## Abstract

At the interface between two distinct materials, desirable properties, such as superconductivity, can be greatly enhanced^{1}, or entirely new functionalities may emerge^{2}. Similar to in artificially engineered heterostructures, clean functional interfaces alternatively exist in electronically textured bulk materials. Electronic textures emerge spontaneously due to competing atomic-scale interactions^{3}, the control of which would enable a top*-*down approach for designing tunable intrinsic heterostructures. This is particularly attractive for correlated electron materials, where spontaneous heterostructures strongly affect the interplay between charge and spin degrees of freedom^{4}. Here we report high-resolution neutron spectroscopy on the prototypical strongly correlated metal CeRhIn_{5}, revealing competition between magnetic frustration and easy-axis anisotropy—a well-established mechanism for generating spontaneous superstructures^{5}. Because the observed easy-axis anisotropy is field-induced and anomalously large, it can be controlled efficiently with small magnetic fields. The resulting field-controlled magnetic superstructure is closely tied to the formation of superconducting^{6} and electronic nematic textures^{7} in CeRhIn_{5}, suggesting that in situ tunable heterostructures can be realized in correlated electron materials.

## Main

The role of interfaces in enhancing or creating functionality is twofold; interfaces exhibit reduced dimensionality, which is known to significantly influence electronic, magnetic and optical properties^{8}. Furthermore, crossed response functions can arise from the interplay of two distinct order parameters at the interface, and lead to entirely new properties. This is successfully utilized in bottom*-*up approaches to device design. For example, semiconductor heterostructures can be grown with clean, atomically flat interfaces, the basis for applications in electronics and quantum optics^{9}. Due to the intrinsic coupling between various order parameters, heterostructures grown from strongly correlated electron materials are a promising path towards new generations of devices, as highlighted by recent discoveries^{1,2,8}. However, despite some impressive initial success, controlling these interfaces remains a significant challenge, precisely due to the underlying complexity^{8}. Interestingly, this complexity is also what holds the key to a top*-*down approach for realizing high-quality interfaces. The complex ground states of strongly correlated electron materials arise from the competition between two or more atomic-scale interactions, often leading to superstructures, which we propose to exploit as intrinsic heterostructures.

We show that heavy electron metals—that is, prototypical strongly correlated electron materials—are exceptional model systems to investigate intrinsic heterostructures. Here a frustrated Ruderman–Kittel–Kasuya–Yosida (RKKY) exchange interaction between localized *f*-electrons, which frequently favours spiral order, directly competes with a substantial easy-axis anisotropy enabled by the large spin–orbit interaction of lanthanide-based materials. The minimal model describing this competition is the axial next-nearest-neighbour Ising (ANNNI) Hamiltonian^{5}, which shows that the conflict of frustration and anisotropy is universally resolved via the formation of modulated superstructures with applications in hard and soft matter.

As illustrated in Fig. 1a–c, in heavy electron metals the formation of a magnetic superstructure may also have important consequences for the electronic ground state. The presence of an additional Kondo interaction favours screening of *f*-electron magnetic moments by conduction electrons, leading to heavy electronic quasiparticles with an enhanced electronic density of states (DOS). Due to this strong coupling between spin and charge, the underlying magnetic superstructure is likely to induce a spatially modulated electronic texture (Fig. 1b, c). Given that the period *λ* of the magnetic superstructure is highly sensitive to external control parameters, our top-down approach offers the advantage that the electronic heterostructure can be tuned in situ.

We demonstrate that a surprisingly small magnetic field of 2 T induces a substantial uniaxial magnetic anisotropy in the magnetically frustrated heavy electron material CeRhIn_{5}, resulting in the formation of a field-tunable magnetic heterostructure. CeRhIn_{5} is a tetragonal antiferromagnet (AFM), with Néel temperature *T*_{N} = 3.8 K at ambient pressure and zero magnetic field. Increasing pressure enhances the Kondo interaction via a growing overlap of neighbouring Ce 4 *f* orbitals, eventually leading to the complete suppression of the Ce magnetic moments at a magnetic quantum critical point (QCP) at *P*_{c = }2.25 GPa, around which a broad superconducting dome emerges (Fig. 1d)^{10}. Remarkably, in CeRhIn_{5}, part of the superconducting phase is textured (TSC in Fig. 1d)^{6}. In a strikingly similar fashion, the AFM phase may also be suppressed by a magnetic field **H**, resulting in a QCP at *H*_{c} = 50 T, regardless of field direction^{11}. Near this QCP, a new phase unstable towards the formation of an electronic nematic texture was recently discovered for *H* > *H*^{*} = 28 T (Fig. 1e). An arbitrarily small in-plane field component breaks the rotational symmetry of the electronic structure, suggesting a surprisingly large nematic susceptibility^{7}.

Interestingly, small in-plane fields also break the rotational symmetry of the AFM state, suggesting that electronic and magnetic textures are indeed related. Due to magnetic frustration arising from competing antiferromagnetic nearest- (NN) and next-nearest-neighbour (NNN) RKKY exchange along the *c*-axis^{12}, the AFM order at low fields (AFM I in Fig. 2a,c) is an incommensurate spin spiral propagating along the *c*-axis with propagation vector **k**_{I} = (1/2 1/2 0.297), which conserves in-plane rotational symmetry^{13}. However, for **H****⊥***c*, a spin-flop transition occurs above the critical field ${H}_{\mathrm{c}}^{\mathrm{III}}=2.1\phantom{\rule{0.3em}{0ex}}\mathrm{T}$ (see Figure 2a)^{14}, where the Ce moments align perpendicular to **H**, forming a commensurate collinear square-wave phase, with propagation vector **k**_{III} = (1/2 1/2 1/4), suggesting a large magnetic-field-induced in-plane easy-axis anisotropy.

To elucidate the role of this field-induced easy-axis anisotropy, we investigate the magnetic interactions of CeRhIn_{5} using neutron spectroscopy. This reveals that the magnetic interactions of CeRhIn_{5} for in-plane fields are remarkably well described by the effective spin model Hamiltonian

which is related to the ANNNI model^{5}. **S**_{
i
} in equation (1) is a spin-1/2 operator representing the effective magnetic moment of the ${\mathrm{\Gamma}}_{7}^{2}$ crystal field doublet. We note that the Hamiltonian in equation (1) is valid for **H** applied in the tetragonal basal plane, and adopt the convention that $\mathit{H}\parallel \left(1\stackrel{\u0304}{1}0\right)$, so that the easy-axis anisotropy is along (110) (Fig. 2c). Our previous *H* = 0 study^{12} revealed that the magnetic excitations are accurately described by $\mathcal{H}$(*δ* = 0,*h* = 0) with only three exchange constants *J*_{
ij
}: a NN exchange in the tetragonal basal plane, *J*_{0}, and two NN and NNN exchange interactions along *c, J*_{1} and *J*_{2}, that, in combination with an easy-plane anisotropy *Δ* > 0 in the basal plane, generate the spiral ground state (see Figure 2c). Two additional ingredients are required to include field dependence: a conventional Zeeman term (final term in equation (1)) and a field-dependent easy-axis exchange anisotropy favouring spin alignment perpendicular to **H**, described by the dimensionless parameter *δ*. The above effective spin-1/2 Hamiltonian can be obtained by projecting the crystal field eigenstates onto the lowest-energy doublet. The exchange anisotropy arises a priori from changes in the orbital character of the Ce 4 *f* electronic wavefunction with *H*, where its strength is expected to be substantial due to the large spin–orbit coupling for Ce and vary as *δ*(*H*) = *I*_{
δ
}*H*^{2}.

In Fig. 3, we show the full spin excitation spectrum of CeRhIn_{5} as measured in the AFM III phase at *H* = 7 T (Fig. 2a,c), along the three principal directions (*h*00), (*hh*0), and (00 *l*), centred at the commensurate magnetic zone centre at **k**_{III} = (1/2 1/2 1/4), with additional fields presented in the Supplementary Information. Comparing data sets at various magnetic fields reveals a clear field-induced increase in the spin gap *Δ*_{S} at **k**_{III}. Figure 4a presents *Δ*_{S} as function of *H* extracted from energy cuts through the spin wave spectra shown in Fig. 3 at **k**_{III}. The dynamic susceptibility *χ*″(**q**,*ω*) (see Figure 3d–f and Supplementary Information), and the corresponding spin-wave dispersion is obtained from a large-*S* expansion:

Here *J*_{1q} and ${J}_{0,2\mathbf{q}}^{A,N}$ are the Fourier transformation of the exchange parameters (Supplementary Eqs. S10–S14 Supplementary Information), each consisting of the exchange integrals *J*_{0}, *J*_{1} and *J*_{2}, easy-plane anisotropy *Δ* > 0, and easy-axis anisotropy *δ*, introduced in equation (1).

The dashed lines in Fig. 3 illustrate exemplary fits of *χ*″(**q**,*ω*) to our data, performed for every *H*, showing that the Hamiltonian in equation (1) describes our data quantitatively (Supplementary Information). Due to the small size of the magnetic Brillouin zone along the *c* direction, Umklapp scattering occurs at the zone boundary, resulting in additional spin wave branches, $\hslash {\omega}_{\mathbf{q}\pm {\mathbf{k}}_{\mathrm{III}}}$^{12}. The easy-plane anisotropy was fixed to *Δ* = 0.82, as determined at *H* = 0^{12}, and assumed to be field-independent; additional fit details are provided in the Methods section. The resulting size of *δ* and exchange integrals as a function of *H* are shown in Fig. 4b, c. Within AFM III, the parameters change smoothly with *H; J*_{0}, *J*_{1} and *J*_{2} decrease, in agreement with the decreasing bandwidth of the spectrum, and *δ* increases in accordance with the growing spin gap. We note that the ratio of *J*_{2}/*J*_{1} remains unchanged for all fields, indicating that the magnetic frustration is not affected by the applied magnetic field. Finally, as demonstrated by the red solid line in Fig. 4b, we find *δ*(*H*) = *I*_{
δ
}*H*^{2} with *I*_{
δ
} = 0.0013(1) (T^{−}^{2}). This implies that the experimental critical exchange anisotropy at ${H}_{\mathrm{c}}^{\mathrm{III}}$ is *δ*_{
c
} = 0.0057(5). By comparison, the critical exchange anisotropy calculated via mean-field modelling of the Hamiltonian in equation (1) (Supplementary Information), ${\delta}_{\mathrm{c}}^{\mathrm{MF}}=0.0091$, agrees well with the experiment, which is remarkable considering that our model assumes *f*-electron localization in CeRhIn_{5} (Supplementary Information), and that the mean field treatment neglects the effects of quantum fluctuations. Although the gap ${\Delta}_{\mathrm{S}}=\hslash {\omega}_{{\mathit{k}}_{\mathrm{III}}}=\sqrt{2\delta \left(2{J}_{0}+{J}_{2}\right)\left[\left(2{J}_{0}+{J}_{2}\right)\left(1+\delta +\Delta \right)-{J}_{1}\Delta \right]}$ is the clearest indicator of increasing uniaxial anisotropy, it is also sensitive to the field-dependent exchange integrals *J*. By inserting interpolated values for the exchange integrals and *δ* we obtain the dashed line in Fig. 4a, demonstrating that our fits to the dynamic susceptibility quantitatively describe the observed spin gap for $H>{H}_{\mathrm{c}}^{\mathrm{III}}$. We note that an unexpected, small spin gap *Δ*_{S} ≈ 0.25 meV was observed at *H* = 0, but likely represents the longitudinal (or Higgs) mode that arises due to Kondo screening of the Ce magnetic moments, as explained in the Supplementary Information(this scenario assumes that there is still a gapless transverse mode). Recent neutron diffraction measurements demonstrate that the Ce magnetic form factor is significantly different from free Ce^{3+}, with a magnetic moment that is reduced by 41% with respect to the expectation from the crystal field ground state, suggesting that the Kondo interaction in CeRhIn_{5} is indeed substantial, in agreement with this scenario^{13}.

As we show now, CeRhIn_{5} exhibits an instability towards the formation of highly-tunable modulated magnetic superstructures. Using the exchange constants shown in Fig. 4c, and ${\delta}_{c}^{\mathrm{MF}}$, we obtain the theoretical temperature versus magnetic field phase diagram for CeRhIn_{5} shown in Fig. 2b, based on our spin Hamiltonian and a mean-field calculation (Supplementary Information). In addition to the remarkable agreement with the experimental phase diagram, it reveals a prominent feature of the ANNNI model, namely that the superstructure period is highly-tunable in proximity to *T*_{N}^{5}. Notably, critical magnetic fluctuations immediately below *T*_{N} compete with the uniaxial anisotropy, which causes a softening of the pinning of the magnetic moments along (110), ultimately leading to a magnetic structure with moments primarily along (110), but with small components parallel to **H**. This high-temperature phase (AFM II) is represented by an elliptical helix in which the size of the moments is modulated (see Figure 2a,c)^{14}. Our model predicts a change of the magnetic propagation vector **k**_{II} = (1/2 1/2 *l*) as a function of both *H* and *T*. For the ANNNI model, the temperature dependence is given by $\mathrm{\Delta}l\left(T\right)\propto -1\u2215\mathrm{ln}\left(T-{T}_{\mathrm{c}}^{\mathrm{III}}\right)$(ref. ^{5}), with *l* = ¼ at $T={T}_{\mathrm{c}}^{\mathrm{III}}$ (critical temperature between AFM II and III), and slowly approaching the value dictated by NN and NNN exchange interactions along *c, l* = 0.297 for *T* ⟶ *T*_{N}. In Fig. 2d we show that *l*(*T*) at *H* = 3.5T, as determined via high-resolution neutron diffraction, indeed changes logarithmically, illustrating the ease with which the superstructure period *λ* = 2π/*k* may be tuned.

The instability towards this highly-tunable magnetic heterostructure is apparent throughout the entire temperature–field–pressure phase diagram, with significant impact on material properties. Transport measurements show that the AFM II phase continues to exist at pressures approaching the QCP^{15}. Further, even for *H* = 0, the magnetic propagation vector changes from **k**_{1} = (1/2 1/2 0.326) to **k**_{2} = (1/2 1/2 0.391) near the phase boundary between textured and bulk superconducting states (indicated by the arrows in Fig. 1d)^{16}. Here the textured superconductivity is suggested to arise due to the coexistence of **k**_{1} and **k**_{2} magnetic domains, where the superconductivity only nucleates in **k**_{2}^{6}. This may be explained via the mechanism shown in Fig. 1a–c, where **k**_{1} and **k**_{2} magnetic superstructures each induce distinct electronic textures; however, with only one of them being compatible with the superconducting order parameter. This notably highlights that the tunable period of the magnetic heterostructure in CeRhIn_{5} enables the control of material properties.

Similarly, invoking the mechanism discussed in Fig. 1a–c for the field-induced nematic phase (Fig. 1e), an underlying modulated magnetic superstructure may generate two-dimensional (2D) electronic layers, where the direction of the local magnetic moments establishes a preferential direction that breaks rotational symmetry within the 2D layers with respect to the underlying lattice. For CeRhIn_{5}, the large field-induced magnetic anisotropy identified here can be accessed by a slight tilting of the magnetic field away from the *c* axis (inset of Fig. 1e) to align the magnetic moments, providing a natural explanation for the observed large nematic susceptibility.

Quantum oscillation measurements report a crossover from a small to a large Fermi surface volume near both QCPs (Fig. 1d, e), suggesting enhanced coupling between spin and charge degrees of freedom due to the Kondo interaction in their vicinity^{17,18}. This may explain why the magnetic superstructures that are omnipresent throughout the entire phase diagram predominantly influence material properties near the QCPs. Finally, the observed large uniaxial anisotropy arises due to changes of the orbital character of the Ce 4 *f* electronic wavefunction with magnetic field. Remarkably, it has been demonstrated previously in the family of materials CeMIn_{5} (M = Co, Rh Ir), to which CeRhIn_{5} belongs, that the orbital character of the 4 *f* wavefunctions can be also controlled via chemical substitution or pressure^{19}. This not only affords an intrinsic mechanism for alternatively tuning the uniaxial anisotropy by pressure, but clarifies the striking similarity of the phase diagrams as functions of *H* and *P* (see Figure 1d,e).

In conclusion, via the quantitative application of an ANNNI-based effective spin model, a notable first for a heavy electron metal, we have identified a simple mechanism to create highly-tunable emergent magnetic heterostructures in CeRhIn_{5} via competing interactions. Through coupling of spin and charge degrees of freedom mediated via the Kondo effect, this mechanism concurrently generates electronic textures that significantly influence material properties. These textures are akin to emergent electronic heterostructures that exhibit clean interfaces and can be tuned with great ease, employing the use of external tuning parameters such as magnetic field or pressure. Our work demonstrates that strongly correlated electron materials are a promising route for top-down approaches to producing tunable and emergent heterostructures. Notably, because frustrated exchange is common to *f-*electron materials, and field-induced uniaxial magnetic anisotropy has been reported in various heavy electron materials^{20,21}, the mechanism identified here may apply universally for heavy electron materials. Furthermore, other classes of strongly correlated electron materials, such as high-*T*_{c} copper oxide, iron pnictide, and ruthenate superconductors all exhibit electronic textures near magnetic QCP^{22,23,24}, many of which exhibit instabilities towards incommensurate modulated magnetism^{25,26,27,28}, where orbital effects^{29} and/or magnetic frustration^{30} have similarly been proposed to be their origin, suggesting intrinsic functional heterostructures may be realized more broadly.

## Methods

### Sample preparation

Neutron scattering measurements were all performed on a mosaic (~2.2 g) of 14 CeRhIn_{5} single crystals grown via the In self-flux method. To mitigate the effects of high neutron absorption by Rh and In, individual crystals were polished to a thickness of < 0.6 mm along the crystallographic *c*-axis and glued to a thin Al plate using a hydrogen-free adhesive (CYTOP). This sample mosaic is well-characterized and was used in our previous neutron spectroscopy study^{12}.

### Neutron spectroscopy

Time-of-flight neutron spectroscopy measurements shown in Fig. 3 and the Supplementary Information were performed on two direct geometry spectrometers: the Cold Neutron Chopper Spectrometer (CNCS)^{31} at the Spallation Neutron Source (SNS), for applied magnetic fields of 5 T and below, with incident neutron energy *E*_{i} = 3.315 meV, and the LET Spectrometer^{32} at the ISIS pulsed neutron and muon source for applied magnetic fields above 5 T, with *E*_{i} = 3.3 meV. Energy resolution in both cases was estimated to be ~0.08 meV. Inelastic slices with subtracted background were generated using Horace and fitted to the theoretical dynamic susceptibility using a least-squares method implemented in NeutronPy (http://neutronpy.github.io/). Background scans were obtained on the CeRhIn_{5} sample at *T* = 20 K. Detailed inelastic neutron scattering measurements of the gap, shown in Fig. 4a, were performed on the Spin Polarized Inelastic Neutron Spectrometer (SPINS), a cold-neutron triple-axis spectrometer at the NIST Center for Neutron Research (NCNR), using a 7 T magnet with a ^{3}He-dipper. Constant-*q* scans were obtained with fixed *E*_{f} = 3.0 meV, 40′ collimation before the sample, a 60′ radial collimator after the sample, and a horizontally focused 11-blade PG(002) analyser. Higher-order neutrons were filtered using a cold Be-filter. Error bars of the gap values reflect the combined fitting error and energy resolution estimated by the quasielastic linewidth as measured on a standard vanadium sample. Error bars shown in Figs. 4b, c reflect the standard errors resulting from least-squares fitting. Diffraction data shown in Fig. 2d were also obtained on SPINS by performing scans along the (00 *l*) direction with *E*_{i} = *E*_{f} = 3.315 meV, 20′-S-10′ collimation in triple-axis mode, a flat monochromator, and a flat 3-blade analyser. Peak positions shown in Fig. 2d were obtained from fitting scans to a single Gaussian with constant background, and error bars represent the combination estimated error and momentum resolution calculated with the Cooper–Nathans method implemented in NeutronPy.

### Calculation of phase diagram

To obtain the phase diagram, we treat spins in equation (1) as classical spins, and then numerically minimize the free energy of equation (1). We first perform numerical annealing using the Markov chain Monte Carlo method^{33}, which minimizes the chance of trapping in a metastable state. Subsequently we use the relaxation method to determine the state with minimal free energy. Because the ordering wavevector is temperature-dependent, we continuously change the system size in the *c*-direction from 4 to 80, and keep the solution with the lowest free energy.

### Data availability statement

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. The neutron spectroscopy raw data from the experiment performed at LET are available at https://doi.org/10.5286/ISIS.E.82355430. Data from experiments carried out at SPINS are available at ftp://ftp.ncnr.nist.gov/pub/ncnrdata/ng5/201610/Fobes/CeRhIn5_22425/ and ftp://ftp.ncnr.nist.gov/pub/ncnrdata/ng5/201509/Fobes/CeRhIn5/.

## Additional information

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## Acknowledgements

We acknowledge useful discussions with R. Baumbach, C. Pfleiderer, M. Garst, M. Votja, P. Böni and J. M. Lawrence. Work at Los Alamos National Laboratory (LANL) was performed under the auspices of the US Department of Energy. LANL is operated by Los Alamos National Security for the National Nuclear Security Administration of DOE under contract DE-AC52-06NA25396. Research supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under the project ‘Complex Electronic Materials’ (material synthesis and characterization) and the LANL Directed Research and Development program (neutron scattering, development of the spin wave model, mean-field computation and development of analysis software). Research conducted at Oak Ridge National Laboratory’s (ORNL) Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy. Experiments at the ISIS Pulsed Neutron and Muon Source were supported by a beam time allocation from the Science and Technology Facilities Council. We acknowledge the support of the National Institute of Standards and Technology, US Department of Commerce, in providing the neutron research facilities used in this work.

## Author information

### Author notes

- Pinaki Das

Present address: Division of Materials Sciences and Engineering, Ames Laboratory, U.S. DOE, Iowa State University, Ames, IA, USA

- N. J. Ghimire

Present address: Argonne National Laboratory, Lemont, IL, USA

### Affiliations

#### MPA-CMMS, Los Alamos National Laboratory, Los Alamos, NM, USA

- D. M. Fobes
- , Pinaki Das
- , N. J. Ghimire
- , E. D. Bauer
- , J. D. Thompson
- , F. Ronning
- , C. D. Batista
- & M. Janoschek

#### Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN, USA

- S. Zhang
- & C. D. Batista

#### T-4, Los Alamos National Laboratory, Los Alamos, NM, USA

- S.-Z. Lin

#### NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD, USA

- L. W. Harriger

#### QCMD, Oak Ridge National Laboratory, Oak Ridge, TN, USA

- G. Ehlers
- & A. Podlesnyak

#### ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Chilton, Didcot, UK

- R. I. Bewley

#### Institute of Crystallography, RWTH Aachen University and Jülich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Garching, Germany

- A. Sazonov
- & V. Hutanu

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### Contributions

N.J.G., P.D. and E.D.B. synthesized the single crystal samples; J.D.T. and F.R. carried out thermal and transport measurements; D.M.F., G.E., A.P., L.W.H., R.I.B., V.H., A.S. and M.J. performed the neutron spectroscopy measurements; D.M.F. wrote the software for analysing the neutron data; DMF and MJ analyzed the neutron data; M.J. supervised the experimental work; S.Z., S.Z.L. and C.D.B. developed the theoretical model and carried out all calculations; D.M.F., S.Z.L., C.D.B. and M.J. proposed and designed this study, and D.M.F., C.D.B. and M.J. wrote the manuscript; all authors discussed the data and commented on the manuscript.

### Competing interests

The authors declare no competing financial interests.

### Corresponding author

Correspondence to M. Janoschek.

## Supplementary information

### Supplementary Information

4 Figures, 14 References

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