NMR Detection of Interlayer Spin Correlations in High-Tc Cuprate Superconductors
【Research Topic】NMR Detection of Interlayer Spin Correlations in High-Tc Cuprate Superconductors
ー Overview ー
Since the discovery of high-temperature superconductivity in the cuprates in 1986, the CuO2 planes have been regarded as the central stage for strongly correlated electron physics. Nuclear magnetic resonance (NMR) has played a particularly important role in probing the low-energy spin dynamics of these systems and contributed significantly to the discovery and characterization of the pseudogap phenomenon.
During the 1990s, however, NMR studies of multilayer cuprates revealed a puzzling behavior. In compounds such as YBa2Cu3O7−δ(YBCO), the spin-lattice relaxation rate 1/T1T decreases below the pseudogap temperature, whereas the Gaussian spin-echo decay rate 1/T2G increases in the same temperature range. Both quantities probe antiferromagnetic spin correlations, yet they appeared to convey contradictory information. This discrepancy became one of the unresolved issues in the NMR community of that period. The problem was later explained in terms of an interlayer spin susceptibility, χ12, describing magnetic correlations between neighboring CuO2 planes. Subsequently, NMR experiments on trilayer cuprates provided direct support for this interpretation. Although the original motivation was to understand anomalous NMR behavior in the pseudogap regime, these studies also addressed a more fundamental question: How can interlayer spin correlations be experimentally detected in strongly correlated layered materials?
In recent years, the discovery of superconductivity in the bilayer nickelate La3Ni2O7 has renewed interest in interlayer exchange interactions J⊥, interlayer pairing, and dimensional crossover phenomena. Modern bilayer t-J-J⊥ models emphasize the importance of interlayer degrees of freedom in determining low-energy electronic properties.
This article revisits the history of interlayer spin correlations in multilayer cuprates from the perspective of NMR, discusses the theoretical and experimental developments that led to their identification, and considers their relevance in the current context of bilayer correlated-electron materials.
1. Introduction
Since the discovery of high-temperature superconductivity in cuprates in 1986, the CuO2 plane has been regarded as the central stage of strongly correlated electron physics. Electronic transport, antiferromagnetic correlations, and superconductivity are largely governed by the electronic states confined within these quasi-two-dimensional planes. Consequently, much of the theoretical understanding of cuprate superconductors has been based on two-dimensional Hubbard or t-J models. Real materials, however, are not strictly two-dimensional. Many cuprates contain two or three neighboring CuO2 planes within a single structural unit.
・Bilayer cuprates: YBa2Cu3O7-δ (YBCO), YBa2Cu4O8 (Y124), Bi2212, Tl2212 etc.
・Trilayer cuprates: HgBa2Ca2Cu3O8 (Hg1223), Bi2223, Tl2223, etc.
This raises a natural question:
Are neighboring CuO2 planes really independent, or do interlayer magnetic correlations play an important role in determining the low-energy properties of multilayer cuprates?
Although interlayer exchange interactions are generally much weaker than in-plane interactions, a number of experimental observations in the 1990s suggested that they could not be ignored. In particular, NMR experiments on multilayer cuprates revealed anomalous behavior that could not be understood within a simple single-layer picture. These observations eventually led to the concept of an interlayer spin susceptibility, χ12, describing magnetic correlations between neighboring CuO2 planes [9–13].
The purpose of this article is to review the historical development of this idea, from early experimental indications and theoretical proposals to its experimental verification, and to discuss its possible relevance to modern bilayer correlated-electron systems such as the nickelate superconductor La3Ni2O7 [14,15].
2. What Does NMR Measure? A Probe of Antiferromagnetic Spin Fluctuations
To understand the problem of interlayer correlations, it is useful to briefly review what information NMR provides in strongly correlated electron systems. Nuclear magnetic resonance is a local probe that detects electronic spin fluctuations through the hyperfine magnetic fields experienced by atomic nuclei. In cuprate superconductors, NMR has served as one of the most powerful tools for investigating low-energy magnetic excitations and the pseudogap phenomenon [1–3]. Two NMR observables are particularly important in the present context.
・Spin-lattice relaxation rate, 1/T1T:
The spin-lattice relaxation time T1 characterizes how rapidly the nuclear spin system returns to thermal equilibrium after being perturbed. The quantity 1/T1T is proportional to low-energy spin fluctuations and is determined by the imaginary part of the dynamic spin susceptibility at the NMR frequency. It therefore probes the dynamic component of antiferromagnetic spin correlations.
・Gaussian spin-echo decay rate, 1/T2G:
The Gaussian component of the spin-echo decay reflects indirect nuclear spin-spin interactions mediated by the electronic spin system. In cuprates, 1/T2G is closely related to the spatial development of antiferromagnetic correlations and depends primarily on the real part of the static spin susceptibility. It therefore probes the static component of spin correlations.
In a simplified picture, both quantities are sensitive to antiferromagnetic correlations, but they emphasize different aspects of the same magnetic system:
・1/T1T: dynamic spin fluctuations
・1/T2G: static spin correlations
This distinction became particularly important in multilayer cuprates, where the two observables exhibited markedly different temperature dependences in the pseudogap regime. The origin of this discrepancy became one of the central questions in NMR studies of high-Tc superconductors during the 1990s [4–5].
3. The Puzzle of the 1990s: Diverging Temperature Dependences of 1/T1T and 1/T2G
During the 1990s, NMR experiments on multilayer cuprates such as YBCO and YBa2Cu4O8 revealed an intriguing anomaly in the pseudogap regime [4–5]. Below the pseudogap temperature,
・1/T1T decreases with decreasing temperature, indicating a suppression of low-energy spin fluctuations.
・In contrast, 1/T2G increases, implying that static antiferromagnetic spin correlations continue to develop.
At first sight, these observations appeared contradictory. Both quantities probe antiferromagnetic spin correlations, and within a simple single-layer picture one would expect them to exhibit broadly similar temperature dependences. Instead, the dynamic component of the spin response was suppressed while the static component became stronger. This discrepancy became one of the major unresolved issues in NMR studies of high-Tc cuprates. It suggested that the conventional two-dimensional description might be incomplete and that an additional degree of freedom beyond intralayer spin correlations could be involved. A theoretical framework capable of reconciling the opposite temperature dependences of 1/T1T and 1/T2G emerged shortly thereafter through the introduction of the interlayer spin susceptibility χ12, which forms the subject of the following sections.
4. Theoretical Approach: Interlayer Correlations Amplify Weak Coupling
To resolve the apparent contradiction between the behaviors of 1/T1T and 1/T2G, attention gradually shifted from purely two-dimensional spin correlations to the possible role of magnetic coupling between neighboring CuO2 planes. Although the interlayer exchange interaction J⊥ is generally much weaker than the intralayer exchange interaction J, several theoretical and experimental studies in the mid-1990s suggested that its effects could be significantly enhanced by strong antiferromagnetic correlations developing within the planes [6–11].
A particularly influential contribution was made by Jun-ichiro Kishine, who investigated magnetically coupled bilayer cuprates in a series of theoretical studies [9–11]. The essential idea was that even a weak interlayer exchange interaction can be strongly amplified by the enhancement of intralayer antiferromagnetic spin fluctuations. As a consequence, interlayer magnetic correlations may become important even when the bare value of J⊥ is relatively small [9–11]. In particular, Jun-ichiro Kishine analyzed the influence of interlayer magnetic coupling on low-energy spin excitations and on the Gaussian spin-echo decay rate 1/T2G [10]. These studies demonstrated that interlayer magnetic correlations can modify the magnetic response of bilayer cuprates in a nontrivial manner and should not be regarded merely as small perturbative corrections to an otherwise two-dimensional system [9–11].
At the same time, experimental evidence for interlayer magnetic coupling was emerging. R. Stern and coworkers employed NQR spin-echo double resonance (SEDOR) measurements on Y2Ba4Cu7O15 and directly detected magnetic coupling between adjacent CuO2 planes. They further showed that the relative importance of the interplane coupling increases upon cooling [7]. Subsequently, A. Suter and coworkers studied interplane electronic spin-polarization transfer in the superconducting state of the same material. Their measurements revealed strong interlayer antiferromagnetic correlations in the normal state and a substantial suppression of interlayer spin-polarization transfer below Tc [8]. These observations provided compelling evidence that interlayer magnetic correlations are a real and experimentally observable aspect of multilayer cuprates [8].
Taken together, these theoretical and experimental developments established the possibility that interlayer magnetic correlations constitute an essential ingredient in understanding the anomalous magnetic behavior of multilayer cuprates. The remaining challenge was to formulate a quantitative framework capable of connecting these correlations directly to experimentally measured NMR relaxation rates. This issue was addressed in the work discussed in the next section [12].
5. The 1998 Theory: Geometrical Effects of Hyperfine Couplings and the Resolution of the NMR Puzzle
Building upon the theoretical work on magnetically coupled bilayer cuprates by Jun-ichiro Kishine and coworkers, the 1998 study by A. Goto and T. Shimizu introduced a quantitative framework for understanding the anomalous NMR behavior observed in multilayer cuprates [12]. The theory addressed how interlayer spin correlations are reflected in experimentally measurable NMR relaxation rates. A crucial aspect of the theory is the role of the hyperfine coupling geometry. When neighboring CuO2 planes fluctuate in an out-of-phase manner, the hyperfine fields at the nuclear sites are affected differently in various NMR observables. As a consequence, the interlayer contribution enters the expressions for 1/T1T and 1/T2G with opposite tendencies.
Specifically,
・The interlayer susceptibility suppresses 1/T1T through geometrical cancellation of the low-energy dynamic hyperfine fields at the nuclear site.
・The same interlayer susceptibility enhances 1/T2G because the static spin correlations contributing to the indirect nuclear spin-spin interaction are added constructively.
The theory therefore provided a natural explanation for the long-standing discrepancy observed experimentally in multilayer cuprates. The apparent contradiction between the decrease of 1/T1T and the increase of 1/T2G does not imply two different magnetic phenomena. Rather, both behaviors originate from the same interlayer magnetic correlations, observed through different hyperfine-coupling channels [12]. An additional strength of the work was that the formalism was extended beyond bilayer systems and generalized to trilayer cuprates. This made it possible to analyze multilayer materials within a unified framework and provided a basis for subsequent experimental tests in Hg1223 and related compounds [12,13]. Perhaps the most important conceptual outcome was that the interlayer spin susceptibility χ12 was transformed from a theoretical quantity into an experimentally accessible observable. By comparing 1/T1T and 1/T2G, it became possible to estimate the strength of interlayer magnetic correlations directly from NMR measurements.
Schematic illustration of intra- and interlayer spin susceptibilities in multilayer cuprates. The interlayer exchange interaction J⊥ gives rise to an interlayer spin susceptibility χ12, which contributes differently to the NMR relaxation rates 1/T1T and 1/T2G and accounts for the characteristic NMR anomaly in multilayer cuprates [12].
6. Experimental Verification in a Trilayer Cuprate
A theoretical explanation, however appealing, requires experimental verification. Such a test became possible in the trilayer cuprate HgBa2Ca2Cu3O8+δ (Hg1223), where the inequivalent CuO2 planes can be distinguished directly by NMR spectroscopy [13]. In a trilayer structure, the CuO2 planes are not equivalent. Two outer planes (OP) surround a crystallographically distinct inner plane (IP), resulting in different local electronic environments and different NMR signals. This unique feature makes trilayer cuprates particularly suitable for investigating interlayer magnetic correlations. Using detailed spin-echo measurements on the inner-plane Cu site, it was found that the enhancement of 1/T2G observed in the pseudogap regime could not be explained solely by intralayer spin correlations. Instead, the experimental results were naturally understood when interlayer spin correlations between the inner and outer CuO2 planes were included in the analysis. The observed angular dependence of the spin-echo decay provided direct evidence that interlayer magnetic correlations contribute significantly to the nuclear spin-spin relaxation process. In particular, the results demonstrated that the same interlayer spin susceptibility responsible for the anomalous behavior of 1/T2G is experimentally detectable through NMR measurements.
These findings provided strong support for the theoretical framework proposed in Ref. [12]. More broadly, they established that interlayer spin correlations are not merely theoretical constructs but experimentally observable quantities in multilayer cuprates. The combination of the 1998 theoretical formulation and the 2002 experimental verification therefore offered a consistent picture of how interlayer magnetic correlations manifest themselves in NMR relaxation phenomena [12,13].
7. Interlayer Physics Revisited: Connections to Bilayer Nickelates
In recent years, interlayer physics has reemerged as a central topic in the field of strongly correlated electron systems following the discovery of superconductivity in the bilayer nickelate La3Ni2O7 under high pressure [14]. In contrast to the cuprates, where interlayer exchange interactions have traditionally been regarded as weak perturbations to the dominant in-plane physics, a growing body of theoretical work has suggested that interlayer coupling may play a fundamentally active role in the nickelates [14,15]. Particular attention has been focused on bilayer t-J-J⊥ models, in which the interplay between intralayer correlations and interlayer exchange interactions can be investigated explicitly [14,15]. These studies indicate that sufficiently strong interlayer exchange coupling J⊥ can stabilize interlayer pairing and even drive a crossover from intralayer superconductivity to interlayer superconductivity [15]. In this sense, interlayer degrees of freedom are no longer viewed merely as secondary corrections to an essentially two-dimensional system, but as potentially essential elements of the low-energy physics itself [14,15].
Of course, cuprates and nickelates are not identical systems. Their electronic configurations, orbital structures, Hund-coupling effects, and the microscopic origins of interlayer interactions differ in important ways. Consequently, the physics of multilayer cuprates cannot simply be transferred to La3Ni2O7 without qualification. Nevertheless, the underlying question is remarkably similar:
Can weak or moderate interlayer interactions fundamentally alter the low-energy properties of a layered correlated-electron system?
This question motivated much of the work on interlayer spin correlations in multilayer cuprates during the 1990s, and it has reappeared today in the context of bilayer nickelates. From this historical perspective, the study of interlayer spin susceptibility χ12 may be viewed not merely as an explanation of a specific NMR anomaly, but as part of a broader effort to understand how dimensionality and interlayer coupling influence emergent quantum states in strongly correlated materials [9–15].
Modern interest in bilayer superconductors therefore provides an opportunity to revisit earlier ideas developed for multilayer cuprates. While the materials themselves are different, the concept that interlayer coupling may constitute an essential degree of freedom rather than a minor perturbation remains highly relevant today [14,15].
8. Concluding Remarks
The studies discussed here were not originally intended to establish a general theory of high-temperature superconductivity. Instead, they were motivated by a specific puzzle in multilayer cuprates: the apparently contradictory temperature dependences of the two NMR observables, 1/T1T and 1/T2G [12,13]. The key outcome of this work was the recognition that interlayer magnetic correlations can manifest themselves in NMR relaxation processes through the interlayer spin susceptibility χ12. By taking into account the geometrical effects of hyperfine couplings, it became possible to understand why 1/T1T decreases while 1/T2G increases in the pseudogap regime of multilayer cuprates [12]. Furthermore, NMR experiments on the trilayer compound Hg1223 provided direct support for this interpretation and demonstrated that interlayer spin correlations are experimentally accessible quantities rather than merely theoretical constructs [13].
From a broader perspective, the significance of interlayer correlations extends beyond a specific NMR anomaly. In real layered materials, weak interlayer interactions may strongly influence low-energy properties and effective dimensionality. In this sense, interlayer correlations should not always be regarded as minor perturbations but may constitute an essential degree of freedom in understanding multilayer correlated-electron systems [9–11]. A complementary viewpoint is provided by the Mermin–Wagner theorem. In an ideal two-dimensional Heisenberg antiferromagnet, long-range magnetic order cannot occur at finite temperature. Real cuprates, however, do exhibit finite-temperature antiferromagnetic ordering in their parent compounds. This implies that finite interlayer couplings, however small, play a physically important role in establishing the effective dimensionality of the system. The existence and magnitude of such couplings therefore become experimentally relevant quantities rather than merely theoretical parameters. In this context, the NMR detection of interlayer spin susceptibility may be viewed as an attempt to quantify how real multilayer materials depart from the strictly two-dimensional limit. Although the layer dependence of the superconducting transition temperature Tc is understood today from multiple perspectives, including inhomogeneous carrier doping and disorder effects rather than interlayer interactions alone, the detection and quantitative evaluation of interlayer spin correlations represented a pioneering attempt to capture the microscopic physical reality of weak interlayer coupling in layered materials. Interlayer spin correlations provide a measurable manifestation of the weak but finite interlayer interactions that connect individual CuO2 planes into a three-dimensional crystal.
The recent emergence of bilayer nickelate superconductors has brought renewed attention to the role of interlayer exchange coupling and interlayer pairing. Although the microscopic physics of cuprates and nickelates differs in important respects, both classes of materials raise a similar question: how strongly can interlayer interactions reshape the low-energy electronic states of a nominally layered system? Modern studies of bilayer t-J-J⊥ models suggest that this question remains highly relevant today [14,15].
References
NMR Basics
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[2] C. P. Slichter, Principles of Magnetic Resonance (Springer).
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NMR Studies of High-Temperature Superconductors
[4] C. P. Slichter, Magnetic Resonance Studies of High Temperature Superconductors, in Handbook of High-Temperature Superconductivity: Theory and Experiment, edited by J. R. Schrieffer and J. S. Brooks (Springer, New York, 2007), pp. 215–256.
[5] H. Alloul, NMR in Correlated Electron Systems: Illustration on the Cuprates, in Quantum Materials: Experiments and Theory, Lecture Notes of the Autumn School Correlated Electrons (Forschungszentrum Jülich, 2016).
[6] R. Stern, M. Mali, J. Roos, and D. Brinkmann, Spin pseudogap and interplane coupling in Y2Ba4Cu7O15: A 63Cu nuclear spin-spin relaxation study, Phys. Rev. B 51, 15478 (1995).
[7] R. Stern, M. Mali, J. Roos, and D. Brinkmann, Interplane coupling in the superconductor Y₂Ba₄Cu₇O₁₅ as revealed by NQR spin-echo double resonance, Phys. Rev. B 52, R15734 (1995).
[8] A. Suter, M. Mali, J. Roos, and D. Brinkmann, Interplane Electronic Spin Polarization Transfer in the Superconducting State of Y₂Ba₄Cu₇O₁₅ as revealed by NQR Spin-Echo Double Resonance, Phys. Rev. Lett. 82, 1309 (1999).
Theoretical Background of Interlayer Correlations
[9] J. Kishine, Spin Fluctuations in Magnetically Coupled Bilayer Cuprates, Ph.D. Thesis, University of Tokyo (1996).
[10] J. Kishine, Nuclear Gaussian Spin-Echo Decay in Magnetically Coupled Bilayer Cuprates, J. Phys. Soc. Jpn. 66, 1229 (1997).
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The Present Work
[12] A. Goto and T. Shimizu, Roles of the interlayer spin correlations in the Cu NMR relaxation rates of bi- and tri-layered high-Tc cuprates, Phys. Rev. B 57, 7977 (1998).
[13] A. Goto, W. G. Clark, P. Vonlanthen, K. B. Tanaka, T. Shimizu, K. Hashi, P. V. P. S. S. Sastry, and J. Schwartz, Origin of the Enhanced Copper Spin Echo Decay Rate in the Pseudogap Regime of the Multilayer High-Tc Cuprates, Phys. Rev. Lett. 89, 127002 (2002).
Modern Developments
[14] Xing-Zhou Qu, Dai-Wei Qu, Jialin Chen, Congjun Wu, Fan Yang, Wei Li, and Gang Su, Bilayer t-J-J⊥ Model and Magnetically Mediated Pairing in the Pressurized Nickelate La3Ni2O7, Phys. Rev. Lett. 132, 036502 (2024).
[15] Yuan Yang, Xin Lu, Yuan Wan, Wei-Qiang Chen, and Shou-Shu Gong, Evolution from intralayer to interlayer superconductivity in a bilayer t-J-J⊥ ladder, Phys. Rev. B 114, 225122 (2026).
Note: This article is a historical perspective on the authors’ NMR studies of multilayer cuprates conducted during the 1990s and early 2000s, viewed in the context of recent developments in bilayer nickelate superconductors. It does not claim a direct historical lineage between cuprate and nickelate research; rather, it discusses both within the common framework of interlayer correlations in strongly correlated electron systems. For scientific citation, readers are encouraged to consult the original publications cited throughout the text.
© 2026 Atsushi Goto. All rights reserved. Last updated: October 2026.
