NMR Investigation of the Quasi One-dimensional Superconductor K$_{2}$Cr$_{3}$As$_{3}$

1.7 MB·English

by  H. Z. Zhi

NMR Investigation of the Quasi One-dimensional Superconductor K Cr As
2 3 3
H. Z. Zhi,1 T. Imai,1,2,∗ F. L. Ning,3,4 Jin-Ke Bao,3,4 and Guang-Han Cao3,4
1Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S4M1, Canada
2Canadian Institute for Advanced Research, Toronto, Ontario M5G1Z8, Canada
3Department of Physics, Zhejiang University, Hangzhou 310027, China
4Collaborative Innovation Centre of Advanced Microstructures, Nanjing University, Nanjing 210093, China
(Dated: March 25, 2015)
Wereport75AsNMRmeasurementsonthenewquasione-dimensionalsuperconductorK Cr As
2 3 3
(T ∼ 6.1 K) [J. K. Bao et al., Phys. Rev. X 5, 011013 (2015)]. We found evidence for strong
5 c
enhancementofCrspinfluctuationsaboveT inthe[Cr As ] double-walledsubnano-tubesbased
1 c 3 3 ∞
on the nuclear spin-lattice relaxation rate 1/T . The power law temperature dependence, 1/T T ∼
0 1 1
T−γ (γ ∼0.25),isconsistentwiththeTomonaga-Luttingerliquid. Moreover,absenceoftheHebel-
2
Slichter coherence peak of 1/T1 just below Tc suggests unconventional nature of superconductivity.
r
a
M PACSnumbers: 74.70.Xa,76.60.-k,71.10.Pm

4
2 The surprising discovery of high Tc superconductiv- (a) (b)
ity in an iron-pnictide LaFeAsO F [1] led to fran- Cr2
1−x x K2 K2 Cr2
] tic search for superconductivity in iron-based pnictides As2 Cr1
n and chalcogenides [2]. The key building unit of these As2 c
o iron-based high T superconductors is the square lattice K1
c c As1
formed by Fe atoms. The mechanism of high T super- As1
- c b
r conductivity in iron-pnictides remains as controversial K1 Cr1 Cr1
p a As2
as that in copper-oxides, which is also comprised of the As1
u (c) (d) Cr2
s squarelatticeofCuatoms. Therecentdiscoveryofsuper-
. conductivityinhelimagneticCrAswithT ∼2.2Kunder
t c
a modestappliedpressureof∼0.7GPa[3,4]expandedthe c
m avenue of superconductivity research to Cr-pnictides. Cr2
- Very recently, Bao et al. discovered superconductiv- As2 Cr1
d
ity in K Cr As with the onset of T =6.1 K in ambient As1
n 2 3 3 c
o pressure[5]. ReplacingK+ ionswithlargerRb+ andCs+
c ionslowerstheT to4.8K[6]and2.2K[7],respectively.
c FIG. 1: (Color online) The crystal structure of quasi 1D
[ In Fig. 1, we present the proposed crystal structure of
3 K2Cr3As3 with the most probable space group P6m2 sAupceormcopnledtuect[Corr3KA2sC3]r∞3ADs3W[5S]T. (ais)iTnotphveiemwidodflfeo.ur(bu)nAitncgellelsd.
v (No. 187) [5]. The key building unit of K2Cr3As3 is the view of a unit cell, and (c)-(d) DWST.
3 one dimensional [(Cr As )2−] double-walled subnano-
3 3 ∞
1 tubes(DWSTs)withtheouterdiameterof0.58nmsepa-
7 ratedbycolumnsofK+ions,instrikingcontrastwiththe the [Cr As ] DWSTs provides us with a unique oppor-
0 3 3 ∞
two dimensional square lattice that forms iron-pnictide tunity to investigate the fundamental physics of a quasi
0
and copper-oxide high T superconductors. Assuming one dimensional (1D) inorganic metal, and its relation
. c
1 the standard ionic valence As3− and ignoring vacancies with superconductivity [5]. Theoretically, it is well es-
0
inthelattice,thenominalvalenceofthetransitionmetal tablishedthatFermionsconfinedin1D,commonlycalled
5
elementisCr2.3+,andhenceeachCrhas3.73delectrons the Tomonaga-Luttinger liquid (TLL), behave very dif-
1
: on average. The short interatomic distance between Cr ferentlyfromthetwoorthreedimensionalanalogues,be-
v atoms, 2.61 to 2.69 ˚A, suggests that the Cr-Cr bond- cause electron-electron interactions completely alter the
Xi ing is metallic, while the bonding of Cr with As3− ions electronic properties near the Fermi energy no matter
may be considered ionic [5]. The bulk physical property how weak the interactions are [9–11]. As a consequence,
r
a measurements on the critical field H [5], the electronic a simple Fermi liquid theory based on Landau’s quasi-
c2
specific[5,6],andthepenetrationdepth[8]pointtoward particle picture breaks down. The peculiar influence of
the presence of a node(s) in the superconducting energy interactions between two particles in the TLL could be
gap. understood intuitively, if we realize that two particles
Besides the generic interest in the mechanism of ex- mustalwaysmeeteachotherheadonin1D;theycannot
otic superconductors, the novel linear chain structure of avoid each other by moving side ways. A fingerprint of
theTLListhepower-lawbehaviorarisingfromthesingu-
larityattheFermienergythatmanifestinvariousphysi-
cal properties. Past explorations of the exotic properties
∗Electronicaddress: [email protected] oftheTLLfocusedonorganicconductors(see,forexam-
2
ple, discussions in [12–14] and references therein), semi-
conductornanostructures[15],carbonnanotubes[16–20] 75As NQR (B = 0) 48 B
andtheirsuperconductivity[21,22], orquasi-1DHeisen- 2
bergmodelsystems[23–25]. Doesthe[Cr3As3]∞ DWST MHz) B
in K2Cr3As3 indeed exhibit the signatures expected for (νQ 1
btheelowTLTLc aalbsooveexTotci?c?If so, is the superconducting state b. units] (a) 200 K A 380 100 200 A 300
ar T (K)
tNaMgIneRotfihntivsheesptvaiegpraestrai,otinwleeonfraKetpu2orCerrto3fAthtseh3ebfiNyrsMtfuRmllyitcetrcaohksncinioqgpuiecasd.7v5WaAnes- o Intensity [ B1 B2
h
probed the electronic and superconducting properties at Ec
different75Assitesseparatelybymeasuringtheirnuclear Spin (b) 6.5 K A B B
1 2
spin-lattice relaxation rate 1/T . We found evidence
1
for strong enhancement of Cr spin fluctuations toward 36 38 40 42 44 46 48 50
T . The dynamical electron spin susceptibility χ(cid:48)(cid:48) of the Frequency [Mhz]
c
DWST obeys a characteristic temperature dependence,
χ(cid:48)(cid:48) ∝ 1/T1T ∼ T−γ (γ = 0.25±0.03). The observed FIG. 2: (Color online) 75As NQR spectra measured at (a)
powerlawbehaviorresemblesthatoftheorganicconduc- 200Kand(b)6.5K.WedividedtherawNQRsignalintensity
tor TTF[Ni(dmit)2]2 [13] and carbon nanotubes [17–20], by the square of the frequency to take into account variation
and is consistent with the TLL. Moreover, we show that of the sensitivity. We marked three distinctive peaks as A,
theHebel-Slichtercoherencepeakof1/T [26],commonly B , and B . Inset: the temperature dependence of the 75As
1 1 2
observed for conventional BCS s-wave superconductors NQR peak frequency, νQ. All solid curves are guides for the
with isotropic energy gaps, is absent in the present case. eye.
In view of the low symmetry of the 75As sites in
K Cr As , the EFG (electric field gradient) at the 75As
2 3 3
sitesoriginatingfromK+, Cr2.3+, andAs3− ionsintheir determinewhichoftheAandBlinesarisefromAs1and
vicinity must be large. Since the nuclear quadrupole in- As2 sites, but none of our discussions below depend on
teractionfrequencyν ofthe75Asnuclearspins(I =3/2, the details of the site assignments.
Q
nuclear gyromagnetic ratio γ /2π = 7.2919 MHz/T) The presence of smaller side peaks and broad contin-
n
is proportional to the EFG, we anticipated that ν in uum suggests the influence of K+ defects on ν , as of-
Q Q
K Cr As shouldbemuchlargerthanν ∼2MHzinthe ten observed in alloys and disordered materials. Accord-
2 3 3 Q
iron-pnictidehighT superconductorBa(Fe Co ) As ing to the energy-dispersive X-ray spectroscopy (EDS)
c 1−x x 2 2
[27]. (The 75As site in Ba(Fe Co ) As is more sym- analysis, the composition of the present material is ac-
1−x x 2 2
metrical and surrounded by Ba2+ and Fe2+ ions.) We tually close to K Cr As [5]. Generally,
1.82±0.19 3 2.99±0.07
first searched for the 75As NMR signals in our randomly defects would locally alter the magnitude of ν through
Q
orientedpowdersampleinhighmagneticfields,andiden- the change of the EFG tensor (see [29] for a recent ex-
tified two distinct 75As NMR signals with ν ∼40 MHz ample of Cu substitution effects on ν in the BaFe As
Q Q 2 2
[28]. Such large values of ν would readily allow us to high T superconductor). Since the side peak of the B
Q c
detect 75As NQR (Nuclear Quadrupole Resonance) be- line is separately observable even at low temperatures,
tween the (nominal) I = ±3/2 and ±1/2 transitions we distinguish the main and side peaks by calling them
z
in zero external magnetic field, B = 0. NQR is advan- B andB ,respectively. Thetemperaturedependenceof
1 2
tageous in probing the intrinsic superconducting prop- the NQR frequency ν is very similar at A, B and B
Q 1 2
erties, because we do not perturb the superconducting sites as shown in the inset to Fig. 2. The decrease of ν
Q
state with the applied magnetic field. from2Kto295Kisanomalouslylarge,3∼5%. Theab-
Armed with the preliminary knowledge of ν , we senceofakinkorjumpinthetemperaturedependenceof
Q
searched for the 75As NQR signals between 33 and 55 νQ, however, rules out the presence of Peierls instability.
MHz. We show representative 75As NQR spectra in Fig. Next, let us examine the electronic properties of the
2. The lineshape observed at 200 K indicates the pres- [(Cr As )2−] DWSTs. In Fig. 3, we plot the tempera-
3 3 ∞
ence of two sets of sharp NQR peaks: the A line near ture dependence of 1/T measured by inversion recovery
1
39 MHz and the B line near 44 MHz, both accompanied techniques [28] at 75As sites in a log-log scale. We also
by smaller side peaks. Since the integrated intensity of present 1/T T in Fig. 4. Quite generally, 1/T T probes
1 1
the NQR signals below 43 MHz is equal to that above the wave-vector q-integral in the first Brillouin zone of
43 MHz, these two sets of NQR signals must arise from theimaginarypartofthedynamicalelectronspinsuscep-
As1 and As2 sites. Notice that the local arrangements tibility, χ(cid:48)(cid:48)(q,ν ), where ν is the NQR frequency used
Q Q
of K+ ions near As1 and As2 sites are different, as read- to measure 1/T . In other words, 1/T T probes the low
1 1
ily seen in Fig.1(a,b). Therefore ν should be somewhat frequency spin dynamics of electrons integrated over the
Q
different,too,betweenAs1andAs2sites. Unlesswecon- Brillouin zone. If the underlying electronic states of the
ductsinglecrystalNMRmeasurements,weareunableto DWSTs may be described by a simple Fermi liquid the-
3
1.2
75As NQR (B = 0) 75As NQR (B = 0) 4
100 1 Tc
T
K) c
~T 0.75 0.8 TT (s1 A
-1K)
10 Tc A ~T 1 -11/TT (s10.6 A 00 100 T (K) 200 300
-1s) B1 0.4 B1
1/T (1 B2 0.2 B2
1 0
0 50 100 150 200 250 300
T (K)
~T 4 FIG. 4: (Color online) 1/T T for A, B , and B sites. The
1 1 2
solid curve through the normal state data of the A sites rep-
0.1
resents the power-law fit, 1/T T ∼ T−γ, with the same γ as
1
inFig.3. Thedottedstraightlinethroughthedatapointsof
1 10 100 1000
the B peak is the Korringa fit, 1/T T =0.27 s−1K−1 above
T (K) 2 1
T . Inset: Temperature dependence of T T for the A peak
c 1
above T , with the best fit, T T ∼T+γ.
c 1
FIG. 3: (Color online) 75As nuclear spin-lattice relaxation
rate 1/T measured by NQR techniques in B =0 for A, B ,
1 1 the temperature range where the electrical resisitivity
and B sites. The solid line above T is the best fit for the A
2 c
sites with a power-law, 1/T ∼T1−γ (γ =0.25±0.03), while shows a linear temperature dependence [30–32]. The
1
the dashed-dotted line below T represents 1/T ∼ T4. The Curie-Weissbehavioristheoreticallyanticipatedfortwo-
c 1
dotted line is the best linear fit of the normal state data for dimensional electron gas systems with antiferromagnetic
the B peak, 1/T =0.27·T s−1. spin correlations [33, 34].
2 1
Incontrastwiththecaseofquasitwo-dimensionaliron-
pnictidehighT superconductors,thefundamentalstruc-
c
ory and the electron-hole pair excitations dominated the tural unit of quasi 1D K2Cr3As3 is a metallic DWST
low energy excitations at the Fermi surface, we expect formed by [Cr3As3]∞. Recent theoretical calculations
1/T ∝ T ·N(E )2, where N(E ) is the electronic den- predicted the existence of two quasi 1D bands and one
1 F F
sity of states at the Fermi energy E . That is, Korringa three dimensional band [35, 36], and very strong an-
F
relation holds, 1/T T ∼ constant, for Fermi liquids with tiferromagnetic correlations along the DWST with the
1
a broad band(s). The strong increase of 1/T T toward nearest-neighbor Cr-Cr exchange interaction as large as
1
T observed for the main A and B sites clearly con- ∼1000 K [36]. We also point out that the DWST’s form
c 1
tradicts with such an expectation. Without relying on a triangular-lattice, as shown in Fig. 1(a). This means
any theoretical assumptions, we conclude that Cr spin that three-dimensional antiferromagnetic couplings be-
fluctuations grow toward T prior to the onset of super- tweentheadjacentDWST’saregeometricallyfrustrated.
c
conductivity at a certain wave vector(s). Thequasi1Dnatureofspincorrelationsisthereforepro-
tected, suggesting the predominantly 1D character of Cr
The magnitude of 1/T T in Fig. 4 is comparable to
1
spin fluctuations enhanced near T .
that of the iron-pnictide high T superconductors such c
c
as Ba(Fe Co ) As [30–32]. If we assume that the Asexplainedabove,the1Delectrongaswithelectron-
1−x x 2 2
hyperfine coupling of 75As nuclear spins with Cr 3d elec- electron interactions forms a TLL that gives rise to a
tron spins in the present case is comparable to that with power law behavior invariousphysicalobservables. Ear-
Fe 3d electron spins in iron-pnictides, our results in Fig. lier theoretical calculations showed that the TLL would
4 imply that the dynamical spin susceptibility of Cr is showapower-lawbehavior,1/T1T ∼T−γ,wheretheex-
enhanced near T as much as the case of the supercon- ponentγ isanon-universalconstantthatdependsonthe
c
ductingBa(Fe Co ) As nearitsT ∼25K[30]. At details of the system, such as the band structure, nest-
0.92 0.08 2 2 c
firstglance,thegradualgrowthof1/T1T (andhenceχ(cid:48)(cid:48)) ing wave vector 2kF, and the strength of interactions
toward T in the present case also seems similar to typ- [12, 17, 19]. Analogous power-law behavior was previ-
c
ical iron-pnictide high Tc superconductors [30–32]. The ously reported for TTF[Ni(dmit)2]2 with γ ∼ 0.7 [13]
growth of χ(cid:48)(cid:48) near T within the FeAs planes of iron- and for single-wall carbon nanotubes with γ ∼0.66 [20].
c
pnicidescouldbesuccessfullyfitwithaCurie-Weisslaw, Close examination of our 1/T data in Fig. 3 indeed
1
1/T T ∼ C/(T +θ), where C and θ are constants, in reveals that all of our 1/T data points for the A (and
1 1
4
B ) sites above T are on a straight line in a log-log ducting state is not an isotropic s-wave state. Instead,
1 c
plot,implyingapower-law,1/T ∼T1−γ,orequivalently, an unconventional superconducting ground state with a
1
χ(cid:48)(cid:48) ∝1/T T ∼T−γ. The best fit yields γ =0.25±0.03. nodeintheenergygapseemsrealizedinK Cr As . This
1 2 3 3
We overlay a corresponding power-law fit with the same conclusion is consistent with other reports of the possi-
γ in Fig. 4, which nicely reproduces the mysteriously ble presence of the nodes in the energy gap based on the
strong divergent behavior of χ(cid:48)(cid:48) near T . Our findings measurements on the bulk properties [5, 6, 8].
c
are similar to earlier reports on quasi 1D materials with Finally, we comment on the nature of the broad side
the TLL behavior [12–14, 18–20, 24, 25]. The observed peak B in Fig. 2. 1/T T at the B sites is comparable
2 1 2
value of γ = 0.25 in the present case suggests that the to that of A and B sites near room temperature, but
1
electron-electroninteractionsarerepulsive,andthedom- remains constant, 1/T T = 0.27 s−1K−1, in the entire
1
inant channel of the spin correlations is antiferromag- temperaturerangeaboveT withnosignatureoftheTLL
c
netic for the wave vector 2k [12]. Our conclusion is behavior. As shown in Fig. 3, all the 1/T data points
F 1
consistentwiththelargeantiferromagneticexchangecou- of the B sites in the superconducting state are below a
2
pling∼1000Kexpectedfornearest-neighborCr-Crspins naiveextrapolationofthecorrespondingT-linearbehav-
along the DWST [36]. We call for additional experimen- ior. This is consistent with a viewpoint that the B sites
2
taltestsoftheTLLbehavioroftheDWSTsinK Cr As sense a small energy gap, suggesting the intrinsic nature
2 3 3
based on ARPES and other techniques. of the B sites. In this scenario, the broad line shape
2
Having established the characteristic quasi 1D behav- of the B sites must be a consequence of the defects in
2
ior of Cr spin fluctuations above T , let us turn our at- their vicinity, which may explain why the TLL behavior
c
tention to the superconducting state. In conventional is suppressed above T and the energy gap is very small
c
isotropic BCS s-wave superconductors, 1/T exhibits a below T . We cannot rule out, however, an alternative
1 c
hump just below T due to the sharp density of states at possibility that the B sites originate from a secondary
c 2
the edge of the energy gap, where the low energy quasi- phasewithslightlydifferentKconcentration,andtheB
2
particleexcitationscontributeconstructivelyto1/T due peak is merely superposed on the broad tail of the B
1 1
to the coherence factor predicted by the BCS theory sites accounting for ∼6 % of the overall intensity.
[26, 37]. The observation of such a Hebel-Slichter co- To summarize, we demonstrated strong enhancement
herence peak of 1/T is a crucial test for the validity of spin fluctuations in K Cr As toward T , obeying a
1 2 3 3 c
of the description of the superconducting state based on characteristic power-law predicted theoretically for the
the conventional isotropic BCS s-wave model. In addi- TLL. The absence of the Hebel-Slichter coherence peak
tion, 1/T decreases exponentially far below T , 1/T ∼ of 1/T just below T is followed by a steep decrease, in
1 c 1 1 c
exp(−∆/k T ), in isotropic BCS s-wave superconduc- analogy with unconventional superconductors in higher
B c
tors, where ∆ is the isotropic energy gap at the Fermi dimensions with point or line nodes in the energy gap.
surface [37, 38]. For contemporary examples of the con- K Cr As exhibits unique quasi 1D properties and be-
2 3 3
ventional BCS s-wave superconductors, see [39, 40]. longs to a league of its own, and deserves further atten-
In the present case, our 1/T data for the A and B tion both as a model system of the TLL and an exotic
1 1
sites in Fig. 3 show a steep drop just below T without superconductor.
c
exhibiting the coherence peak. Moreover, the tempera- Acknowledgement: H. Z. and T. I. thank S. Forbes
ture dependence of 1/T below T is consistent with a and Y. Mozharivskyj for their assistance with their
1 c
power law, 1/T ∼ Tn with n ∼ 4. These results below glovebox, and P. Dube for SQUID measurements. The
1
T aresimilartothecaseofunconventionalsuperconduc- crystal structure in Fig. 1 was plotted with Vesta [42].
c
tors, such as the high T superconductor YBa Cu O The work at McMaster was supported by NSERC and
c 2 3 7−δ
with d-wave pairing symmetry [41]. We are not aware of CIFAR. The work at Zhejiang was supported by the
any theoretical prediction of 1/T below T for the TLL; Natural Science Foundation of China (No. 11190023
1 c
ifweapplytheconventionalwisdomforexoticsupercon- and No. 11274268) and the National Basic Research
ductivityintwo-orthree-dimensionalcorrelatedelectron Program (No. 2011CBA00103 and No.2014CB921203).
systems,ourfindingsstronglysuggestthatthesupercon-
[1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, Y.-L. Sun, H.-F. Zhai, H. Jiang, H. Bai, C.-M. Feng,
J. Amer. Chem. Soc. 130, 3296 (2008). et al., Phys. Rev. X 5, 011013 (2015).
[2] D. Johnston, Advances in Physics 59, 803 (2010). [6] Z.-T. Tang, J.-K. Bao, Y. Liu, Y.-L. Sun, A. Ablimit,
[3] W. Wu, J. Cheng, K. Matsubayashi, P. Kong, F. Lin, H.-F. Zhai, H. Jiang, C.-M. Feng, Z.-A. Xu, and G.-H.
C. Jin, N. Wang, Y. Uwatoko, and J. Luo, Nat. Comm. Cao, arXiv:1412.2596.
5, 5508 (2014). [7] Z.-T. Tang, J.-K. Bao, Z. Wang, H. Bai, H. Jiang,
[4] H.Kotegawa,S.Nakahara,H.Tou,andH.Sugawara,J. Y. Liu, H.-F. Zhai, C.-M. Feng, Z.-A. Xu, and G.-H.
Phys. Soc. Jpn. 83, 093702 (2014). Cao, arXiv:1412.2596.
[5] J.-K.Bao,J.-Y.Liu,C.-W.Ma,Z.-H.Meng,Z.-T.Tang, [8] G.M.Pang,M.Smidman,W.B.Jiang,J.K.Bao,Z.F.
5
Weng,Y.F.Wang,L.Jiao,J.L.Zhang,G.H.Cao,and Kuhns,M.M.Turnbull,C.P.Landee,andH.-H.Klauss,
H. Q. Yuan, arXiv:1501.01880. Phys. Rev. B 83, 100407(R) (2011).
[9] S. Tomonaga, Prog. Theor. Physics 5, 544 (1950). [26] L. C. Hebel and C. P. Slichter, Phys. Rev. 113, 1504
[10] J. M. Luttinger, J. Math. Phys. 4, 1154 (1963). (1959).
[11] D. C. Mattis and E. H. Lieb, J. Math. Phys. 6, 304 [27] F.L.Ning,K.Ahilan,T.Imai,A.S.Sefat,R.Jin,M.A.
(1965). Mcguire,B.C.Sales,andD.Mandrus,J.Phys.Soc.Jpn.
[12] C. Bourbonnais, F. Creuzet, D. Jerome, K. Bechgaard, 77, 103705 (2008).
andA.Moradpour,J.dePhysiqueLett.45,L755(1984). [28] See the accompanying supplementary material.
[13] C. Bourbonnais, P. Wzietek, D. Jerome, F. Creuzet, [29] H. Takeda, T. Imai, M. Tachibana, J. Gaudet, B. D.
L.Valade,andP.Cassoux,Europhys.Lett.6,177(1988). Gaulin, B. I. Saparov, and A. S. Sefat, Phys. Rev. Lett.
[14] W. Wu, P. M. Chaikin, W. Kang, J. Shinagawa, W. Yu, 113, 117001 (2014).
and S. E. Brown, Phys. Rev. Lett. 94, 097004 (2005). [30] F. L. Ning, K. Ahilan, T. Imai, A. S. Sefat, M. A.
[15] M.Grayson,D.C.Tsui,L.N.Pfeiffer,K.W.West,and McGuire, B. C. Sales, D. Mandrus, P. Cheng, B. Shen,
A. M. Chang, Phys. Rev. Lett. 80, 1062 (1998). and H.-H. Wen, Phys. Rev. Lett. 104, 037001 (2010).
[16] M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. [31] Y. Nakai, T. Iye, S. Kitagawa, K. Ishida, H. Ikeda,
Smalley,L.Balents,andP.L.McEuen,Nature397,598 S.Kasahara,H.Shishido,T.Shibauchi,Y.Matsuda,and
(1999). T. Terashima, Phys. Rev. Lett. 105, 107003 (2010).
[17] H. Yoshioka, J. Phys. Chem. Solids 63, 1281 (2002). [32] T.Oka,Z.Li,S.Kawasaki,G.F.Chen,N.L.Wang,and
[18] P. M. Singer, P. Wzietek, H. Alloul, F. Simon, and G. Q. Zheng, Phys. Rev. Lett. 108, 047001 (2012).
H. Kuzmany, Phys. Rev. Lett. 95, 236403 (2005). [33] A. J. Millis, H. Monien, and D. Pines, Phys. Rev. B 42,
[19] B. Dora, M. Gulacsi, F. Simon, and H. Kuzmany, Phys. 167 (1990).
Rev. Lett. 99, 166402 (2007). [34] T. Moriya, Y. Takahashi, and K. Ueda, J. Phys. Soc.
[20] Y.Ihara,P.Wzietek,H.Alloul,M.H.Rummeli,T.Pich- Jpn. 59, 2905 (1990).
ler, and F. Simon, Euro. Phys. Lett 90, 17004 (2008). [35] H. Jiang, G. Cao, and C. Cao, arXiv:1412.1309.
[21] Z.K.Tang,L.Zhang,N.Wang,X.X.Zhang,G.H.Wen, [36] X. Wu, C. Le, J. Yuan, H. Fan, and J. Hu,
G.D.Li,J.N.Wang,C.T.Chan,andP.Sheng,Science arXiv:1501.00412.
292, 2462 (2001). [37] D. E. MacLaughlin, Solid State Physics 31, 1 (1976).
[22] W. Shi, Z. Wang, Q. Zhang, Y. Zheng, C. Ieong, M. He, [38] Y. Masuda and A. C. Redfield, Phys. Rev. 125, 159
R. Lortz, Y. Cai, N. Wang, T. Zhang, et al., Scientific (1962).
Reports 2, 625 (2012). [39] H. Kotegawa, K. Ishida, Y. Kitaoka, T. Muranaka, and
[23] M. Klanjsek, H. Mayaffre, C. Berthier, M. Horvatic, J. Akimitsu, Phys. Rev. Lett. 87, 127001 (2001).
B. Chiari, O. Piovesana, P. Bouillot, C. Kollath, [40] P. M. Singer, T. Imai, T. He, M. A. Hayward, and R. J.
E.Orignac,R.Citro,etal.,Phys.Rev.Lett.101,137207 Cava, Phys. Rev. Lett. 87, 257601 (2001).
(2008). [41] T. Imai, T. Shimizu, H. Yasuoka, Y. Ueda, and K. Ko-
[24] H.Kuhne,H.H.Klauss,S.Grossjohann,W.Brenig,F.J. suge, J. Phys. Soc. Jpn. 57, 2280 (1988).
Litterst, A. P. Reyes, P. L. Kuhns, M. M. Turnbull, and [42] K.MommaandF.Izumi,J.Appl.Crystallogr.44,1272
C. P. Landee, Phys. Rev. B 80, 045110 (2009). (2011).
[25] H.Kuhne,A.A.Zvyagin,M.Gunther,A.P.Reyes,P.L.