Exotic Superconducting States in FeSe-based Materials
JJournal of the Physical Society of Japan
FULL PAPERS
Exotic Superconducting States in FeSe-based Materials
Takasada Shibauchi ∗ Tetsuo Hanaguri † , and Yuji Matsuda ‡ , Department of Advanced Materials Science, University of Tokyo, Kashiwa 277-8561, Japan RIKEN Center for Emergent Matter Science, Wako 351-0198, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan
High-temperature superconductivity and a wide variety of exotic superconducting states discovered in FeSe-based ma-terials have been at the frontier of research on condensed matter physics in the past decade. Unique properties originatedfrom multiband electronic structure, strongly orbital-dependent phenomena, extremely small Fermi energy, electronicnematicity, and topological aspects, give rise to many distinct and fascinating superconducting states. Here, we providean overview of our current understanding of the superconductivity of bulk
FeSe-based materials, focusing on FeSe andthe isovalent substituted FeSe − x S x and FeSe − x Te x . We discuss the highly non-trivial superconducting properties inFeSe, including extremely anisotropic pairing states, crossover phenomena from Bardeen-Cooper-Schrie ff er (BCS) toBose-Einstein-condensation (BEC) states, novel field-induced superconducting phase, and broken time reversal symme-try. We also discuss the evolution of the superconducting gap function with sulfur and tellurium doping, paying particularattention to the impact of quantum critical nematic fluctuations and the topological superconductivity. FeSe-based ma-terials provide an excellent playground to study various kinds of exotic superconducting states. KEYWORDS: Iron-based superconductors, Electronic nematicity, BCS-BEC crossover, Quantum criticality, FFLOstate, Time-reversal symmetry breaking, Topological superconductivity
CONTENTS1. Introduction2. Electronic Structure and Phase Diagram
3. Superconducting Gap Structure / Spectroscopy (STM / STS)
4. BCS-BEC Crossover − x S x
5. Exotic Superconducting State Induced by MagneticField ∗ [email protected] † [email protected] ‡ [email protected]
6. Superconductivity near the Nematic Critical Point
7. Time Reversal Symmetry Breaking (TRSB) ff ect of nematic twin boundary7.2 Evidence from gap structure7.3 Evidence from muon spin rotation ( µ SR)7.4 TRSB in the bulk
8. Topological Superconducting State − x Te x − x Te x
9. Summary1. Introduction
The discovery of high-temperature superconductivity iniron-pnictide compounds has been a significant breakthroughin the condensed matter community. In 2006, H. Hosono’sresearch group discovered superconductivity at supercon-ducting transition temperature T c ≈ By replacing phosphorus with arsenic and bypartially substituting oxygen with fluorine, T c increases to26 K. By replacing La with other rare earth elements, T c raises up to 56 K. The iron-chalcogenide superconductorshave also been extensively studied. In particular, iron-selenide a r X i v : . [ c ond - m a t . s up r- c on ] M a y . Phys. Soc. Jpn. FULL PAPERS
FeSe with T c ≈ has drawn considerable attention because T c increasesto 37 K under pressure. Moreover, T c increases more than50 K in monolayer FeSe thin films grown on SrTiO . Thusiron-pnictides / chalcogenides became a new class of high- T c superconductors, knocking the cuprates o ff their pedestalas a unique class of high- T c superconductors. There is al-most complete consensus that high- T c superconductivity inthese iron-based superconductors cannot be explained theo-retically by the conventional electron-phonon pairing mech-anism. Thus, the origin of superconductivity is unconven-tional. Iron-based superconductors are two-dimensional (2D) lay-ered materials with metallic Fe-pnictogen / chalcogen tetrahe-dral FeAs (or P / Se / Te) layers.
The pnictogen / chalcogenatoms reside above and below the Fe layers, alternately, andare located at the center of the Fe-atom squares. There are sev-eral di ff erent types of iron-pnictide superconductors, whichare often abbreviated by the ratio of the elements in theirparent compositions and are known, such as 111, 122, 1111types, and iron-chalcogenide superconductors, such as 11 and122 types. The parent compounds are metals, in contrast toMott insulators as parent compounds of the cuprates. More-over, whereas in cuprates the physics is captured by a singleband originating from one 3 d x − y -orbital per Cu site, iron-based superconductors have about six electrons occupying thenearly degenerate 3 d Fe-orbitals. Therefore the systems areintrinsically multiband / multi-orbital systems, and the inter-orbital Coulomb interaction plays an essential role. The Fermisurface in these materials mainly consists of d xy , d yz and d xz orbitals, forming well-separated hole pockets near the centerof the Brillouin zone and electron pockets near the zone cor-ners.The parent compound of iron-pnictides is a spin-density-wave (SDW) metal, which exhibits a transition at T N . Be-low T N , a stripe-type long-range antiferromagnetic (AFM) or-der sets in, which breaks the lattice four-fold ( C ) rotationalsymmetry. The high- T c superconductivity appears when theSDW is suppressed either by chemical substitution or by pres-sure. The highest T c is often achieved in the vicinity of anAFM quantum critical point (QCP), where the SDW transi-tion vanishes. Therefore, a pairing mechanism mediatedby the exchange of AFM fluctuations, which stem from theintra-atomic Coulomb repulsion associated with the quasi-nesting between electron and hole pockets, has been widelydiscussed. This scenario predicts the so-called s ± order pa-rameter, in which the sign of the superconducting gap changesbetween the electron and hole pockets of the Fermi sur-face. On the other hand, the orbital degrees of freedom in iron-pnictides give rise to various phenomena. Almost all familiesof iron-pnictides exhibit a tetragonal-to-orthorhombic struc-tural transition at T s , which is accompanied by the orbital or-dering that splits the degenerate d xz and d yz orbitals. Thistransition that breaks C symmetry of the crystal lattice is re-ferred to as an electronic nematic transition. This nematictransition is believed to be a result of intrinsic electronic insta-bility because the e ff ect on the electronic properties is muchlarger than expected based on the observed structural distor-tion. This nematic transition either precedes or is coincidentwith the SDW transition, and the endpoint of the nematic tran- Fig. 1.
Crystal structure of FeSe. Adopted from Ref. 4. sition is located very close to the AFM QCP. Moreover, ithas been reported that the electronic nematic order persistseven above the superconducting dome in the tetragonal latticephase in some of the iron-based superconductors.
It hasbeen shown that the nematic susceptibility, which is measuredas an induced resistivity anisotropy in response to an exter-nal strain, exhibits a divergent behavior at T → Consequently, an alterna-tive scenario of the pairing mechanism, which is mediated byorbital fluctuations, has been proposed.
This scenario canhardly support the s ± gap function.Thus a question as to whether the nematic order is drivenby spin and / or orbital degrees of freedom has been a topicof intense research, which is intimately related to the drivingmechanism of iron-based superconductivity. The nematic cor-relations intertwined with AFM order in iron-pnictides pre-vent us from identifying the essential role of nematic fluctua-tions, raising a fundamental question as to which fluctuationsare the main driving force of the Cooper pairing. Identifyingthe relationship between nematic and SDW orders presents a“chicken-or-egg” problem: Does the SDW order induces thenematic order, or does the nematic order facilitate the SDWorder? Although an intricate coupling between magnetic andorbital degrees of freedoms is crucial to understand the under-lying physics responsible for their wide variety of exotic prop-erties of iron-pnictides, these questions have been the topic ofmany debates. The nematic order is directly linked tothe superconductivity because nematic instability is a charac-teristic feature of the normal state, upon which superconduc-tivity emerges at lower temperatures. Moreover, the nematic-ity has now been extensively discussed not only in iron-basedsuperconductors but also in cuprates.
Recently iron-chalcogenide FeSe ( T c ≈ − x S x and FeSe − x Te x have becomea central system in the research of exotic superconductingstates. FeSe is structurally the simplest among the iron-basedsuperconductors, which consists only of a stack of 2D FeSelayers weakly coupled by the van der Waals interaction (seeFig. 1). The experimental progress has been largely acceler-ated owing to the significant advances in the material qualityof FeSe. In particular, the chemical vapor transport techniquehas enabled us to grow high-quality millimeter-sized singlecrystals of FeSe free of impurity phases, making detailedstudies of the intrinsic bulk properties of FeSe possible. FeSeis a strongly correlated semimetal, as revealed by the quasi-particle e ff ective masses determined by the quantum oscilla-tions, angle-resolved photoemission spectroscopy (ARPES)
2. Phys. Soc. Jpn.
FULL PAPERSFig. 2. (a) Temperature dependence of relative length changes along thethree axes. Tiny orthorhombic lattice distortion of ∼ .
2% develops belowthe nematic phase transition at T s ≈
90 K. This is too small to account forthe very large electronic anisotropy discussed in this review, and thus thenematicity is not due to the lattice instability but electronic in origin. Adoptedfrom 35. (b) Temperature dependence of the resistivity of FeSe for H (cid:107) c . Below T s , very large magnetoresistance is observed, which is a signature oflong mean free path in compensated semimetals. Inset illustrates appearanceof the electronic nematicity. and heat capacity measurements, which are strongly enhancedfrom those calculated by density functional theory (DFT).FeSe-based materials provide a unique opportunity to ex-plore the e ff ect of nematicity. FeSe also exhibits a nematictransition at T s ≈
90 K, as shown in Fig. 2. In contrast to iron-pnictides, however, FeSe exhibits no magnetic order down to T → T s , and still its ground state is anunconventional superconducting state. The nematicity is re-markably tunable by hydrostatic pressure P and chemical sub-stitution. In FeSe, the structural (nematic) transition is rapidlysuppressed by pressure, and T s goes to zero at P ≈ T s and nematicity appears to coexist with the AFM or-der. Nuclear magnetic resonance (NMR) and inelastic neutronscattering experiments reveal that the nematicity and mag-netism are still highly entangled in FeSe.The nematicity in FeSe − x S x is also strongly suppressedwith sulfur doping, and T s goes to zero at x ≈ .
17. The ne-matic fluctuations are strongly enhanced with sulfur doping,and the nematic susceptibility diverges towards absolute zero,revealing the presence of a nematic QCP at x ≈ .
17 .
Nearthe nematic QCP, no sizable AFM fluctuations are observed,indicating that the nematicity is disentangled from magneticorder.What makes FeSe-based materials distinguished from othersuperconductors is the unique electronic structure, particu-larly the extremely shallow Fermi surface associated withvery small carrier number, multiband nature and orbital de-pendent electron correlations. Remarkably, the superconduc-tivity, magnetism and nematicity, all of these most fundamen-tal properties can be largely tuned. Because of these proper-ties, FeSe-based materials serve as not only a model systemfor the understanding of the influence of the nematicity on thenormal and superconducting states, but also a new playgroundfor exotic pairing states, which have been a long standing is-sue of superconductivity.In this review article, we shall attempt to bring the readersup to date in the rapidly expanding field of research on the su-perconductivity of FeSe-based materials, focusing on the bulk properties. Excellent reviews of research on atomic layer thinfilms of FeSe were recently published. In §
2, we discussthe electronic structure and phase diagram of FeSe briefly. Wethen discuss several topics of active research among exoticsuperconducting states in FeSe-based materials, such as su-perconducting gap structure ( § ff er (BCS) to strongcoupling Bose-Einstein condensation (BEC) ( § § § § §
2. Electronic Structure and Phase Diagram
First of all, we will not attempt to give an exhaustive sur-vey of current research on the band structure of FeSe in thelimited space here. We will direct the reader to more exten- E ( m e V ) k x k y π π Fig. 3.
Schematic energy dispersion of the hole and electron pockets in thenematic phase including spin-orbit interaction. The orbital-dependent energyshift is taken into account so that each band can be fitted to the ARPES data.By courtesy of Y. Yamakawa and H. Kontani.3. Phys. Soc. Jpn.
FULL PAPERS sive reviews.
As FeSe is a compensated semimetal withequal numbers of electron and hole carriers, it is essentially amultiband superconductor. The Fermi surface consists of wellseparated hole pockets near the center of the Brillouin zoneand electron pockets near the zone corners. Compared withthe Fermi surface obtained by the DFT calculations, the ac-tual Fermi surface in the tetragonal phase above T s is muchsmaller and dispersions are significantly renormalized. In the tetragonal phase, the DFT calculations show threehole pockets, but ARPES measurements report two pock-ets, showing that one hole band shifts below the Fermi level.Below the nematic transition at T s ≈
90 K, the orbital order-ing lifts the degeneracy of d xz and d yz orbitals. The splitting ofthe d xz and d yz energy bands is ∼
50 meV at the M x point inthe unfolded Brillouin zone. Then one hole pocket in thetetragonal phase is shifted below the Fermi level below T s . Asa result, there is only one quasi-2D hole pocket around Γ pointin the nematic phase. The hole pocket is strongly distorted toelliptical shape, which consists of d yz orbital along the longeraxis and d xz orbital along the shorter axis of the ellipse.Although the shape and orbital character of the hole pocketappear to be well understood, the consensus about the shapeof the electron pockets has not yet been reached. This ismainly because both hole and electron pockets are extremelyshallow and therefore high-resolution ARPES measurementsare prerequisite to determine the detailed structure of theFermi surfaces. Unfortunately, such high-resolution ARPESmeasurements are available only in a limited momentumrange around the Γ point. The schematic energy dispersionof the hole and electron pockets in the nematic phase includ-ing spin-orbit interaction is shown in Fig. 3. As a result ofthe finite spin-orbit interaction, Dirac cones around M x pointhave a small gap, forming massive Dirac cones. Moreover,the degeneracy at the Γ point of the hole bands is lifted. Inthis energy shift, there is only one hole pocket around Γ pointand one electron pocket around M y point.Figures 4(a)-(d) illustrate four possible Fermi surface struc-tures proposed by ARPES and quantum oscillations. Fig-ures 4(a) and (b) show the Fermi surface reported by someARPES measurements, where the electron pocket has a bow-tie-shape around the M x point. On the other hand,quantum oscillation measurements report the presence of tinypocket having a non-trivial Berry phase.
This suggests thatelectron pocket around the M x point is disconnected, formingthe Dirac-core-like band structure, as shown in Fig. 4(c). Asshown in Fig. 3, the electron pockets around M x point shownby red has a fork-tailed shape near the Fermi level. Therefore,the shape of electron pocket strongly depends on the positionof the Fermi level, which is expected to be sensitive to thecarrier number. Then a slight deviation from the compensa-tion condition may change the shape of the electron pocketdramatically.Besides these Fermi surfaces, petal-like electron pocketsboth at M x and M y points has also been proposed by some ofthe ARPES experiments. However, as these ARPESmeasurements have been performed by using heavily twinnedcrystals, careful interpretation may be necessary. Recent ex-periments by using nano-ARPES on twinned crystal reportsingle electron pocket at M x point shown in Fig. 4 (a) and(b). The electron pocket at M y point is also controversial be-tween ARPES measurements using detwinned crystals. While Fig. 4.
Four possible Fermi surface structures proposed by ARPES andquantum oscillations. Solid and dotted black lines represent the unfolded andfolded Brillouin zone boundaries, respectively. In all the possibilities, thereis one hole pocket around the Γ -pocket. For (a), (b) and (c), the hole pocketconsists of d yz and d xz orbitals, while for (d), it consists only of d xz orbital.The electron pocket around M x point has a bow-tie shape for (a) and (b),while electron pocket around M y point is present for (a) and absent for (b).For (c), the electron pocket around M x point is disconnected, forming a dou-ble Dirac-core-like structure. For (d), petal-like electron pockets appear bothat M x - and M y points. Note that here the axes are defined as b < a < c , whichis di ff erent from some experimental definition. a very tiny electron pocket at M y point is reported as illus-trated in Fig. 4(a), the absence of such a pocket is reported as shown in Fig. 4(b). It has been pointed out that the presenceor absence of electron pocket at M y point strongly depends onthe level of d xy orbital. Besides the above four types of theFermi surface, an additional inner hole pocket with d xz orbitalcharacter has been suggested very recently. However, weassume that there is only one hole pocket in the following ar-guments.Quantum oscillations of the resistivity, known asShubnikov-de Haas (SdH) e ff ect, have been reported byseveral groups. Four branches are observed in the SdHoscillations. It has been suggested that two of the branchescorrespond to the extremal orbits of the hole pocket and theother two of the branches correspond to those of electronpocket. The cross section of each Fermi surface is extremelysmall, which occupies at most 2-3% of the Brillouin zone.The electronic specific heat coe ffi cient γ estimated from thee ff ective masses and Fermi surface volume assuming 2Dcylindrical Fermi surface is close to the observed γ -value,suggesting that most parts of the Fermi surface are mappedout by the SdH measurements. Pressure and chemical substitution are non-thermal controlparameters, which provide a continuous means to modify theelectronic structure. The ground state of iron-based materialscan be significantly tuned by these parameters, which havebeen widely employed to access the QCP.
The hydrostaticpressure experiments on FeSe have been performed by manygroups.
As shown in Fig. 5, the structural (nematic)transition at T s is rapidly suppressed by pressure ( P <
4. Phys. Soc. Jpn.
FULL PAPERSFig. 5.
Pressure-temperature ( P - T ) phase diagram of FeSe. The structuralor nematic ( T s , blue), SDW ( T m , green), and superconducting transition tem-peratures ( T c , red) as functions of hydrostatic pressure determined by theresistive anomalies measured in the piston-cylinder cell (PCC, open circles),clamp-type cubic anvil cells (CAC, closed circles), and constant-loading typeCAC (closed squares). Color shades for the nematic, SDW, and supercon-ducting (SC) states are guides to the eyes. Adopted from Ref. 63. Before the complete suppression of T s , the static magneticorder is induced at T m , suggesting the presence of an over-lap region of nematic and pressure-induced magnetic phases.The presence of a static long-range magnetic order has beenconfirmed by the muon spin rotation ( µ SR), M¨ossbauer spec-troscopy and NMR measurements, and the magnetic phase ismost likely to be an SDW phase of stripe type similar to theone observed in iron-pnictides. With increasing pressure, T m increases and intersects with T s at ∼ ∼ . T m peaks and vanishes abruptly at ∼ . P - T phase diagram. It has been reported that superconductivity isfilamentary rather than a bulk phenomenon inside the mag-netic dome, implying that the ground state in this dome pres-sure range is likely to be AFM metal. Near the end point ofmagnetic order, T c is sharply enhanced up to ∼
38 K.Electronic nematicity is a ubiquitous property of the iron-based superconductors. There are two scenarios for the driv-ing mechanism of the nematicity. One route to the nematicityis via critical magnetic (spin) fluctuations and the other is viathe critical orbital fluctuations. In iron-pnictides, where thenematicity is always accompanied by the AFM order, strongAFM fluctuations are observed above T s . Then, the criti-cal spin fluctuations have been suggested to be responsiblefor the nematicity.
It has been argued that this spin nematicscenario envisaged in iron-pnictides may be still applicableto FeSe.
In fact, the magnitude of the lattice distortion,elastic softening, and elasto-resistivity associated with thestructural transition in FeSe are comparable with those of Fe-pnictides.
It has been shown that the nematic order couldbe driven by the AFM spin fluctuations without the require-ment of magnetic order.
Spin excitations measured bythe inelastic neutron scattering experiments show that the dy-namic susceptibility χ ( q , ω ) peaks at q = ( π, q is thescattering vector and (cid:126) ω is the energy change between inci-dent and outgoing neutrons. The results suggest the presence of both stripe and N´eel spin fluctuations over a wide energyrange even above T s . In the nematic state well below T s ,the NMR spin-lattice relaxation rate 1 / T shows the presenceof strong AFM fluctuations down to T c . There is an alternative scenario where charge or orbitaldegrees of freedom play a more predominant role thanspins.
In FeSe, where the nematic transition occurswithout magnetic order, no sizable low energy spin fluctua-tions are observed above T s by NMR, in contrast toiron-pnictides. It has been shown that the orbital ordering isunequivocally the origin of the nematic order in FeSe. In the nematic phase where the splitting of the d xz and d yz energy bands occurs, a momentum-dependent orbital polar-ization has been found in ARPES measurements, indicatingthat the nematicity is most likely to be orbital in origin. Moreover, NMR measurements report that the di ff erence ofthe static internal field in the ab -plane in the orthorhom-bic phase at the Se-nucleus is predominantly from the Fe-ion 3 d electron orbitals, not from the electron spins. Re-cently, symmetry-resolved electronic Raman scattering mea-surements provide a direct experimental observation of crit-ical fluctuations associated with electronic charge or orbitalnematicity near T s . These results put into question the spinnematic scenario for the nematicity in FeSe.The appearance of the AFM order induced at low pressureis consistent with the fact that the nematic order is close to themagnetic instability as suggested by ab initio calculations.
It has been suggested that the pressure changes the Fermi sur-face topology of FeSe. A hole band with d xy orbital characteris nearly 10 meV below the Fermi level around the ( π, π ) pointin the unfolded Brillouin zone at ambient pressure. The top ofthe hole band crosses the Fermi energy under pressure, giv-ing rise to a hole pocket. This hole pocket largely enhancesthe AFM nesting properties, leading to an appearance of thestatic AFM order. Therefore the spin and orbital degreesof freedom are highly entangled even in nonmagnetic FeSe,which makes it di ffi cult to pin down the driving mechanismof the nematicity and superconductivity.Until now, most studies have been conducted in thermalequilibrium, where the dynamical property and excitation canbe masked by the coupling with the lattice. Recently, by us-ing femtosecond optical pulse, the ultrafast dynamics of elec-tronic nematicity has been detected. A short-life nematicityoscillation, which is related to the imbalance of Fe d xz and d yz orbitals, has been reported. Such real-time observations ofthe electronic nematic excitation that is instantly decoupledfrom the underlying lattice would be important for the futureinvestigation of the nematicity. Next we discuss that FeSe − x S x is a more suitable systemto explore the e ff ect of nematicity disentangled from that ofthe static AFM order. The nematicity can be tuned contin-uously by isoelectronic sulfur substitution and T s vanishesat x ≈ .
17. The compensated metal character should beuna ff ected by the isovalent substitution of Se. Moreover, incontrast to the application of pressure, the sulfur substitutionappears not to change the Fermi surface topology. TheNMR 1 / T measurements report that the AFM fluctuationsare slightly enhanced at small x but strongly suppressed byfurther sulfur substitutions, leading to no sizable AFM fluc-
5. Phys. Soc. Jpn.
FULL PAPERSFig. 6.
Temperature-pressure-concentration ( T - P - x ) phase diagram in FeSe − x S x . The structural or nematic ( T s , blue squares), SDW ( T m , green triangles)and superconducting transition temperatures ( T c , red circles) determined by the resistivity anomalies are plotted against hydrostatic pressure P and sulfurcontent x . T c is also determined by the magnetic susceptibility. Adopted from Ref. 64. tuations near the nematic QCP. Figure 6 displays thetemperature-pressure-concentration ( T - P - x ) phase diagram inFeSe − x S x in wide ranges of pressure up to 8-10 GPa and sul-fur content (0 ≤ x ≤ . T s and T m are determinedby the resistivity anomalies. The T c values are determinedfrom the zero resistivity as well as the magnetic susceptibil-ity measurements. As shown in Fig. 6, the magnetic domeshrinks with increasing x , and disappears at x ≈ . Thisindicates that in contrast to the pressure, sulfur substitutionmoves the magnetic instability away from the nematic order.Then an important issue is how the nematic fluctuationsevolve with the sulfur substitution. An elegant way to eval-uate experimentally the nematic fluctuations has been de-veloped by the Stanford group, that is based on the elasto-resistivity measurements by using a piezoelectric device.
Nematic susceptibility is defined as χ nem ≡ d η/ d (cid:15) , where η = ( ρ xx − ρ yy ) /ρ ∼ ∆ ρ/ρ is the change of the resistiv-ity induced by the lattice strain (cid:15) . The nematic susceptibilitycan probe fluctuations associated with the phase transition,which bears some resemblance to the magnetic susceptibility χ mag = d M / d H in the magnetic system. It has been reportedthat χ nem of FeSe − x S x exhibits the Curie-Weiss like tempera-ture dependence as, χ nem ( T ) = aT − T θ + χ , (1)where a and χ are constants and T θ corresponds to the Curie-Weiss temperature of the electronic system. As shown inFig. 7, the nematic fluctuations are strongly enhanced withsulfur content x . Near x ≈ . T θ goes to zero, indicatingthat the nematic susceptibility diverges towards absolute zero.Moreover, quantum oscillations and quasiparticle interference(QPI) measurements reported that the Fermi surface changessmoothly when crossing the nematic QCP. These results re- veal the presence of a nematic QCP at x ≈ . In the nematic phase, T c increases gradually, peaks at x ∼ .
08 and decreases gradually with x . Around the ne-matic QCP, T c decreases and reaches ∼ Fig. 7.
Phase diagram and quantum criticality in FeSe − x S x . Temperaturedependence of the nematic transition ( T s , green diamonds) and the super-conducting transition temperature ( T c , orange circles) determined by thezero resistivity criteria. The Curie-Weiss temperature is also plotted ( T θ , redhexagons). The magnitude of χ nem in the tetragonal phase is superimposedon the phase diagram by a color contour (see the color bar for the scale). Thelines are the guides for the eyes. Adopted from Ref. 36.6. Phys. Soc. Jpn. FULL PAPERS nal phase.
Very recently, linear-in-temperature resistiv-ity, which is a hallmark of the non-Fermi liquid property, isreported at x ≈ .
17, indicating that the nematic critical fluc-tuations emanating from the QCP have a significant impacton the normal-state electronic properties.
We will show thatthe nematicity also strongly influences on the superconductiv-ity in §
3. Superconducting Gap Structure
One of the most important properties of unconventionalsuperconductors is the characteristic structure of the super-conducting gap ∆ ( k ), which is intimately related to the pair-ing mechanism. In the phonon-mediated conventional su-perconductivity, the momentum-independent pairing interac-tion leads to BCS s -wave superconductivity with a constant ∆ = . k B T c , and thus the bulk physical quantities that arerelated to quasiparticle excitations show exponential temper-ature dependence at low temperatures. For unconventionalsuperconductors, however, the pairing interaction may havestrong dependence on momentum k leading to anisotropic ∆ ( k ), which sometimes have zeros (nodes) at certain k di-rections. In such cases, the existence of low-lying excitationsin the quasiparticle energy spectrum changes the exponentialtemperature dependence to power-law behaviors. Thereforethe low-temperature measurements of bulk quantities sensi-tive to low-energy quasiparticle excitations, such as magneticpenetration depth, specific heat, and thermal conductivity, arequite important to study the pairing mechanism of supercon-ductors.The temperature dependence of London penetration depth λ ( T ), which is directly related to the number of superconduct-ing electrons, is one of the sensitive probes of thermally ex-cited quasiparticles. When the gap ∆ ( k ) has line (point) nodes,the low-energy quasiparticle excitation spectrum depends onenergy E as ∝ E ( ∝ E ), and thus ∆ λ ( T ) = λ ( T ) − λ (0)is proportional to T ( T ) at low temperatures unless the su-percurrent direction is always perpendicular to the nodal di-rections. The precision measurements of penetration depth inthe Meissner state by using the tunnel diode oscillator tech-nique at 13 MHz have shown that ∆ λ ( T ) in high-quality sin-gle crystals of FeSe has non-exponential, quasi-linear temper-ature dependence ( ∼ T . ) at low temperatures below ∼ . T c ,as shown in Fig. 8(a). This result suggests the presence ofline nodes in the superconducting gap. The deviation from the T -linear dependence may be attributed by the impurity scat-tering or multiband e ff ect (combining nodal and full-gappedbands), which can increase the exponent α from unity in thepower-law temperature dependence T α .A surface impedance study at higher frequencies of 202and 658 MHz by using a cavity perturbation technique alsoreports strong temperature dependence of superfluid density ρ s ( T ) = λ (0) /λ ( T ), but as shown in Fig. 8(b), it exhibits aflattening at the lowest temperatures in contrast to the nodalgap behavior. The data can be fitted to two gaps with asmall minimum gap ∆ min ≈ . k B T c in one band. The im-plications of the presence of two di ff erent results, nodal andgap minima, will be discussed later.The heat capacity is the most fundamental thermodynamicquantity that can also probe the quasiparticle excitations in the superconducting state. In the analysis of specific heat C ( T ),the contribution from the phonons that usually depends ontemperature as C ∼ T at low temperatures needs to be sub-tracted to extract the electronic contribution. This can be doneby comparing the data at zero field and above the upper criti-cal field H c2 in the normal state. The temperature dependenceof electronic specific heat divided by temperature, C e / T ( T ),also shows exponential behavior in fully gapped supercon-ductors and T -linear behavior for the gap structure with linenodes, as in the case of ∆ λ ( T ). While the low-temperaturedata of C e / T ( T ) show sometimes multigap behaviors sug-gesting the presence of a small full gap, the most recentmeasurements using high-quality, vapor-grown single crys-tals clearly indicate the T -linear behavior at low temperaturesdown to ∼ . as shown in Figs. 8(c) and (d). This alsosuggests the presence of line nodes or tiny gap minima in thesuperconducting gap in FeSe, which is in a similar situationas in the penetration depth studies.Another sensitive probe of quasiparticle excitations is thethermal conductivity κ , which can be measured in the su-perconducting state. The temperature dependence of thermalconductivity also has phononic and electronic components,but in the zero-temperature limit κ/ T gives a very impor-tant information on the superconducting gap structure. As κ/ T is proportional to C / T as well as to the mean free path (cid:96) , κ/ T ( T →
0) always vanishes for fully gapped super-conductors. This is understood by the fact that in the zero-temperature limit, C / T vanishes while (cid:96) is limited by the im-purity scattering. In contrast, the presence of nodes in thegap leads to small but finite low-energy states due to impu-rity scattering, thus giving rise to the finite residual κ / T ≡ κ/ T ( T → κ / T , which is called universal residual thermal conductiv-ity in nodal superconductors. In FeSe bulk crystals, there arealso two kinds of reports with di ff erent conclusions as shownin Figs. 8(e) and (f); one shows a sizable κ / T suggesting anodal state, and the other shows a much smaller resid-ual values of κ / T from which the authors concluded a fullygapped state with small gap minima. Not only the temperature dependence but also the field de-pendence give important clues on the presence or absence ofthe superconducting gap nodes. When the magnetic field H is applied to induce vortices inside nodal superconductors,the supercurrent flowing around a vortex a ff ects the quasipar-ticle energy spectrum through the Doppler shift mechanism E ( k ) → E ( k ) − (cid:126) k · v s (where v s is the supercurrent veloc-ity around the vortex), and enhance the low-energy density ofstates (DOS). This Doppler shift e ff ect (or Volovik e ff ect) canbe seen by a strong increase of specific heat and thermal con-ductivity in the zero-temperature limit with increasing mag-netic field, and in the case of line nodes √ H dependence of C e / T and κ / T is expected. In the thermal conductivity studythat reports on the absence of κ / T , one of the measured crys-tal clearly shows a strong increase of κ/ T with field at lowtemperatures. This indicates that the superconducting gap ∆ ( k ) has very strong momentum dependence even though thegap does not have any zeros; in other words, the minimum
7. Phys. Soc. Jpn.
FULL PAPERSFig. 8.
Temperature dependence of bulk quantities in FeSe single crystals. (a) Temperature dependence of magnetic penetration depth at low temperatures,measured in the Meissner state by a tunnel diode oscillator technique. Adopted from Ref. 91. The inset shows the temperature dependence of superfluid density.(b) Superfluid density measured by a cavity perturbation technique. The data are fitted by a two-gap model (lines). Adopted from Ref. 92. (c) Electronic specificheat divided by temperature C e / T as a function of temperature in the superconducting state at zero field and in the normal state at 14 T. The inset is an expandedview at low temperatures. Adopted from Ref. 93. (d) The di ff erence of C e / T between the superconducting and normal states, fitted by a two-gap model (redline). The inset is the temperature dependence of entropy change. Adopted from Ref. 94. (e) Thermal conductivity divided by temperature κ/ T as a functionof temperature. The inset is an expanded view at low temperatures. Adopted from Ref. 91. (f) κ/ T as a function of T at zero and low fields. Here the residual κ / T at zero field is an order of magnitude smaller than that in (e). Adopted from Ref. 97. gap is very small. In the other study showing the presenceof κ / T , however, the κ / T actually decreases with H at lowfields. This unusual behavior can be explained by the re-duction of mean free path dominating over the increase ofDOS, suggesting that the quasiparticles are scattered by vor-tices. Such scattering induced by vortices may be seen in veryclean single crystals with very long (cid:96) , and in fact a similar re-duction of κ / T at low fields has been observed in very cleancrystals of CeCoIn . In S-substituted crystals of FeSe − x S x ,where mean free path is naturally suppressed by the chemicalsubstitution, the low-field κ/ T at low temperatures exhibits √ H behavior, consistent with the presence of line nodes. Inthe recent field-dependence measurements of specific heat inFeSe, clear √ H dependence of C e / T is also resolved. Summarizing these bulk measurements, one can safelyconclude that the superconducting gap structure of bulk FeSehas very strong k dependence with line nodes or deep gapminima. The fact that some measurements suggest fullygapped behaviors although most results are consistent withthe presence of line nodes, implies that the nodes are unlikelyprotected by symmetry, but are accidental ones. The symme-try protected nodes are robust against impurity scattering, butthe accidental nodes may be lifted by various perturbationssuch as disorder. This is most consistent with the s -wave A g symmetry of the superconducting order parameter, either s ± or s ++ , with strong anisotropy in at least one of the multi-bands. This strong anisotropy confirms the unconventional nature of superconductivity in FeSe. The ARPES measurements have a strong advantage overother techniques, namely it can provide direct information onthe momentum dependence of the energy spectrum. In thesuperconducting state, the momentum dependence of super-conducting gap ∆ ( k ) can be mapped out, and indeed the d -wave superconducting gap has been clearly found in cupratesuperconductors. In FeSe with relatively low T c , however, avery high energy resolution is required to resolve the rela-tively small energy gap, and a high momentum resolution isalso needed to resolve k dependence around the very smallFermi surfaces. Such high resolution ARPES measurementsare available recently by using a laser light source with ∼ Γ point), and the electron bands near the zone edgecannot be explored. The laser ARPES results for single do-main samples of FeSe, independently obtained in Institute forSolid State Physics, University of Tokyo, and Institute ofPhysics, Chinese Academy of Sciences, show that ∆ ( k ) inthe hole band near the Γ point is strongly anisotropic. Thehole Fermi surface has an ellipsoidal shape (see Fig. 3), and ∆ ( k ) is also two-fold symmetric. They found that along thelong axis of the ellipsoid, the gap becomes almost zero sug-gesting the presence of nodes near this direction, as shown in
8. Phys. Soc. Jpn.
FULL PAPERSFig. 9.
Momentum dependence of the superconducting gap determined byARPES measured at 1.6 K. (a) and (b) Symmetrized energy distributioncurves at di ff erent Fermi momenta along the hole Fermi surface. A phe-nomenological gap formula (red curves) is used to extract ∆ ( k ). (c) The lo-cation of the Fermi momentum is defined by the Fermi surface (FS) angle θ .(d) Momentum dependence of the superconducting gap, with averaging overthe four quadrants. The measured gap (empty circles) is fitted by several gapforms. Adopted from Ref. 100. Figs. 9(a)-(d). In the two-fold nematic phase, the s -wave and d -wave components can mix in the A g symmetry, and theobserved two-fold anisotropic ∆ ( k ) suggests that the s -waveand d -wave components are very close in magnitude, a pair-ing state with nascent nodes. Although the gap structureof the electron band is not clear from ARPES measurements,this strong anisotropy in the hole band is consistent with thebulk measurements mentioned above. / Spectroscopy (STM / STS)
The tunneling experiment is a quite powerful probe of su-perconducting gap structure, because it can extract the quasi-particle DOS as a function of energy in a direct way. Ow-ing to the recent advances of scanning tunneling microscopy / spectroscopy (STM / STS) techniques, one can obtain highlyreliable DOS data with a very high energy resolution at verylow temperatures. The first evidence from STS for the pres-ence of nodes is reported by Xue group from Tsinghua inthin films of FeSe.
They found a V-shaped DOS in theenergy dependence of tunneling conductance near zero en-ergy, consistent with the line nodes in the gap. In bulk vapor-grown single crystals, similar V-shaped tunneling spectra areobserved, again suggesting line nodes (Fig. 10(a)).At higher energies, the spectra exhibit at least two distinctfeatures of superconducting gaps at ∆ l ≈ . ∆ s ≈ . ∆ ( k ).However, by analyzing the interference of standing waves in-
90 180 270 , (deg) -1.001.02.0 ∆ k ( m e V ) ∆∆ | ∆ | k y k x ( π /a Fe , π /b Fe )( - π /a Fe , π /b Fe ) ( - π /a Fe , - π /b Fe ) ( π /a Fe ,- π /b Fe ) T unne li ng c ondu c t an c e ( n S ) -6 -4 -2 0 2 4 6Sample bias (mV) (a) (b)(c) Fig. 10.
Superconducting gap structure determined by STS measurements.(a) Typical conductance spectrum measured at 0.4 K at a cleaved surfaceof FeSe single crystal. The bottom of the gap is V-shaped and there are atleast two features at the gap edges as indicated by arrows. Data adopted fromRef. 104. (b) and (c) Momentum dependence of the superconducting gapderived from the QPI analysis. The red and blue colors indicate the di ff erentsigns of the two gap functions at the hole and electron bands. One of the elec-tron bands has not been deduced from this analysis (thin ellipsoids). Adoptedfrom Ref. 105. duced by impurity scattering, which is called as BogoliubovQPI imaging technique, the energy-dependent Fermi surfacestructures in the scattering wave vector plane can be mapped.From these analyses the superconducting gap structures ∆ ( k )of FeSe are extracted, which reveal very strong anisotropiesof the gap in the hole band near the zone center as well asone of the electron bands near the zone edge (Fig. 10(b)). In addition, by comparing detailed energy dependence of QPImapping with theoretical calculations, it is concluded thatthe sign changing order parameter between the hole and elec-tron bands in FeSe. Although the extracted gap structure hasno nodes but deep minima, the suggested pairing state is astrongly anisotropic s ± state, which has been discussed interms of orbital-dependent pairing.Previously, the QPI evidence for the sign-changing s ± state is found in FeSe − x Te x , where the e ff ect of magneticfield on QPI is analyzed. In FeSe − x Te x , the tunnelingspectra are more U-shaped and the gap is more isotropic.This non-universality of superconducting gap structure is alsofound in iron-pnictides; optimally doped Ba − x K x Fe As exhibits a fully gapped state while BaFe (As − x P x ) showsclear signatures of line nodes. It has also been revealed thatin BaFe (As − x P x ) system, in which the disorder can becontrolled by electron irradiation, the nodes can be liftedby impurity scattering and that the observed non-monotonicchanges of low-energy excitations with disorder indicatesa sign-changing s ± state after node lifting. Such non-universal superconducting gap structure with A g symmetrymay indicate the presence and importance of multiple pairingmechanisms in iron-based superconductors, which are mostlikely based on spin fluctuations that favor s ± symmetry andorbital fluctuations that favor s ++ state.
4. BCS-BEC Crossover
An ideal gas consisting of non-interacting Bose particlescan exhibit a phase transition, called BEC; below some crit-
9. Phys. Soc. Jpn.
FULL PAPERS ical temperature T B , a macroscopic fraction of the bosonsis condensed into one single ground state. The BEC occurswhen the thermal de Broglie wave length ( ∝ / √ T ) becomescomparable to the inter-particle distance n − / at low tem-peratures, where n B is the number of Bose particles. Below T B , wavefunction interference becomes apparent macroscop-ically. The superfluidity of the system is a consequence ofBEC.In Fermi systems, attractive interactions between fermionsare needed to form bosonic-like molecules (Cooper pairs),which are driven to BEC. There are two limiting cases, weak-coupling BCS and strong-coupling BEC limits, where at-tractive interactions are weak and strong, respectively. Thephysics of the crossover between the BCS and BEC limitshas been of considerable interest in the fields of condensedmatter, ultracold atoms and nuclear physics, giving a uni-fied framework of quantum superfluid states of interactingfermions. The crossover has hitherto been realized ex-perimentally in ultracold atomic gases.
On the other hand,in solid, almost all superconductors are in the BCS regime.In this section, a unique feature of the superconductivity ofFeSe is covered. There is growing evidence that FeSe andFeSe − x S x are in the BCS-BEC crossover regime. FeSe-based superconductors may provide new insights intofundamental aspects of the physics of the crossover.In the BCS limit where the attraction is weak, the Cooperpairing is described as a momentum space pairing. The pair-wise occupation of states ( k ↑ , − k ↓ ) with zero center-of-mass momentum for bosonic-like pairs leads to a profoundrearrangement of the Fermi surface, leading to the forma-tion of energy gap ∆ , which corresponds to the pair conden-sation energy. The size of the Cooper pairs, i.e. coherencelength ξ , is much larger than the average inter-electron dis-tance ∼ k − , where k F is the Fermi momentum, indicatingthat Cooper pairs are strongly overlapped, k F ξ (cid:29)
1. Thiscorresponds to ∆ /ε F (cid:28)
1, i.e. the pair condensation energyis much smaller than the Fermi energy. In this regime, thecondensation occurs simultaneously with the pair formation.In the BEC limit where the attraction is strong, two elec-trons are tightly bounded, forming a bound molecule, and theCooper pairs behave as independent bosons. In this limit, thesize of the composite bosons is much smaller than the aver-age inter-electron distance, k F ξ (cid:28)
1; composite bosons arenon-overlapping, which can be regarded as a real-space pair-ing. Even when the pair formation occurs at T ∗ , the thermalde Broglie wave length is still much shorter than the inter-electron distance. As a result, the BEC transition occurs at T c ,a temperature much lower than the pair formation temperature( T ∗ (cid:29) T c ). T c = π ( ζ (3 / / (cid:126) ( n / / m = . T F , (2)where n is the number of Fermion particles, ζ ( z ) is the Rie-mann zeta function ( ζ (3 / = . T F is the Fermitemperature. Thus, in contrast to the BCS limit, the preformedCooper pair regime extends over a wide range of temperature(Fig. 11). The appearance of the preformed pairs has beensuggested to lead to the pseudogap formation, which is theprecursor of the well-developed superconducting gap. The es-sential defining feature of the real space pairing is that thechemical potential becomes negative ( µ < crossover regime T / T F condensationFermi liquid Bose liquid T c Attraction
Preformed pairs T * BECBCS crossover regime
Fig. 11.
Canonical phase diagram of the BCS-BEC crossover. With in-creasing attractive interaction, the superconducting condensation temperature T c increases in the BCS regime, and becomes independent of interaction inthe BEC regime. The dashed white line represents the pairing temperature T ∗ , where the preformed pairs appear. In the crossover regime, T c exhibitsa broad maximum. As the pairing strength is increased, T c and T ∗ are sepa-rated. The pseudogap is expected at T c < T < T ∗ . the bottom of the band.The bound energy of a Cooper pair is given by the super-conducting gap in the BCS regime, while it is given by thechemical potential in the BEC regime. The excitation energyof the quasiparticles (Bogoliubov quasiparticles) in the super-fluid phase of the fermionic condensate is given by E k = ± (cid:113) ( ε k − µ ) + ∆ , (3)where ε k = (cid:126) k / m is the electron energy dispersion. Inthe BCS regime, the chemical potential coincides with theFermi energy at T = µ = ε F ( = (cid:126) k m ). The minimum ofthe spectral gap E k = ∆ opens at ε = µ corresponding to | k | = k F , as displayed in Fig. 12(a). In the BEC regime, where µ is negative, the minimum spectral gap locates at k = | µ | (cid:29) ∆ , E k = (cid:112) µ + ∆ ≈ | µ | . Thus the minimum energy that breaksthe Cooper pairs is 2 ∆ and 2 | µ | for BCS and BEC regimes,respectively.The BCS state with cooperative Cooper pairing and BECstate with composite bosons share the same kind of sponta-neous symmetry breaking. The change between the two statesis a continuous crossover at T = and at finite tempera-ture connected through a progressive reduction of the sizeof electron pairs involved as fundamental entities in both phe-nomena. This crossover goes across the intermediate regimewhere the size of the pairs is comparable with the averageinter-particle distance, k F ξ ∼
1. The BCS-BEC crossover hasbeen extensively studied in the ultracold atomic systems, inwhich the attractive interaction can be controlled experimen-tally by a Feshbach resonance, but is extremely di ffi cult tobe realized for electrons in solids. The interest in the BCS-BEC crossover has also grown up in high- T c cuprate, inwhich the size of the pairs appears to be comparable to theinter-particle spacing. In particular, the problem of the pre-formed pairs has attracted considerable attention in high- T c cuprates as an origin of the pseudogap formation in the un-derdoped regime. However, the pseudogap and pre-
10. Phys. Soc. Jpn.
FULL PAPERS formed pairs of cuprates remain highly controversial and un-resolved issues.
In FeSe, values of 2 ∆ l / k B T c ≈ ∆ s / k B T c ≈ . The high-quality single crystals of FeSe enable us to es-timate the Fermi energies ε eF and ε hF from the band edges ofelectron and hole sheets, respectively, by using several tech-niques. All of them consistently point to extremely smallFermi energies. First we discuss the absolute value of pen-etration depth in FeSe.In 2D systems ε F is related to the London penetration depth λ (0) as ε F = π (cid:126) d µ e λ − (0), where d is the interlayer distance. For FeSe, λ (0) ≈
400 nm.
As the Fermi surface consistsof one hole sheet and one (compensating) electron sheet, λ can be written as 1 /λ = / ( λ e ) + / ( λ h ) , where λ e and λ h represent the contribution from the electron and hole sheets,respectively. Assuming that two sheets have similar e ff ectivemasses, ε hF ∼ ε eF ≈ The magnitude ofthe Fermi energy can also be inferred from the thermoelectricresponse in the normal state. From the Seebeck coe ffi cient, the upper limit of ε eF is deduced to be ∼
10 meV. These resultsindicate that both of the Fermi energies of the hole and elec-tron pockets are extremely small.To place FeSe in the context of other superconductors, T c is plotted as a function of Fermi temperature T F ≡ ε F / k B or an equivalent critical temperature T B for BEC of elec-tron pairs for several materials including FeSe (Uemura plot,Fig. 13). Because the relevant Fermi surface sheets are
Fig. 12.
Dispersion of the Bogoliubov quasiparticle (a) in the BCS and (b)in the BEC regimes for the parabolic electron band in the normal state. Inten-sity of color represents the spectral weight of the Bogoliubov quasiparticle. Inthe BCS regime, the minimum of the spectral gap E k = ∆ occurs at | k | = k F .In the BEC regime, the minimum spectral gap occurs at k =
0. The minimumenergy that breaks the Cooper pairs is 2 ∆ and 2 | µ | for BCS and BCE regimes,respectively. nearly cylindrical, T F for 2D systems may be estimated di-rectly from λ (0) via the relation T F = π (cid:126) n k B m ∗ ≈ (cid:16) π (cid:126) d µ k B e (cid:17) λ − (0),where n is the carrier concentration within the supercon-ducting planes and d is the interlayer distance. For three-dimensional (3D) systems, T F = ( (cid:126) / π n ) / / k B m ∗ . Thedashed line corresponds to the BEC temperature for an ideal3D Bose gas, T B = (cid:126) π m ∗ k B (cid:16) n ζ (3 / (cid:17) / . In a quasi-2D system,this value of T B provides an estimate of the maximum con-densate temperature. Notably, the magnitude of T c / T F ≈ . He. It has been shown that T c / T F inBaFe (As − x P x ) is strikingly enhanced near an AFM QCPat x c ≈ .
30 due to the enhancement of m ∗ . We note that T c / T F of FeSe is even larger than that of BaFe (As − x P x ) at x c . Thus Fig. 13 indicates that FeSe is located closer to theBEC line than any other superconductors. As discussed above, in the BCS-BEC regime, the Bogoli-ubov quasiparticles exhibit a characteristic flat band disper-sion near k =
0, which is distinctly di ff erent from the back-bending behavior at k F expected in the BCS regime. Such aquasiparticle dispersion can be directly observed by ARPES.A signature of the BCS-BEC crossover has been reportedby ARPES measurements in several iron-based supercon-ductors. In Ba − x K x Fe As and LiFe − x Co x As, thecrossover condition ∆ /ε F ∼ ∆ /ε F (cid:28)
1. In Fe + y Se x Te − x , the ∆ /ε F increases and Bo-goliubov quasiparticles band changes from BCS-like to flatband like by changing the concentration of excess Fe. However, in this system, due to the crystal imperfection and (cid:1)(cid:2)(cid:3)(cid:2)
Fig. 13.
Uemura plot. T c is plotted as a function of Fermi temperature T F evaluated from 1 /λ (0) for various 2D and 3D superconductors, includ-ing the conventional superconductors such as Nb, high- T c cuprates such asLa − x Sr x CuO (214), YBa Cu O − δ (123), and Bi Sr Ca Cu O y (2223), or-ganic and heavy fermion compounds. The dashed line is the BEC temperature T B for the ideal 3D Bose gas.11. Phys. Soc. Jpn. FULL PAPERS excess Fe-atoms, the superconducting gap is spatially inho-mogeneous, compared with FeSe − x S x . In addition, the Fermienergy of electron band is not well known.Very recently, flat energy dispersions, which are character-istic of the crossover regime, have been reported at the holepocket in the superconducting state of FeSe. As the Fermienergy of electron pocket in FeSe is much smaller than thatof hole pocket, all the bands satisfy the crossover condi-tion. Moreover, it has been reported that the crossoversignature is more pronounced with sulfur substitution. In par-ticular, in FeSe − x S x with x = .
18 in the tetragonal regime,an unusual quasiparticle dispersion, which is close to that ex-pected in the BEC regime displayed in Fig. 12(b), has beenobserved.
STM / STS can also be used to investigate the electronic dis-persions through the QPI e ff ect. The QPI patterns are noth-ing but the electronic standing waves scattered o ff defects andappear in the energy-dependent conductance images. Fouriertransformation of the conductance images allows us to de-termine the energy-dependent scattering vectors q ( E ), fromwhich one can infer the quasiparticle dispersions in momen-tum space. Unlike ARPES, QPI can access not only the filledstate but also the empty state above ε F , being useful to explorethe electron bands in FeSe.The QPI patterns of FeSe (Fig. 14) are highly anisotropicdue to nematicity. The obtained QPI dispersions consist ofone electron branch and multiple hole branches (Fig. 14),and the electron and hole branches disperse along orthogonalaxes. There are at least two hole-like QPI branchesthat cross ε F , while there is only one hole band at ε F ( § . ff erent k z states at k z = k z = π . These QPI signals may beassociated with the scattering vectors that correspond to theminor axes of the cross sections of the nematicity-deformedhole and electron Fermi pockets.
Besides these features, one can estimate Fermi energies andFermi momenta from the QPI dispersions, which faithfullyrepresent the band dispersions. The top and bottom of thehole and electron branches correspond to ε hF and ε eF , respec-tively, allowing us to estimate ε hF ∼ ε eF ∼ k hF ≈ . − . − and k eF ≈ . − for hole and electroncylinders, respectively. These correspond to the minor axes ofthe deformed Fermi cylinder and the k F ’s along the major axesmay be a few times larger. Such shallow bands are consistentwith those reported by the quantum oscillations and ARPESmeasurements. STM / STS can also directly evaluate the superconducting-gap size from the tunneling spectrum. As discussed in § . ∆ l ≈ . ∆ s ≈ . ff erent Fermi surfaces. It is still unclearwhich gap opens on which Fermi surface. Nevertheless, since ε F (cid:46)
20 meV and ∆ (cid:38) . ∆ /ε F must belarger than 0.1 for both bands, placing FeSe in the BCS-BEC crossover regime. Additional strong support of the BCS-BEC crossover is provided by extremely small k F ξ . Since ξ ab , q a (2 π / a ) q a (2 π / a ) -0.2 -0.1 0.0 0.1 0.2-0.2 -0.1 0.0 0.1 0.2+15 meV-40-2002040 E ne r g y ( m e V ) -0.2 -0.1 0.0 0.1 0.2-0.2-0.10.00.10.2-0.2 -0.1 0.0 0.1 0.2-15 meV(a) (b)(c) (d) q a (2 π / a ) q b (2 π / b ) q b ( π / b ) Fig. 14.
QPI patterns of FeSe at a low temperature (1.5 K). Magnetic field µ H =
12 T was applied along the c axis to suppress superconductivity.(a) and (b) Fourier-transformed normalized conductance images at -15 meVand +
15 meV, respectively. Uniaxial patterns are observed due to nematicity.Here, the axes of the orthorhombic unit cell is defined to be a < b < c . (c) and(d) QPI dispersions obtained by taking line cuts from the energy-dependentFourier-transformed QPI patterns along principle axes q a and q b in scatteringspace. Electron-like and hole-like dispersions are identified along q a and q b ,respectively. Adopted from Ref. 88 which is an average value in 2D plane, determined from theupper critical field ( ∼
17 T) in perpendicular field ( H (cid:107) c )is roughly 5 nm, k F ξ should be of the order of unity, againindicating the BCS-BEC crossover superconductivity. The large value of ∆ /ε F should give rise to novel featuresin the vortex core. Since the vortex core is a sort of poten-tial well, quantized bound states (Caroli-de Gennes-Matriconstates) should be formed inside, as schematically shown inFig. 15(a). Such vortex-core states can be investigated bySTM / STS, in principle. The energies of these states are givenby ± µ c ∆ /ε F , where µ c = / , / , / , · · · is the quantumnumber that represents the angular momentum. In the BCSlimit, owing to the small ratio of ∆ /ε F , ∆ /ε F is order of µ eVfor most of superconductors so far known. The number ofthe bound state is roughly ε F / ∆ , which is usually very large,more than 1000. Therefore, because of inevitable smearinge ff ects (e.g. thermal broadening), it is almost impossible toobserve the individual Caroli-de Gennes-Matricon states bySTM / STS. Instead, a large number of bound states overlap toform a broad particle-hole symmetric peak at zero energy inthe tunneling spectrum at the vortex center. With increasingdistance from the center, this zero-energy peak splits and con-tinuously approaches to ± ∆ . By contrast, in the BCS-BEC-crossover regime, ε F / ∆ should be of the order of unity and thus the vortex core accom-modates only a few levels, resulting in a so-called quantum-limit vortex. Here, ∆ /ε F can become large enough andeach bound state may be resolved by STM / STS. Spatial evo-lution of the bound states should be no longer continuous but
12. Phys. Soc. Jpn.
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E r ∆ ξ centerfaraway T unne li ng c ondu c t an c e ( n S ) -6 -4 -2 0 2 4 6Sample bias (mV)50403020100 -4 -2 0 2 4Sample bias (mV)-30-20-100102030 D i s t an c e ( n m ) (a) Fig. 15. (a) Schematic energy E diagram of an s -wave superconductor as afunction of the distance r from the center of the vortex core. Superconductinggap ∆ (red) recovers over the coherence length ξ and the discrete Caroli-de Gennes-Matricon states (blue) are formed in the core region. (b) Zero-energy conductance map showing a single vortex at zero energy at 0.4 K in amagnetic field of 0.25 T along the c axis. (c) Tunneling spectra taken at thevortex center (red) and away from vortices (blue). Inset shows a magnifiedspectrum at the vortex center. (d) Spatial evolution of tunneling spectra alongthe dashed in (b). Adopted from Ref. 104 shows the Friedel-like oscillations. Such characteristic signatures of BCS-BEC crossover hasbeen reported in FeSe.
Figure 15(b) shows the vortex im-age of FeSe in a magnetic field of 0.25 T along the c axis.The vortex is elongated due to the nematicity. As shownin Fig. 15(c), the lowest-energy local-DOS (LDOS) peak ofFeSe is not at zero energy, representing the lowest bound statein the quantum-limit vortex core. The spatial evolution of thebound states exhibits an oscillatory behavior. The wavelength of such spatial oscillations corresponds to π/ k F , beingconsistent with the theoretical prediction of the Friedel-likeoscillations. It is well known that Cooper pairs can survive even above T c as thermally fluctuating droplets. These fluctuations arisefrom amplitude fluctuations of the superconducting order pa-rameter and have been investigated for many decades. Ithas been shown that their e ff ects on thermodynamic, trans-port, and thermo-electric quantities in most superconductorsare well understood in terms of the standard Gaussian fluctua-tion theories. However, in the presence of preformed pairs as-sociated with the BCS-BEC crossover, superconducting fluc-tuations are expected to be strikingly enhanced compared tothe Gaussian theories due to additional phase fluctuations. Of particular interest is the pseudogap formation, which is thecentral enigma of the underdoped cuprates.
The origin ofthe pseudogap has been discussed in terms of the preformedpairs associated with the crossover phenomenon, which canlead to a partial reduction of the DOS near the Fermi level.
However, it is still highly controversial. Another important is-sue associated with the BCS-BEC crossover is the breakdownof Landau’s Fermi-liquid theory due to the strong interac-tion between fermions and fluctuating bosons. In ultracold-atomic systems, a Fermi-liquid-like behavior has been re-ported in thermodynamics even in the crossover regime, butmore recent photoemission experiments have suggested a siz-able pseudogap opening and a breakdown of the Fermi-liquiddescription.
Thus the superconducting fluctuations and pseudogap for-mation in FeSe are highly intriguing. Superconducting fluctu-ations give rise to the reduction of the normal-state resistivity.In zero field, d ρ xx / d T of FeSe shows a minimum at around T ∗ ∼
20 K, which can be attributed to the appearance of the ad-ditional conductivity due to the fluctuation of the order param-eter (paraconductivity) below T ∗ . However, a quantitativeanalysis of the paraconductivity is di ffi cult to achieve becauseits evaluation strongly depends on the extrapolation of thenormal-state resistivity above T ∗ to lower T . The fluctuation-induced magnetoresistance of FeSe is also di ffi cult to analyzeowing to a large and complicated magnetoresistance, which ischaracteristic of compensated semimetals (see Fig. 2).The superconducting fluctuations in FeSe have been ex-amined through the magnetic measurements by severalgroups. The diamagnetic response due to supercon-ducting fluctuations is clearly observed in the magnetization M ( H ) for H (cid:107) c , which exhibits a pronounced decreasebelow T ∗ . A crossing point in the diamagnetic response inmagnetization, where M dia ( T , H ) exhibits a field indepen-dent value, is observed near T c . Such a crossing behav-ior is also observed in cuprates, which has been pointed outto be a signature of large superconducting fluctuations.
The fluctuation-induced diamagnetic susceptibility of mostsuperconductors including multiband systems can be well de-scribed by the standard Gaussian type (Aslamasov-Larkin)fluctuation susceptibility χ AL , which is given by χ AL = − π k B T c Φ ξ ab ξ c (cid:114) T c T − T c , (4)in the zero-field limit, where Φ is the flux quantum, and ξ ab and ξ c are the coherence lengths parallel and perpendicular tothe ab plane at zero temperature, respectively. In the multi-band case, the behavior of χ AL is determined by the shortestcoherence length of the main band, which governs the orbitalupper critical field.The low-field diamagnetic response in FeSe, which is mea-sured by the magnetic torque τ = µ V M × H (where V isthe sample volume), has been reported to exhibit highly un-usual nature of the superconducting fluctuations. From thetorque measurements, the di ff erence between the c axis and ab plane susceptibilities, ∆ χ = χ c − χ ab , can be determined.It has been reported that ∆ χ ( T ) is strongly enhanced with de-creasing T and exhibits divergent behavior near T c , indicat-ing the presence of the superconducting fluctuation-induceddiamagnetic contribution. The Gaussian type fluctuationsusceptibility given by Eq. 4 indicates that diamagnetic re-sponse of the magnetization is H -linear. In contrast, the ob-served diamagnetic response of FeSe contains both H -linearand non-linear contributions of the magnetization. Fig-ure 16 and its inset show the temperature dependence of non-
13. Phys. Soc. Jpn.
FULL PAPERSFig. 16.
Temperature dependence of the non-linear diamagnetic responseat µ H = . T ∗ this diamagnetic response is largely enhanced. Blueline represents | ∆ χ AL | in the standard Gaussian fluctuations theory calculatedfrom Eq.(4). The inset shows | ∆ χ nldia | plotted in a semi-log scale at low tem-peratures. Adopted from Ref. 134. linear part ∆ χ nldia , which is estimated by subtracting the H -linear contribution obtained at the highest field as ∆ χ nldia ( H ) ≈ ∆ χ ( H ) − ∆ χ (7 T). In Fig. 16 and its inset, the contribu-tion expected from the Gaussian fluctuation theory given by ∆ χ AL ≈ − π k B T c Φ (cid:18) ξ ab ξ c − ξ c (cid:19) (cid:113) T c T − T c is also plotted, where weuse ξ ab = . ξ c = . T c , ∆ χ nldia at 0.5 T isnearly 10 times larger than ∆ χ AL .The above results provide evidence that the amplitude ofthe diamagnetic fluctuations of FeSe is by far exceeding thatexpected in the standard Gaussian theory, implying that thesuperconducting fluctuations in FeSe are distinctly di ff erentfrom those in conventional superconductors. The supercon-ducting fluctuations in FeSe have been further examined byseveral groups. The NMR relaxation rate divided by the tem-perature ( T T ) − , which increases below T s , starts to be sup-pressed below T ∗ . The NMR results report the presence ofthe superconducting fluctuations that deviate from the stan-dard Gaussian theory. On the other hand, the magnetizationmeasurements using by a superconducting quantum interfer-ence device (SQUID) did not observe such a large supercon-ducting fluctuation signal. Moreover, the magnetic torquemeasurements by using an optical detection technique reporta considerably smaller fluctuation signal originating from thevortex liquid.
The discrepancy between these measure-ments may be due to the sample quality. In fact, the crystalsin which a giant superconducting fluctuation is observed ex-hibit very large magnetoresistance and distinct quantum os-cillations at high magnetic fields.Although giant superconducting fluctuations are observedin the diamagnetic response in FeSe, the jump of the heat ca- pacity at T c is still an ordinary mean-field BCS-like as shownin Fig. 8(c). However, recent detailed analysis of the heat ca-pacity near T c suggests the presence of superconducting fluc-tuations that substantially exceed Gaussian fluctuations. The heat capacity measurement report significant fluctuatione ff ects not only in zero and but also in the vortex state withmagnetic field applied both parallel and perpendicular to the ab plane.Thermal fluctuations have a dramatic e ff ect on the vor-tex system in type-II superconductors. One of the mostprominent e ff ect is the vortex lattice melting. The vortex lat-tice melts when the thermal displacement of the vortices isan appreciable fraction of the distance between vortices. Inquasi-2D high- T c cuprates, the magnetic field at which themelting transition occurs is much lower than the mean up-per critical field H c2 . The strength of the thermal fluctuationsis quantified by the dimensionless Ginzburg number, G i = [ (cid:15) k B T c / H (0) ξ ab ] /
2, which measures the relative size of thethermal energy k B T c and the condensation energy within thecoherence volume. Here (cid:15) ≡ λ c /λ ab ( λ c is the penetrationdepth for screening current perpendicular to the ab plane)is the anisotropy ratio and H c = Φ / √ λ ab ξ ab is the ther-modynamic critical field. The large G i leads to the reductionof the vortex lattice melting temperature T melt . G i is roughlyproportional to ( ∆ /ε F ) . In conventional low- T c superconduc-tors, G i ranges from 10 − to 10 − , while in FeSe with large ∆ /ε F , G i is estimated to be as large as 10 − , which is compa-rable or even larger than that of YBa Cu O . We therefore ex-pect a sizable separation of the melting transition below T c ( H )over a large portion of the phase diagram, as it is the case inthe high- T c cuprates. Very recently, the vortex lattice melt-ing transition has been reported by heat capacity measure-ments. It has been reported that the melting line mergeswith H c2 at a finite temperature, which is consistent with thetheory of the vortex lattice melting in strongly Pauli-limitedsuperconductors. An important question related to the preformed pairs is theformation of the pseudogap, which is a characteristic signa-ture of BCS-BEC crossover other than the giant supercon-ducting fluctuations. The pseudogap formation in FeSe is anontrivial issue because FeSe is a compensated semimetalwith hole and electron pockets, which may give rise to morecomplicated phenomena than in the single-band case.
Be-low T ∗ , the NMR ( T T ) − is suppressed and exhibits a broadmaximum at T p ( H ), which bears a resemblance to the pseudo-gap behavior in optimally doped cuprate superconductors. It has been reported that T ∗ and T p ( H ) decrease in the samemanner as T c ( H ) with increasing H . This suggests that thepseudogap behavior in FeSe is ascribed to superconductingfluctuations, which presumably originate from the theoreti-cally predicted preformed pairs. Spectroscopic signature for the pseudogap formation above T c was first reported by the STM / STS measurements.
However, as shown in Fig. 17(a), the subsequent STM / STSmeasurements reported the absence of the pseudogap.
The absence of the spectroscopic pseudogap despite a large ∆ /ε F has been discussed in terms of the multi-band characterand its compensated semimetal nature of FeSe. In the BCS-BEC crossover superconductivity of a single band system, the
14. Phys. Soc. Jpn.
FULL PAPERS T unne li ng c ondu c t an c e ( n S ) -8 -6 -4 -2 0 2 4 6 8Sample bias (mV) 0.4 K1.5 K2.5 K5.0 K7.5 K10.0 K12.5 K15.0 K20.0 K25.0 K Fig. 17.
Tunneling spectra taken at di ff erent temperatures. Each curve isshifted by 5 nS for clarity. Adopted from Ref. 104. chemical potential is shifted outside of the band edge. How-ever, if there are hole and electron bands, which nearly com-pensate each other and have strong interband interactions, thechemical potential should be pinned at the original energy po-sition. In the case of perfectly symmetrical electron and holebands, chemical potential should always be pinned at zero en-ergy because µ e + µ h = µ e and µ h are chemical potentials of theelectron and hole bands, respectively. It has been pointed outthat in such a case, the splitting between T ∗ and T c is largelyreduced, leading to the suppression of the pseudogap for-mation even in the BCS-BEC crossover regime.In a real material, however, there is a certain asymmetrybetween the electron and hole bands, which should cause ashift of the chemical potential as a function of temperature.The multiband character brings about another e ff ect that sup-presses the pseudogap formation. If there are electron andhole pockets, there should appear two pairing channels asso-ciated with the interband and intraband interactions. In thecase of FeSe, superconductivity occurs in the nematic phaseand the superconducting gap possesses strong anisotropy asdiscussed in §
3. A possible scenario is that the anisotropyis caused by the mixture of s -wave and d -wave symmetriesand the former and the latter are caused by interband andintraband interactions, respectively. Theoretical calculationsbased on such a model have been performed to estimate thepair-formation and the superconducting-transition tempera-tures. It has been shown that if the interband pairing isstronger than the intraband pairing, which may be the case forFeSe, the pair-formation and the superconducting-transitiontemperatures do not split despite of large ∆ /ε F . The BCS-BEC crossover in the multiband system may be di ffi cult torealize in ultracold atomic systems. Therefore, FeSe is uniqueand can open a new research field for the BCS-BEC crossover. − x S x Very recently, the evolution of the BCS-BEC crossoverproperties with sulfur substitution has been investigated inFeSe − x S x . ARPES measurements reported that ∆ /ε F de- creases with x , but it still remains large. Therefore, itis expected that the crossover nature is less pronouncedin FeSe − x S x . Contrary to this expectation, it has beenfound that the crossover nature becomes more significant inFeSe − x S x . The ARPES experiments report that flatdispersion of the Bogoliubov quasiparticles is more pro-nounced with x in the nematic regime. Surprisingly, on en-tering the tetragonal regime beyond the nematic QCP, flat dis-persion changes to BEC-like one, which shows the gap mini-mum at k = The heat capacity measurements reveal thehighly unusual BEC-like transition with strong fluctuations intetragonal FeSe − x S x , which appears to be consistent withthe ARPES measurements.However, the formation of the spectroscopic pseudogap inthe tetragonal FeSe − x S x is controversial. Although ARPESmeasurements report the distinct pseudogap-like reduction ofthe DOS in the hole pocket above T c , no discernible re-duction of the DOS is observed in STS measurements. Theevolutions of the BCS-BEC crossover behavior in FeSe − x S x again suggest that a multiband system may possess a uniquefeature that is absent in a single band system. Thus FeSe − x S x o ff ers a unique playground to search for as-yet-unknownnovel phenomena in strongly interacting fermions, which de-serves future attention.
5. Exotic Superconducting State Induced by MagneticField
The emergence of a novel superconducting phase at highmagnetic fields, whose pairing state is distinctly di ff erentfrom that of the low-field phase, has been a longstanding issuein the study of superconductivity. One intriguing issue relatedwith this field-induced phase concerns whether the spin im-balance or spin polarization will lead to a strong modificationof the properties of the electron systems. This problem hasbeen of considerable interest not only for superconductors insolid state physics, but also in the studies of neutral Fermionsuperfluid in the field of ultracold atomic systems and for thecolor superconductivity in high-energy physics. Among sev-eral possible exotic states associated with the spin imbalance,a spatially nonuniform superconducting state caused by theparamagnetism of conduction electrons has been one of themost intensively studied topics in the past half-century af-ter the pioneering work by Fulde and Ferrell (FF) as well asLarkin and Ovchinnikov (LO). In the FFLO state, aninhomogeneous superconducting state with modulated super-conducting order parameter is formed.It has been reported that in FeSe, a new superconduct-ing phase appears at the low-temperature / high-field cornerin the superconducting state of the H - T phase diagram forboth H (cid:107) ab plane and H (cid:107) c axis. It has beendiscussed that the field-induced phase for H (cid:107) ab is atleast consistent with the FFLO phase. However, even if theFFLO state is realized in FeSe, its physical properties are ex-pected to be very di ff erent from those of the originally pre-dicted FFLO state in several aspects, such as the extremelyhighly spin polarized state, coexistence of the FFLO andthe Abrikosov vortex states, strongly orbital-dependentpairing interaction, nontrivial Zeeman e ff ect due to spin-orbit coupling, and multiband electronic structure. In par-ticular, the magnetic field-induced superconducting phase in
15. Phys. Soc. Jpn.
FULL PAPERSFig. 18. (a) High-field phase diagram of FeSe for H (cid:107) c . Solid blue and open red circles represent the irreversible field H irr determined by the resistivity andthe magnetic torque, respectively. The mean-field upper critical field is above H irr . Solid red circles represent H ∗ determined by the cusp of the field dependenceof the thermal conductivity. High-field superconducting B-phase separated from the low-field A phase, which is the BCS pairing ( k ↑ , − k ↓ ) phase, has beenproposed. Adopted from Ref. 91. (b) Phase diagram for H (cid:107) ab plane. Blue circles and green crosses show H irr and H p determined by resistivity measurements.Orange and yellow circles show H k and H ∗ determined by thermal-conductivity measurements, respectively. Above the first-order phase transition field H ∗ a distinct field-induced superconducting phase emerges at low temperatures. High-field phase has been attributed to the FFLO pairing ( k ↑ , − k + q ↓ ) state.Adopted from Ref. 151. FeSe provides insights into previously poorly understood as-pects of the highly spin-polarized Fermi liquid in the BCS-BEC crossover regime.
The presence of the high-field superconducting phase sep-arated from the low-field one has been reported by severalmeasurements, including resistivity, magnetic torque, heat ca-pacity and thermal transport measurements. The phase dia-gram in magnetic field applied parallel to the c axis ( H (cid:107) c ) isshown in Fig. 18(a). At a field H ∗ , the thermal conductivityexhibits a cusp-like feature. As the Cooper pair condensatedoes not contribute to heat transport, the thermal conductiv-ity can probe quasiparticle excitations out of the supercon-ducting condensate. Moreover, the thermal conductivityhas no fluctuation corrections, the cusp of κ/ T usually cor-responds to a mean-field phase transition. The presence of H ∗ has also been reported by a distinct kink anomaly of the ther-mal Hall conductivity κ xy . The analysis of the thermal Hallangle κ xy /κ indicates a change of the quasiparticle scatteringrate at H ∗ . Very recently, the anomaly at H ∗ is also con-firmed by heat capacity measurements.The irreversibility field H irr caused by the vortex pinningis determined by the magnetic torque, which measures thebulk properties, and by the resistivity. The irreversibility lineat low temperatures extends to high fields well above H ∗ ,demonstrating that H ∗ is located inside the superconductingstate. These results suggest the presence of a field-inducedsuperconducting phase ( B -phase in Fig. 18(a)). However,the presence of B -phase is controversial between di ff erentgroups. The H - T phase diagram of FeSe for H (cid:107) ab has also beenstudied recently by several groups via the measurements of in-plane electrical resistivity, thermal conductivity, magne-tocaloric e ff ect and heat capacity up to 35 T. All mea-surements appear to consistently show the presence of field-induced superconducting phase in this geometry. Figure 18(b) Fig. 19.
Magnetic field dependence of thermal conductivity in the highfield regime at low temperatures in FeSe for H (cid:107) ab . A discontinuous down-ward jump appears at µ H ∗ ≈
24 T inside the superconducting state as indi-cated by black arrows. Adopted from Ref. 151. displays the H - T phase diagram. The anomaly in the super-conducting state has been first reported by the magnetocalorice ff ect. The most remarkable anomaly has been reportedby the thermal conductivity measurements on a twinned crys-tal in H applied along the diagonal direction in the ab plane( H (cid:107) [110] O , in orthorhombic notation). As displayed inFig. 19, κ ( H ) exhibits a discontinuous downward jump at µ H ∗ ≈
24 T inside the superconducting state. At H ∗ , κ ( H )shows a large change of the field slope and increases steeplywith H above H ∗ . It should be stressed that the jump of κ ( H ),which is caused by a jump in entropy, is a strong indication ofa first-order phase transition, as reported for CeCoIn andURu Si . In the H - T phase diagram, the irreversibility field, H irr ,determined by the onset field of non-zero resistivity is alsoshown. Below ∼ H irr exhibits an anomalous upturn. It
16. Phys. Soc. Jpn.
FULL PAPERS should be stressed that H ∗ is deep inside the superconduct-ing state at low temperatures, as evidenced by the fact that H ∗ is well below H irr . No discernible anomaly of κ ( H ) is ob-served above about 2 K, indicating that the first-order transi-tion occurs only within the superconducting state. It has beenreported that the resistive transition under magnetic fields ex-hibits a significant broadening at high temperatures. This isattributed to a strongly fluctuating superconducting order pa-rameter, which gives rise to the drift motion of vortices inthe liquid state. On the other hand, below ∼ Thus, there is a distincthigh-field superconducting phase, which is well separated bya first-order phase transition from the low-field phase, alsofor H (cid:107) ab . The quantum oscillation measurements excludethe possibility that the high-field superconducting phases ob-served for both H (cid:107) c and H (cid:107) ab are AFM ordered phases. Superconductivity is destroyed by external magnetic fieldthrough the orbital and Pauli pair breaking e ff ects. The formere ff ect is associated with the Lorentz force acting on electrons,which results in the formation of vortices. This orbital pair-breaking field is given as H orb = Φ / πξ . The latter e ff ect isassociated with the spin paramagnetic e ff ect that tries to alignthe spin of the original singlet Cooper pairs through the Zee-man e ff ect. This Pauli pair-breaking limit takes place whenthe paramagnetic energy in the normal state E P = χ n H ,where χ n = g µ N ( ε F ) is the normal-state spin susceptibil-ity, where g is the g -factor and µ B the Bohr magneton, co-incides with the superconducting condensation energy E s = N ( ε F ) ∆ , which yields H P = ∆ / √ g µ B . The ratio of H orb and H P , which is called Maki parameter, is given by α M ≡ √ H orb H P ≈ m ∗ m ∆ ε F , (5)where m ∗ is the e ff ective mass of the conduction electronand m is the free electron mass. The Maki parameter is usu-ally much less than unity, indicating that the influence of theparamagnetic e ff ect is negligibly small in most superconduc-tors. However, in quasi-2D layered superconductors (for par-allel fields) and heavy fermion superconductors, α M is largelyenhanced owing to large m ∗ / m values, and thus the super-conductivity may be limited by Pauli paramagnetic e ff ect. Itshould be stressed that in superconductors in the BCS-BECcrossover regime, large ∆ /ε F leads to the enhancement of α M value.FFLO proposed that when the superconductivity is lim-ited by the Pauli paramagnetic e ff ect, the upper critical fieldcan be enhanced by forming an exotic pairing state. Incontrast with the ( k ↑ , − k ↓ ) pairing in the traditional BCSstate, as shown in Fig. 20(a), the Cooper pair formation inthe FFLO state occurs between Zeeman splitted parts of theFermi surface leading to a new type of ( k ↑ , − k + q ↓ ) pair-ing with | q | ∼ g µ B H / (cid:126) υ F ( υ F is the Fermi velocity), as shownin Fig. 20(b); the Cooper pairs have finite center-of-mass mo-menta. Because of the finite q ≡ | q | , the superconducting or-der parameter ∆ ( r ) ∝ (cid:104) ψ †↓ ( r ) ψ †↑ ( r ) (cid:105) has an oscillating com-ponent exp(i q · r ). The FF superconducting state has a spon-taneous modulation in the phase of the order parameter, while the LO state has a spatial modulation of Cooper pairdensity. It is generally found that the LO states are fa-
Fig. 20. (a) Schematic illustration of Cooper pairing ( k ↑ , − k ↓ ) in the BCSstate. (b) Pairing state with ( k ↑ , − k + q ↓ ) in the FFLO state. (c) Schematicillustration of the superconducting order parameter ∆ in real space and seg-mentation of the magnetic flux lines by planar nodes. (d) Schematic elec-tronic structures of hole and electron pockets at fields around H ∗ in FeSe.Both Fermi surfaces are highly spin imbalanced. Adopted from Ref. 151. vored over the FF states, but henceforth both states are simplyrefer to as FFLO state. In the FFLO state, spatial symmetrybreaking originating from the appearance of the q -vector ap-pears, in addition to gauge symmetry breaking. A fascinatingaspect of the FFLO state is that it exhibits inhomogeneoussuperconducting phases with a spatially oscillating order pa-rameter. In its simplest form, order parameter is modulatedas ∆ ( r ) ∝ sin( q · r ), and periodic planar nodes appear per-pendicular to the magnetic field, leading to a segmentation ofthe vortices into pieces of length Λ = π/ q , as illustrated inFig. 20(c).Despite considerable research e ff orts in the search for theFFLO states in the past half century, the FFLO state still con-stitutes a challenge for the researchers. Very stringent con-ditions are required for the realization of the FFLO state. Inreal bulk type-II superconductors, the orbital e ff ect is invari-ably present, which is detrimental to the formation of theFFLO state. The FFLO state can exist at finite temperaturesif α M is larger than 1.8, but the FFLO region shrinks consid-erably from that in the absence of the orbital e ff ect. More-over, the FFLO state is highly sensitive to disorder. Despitetremendous studies of the FFLO state, its firm experimentalconfirmation is still lacking. Some signatures of the FFLOstate have been reported in only a few candidate materials,including heavy fermion and quasi-2D organic supercon-ductors. Among them, organic κ -(ET) Cu(NCS) andheavy fermion CeCoIn have been studied most extensively.In both systems, a thermodynamic phase transition occursbelow upper critical fields and a high-field superconductingphase emerges at low temperatures. In the former, eachsuperconducting layer is very weakly coupled via the Joseph-son e ff ect. A possible FFLO state is reported in a magneticfield applied parallel to the layers, where the magnetic flux isconcentrated in the regions between the layers forming core-less Josephson vortices. However, the position of the firstorder transition in H - T phase diagram has been controver-
17. Phys. Soc. Jpn.
FULL PAPERS sial, depending on the measurement method. Moreover, it hasbeen pointed out that vortex phase transitions have given riseto considerable ambiguity in the interpretation of the exper-imental data. The presence of the FFLO phase in CeCoIn remains a controversial issue. In fact, the H - T phase dia-gram of CeCoIn is very di ff erent from that expected in theoriginal FFLO state. Moreover, the magnetic order occurs si-multaneously at the putative FFLO transition, indicatingthat this phase is not a simple FFLO phase. PossibleFFLO states have also been discussed recently in other sys-tems, including heavy fermion CeCu Si and iron-pnictideKFe As . FeSe may satisfy some of the prerequisites for the realiza-tion of the FFLO state. In FeSe in the vicinity of the BCS-BEC crossover regime, an estimate gives ∆ /ε F ∼ . ∆ /ε F for the electron band. By using m ∗ ≈ m (4 m ) for hole (electron) pocket deter-mined by SdH oscillation experiments, α M is found to beas large as ∼ ∼ .
5) for the electron (hole) pocket. This ful-fills a requirement for the formation of the FFLO state. More-over, the analyses of magnetoresistance and quantum oscilla-tions show that high-quality single crystals of FeSe, obtainedthrough flux / vapor-transport growth techniques, are in theultra-clean limit with extraordinary long mean free path (cid:96) .On the other hand, there are several unique aspects in FeSe,which have not been taken into account in the original idea ofthe FFLO state. They arise from the extremely shallow pock-ets and multi-band character. • Extremely large spin imbalance. The BCS-BECcrossover nature in FeSe gives rise to the large spinimbalance near the upper critical field, which will bediscussed in the next subsection. • Strong spin-orbit coupling, λ so ∼ ε F , which yieldslargely orbital-dependent Zeeman e ff ect. It has been sug-gested that this seriously modifies the Pauli limiting fieldthrough the g -factor. • Orbital-dependent pairing interaction, which is expectedto a ff ect seriously the q -vector. The FFLO pairing mayalso be orbital dependent.It has been discussed that the high-field phase for H (cid:107) ab can be associated with an FFLO phase for the following rea-sons. First, in the H - T phase diagram shown in Fig. 18(b),the steep enhancement of H ab c2 at low temperatures and thefirst-order phase transition at a largely T -independent H ∗ areconsistent with the original prediction of the FFLO state. Sec-ond, planar nodes perpendicular to H are expected as the mostoptimal solution for the lowest Landau level. In the presentgeometry, where the thermal current density j T (cid:107) H , quasi-particles that conduct heat are expected to be scattered by theperiodic planar nodes upon entering the FFLO phase. Thisleads to a reduction of κ ( H ) just above H ∗ , which is consis-tent with the present results. Third, as the c -axis coherencelength ( ξ c ≈ . one-dimensional (1D) tube-like Abrikosovvortices are formed even in a parallel field. In this case, theplanar node formation leads to a segmentation of the vorticesinto pieces of length Λ . The pieces are largely decoupled and,hence, better able than conventional vortices to position them-selves at pinning centers, leading to an enhancement of the pinning forces of the flux lines in the FFLO phase. This is con-sistent with the observed sharp resistive transition above H ∗ .Fourth, as will be discussed in the next subsection, the elec-tron pocket is extremely spin polarized near the upper criti-cal fields. Therefore, it is questionable that superconductingpairing is induced in the electron pocket in such a stronglyspin-imbalanced state. It has been shown that the FFLO in-stability is sensitive to the nesting properties of the Fermisurface. When the Fermi surfaces have flat parts, the FFLOstate is more stabilized through nesting. As the portion of thehole pocket derived from the d yz orbital forms a Fermi-surfacesheet that is more flattened than the other portion of the Fermisurface, this 1D-like Fermi sheet is likely to be responsiblefor the FFLO state.In contrast to H (cid:107) ab , the high field phase for H (cid:107) c re-mains elusive and its identification is a challenging issue. Itcannot be simply explained by the FFLO state, although the H - T phase diagram has some common features with that for H (cid:107) ab . The q -vector, which is always in the ab plane, doesnot stabilize FFLO state for H (cid:107) c . Therefore, an FFLO statemay be di ffi cult to be formed due to the lack of q -vector for H applied perpendicular to the quasi-2D Fermi surface of FeSe.According to the calculation of the e ff ective g -factor obtainedby the orbitally projected model, the formation of FFLOstate is more favored for H (cid:107) ab than for H (cid:107) c . Recentlya possible FFLO state has been proposed even for H (cid:107) c . The field-induced superconducting phase provides insightsinto previously poorly understood aspects of the highly spin-polarized Fermi liquid in the BCS-BEC crossover regime. Inthe standard BCS theory of spin singlet pairing, the pairingoccurs between the fermions with opposite spins. The ques-tion of what happens if a large fraction of the spin-up fermionscannot find spin-down partners has been widely discussed byresearchers from di ff erent aspects. In conventional supercon-ductors, however, a large unequal population of spin-up anddown electrons is very di ffi cult to be realized, essentially be-cause superconductivity is usually destroyed by the orbitalpair-breaking e ff ects. Even when the superconductivity is de-stroyed by Pauli paramagnetic e ff ect, such a spin imbalanceis usually negligibly small.In paramagnetic metals, the spin imbalance is caused bythe Zeeman splitting in magnetic field, as shown in Fig. 20(d).The magnitude of the spin imbalance P spin = ( N ↑ − N ↓ ) / ( N ↑ − N ↓ ), where N ↑ and N ↓ are the numbers of up and down spins,respectively, is roughly estimated as P spin ≈ µ B H /ε F . There-fore, P spin is estimated to be P spin ≈ ∆ /ε F at H P in Pauli-limited superconductors. In orbital-limited superconductors,where H orb < H P , P spin at H orb is smaller than that expectedin Pauli-limited superconductor. Therefore, in almost all su-perconductors, P spin is usually negligibly small, P spin < − ,near the upper critical field, i.e. the e ff ect of the spin imbal-ance is not taken into account in the original FFLO proposal.One intriguing issue concerns whether the large spin imbal-ance will lead to a strong modification of the properties of thecorrelated Fermi systems. Although highly spin-imbalancedFermi systems have been realized in ultracold atomic gases,the nature of the spin imbalanced superfluid remains unex-ploited experimentally due to the di ffi culty in cooling the sys-
18. Phys. Soc. Jpn.
FULL PAPERSFig. 21. (a) Field dependence of specific heat at low temperatures below ∼ . − x S x covering orthorhombic ( x = , . , .
13) and tetragonal( x = .
20) phases. (b-e) Field dependence of thermal conductivity at low temperatures below ∼ . x = .
08 (b), 0.13 (c), 0.16 (d), andtetragonal 0.20 (e). The insets are the same data plotted against H / . The dashed lines are the fits to √ H dependence. Adopted from Ref. 89. tems to su ffi ciently low temperatures. In FeSe in the BCS-BEC crossover regime, the Zeeman e ff ect is particularly ef-fective in shrinking the Fermi volume associated with the spinminority, giving rise to a highly spin-imbalanced phase where ε F ∼ ∆ ∼ µ B H c2 near the upper critical fields. For H (cid:107) ab ,an estimate yields P spin ∼ . g = This indicates thatelectron pockets are extremely highly polarized. Therefore inthe high-field phase of FeSe, a large fraction of the spin-upfermions cannot find spin-down partners.The presence of a possible FFLO phase in FeSe shouldstimulate considerable further work in understanding and ex-ploiting strongly interacting Fermi liquids near the BCS-BEC crossover regime, which remains largely unexplored andmight bridge the areas of condensed-matter and ultracold-atom systems.
6. Superconductivity near the Nematic Critical Point
As discussed in § .
3, the electronic nematic phase in FeSecan be suppressed by isovalent S substitution for Se site. Nearthe nematic QCP ( x c ≈ . and the transport properties show non-Fermi liquid properties. The impact of such nematicquantum criticality on superconductivity is an important sub-ject in the field of condensed matter physics.
Inside the ne-matic phase ( x < x c ), the superconducting transition tempera-ture T c shows a broad peak at x ∼ .
08 (see Fig. 7). At x = x c , T c jumps from ∼ ∼ As can be seen in Fig. 21, theupper critical field for H (cid:107) c is also strongly suppressed from ∼
10 T ( x = . < x c ) to ∼ x = . > x c ). Thesesignificant changes of the superconducting properties at thenematic critical concentration implies that the nematicity, orrotational symmetry breaking, a ff ects strongly on supercon-ductivity.The superconducting gap structure is changed abruptly atthe nematic QCP, which is evidenced by the field dependencestudies of specific heat C and thermal conductivity κ . In-side the nematic phase, the field dependence of both C / T and κ/ T shows the √ H behavior at low fields as expected in thenodal superconductors, while it deviates from √ H at fieldsmuch lower than the upper critical field H c2 , as shown inFigs. 21(a)-(d). This deviation can be explained by the multi-gap e ff ect, and the deviation field H ∗ has been attributed to thevirtual upper critical field of the smaller gap. In contrast, in thetetragonal samples outside the nematic phase, such a multigapbehavior is not observed, and as shown in Figs. 21(a) and (e),the field dependence of C / T and κ/ T can be fitted to √ H de-pendence in the entire field range up to H c2 . Near the nematicQCP, charge fluctuations of d xz and d yz orbitals are enhancedequally in the tetragonal side, while they develop di ff erentlyin the nematic phase. From these results, it has been suggestedthat the orbital-dependent nature of the nematic fluctuationshas a strong impact on the superconducting gap structure andhence on the pairing interaction. This drastic change of the superconducting gap has beencorroborated by the systematic studies of scanning tunneling
19. Phys. Soc. Jpn.
FULL PAPERSFig. 22.
Evolutions of electronic structure and low-energy excitations with S-substitution determined by QPI and STS measurements in FeSe − x S x . (a)Schematics of the 3D constant energy surface of hole and electron bands, and definitions of characteristic scattering wave vectors q h1 and q h2 . (b) Evolutionsof the scattering wave vectors as functions of sulfur content x . Lines are the guides to the eyes. (c) Evolution of the Fermi velocity with x . (d) Averaged tunnelingconductance spectra of FeSe − x S x for 0 ≤ x ≤ .
25. Each curve is shifted vertically for clarity. Blue (red) curves are for the orthorhombic (tetragonal) phase. (e)Energy second derivative of averaged tunneling spectra. Each curve is shifted vertically clarity. (f) Evolutions of the apparent gap amplitude (green diamond)and the zero-energy spectral weight normalized by the weights at the gap-edge energies (black stars). Adopted from Ref. 88. spectroscopy, which are summarized in Fig. 22. The tun-neling conductance remains essentially unchanged with in-creasing sulfur content x inside the nematic phase, but oncethe nematicity vanishes at x > x c ≈ .
17 the superconducting-gap spectrum shows a dramatic change (see Figs. 22(d)-(f)).Below x c , clear quasiparticle peaks are observed at energies ∼ ± . x c , however, quasiparticle peaks are strongly dumpedwith a reduced gap size below ∼ . ff erence ofthe zero-bias conductance is also consistent with the thermo-dynamic properties, and the specific heat data shows a largeresidual DOS only for x > x c . By using the QPI tech-nique, the evolution of normal-state electronic structure with x has been also studied by the same specimens (see Figs. 22(a)-(c)), which revealed that the Fermi surface structure changessmoothly across the nematic QCP. This implies that the abruptchange of the superconducting properties is not linked to thestrength of nematicity, but the presence or absence of nematic-ity results in two distinct pairing states separated by the ne-matic QCP. An intriguing theoretical proposal that may account forthese anomalous superconducting states in FeSe − x S x hasbeen recently made by Setty et al. This is based onthe recent developed notion on a novel superconducting state,dubbed Bogoliubov Fermi surface.
In unconventional superconductors, the superconductinggap is anisotropic in the momentum space, and often exhibitsnodes at certain k points. Thus there are three possible types of superconducting gaps; the gap is nodeless, it has pointnodes, or it has line nodes. It has been shown theoretically thatwhen the even-parity superconductors break the time reversalsymmetry (TRS), there is a possibility to have the fourth typehaving a surface of nodes in some circumstances. Such anovel state with Bogoliubov Fermi surface (see Fig. 23(a) asan example), which may also be called as a topological ul-tranodal pair state, can be realized when the Pfa ffi an of theBogoliubov-de Gennes Hamiltonian, which is non-negativefor TRS preserved states, becomes negative.In FeSe, as discussed in §
3, the gap structure has linenodes or deep minima, and the pairing is likely to be spinsinglet. The theoretical calculations show that when the rela-tive strength of intraband to interband pairing interactions isaltered as a function of sulfur substitution, the Pfa ffi an maychange sign to negative, providing that the TRS is broken,which gives rise to exotic superconducting states with Bo-goliubov Fermi surfaces as shown in Figs. 23(b)-(e). Ifthis condition is realized in the tetragonal phase of FeSe − x S x ,then the presence of Bogoliubov Fermi surface changes dra-matically the zero-energy DOS, which is consistent with theexperimental observations of the substantially large residuallow-energy states in tunneling spectra as well as in specificheat (see Figs. 23(f) and (g)).To realize such a state, the superconducting state mustbreak TRS, and thus the question is whether the FeSe-basedsuperconductors have TRS breaking states or not. The cur-rent experimental situation on TRS breaking is reviewed inthe next section.
20. Phys. Soc. Jpn.
FULL PAPERSFig. 23.
Exotic superconducting state with Bogoliubov Fermi surfaces. (a) Schematics of the Bogoliubov Fermi surfaces (yellow region), which may appearnear the point and line nodes (red points and lines) when time reversal symmetry is broken. Adopted from Ref. 172. (b)-(e) Schematic Fermi surfaces ofFeSe-based superconductors in the normal (red) and superconducting states (green and blue patches), for di ff erent interband and intraband gap anisotropyparameters. (f) Corresponding temperature dependence of specific heat divided by temperature C / T . (g) Corresponding tunneling conductance spectra at lowenergies. Panels (b)-(g) are adopted from Ref. 170.
7. Time Reversal Symmetry Breaking ff ect of nematic twin boundary Time reversal is simply equivalent to complex conjugationof the wave functions for a spinless system. Thus in spin sin-
Fig. 24.
Possible complex order parameter induced near nematic twinboundaries. (a) Schematic view of the crystal structure of FeSe near atwin boundary. Adopted from Ref. 103. (b) Possible position dependence of d -wave component near a twin boundary with finite imaginary part. The realpart should change sign across the boundary, while the magnitude of the gapremains finite owing to the induced imaginary part. (c) Schematic trajectoryof the gap function as a function of position in the complex plane. glet superconductors, a TRS breaking state can be describedby a complex order parameter ∆ = ∆ + i ∆ , whose timereversal ∆ ∗ = ∆ − i ∆ is not identical to ∆ . In a tetrago-nal D h system, the s -wave and d -wave even-parity pairingstates belong to di ff erent irreducible representations, and thusthese states in general have di ff erent transition temperatures.Usually one of the transition temperatures wins over the otherone, but when the pairing interactions that drive these di ff er-ent states are comparable, these two states may mix in a formof s + i d state that onsets at a temperature lower than the actual T c .In the nematic phase with orthorhombic symmetry, the s -wave and d -wave states no longer belong to di ff erent ir-reducible representations, and they can mix in a real form s + d . This can be easily understood by the fact that thenematicity is characterized by the two-fold symmetry in theplane, and thus the superconducting order parameter shouldalso be two-fold symmetric, as evidence by the observationof the elongated ellipsoidal shape of vortices, whichcan be described by the sum of fourfold s -wave and twofold d -wave components.In the nematic phase, another important aspect is the for-mation of domains with di ff erent nematic directions. Across aboundary of the two domains, namely nematic twin boundary,the crystal structure is rotated by 45 degrees as schematicallyshown in Fig. 24(a). Thus the twofold d -wave component ofthe superconducting order parameter must change sign acrossa twin boundary, i.e. one domain has an s + d state and theother domain has an s − d state. Then the question is how toreverse the sign near the twin boundary. One possibility is tochange sign with keeping the order parameter real, and themagnitude of d -wave component shrinks when approachingthe boundary and becomes zero at the twin boundary. Another
21. Phys. Soc. Jpn.
FULL PAPERSFig. 25.
Evolution of STM conductance spectra across nematic twin boundaries in FeSe. (a) STM topographic image in an area containing two vertical twinboundaries separated by ∼
33 nm. (b) Intensity plot for the position dependence of conductance spectra along the yellow dashed line in (a). (c) Tunnelingspectra at 4 positions indicated in (a). Each curve is shifted vertically for clarity. (d) An expanded view of spectra at low energies. The solid lines are the fitsto power-law energy dependence | E | α . (e) Position dependence of the exponent α obtained by the power-law fitting (blue circles), compared with the result forsingle twin boundary case. Adopted from Ref. 103. possibility is to have an imaginary component to avoid van-ishing order parameter as shown in Fig. 24(b); as a functionof position across the boundary, the order parameter followsan arch trajectory in the complex plane from s − d to s + d through s + i d state (see Fig. 24(c)). Such a problem has beenfirst considered by Sigrist et al. , who developed a theory for a d + s order parameter in the orthorhombic YBa Cu O − δ high- T c superconductor with twin boundaries. They found thatthe imaginary component may appear near twin boundaries ina length scale much longer than the coherence length ξ . One of the consequences of the presence of imaginarycomponent in the superconducting order parameter ∆ ( k ) =∆ ( k ) + i ∆ ( k ) is that the low-energy quasiparticle excitationsgiven by E k = (cid:113) ( ε k − µ ) + | ∆ ( k ) | = (cid:113) ( ε k − µ ) + ∆ ( k ) + ∆ ( k ) (6)are strongly modified. In general, the momentum dependenceof imaginary part ∆ ( k ) is di ff erent from that of real part,so the low-lying excitations that are determined by nodes in ∆ ( k ) are expected to be gapped out.The detailed position-dependent STM / STS studies in FeSeclean crystals have shown that the nematic twin boundariescan be considered as ideal interfaces with no noticeable struc-tural distortion in an atomic scale.
The low-energy con-ductance spectra at positions far away from the boundarieshave a V shape, indicating the presence of low-lying quasi-particle excitations. In contrast, the STS conductance curves near a twin boundary show a flattening behavior at low ener-gies, without exhibiting a zero-energy conductance peak thatmay be expected when the order parameters at neighboringdomains change sign without having the imaginary part. Asshown in Fig. 25, this flattening behavior is more pronouncedat the positions in between two twin boundaries, and the thepresence of the finite excitation gap is clearly resolved. Theseobservations of the twin-boundary-induced gap opening areconsistent with the complex order parameter near the twinboundaries, and the essential features of conductance spec-tra have been reproduced by theoretical calculations assumingthe presence of imaginary component near boundaries.
The low-energy flattening behaviors are found over an ex-tended length scale of (cid:38)
50 nm (see Fig. 25), an order ofmagnitude longer than the averaged in-plane coherence length ξ ab ≈ ξ can be much longer than the coherence length anddiverges when approaching the phase boundary between thetime-reversal symmetric s + d state and the TRS broken s + i d state in the bulk. This phase boundary is determined by thecloseness of transition temperatures of s and d states and bythe amount of orthorhombicity. In YBa Cu O − δ , the super-conducting order parameter is dominated by the d -wave com-ponent and the orthorhombicity-induced s -wave componentis much smaller. Thus the onset temperature of TRS-breakingstate may be much lower than the actual transition temper-ature, and there have been no report showing clear evidencefor such a TRS-breaking state near twin boundaries. This mayalso be related to the fact that the STM / STS measurements
22. Phys. Soc. Jpn.
FULL PAPERSFig. 26.
ARPES measurements of the superconducting gap for the holeband in single-domain and multiple-domain samples of FeSe. (a) Momen-tum dependence of the gap along the hole Fermi surface ellipsoid in single-domain (green triangles) and multiple-domain (red circles) samples. Solidline is a fit for ∆ ( k ) in multi-domain sample. (b) Possible positions of nodesfor a spin-triplet p -wave state. (c) Possible positions of nodes for a spin-singlet s + d state. Adopted from Ref. 99 in YBa Cu O − δ single crystals are quite challenging due tothe di ffi culties of cleavage. In FeSe, in contrast, the supercon-ducting order parameter has comparable s and d componentsas discussed in §
3, which may lead to the observation of sucha state.In a laser-ARPES study, the angle dependence of supercon-ducting gap in the hole band near the Γ point has been com-pared between almost single-domain and multiple-domainsamples. In the former, the anisotropic gap reaches almostzero along the long axis of the ellipsoidal underlying Fermisurface, but in the latter the nodes are lifted and the gap min-ima are found as shown in Fig. 26. This di ff erence in gapstructure between samples with di ff erent domain structureshas been interpreted as another piece of experimental evi-dence that the finite gap opens near twin boundary. This ob-servation is quite consistent with the STS results, althoughanother laser-ARPES study from a di ff erent group does notconfirm such a node lifting behavior in samples with multipledomains. The results in multiple-domain samples may de-pends on the density of twin boundary, and further studies arerequired to fully understand how ∆ ( k ) evolves as a functionof position near the boundaries. µ SR)
The studies of gap structure can provide only indirect infor-mation on the TRS breaking in the superconducting state. Amore direct consequence of the TRS breaking is that a finitemagnetic field is induced inside the superconducting sample.
Fig. 27.
Phase diagram of TRS-breaking (TRSB) state and correspondingphase di ff erence between d and s components. (a) Temperature versus or-thorhombic distortion ( ε ) phase diagram of TRSB states calculated for a d + s state in which s -wave transition temperature is assumed as 0 . T c . (b) Phasedi ff erence θ between d and s components in a d + e i θ s state as a function ofposition near a twin boundary. Three curves correspond to the three phasesin (a). In the TRSB state in bulk, phase (2) in (a), the characteristic lengthscale ˜ ξ diverges and a finite phase di ff erence is present deep in the bulk butthe dependence near the boundary is similar to that in phase (1)-(II). Adoptedfrom Ref. 174. This is related to the fact that the s + i d and s − i d statesare energetically degenerate, forming chiral domains, whichis analogous to the magnetic domain formation at zero field inferromagnets. Near the boundaries and the impurities, smallbut finite magnetic field is induced, which can be detected byexperimental probes. The µ SR at zero external field is oneof the very sensitive magnetic probes that have been used asdirect probes of TRS breaking states in unconventional su-perconductors. Very recent zero-field µ SR measurements invapor-grown single crystals of FeSe have shown that whilethe muon relaxation rate is almost independent of tempera-ture above T c , which is consistent with the absence of mag-netic order in FeSe, it starts to develop just below T c ≈ ∼ These results also provide strong evidence for a TRS break-ing state in FeSe. The onset temperature of the magnetic in-duction is very close to T c , which may be explained by thecomparable magnitudes of s and d -wave components of theorder parameter. The magnitude of the magnetic induction atlow temperatures is estimated as small as ∼ .
15 G. Whetherthis TRS breaking state occurs only in the vicinity of nematictwin boundaries or not cannot be concluded by this µ SR resultin FeSe alone. This motivates similar experiments in tetrago-nal FeSe − x S x , which we discuss below.
23. Phys. Soc. Jpn.
FULL PAPERS
In the tetragonal phase of FeSe − x S x ( x > . µ SR measure-ments in tetragonal FeSe − x S x can test TRS breaking insidethe bulk. Most recent data of zero-field µ SR for x (cid:38) .
20 showsimilar enhancements of relaxation rate just below T c ≈ This immediately impliesthat tetragonal FeSe − x S x has a superconducting state withbroken TRS in the bulk. In addition, the observations of fi-nite induced magnetic fields with similar magnitudes in or-thorhombic and tetragonal samples suggest that in orthorhom-bic FeSe TRS breaking occurs not only near the nematictwin boundaries but also deep in the single domains. Indeed,the phase diagram in Fig. 27(a) studied by Sigrist et al. fororthorhombic superconductors indicates that the bulk TRSbreaking state exists in a wide range of parameters at tem-peratures below the state of broken TRS only near the twinboundaries. We note that the di ff erence between these twoTRS breaking states in the orthorhombic phase is character-ized by the presence or absence of small phase di ff erence be-tween s and d components away from the boundaries, and inboth cases the position dependence of the phase di ff erenceis similarly significant near the twin boundaries as shown inFig. 27(b). The presence of small phase di ff erence deep inthe single domain implies that the tiny gap opening may bepresent in low-energy quasiparticle excitations. The observa-tion of such a tiny gap requires measurement techniques hav-ing very high energy resolutions at very low temperatures.The TRS breaking in FeSe − x S x superconductors fulfillsone of the strong requirements to realize a novel ultran-odal superconducting state with Bogoliubov Fermi surfaceintroduced in the previous section. The µ SR experimentsunder magnetic fields provide further support for this. Thetransverse-field µ SR measurements in type-II superconduc-tors can provide quantitative information on the magneticpenetration depth that characterizes the field distributionsaround superconducting vortices. The magnitude of penetra-tion depth λ (0) is directly related to the density of supercon-ducting electrons, which participate in the supercurrent flowsthat screen the magnetic field. It is found that the magnitudeof λ (0) is larger in tetragonal FeSe − x S x than in orthorhombicFeSe. This shows an opposite trend to that expected from theincrease of Fermi surface with S substitution found in quan-tum oscillations. This immediately indicates that the den-sity of normal electrons is larger in tetragonal side, whereasthe density of superconducting electrons is smaller. This sur-prising result can be consistently explained by the presenceof Bogoliubov Fermi surface in the superconducting groundstate for tetragonal side of FeSe − x S x , which reduces the den-sity of superconducting electrons from the normal one. Thisis also consistent with the large residual density of states ob-served in the STS and thermodynamic measurements. It has been widely established that in unconventional su-perconductors with line nodes, the application of magneticfield gives rise to a rapid increase of low-energy density ofstate as discussed in § ff ect. The reason why such a state can appear only in the tetrago-nal side of FeSe − x S x deserves further theoretical and experi-mental investigations. It is also intriguing to study how this isrelated to BEC-like superconducting state found in tetragonalFeSe − x S x as discussed in §
8. Topological Superconducting State
This section reviews a rather di ff erent aspect of the FeSe-family compound, namely the topological nature. Topologi-cal quantum physics is one of the most active areas in recentcondensed-matter research, which was ignited by the theoreti-cal prediction of quantum spin-Hall insulator followedby the experimental verification. Quantum spin-Hall insu-lator, or 2D topological insulator (TI), is a 2D insulator witha band gap across which the order of the energy bands areinverted from that expected from the energies of the corre-sponding atomic orbitals. Such a situation can be caused bythe spin-orbit interaction. Unlike ordinary insulators, quan-tum spin-Hall insulators always host spin-polarized helicaledge states with a linear energy-momentum dispersion, whichcan be described by a massless Dirac equation. Such edgestates are not related to chemical and structural properties ofthe edge but are robust and nontrivial in the sense that theyare associated with the topological nature of the Bloch wave-functions due to the band inversion.The topological nature can be classified by the topologicalinvariant Z = { , } . The Z invariants in ordinary andquantum spin-Hall insulators are 0 and 1, respectively. Theconcept of the quantum spin-Hall insulator can be extendedin three dimensions, where the topological nature is classifiedby a set of four Z invariants. There are two types of3D TIs, namely the weak TI and the strong TI. The strong TIis an insulator in the bulk but possesses spin-polarized Diracsurface states all over the surfaces, irrespective of the chemi-cal and structural properties of the surface, corresponding tothe quantum spin-Hall insulator with the Dirac edge states. Inweak TIs, such a Dirac surface state appears only on particu-lar surfaces.Experimental realization of the quantum spin-Hall insula-tor was achieved in a HgTe quantum well in 2007, and Bi-Sb alloy was identified as a 3D strong TI in 2008.
This wasjust around the same time when iron-based superconductivitywas discovered. Since then, iron-based superconductivity andtopological quantum physics have been investigated activelybut in parallel, and thus there have been little interaction be-tween them. Nevertheless, in principle, if the concept of topol-ogy is applied to superconductors, unique and useful phe-nomena may emerge. Here the prerequisite is that the super-conducting state should be topologically nontrivial. There isgrowing evidence that Te substitution for Se in FeSe gives riseto topological superconductivity at the surface. In the follow-ing we describe the topological phenomena in FeSe − x Te x , inparticular paying attention to the Majorana quasiparticle in avortex core. We start by briefly introducing topological superconduc-tivity. For details, text books and a review article are available. A topological superconductor can be viewed
24. Phys. Soc. Jpn.
FULL PAPERS as a superconducting counterpart of TI, where the Cooper-pair wavefunction possesses topologically nontrivial nature.Here, superconducting gap corresponds to the band gap in TI.As similar to the case of TIs, gapless boundary states withlinear quasiparticle dispersion appears in topological super-conductors. Such quasiparticles can take exactly zero energy,whereas all the quasiparticle states in an ordinary supercon-ductor appear at finite energies. Irrespective of the topolog-ical nature, quasiparticle states in the superconducting stateare coherent superpositions of electron and hole states. Thezero-energy state in the topological superconductor is uniquebecause electron and hole weights are exactly equal. Such ahalf-electron half-hole state can be regarded as an quasiparti-cle that is its own antiparticle, which is known as the Majo-rana quasiparticle. Majorana quasiparticles can be used as afundamental building block for future fault-tolerant quantumcomputing, and thus are attracting much attention.
Unfortunately, topological superconductivity and thereforeMajorana quasiparticles have been elusive. The early propos-als mostly focused on chiral p -wave superconductors, butsuch superconductors have yet to be established experimen-tally. To overcome this situation, there have been proposedseveral ways to realize e ff ective chiral p -wave superconduc-tivity in artificial systems. These include 1D Rashba semi-conductor nanowires and magnetic-atom chains inwhich superconductivity is induced via the proximity e ff ectfrom an ordinary s -wave superconductor attached.In 2008, Fu and Kane proposed a novel way to realize2D topological superconductivity using a heterostructure thatconsists of ordinary s -wave superconductor and 3D strongTI. If superconductivity is induced in the spin-polarizedDirac surface state of TI via the proximity e ff ect, it can bee ff ectively regarded as a chiral p -wave superconducting stateowing to the spin polarization of the Dirac surface state. Magnetic field applied perpendicular to the interface gener-ates quantized superconducting vortices, in which Majoranaquasiparticles would be localized to form Majorana boundstate (MBS).
Since the MBS has exactly zero energy, itmay show up as a zero-bias peak (ZBP) in the LDOS spectrameasured by STM / STS, in principle.However, there are two immediate issues to be addressedto implement actual experiments. First, in the Fu-Kane pro-posal, topological superconductivity emerges at the interface,which is buried beneath either TI or s -wave superconductorfilms. Therefore, surface sensitive probes such as STM / STScannot directly access to the MBS, even if it exists. Second,even though the topologically trivial vortex bound states areexpected to appear only at finite energies, their lowest energy ∼ ∆ /ε F is generally very small ∼ µ eV (see § / STS were madeon heterostructures where Bi Te (TI) thin films were epitax-ially grown on the NbSe ( s -wave superconductor) substrateby molecular-beam epitaxy. The superconducting-gap sizeobserved at the Bi Te surface decayed exponentially with in-creasing film thickness, being consistent with the proximity-induced superconductivity. This suggests that the observedLDOS spectra include information at the interface although it is indirect. The above mentioned energy-resolution issue re-mained but other characters, such as spatial dependence of theLDOS spectrum and the spin polarization, were investigatedto argue the features that may signify the MBS. Obviously, it is desirable to investigate a system where theMBS is exposed at the surface and is energetically well sep-arated from other trivial vortex bound states. To have topo-logical superconductivity at the surface, one can think of asuperconductor that has TI-like character in its normal-statebulk band structure. This is possible if the superconductor isa certain semimetal. As in the case of insulators, semimetalspossess a band gap although it meanders in the Brillouin zoneand turns out to place the highest energy of the valence bandabove the lowest energy of the conduction band. If such a me-andering band gap is topologically nontrivial, spin-polarizedDirac surface state should emerge and can cross ε F . Further,if the bulk of such a semimetal is an s -wave superconduc-tor, the spin-polarized Dirac surface state may turn out to ex-hibit topological superconductivity due to the self-proximitye ff ect from the bulk. This is the natural realization of the Fu-Kane proposal at the exposed surface, providing a platformto investigate the MBS by surface sensitive probes. Thereare several candidate materials for such connate topologicalsuperconductors , such as β -PdBi and PbTaSe . FeSe − x Te x can also be categorized as a connate topologicalsuperconductor. − x Te x Before discussing topological nature, we briefly summarizebasic properties of FeSe − x Te x . As in the case of S substi-tution, Te substitution keeps the compensation condition be-cause Te is isovalent to Se. Sample preparation in the lowTe-concentration regime has been challenging because of thepossible miscibility gap in 0 . (cid:46) x (cid:46) . Very few at-tempts have been reported in this regime using pulsed laserdeposition and flux growth.
Single crystals with higher x can be obtained by melt-growth technique but they tend tocontain excess iron atoms at the interstitial sites, which af-fect various properties. Subsequent annealing processis generally required to remove the excess irons.
In short,high quality single crystals are more di ffi cult to prepare inFeSe − x Te x than in FeSe − x S x and the samples so far avail-able inevitably contain certain amount of disorders. As in the case of of S substitution, nematicity tends to besuppressed upon Te substitution and diminishes above x ∼ . T c is rather insensitive to x between 0 . (cid:46) x (cid:46) . − x S x with highly anisotropic superconductinggaps, rather isotropic superconducting gap ∆ ∼ . − x Te x . A double-stripe-type long-range magnetic order appears at x (cid:38) . Besides topological nature discussed in the next subsec-tion, FeSe − x Te x is advantageous for the MBS search becauseit is in a BCS-BEC crossover regime (see § ε hF ∼ ε eF ∼
10 meV.
Consid-ering ∆ ∼ . we can estimate the lowest trivialbound-state energy ∆ / ε F to be as large as 100 µ eV. This isstill small but enough to distinguish the MBS from the trivialstates, if the highest energy resolution STM / STS technologyis employed. Therefore, except for the issue associated with
25. Phys. Soc. Jpn.
FULL PAPERSFig. 28. (a) and (b) Calculated band structures without the spin-orbit inter-action for FeSe and for FeSe . Te . , respectively. The size of the red circledenotes the weight of the chalcogen p z orbitals. (c) and (d) Calculated bandstructure for FeSe . Te . with the spin-orbit interaction. A gap opens at oneof the band crossing points along the Γ − Z direction, giving rise to the me-andering band gap indicated by the red dashed line. Adopted from Ref. 208. the disorders, FeSe − x Te x provides an excellent platform tosearch for the MBS. − x Te x Early experimental signature that suggested topological na-ture of FeSe − x Te x was the ZBP found in the tunneling spectraat the interstitial excess irons. The ZBP was robust in thesense that it did not split nor shift to finite energies, even if theSTM tip was moved away from the excess iron site and evenif magnetic fields were applied.
Such an apparent robust-ness triggered theoretical analyses of the topological natureof FeSe − x Te x , including a proposal of quantum anomalousvortex generated by the magnetic excess iron atom and thespin-orbit interaction. First principles band-structure calculations for FeSe andFeSe . Te . revealed that topological nature indeed emergesupon Te substitution. As discussed in §
2, band structureof FeSe is complicated but trivial from band topology pointof view. Te substitution alters this original band structurethrough the following two e ff ects. First, Te 5 p orbital is moreextended than Se 4 p orbital. This brings about stronger cou-pling between chalcogen p z orbitals and thus larger band dis-persion along the Γ − Z direction for the associated band(Fig. 28(b)). This gives rise to additional band crossings alongthe Γ − Z , resulting in the band inversion at the Z point. Sec-ond, because Te is heavier than Se, stronger spin-orbit inter-action is expected. Indeed, the spin-orbit interaction opens agap at one of the additional band crossing points along the Γ − Z and brings about the meandering band gap that is topo-logically nontrivial (Figs. 28(c) and (d)). The topologicallynontrivial nature has been pointed out from di ff erent pointsof view in FeSe − x Te x and also discussed in other iron-based superconductors. The above band structure results in the spin-polarized Diracsurface states, which are a hallmark of the topological na-ture and provide an important platform to host topologicalsuperconductivity. Experimental observation of such surface (b)(a) (c) (d)(e)
Fig. 29. (a) ARPES intensity map showing the Dirac dispersion at the(001) surface of FeSe . Te . . (b) and (c) Spin-resolved energy distributioncurves and their di ff erence, respectively, taken at one side of the Dirac cone.Spin polarizations are illustrated in the inset of (c). (d) and (e) Same as (b)and (c) but taken at the other side of the cone. Spin polarizations are reversedas being consistent with the helical spin structure. Adopted from Ref. 212. (c) d I / d V ( a . u . ) -4 -2 0 2 4Energy (meV) (b) -4 -2 0 2 4Energy (meV)1050-5-10 D i s t a n c e ( n m ) (a) ab LowHighLowHigh Fig. 30. (a) An image of the vortex obtained by mapping the zero-bias tun-neling conductance. Magnetic field of 0.5 T was applied perpendicular to theobserved (001) surface. (b) A line profile of the tunneling conductance takenalong the black dashed arrow in (a), showing a non-splitting ZBP. (c) A wa-terfall plot of the same data shown in (b). The spectrum taken at the vortexcenter is shown in black. Adopted from Ref. 213. states was challenging because they are in the close vicinityof ε F and a bulk band, demanding high energy resolution forARPES along with the spin resolution. Later on, ultra-highresolution laser-based spin ARPES was utilized and the spin-polarized Dirac surface state was successfully observed at the(001) surface of a FeSe . Te . single crystal (Fig. 29). Given the observation of the spin-polarized Dirac surfacestate in FeSe − x Te x , there is an enough hope that MBS wouldbe formed in the vortex cores. The first STM / STS experimentin this context succeeded in detecting the ZBP in the vor-tex cores of FeSe . Te . (Fig. 30). Unlike trivial vortex
26. Phys. Soc. Jpn.
FULL PAPERS B = 1 T g ( n S ) d I / d V ( n S ) -1.0 -0.5 0.0 0.5 1.0 Bias voltage (mV) d I / d V ( n S ) -1.0 -0.5 0.0 0.5 1.0 Bias voltage (mV) (a) (b) (c)
Fig. 31. (a) Zero-energy conductance map g showing of the disordered vortex lattice in FeSe . Te . under magnetic field 1 T applied perpendicular to theobserved (001) surface. (b) A high-energy resolution tunneling spectrum taken at the center of the vortex labeled as 1 in (a). A ZBP is observed as indicatedby the red arrow. (c) A high-energy resolution tunneling spectrum taken at the center of the vortex labeled as 2 in (a). No ZBP is observed. Adopted fromRef. 204. Fig. 32. (a)-(e) Series of zero-energy conductance g map showing the vortex lattice in FeSe . Te . under di ff erent magnetic fields. (a) 1 T, (b) 2 T, (c) 3 T,(d) 4 T, (e) 6T. (f)-(j) Fourier-transformed images from (a)-(e), respectively. Ring-like features mean that there is a distance correlation while the long-rangeorientation order is lost. (k)-(o) The respective histograms of the appearance frequency of the conductance peaks at given energies. The probability to find theZBP decreases with increasing magnetic field. Adopted from Ref. 204. bound states, the observed ZBP stays at zero energy over cer-tain distance from the vortex center and its intensity evolutionagrees with that expected from the theoretical spatial profileof the MBS. Similar ZBP has also been observed in otheriron-based superconductors such as (Li . Fe . )OHFeSe and CaKFe As . The early experiments were done at about 0.5 K with anenergy resolution of ∼ µ eV, which is somewhatlarger than the estimated energy of the lowest trivial boundstates in the vortex core ( ∼ µ eV). Therefore, there still re-mained an ambiguity whether the observed ZBP indeed rep-resents the MBS or it is a bundle of trivial vortex bound statesthat are thermally broadened to form an apparent peak at zeroenergy. In addition, there is a puzzle that the ZBP has been ob-served only in a fraction of vortices and the rest of vortices donot host the ZBP. The problem is that FeSe − x Te x sam-ples inevitably contain various chemical and electronic disor- ders as mentioned above. It is important to clarify what kindof disorder governs the ZBP.Subsequent STM / STS experiments addressed these is-sues.
A dilution-fridge-based STM was employed toreach ultra-low temperatures below 90 mK. As a result, theenergy resolution as high as ∼ µ eV was achieved. This isenough to distinguish the ZBP from the finite-energy trivialbound states and gave a strong constraint that the origin ofthe ZBP is the MBS (Fig. 31). The correlations betweenvortices with and without the ZBP and various quenched dis-orders were also investigated systematically. Interestingly,any chemical and electronic disorders preexisting in the sam-ple do not a ff ect the presence or absence of the ZBP. Mean-while, it was found that the fraction of vortices with the ZBPdecreases with increasing applied magnetic field, namely in-creasing vortex density (Fig. 32). This suggests that in-teractions among the MBSs in di ff erent vortices may be re-
27. Phys. Soc. Jpn.
FULL PAPERS sponsible for the diminishing ZBP at higher fields. Moreover,since there are two kinds of vortices with and without theZBP, and the quenched disorders do not play any roles forthis distinction, one can infer that the disorder in the vortex-lattice structure may a ff ect the ZBP. Large scale theoreticalsimulations have been performed to confirm this idea. Theemployed model includes the Majorana-Majorana interactionand disorder in the vortex lattice. The results reproduced thebasic features of the experimental observations.
Very re-cently, alternative theoretical model based on the spatially-inhomogeneous Zeeman e ff ect has been proposed. Strictly speaking, the ZBP is nothing more than one of thenecessary conditions for the MBS. Further challenges to de-tect the features that are unique to the MBS have been done.The detailed energy spectrum in the vortex core should pro-vide an important clue. As described in §
4, quantized energiesof the low-lying vortex bound states are given by ± µ c ∆ /ε F ,where µ c are half-odd integers. This is actually the case forthe vortices in topologically trivial superconductors. In thecase of the topological vortex with the MBS, the quantizedsequence becomes ± µ t ∆ /ε F , where µ t are integers. Thereis a 1 / µ t = − x Te x with and without theZBP. Half-odd integer and integer level sequences are re-ported in the former and the latter vortices, respectively, sug-gesting that FeSe − x Te x hosts both topologically trivial andnon-trivial vortices depending on the location. Apparently,this is incompatible to the observation that the quenched dis-orders have nothing to do with the ZBP.
The LDOS spec-trum and its spatial evolution are di ff erent from vortex to vor-tex in FeSe − x Te x , making it di ffi cult to reachclear conclusions. Experiments on more homogeneous sam-ples are highly desired.Another signature that is expected to be unique for theMBS is the quantization of the tunneling conductance. It hasbeen theoretically predicted that if the Majorana quasipar-ticles are involved in the tunneling process, induced reso-nant Andreev reflections may quantize the tunneling conduc-tance to be 2 e / h . This has been experimentally tested us-ing the Majorana state formed at the superconducting InSbnanowire covered with a superconductor (Al) shell.
In thecase of the MBS in the vortices, STM / STS is a powerful toolbut the challenge is that extra-ordinary high tunneling con-ductance must be achieved to see the expected quantization.Such experiments have been performed in FeSe − x Te x andrelated compound (Li . Fe . )OHFeSe. Plateau-like be-haviors in the tunneling conductance have indeed been ob-served. However, the quantization behavior is not so clear yet.Moreover, multiple tunneling paths can exist because of thecontributions from the multiple bulk bands. This should a ff ectthe quantization condition. Further experimental and theoret-ical e ff orts are anticipated.The vortex core at the surface of FeSe − x Te x can be re-garded as a zero-dimensional boundary in the 2D topologi-cal superconductor. Extended 1D boundary, namely the edge,may also host Majorana quasiparticles that can move alongthe edge. STM / STS has been utilized to detect such dispersingMajorana quasiparticles in FeSe − x Te x . The platform was a novel naturally-formed domain boundary across which thecrystal lattice exhibits a half-unit-cell shift in its structure.
The LDOS spectrum observed at the domain boundary is con-stant as a function of energy, just like an LDOS spectrum ofnormal metals. It has been argued that this behavior is con-sistent with the Majorana quasiparticles moving along the 1Dchannel because of the linear energy-momentum dispersion.The actual edge of the superconducting nano-island may pro-vide more direct opportunity to investigate the details of thedispersing Majorana quasiparticles, and the development ofsample fabrication technique is awaited.
9. Summary
In this review, we have discussed a wide variety of exoticsuperconducting states observed in bulk FeSe-based super-conductors. What makes this system unique from other su-perconductors lies in its peculiar electronic structure, in par-ticular the extremely small Fermi energy, multiband natureand orbital-dependent electron correlations. Because of theseproperties, spin and orbital degrees of freedom, i.e. mag-netism and nematicity, both of which are intimately relatedto the electron pairing, can be largely tuned by non-thermalparameters, such as pressure, chemical substitution and mag-netic field. Therefore, it is natural to consider that many dif-ferent pairing states emerge as a result of this large tunability.While more experimental and theoretical works are clearlyneeded to arrive at a more quantitative description of thedata, we feel confident that FeSe-based materials serve as anovel platform of many kinds of exotic pairing states, someof which have never been realized in any other superconduc-tors so far. However, we believe that the following questionsregarding the superconducting states in FeSe-based materialsremain to be answered. • The electron Fermi surface still remains for thoroughand comprehensive investigations. The precise shape ofthe Fermi surface and the detailed superconducting-gapproperties are yet to be determined. • The most fundamental question is whether the prevailing s ± pairing state with the sign reversal between electronand hole Fermi surfaces is realized even in FeSe − x S x near the nematic QCP, where no sizable spin fluctuationsare observed and nematic fluctuations are strongly en-hanced. • Closely related to the above issue, it is an open ques-tion why the superconducting gap function dramati-cally changes at the nematic QCP. In addition, it isalso intriguing to clarify whether the highly unusual su-perconducting gap function in the tetragonal phase ofFeSe − x S x at x ≥ .
17 is related to the Bogoliubov Fermisurface. • It appears that the BCS-BEC crossover properties arelargely modified by the multiband and orbital dependentnature. However, we still lack a quantitative descriptionwhy the pseudogap is hardly observed despite of giantsuperconducting fluctuations. • The field-induced superconducting phase can be at-tributed to the FFLO state. However, it is not a conven-tional FFLO state because of a large spin polarizationand orbital dependent pairing.
28. Phys. Soc. Jpn.
FULL PAPERS • To confirm the time reversal symmetry breaking, moredirect measurements, such as the observation of chiraldomains in the superconducting state, are desired. • The correspondence between the zero-energy conduc-tance peak in the vortex core of FeSe − x Te x and the Ma-jorana bound state should be examined further to findthe features that represent ”Majorananess”. The obviousgoal is to manipulate the Majorana state. More homoge-neous samples are indispensable for this purpose.Evidence for exotic superconducting states of FeSe-basedmaterials have continued to motivate researchers to furtherinvestigate and develop novel type superconducting states,which are at the forefront of modern research. We hope thatthis overview presented here is helpful. Acknowledgments
The authors acknowledge the collaboration of S. Arseni-jevi´c, A. E. B¨ohmer, J.-G. Chen, A. I. Coldea, I. Eremin,N. Fujiwara, T. Fukuda, A. Furusaki, Y. Gallais, S. K. Goh,T. Hashimoto, S. Hosoi, N. E. Hussey, K. Ishida, K. Ishizaka,K. Iwaya, Y. Kasahara, W. Knafo, Y. Kohsaka, M. Kon-czykowski, S. Licciardello, H. v. L¨ohneysen, T. Machida,K. Matsuura, C. Meingast, A. H. Nevidomskyy, K. Okazaki,T. Sasagawa, Y. Sato, T. Shimojima, S. Shin, M. Sigrist,T. Tamegai, T. Terashima, Y. Tsutsumi, Y. J. Uemura, Y. Uwa-toko, T. Watanabe, T. Watanuki, T. Watashige, M. D. Wat-son, T. Wolf, and J. Wosnitza. We especially thank the long-term collaboration with S. Kasahara and Y. Mizukami. We aregrateful to D. Agterberg, A. V. Chubkov, P. Dai, R. M. Fer-nandes, P. J. Hirschfeld, R. Ikeda, J. S. Kim, S. A. Kivelson,H. Kontani, E.-G. Moon, S. Onari, R. Prozorov, J. Schmalian,J. Wang, Y. Yamakawa, and Y. Yanase for valuable discus-sions. This work was supported by Grants-in-Aid for Scien-tific Research (KAKENHI) (Nos. JP19H00649, JP18H05227,JP15H02106, JP16H04024, JP25220710, JP25610096, andJP24244057) and Innovative Areas “Quantum LiquidCrystals” (No. JP19H05824), “Topological Material Sci-ence” (No. JP15H05852), and “3D Active-Site Science”(No. 26105004) from Japan Society for the Promotion ofScience (JPSJ), and JST CREST (Nos. JPMJCR18T2, JP-MJCR16F2).
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