Giant superconducting fluctuations in the compensated semimetal FeSe at the BCS-BEC crossover

Abstract

The physics of the crossover between weak-coupling Bardeen-Cooper-Schrieffer (BCS) and strong-coupling Bose-Einstein condensate (BEC) limits gives a unified framework of quantum-bound (superfluid) states of interacting fermions. This crossover has been studied in the ultracold atomic systems, but is extremely difficult to be realized for electrons in solids. Recently, the superconducting semimetal FeSe with a transition temperature T_c =8.5 K has been found to be deep inside the BCS-BEC crossover regime. Here we report experimental signatures of preformed Cooper pairing in FeSe, whose energy scale is comparable to the Fermi energies. In stark contrast to usual superconductors, large non-linear diamagnetism by far exceeding the standard Gaussian superconducting fluctuations is observed below T*∼20 K, providing thermodynamic evidence for prevailing phase fluctuations of superconductivity. Nuclear magnetic resonance and transport data give evidence of pseudogap formation at ∼T*. The multiband superconductivity along with electron-hole compensation in FeSe may highlight a novel aspect of the BCS-BEC crossover physics.

ARTICLE

Received 8 Jun 2016 | Accepted 3 Aug 2016 | Published 30 Sep 2016

S. Kasahara1, T. Yamashita1, A. Shi1, R. Kobayashi2, Y. Shimoyama1, T. Watashige1, K. Ishida1, T. Terashima2,

T Wolf3, F. Hardy3, C. Meingast3, H. v. Lhneysen3, A. Levchenko4, T. Shibauchi5 & Y. Matsuda1

The physics of the crossover between weak-coupling BardeenCooperSchrieffer (BCS) and strong-coupling BoseEinstein condensate (BEC) limits gives a unied framework of quantum-bound (superuid) states of interacting fermions. This crossover has been studied in the ultracold atomic systems, but is extremely difcult to be realized for electrons in solids. Recently, the superconducting semimetal FeSe with a transition temperature Tc 8.5 K has

been found to be deep inside the BCSBEC crossover regime. Here we report experimental signatures of preformed Cooper pairing in FeSe, whose energy scale is comparable to the Fermi energies. In stark contrast to usual superconductors, large non-linear diamagnetism by far exceeding the standard Gaussian superconducting uctuations is observed below T*B20 K, providing thermodynamic evidence for prevailing phase uctuations of superconductivity. Nuclear magnetic resonance and transport data give evidence of pseudogap formation at BT*. The multiband superconductivity along with electronhole compensation in FeSe may highlight a novel aspect of the BCSBEC crossover physics.

DOI: 10.1038/ncomms12843 OPEN

Giant superconducting uctuations in the compensated semimetal FeSe at the BCSBEC crossover

1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan. 2 Research Center for Low Temperature and Materials Sciences, Kyoto University, Kyoto 606-8501, Japan. 3 Institute of Solid State Physics, Karlsruhe Institute of Technology, Karlsruhe D-76021, Germany. 4 Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA. 5 Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan. Correspondence and requests for materials should be addressed to T.S. (email: mailto:[email protected]

Web End [email protected] ) or to Y.M. (email: mailto:[email protected]

Web End [email protected] ).

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In the BardeenCooperSchrieffer (BCS) regime, weakly coupled pairs of fermions form the condensate wave function, while in the BoseEinstein condensate (BEC) regime, the

attraction is so strong that the fermions form local molecular pairs with bosonic character. The physics of the crossover is described by two length scales, the average pair size or coherence length xpair and the average interparticle distance 1/kF, where kF is the Fermi wave number. In the BCS regime, the pair size is very large and kFxpairc1, while local molecular pairs in the BEC regime lead to kFxpair51. The crossover regime is characterized by kFxpairB1, or equivalently the ratio of superconducting gap to Fermi energy D/eF of the order of unity. In this crossover regime, the pairs interact most strongly and new states of interacting fermions may appear; preformed Cooper pairing at much higher temperature than Tc is theoretically proposed1,2. Experimentally, however, such preformed pairing associated with the BCSBEC crossover has been controversially debated in ultracold atoms3,4 and cuprate superconductors58. Of particular interest is the pseudogap formation associated with the preformed pairs that lead to a suppression of low-energy single-particle excitations. Also important is the breakdown of Landaus Fermi liquid theory due to the strong interaction between fermions and uctuating bosons. In ultracold atomic systems, this crossover has been realized by tuning the strength of the interparticle interaction via the Feshbach resonance. In these articial systems, Fermi liquid-like behaviour has been reported in thermodynamics even in the middle of crossover3, but more recent photoemission experiments have suggested a sizeable pseudogap opening and a breakdown of the Fermi liquid description4.

On the other hand, for electron systems in bulk condensed matter, it has been extremely difcult to access the crossover regime. Perhaps, the most frequently studied systems have been underdoped high-Tc cuprate superconductors58 with substantially shorter coherence length than conventional superconductors. In underdoped cuprates, pseudogap formation and non-Fermi liquid behaviour are well established, and unusual superconducting uctuations have also been found above Tc (refs 6,7). However, the pseudogap appears at a much higher temperature than the onset temperature of superconducting uctuations8. It is still unclear whether the system is deep inside the crossover regime and to what extent the crossover physics is relevant to the phase diagram in underdoped cuprates. It has been also suggested that in iron-pnictide BaFe2(As1 xPx)2, the system may approach the

crossover regime in the very vicinity of a quantum critical point9,10, but the ne-tuning of the material to a quantum critical point by chemical substitution is hard to accomplish. Therefore, this situation calls for a search of new systems in the crossover regime.

Among different families of iron-based superconductors, iron chalcogenides FeSexTe1 x exhibit the strongest band renormaliza

tion due to electron correlations, and recent angle-resolved photoemission spectroscopy studies for x 0.35 0.4 have shown

that some of the bands near the Brillouin zone centre have very small Fermi energy, implying that the superconducting electrons in these bands are in the crossover regime11,12. Among the members of the iron chalcogenide series, FeSe (x 0) with the

simple crystal structure formed of tetrahedrally bonded layers of iron and selenium is particularly intriguing. FeSe undergoes a tetragonalorthorhombic structural transition at TsE90 K, but in contrast to other Fe-based superconductors, no long-range magnetic ordering occurs at any temperature. Recently, the availability of high-quality bulk single crystals grown by chemical vapour transport13 has reopened investigations into the electronic properties of FeSe. Several experiments performed on these crystals have shown that all Fermi surface bands are very shallow1416; one or two electron pockets centred at the Brillouin zone corner with

Fermi energy eeF 3 meV, and a compensating cylindrical

hole pocket near the zone centre with ehF 10 meV. FeSe is a

multigap superconductor with two distinct superconducting gaps D1E3.5 and D2E2.5 meV (ref. 14). Remarkably, the Fermi energies are comparable to the superconducting gaps; D/eF is

B0.3 and B1 for hole and electron bands, respectively14. These large D/eF(E1/(kFxpair)) values indicate that FeSe is in the

BCSBEC crossover regime. In fact, values of 2D1/kBTcE9 and

2D2/kBTcE6.5, which are signicantly enhanced with respect to the weak-coupling BCS value of 3.5, imply that the attractive interaction holding together the superconducting electron pairs takes on an extremely strong-coupling nature, as expected in the crossover regime. Moreover, the appearance of a new high-eld superconducting phase when the Zeeman energy is comparable to the gap and Fermi energies, m0HBDBeF, suggests a peculiar superconducting state of FeSe (ref. 14). Therefore, FeSe provides a new platform to study the electronic properties in the crossover regime.

Here we report experimental signatures of preformed Cooper pairing in FeSe below T*B20 K. Our highly sensitive magneto-metry, thermoelectric and nuclear magnetic resonance (NMR) measurements reveal an almost unprecedented giant diamagnetic response as a precursor to superconductivity and pseudogap formation below T*. This yields profound implications on exotic bound states of strongly interacting fermions. Furthermore, the peculiar electronic structure with the electronhole compensation in FeSe provides a new playground to study unexplored physics of quantum-bound states of interacting fermions.

ResultsGiant superconducting uctuations. It is well known that thermally uctuating droplets of Cooper pairs can survive above Tc.

These uctuations arise from amplitude uctuations of the superconducting order parameter and have been investigated for many decades. Their effect on thermodynamic, transport and thermoelectric quantities in most superconductors is well understood in terms of standard Gaussian uctuation theories17. However, in the presence of preformed pairs associated with the BCSBEC crossover, superconducting uctuations are expected to be strikingly enhanced compared with Gaussian theories due to additional phase uctuations. Moreover, it has been suggested that such enhanced uctuations can lead to a reduction of the density of states (DOS), dubbed the pseudogap1,2.

Quite generally, superconducting uctuations give rise to an enhancement of the normal-state conductivity, which manifests itself as a downturn towards lower T of the resistivity versus temperature curve above Tc. The high-eld magnetoresistance of compensated semimetals is essentially determined by the product of the scattering times of electron and hole bands14. The large, insulating-like upturn in rxx(T) at high elds is thus an indication of the high quality of our crystals (Fig. 1a). At low temperatures, however, the expected downturn behaviour is observed, implying large superconducting uctuations. Even at zero eld, drxx(T)/dT

shows a minimum around T*B20 K (Fig. 1b), indicating the appearance of excess conductivity below BT*. However, a quantitative analysis of this excess conductivity is difcult to achieve because it strongly depends on the extrapolation of the normal-state resistivity above T* to lower T. In addition, the resistivity may be affected by a change of the scattering time when a pseudogap opens at T* as observed in underdoped cuprates18.

We therefore examine the superconducting uctuations in FeSe through the diamagnetic response in the magnetization. The magnetization M(T) for magnetic eld H parallel to the c axis (Supplementary Fig. 1) exhibits a downward curvature below BT*. This pronounced decrease of M(T) can be attributed to the

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Figure 1 | Excess conductivity and diamagnetic response of a high-quality single crystal of FeSe. (a) T dependence of rxx in magnetic elds (H||c). The structural transition occurs at Ts 90 K, which is accompanied by a kink in rxx(T). Inset shows the crystal structure of FeSe. (b) T dependence of rxx (red)

and drxx/dT (grey). Below T* shown by arrow, rxx shows a downward curvature. The blue dashed line represents rxx(T) r0 ATa with r0 7 mO cm

A 0.6 mO cm K 2 and a 1.2. (c) Diamagnetic response in magnetization Mdia for H||c. The inset shows the diamagnetic susceptibility wdia at 8 T (blue)

compared with the estimated wAL in the standard Gaussian uctuations theory (red).

diamagnetic response due to superconducting uctuations. Figure 1c shows the diamagnetic response in the magnetization Mdia between 0 and 40 K for m0H 4, 8 and 12 T, obtained by

subtracting a constant M as determined at 30 K. Although there is some ambiguity due to weakly temperature-dependent normal-state susceptibility, we nd a rough crossing point in Mdia(T, H)

near Tc. Such a crossing behaviour is considered as a typical signature of large uctuations and has been found in cuprates19. The thermodynamic quantities do not include the Maki Thompson-type uctuations. Hence, the uctuation-induced diamagnetic susceptibility of most superconductors including multiband systems can be well described by the standard Gaussian-type (AslamasovLarkin, AL) uctuation susceptibility wAL (refs 2022), which is given by

wAL

2p2

3

r 1

in the zero-eld limit23. Here F0 is the ux quantum and xab (xc) is the effective coherence lengths parallel (perpendicular) to the ab plane at zero temperature. In the multiband case, the behaviour of wAL is determined by the shortest coherence length of the main band, which governs the orbital upper critical eld. The diamagnetic contribution wAL is expected to become smaller in magnitude at higher elds, and thus |wAL| yields an upper bound for the standard Gaussian-type amplitude uctuations. In the inset of Fig. 1c, we compare wdia at 8 T with wAL, where we use

xab ( 5.5 nm) and xc ( 1.5 nm)14,15. Obviously the amplitude

of wdia of FeSe is much larger than that expected in the standard theory, implying that the superconducting uctuations in FeSe are distinctly different from those in conventional superconductors.

The highly unusual nature of superconducting uctuations in FeSe can also be seen in the low-eld diamagnetic response. Since the low-eld magnetization below 2 T is not reliably obtained

from conventional magnetization measurements, we resort to sensitive torque magnetometry. The magnetic torque t m0VM H is a thermodynamic quantity that has a

high sensitivity for detecting magnetic anisotropy. Here V is the sample volume, M is the induced magnetization and H is the external magnetic eld. For our purposes, the most important advantage of this method is that an isotropic Curie contribution from impurity spins is cancelled out24.

At each temperature and eld, the angle-dependent torque curve t(y) is measured in H rotating within the ac (bc) plane, where y is the polar angle from the c axis. In this geometry, the difference between the c axis and ab plane susceptibilities, Dw wc wab,

yields a p-periodic oscillation term with respect to y rotation, t2y

T; H; y 12 m0H2VDw T; H sin 2y (Fig. 2a; Supplementary

Fig. 2; Supplementary Note 1)25,26. In the whole measurement range, Dw is negative, that is, wab4wc, which is consistent

with magnetic susceptibility27 and NMR Knight-shift measurements28,29. Figure 2b shows the T dependence of Dw at 7 T, which is determined by the amplitude of the sinusoidal curve. At Ts, Dw(T) exhibits a clear anomaly associated with the tetragonalorthorhombic structural transition. On approaching Tc, Dw shows a diverging behaviour. Figure 2c,d depicts the T and H dependence of |Dw|(T,H), respectively. Above T*B20 K, |Dw|(T, H) is nearly eld independent. Below T*, however, |Dw|(T,H) increases with decreasing H, indicating nonlinear H dependence of M. This non-linearity increases steeply with decreasing temperature. Since |Dw| points to a diverging behaviour in the zero-eld limit on approaching Tc (Fig. 2d), this strongly non-linear behaviour is clearly caused by superconducting uctuations.

Thus, the diamagnetic response of FeSe contains H-linear and non-linear contributions to the magnetization; Dw(T) can be written as Dw Dwnldia Dwldia DwN, where Dwnldia and

Dwldia represent the diamagnetic contributions from non-linear

kBTc

F20

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xc

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Figure 2 | Diamagnetic response detected by magnetic torque measurements above Tc. (a) The magnetic torque t as a function of y. Torque curves measured by rotating H in clockwise (red) and anticlockwise (blue) directions coincide (the hysteresis component is o0.01% of the total torque).

(b) Anisotropy of the susceptibility between the c axis and ab plane, Dw, at 7 T. The inset is schematics of the y-scan measurements. (c) The T dependence of |Dw| at various magnetic elds. (d) The H dependence of |Dw| at xed temperatures. (e) Temperature dependence of the non-linear diamagnetic response at m0H 0.5 T (red) and 1 T (blue) obtained by Dwnldia

DwH Dw7T. Blue line represents the estimated |DwAL| in the standard Gaussian

uctuations theory. (f) Dwnldia

plotted in a semi-log scale at low temperatures. Error bars represent s.d. of the sinusoidal t to the t(y) curves.

and linear eld dependence of magnetization, respectively, and DwN is the anisotropic part of the normal-state susceptibility, which is independent of H. Since Dw(T) is almost H independent at high elds (Fig. 2d), Dwnldia is estimated by subtracting H-independent terms from Dw. In Fig. 2e, we plot

Dwnldia

Dwnldia

in the zero-eld limit should be much larger than Dwnldia

at 0.5 T. Thus, the non-linear diamagnetic response dominates the superconducting uctuations when approaching Tc in the zero-eld limit. We note that, although the AL diamagnetic contribution contains a non-linear term visible at low elds, this term is always smaller than the AL uctuation contribution at zero eld2022.

Our magnetization and torque results provide thermodynamic evidence of giant superconducting uctuations in the normal

state of FeSe by far exceeding the Gaussian uctuations. We stress that, since the energy scale of kBT*B2 meV is comparable to eeF, it is natural to attribute the observed uctuations to preformed pairs associated with the BCSBEC crossover. In the presence of those pairs, superconducting phase uctuations5 arising from the mode coupling of uctuations are expected to be signicantly enhanced and to produce a highly non-linear diamagnetic response, as observed in the experiments. This non-linear response with large amplitude is profoundly different from the Gaussian behaviour in conventional superconductors.

Pseudogap formation. Next, we discuss the possible pseudogap formation associated with the preformed pairs, which suppresses the DOS and hence leads to a change in quasiparticle scattering. We have measured the relaxation time T1 of 77Se NMR spectroscopy in FeSe single crystals (Supplementary Fig. 3) at different elds applied along the c axis. At 14.5 T close to the upper critical eld, the temperature dependence of 1/T1T, which is dominated by the dynamical spin susceptibility w(q) at the antiferromagnetic wave vector q (p, p), can be tted well by a

estimated from DwnldiaH DwH Dw7T, which we

compare with the expectation from the Gaussian uctuation

theory at zero eld given by DwAL 2p23 kBTcF20

x2ab xc

TcT Tc

xc

q

.

Near Tc, Dwnldia at 0.5 T is nearly 10 times larger than DwAL. It should be noted that since Dwnldia

increases with decreasing H,

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Figure 3 | Possible pseudogap formation below T* evidenced by NMR and transport measurements. (a) Temperature dependence of the NMR relaxation rate divided by temperature 1/T1T. Inset: at 14.5 T, the temperature dependence of 1/T1T between B10 and 70 K is tted to a CurieWeiss law p(T 16 K) 1 (dashed line). Main panel: the difference between the CurieWeiss t and the low-eld data D(1/T1T) is plotted as a function of

temperature. (b) Hall coefcient, RH. (c) Seebeck coefcient, S. (d) Nernst coefcient, n, in the zero-eld limit as functions of temperature. Inset in d is a schematic of the measurement set-up of the thermoelectric coefcients.

CurieWeiss law in a wide temperature range below Ts (Fig. 3a, inset). At low elds of 1 and 2 T, however, 1/T1T(T) shows a noticeable deviation from this t (dashed line in Fig. 3a, inset), and the difference between the t and the low-eld data D(1/T1T) starts to grow at BT* (Fig. 3a, main panel). As the superconducting diamagnetism is an orbital effect that is dominated at q 0, the spin susceptibility w(p, p) is not

inuenced by the orbital diamagnetism. Therefore, the observed deviation of 1/T1T(T) is a strong indication of a depletion of the

DOS, providing spectroscopic evidence for the psedugap formation below BT *. The onset temperature and the eld dependence of the non-linear contribution of 1/T1T(T) bear a certain similarity to the features of the diamagnetic susceptibility, pointing to the intimate relation between the pseudogap and preformed pairs in this system.

The pseudogap formation is further corroborated by the measurements of Hall (RH), Seebeck (S) and Nernst (n)

coefcients (Fig. 3bd). The negative sign of the Hall and Seebeck data indicates that the transport properties are governed mainly by the electron band, which is consistent with the previous analysis of the electronic structure in the orthorhombic phase below Ts (ref. 16). Obviously, at T *B20 K, all the coefcients show a minimum or maximum. Since the Hall effect is insensitive to superconducting uctuations, the minimum of RH(p(sh se)/(sh se)), where se(h) is the

conductivity of electrons (holes), suggests a change of the carrier mobility at BT *. The thermomagnetic Nernst coefcient consists of two contributions generated by different mechanisms: n nN nS. The rst term represents the contribution of

normal quasiparticles. The second term, which is always positive, represents the contribution of uctuations of either amplitude or phase of the superconducting order parameter. On approaching Tc, nS is expected to diverge30. As shown in Fig. 3d, however, such a divergent behaviour is absent. This is because in the present very clean system, nN is much larger than nS (Supplementary Fig. 4a; Supplementary Note 2). Since nN

and S are proportional to the energy derivatives of the Hall angle and conductivity at the Fermi level, respectively, nN /

@ tan yH=@eeeF and S / @ ln s=@eeeF both sensitively detect

the change of the energy dependence and/or anisotropy of the scattering time at the Fermi surface (see also Supplementary Fig. 4b,c for n/T(T) and S/T(T)). Therefore, the temperature dependence of the three transport coefcients most likely implies a change in the quasiparticle excitations at T *, which is consistent with the pseudogap formation. We also note that anomalies at similar temperatures have been reported for the temperature dependence of the thermal expansion13 as well as of Youngs modulus29. Recent scanning tunnelling spectroscopy data also suggest some suppression of the DOS at low energies in a similar temperature range31.

DiscussionFigure 4 displays the schematic HT phase diagram of FeSe for H||c. The uctuation regime associated with preformed pairing is determined by the temperatures at which drxx(H)/dT shows a

minimum and n(H) shows a peak (Supplementary Fig. 5a,b; Supplementary Note 3) in magnetic elds, as well as by the onset of D(1/T1T) (Fig. 3a). The diamagnetic signal, NMR relaxation rate and transport data consistently indicate that the preformed pair regime extends over a wide range of the phase diagram. The phase uctuations dominate at low elds where the non-linear diamagnetic response is observed (Fig. 2d). This phase-uctuation region continuously connects to the vortex liquid regime above the irreversibility eld Hirr, where a

nite resistivity is observed with a broad superconducting transition (Fig. 1a).

Let us comment on the electronic specic heat, which is another thermodynamic quantity related to the DOS of quasiparticles. The specic heat C at comparatively high temperatures, however, is dominated by the phonon contribution pT3 (refs 29,32), which makes it difcult to resolve the pseudogap anomaly. Also, the reduction of C/T may partly be cancelled with the increase by the

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Figure 4 | HT phase diagram of FeSe for H||c. Solid line is the irreversibility line Hirr(T) (ref. 14). The colour represents the magnitude of Dw (in 10 5, scale shown in the colour bar) from magnetic torque measurements (Fig. 2c). Preformed pair regime is determined by the minimum of drxx(H)/dT (blue circles), the peak of Nernst coefcient npeak (green circles) and the onset of D(1/T1T) in the NMR measurements(red circles).

strong superconducting uctuations found in the present study. It should be also stressed that FeSe exhibits a semimetallic electronic structure with the compensation condition, that is, the electron and hole carrier densities should be the identical. Such a compensated situation of the electronic structure may alter signicantly the chemical potential shift expected in the BEC theories for a single-band electronic structure. How the entropy in crossover semimetals behaves below T * is a fundamentally new problem, which deserves further theoretical studies.

Finally, we remark that the preformed Cooper pairs and pseudogap develop in the non-Fermi liquid state characterized by a linear-in-temperature resistivity, highlighting the highly unusual normal state of FeSe in the BCSBEC crossover regime. The resistivity above T * can be tted up to B50 K as rxx(T)

rxx(0) ATa with a 1.1 1.2, where the uncertainty arises

from the fact that rxx(0) is unknown (Fig. 1b). Thus, the exponent a close to unity indicates a striking deviation from the Fermi liquid behaviour of a 2. This non-Fermi liquid behaviour in

FeSe is reminiscent of the anomalous normal-state properties of high-Tc cuprate superconductors. The main difference between these systems and FeSe is the multiband nature of the latter34,35;

the Fermi surface consists of compensating electron and hole pockets. The present observation of preformed pairs together with the breakdown of Fermi liquid theory in FeSe implies an inherent mechanism that brings about singular inelastic scattering properties of strongly interacting fermions in the BCSBEC crossover.

Methods

Sample preparation and characterization. High-quality single crystals of tetragonal b-FeSe were grown by low-temperature vapour transport method at Karlsruhe Institute of Technology and Kyoto University13. As shown in Fig. 1b, taking rxx(Tc )E10 mO cm as an upper limit of the residual resistivity leads to the residual resistivity ratio (RRR)440. The large RRR value, large magnetoresistance below Ts, quantum oscillations at high elds15,16, a very sharp 77Se NMR line width29, and extremely low level of impurities and defects observed by scanning tunnelling microscope topographic images14,33, all demonstrate that the crystals

used in the present study are very clean. The tetragonal structure is conrmed by single-crystal X-ray diffraction at room temperature. The tetragonal [100]T/[010]T is along the square edges of the crystals, and below the structural transition, the orthorhombic [100]O/[010]O along the diagonal direction.

Magnetization and magnetic torque measurements. The magnetization was measured using a vibrating sample option (VSM) of the Physical Properties Measurement System by Quantum Design. Supplementary Figure 1 shows temperature dependence of the magnetization in a single crystal of FeSe for several different elds. We obtained the diamagnetic response in the magnetization, Mdia, by shifting the curves to zero at 30 K, that is, by subtracting a constant representative of the normal-state magnetization ignoring the small paramagnetic CurieWeiss contribution.

Magnetic torque is measured by using a micro-cantilever method25,26. As illustrated in the inset of Fig. 2b, a carefully selected tiny crystal of ideal tetragonal shape with 200 200 5 mm3 is mounted on to a piezo-resistive cantilever. The

crystals contain orthorhombic domains with typical size of B5 mm below Ts.

Supplementary Figure 2af shows the magnetic torque t measured in various elds, where the eld orientation is varied within a plane including the c axis (y 0,180)

10 and the eld strength H |H| is kept during the rotation. The torque curves at 0.5

and 1T (Supplementary Fig. 2a and b) are distorted at 8.5 K, which is expected in the superconducting state of anisotropic materials36 whereas those above 9 K are perfectly sinusoidal.

NMR measurements. 77Se NMR measurements were performed on a collection of several oriented single crystals, and external elds (1, 2 and 14.5 T) are applied parallel to the c axis. Since 77Se has a nuclear spin I 1/2, and thus no electric

quadrupole interactions, the resonance linewidth of the NMR spectra are very narrow with full width at half maximum of a couple of kHz (Supplementary Fig. 3). The nuclear spin-lattice relaxation rate 1/T1 is evaluated from the recovery curve

R(t) 1 m(t)/m(N) of the nuclear magnetization m(t), which is the nuclear

magnetization at a time t after a saturation pulse. R(t) can be described by R(t)pexp( t/T1) with a unique T1 in the whole measured region, indicative of a

homogeneous electronic state. In general, 1/T1 for H||c is related to the imaginary part of the dynamical magnetic susceptibility w(q, o) by the relation

1T1T / X

q

o ; 2

where A(q) is the transferred hyperne coupling tensor along the c axis at the Se site and o gn/H with gn/(2p) 8.118 MHzT 1 is the NMR frequency. 1/T1T at

the Se site is mainly governed by the magnetic uctuations at the Fe sites,that is, particularly in FeSe, the short-lived stripe-antiferromagnetic correlations at q (p, p) in the tetragonal notation. It should be noted that the superconducting

diamagnetism is an orbital effect that is dominated at q 0 and thus it does not

affect the dynamical spin susceptibility at q (p, p).

Thermoelectric measurements. The thermoelectric coefcients were measured by the standard d.c. method with one resistive heater, two Cernox thermometers and two lateral contacts (Fig. 3d, inset). The Seebeck signal S is the transverse electric eld response Ex (||x), while the Nernst signal N is a longitudinal response

Ex (||x) to a transeverse temperature gradient rxT(||x) in the presence of a

magnetic eld Hz (||z), that is, S Ex/( rxT) and N Ey/( rxT), respectively.

The Nernst coefcient is dened as n N/m0H.

Data availability. The data that support the ndings of this study are available on request from the corresponding authors (T.S. or Y.M.).

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Acknowledgements

We thank K. Behnia, I. Danshita, H. Kontani, A. Perali, M. Randeria and Y. Yanase for fruitful discussions. This work was supported by Grants-in-Aid for ScienticResearch (KAKENHI) (nos 25220710, 15H05745, 15H02106 and 15H03688) and on Innovative Areas Topological Material Science (no. 15H05852), and J-Physics(nos. 15H05882, 15H05884 and 15K21732). The work of A.L. was supported by NSF grants no. DMR-1606517 and no. ECCS-1560732, and in part by Wisconsin Alumni Research Foundation.

Author contributions

S.K. and T.Wo. prepared the samples. S.K., T.Y., A.S., R.K, Y.S., T.Wa., K.I., T.T., F.H. and C.M. carried out the measurements. S.K., K.I., H.v.L., A.L., T.S. and Y.M. interpreted and analysed the data. T.S., Y.M., H.v.L., S.K. and A.L. wrote the manuscript with inputs from all authors.

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How to cite this article: Kasahara, S. et al. Giant superconducting uctuations in the compensated semimetal FeSe at the BCSBEC crossover. Nat. Commun. 7:12843doi: 10.1038/ncomms12843 (2016).

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