## Abstract

Condensed-matter systems that are both low-dimensional and strongly interacting often exhibit unusual electronic properties. Strongly correlated electrons with greatly enhanced effective mass are present in heavy fermion compounds, whose electronic structure is essentially three-dimensional. We realized experimentally a two-dimensional heavy fermion system, adjusting the dimensionality in a controllable fashion. Artificial superlattices of the antiferromagnetic heavy fermion compound CeIn_{3} and the conventional metal LaIn_{3} were grown epitaxially. By reducing the thickness of the CeIn_{3} layers, the magnetic order was suppressed and the effective electron mass was further enhanced. Heavy fermions confined to two dimensions display striking deviations from the standard Fermi liquid low-temperature electronic properties, and these are associated with the dimensional tuning of quantum criticality.

Heavy fermion materials are metallic compounds with extremely large effective electron masses, and they typically contain a rare-earth element. In these materials, the electrons populating the 4f orbitals are, at high temperatures, essentially localized with well-defined magnetic moments. As the temperature is lowered, the localized moments are screened by conduction electrons (s, p, and d* *orbitals), forming a nonmagnetic state by virtue of the Kondo effect (*1*). At yet lower temperatures, the f-orbital electrons dressed by conduction electron clouds (Kondo clouds) become itinerant, forming a very narrow conduction band that is characterized by a heavy effective quasiparticle mass. On the other hand, the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which is an intersite exchange interaction between the localized f-orbital moments, promotes magnetic ordering. Thus, the ground state of these compounds is either a nonmagnetic metal or a magnetically ordered state, as determined by the competition of the above two effects. Generally, in reduced spatial dimensions, many-body correlation effects due to the Coulomb interaction between electrons become more relevant. Moreover, thermal and quantum fluctuations are significantly enhanced, extending critical regimes with no long-range ordering to a wide temperature range. Therefore, many-body effects that are not observed in three dimensions are expected to arise in two-dimensional (2D) heavy fermion systems.

Quantum criticality is a central research issue in the physics of highly correlated matter (*2*, *3*). In conventional metals, interacting electrons (quasiparticles) are well described by Landau’s Fermi liquid theory. Near the quantum critical point (QCP), where a second-order phase transition occurs at zero temperature, low-lying spin fluctuations give rise to a serious modification of the quasiparticle mass and the scattering cross section of the Fermi liquid. This results in a strong deviation of physical properties from the standard Fermi liquid behavior. In heavy fermion metals, quantum criticality can be tuned by external parameters, such as doping, pressure, and magnetic fields (*2*). Fabricating superlattice heterostructure provides another way to control the quantum criticality through “dimensional tuning”; however, the epitaxial growth of heavy fermion thin films has been challenging (*4*–*6*).

There have been attempts to realize heavy fermion systems with low dimensions. One is the bulk crystals of Ce*T*In_{5} (where *T *= Rh, Co, or Ir), whose crystal structure yields alternating layers of CeIn_{3} and *T*In_{2} (*7*, *8*). However, the largely corrugated Fermi surface (*9*), the small anisotropy of upper critical fields (*10*), and the strong deviations from 2D antiferromagnetic spin fluctuations (*11*) all indicate that the electronic and magnetic properties of Ce*T*In_{5} are anisotropic 3D rather than 2D. Another example is the bilayer 2D films of ^{3}He fluid (*12*), where the mass enhancement is observed near a QCP. Here the controlled parameter is the ^{3}He density of the second layer. However, the dimensional tuning from 3D to 2D heavy fermions is still lacking.

The heavy fermion compound CeIn_{3} appears to be a good candidate for addressing the key issues of QCP physics. Bulk cubic CeIn_{3} exhibits a 3D antiferromagnetic ordering at Neel temperature *T*_{N }= 10 K (*13*, *14*) that is destroyed in a quantum phase transition accessed by applying pressure. Near the critical pressure of *p*_{c} ~ 24 kbar, in the vicinity of the QCP, CeIn_{3} undergoes a transition into an unconventional superconducting state, and a remarkable deviation from the Fermi liquid behavior is reported (*14*–*16*). To adjust the dimensionality in a controllable way, we used molecular beam epitaxy to grow the CeIn_{3}/LaIn_{3} superlattices (Fig. 1A): *m* layers of CeIn_{3} and *n* layers of isomorphic LaIn_{3} were grown alternately, forming an (*m*:*n*) heterostructure, where *m* and *n* are integers. The clearly observed streak pattern of the reflection high-energy electron diffraction (RHEED) (Fig. 1B) indicates the epitaxial growth of each layer with atomic-scale flatness. The cross-sectional transmission electron microscope (TEM) image (Fig. 1C, lower panel), which shows bright spots corresponding to Ce atoms, demonstrates that the CeIn_{3} layers are continuous even for the *m *= 1 structures (*17*). The electron diffraction pattern with clear superspots (Fig. 1C, upper panel), the x-ray diffraction pattern (Fig. 1D), and the simulation results (Fig. 1E) all demonstrate the realization of an epitaxial superlattice structure. These indicate the successful confinement of f-orbital electrons to the 2D spaces.

We investigated the transport properties of the (*m*:4) superlattices by varying *m*. Because the RKKY interaction decays as 1*/r*^{3}, where *r* is the distance between the interacting magnetic moments, the magnetic interaction between the Ce ions in different layers reduces to less than 1% of that between the neighboring Ce ions within the same layer. Therefore, the interlayer magnetic interactions are negligible. Resistivity ρ of the superlattices at room temperature is of the same order as for CeIn_{3} (Fig. 2B, inset). CeIn_{3} exhibits a ρ(*T*) behavior that is typical of heavy fermion compounds. Below ~200 K, ρ(*T*) increases because of the Kondo scattering and shows a maximum ρ(*T*) at *T*_{peak }~ 50 K. At *T*_{N }= 10 K, ρ(*T*) shows a distinct cusp, below which it decreases rapidly as the magnetic scattering is suppressed. In the superlattices, the hump structure of ρ(*T*) at *T*_{peak }~ 50 K becomes less pronounced with decreasing *m* (Fig. 2A). Below *T*_{peak}, ρ(*T*) increases again with decreasing temperature. Such behavior is caused by the interplay of the Kondo interaction with the crystal field effect (*18*), whose energy scale is 123 K in CeIn_{3} (*19*). At lower temperatures, ρ(*T*) shows a pronounced peak structure, as indicated by arrows for *m *= 8, 6, and 3 (Fig. 2B). For *m *= 2 and 1, ρ(*T*) decreases gradually without exhibiting a pronounced peak below a characteristic temperature *T*_{coh }~ 1.6 K, which marks the onset of coherent electron conduction.

The residual resistivity ρ_{0} of the superlattices is larger than that of CeIn_{3} (Fig. 2A). This is probably due to the inevitable scattering at the interfaces between the CeIn_{3} and LaIn_{3} layers, where Kondo holes that were created by possible Ce/La disorder act as strong impurity-scattering centers. Residual resistivity in the 2D planes can be enhanced with small impurity concentration by quantum fluctuations via enlarged effective cross section of local impurity scattering (*20*, *21*).

The observed distinct anisotropy in the magnetoresistance (Fig. 3A) indicates the substantial contribution from the 2D electrons confined within the layers, which is in sharp contrast to the 3D isotropic Ce(La)In_{3}. The negative magnetoresistance (Fig. 3A, inset) is consistent with the view that antiferromagnetic fluctuations that are responsible for the scattering can be suppressed by the magnetic field. The Hall coefficient *R*_{H}(*T*) exhibits a cusp-like minimum in bulk CeIn_{3} and superlattices for *m *= 8, 6, and 3 (Fig. 3B). These cusp temperatures coincide with the temperatures at which ρ(*T*) exhibit cusps (Fig. 2B, arrows). Therefore, we surmise that the observed transport anomalies for *m *= 8, 6, and 3 appear as a result of magnetic ordering, as in CeIn_{3}. The low-temperature behavior of the transport properties for *m *= 2 and 1 are essentially different from those for *m *> 2. For *m *≤ 2, ρ(*T*) does not exhibit a pronounced peak, and *R*_{H}(*T*) decreases monotonically with decreasing temperature without showing the upturn. These results indicate that the magnetic order is suppressed when the CeIn_{3} layer thickness decreases, and is vanishing near *m *= 2. This conclusion is reinforced by the results shown in Fig. 4A, where *T*_{N} decreases nearly linearly with 1/*m* and goes to zero in the vicinity of *m *= 2. Therefore, it is natural to conclude that the enhanced antiferromagnetic fluctuations associated with two-dimensionality are responsible for destroying the magnetic order.

The vanishing magnetic order implies that a quantum phase transition occurs close to *m *= 2. Close to a QCP, the resistivity strongly deviates from the Fermi liquid behavior ρ(*T*) = ρ_{0 }*+ AT*^{α}, with α = 2, where *A* is the Fermi liquid coefficient. In Fig. 4B, the exponent α is mapped in the *T* versus 1/*m* diagram. The *T*^{2}-dependence is observed at low temperatures for *m *> 2. In contrast, for *m *= 2, a marked deviation from such behavior is observed (Fig. 4C): A *T*-linear dependence with α = 1.01 ± 0.02 is seen below 0.5 *T*_{coh}, which corresponds to the Fermi temperature of heavy fermions. At even lower temperatures, below ~140 mK, ρ(*T*) shows a faster-than-linear decrease, which is suppressed by small magnetic fields (Fig. 4C, upper inset). We conclude that for *m *= 2, the lowered dimensionality suppresses the magnetic order down to *T*_{N }~ 140 mK, which can be further suppressed by a small field (*17*). The *T*-linear behavior is consistent with scattering by the 2D antiferromagnetic fluctuations (*22*) that are enhanced by the QCP. This is in contrast to the 3D case, where α = 1.5 is expected (*22*), and indeed α ~ 1.6 is observed near the pressure-induced QCP in the bulk CeIn_{3} (*14*, *15*).

Even for *m *> 2, where *T*^{2} behavior is observed, the initial slope of ρ − ρ_{0} as *T *approaches 0 K becomes larger with decreasing *m*, and *A* is enhanced toward *m *= 2 (Fig. 4A). Because *A* is related to the Sommerfeld coefficient of specific heat γ as *A *= *a*_{0}γ^{2}, with *a*_{0} close to the universal value of 10^{−5} μΩ∙cm (K·mol/mJ)^{2} (*14*, *21*), the large *A* immediately indicates the enhanced quasiparticle mass. At temperatures below ~1 K, the resistivity of LaIn_{3} is much smaller than that of superlattices, and its temperature dependence is negligible. Thus the observed large *A* values indicate that the temperature-dependent part of the resistivity is governed by the 2D heavy fermion layers. In fact, the γ value estimated from *A* is enhanced from the bulk CeIn_{3} value of ~120 mJ/(K^{2 }mol) and reaches ~350 mJ/(K^{2 }mol) for *m *= 3, which corresponds to a quasiparticle mass at least several hundred times larger than the free electron mass. Because the *T*-linear behavior observed for *m *= 2 suggests further enhancement of *A*, it is tempting to associate this result with diverging quasiparticle mass because of the quantum fluctuations enhanced upon approaching a QCP. For *m *= 1, the determination of *A* is ambiguous because of the weak temperature dependence of ρ. The diverging behavior of γ near a QCP is also reported in bilayer films of ^{3}He systems (*12*).

Quantum criticality is reported to be strongly modified by magnetic fields in strongly correlated electron systems (*2*, *9*, *23*, *24*). The quantum criticality for *m *= 2 is removed by strong magnetic fields, and the Fermi-liquid properties with α = 2 are recovered (Fig. 4C, lower inset). Simultaneously, the *A* value is quickly suppressed with increasing magnetic field *H* (Fig. 4A). Thus, both temperature and field dependencies of the transport properties provide strong support for the presence of the quantum phase transition in the vicinity of *m *= 2, which is tuned by the dimensionality parameter. The successful growth of epitaxial Ce-based superlattices promises to provide a setting in which to explore the fundamental physics of strongly correlated electron systems such as the 2D Kondo lattice and, potentially, 2D superconductivity near a QCP.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/327/5968/980/DC1

Materials and Methods

SOM Text

Figs. S1 to S5

## References and Notes

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- We thank M. Izaki, N. Kawakami, H. Kurata, K. Miyake, and A. Tanaka for discussion. This work was supported by Grants-in-Aid for Scientific Research (KAKENHI) from the Japan Society for the Promotion of Science and by a Grant-in-Aid for the Global Centers of Excellence program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology.