Controlling unconventional superconductivity in artificially engineered f-electron Kondo superlattices
CControlling unconventional superconductivity in artificially engineered f -electronKondo superlattices M. Naritsuka ∗ , T. Terashima, and Y. Matsuda Department of Physics, Kyoto University, Kyoto 606-8502 Japan
Unconventional superconductivity and magnetism are intertwined on a microscopic level in a wideclass of materials, including high- T c cuprates, iron pnictides, and heavy-fermion compounds. A newapproach to this most fundamental and hotly debated subject focuses on the role of interactionsbetween superconducting electrons and bosonic fluctuations at the interface between adjacent layersin heterostructures. A recent state-of-the-art molecular-beam-epitaxy technique has enabled usto fabricate superlattices consisting of different heavy-fermion compounds with atomic thickness.These Kondo superlattices provide a unique opportunity to study the mutual interaction betweenunconventional superconductivity and magnetic order through the atomic interface. Here, we designand fabricate hybrid Kondo superlattices consisting of alternating layers of superconducting CeCoIn with d -wave pairing symmetry and nonmagnetic metal YbCoIn or antiferromagnetic heavy fermionmetals, such as CeRhIn and CeIn . In these Kondo superlattices, superconducting heavy electronsare confined within the two-dimensional CeCoIn block layers and interact with the neighboringnonmagnetic or magnetic layers through the interface. In CeCoIn /YbCoIn superlattices, thesuperconductivity is strongly influenced by the local inversion symmetry breaking at the interface.In CeCoIn /CeRhIn and CeCoIn /CeIn superlattices, the superconducting and antiferromagneticstates coexist in spatially separated layers, but their mutual coupling via the interface significantlymodifies the superconducting and magnetic properties. The fabrication of a wide variety of hybridsuperlattices paves a new way to study the relationship between unconventional superconductivityand magnetism in strongly correlated materials. I. INTRODUCTION
Superconductivity found in several classes of stronglycorrelated electron systems, including cuprates[1, 2],iron pnictides/chalcogenides[3–5] and heavy fermioncompounds[6, 7] has attracted researchers over the pastthree decades. There is almost complete consensus thatsuperconductivity in these systems cannot be explainedby the conventional electron-phonon attractive pairinginteractions [8, 9]. As the superconductivity occurs in thevicinity of the magnetic order, it is widely believed thatmagnetic fluctuations, which arises from purely repul-sive Coulomb interactions, act as the source of electronpairing. Moreover, the highest superconducting transi-tion temperature T c is often found near a quantum crit-ical point (QCP), at which a magnetic phase vanishes inthe limit of zero temperature, indicating that prolifera-tion of critical magnetic excitations resulting from theQCP plays a significant role in determining supercon-ducting properties [10–16]. In these materials, a micro-scopic coexistence of superconducting and magneticallyordered phases both involving the same charge carriers isa striking example of unusual emergent electronic phases.Despite tremendous research, however, the relationshipbetween superconductivity and magnetism has remainedlargely elusive.The strongest electron correlation is realized in heavy-fermion compounds, containing f electrons (4 f for lan- ∗ Present address: School of Physics and Astronomy, University ofSt Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK. thanide and 5 f for actinide), especially in materials con-taining Ce, Pr, U and Pu atoms [17–23]. At high tem-perature, f electrons are essentially localized with well-defined magnetic moments. As the temperature is low-ered, the f electrons begin to delocalize due to the hy-bridization with conduction electron band ( s , p , d or-bital), and Kondo screening. At yet lower temperature,the f electrons become itinerant, forming a narrow con-duction band with heavy effective electron mass (up to afew hundred to a thousand times the free electron mass).Strong Coulomb repulsion within a narrow band and themagnetic interaction between remnant unscreened 4 f or5 f moments leads to notable many-body effects, and su-perconductivity mediated by magnetic fluctuations. Thehybridized f electrons are not only responsible for long-range magnetic order, but are also involved in supercon-ductivity. Therefore, the heavy-fermion compounds offera fascinating playground where magnetism and uncon-ventional superconductivity can both compete and coex-ist.Among heavy fermion compounds, Ce M In (where M = Co, and Rh) with layered structure are ideal modelsystems due to their rich electronic phase diagrams inwhich an intricate interplay between superconductivityand magnetism is observed [24–26]. At ambient pressure,CeCoIn is a superconductor ( T c = 2.3 K) with d x − y -wave symmetry [27–31]. The normal state displays non-Fermi-liquid properties associated with a nearby underly-ing QCP [32, 33]. In contrast, CeRhIn orders antiferro-magnetically at ambient pressure ( T N = 3.8 K) [34]. Itsmagnetic transition is suppressed by applying pressureand the ground state becomes a purely superconductingstate at P > P ≈ . a r X i v : . [ c ond - m a t . s up r- c on ] F e b of a pressure-induced QCP [11, 13, 35–39].It has been shown that interactions between super-conducting electrons and bosonic excitations throughan atomic interface may have a profound influence onCooper pairing. For example, when a monolayer FeSefilm grown on a SrTiO substrate, the coupling betweenthe FeSe electrons and SrTiO phonons enhances theCooper pairing, giving rise to the highest T c among allknown iron-based superconductor, which is almost an or-der of magnitude higher than that of the bulk FeSe [5, 40–43]. This raises the possibility of a magnetic analog inwhich the pairing interaction is influenced by magneticfluctuations though an interface between an unconven-tional superconductor and a magnetic metal. This con-cept is illustrated schematically in Figs. 1(a) and 1(b).Besides allowing a new approach to revealing the entan-gled relationship between magnetism and unconventionalsuperconductivity, this concept has the advantage thatmagnetic excitations are tunable as a magnetic transi-tion is driven toward zero temperature, unlike phononexcitations in SrTiO .In this review, we discuss the recent advances of Kondosuperlattices which consist of alternating layers of heavy-fermion superconductor and heavy-fermion antiferromag-net. We focus on mutual interactions between d -wave su-perconductivity and magnetic order through the atomicinterface, in particular paying attention to how the pair-ing interaction is influenced by magnetic fluctuations in-jected from the neighboring layers through the atomicinterface. For this purpose, we have designed and fab-ricated two types of superlattices formed by alternat-ing atomically thick layers of CeCoIn and (1) conven-tional non-mangetic metal YbCoIn and (2) antiferro-magnetic (AFM) heavy fermion metals, such as CeRhIn and CeIn . In these Kondo superlattices, supercon-ducting heavy electrons are confined within the two-dimensional (2D) CeCoIn block-layers (BLs) and inter-act with the neighboring nonmagnetic or magnetic lay-ers through the interface. In CeCoIn /YbCoIn super-lattices, local inversion symmetry breaking at the inter-face enables Rashba spin-orbit coupling to play a key rolein superconductivity [44–47]. In CeCoIn /CeRhIn andCeCoIn /CeIn Kondo superlattices, the superconduct-ing and AFM states coexist in spatially separated layers[48, 49]. The AFM ordering temperature of CeRhIn and CeIn BLs can be tuned to zero by applying hy-drostatic pressure, leading to a magnetic QCP. In thesesuperlattices, we show that the superconducting and non-
FIG. 1: (a) Schematic figure of the interaction between d -wave superconducting (SC) state and static AFM order viathe interface. (b) Interaction between two competing ordersunder pressure near a QCP, where AFM order disappears. superconducting magnetic layers can interact with eachother. In particular, in CeCoIn /CeRhIn superlattices,upon suppressing the AFM order, the force binding su-perconducting electron pairs acquires an extreme strongcoupling nature superlattices, demonstrating that super-conducting pairing can be tuned nontrivially by magneticfluctuations injected through the interface. II. KONDO SUPERLATTICEA. Kondo superlattice
Recently, the state-of-the-art molecular beam epitaxy(MBE) technique enables the realization of high qual-ity hetero-interface of heavy fermion systems throughthe fabrication of Ce-based compounds [44, 50, 51]. Su-perlattices consisting of alternating layers of supercon-ductor CeCoIn , nonmagnetic metals, such as YbCoIn and YbRhIn , and AFM heavy fermion metals, such asCeRhIn and CeIn [48, 49] with atomic layer thicknesseshave been fabricated.Here we design and fabricate three kinds of super-lattices formed by alternating atomically thick layersof CeCoIn and (1)YbCoIn , (2)CeRhIn and (3)CeIn .YbCoIn and Ce M In ( M = Co and Rh) crystalize in thetetragonal HoCoGa structure. Ce M In can be viewedas alternating layers of CeIn and M In stacked sequen-tially along the tetragonal c -axis. CeIn has a cubicAuCu -type structure with a 3D Fermi surface. It shouldbe noted that as disorder may greatly influence physicalproperties, especially near a QCP, there is a great benefitin examining quantum critical systems that are stoichio-metric and hence relatively disorder free; these heavyfermion compounds are examples of a small number ofsuch systems. B. Layered heavy-fermion CeCoIn and CeRhIn CeCoIn is a heavy-fermion superconductor with T c = 2.3 K, which is the highest among Ce-based heavy-fermion superconductors [52]. Related to the layeredstructure, de Haas–van Alphen experiments on CeCoIn reveal a corrugated cylindrical Fermi surface [52, 53]. Inaddition, nuclear magnetic resonance (NMR) relaxationrate T measurements indicate the presence of anisotropic(quasi-2D) AFM spin fluctuations in the normal state[54, 55]. A large Sommerfeld constant γ = C/T = 290mJ mol − K − is observed just above T c [56]. The nor-malized jump in heat capacity ∆ C/γT c ∼ exhibits very strong coupling superconduc-tivity compared with the BCS value of 1.43 [57]. The nor-mal state possesses non-Fermi-liquid properties in zerofields, including T -linear resistivity, indicating a nearbyunderlying QCP [32, 58]. It is well established by severalexperiments that the superconducting gap has d x − y -wave symmetry, which is a strong indication for magnet- T ( K ) P (GPa) 6420 T ( K ) P (GPa) 129630 T ( K ) P (GPa) P * SC AFM SC
AFM+SC
AFM SCCeCoIn CeRhIn CeIn (a) (b) (c) FIG. 2: Schematic pressure-temperature phase diagrams of(a) CeCoIn , (b) CeRhIn , and (c) CeIn . Regions of antifer-romagnetic order and superconducting state are indicated byAFM and SC, respectively. ically mediated Cooper pairing [27–31]. Inelastic neutronscattering measurements detected the presence of mag-netic fluctuations at the incommensurate wavevector q =(0.45, 0.45, 0.5) [59].The magnetic field destroys the superconductivity intwo distinct ways, the orbital pair-breaking effect (vortexformation) and the Pauli paramagnetic effect, a breakingup of pairs by spin polarization. In CeCoIn , the uppercritical field H c for both H (cid:107) ab and H ⊥ ab is limitedby extremely strong Pauli pair-breaking [13, 27, 60]. ThePauli-limited upper critical field H Pauli c is given by [61] H Pauli c = √ /gµ B , (1)where g is the gyromagnetic ratio, which is determined bythe Ce crystalline electric field levels for CeCoIn , and ∆is the superconducting gap energy. As a result of strongPauli effect on the superconductivity, a possible Fulde–Ferrel–Larkin–Ovchinnikov state has been suggested [25,62–67].At ambient pressure, superconductivity in CeCoIn emerges from a non-Fermi liquid state. The normalstate resistivity shows a linear temperature dependence , ρ ( T ) ∝ T over a wide temperature range [32]. The elec-tronic heat capacity in magnetic field, which is enoughto suppress the superconductivity, exhibits a logarith-mic increase , C e ( T ) /T ∝ − ln T [52]. These have beendiscussed in terms of the hallmarks of non-Fermi liquidbehavior near the 2D AFM QCP [68]. With apply pres-sure, the Fermi liquid behavior is recovered in resistivityand heat capacity [52, 57, 69]. The pressure-temperature( P – T ) phase diagram is displayed in Fig. 2(a).CeRhIn has a smaller quasi-2D Fermi surface thanthat of CeCoIn with reflecting localized nature of f -electrons. At ambient pressure, CeRhIn undergoes anAFM transition at N´eel temperature T N = 3.8 K, withthe ordered magnetic moment of 0.75 µ B , and with anincommensurate wave vector q = (0.5, 0.5, 0.297) helicalin the c -axis direction [34]. The In nuclear quadrupoleresonance (NQR) measurements suggested that in theordered state the magnetic moments lie strictly in theCeIn plane and a spiral spin structure is realized alongthe c -axis direction [70]. With increasing pressure P , T N increases and shows a maximum at P ∼ T c increases with pressure but T N decreases [71]. Abovethe pressure P ∗ where T c = T N , the AFM order sud-denly disappears and the pressure-induced transition tosuperconductivity appears to be first order, thus avoidinga QCP. Apparent deviations from Fermi liquid behaviorhave been observed in ρ ( T ) over a wide temperature andpressure range [37]. The P – T phase diagram is displayedin Fig. 2(b). C. Antiferromagnetic heavy fermion CeIn At ambient pressure, CeIn undergoes an AFM tran-sition at T N = 10.1 K with an ordered magnetic momentof 0.48 µ B and a commensurate wave vector q = (0.5,0.5, 0.5) [72, 73]. A dome-like superconducting phaseappears with T max c = 0.2 K around the critical pressure P c = 2.6 GPa, which indicates that the superconductivityis believed to be mediated by quantum critical spin fluc-tuations [10, 73]. The normal state resistivity near thecritical pressure shows non-Fermi liquid behavior [74], ρ ( T ) ∼ T α with α = 1 .
6, which strongly deviate fromthe Fermi liquid value of α = 2. This critical exponent α near P c is close to 3/2 reflecting 3D AFM magnetic fluc-tuations [75], indicating the existence of 3D AFM QCP.The P − T phase diagram is displayed in Fig. 2(c). D. Non-magnetic metal YbCoIn The Yb-ions in YbCoIn are divalent and form theclosed-shell 4 f configuration [76]. As a result, YbCoIn isa nonmagnetic compound, showing conventional metallicbehavior in resistivity and magnetic susceptibility [77].No superconducting transition has been reported in bulkand thin film YbCoIn at ambient and under pressure[48, 77]. III. EXPERIMENTAL METHODA. Molecular beam epitaxy systems
MBE is essentially a refined ultra-high-vacuum evapo-ration method, which helps to prevent contamination ofthe surface and oxidation of elements such as Ce. Thus,high-quality thin films of Ce based compounds can begrown using MBE. MBE enables a slow growth rate of0.01–0.02 nm/s that permits very precise control of layerthickness. Consequently, abrupt material interfaces canbe achieved, enabling the fabrication of heterostructuressuch as superlattices. The typical pressure in the MBEchamber is maintained at < − Pa during the fabri-cation of thin films, enabling powerful diagnostic tech-niques such as reflection high-energy electron diffraction(RHEED) for in situ monitoring of thin films’ growthwithout the complication of surface degradation.Magnesium fluoride MgF is used as the substrate.MgF has a rutile-type tetragonal structure with a latticeparameter a = 0.462 nm, which matches the lattice pa-rameters a = 0.468, 0.453, 0.461 and 0.465 nm for CeIn ,YbCoIn , CeCoIn and CeRhIn , respectively. Further-more, because MgF does not contain oxygen, the oxida-tion of Ce compounds during the growth can be avoided.Thus, single-crystal MgF is a suitable substrate materialto support the epitaxial growth of CeIn and CeCoIn thin films. To relax the lattice mismatch and improvethe quality of the superlattice, we first grow buffer lay-ers. Initially, 30 nm of CeIn buffer layers were grownat 450 ° C. Subsequently, 15 nm of YbCoIn buffer layerswere grown at 550 ° C. On top of these, superlattice layerswere grown. For CeCoIn /CeIn superlattices, ∼ n -UCT CeCoIn and m -UCT CeRhIn as CeCoIn ( n )/CeRhIn ( m ).The left panel of Fig. 3 displays high-resolution cross-sectional transmission electron microscope (TEM) imageof CeCoIn (1)/YbCoIn (5) superlattice, where 1-UCTCeCoIn layer are sandwiched by 5-UCT YbCoIn lay-ers. The bright dot arrays are identified as the Ce layersand the less bright dots are Yb atoms, which is consis-tent with the designed superlattice structure. As shownin the right panel of Fig. 3 , the intensity integrated overthe horizontal width of the image plotted against the ver-tical position indicates a clear difference between the Ceand Yb layers, showing no discernible atomic interdiffu-sion between the neighboring Ce and Yb layers.Figures 4(a) and 4(b) display high resolution cross-sectional TEM images of CeCoIn (5)/CeRhIn (5) andCeCoIn (7)/CeIn (13) superlattices, respectively. Aclear interface between CeCoIn and CeRhIn or CeIn layers is observed. Figures 4(c) and 4(d) displayelectron energy loss spectroscopy (EELS) images ofCeCoIn (5)/CeRhIn (5) and CeCoIn (7)/CeIn (13) su- Integrated intensity (a.u.)1 uctYbCeCoIn
FIG. 3: Cross-sectional TEM image of theCeCoIn (1)/YbCoIn (5) superlattice. The bright dotarrays are identified as the Ce layers and the less brightdots are Yb atoms. The right panel represents the intensityintegrated over the horizontal width of the TEM imageplotted against the vertical position. perlattices, respectively. The EELS images clearly re-solve CeCoIn , CeRhIn or CeIn BLs, demonstratingsharp interfaces with no atomic interdiffusion betweenthe neighboring BLs. (a)
CeCoIn CeRhIn Ce L Rh M In M Co L (c) CeIn Co L Ce L In M (b) (d) FIG. 4: (a) and (b) represent high-resolution cross-sectional TEM and images for CeCoIn (5)/CeRhIn (5) andCeCoIn (7)/CoIn (13) superlattices, respectively. (c) EELSimages measured in the boxed area in the TEM image of (a)for In M , Rh M , Ce L , and Co L edges. (d) EELS imagefor the CeCoIn (7)/CoIn (13) superlattice with the electronbeam alined along the (100) direction. The EELS images weretaken for Co L , Ce L , and In M edges. For all the above Kondo superlattices, streak patternsof the RHEED image were observed during the wholegrowth of the superlattices, indicating good epitaxy. Inaddition, the atomic-force-microscope measurements re-veal that the surface roughness of both superlattices iswithin ± c axis of the constituents. Because atomically flatregions extend over distances of ∼ µ m, it can be ex-pected that transport properties are not seriously influ-enced by the roughness. Superlattice structures were alsoconfirmed by the satellite peaks of the x-ray diffractionpatterns for all superlattices. These results demonstratethe successful fabrication of epitaxial superlattices withsharp interfaces, revealing that the MBE technology iswell suited for achieving our goal of designing Kondo su-perlattices. B. Pressure experiments
In-plane resistivity measurements under pressure wereperformed with a commercial piston-cylinder-type high-pressure cell (CT Factory, Ltd.) [78]. We usedDaphne7373 oil as the pressure medium. We applieda load with a 20 tons hydraulic pressure machine atroom temperature. The pressure inside the pressure cellwas determined by the superconducting transition of lead(Pb), which was obtained by the quasi-four terminal re-sistivity measurement. A long and narrow shaped Pbinstalled inside the sample space is sensitive to the pres-sure gradient for the axial direction of the cell, which canbe known by examining the width of the superconductingtransition. In this study, the transition width was within20 mK, indicating a good hydrostatic condition.
IV. HEAVY FERMION SUPERCONDUCTIVITYAT THE METALLIC INTERFACEA. 2D confinement of heavy fermionsuperconductivity
In CeCoIn /YbCoIn superlattices, the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between theCe atoms in neighboring CeCoIn BLs is substantiallyreduced [79]. Moreover, the superconducting proxim-ity effect between CeCoIn and YbCoIn layers is neg-ligibly small due to the large Fermi velocity mismatch[80]. Then an important question is whether the super-conducting electrons in the superlattices are heavy andif so what their dimensionality is. When the thicknessof the CeCoIn BL is comparable to the perpendicularcoherence length ξ c (about 2.1 nm for CeCoIn ), andthe separation of superconducting layers ( ∼ ( n )/YbCoIn (5) superlattices) exceeds ξ c , eachCeCoIn BL acts as a 2D superconductor [44, 45, 81].Figure 5(a) depicts the magnetic-field dependence ofresistivity for CeCoIn (3)/YbCoIn (5) superlattice atseveral field angles from θ = 0 ( H ⊥ ab ) to 90 ( H (cid:107) a ) at T =150 mK. Figure 5(b) shows the anisotropy ofupper critical field H c (cid:107) /H c ⊥ , where H c (cid:107) and H c ⊥ are critical field parallel and perpendicular to the ab plane, as a function of reduced temperature T /T c forCeCoIn ( n )/YbCoIn (5) superlattices with n =3,5 and7 and for the bulk CeCoIn . H c is determined bythe mid-point of the resistive transition. Unlike the al-most T -independent anisotropy seen in single crystal ofCeCoIn , anisotropy in the superlattice shows a divergentincrease toward T c . This diverging anisotropy is charac-teristic of 2D superconductivity, in which H c (cid:107) increasesas √ T c − T due to the Pauli paramagnetic limiting, but H c ⊥ increases as T c − T due to orbital limiting near T c .The 2D superconductivity is also confirmed from theangle variation of H c ( θ ) shown in Fig. 5(c). For 3Danisotropic mass model, H c ( θ ) is represented as [82] H c ( θ ) = H c (cid:107) / (sin θ + γ m cos θ ) / , (2)where γ m = H c (cid:107) /H c ⊥ . In this model, H c variessmoothly with field orientation. We note that when H c is limited by Pauli paramagnetic effect as represented byEq. (1), H c also varies smoothly as a function of θ . On FIG. 5: (a) Field dependence of resistivity forCeCoIn (3)/YbCoIn (5) at several field angles from H ⊥ ab ( θ = 0) to H (cid:107) a ( θ = 90) at T =150 mK. (b) H c (cid:107) /H c ⊥ vs. T /T c , for the n = 3, 5, and 7 superlattices and for the bulkCeCoIn . (c) Angular dependence of the upper critical field, H c ( θ ). The blue and red lines represent fits to the data usingthe 3D anisotropic mass model (Eq.(2)) and the 2D Tinkhammodel (Eq.(3)), respectively.FIG. 6: H – T phase diagram of n =3, 5 and 7 superlatticesin magnetic field parallel (open symbols) and perpendicu-lar (closed symbols) to the ab plane, compared to the bulkCeCoIn data. the other hand, for 2D superconductor and Josephsoncoupled layered superconductors, H c ( θ ) is representedby Tinkham’s formula as [83] | H c ( θ ) cos θ/H c ⊥ | + | H c ( θ ) sin θ/H c (cid:107) | = 1 . (3)At θ = 90 ◦ , H c ( θ ) exhibits a sharp cusp. The solidblue and red lines in Fig. 5(c) are the fits to Eq. (2) andEq. (3), respectively. A clear cusp at θ = 90 ◦ is observedat T =0.8 and 1.0 K, indicating the 2D superconductiv-ity. The cusp-like behavior of H c ( θ ) becomes less pro-nounced well below T c , which is the opposite trend to the H c ( θ ) behavior of conventional multilayer systems. Thissuggests that H c ( θ ) at low temperatures is dominatedby the Pauli effect in any field directions.Figure 6 shows H – T phase diagram ofCeCoIn ( n )/YbCoIn (5) superlattices with n =3, 5and 7 in magnetic field parallel (open symbols) andperpendicular (closed symbols) to the ab plane. Forthe comparison, the results of the bulk CeCoIn arealso plotted. At low temperatures H c (cid:107) and H c ⊥ ofthe superlattices are significantly larger than those inconventional superconductors with similar T c . Thezero-temperature value of the orbital upper criticalfield in perpendicular field H orb c ⊥ (0) reflects the effectiveelectron mass in the plane m ∗ ab , H orb c ⊥ (0) ∝ m ∗ ab . Here, H orb c ⊥ (0) is determined from the initial slope of H c ⊥ ( T )at T c through the relation [84], H orb c ⊥ ≈ . T c ( − dH c ⊥ /dT ) T c . (4) H orb c ⊥ (0) is estimated to be 6, 11 and 12 T forCeCoIn ( n )/YbCoIn (5) with n = 3, 5 and 7, respec-tively. These magnitudes are comparable with or of thesame order as H orb c ⊥ (0) (=14 T) in bulk CeCoIn [44].Based on these results, we conclude the 2D confine-ment of the superconducting ‘heavy’ electrons in theCeCoIn /YbCoIn superlattices. B. Local inversion symmetry breaking
In Fig. 7, H c ⊥ normalized by H orb c ⊥ (0) forCeCoIn ( n )/YbCo (5) superlattices with n = 3, 5and 7 is plotted as a function of the normalized tem-perature T /T c . Two extreme cases, i.e., the resultof the bulk single crystal of CeCoIn dominated byPauli paramagnetic effect and the Werthamer-Helfand-Hohenberg (WHH) curve with no Pauli effect [84], arealso shown. For all superlattices, H c ⊥ /H orb c ⊥ (0) is muchlarger than that of single crystal CeCoIn , indicatingthat Pauli paramagnetic pair breaking effect is reducedin these superlattices. More importantly, H c ⊥ /H orb c ⊥ (0)is strikingly enhanced with decreasing n .Recently, it has been suggested that the inver-sion symmetry breaking (ISB), together with strongspin-orbit interaction, can dramatically affect thesuperconductivity[85–87]. It has also been pointed outthat such phenomena are more pronounced in stronglycorrelated electron systems. The inversion symmetry im-poses important constraints on the pairing states: In thepresence of inversion symmetry, Cooper pairs are clas-sified into a spin-singlet or triplet state, whereas in theabsence of inversion symmetry, an asymmetric potentialgradient ∇ V yields a spin-orbit interaction that breaksparity, and the admixture of spin-singlet and tripletstates is possible. For instance, asymmetry of the po-tential in the direction perpendicular to the 2D plane ∇ V (cid:107) [011] induces Rashba spin-orbit interaction α R g ( k ) · σ ∝ ( k × ∇ V ) · σ , (5)where g = ( − k y , k x , /k F , k F is the Fermi wave number,and σ is the Pauli matrix. Rashba interaction splits theFermi surface into two sheets with different spin struc-tures [87–89]. The energy splitting is given by α R , andthe spin direction is tilted into the plane, rotating clock-wise on one sheet and anticlockwise on the other. When the Rashba splitting exceeds the superconducting gap en-ergy ( α R > ∆), the superconducting properties are dra-matically modified. As the spin-orbit interaction is gen-erally significant in Ce-based superconductors, the in-troduction of ISB makes the systems a fertile groundfor observing exotic properties. Moreover, theoreticalstudies suggest that when the interlayer hopping integralis comparable to, or smaller than, the Rashba splitting( t c ≤ α R ) , the local ISB plays an important role in de-termining the nature of the superconducting state[86].This appears to be the case for the CeCoIn /YbCoIn superlattices.Although these superlattices maintain centrosymme-try, it has been suggested that the local ISB at the in-terface between two compounds influences the supercon-ducting. Figure 8(a) represents the schematic representa-tion of CeCoIn ( m )/YbCoIn (5) superlattice. The mid-dle CeCoIn layer in a given CeCoIn BL indicated by thegray plane is a mirror plane. The green (small) arrowsrepresent the asymmetric potential gradient associatedwith the local ISB, −∇ V local . The Rashba splitting oc-curs at the interface between the CeCoIn and YbCoIn due to the local ISB. The spin direction is rotated in the ab plane and is opposite between the top and bottomCeCoIn layers. Because the fraction of noncentrosym-metric interface layers increases with decreasing n , theobserved remarkable enhancement of H c ⊥ /H orb c ⊥ (0) withdecreasing n shown in Fig. 7 is attributed to the increasedimportance of the local ISB.It has been reported that the local ISB also seriously H c ⊥ / H c ⊥ o r b ( ) T/T c WHHCeCoIn single crystal0 GPa CeCoIn ( n )/CeRhIn ( n ) n = 7 5 3 CeCoIn ( n )/YbCoIn (5) n = 7 5 3 FIG. 7: Out-of-plane upper critical field H c ⊥ nor-malized by the orbital-limited upper critical field at T = 0 K, H c ⊥ /H orbc ⊥ (0), for CeCoIn ( n )/YbCoIn (5) andCeCoIn ( n )/CeRhIn ( n ) superlattices with n = 7, 5, and3 are plotted as a function of the normalized tempera-ture T /T c . Two extreme cases, i.e., the result of thebulk CeCoIn dominated by Pauli paramagnetic effect andthe WHH curve with no Pauli effect, are also shown. InCeCoIn ( n )/YbCoIn (5) , H c ⊥ /H orbc ⊥ (0) is enhanced withdecreasing n , indicating the importance of the local ISB. Incontrast, in CeCoIn ( n )/CeRhIn ( n ), H c ⊥ /H orbc ⊥ (0) is inde-pendent of n . BABA
CeCoIn ( m )YbCoIn (5) CeCoIn ( n )YbCoIn (3)YbRhIn (3) BABACCA globallocal (a) (b)
FIG. 8: (a) Schematic representation of bicolor Kondo super-lattice CeCoIn ( m )/YbCoIn (5). The center of a CeCoIn BL(ash plane) is a mirror plane. The green (small) arrows rep-resent the asymmetric potential gradient associated with thelocal ISB, −∇ V local . The Rashba splitting occurs at the inter-face between the CeCoIn and YbCoIn BLs due to the localISB. The spin direction is rotated in the ab plane and is oppo-site between the top and bottom CeCoIn BLs. (b) Schematicrepresentation of noncentrosymmetric tricolor Kondo super-lattices YbCoIn (3)/CeCoIn ( n )/YbRhIn (3). The orange(large) arrows represent the asymmetric potential gradient −∇ V global . In the tricolor superlattices, all layers are not themirror planes. The orange arrows represent the asymmetricpotential gradient −∇ V global due to the global broken inver-sion symmetry. The amplitude of Rashba splitting at the toplayer of CeCoIn BL is larger than that at the bottom of theBL, owing to the presence of −∇ V global shown by the greensmall arrows. influence the magnetic properties in non-superconductingKondo superlattices. In CeRhIn /YbRhIn superlat-tices, with reducing the thickness of magnetic CeRhIn BLs, the N´eel temperature is suppressed and the quasi-particle mass is strongly enhanced, implying dimensionalcontrol toward a magnetic QCP [90].
C. Tricolor Kondo superlattices
Recently, it has been reported that the magnitudeof the Rashba spin-orbit interaction arising from theISB is controllable by fabricating two types of Kondosuperlattices[46, 91]. One is the introduction of the thick-ness modulation of YbCoIn BLs that breaks the inver-sion symmetry centered at the superconducting blockof CeCoIn . The other is the ‘tricolor’ superlattices,in which CeCoIn BLs are sandwiched by two differ-ent nonmagnetic metals, YbCoIn and YbRhIn , as il-lustrated in Fig. 8(b). In these two types of Kondo su-perlattices, the weakening of the Pauli paramagnetic pairbreaking effect is more pronounced than that in ‘bicolor’CeCoIn /YbCoIn superlattices, as revealed by the fur-ther enhancement of H c ⊥ /H orb c ⊥ (0).In particular, in the tricolor Kondo superlattices, theRashba spin-orbit- interaction induced global inversionsymmetry breaking is largely tunable by changing thelayer thicknesses of YbCoIn and YbRhIn , leading toprofound changes in the superconducting properties of2D CeCoIn BLs. Remarkably, the temperature depen- dence of H c (cid:107) of YbCoIn (3)/CeCoIn ( n )/YbRhIn (3),in which 3-UTC YbCoIn , n -UCT CeCoIn ( n = 5 and 8)and 3-UCT YbRhIn are stacked alternatively, in-planeupper critical field exhibits an anomalous upturn at lowtemperatures, which is attributed to a possible emergenceof a helical or stripe superconducting phase [91]. Theseresults demonstrate that the tricolor Kondo superlatticesprovide a new playground for exploring exotic supercon-ducting states in the strongly correlated 2D electron sys-tems with the Rashba effect.The fabrication of tricolor superlattices containing d -wave superconducting layers offers the prospect ofachieving even more fascinating pairing states than bulkCeCoIn , such as helical and stripe superconductingstates [92], a pair-density-wave state [93], complex stripestate [94], a topological crystalline superconductivity[95, 96], and Majorana fermion excitations[97–101], instrongly correlated electron systems. V. TUNING THE PAIRING INTERACTIONTHROUGH THE INTERFACEA. CeCoIn /CeRhIn Kondo superlattices (d)(e) T ( K ) P (GPa) CeCoIn SC T c single crystalthin film T ( K ) P (GPa) CeRhIn AFM T c T N single crystalthin film ( µ Ω c m ) ( µ Ω c m ) ( µ Ω c m ) T (K) d / d T ( µ Ω c m / K ) d / d T ( µ Ω c m / K ) CeCoIn thin filmCeRhIn thin filmCeCoIn (5)/CeRhIn T N (a)(b)(c) SC (5) T N P* Pc
FIG. 9: (a) Temperature dependence of the resistivityof CeCoIn thin film at ambient pressure and at P =2.1 GPa. (b), (c) Temperature dependence of the resis-tivity (solid lines, left axes) and its temperature derivative dρ ( T ) /dT (dotted lines, right axes) for CeRhIn thin film andCeCoIn (5)/CeRhIn (5) superlattice at ambient pressure andat P = 2.1 GPa, respectively. The peak of dρ ( T ) /dT corre-sponds to AFM transition. (d), (e) P − T phase diagrams ofthin films and single crystals of (d) CeCoIn and (e) CeRhIn . Figures 9(a), (b) and (c) depict the temperature de-pendence of the resistivity and its temperature deriva-tive, dρ/dT , for CeCoIn and CeRhIn thin films andCeCoIn (5)/CeRhIn (5) superlattice at ambient and un-der pressure ( P = 2 . dρ/dT corresponds to the AFM transition. The pressuredependence of T c and T N for CeCoIn and CeRhIn thinfilms, along with those for single crystals, are shown inFigs 9(d) and 9(e). The P − T phase diagrams of bothfilms are essentially similar to those of single crystals.However, T c (= 2.0 K) in the CeCoIn thin film is slightlyreduced from the bulk value, whereas T N (= 3.7 K) ofCeRhIn thin film is almost the same as that in a sin-gle crystal. With applying pressure, T c of the CeCoIn thin film increases and shows a broad peak near P ∼ single crystals [13, 102],superconductivity in the thin films develops at P (cid:38) single crystals, there appears to be a purelysuperconducting state at P (cid:38) P -dependence of T c and T N determined by the peak in dρ ( T ) /dT forCeCoIn (5)/CeRhIn (5) superlattice. At P ∼ T c is at a maximum, forming a dome-shaped P -dependence.With pressure, T N gradually decreases at low P anddecreases sharply when it exceeds P (cid:38) P (cid:38) P c , a simpleextrapolation of T N gives P c ∼ T c has amaximum. Furthermore, this critical value is very closeto the P c of CeRhIn single crystal.The T c and T N of the hybrid superlattice are lowerthan those of the CeCoIn and CeRhIn thin films,suggesting that the reduction in dimensions affectedthe electronic structure. However, these values arehigher than the corresponding CeCoIn /YbCoIn andCeRhIn /YbRhIn , indicating the importance of inter-action between CeCoIn and CeRhIn BLs.
B. Superconductivity and antiferromagnetism inspecially separated layers
We show that 2D superconductivity is realized inthe CeCoIn BLs in the whole pressure regime inCeCoIn (5)/CeRhIn (5) superlattice. Figure 10(b) de-picts the T -dependence of the upper critical field de-termined by the midpoint of the resistive transition ina magnetic field H applied parallel ( H c (cid:107) ) and per-pendicular ( H c ⊥ ) to the ab plane. Figure 10(c) showsthe T -dependence of the anisotropy of upper criticalfields, H c (cid:107) /H c ⊥ . Unlike the almost T -independentanisotropy seen in single crystals and thin films ofCeCoIn , anisotropy in the superlattice shows a divergentincrease toward T c . This diverging anisotropy is charac-teristic of 2D superconductivity, in which H c (cid:107) increasesas √ T c − T due to the Pauli paramagnetic limiting, but H c ⊥ increases as T c − T due to orbital limiting near T c .Considering this result and the fact that the 5-UCT ofthe CeCoIn BL is comparable to the superconductingcoherence length in the c -axis direction ξ ⊥ ∼ BL effectively act as a 2D super-conductor [44]. The 2D superconductivity is also con-firmed from the angle variation of H c ( θ ). Figure 10(d)and its inset show H c ( θ ) below and above P ∗ . For bothpressures, at temperature well below T c , H c ( θ ) in theregime | θ | (cid:46) ◦ is enhanced with decreasing | θ | and ex-hibits a sharp cusp at θ = 0. This cusp behavior is typicalof Josephson-coupled layered superconductors [82]. H c ( T ) -90 -45 0 45 90 (degree) T c T c H c || / H c ⊥ T/T c H c ( T ) T (K) H || a H || c T ( K ) P (GPa) 7531 H c ⊥ / T c ( T / K ) T N T c p c CeCoIn (5)/CeRhIn (5) CeCoIn single crystal (a) (b)(c) (d) θ H ac FIG. 10: (a) P − T phase diagram of CeCoIn (5)/CeRhIn (5)superlattice. Out-of-plane upper critical field H c ⊥ normal-ized by T c , H c ⊥ /T c , measures the coupling strength of thesuperconductivity. (b) Temperature dependence of in-planeand out-of-plane upper critical fields at ambient pressure andat P = 1.8 and 2.1 GPa. (c) Anisotropy of upper critical field, H c (cid:107) /H c ⊥ , near T c of superlattices at ambient pressure andat 2.1 GPa, along with the data of CeCoIn thin film. (d)Angular dependence of upper critical field of superlattice at P = 1.8 and 2.1 GPa. (Inset) An expanded view of low angleregion. It should be noted that in contrast to single crys-tals and thin films of CeRhIn , the CeRhIn layers inCeCoIn /CeRhIn hybrid superlattices do not becomefully superconducting even under pressure where AFMorder is suppressed. As a result, 2D superconductivityoccurs in a wide pressure regime. In fact, as shown inFig. 10(d), at P = 1 . thin film doesnot show bulk superconductivity, the hybrid superlatticeshows an angular dependence with a cusp structure near θ = 0. Essentially similar cusp-like behavior is observedat P = 2 . P c , suggesting that 2D supercon-ductivity derived from the CeCoIn BLs is realized belowand above P c .When the number of BL thickness is reduced, super-conductivity survives in CeCoIn , but is suppressed inCeRhIn . This difference may be related to the order-ing vector q = (0 . , . , . . In CeCoIn , onthe other hand, the AFM fluctuations are dominated by q = (0 . , . , .
5) [59]. This commensurability alongthe c -axis would match well with the superlattice struc-ture, and as a result, the superconductivity is robustagainst the decrease in the BL thickness [47, 103]. Recentsite-selective NMR measurements on CeCoIn /CeRhIn superlattice have shown that AFM order is not inducedin the CeCoIn BLs [104]. The pressure suppresses mag-netic order in CeRhIn and CeCoIn approaches theFermi liquid state, so it is unlikely that AFM order isinduced in the CeCoIn BLs in the superlattice underpressure. We comment on the reversal of H c of theCeCoIn (5)/CeRhIn (5) superlattice at low temperatureunder pressure(Fig. 10(b)). Such a reversed anisotropy of H c can be seen in CeRhIn single crystal in a high pres-sure region where AFM order is completely suppressed.However, similar reversed anisotropy ( H c ⊥ > H c (cid:107) ) ispreserved at P = 1 . H c (cid:107) exceeds H c ⊥ inCeRhIn single crystal and thin film. This result sug-gests that the reversal of H c occurs in 5-UCT CeCoIn BLs. From the above results, we conclude that 2D super-conductivity of CeCoIn coupled by the Josephson effectwithin a BL is realized in the whole pressure regime.Figure 7 displays H c ⊥ ( T ) /H c ⊥ (0) ofCeCoIn ( n )/CeRhIn ( n ) and CeCoIn ( n )/YbCoIn (5)superlattices plotted as a function of T /T c . Here H orb c ⊥ (0)is calculated by the initial slope of H c ⊥ ( T ) at T c byusing Werthamer-Helfand-Hohenberg (WHH) formula, H orb c ⊥ (0) = − . T c ( dH c ⊥ /dT ) T c . For comparison,we also include two extreme cases: H c ⊥ /H orbc ⊥ (0) forbulk CeCoIn [105], in which H c is dominated byPauli paramagnetism, and the WHH curve with noPauli effect. In CeCoIn ( n )/YbCoIn (5), H c ⊥ /H orb c ⊥ (0)increases with decreasing n . This is because the localinversion symmetry breaking suppresses the Paulipair-breaking effect at the interfaces between BLs. As n decreases, the contribution of the interface increases andthe relative importance of orbital pair-breaking effectcompared with Pauli pair-breaking effect increases. Onthe other hand, H c ⊥ /H orb c ⊥ is almost independent on n in CeCoIn ( n )/CeRhIn ( n ), suggesting that the localinversion symmetry breaking in not important in thesuperlattices in which both substances constituting thesuperlattice are Ce-based compounds. C. Enhancement of superconducting pairingstrength
The superconducting properties of the hybrid superlat-tice change dramatically when pressure is applied. Fig-ure 11(a) depicts the T -dependence of H c ⊥ / H orb c ⊥ (0) ofCeCoIn (5)/CeRhIn (5) for several pressures. Remark-ably, near the critical pressure of P c ∼ H c ⊥ /H orb c ⊥ almost coincides with WHHcurve, indicating that H c ⊥ is determined only by theorbital pair-breaking effect.The fact that H c ⊥ reaches the orbital limit has im-portant implications for the superconducting propertiesof the hybrid superlattice. In CeCoIn /YbCoIn , whereYbCoIn is a conventional metal, Pauli pair-breaking ef- fect is weakened in the superlattice compared with thebulk due to local inversion symmetry breaking at the in-terfaces, where the Fermi surface splits with spin momen-tum locking due to anisotropic Rashba spin-orbit interac-tion. This leads to anisotropic suppression of the Zeemaneffect which may be partly responsible for the observedreversed anisotropy H c (cid:107) /H c ⊥ < ( n )/CeRhIn ( n ) superlattices compared withCeCoIn /YbCoIn , which is evidenced by the fact that H c ⊥ /H orb c ⊥ (0) does not strongly depend on n (Fig. 7).Furthermore, such an effect is not expected to show sig-nificant pressure dependence. Therefore, there must bea different mechanism that significantly enhances H Pauli c ⊥ given by Eq. (1).An enhancement of H Pauli c ⊥ is not due to a dramaticsuppression of g . As g is enhanced by pressure in bothCeCoIn and CeRhIn [102], g is expected to be enhancedwith pressure in the superlattice. Therefore, a significantincrease in the superconducting gap is thought to be theorigin of the increase in H Pauli c ⊥ . This is also supportedby the sharp increase in H c ⊥ /T c upon approaching P c shown in Fig. 11(a). Because H c ⊥ ≈ H Pauli c ⊥ (cid:28) H orb c ⊥ (0)in the low P regime and H c ⊥ ≈ H orb c ⊥ (0) (cid:28) H Pauli c ⊥ near P ∼ p c , the enhancement of H c ⊥ /k B T c directly indi-cates an enhancement of H Pauli c ⊥ /T c and hence ∆ /k B T c .This behavior is significantly different from CeCoIn sin-gle crystal, in which H c ⊥ /T c monotonically decreaseswith pressure, approaching the Fermi liquid state. Theenhancement of ∆ /k B T c is caused as a consequence ofthe enhancement of pairing interaction. In the spinfluctuation mediated mechanism, the pairing interactionis brought about by high-energy spin fluctuations wellabove ∆, while low-energy fluctuations cause the pair-breaking. High-energy fluctuations have the effect of in-creasing T c , while low-energy fluctuations decrease T c ,so that the enhancement of pairing interaction can giverise to increase in ∆ /k B T c without accompanying a largeenhancement of T c . Therefore, these results demon-strate that the critical magnetic fluctuations developedin CeRhIn BLs near its critical pressure are injectedinto CeCoIn BLs through the interface and enhance thepairing interaction of the CeCoIn BLs.It has been established that normal and superconduct-ing properties are greatly affected by quantum fluctua-tions in many classes of unconventional superconductors.The common behavior is that the effective mass of quasi-particle diverges as the system approaches a QCP, as re-ported in cuprates, pnictides and heavy-fermion systems[16, 38, 106]. Such an increase in effective mass gives riseto a corresponding enhancement H orb c , which is propor-tional to ( m ∗ ∆) . the CeCoIn /CeRhIn superlatticesshow different behavior. In contrast to the CeRhIn 5 sin-gle crystal, which shows a sharp peak at the critical pres-sure, the H orb c ⊥ of the CeCoIn ( n )/CeRhIn ( n ) superlat-tices with n = 4 and 5 does not show much P -dependentbehavior, and there is no anomaly in P c . Compared toa monotonic decrease of effective mass in CeCoIn single0crystal, the result of the hybrid superlattice is consistentwith an enhancement of ∆, indicating that there is nomass enhancement in CeCoIn BLs. Such behavior is incontrast to what is expected for usual quantum critical-ity, and is a subject for future research. H c o r b ( ) ( T ) P (GPa) CeCoIn (5)/CeRhIn (5) CeCoIn (4)/CeRhIn (4) CeRhIn CeCoIn H c ⊥ / H c ⊥ o r b ( ) T/T c CeCoIn (5)/CeRhIn (5) WHH
CeCoIn single crystal0 GPa (a) (b) p c FIG. 11: (a) Out-of-plane upper critical field H c ⊥ normal-ized by the orbital-limited upper critical field at T = 0 K, H c ⊥ /H orb c ⊥ (0), for CeCoIn (5)/CeRhIn (5) superlattice isplotted as a function of the normalized temperature T /T c .Two extreme cases, i.e., the result of the bulk CeCoIn domi-nated by Pauli paramagnetic effect and the WHH curve withno Pauli effect, are also shown. (b) Pressure dependence of H orb c (0) of CeCoIn ( n )/CeRhIn ( n ) superlattices with n = 4and 5 for H (cid:107) c . For comparison, H orb c (0) of CeRhIn sin-gle crystals for H (cid:107) a and that of CeCoIn single crystal for H (cid:107) c are shown. Solid and dashed arrows represent P c forCeCoIn ( n )/CeRhIn ( n ) superlattices and CeRhIn singlecrystal, respectively. VI. COUPLING BETWEENSUPERCONDUCTIVITY ANDANTIFERROMAGNETISM
To further examine how d -wave superconductors andantiferromagnets interact through the interface, we de-signed another hybrid superlattice using a different AFMmetal CeIn [49]. The cubic CeIn forms 3D AFM orderwith the ordered magnetic moment of 0.48 µ B occurs witha commensurate wave vector q = (0.5, 0.5, 0.5) at T N =10 K, where µ B is the Bohr magneton [72]. This is con-trast to CeRhIn , which forms an incommensurate helicalAFM order with q = (0.5, 0.5, 0.239) at T N = 2.3 K. Onthe other hand, both are Ce-based heavy-fermion AFMmetal with AFM QCP under pressure [10, 74]. Therefore,it becomes possible to investigate the effect of differenttypes of antiferromagnetism on d -wave superconductiv-ity by measuring the H c for CeCoIn /CeIn superlatticeunder pressure, as have done for CeCoIn /CeRhIn . A. Robust magnetism against thickness reduction
Figure 12(a) depicts the temperature dependence ofthe resistivity ρ of CeCoIn (7)/CeIn( n ) superlatticeswith n = 3, 4, 6 and 13. We also show ρ of CeCoIn andCeIn thin films grown by MBE. The mean free path ofthese superlattices is difficult to estimate because of theparallel conductions of CeCoIn and CeIn BLs. How-ever, the mean free path in each BL is expected to beshorter than the atomically flat regions extending overdistances of ∼ . µ m, because of the following rea-sons. In CeCoIn and CeIn single crystals, the meanfree path determined by the de Haas-van Alphen oscil-lations is ∼ . µ m [107, 108]. The residual resistivityratio of CeCoIn and CeIn thin films with 100 nm thick-ness is 4–5 times smaller than that of the single crys-tals. Therefore, the mean free path of CeCoIn andCeIn BLs in the superlattices is expected to be muchshorter than 0.1 µ m, suggesting that the transport prop-erties are not seriously influenced by the surface rough-ness. The resistivity of CeCoIn (7)/CeIn( n ) superlat-tices follows the typical heavy-fermion behavior. Withdecreasing temperature, ρ ( T ) increases below ∼
150 Kdue to the Kondo scattering but then begins to decreasedue to strong c - f hybridization between f -electrons andconduction ( c ) band electrons, leading to the narrow f -electron band at the Fermi level. The Kondo coher-ence temperature T coh , at which the formation of heavy-fermion occurs, is estimated from the maximum in ρ ( T ).As shown in Fig. 12(a), T coh of CeCoIn (7)/CeIn( n ) su-perlattices is nearly independent of n and is closer to T coh of CeCoIn thin film than T coh of CeIn thin film, sug-gesting that T coh is mainly determined by CeCoIn BLs.Figure 12(b)-(f) depict ρ ( T ) at low temperatures. All su-perlattices show the superconducting transition at T ≈ n = 3- and 4-superlattices, ρ ( T ) decreaseswith increasing slope, dρ ( T ) /dT , as the temperature islowered below 12 K down to T c .The lattice parameters along the a -axis of CeCoIn ,CeRhIn , and CeIn is 4.613, 4.653, and 4.690 ˚A, respec-tively. Therefore, a large tensile strain along the a -axis isexpected in CeCoIn BLs of CeCoIn /CeIn compared toCeCoIn /CeRhIn . It has been shown that the uniaxialpressure dependence of T c along the a -axis for CeCoIn isd T c /d P a = 290 mK/GPa ( [109]), indicating that T c de-creases by tensile strain. However, T c of CeCoIn /CeIn ( T c ∼ /CeRhIn ( T c ∼ c -axis for CeCoIn BLs in CeCoIn /CeIn , whichis estimated from x-ray diffraction, well coincides withthat in CeCoIn /CeRhIn . These results suggest thatthe strain effect at the interfaces is not important fordetermining T c . Figure 12(g)–(k) display the tempera-ture derivative of the resistivity dρ ( T ) /dT . As shown bythe arrows in Fig. 12(g), dρ ( T ) /dT of CeIn thin film ex-hibits a distinct kink at T N = 10 K [72]. Similar kinkstructures are observed in all superlattices at the tem-peratures indicated by arrows, showing the AFM tran-1 FIG. 12: (a) Temperature dependence of the resistivity ρ ( T )in CeCoIn (7)/CeIn ( n ) superlattices for n = 3, 4, 6, and13, along with ρ ( T ) for CeIn (black solid line) and CeCoIn (black dashed line) thin films. Inset illustrates the schemat-ics of CeCoIn (7)/CeIn ( n ) superlattice. (b)-(f) ρ ( T ) at lowtemperatures. (g)-(f) Temperature derivative of the resistiv-ity, dρ ( T ) /dT , as a function of temperature. The arrows in-dicate the N´eel temperature T N . sition. Figure 13 shows the thickness dependence of T N FIG. 13: The N´eel temperature T N for CeCoIn (7)/CeIn ( n )as a function of n . For comparison, T N for CeIn ( n )/LaIn (4)and CeCoIn ( n )/CeRhIn ( n ) are shown. Open square andtriangle are T N of bulk CeIn and CeRhIn single crystals,respectively. of the CeCoIn (7)/CeIn ( n ) superlattices. For compar-ison, the data sets of CeIn (4)/LaIn ( n ), where LaIn is a nonmagnetic conventional metal with no f -electrons[50], and CeCoIn ( n )/CeRhIn ( n ) are also included inFig. 13. Remarkably, the observed thickness dependenceof T N in CeCoIn /CeIn is in striking contrast to thatin CeIn /LaIn ; While T N is strongly suppressed withdecreasing n and vanishes at n = 2 in CeIn /LaIn , T N is nearly independent of n in CeCoIn (7)/CeIn ( n ).This suggests that CeIn BLs are coupled weakly by theRKKY interactions through the adjacent LaIn BL, butthey can strongly couple through the adjacent CeCoIn BL. This is even more surprising, as the distance betweendifferent CeIn BLs is larger in the CeCoIn (7)/CeIn ( n )superlattices than in the CeIn ( n )/LaIn (4) superlat-tices. We thus conclude that small but finite magneticmoments are induced in CeCoIn BLs in CeCoIn /CeIn ,which mediate the RKKY-interaction. On the otherhand, because of the absence of strongly interacting f -electrons in LaIn , which can form magnetic moments,the RKKY interaction in CeIn /LaIn can be expectedto be much weaker. To clarify this, a microscopic probe ofmagnetism, such as NMR measurements, is required. Wenote that as shown in Fig. 13, the reduction of T N is alsoobserved in CeCoIn ( n )/CeRhIn ( n ) superlattices [48],suggesting that the RKKY interaction between CeRhIn BLs through adjacent CeCoIn BL is negligibly small.This is supported by the recent site-selective NMR mea-surements which report no discernible magnetic momentsinduced in the CeCoIn BLs while magnetic fluctuationsare injected from CeRhIn BLs into one or two layers ofCeCoIn BLs in CeCoIn /CeRhIn [104]. B. Tuning AFM fluctuations via pressure
The pressure dependence of the superconductingand magnetic properties provide crucial informationon the mutual interaction between superconductivityand magnetism through the interface. Figure 14(a)depicts the pressure dependence of T N and T c forCeCoIn (7)/CeIn ( n ) superlattices for n = 6 and 13.With applying pressure, T N decreases rapidly. For com-parison, T N of a single crystal CeIn is also shown by thesolid line [10]. The pressure dependence of T N of both su-perlattices is very similar to that of the bulk CeIn singlecrystal. In bulk CeIn crystal, the AFM QCP is locatedat P c ≈ . ρ ( T ) = ρ + AT ε . (6)Figure 14(b) shows the pressure dependence of ε obtainedfrom d ln ∆ ρ/d ln T , where ∆ ρ = ρ ( T ) − ρ . The magni-tude of ε decreases with pressure. In bulk CeIn singlecrystal, ε decreases with pressure and exhibits a mini-mum at the AFM QCP [10, 74]. On the other hand,applying pressure to CeCoIn leads to an increase of ε ,which is attributed to the suppression of the non-Fermiliquid behavior, ρ ( T ) ∝ T , and the development of aFermi liquid state with its characteristic ρ ( T ) ∝ T de-pendence [32, 33]. Therefore, the reduction of ε with2pressure arises from the CeIn BLs, indicating that theCeIn BLs approach the AFM QCP.As shown in Fig 14(a), T c increases, peaks at ∼ bulk single crystals [32]. An analysis of theupper critical field provides important information aboutthe superconductivity of CeCoIn BLs. Figure 15 depictsthe temperature dependence of the upper critical fielddetermined by the midpoint of the resistive transition ina magnetic field H applied parallel ( H c (cid:107) ) and perpen-dicular ( H c ⊥ ) to the layers. The inset of Fig 15 showsthe anisotropy of the upper critical fields H c (cid:107) /H c ⊥ atambient pressure. The 2D feature is revealed by the di-verging anisotropy of H c (cid:107) /H c ⊥ of the superlattice onapproaching T c , in sharp contrast to the CeCoIn thinfilm. Thus, each CeCoIn BL in CeCoIn /CeIn super-lattice effectively behaves as a 2D superconductor. FIG. 14: (a) Pressure dependence of T N and T c ofCeCoIn (7)/CeIn ( n ) superlattices for n = 13 and 6. Forcomparison, T N of CeIn and T c of CeCoIn single crystalsare shown by solid lines. (b) Pressure dependence of the ex-ponent ε in ρ ( T ) = ρ + AT ε , obtained from d ln ∆ ρ/d ln T (∆ ρ = ρ ( T ) − ρ ), for the CeCoIn (7)/CeIn ( n ) superlatticesfor n = 13 and 6. For comparison, ε for bulk CeIn andCeCoIn single crystals is shown. C. Effect of magnetic fluctuations onsuperconductivity
It has been revealed that the T dependence of H c ⊥ provides crucial information on the effect of interfaceson the superconductivity of CeCoIn BLs. In particular,the modification of the Pauli paramagnetic effect in thesuperlattice, which dominates the pair breaking in bulkCeCoIn single crystals, provide valuable clues [45, 46,48, 91]. Figure 16(a) and (b) depict the T dependence ofthe H c ⊥ of CeCoIn (7)/CeIn (13) superlattice at am-bient pressure Fig. 16(a) and under pressure Fig. 16(b),normalized by the orbital-limited upper critical field atzero temperature, H orb c ⊥ (0), which is obtained from theWHH formula, H orb c ⊥ (0) = − . T c ( dH c ⊥ /dT ) T c [84]. FIG. 15: Temperature dependence of upper critical fieldsin magnetic fields parallel ( H c (cid:107) , open symbols) and per-pendicular ( H c ⊥ , closed symbols) to the ab -plane forCeCoIn (7)/CeIn (13) superlattice at ambient pressure andat 2.1 and 2.4 GPa. The inset shows anisotropy of the uppercritical field, H c (cid:107) /H c ⊥ . The data of CeCoIn thin film atambient pressure is shown by dotted line. In figure 16(a) and 16(b), two extreme cases are alsoincluded; the WHH curve with no Pauli pair-breakingand H c /H orb c ⊥ (0) for bulk CeCoIn single crystal [105].For comparison, H orb c ⊥ (0) for CeCoIn /YbCoIn andCeCoIn /CeRhIn are also shown [44, 48]. FIG. 16: (a) Upper critical field in perpendicular fieldnormalized by the orbital limiting upper critical field, H c ⊥ /H orb c ⊥ (0), plotted as a function of T /T c (a) atambient pressure and (b) under pressure about 2 GPafor CeCoIn (7)/CeIn (13) superlattices. For compar-ison, H c ⊥ /H orb c ⊥ (0) for bulk CeCoIn single crystal,CeCoIn (5)/YbCoIn (5) and CeCoIn (5)/CeRhIn (5) areshown. Orange dotted lines represent the WHH curve, whichis upper critical field for purely orbital limiting. At ambient pressure, H c ⊥ /H orb c ⊥ (0) is significantlyincreased from bulk CeCoIn single crystal in bothCeCoIn /YbCoIn and CeCoIn /CeRhIn , indicatingthe suppression of the Pauli paramagnetic pair-breakingeffect. However, we point out that the mechanisms of this3suppression in these two systems are essentially different.In CeCoIn /YbCoIn , as discussed in section IV-B, theenhancement of H c ⊥ /H orb c ⊥ (0) is caused by the local ISBat the interface [45, 86]. At P = 2.2 GPa, H c ⊥ /H orb c ⊥ (0)of CeCoIn /YbCoIn nearly coincides with the WHHcurve, indicating that H c ⊥ is dominated by the orbitalpair breaking most likely due to the suppression of thePauli paramagnetic pair-breaking effect by the Rashbasplitting.As shown in Fig. 7, in stark contrast toCeCoIn ( n )/YbCoIn ( n ), where H c ⊥ /H orb c ⊥ (0) isstrongly enhanced with decreasing n , H c ⊥ /H orb c ⊥ (0)in CeCoIn ( n )/CeRhIn ( n ) is independent of n . Thisindicates that the effect of the local ISB on H c ⊥ is much less important in CeCoIn /CeRhIn thanCeCoIn /YbCoIn , possibly owing to smaller asym-metric potential gradient at Ce/Ce interface comparedwith that at Ce/Yb one [48]. It has been proposedthat magnetic fluctuations in CeRhIn BLs injectedthrough the interface dramatically enhance the pairinginteraction in CeCoIn BLs, leading to the enhancementof ∆. As a result, H Pauli c ⊥ is enhanced. This raises therelative importance of the orbital pair-breaking effect,giving rise to the enhancement of H c ⊥ /H orb c ⊥ (0) [48].At P = 2.1 GPa, which is close to the AFM QCP ofCeRhIn BLs, H c ⊥ /H orb c ⊥ (0) nearly coincides with theWHH curve. This has been attributed to the enhancedPauli limiting field that well exceeds the orbital limitingfield ( H Pauli c ⊥ (cid:29) H orb c ⊥ ).In contrast to CeCoIn /YbCoIn andCeCoIn /CeRhIn , H c ⊥ /H orb c ⊥ (0) only show a slightincrease from CeCoIn (7)/CeIn (13) superlattice atambient pressure from that of bulk CeCoIn singlecrystal. This indicates that H c ⊥ is dominated byPauli paramagnetic effect, i.e. H c ⊥ ≈ H Pauli c ⊥ (cid:28) H orb c ⊥ .This implies that the effect of local inversion symmetrybreaking on the superconductivity in CeCoIn /CeIn is weak compared with CeCoIn /YbCoIn . The localinversion symmetry is broken for the CeCoIn /YbCoIn on the CoIn-layer while it is broken on the Ce layerfor CeCoIn /CeIn and CeCoIn /CeRhIn . Therefore,the present results suggest that the inversion symmetrybreaking on the CoIn-layer induces a larger local electricfield gradient. Moreover, superconducting electrons inCeCoIn BLs are not strongly influenced by the AFMorder in CeIn BLs compared with CeCoIn /CeRhIn .When superconductivity is dominated by the Pauli-limiting effect ( H c ⊥ ≈ H Pauli c ⊥ ), 2∆ /k B T c is obtainedfrom Eq. (1) as 2∆ k B T c ≈ √ gµ B H c ⊥ k B T c . (7)where µ B is Bohr magneton and g is the g -factor of elec-tron. In Fig. 17, g = 2 is assumed. Figure 17 depictsthe pressure dependence of q = √ gµ B H c ⊥ /k B T c forCeCoIn /CeRhIn and CeCoIn /CeIn , along with q forbulk CeCoIn single crystal. Here g = 2 is assumed. Al-though this simple assumption should be scrutinized, the fact that q = 4 . is larger than theBCS value of q = 3.54 is consistent with the strong cou-pling superconductivity, which is supported by the spe-cific heat measurements that report 2∆ /k B T c ≈ q with pressure in CeCoIn /CeRhIn im-plies the increase of 2∆ /k B T c . This increase has beenattributed to an enhancement of the force binding su-perconducting electron pairs. In this case, an increaseof 2∆ /k B T c occurs without accompanying a large en-hancement of T c , which is consistent with the results ofCeCoIn /CeRhIn [48]. Thus, the critical AFM fluctu-ations that develop in CeRhIn BLs near the QCP areinjected into the CeCoIn BLs through the interface andstrongly enhance the pairing interaction in CeCoIn BLs.
FIG. 17: Pressure dependence of q = √ gµ B H c ⊥ /k B T c ≈ /k B T c for CeCoIn (7)/CeIn (13) superlattice. Forcomparison, q of bulk CeCoIn single crystal andCeCoIn (5)/CeRhIn (5) are plotted. In stark contrast to CeCoIn /CeRhIn superlattices, q decreases with pressure in bulk CeCoIn single crystal.This implies that the pairing interaction is weakened byapplying pressure, which is consistent with the fact thatthe pressure moves the system away from the QCP ofCeCoIn . The reduction of 2∆ /k B T c with pressure inbulk CeCoIn single crystals is confirmed by the jump ofthe specific heat at T c [57]. It should be stressed thatthe pressure dependence of q in CeCoIn (7)/CeIn (13)is very similar to that of bulk CeCoIn . This stronglyindicates that the pairing interactions in CeCoIn BLsare barely influenced by AFM fluctuations injected fromthe adjacent CeIn BLs through the interface even whenCeIn BLs are located near the AFM QCP.The most salient feature in the CeCoIn /CeIn super-lattices is that the superconductivity of CeCoIn BLs islittle affected by the critical AFM fluctuations in CeIn BLs, despite the fact that AFM fluctuations are injectedfrom the adjacent CeIn BLs into CeCoIn BLs, as evi-denced by the AFM order in CeCoIn /CeIn demonstrat-ing that different CeIn BLs are magnetically coupled by4the RKKY interaction through adjacent CeCoIn BLs.Even in the vicinity of the AFM QCP of the CeIn BLs,the superconducting state in the CeCoIn BLs is verysimilar to that of CeCoIn bulk single crystals. This indi-cates that the AFM fluctuations injected from CeIn BLsdo not help to enhance the force binding the supercon-ducting electron pairs in CeCoIn BLs. This is in starkcontrast to CeCoIn /CeRhIn , in which the pairing forcein CeCoIn BL is strongly enhanced by the AFM fluctu-ations in CeRhIn BLs [48], although the CeRhIn BLsare magnetically only weakly coupled through CeCoIn BLs.We note that the superconducting phase appears un-der pressure in CeRhIn single crystals and epitaxial thinfilms. On the other hand, in the CeRhIn /YbRhIn , zeroresistivity is not attained even under pressure [48]. Thisresult indicates that the superconductivity of CeRhIn issuppressed when the thickness of the BLs was reduced.Similarly, in CeCoIn /CeRhIn superlattices, 2D super-conductivity derived from the CeCoIn BLs is thought tobe realized from ambient pressure to under pressure nearQCP.
D. Contrasting behaviors between CeCoIn /CeIn and CeCoIn /CeRhIn superlattices As discussed in the previous sections, CeCoIn /CeIn and CeCoIn /CeRhIn superlattices exhibit contrastingsuperconducting and magnetic properties. We point outthat there are two possible important factors that deter-mine whether magnetic fluctuations are injected throughthe interface; One is the magnetic wave vector and theother is the matching of the Fermi surface between twomaterials.For CeCoIn , the Fermi surface is 2D-like and AFMfluctuations with wave vector q = (0.45, 0.45, 0.5) aredominant [59]. The magnetic wave vector in the orderedphase of CeIn is commensurate with q = (0.5, 0.5, 0.5)[72]. The evolution of the ordered moment below T N isconsistent with mean field theory. While the wave num-ber along the c axis, q c , of CeIn is the same as thatof CeCoIn , the 3D Fermi surface of CeIn is very dif-ferent from the 2D Fermi surface of CeCoIn . On theother hand, for CeRhIn , q in the ordered phase is in-commensurate q = (0.5, 0.5, 0.297) at low pressure [34]and changes to q = (0.5, 0.5, 0.4) above ∼ . q c of CeRhIn is different from the q c of CeCoIn . The evolution of the ordered moment below T N deviates from mean field behavior, likely due to 2Dfluctuations. However, the 2D Fermi surface of CeRhIn bears a close resemblance to that of CeCoIn .The equality between the c axis component of q inCeCoIn and CeIn would explain why the magneticcoupling between CeIn BLs through a CeCoIn BL isstronger than that between CeRhIn BLs. Thus, AFMorder is formed in CeCoIn (7)/CeIn ( n ) even for small n , for which the AFM order has already vanished in CeCoIn ( n )/CeRhIn ( n ). In magnetically mediated su-perconductors, the pairing interaction is expected to bestrongly wave number dependent. Considering the goodresemblance of the Fermi surface and the same d x − y superconducting gap symmetry of CeCoIn and CeRhIn [102], it is likely that the pairing interaction in both com-pounds has 2D character and peaks around the samewave number on the Fermi surface. Furthermore, it hasbeen assumed that 2D magnetic fluctuations are strongin CeRhIn . Thus, superconductivity in the CeCoIn BLs of CeCoIn ( n )/CeRhIn ( n ) is strongly influenced.On the other hand, in CeIn with 3D Fermi surface,2D AFM fluctuations are expected to be very weak.AFM fluctuations having 3D character in CeIn maynot play an important role for the pairing interaction inCeCoIn , resulting in little change of the superconduc-tivity in CeCoIn /CeIn . VII. CONCLUSION
We reviewed the most recent advances of Kondo su-perlattices containing atomic layers of strongly corre-lated heavy fermion superconductor CeCoIn with d -wave symmetry, CeCoIn /YbCoIn , CeCoIn /CeRhIn and CeCoIn /CeIn grown by using a state-of-the-artMBE technique. In these Kondo superlattices, super-conducting heavy electrons are confined within the 2DCeCoIn block layers and interact with the neighboringnonmagnetic or magnetic layers through the interface.In CeCoIn /YbCoIn superlattices, the superconduc-tivity is strongly influenced by the local ISB at theinterface, which seriously reduces the Pauli paramag-netic pair-breaking effect. Our results demonstrate thatthe tricolor Kondo superlattices provide a new play-ground for exploring exotic superconducting states in thestrongly correlated 2D electron systems with the Rashbaeffect.CeCoIn /CeRhIn and CeCoIn /CeIn superlattices,the superconducting and antiferromagnetic states coexistin spatially separated layers, but their mutual couplingvia the interface significantly modifies the superconduct-ing and magnetic properties. In CeCoIn /CeRhIn su-perlattices, the superconductivity in the CeCoIn BLs isprofoundly affected by AFM fluctuations in the CeRhIn BLs. Upon suppressing the AFM order by applied pres-sure, the force binding superconducting electron pairsacquires an extreme strong coupling nature, highlightingthat the pairing interaction can be maximized by the crit-ical fluctuations emanating from the magnetic QCP. InCeCoIn /CeIn superlattices, each CeIn BL is magneti-cally coupled by RKKY interaction through the adjacentCeCoIn BLs. In stark contrast to CeCoIn /CeRhIn superlattices, the superconductivity in the CeCoIn BLsin CeCoIn /CeIn superlattices is barely influenced bythe AFM fluctuations in the CeIn BLs, even when theCeIn BLs are tuned in the vicinity of the AFM QCP bypressure. The striking difference between the two Kondo5superlattices provides direct evidence that 2D AFM fluc-tuations are essentially important for the pairing inter-actions in CeCoIn .Finally, we describe the future prospect of the Kondosuperlattices. Cd- and Hg-doped CeCoIn are known toshow anomalous antiferromagnetic order at small dopinglevels[111–113]. The fabrication of superlattices consist-ing of alternating layers of Cd- and Hg-doped CeCoIn and pure CeCoIn is expected to provide important infor-mation on the interplay between superconductivity andunusual magnetic order. The advantage of these super-lattices is that the mismatch of the lattice constant isvery small. Recently, strongly correlated electron sys-tems with strong Rashba interaction are attracting muchattention[114–116]. Thus fabricating superlattices in-cluding the noncentrosymmetric heavy fermion supercon-ductors CePt Si[117] and pressure-induced superconduc-tors CeRhSi [118], CeIrSi [119], and heavy fermion su-perconductors whose inversion symmetry is broken lo-cally around the Ce site CeRh As [120] will give im-portant information of the interplay between supercon-ductivity and strong Rashba interaction. We also notethat the Dzyaloshinskii–Moriya interaction plays an im-portant role for the Kondo effect and magnetic order atthe interface between different materials. Theoretical re-search in this direction appears to be ongoing[121, 122],but the impact on the f -electron superlattice is not clearat the moment.Besides the Kondo superlattices, the present MBEtechnique enables us to grow monolayer CeCoIn andCeRhIn on the top of the nonmagnetic YbCoIn thinfilm. It has been proposed theoretically that monolayerCeCoIn forms a topological superconducting state owingto the strong Rashba interaction[98, 99, 123, 124]. Mostof the past research on topological superconductivity hasfocused on weakly correlated materials. The topologi-cal superconductivity in monolayer CeCoIn will give anopportunity to study topological excitations, includingMajorana fermions in strongly correlated d -wave super- conductors. Moreover, it is an open question whethermonolayer CeRhIn exhibits magnetic order. It is highlychallenging to study these issues by using the scanningtunneling microscopy measurements.The fabrication of a wide variety of Kondo superlat-tices paves a new way to study the entangled relation-ship between unconventional superconductivity and mag-netism in strongly correlated electron systems, offering aroute to exploring the emergence of novel superconduct-ing systems and the roles of their interface. Acknowledgments
The authors acknowledge collaborations withYongkang Luo, P. F. S. Rosa, F. Ronning, J. D.Thompson, K. Ishida, T. Yamanaka, G. Nakamine, S.Miyake, S. Nakamura, T. Ishii, R. Peters, Y. Tokiwa,Y. Kasahara, and T. Shibauchi. We thank H. Kon-tani and Y. Yanase for valuable discussion. Thiswork was supported by Grants-in-Aid for ScientificResearch (KAKENHI) (No. 25220710, No. 18J10553,and No. 18H05227) and Innovative Areas ”TopologicalMaterial Science” (No. JP15H05852) and ”3D Active-Site Science” (No. 26105004) from the Japan Societyfor the Promotion of Science (JPSJ) and JST CREST(No. JP-MJCR18T2).
Data Availability
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