=1 was evaluated from the
Knight shift at the As site parallel or perpendicular to
racy.45) Butweusedthefollowingapproximationforthe the ab-plane and the magnetic susceptibility parallel or
other x’s since the NMR spectra are much broader than perpendicular to the ab-plane by assuming the following
that of x=1. formula,
ν
H≃−γ(1+Kab)HextIz+ 6Q (cid:0)3Ix2−I(I +1)(cid:1) (H//a(b5)), Ki(T)= NAAµdi Bχdi(T)+ NAAiµ0Bχi0, (7)
ν χ (T)=χd(T)+χ , i=aborc. (8)
H≃−γ(1+K )H I + Q 3I2−I(I +1) (H//c). i i i0
c ext z 6 z
(cid:0) (cid:1) (6) Here, NA represents Avogadro’s number. χdi represents
T dependentspinsusceptibilityoriginatingfrom3delec-
Even with the approximation,we cannot determine K
ab
tron of Fe. χ represents T independent terms arising
for x = 0.58, 0.64, and 0.69 with sufficient accuracy be- i0
from orbital contribution. The evaluated hyperfine cou-
cause all the center and satellite lines are broad and the
plingconstantsAd andAd originatingfromthetransfer
position of the center line is affected by both the Kab ab c
from 3d conduction electron to 75As nuclear spin of Fe
and ν : the FWHM of the center peak for these com-
Q
are +26(5)and +13(4) kOe/µ , respectively. These val-
positionisapproximately4-8timesbroaderthanthatfor B
ues arequite similarto thoseobtainedfor x=0.44) This
x=1.Therefore,thesolidlinesinFig.2(a)forx=0.58,
indicates that the hyperfine coupling constantnearly re-
0.64, and 0.69 are just reference containing large fitting
mains constant with hole doping in this system.
error. We can determine the K for x = 0.58, 0.64, and
c
In the inset of Fig. 3(b), we show the temperature de-
0.69withreasonablefittingerrorsincethepositionofthe
J.Phys.Soc.Jpn. FullPaper AuthorName 5
Fig. 3. (color online) The temperature dependence of Knight
shift of BKFA with magnetic field (a) parallel and (b) perpen-
dicular to the ab plane. The data for x = 0 was quoted from Fig. 4. (coloronline)Temperaturedependence ofspin-latticere-
ref. 44. In the inset, the temperature dependence of Kc below laxation rate inversely multiplied with temperature 1/T1T of
30Kisshownforx=0.39,0.64,0.69. 75As of BKFA with magnetic field (a) parallel and (b) perpen-
dicular to the ab plane. The data for x = 0 was quoted from
ref.44. The inset shows the typical recovery curve for x=0.69
obtainedat1.45,44,267K.
pendence ofK below 30 K for x = 0.39,0.64,0.69.The
c
K exhibitscleardecreasebelowT (H)forboththesam-
c c
ples. Since the Knight shifts correspond to the spin sus- of the 75As nucleus,50)
ceptibility,thisresultindicatesspin-singletsuperconduc-
m(t) t 6t
tivity in BKFA for x = 0.39, 0.64, 0.69. Combined with 1− =0.1exp − +0.9exp − , (9)
the results for x = 1,45) even in the heavily hole-doped m0 (cid:18) T1(cid:19) (cid:18) T1(cid:19)
region,the spin partof superconducting pairingsymme- where m(t) and m are nuclear magnetizations after a
0
try is singlet as realized in other iron-pnictide supercon- time t andenoughtime fromthe NMR saturationpulse.
ductors21,38,48,49)whichareclosetoparentcompensated Inthe insetofFig.4,weshowthe typicalrecoverycurve
metals. Note that we could not find any anomaly of Kc for x = 0.69 obtained at 1.45, 44, 267 K. Clearly, the
associated with multiple SC gap structures. dataarewellfittedtotheaboveidealcurvewithasingle
T component.We tried to use this formula for x = 0.27
1
3.4 Spin lattice relaxation rate 1/T1 in normal state belowabout100Kundermagneticfieldparalleltotheab
InFig.4,weshowthetemperaturedependenceofspin- plane, but could not obtain good fitting results. This is
lattice relaxation rate divided by temperature 1/T1T of duetomorethantwoT1 componentsbelowthistemper-
75As of BKFA in the normal state with magnetic field ature because the center line shownin Fig. 2(b) consists
(a) parallel and (b) perpendicular to the ab plane. The of the paramagnetic and AF states. For this composi-
nuclear magnetization recovery curve was fitted by the tion parallel to the c axis we could obtain T down to
1
followingdouble-exponentialfunctionasexpectedforthe T because the center line consists of only paramagnetic
c
center line of the spectrum of the nuclear spin I = 3/2 state.
For all the composition, the 1/T T monotonically in-
1
creases down to approximately 100 K. However, be-
6 J.Phys.Soc.Jpn. FullPaper AuthorName
low this temperature there is clear difference. While distance from the AF instability point. The value of χ
0
the 1/T T continuously increases down to T for x = strongly influences evaluation of θ . In the present
1 c CW
0.27, 0.31, 0.39, and 1, it saturates at around 60-80 K, case, we determined χ as a product of 1/T T at 300 K
0 1
and decreases below this temperature for x = 0.58, and coefficient ε(x). ε(x) reflects convergence of T T at
1
0.64, 0.69. This gap-like behavior for x = 0.58, 0.64, hightemperatures:typicalvaluesofε(x)are0.8and0.55
0.69 is remembrance of that in electron-doped system for x = 0 and 1, respectively. In order to discuss the x
Ba(Fe1−xCox)2As2.48)Inthesystemthetemperaturede- dependenceofθCW,weshouldperformthefittingforthe
pendence of 1/T T is interpreted as thermal excitation samerangeoftemperature.However,asdiscussedabove,
1
originatingfromspecificbandstructureowingtothedis- gap-like behavior was observed at around 50-100 K for
appearance of the hole Fermi surfaces by electron dop- x = 0.58, 0.64, and 0.69. Therefore, it is adequate to
ing.51) In BKFA, Fermi surfaces change with increasing discuss spin fluctuation in this system using higher tem-
x:holeFermisurfacesataroundtheΓpointintheBZbe- perature data. We performed the fitting using the data
comes larger, and electron Fermi surfaces at around the above 200 K. In Fig. 5, we summarized x dependence of
X point shrinks significantly or disappears. This Fermi θ obtained from the fitting and that of T . Because
CW c
surface change is reported by recent ARPES measure- we adopted the eq. (11) at high temperatures and the
ments,38) and it worsen the nesting condition in BKFA. absolute value of θ has ambiguity due to experimen-
CW
Since the parent compound BaFe As is a compensated tal error of T , the relative values and overall feature
2 2 1
metal,itisnaturaltoconsiderthatthegap-likebehavior ofthe xdependence ofθ areimportantto discussthe
CW
inthissystemhasthesameoriginexplainedforelectron- spinfluctuationinthissystem.Thexdependenceofθ
CW
dopedsystemBa(Fe1−xCox)2As2.However,the detailed indicates that the AF fluctuation becomes weaker with
change of band structure in hole-doped system is some- increasing x and that the spin fluctuation remains even
what different from that in electron-doped system.52) In at the end composition x = 1. The tendency that the
Ba(Fe1−xCox)2As2, hole Fermi surfaces disappear with AF spinfluctuation becomes weakerwith increasingxis
electron doping, and their density of states near Fermi attributabletothepoorernestingconditionofFermisur-
level also decrease with further doping. In contrast, in face with increasing x. This is consistent with recent re-
BKFA electron Fermi surfaces do not completely disap- portsoninelasticneutronscatteringofBKFA.35,57) The
pear with hole doping, but their density of states near θ obtainedathighertemperaturescorrespondstothe
CW
Fermi level still exist with further doping until x =1. potentialAFfluctuationathighenergyexcitation,which
This is confirmed by ARPES measurements36,38,53–56) is responsible forsuperconductivity,comparedwith that
and also by band structure calculation.17,35) The recov- obtainedatlowertemperatures.These results showthat
ery of increase of the 1/T T at low temperatures in x = there is a correlation between attracting interaction of
1
1 is also indirectevidence for the specific band structure superconductivity and AF spin fluctuation to some ex-
of hole-doped system. tent.
TheAFspinfluctuationisstrongnearoptimallydoped Although overall feature of temperature dependence
region (x = 0.3-0.4). This is confirmed by our previ- of 1/T is quite similar between 1/T with magnetic
1 1
ousNMRmeasurementsusingpolycrystallinesamples40) field parallel to ab plane and that parallel to the c axis.
andalsobyneutrondiffractionmeasurements.57)TheAF However, 1/T (T) is always greater than 1/T (T)
1ab 1c
spin fluctuation also exists in heavily overdoped region at every temperature for all x’s. In order to discuss
for x = 0.94 and 1. Because T independent dynamical this tendency which is related with the dynamical spin
susceptibility becomes larger with increasing x, relative susceptibility χ(~q,ω), we adopt the same procedure
1/T T for x = 0.94 and 1 is larger than that for x = described in ref. 59. As described in eq. (10), 1/T T
1 1
0.3-0.4.From the temperature dependence of 1/T T, we is related with the ~q-dependent hyperfine coupling
1
can discuss the ~q-dependent spin fluctuation of system. constant and dynamical susceptibility. This equation
1 χ′′(~q,ω ) can be derived by using fluctuation-dissipation theorem.
∝ |Aq~|2 ⊥ 0 (10) The combination of the hyperfine coupling constant
T T ω
1 Xq~ 0 and dynamical susceptibility is intrinsically brought out
′′ from hyperfine field at As site. By using hyperfine field,
Here, A and χ (~q,ω) are ~q-dependent hyperfine cou-
q~ ⊥
1/T can be rewritten as follows:
plingconstantandperpendicularcomponentagainstthe 1
quantizationaxisofimaginarypartofdynamicalsuscep-
tibility, respectively. Based on the self consistent renor- 1 (µ γ )2 +∞
0 N
= dt(hH (t),H (0)i
malization theory (SCR) assuming two dimensional AF (cid:18)T1(cid:19)z 2 Z−∞ hf,x hf,x
fluctuation, this general expression for 1/T T can be
written as follows.58) 1 +hHhf,y(t),Hhf,y(0)i)eiω0t(12)
1 C = (µ γ )2 |H (ω )|2+|H (ω )|2(1.3)
=χ + (11) 0 N hf,x 0 hf,y 0
T T 0 T −θ (cid:0) (cid:1)
1 CW Here,directionz correspondstothedirectionofexternal
Here, χ0, C, and θCW are constant term of dynamical field. Hhf,x and Hhf,y are hyperfine fields perpendicular
susceptibility,Curieconstant,andCurie-Weisstempera- to the z direction. We assume that the hyperfine field
ture, respectively. χ0 includes all of the temperature in- H~hAfs at As site is brought out from spin S~ originating
dependencetermsofdynamicalsusceptibility.TheCurie- from the nearest neighboring four Fe sites through the
Weiss temperature in this definition corresponds to the
J.Phys.Soc.Jpn. FullPaper AuthorName 7
hyperfine interaction tensor A˜.
4
H~As = B˜ S~ =A˜S~ (14)
hf i i
Xi=1
A D B
a 1
A˜= D Ab B2 (15)
B B A
1 2 c
 
Here,S~ andB˜ arespinatthei-thFesiteandhyperfine
i i
coupling interaction tensor between the i-th Fe site and
As site, respectively. A (i = a,b,c), B (i = 1,2), and
i i
D arediagonalcomponentofA˜alongthe i direction,off
diagonal component related with wave vector ~q = (π,0)
or (0,π), and off diagonal component related with wave
vector ~q = (π,π), respectively.60) Note that the wave
vector ~q = (π,0) or (0,π) used here corresponds to the
wave vector ~q = (π,π) which was experimentally found
in inelastic neutron diffraction study.35) By using these Fig. 5. (color online) Phase diagram of BKFA andthe xdepen-
equationsandassumingthatdiagonalcomponentsofthe denceofobtainedCurie-WeisstemperatureθCW fromthefitting
describedinthetext.Redboldlineisguidetotheeye.
tensor and spin are equivalent within ab plane (A =
ab
A = A , S = S = S ), we obtain relation between
a b ab a b
the ratio R≡T /T and spin S~.
1c 1ab
A S 2
c c
0.5+0.5 :nocorrelation (16a)
 (cid:18)A S (cid:19)
ab ab
R=  0.5+ Sab 2 :(π,0)or(0,π) (16b)
(cid:18) S (cid:19)
c
 0.5 :(π,π) (16c)
As discussed in the subsection of Knight shifts, there is
a tendency that A and K (T) are greater than A
ab ab c
and K (T) for entire x’s, respectively. Hence, we may
c
assume that S is also greater than S . This leads that
ab c
the value of eq. (16a) is less than 1 and that the value
of eq. (16b) is greater than 1.5. In Fig. 6, we plotted
temperaturedependenceoftheratioR.Risgreaterthan
1 for all x’s within error bar. This result indicates that
spin fluctuation with the wave vector~q =(π,0) or (0,π)
orwavevectornearsuchwavevector,whichcorresponds
tothewavevector~q =(π,π)withthenotationintheBZ
Fig. 6. (color online) Temperature dependence of the ratio of
for tetragonal I4/mmm structure, is dominant in entire
1/T1ab to1/T1c.
x composition of BKFA. Note that the current result
anisotropyof 1/T for optimally doped composition x=
1
0.39 is consistent with the report for x = 0.32 by Li et 1/T forx=0.69under5.1TvariesproportionaltoT
1
al.61) Moreover, conclusion obtained in this subsection atlowesttemperatures.Withincreasingexternalfield,T
is consistentwith the recentinelastic neutrondiffraction linearregionexpands.AndT linearbehaviordisappeared
studies.35,57) under 2.4 T. 1/T for x = 0.69 is consistent with the
1
report of Zhang et al.,62) although steplike feature just
3.5 1/T in superconducting state
1 below T was not observed.
c
InFig.7,weshowT dependence of1/T1 forx =0.39, In order to understand the change of gap structure,
0.58,0.69 and1.Eacharrowindicates Tc under external we analyze 1/T1 in SC state with two gap model. For a
field. 1/T1 has no coherence peak just below Tc and de- simplicity we assume that both gaps have same symme-
creases rapidly. T dependence of 1/T1 from Tc to 0.8Tc try suchasfull gappeds± orline node.Here,weassume
is approximately proportional to T7 for x = 0.39, T3.2 line node model for s±-wave with line node or d-wave
for x = 0.58, T2.2 for x = 0.69. The power-law behavior symmetries. The 1/T in SC state is expressed as,
1
was also observed at lower temperature. The exponent
∞
1
xα =of p0.o3w9erc-oluawldTnαotdbecereoabsteasinweidthatinlcorweaersintegmxp.e1r/aTtu1rfeosr, T1 ∝iX=1,2n2i Z0 {NSi(E)2+MSi(E)2}f(E){1−f(E)}dE,
because the signal intensity dramatically decreases and
where Ni(E)2, Mi(E)2, f(E) are the density of states
T becomes longer in SC state. S S
1
8 J.Phys.Soc.Jpn. FullPaper AuthorName
Fig. 7. (coloronline)Temperaturedependence ofspin-latticere-
laxationrate1/T1 of75AsofBKFA. Figm.a8li.zed(csoplionr-olantltiincee)rNeloarxmatailoiznedratteem1p/Ter1atautrTecdoepfe7n5AdesnocfeBoKfnFoAr-.
Black dashed lines and brown solidlines are calculated 1/T1 of
fullygappeds± modelandlinenodemodel,respectively.
(DOS), the anomalous DOS arising from the coherence
effect of Cooper paris, and the Fermi distribution func-
tions, respectively. ni presents the fraction of DOS of fitting at low temperatures is worse than that at higher
thei-thgapandn1+n2 =1.Thisisthesameprocedure temperatures.
utilized in Refs. 20 and 21. InFig.9,weshowthexdependenceofanalyzedsuper-
conductinggapparameter∆dividedbyT ofBKFA.The
c
x dependence of 2∆ /T for both i’s shows monotonous
i c
Table I. Fittedparameters decrease with increasing x. The smaller gap 2∆2/Tc
Gap type x 2∆1(0)/Tc 2∆2(0)/Tc n1 δi/∆i rapidly decreases from x = 0.39 to 0.58. This proba-
s± 0.39 9.6 2.5 0.76 0.1 bly originates from the poorer nesting condition which
s± 0.58 5.8 0.92 0.75 0.1 arises from the change of band structure by hole dop-
s± 0.69 4.8 0.62 0.65 0.1 ing. Smaller gap values above x = 0.58 were strongly
s± 1.0 4.0 0.54 0.51 0.1 suppressed. It corresponds to concentrations where the
d-wave 0.39 14.2 4.4 0.80 - gapbehaviorisobserved.Thexdependenceof2∆ /T is
2 c
d-wave 0.58 8.1 0.91 0.75 - consistentwiththerecentresultsofARPESofBKFA.56)
d-wave 0.69 5.6 0.66 0.64 - The larger gap 2∆ /T is related with the inclination of
1 c
d-wave 1.0 4.2 0.48 0.50 - 1/T just below T . Compared with the x dependence
1 c
of 2∆ /T , the x dependence of 2∆ /T is more gradual
2 c 1 c
andthemagnitudedecreasescontinuously.Therefore,we
may conclude that there is no SC symmetry change in
In Fig. 8, we show experimentally obtained 1/T and
1 BKFAand thatthe nodal-line SC gapstructure realized
calculated 1/T . Each 1/T is normalized by the value
1 1 in x = 1 should be explained with the same SC sym-
at T in order to clarify x dependence of 1/T . The ex-
c 1 metry as that realized in optimally doped region x ∼
perimentaldata for x=1 was quotedfrom Ref.27) using
0.4.We may speculate that nodal-line structure emerges
poly crystals,becausewe couldnotperformNMR/NQR
from x∼ 0.7 and that it develops gradually without the
measurements using single crystals owing to the drastic
change of SC gap symmetry in this system. However,
reductionofNMRsignalsinSCstate.Blackdashedlines
we cannotcompletely deny the possibility of the SC gap
areobtainedwithfullygappeds± modelandbrownsolid symmetry change at around x∼ 0.7 since the numerical
lineswithlinenodemodel.Thebestfittedparametersare
calculationsuggeststhattheSCcondensationenergyfor
listed in Table. I. Although we neglected the impurity
d-wave nodal-line gap symmetry is slightly lower than
effect,theobtainedgapparametersforx=0.39arecon-
that for fully-gapped s-wave gap symmetry and that T
c
sistentwith the previousreportonself-flux-grownsingle
andtheSCgap2∆ /T changecontinuouslywithdoping
i c
crystal for x = 0.32.61) Both models can well reproduce in such case.19,34)
the observed 1/T behaviors in SC state, although the
1 The gap symmetry cannot be determined solely with
J.Phys.Soc.Jpn. FullPaper AuthorName 9
doping. Knight shift in SC state exhibits decease below
T (H) for x = 0.39, 0.64, 0.69 and the spin part of SC
c
paringsymmetryisalsosingletintheheavilyhole-doped
region.
T dependence of 1/T T indicates that AF spin fluc-
1
tuation exist for all x. The gap-like temperature depen-
dence for x = 0.58, 0.64, 0.69 is also observed.The gap-
like behavior has probably the same originexplained for
electron-dopedsystem.Inordertoestimatethe strength
of spin fluctuation, we performed the fitting from 200to
300 K using 2D SCR theory. The x dependence of θ
CW
indicates that the spin fluctuation becomes weaker with
increasingx andthat the spine fluctuationremains even
at x = 1. The spin fluctuation tends to become weaker
with increasing x. This tendency also relates with the
x dependence of T . The ratio experimental (1/T ) to
c 1 ab
(1/T ) is greater than 1 in all T for all x. This indi-
1 c
cates that the wave vector ~q = (π,π) with the notation
Fig. 9. (color online) The x dependence of analyzed supercon-
in the BZ for tetragonal structure is dominant in entire
ducting gap parameter ∆ divided by Tc of Ba1−xKxFe2As2.
x. Therefore, we may conclude that there is correlation
Solid triangles, open triangles, solid diamonds, and open dia-
monds denote 2∆1/Tc for full gap model, 2∆2/Tc for full gap between attracting interaction of superconductivity and
model,2∆1/Tc fornodal-linegapmodel,and2∆2/Tc fornodal- AF spin fluctuation to some extent.
line gap model, respectively. Black dashed line shows zero of T dependence of 1/T in SC state has no coherence
1
2∆i/Tc. peakjustbelowT anddecreasesrapidly.Inordertoun-
c
derstand the change of gap structure, we calculate 1/T
1
in SC state with two gap model with s± and nodal-
NMR measurements, which can be clearly understood
line symmetry cases.Both symmetry model can well ex-
fromthecalculatedresults.Todeterminegapsymmetry,
plaininthissystem,althoughSCsymmetrycouldnotbe
not only NMR/NQR experiments but also other exper-
determined by only NMR experiments. Both obtained
iments such as specific heat, low temperature ARPES,
larger gap ∆ /k T and smaller gap ∆ /k T rapidly
arenecessaryassuggestedin ourpreviousreport.27) Re- 1 B c 2 B c
decreasefromx= 0.39to 0.58.This probablyoriginates
cent ARPES experiment of KFA observed vertical node
from the poor nesting condition arising from the change
on zone central hole Fermi surfaces.32) Since BKFA has
ofbandstructure.Furthermore,∆ /k T isstronglysup-
complicated band and gap structures, theoretical inter- 2 B c
pressed above x = 0.58. The x dependence of ∆ /k T
pretationswhichcanexplaintheseresultswithoutincon- 1 B c
isdecreasescontinuouslywithincreasingx.Therefore,it
sistency are required.
can be concluded that there is basically no SC symme-
Finally,wediscussedtheoriginofT linearbehaviorfor
try change in BKFA. The nodal-line SC gap structure
x = 0.69 with increasing magnetic field at lowest mea-
may be explained with the same SC symmetry in opti-
surement temperature. According to the obtained gap
mum doped region. According to the analysis for 1/T
values, ratio of ∆ to ∆ is nearly 8. When we assume 1
1 2 forx=0.69undermagneticfields,wealsoconcludethat
the upper critical field of SC (H ) is proportional to
c2 thesmallergapiscollapsedbytheexternalfieldatlower
T (∼20 K), H of ∆ would be 20 T or more than this
c c2 1 temperatures.
value. Thus H of ∆ becomes equal to or more than
c2 2
approximately 2.5 T. By this estimation it is natural to Acknowledgments
considerthatthesmallergapiscollapsedbytheexternal
The authors thank K. Ohishi, Y. Ishii, I. Watanabe,
magnetic field and that T linear behavior under higher
K. Okazaki, W. Malaeb, Y. Oota, S. Shin, S. Kittaka,
magnetic fields at lower temperatures in SC state was
Y. Aoki, and T. Sakakibara for fruitful discussion and
observed.
permitting the authors to refer their unpublished data.
4. Conclusion This work is supported by Grants-in-Aid for Scientific
Research(Nos.21540351&22684016)fromtheMinistry
In summary, we performed NMR/NQR measurement
of Education, Culture, Sports, Science and Technology
for BKFA. The x dependence of the experimental ν
Q (MEXT) and the Japan Society for the Promotion of
is linear increase with increasing concentration x. This
Science(JSPS),andInnovativeAreas“HeavyElectrons”
tendencyisconsistentwithobtainedν bytheelectronic
Q (Nos. 20102005 & 21102505) from MEXT, Global COE
structure calculation using WINE2K code.
andAGGST financialsupportprogramfromChibaUni-
T dependence of NMR spectra for x = 0.27 revealed
versity. This work at Ames Laboratory was supported
that the paramagnetic region and antiferromanetically
by the Division of Material Sciences and Engineering,
ordered region are phase-separated. Knight shift in nor-
Office of BasicEnergySciences,U.S. Department ofEn-
mal state exhibits weak temperature dependence for all
ergy. Ames Laboratory is operated for US Department
x. With increasing x, Knight shift tends to increase due
of Energy by Iowa State University under contract No.
to the increase of DOS at the Fermi level with hole
DE-AC02-07CH11358.
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