Coupling between heavy fermion superconductor CeCoIn$_5$ and antiferromagnetic metal CeIn$_3$ through the atomic interface

TL;DR

This study investigates the interaction between superconductivity and antiferromagnetic order in Kondo superlattices of CeCoIn$_5$ and CeIn$_3$, revealing that AFM fluctuations have minimal impact on superconductivity, emphasizing the importance of 2D AFM fluctuations for pairing.

Contribution

It demonstrates that AFM fluctuations injected through the interface do not significantly influence superconductivity in CeCoIn$_5$, highlighting the role of 2D AFM fluctuations in pairing mechanisms.

Findings

AFM order in CeIn$_3$ layers is magnetically coupled via RKKY interaction.

Superconductivity in CeCoIn$_5$ layers is barely affected by AFM fluctuations in CeIn$_3$ layers.

AFM fluctuations are crucial for pairing, but their influence is limited across the interface.

Abstract

To study the mutual interaction between unconventional superconductivity and magnetic order through an interface, we fabricate Kondo superlattices consisting of alternating layers of heavy-fermion superconductor CeCoIn$_5$ and antiferromagnetic (AFM) heavy-fermion metal CeIn$_3$. The strength of the AFM fluctuations is tuned by applying hydrostatic pressure to CeCoIn$_5(m)$/CeIn$_3(n)$ superlattices with $m$ and $n$ unit-cell-thick layers of CeCoIn$_5$ and CeIn$_3$, respectively. Superconductivity in CeCoIn$_5$ and AFM order in CeIn$_3$ coexist in spatially separated layers. At ambient pressure, N\'{e}el temperature $T_N$ of the CeIn$_3$ block layers (BLs) of CeCoIn$_5$(7)/CeIn$_3(n)$ shows little dependence on $n$, in contrast to CeIn$_3(n)$/LaIn$_3$(4) superlattices where $T_N$ is strongly suppressed with decreasing $n$. This suggests that each CeIn$_3$ BL is magnetically coupled by the RKKY interaction through the adjacent CeCoIn$_5$ BL and a 3D magnetic state is formed. With applying pressure to CeCoIn$_5$(7)/CeIn$_3$(13), $T_N$ of the CeIn$_3$ BLs is suppressed up to 2.4 GPa, showing a similar pressure dependence as CeIn$_3$ single crystals. An analysis of upper critical field reveals that the superconductivity in the CeCoIn$_5$ BLs is barely influenced by the AFM fluctuations in the CeIn$_3$ BLs, even when the CeIn$_3$ BLs are in the vicinity of the AFM quantum critical point. This is in stark contrast to CeCoIn$_5$/CeRhIn$_5$ superlattice where the superconductivity in the CeCoIn$_5$ BLs is profoundly affected by AFM fluctuations in the CeRhIn$_5$ BLs. The present results show that although AFM fluctuations are injected into the CeCoIn$_5$ BLs from the CeIn$_3$ BLs through the interface, they barely affect the force which binds superconducting electron pairs. These results demonstrate that 2D AFM fluctuations are essentially important for the pairing interactions in CeCoIn$_5$.

Equations

$$\rho (T)={\rho }_0+A{T}^{\epsilon }.$$

$$\frac{2\Delta }{k_B{T}_c}\approx \sqrt{2}\frac{g{\mu }_B{H}_{c2\perp }}{k_B{T}_c}.$$

Full text

††preprint:

I Introduction

It is well established that in several compound families, such as high-$T_{c}$ cuprates, iron pnictides/chalcogenides, and heavy-fermion compounds, Cooper pairs are not bound together through phonon exchange but instead through exchange of some other kind, such as spin fluctuationsSigrist ; Bennemann ; Thalmeier ; Scalapino ; Stewart2017 ; Keimer2015 ; Hirschfeld ; Kontani . Despite tremendous efforts, however, the interplay between unconventional superconductivity and magnetism still remains largely unexplored in these systems. This includes fascinating electronic phases, where superconductivity and antiferromagnetic (AFM) order, involving the same charge carriers, coexists, and the important question why superconductivity is often strongest near a quantum critical point (QCP) where the AFM order vanishes in the zero temperature limit and spin fluctuations become singular Mathur1998 ; Shibauchi2014 ; Hashimoto2012 ; Knebel2008 ; Park2006 .

By using a recent state-of-the-art molecular beam epitaxy (MBE) technique, we grow artificial Kondo superlattices with alternating layers of heavy-fermion superconductors and conventional metals or heavy-fermion AFM compounds Shishido2010 ; Shimozawa2016 . These Kondo superlattices provide unique opportunity to study the mutual interactions between the unconventional superconducting state and magnetically ordered- or conventional metallic-states through the atomic interface and thereby seek answers to the above-mentioned questions. Until now, several types of Kondo superlattices containing the heavy-fermion superconductor CeCoIn5 Petrovic2001 with a layered structure have been fabricated Mizukami2011 ; Goh2012 ; Shimozawa2014 ; Ishii2016 ; Naritsuka2017 ; Naritsuka2018 . CeCoIn5 has a quasi-two dimensional (2D) Fermi surfaceSettai2007 and the presence of quasi-2D AFM fluctuations has been reported in the normal state Kawasaki2003 ; Raymond2015 . Furthermore, a superconducting gap with $d_{x^{2}-y^{2}}$-wave symmetry has been observed by a variety of experiments Izawa2001 ; An2010 ; Matsuda2006 ; Stock2008 ; Zhou2013 ; Zhou2013 ; Allan2013 . The superconducting state is strongly Pauli limited, as demonstrated by a first-order phase transition at upper critical fields for directions parallel and perpendicular to the $ab$ plane Izawa2001 ; Tayama2002 ; Bianchi2002 ; Shimahara . It is a prototypical system, in which non-Fermi liquid behaviors in the normal state and unconventional superconductivity are thought to arise from the proximity to a AFM QCP Sidorov2002 ; Nakajima2007 ; Sarro2007 . Under pressure, CeCoIn5 moves away from the QCP and Fermi liquid behavior is recovered.

It has been shown that in superlattices consisting of alternating layers of CeCoIn5 and the conventional metal YbCoIn5 with atomic layer thicknesses (Fig. 1a), the Pauli pair-breaking effect is strongly suppressed from that in the bulk of CeCoIn5 single crystalsGoh2012 ; Shimozawa2014 . Site-selective nuclear magnetic resonance (NMR) measurements on CeCoIn5/YbCoIn5 superlattices have reported that AFM fluctuations in the CeCoIn5 block layers (BLs), particularly in the vicinity of the interface, are weakenedYamanaka2015 . These results have been attributed to the local inversion symmetry breaking at the interface, which results in spin-split Fermi surfaces and thus effectively suppresses the Zeeman effectGoh2012 ; Maruyama2012 ; Shimozawa2014 .

In superlattices consisting of alternating layers of CeCoIn5 and the heavy-fermion AFM metal CeRhIn5 (Fig. 1b), the superconducting- and AFM-states coexist in spatially separated layers. In these superlattices, the influence of the local inversion symmetry breaking at the interface has been shown to be less important compared to CeCoIn5/YbCoIn5. In sharp contrast to CeCoIn5/YbCoIn5, NMR measurements have revealed that magnetic fluctuations in CeCoIn5 BLs of CeCoIn${}_{5}/$CeRhIn5 superlattices are enhanced compared to bulk CeCoIn5 single crystal, highlighting the importance of the magnetic proximity effect Nakamine2019 . In particular, it has been pointed out that in the vicinity of the QCP of CeRhIn5 BLs, AFM fluctuations are enhanced and the force binding superconducting electron-pairs acquires an extremely strong-coupling nature. This indicates that superconducting pairing can be manipulated by magnetic fluctuations injected through the interface Naritsuka2018 .

To obtain further insight into the mutual interactions between unconventional superconductivity and magnetic order, we here fabricate superlattices consisting of alternating layers of CeCoIn5 and the AFM metal CeIn3 (Fig. 1c). CeIn3 is an isotropic Kondo lattice material with cubic crystal structure. In bulk CeIn3 single crystals, AFM order with ordered magnetic moment of 0.48 $\mu_{B}$ occurs at $T_{N}$=10 K, where $\mu_{B}$ is the Bohr magneton Benoit1980 . With applying pressure, $T_{N}$ decreases and vanishes at $\sim$2.6 GPa, indicating an AFM QCP. Superconductivity with a maximum $T_{c}\approx$200 mK is induced in a very narrow pressure range around the QCP Mathur1998 ; Knebel2001 .

Our results reveal that, similar to CeCoIn5/CeRhIn5 but in contrast to CeCoIn5/YbCoIn5 superlattices, the local inversion symmetry breaking at the interface has only little effect on the superconductivity in CeCoIn5/CeIn3 superlattices. However, we find that the magnetic and the superconducting properties in CeCoIn5/CeIn3 are in marked contrast to those in CeCoIn5/CeRhIn5 superlatticesNaritsuka2018 . Although the AFM fluctuations are injected to the CeCoIn5 BLs from the CeIn3 BLs through the interfaces, they barely affect the electron pairing interactions in the CeCoIn5 BLs. These results provide compelling evidence that 2D AFM fluctuations are essentially important for the superconductivity in CeCoIn5.

II Experimental Details

The hybrid superlattices CeCoIn5(7)/CeIn3($n$) ($n$=3, 4, 6 and 13) with $c$ axis oriented structure are grown on a MgF2 substrate by MBE technique Shishido2010 ; Shimozawa2016 . We first grow $\sim$20 unit-cell-thick (UCT) CeIn3 ($\sim$10 nm) as a buffer layer on MgF2. Then 7-UCT CeCoIn5 and $n$-UCT CeIn3 ($n$=3, 4, 6 and 13) are grown alternatively with total thicknesses of approximately 200 nm. As the epitaxial growth temperature of CeCoIn5 and CeIn3 layers are different, CeCoIn5 and CeIn3 BLs were grown at 570 and 420 °C, respectively. The superlattice is capped with $\sim$5 nm Co to prevent oxidation. Streak pattern of the reflection high-energy electron diffraction (RHEED) image shown in Fig. 2(a) have been observed during the whole growth of the superlattices, indicating good epitaxy. The atomic force microscope measurements reveal that the surface roughness is within $\pm$1 nm, which is comparable to 1-2 UCT along the $c$ axis of the constituents. Because atomically flat regions extend over distances of $\sim$0.1 $\mu$m, it can be expected that transport properties are not seriously influenced by the roughness. Figure 2(b) displays a high-resolution cross-sectional transmission electron microscope (TEM) image along the (100) direction for the CeCoIn5(7)/CeIn3(13) superlattice. A clear interface between the CeCoIn5 and the CeIn3 layers is observed. Figure 2(c) displays an electron energy loss spectroscopy (EELS) image of the same superlattice. The EELS images clearly resolve the 7-UCT CeCoIn5 and the 13-UCT CeIn3 BLs, demonstrating sharp interfaces with no atomic interdiffusion between the neighboring CeCoIn5 and CeIn3 BLs. Figure 2(d) shows the X-ray diffraction patterns for CeCoIn5/CeIn3($n$) superlattices. The shoulder structure shown by the red arrows near the [003] peak of CeCoIn5 (blue arrows) is consistent with the superlattice structure. These results demonstrate the successful fabrication of epitaxial superlattices with sharp interfaces. High-pressure resistivity measurements have been performed under pressure up to 2.4 GPa using a piston cylinder cell with Daphne oil 7373 as the pressure transmitting medium. The pressure has been measured by the $T_{c}$ of Pb.

III Results

Figure 3(a) depicts the temperature dependence of the resistivity $\rho$ of CeCoIn5(7)/CeIn($n$) superlattices with $n$=3, 4, 6 and 13. We also show $\rho$ of CeCoIn5 and CeIn3 thin films grown by MBE. The resistivity of CeCoIn5(7)/CeIn($n$) superlattices follows the typical heavy-fermion behavior. With decreasing temperature, $\rho(T)$ increases below $\sim$150 K due to the Kondo scattering but then begins to decrease due strong $c$-$f$ hybridization between $f$-electrons and conduction ($c$) band electrons, leading to the narrow $f$-electron band at the Fermi level. Figures  3(b)-3(f) depict $\rho(T)$ at low temperatures. All superlattices show the superconducting transition at $\approx$1.5 K. For the $n$=3- and 4-superlattices, $\rho(T)$ exhibit a slight downward curvature. Figures 3(g)-3(k) display the temperature derivative of the resistivity $d\rho(T)/dT$. As shown by the arrows in Fig. 3(g), $d\rho(T)/dT$ of CeIn3 thin film exhibits a distinct kink at $T_{N}$=10 K Benoit1980 . Similar kink structures are observed in all superlattices at the temperatures indicated by arrows, showing the AFM transition.

Figure 4 shows the thickness dependence of $T_{N}$ of the CeCoIn5(7)/CeIn3($n$) superlattices. For comparison, the data sets of CeIn${}_{3}(4)$/LaIn3($n$), where LaIn3 is a nonmagnetic conventional metal with no $f$-electrons Shishido2010 , and CeCoIn5($n$)/CeRhIn5($n$) are also included in the figure. Remarkably, the observed thickness dependence of $T_{N}$ in CeCoIn5/CeIn3 is in striking contrast to that in CeIn3/LaIn3; While $T_{N}$ is strongly suppressed with decreasing $n$ and vanishes at $n$=2 in CeIn3/LaIn3, $T_{N}$ is nearly independent of $n$ in CeCoIn5(7)/CeIn3($n$). This suggests that CeIn3 BLs are coupled weakly by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions through the adjacent LaIn3 BL, but they can strongly couple through the adjacent CeCoIn5 BL. This is even more surprising, as the distance between different CeIn3 BLs is larger in the CeCoIn5(7)/CeIn3($n$) superlattices than in the CeIn3($n$)/LaIn3(4) superlattices. We thus conclude that small but finite magnetic moments are induced in CeCoIn5 BLs in CeCoIn5/CeIn3, which mediate the RKKY-interaction. On the other hand, because of the absence of strongly interacting $f$-electrons in LaIn3, which can form magnetic moments, the RKKY interaction in CeIn3/LaIn3 can be expected to be much weaker. To clarify this, a microscopic probe of magnetism, such as NMR measurements, is required. We note that as shown in Fig. 4, the reduction of $T_{N}$ is also observed in CeCoIn5($n$)/CeRhIn5($n$) superlattices Naritsuka2018 , suggesting that the RKKY interaction between CeRhIn5 BLs through adjacent CeCoIn5 BL is negligibly small. This is supported by the recent site-selective NMR measurements which report no discernible magnetic moments induced in the CeCoIn5 BLs in CeCoIn5/CeRhIn5 Nakamine2019 .

The pressure dependence of the superconducting and magnetic properties provide crucial information on the mutual interaction between superconductivity and magnetism through the interface. Figures 5(a) and 5(b) and their insets show the temperature dependence of $\rho(T)$ under pressure for CeCoIn5(7)/CeIn3($n$) for $n$=13 and 6, respectively. With the application of pressure, the temperature at which $\rho(T)$ shows its maximum increases due to the enhancement of the $c$-$f$ hybridizationNakajima2007 . As shown in the insets, both superlattices undergo a superconducting transition under pressure. Figures 5(c)-5(e) and 5(f)-5(h) show $d\rho(T)/dT$ under pressure for $n$=13 and 6, respectively. Clear kink structure associated with the AFM transition can be seen in the data.

Figure 6(a) depicts the pressure dependence of $T_{N}$ and $T_{c}$ for CeCoIn5(7)/CeIn3($n$) superlattices for $n$=6 and 13. With applying pressure, $T_{N}$ decreases rapidly. For comparison, $T_{N}$ of a bulk single crystal CeIn3 is also shown by the solid line Mathur1998 . The pressure dependence of $T_{N}$ of both superlattices are very similar to that of the bulk CeIn3 single crystal. In bulk CeIn3 crystal, the AFM QCP is located at $p_{c}\approx 2.6$ GPa. It is natural to expect, therefore, that the AFM QCP of the superlattices is close to 2.6 GPa. Thus, at 2.4 GPa, the superlattices are in the vicinity of the AFM QCP. This is supported by the temperature dependence of the resistivity under pressure. The resistivity can be fitted as

[TABLE]

Figure 6(b) shows the pressure dependence of $\varepsilon$ obtained from $d\ln\Delta\rho/d\ln T$, where $\Delta\rho=\rho(T)-\rho_{0}$. The magnitude of $\varepsilon$ decreases with pressure. In bulk CeIn3 single crystal, $\varepsilon$ decreases with pressure and exhibits a minimum at the AFM QCPMathur1998 ; Knebel2001 . On the other hand, applying pressure to CeCoIn5 leads to an increase of $\varepsilon$, which is attributed to the suppression of the non-Fermi liquid behavior, $\rho(T)\propto T$, and the development of a Fermi liquid state with its characteristic $\rho(T)\propto T^{2}$ dependenceSidorov2002 ; Nakajima2007 . Therefore, the reduction of $\varepsilon$ with pressure arises from the CeIn3 BLs, indicating that the CeIn3 BLs approach the AFM QCP.

As shown in Fig. 6(a), $T_{c}$ increases, peaks at $\sim$ 1.8 GPa, and then decreases when applying pressure. This pressure dependence bears resemblance to that of CeCoIn5 bulk single crystals Sidorov2002 . An analysis of the upper critical field provides important information about the superconductivity of CeCoIn5 BLs. Figure 7 depicts the temperature dependence of the upper critical field determined by the midpoint of the resistive transition in a magnetic field $\bm{H}$ applied parallel ($H_{c2\parallel}$) and perpendicular ($H_{c2\perp}$) to the layers. The inset of Fig. 7 shows the anisotropy of the upper critical fields $H_{c2\parallel}/H_{c2\perp}$ at ambient pressure. The anisotropy diverges on approaching $T_{c}$. This is in sharp contrast to the CeCoIn5 thin film, whose anisotropy is nearly temperature independent up to $T_{c}$. The observed diverging anisotropy indicates that the superconducting electrons are confined in the 2D CeCoIn5 BLs. In fact, in 2D superconductivity, $H_{c2\parallel}$ is limited by Pauli paramagnetic pair breaking and increases as $\sqrt{T_{c}-T}$, while $H_{c2\perp}$ increases as $T_{c}-T$ due to the orbital pair breaking near $T_{c}$ Mizukami2011 . Moreover, the thickness of the CeCoIn5 BL is comparable to the coherence length perpendicular to the layer, $\xi_{c}\sim 4$ nm. Thus each 7-UCT CeCoIn5 BL effectively behaves as a 2D superconductor.

IV Discussion

It has been revealed that the temperature dependence of $H_{c2\perp}$ provides crucial information about the impact of the interface on the superconductivity in CeCoIn5 BLs. In particular, the modification of the Pauli paramagnetic effect in the superlattice, which dominates the pair breaking in bulk CeCoIn5 single crystals, gives valuable cluesGoh2012 ; Shimozawa2014 ; Naritsuka2017 ; Naritsuka2018 . Figure 8(a) and 8(b) depict the $T$ dependence of the $H_{c2\perp}$ of CeCoIn5(7)/CeIn3(13) superlattice, normalized by the orbital-limited upper critical field at zero temperature, $H_{c2\perp}^{orb}(0)$, which is obtained from the Werthamer-Helfand-Hohenberg (WHH) formula, $H_{c2\perp}^{orb}(0)=-0.69T_{c}(dH_{c2\perp}/dT)_{T_{c}}$WHH . In Figs. 8(a) and 8(b), two extreme cases are also included; the WHH curve with no Pauli pair-breaking and $H_{c2}/H_{c2\perp}^{orb}(0)$ for bulk CeCoIn5 single crystal Tayama2002 . For comparison, $H_{c2\perp}^{orb}(0)$ for CeCoIn5/YbCoIn5 and CeCoIn5/CeRhIn5 are also shownMizukami2011 ; Naritsuka2018 .

At ambient pressure, $H_{c2\perp}/H_{c2\perp}^{orb}(0)$ of CeCoIn5/YbCoIn5 and CeCoIn5/CeRhIn5 are strongly enhanced from that of CeCoIn5 bulk single crystals, indicating the suppression of the Pauli paramagnetic pair-breaking effect. However, it has been pointed out that the mechanisms of this suppression in these two systems are essentially different. In CeCoIn5/YbCoIn5, the enhancement of $H_{c2\perp}/H_{c2\perp}^{orb}(0)$ is caused by the local inversion symmetry breaking at the interface Goh2012 ; Maruyama2012 . The asymmetry of the potential perpendicular to the 2D plane of the superlattice, $\nabla V\parallel$[001], induces the Rashba spin-orbit interaction $\alpha_{R}={\bm{g}}({\bm{k}})\cdot{\bm{\sigma}}\propto({\bm{k}}\times\nabla V)\cdot{\bm{\sigma}}$, where ${\bm{g}}({\bm{k}})=(k_{y},-k_{x},0)/k_{F}$, $k_{F}$ and ${\bm{\sigma}}$ are the Fermi wave number and the Pauli matrices, respectively. The Rashba spin-orbit interaction splits the Fermi surface into two sheets with different spin texturesBauer2012 . The energy splitting is given by $\alpha_{R}$, and the spin direction is tilted into the 2D plane, rotating clockwise on one sheet and anticlockwise on the other. When the Rashba splitting exceeds the superconducting gap energy ($\alpha_{R}>\Delta$), the superconducting state is dramatically modifiedMaruyama2012 ; Bauer2012 ; Fujimoto2007 . In particular, when the magnetic field is applied perpendicular to the 2D plane, the magnetic field does not couple to the spins, leading to a suppression of the Pauli pair-breaking effect. At $p$=2.2 GPa, $H_{c2\perp}/H_{c2\perp}^{orb}(0)$ of CeCoIn5/YbCoIn5 nearly coincides with the WHH curve. This indicates that $H_{c2\perp}$ is dominated by the orbital pair breaking most likely due to the suppression of the Pauli paramagnetic pair-breaking effect by the Rashba splitting.

On the other hand, in CeCoIn5/CeRhIn5 superlattices, it has been shown that the effect of the local inversion symmetry breaking on $H_{c2\perp}$ is less important compared with CeCoIn5/YbCoIn5 Naritsuka2018 . It has been proposed that magnetic fluctuations (paramagnons) in CeRhIn5 BLs injected through the interface dramatically enhance the force binding superconducting electron pairs in CeCoIn5 BLs, leading to the enhancement of $\Delta$. As a result, the Pauli limiting field $H_{c2\perp}^{Pauli}(=\sqrt{2}\Delta/g\mu_{B}$) is enhanced, where $g$ is the $g$-factor of the electrons. This increases the relative importance of the orbital pair-breaking effect, giving rise to the enhancement of $H_{c2\perp}/H_{c2\perp}^{orb}(0)$ Naritsuka2018 . At $p$=2.1 GPa, which is close to the AFM QCP of CeRhIn5 BLs, $H_{c2\perp}/H_{c2\perp}^{orb}(0)$ nearly coincides with the WHH curve. This has been attributed to the enhanced Pauli limiting field that well exceeds the orbital limiting field ($H_{c2\perp}^{Pauli}\gg H_{c2\perp}^{orb}$) Naritsuka2018 .

In contrast to CeCoIn5/YbCoIn5 and CeCoIn5/CeRhIn5, $H_{c2\perp}/H_{c2\perp}^{orb}(0)$ is only slightly enhanced in CeCoIn5(7)/CeIn3(13) superlattice at ambient pressure from that of bulk CeCoIn5 single crystal. This indicates that $H_{c2\perp}$ is dominated by Pauli paramagnetic effect, i.e. $H_{c2\perp}\approx H_{c2\perp}^{Pauli}\ll H_{c2\perp}^{orb}$. This implies that the effect of local inversion symmetry breaking on the superconductivity in CeCoIn5/CeIn3 is weak compared with CeCoIn5/YbCoIn5. The local inversion symmetry is broken for the CeCoIn5/YbCoIn5 on the CoIn-layer while it is broken on the Ce layer for CeCoIn5/CeIn3 and CeCoIn5/CeRhIn5. Therefore, the present results suggest that the inversion symmetry breaking on the CoIn-layer induces a larger local electric field gradient. Moreover, superconducting electrons in CeCoIn5 BLs are not strongly influenced by the AFM order in CeIn3 BLs compared with CeCoIn5/CeRhIn5.

When superconductivity is dominated by the Pauli-limiting effect ($H_{c2\perp}\approx H_{c2\perp}^{Pauli}$), $2\Delta/k_{B}T_{c}$ is estimated as

[TABLE]

Figure 9 depicts the pressure dependence of $q=\sqrt{2}g\mu_{B}H_{c2\perp}/k_{B}T_{c}$ for CeCoIn5/CeRhIn5 and CeCoIn5/CeIn3, along with $q$ for bulk CeCoIn5 single crystal. Here $g$=2 is assumed. We note that $q=4.2$ of the bulk CeCoIn5 is smaller than the value determined by the specific heat measurements $2\Delta/k_{B}T_{c}\approx$ 6 Petrovic2001 , but is larger than the BCS value of $q=3.54$, which is consistent with the strong coupling superconductivity. The increase of $q$ with pressure in CeCoIn5/CeRhIn5 implies the increase of $2\Delta/k_{B}T_{c}$. This increase has been attributed to an enhancement of the force binding superconducting electron pairs. In spin fluctuation mediated superconductors, the pairing interaction is mainly provided by high-energy fluctuations while low-energy fluctuations act as pair breaking. In this case, an increase of $2\Delta/k_{B}T_{c}$ occurs without accompanying a large enhancement of $T_{c}$, which is consistent with the results of CeCoIn5/CeRhIn5 Naritsuka2018 . Thus, the critical AFM fluctuations that develop in CeRhIn5 BLs near the QCP are injected into the CeCoIn5 BLs through the interface and strongly enhance the pairing interaction in CeCoIn5 BLs.

In stark contrast to CeCoIn5/CeRhIn5 superlattices, $q$ decreases with pressure in bulk CeCoIn5 single crystal. This implies that the pairing interaction is weakened with applying pressure, which is consistent with the fact that the pressure moves the system away from the QCP of CeCoIn5. The reduction of $2\Delta/k_{B}T_{c}$ with pressure in bulk CeCoIn5 single crystals is confirmed by the jump of the specific heat at $T_{c}$ Knebel2004 . It should be stressed that the pressure dependence of $q$ in CeCoIn5(7)/CeIn3(13) is very similar to that of bulk CeCoIn5. This strongly indicates that the pairing interactions in CeCoIn5 BLs are barely influenced by AFM fluctuations injected from the adjacent CeIn3 BLs through the interface even when CeIn3 BLs are located near the AFM QCP.

The most salient feature in the CeCoIn5/CeIn3 superlattices is that the superconductivity of CeCoIn5 BLs is little affected by the critical AFM fluctuations in CeIn3 BLs, despite the fact that AFM fluctuations are injected from the adjacent CeIn3 BLs into CeCoIn5 BLs, as evidenced by the AFM order in CeCoIn5/CeIn3 demonstrating that different CeIn3 BLs are magnetically coupled by the RKKY interaction through adjacent CeCoIn5 BLs. Even in the vicinity to the AFM QCP of the CeIn3 BLs, the superconducting state in the CeCoIn5 BLs is very similar to that of CeCoIn5 bulk single crystals. This indicates that the AFM fluctuations injected from CeIn3 BLs do not help to enhance the force binding the superconducting electron pairs in CeCoIn5 BLs.

This is in stark contrast to CeCoIn5/CeRhIn5, in which the pairing force in CeCoIn5 BL is strongly enhanced by the AFM fluctuations in CeRhIn5 BLsNaritsuka2018 , although the CeRhIn5 BLs are magnetically only weakly coupled through CeCoIn5 BLs. We point out that these contrasting behaviors can be attributed to the differences of the magnetic and electronic properties of CeRhIn5 and CeIn3. The magnetic wave vector in the ordered phase of CeIn3 is commensurate $\bm{q}_{0}$=(0.5,0.5,0.5)Benoit1980 . The evolution of the ordered moment below $T_{N}$ is consistent with mean field theory. On the other hand, the magnetic wave vector in the ordered phase of CeRhIn5 is incommensurate $\bm{q}_{0}$=(0.5,0.5,0.297)Bao2000 . The evolution of the ordered moment below $T_{N}$ deviates from mean field behavior, likely due to 2D fluctuations. In CeCoIn5, AFM fluctuations with wave vector $\bm{q}_{f}$=(0.45, 0.45, 0.5) are dominantRaymond2015 . Thus, the $c$ axis component of $\bm{q}_{f}$ in CeCoIn5 is commensurate and has the same value as that of $\bm{q}_{0}$ in CeIn3. On the other hand, the $c$ axis component of $\bm{q}_{0}$ in CeRhIn5 is incommensurate and very different from that of $\bm{q}_{0}$ in CeCoIn5.

The equality between the $c$ axis component of $\bm{q}_{f}$ in CeCoIn5 and $\bm{q}_{0}$ in CeIn3 would explain why the magnetic coupling between CeIn3 BLs through CeCoIn5 BL is stronger than that between CeRhIn5 BLs. Thus, AFM order is formed in CeCoIn5(7)/CeIn3($n$) even for small $n$, for which the AFM order has already vanished in CeCoIn5($n$)/CeRhIn5($n$). In magnetically mediated superconductors, the pairing interaction is expected to be strongly wave number dependent. Considering that the quasi-2D Fermi surface of CeCoIn5 bears a close resemblance to that of CeRhIn5 and the superconducting pairing state of both compounds is $d_{x^{2}-y^{2}}$Park2008 , it is likely that the pairing interaction in both compounds has 2D character and peaks around the same wave number. Furthermore, it has been assumed that 2D magnetic fluctuations are strong in CeRhIn5. Thus, superconductivity in the CeCoIn5 BLs of CeCoIn5($n$)/CeRhIn5($n$) is strongly influenced. On the other hand, AFM fluctuations having 3D character in CeIn3 may not play an important role for the pairing interaction in CeCoIn5, resulting in little change of the superconductivity in CeCoIn5/CeIn3.

V Summary

A state-of-the-art MBE technique has enabled us to fabricate superlattices consisting of different heavy-fermion compounds. These Kondo superlattices provide a unique opportunity to study the mutual interaction between unconventional superconductivity and magnetic order through the atomic interface. In hybrid Kondo superlattice CeCoIn5/CeIn3, the superconductivity in CeCoIn5 BLs and AFM order in CeIn3 BLs coexist in spatially separated layers. We find that each CeIn3 BL is magnetically coupled by the RKKY interaction through adjacent CeCoIn5 BLs. An analysis of the upper critical field under pressure reveals that the superconductivity in CeCoIn5 BLs is little influenced even in the presence of abundant AFM fluctuations in the vicinity of the AFM QCP of adjacent CeIn3 BLs. Thus, although the AFM fluctuations are injected to the CeCoIn5 BLs from the CeIn3 BLs through the interfaces, they barely influence the force binding superconducting electron pairs. This is in sharp contrast to CeCoIn5/CeRhIn5, in which the superconductivity in the CeCoIn5 BLs are strongly influenced by quantum critical AFM fluctuations in CeRhIn5 BLs.

It has been widely believed that 2D AFM fluctuations are important for the pairing interaction in CeCoIn5. However, direct evidence was lacking. The present results provide strong support that 2D AFM fluctuations are essentially important for the unconventional superconductivity in CeCoIn5.

Acknowledgements

We thank K. Ishida, H. Kontani, and Y. Yanase for fruitful discussions. This work was supported by Grants-in-Aid for Scientific Research (KAKENHI) (Nos. 25220710, 15H02014, 15H02106, 17K18753, 18H05227, 18J10553), and on Innovative Areas ‘Topological Material Science’ (No. JP15H05852) and ‘3D Active-Site Science’ (No. 26105004) from Japan Society for the Promotion of Science (JSPS). M. N. also acknowledges support from a JSPS Fellows.