Abstract
In the Kondo lattice model, the interaction of a lattice of localized magnetic moments with a sea of conduction electrons induces rich quantum phases of matter, including Fermi liquids with heavily renormalized electronic quasiparticles, quantum critical non-Fermi liquid metals and unconventional superconductors. The recent demonstration of moiré Kondo lattices has presented an opportunity to study the Kondo problem with continuously tunable parameters. Although a heavy Fermi liquid phase has been identified, the magnetic phases remain unexplored in moiré Kondo lattices. Here we report a density-tuned Kondo breakdown in MoTe2/WSe2 moiré bilayers by combining magnetotransport and optical studies. At a critical density, we observe a heavy Fermi liquid to insulator transition and a nearly concomitant emergence of ferromagnetic order. The observation is consistent with the emergence of a ferromagnetic Anderson insulator and suppression of the Kondo screening effect below the critical density. Our results suggest a path for realizing other quantum phase transitions in moiré Kondo lattices.
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Acknowledgements
This work was supported by the Air Force Office of Scientific Research under award no. FA9550-19-1-0390 (transport measurements) and the National Science Foundation (NSF) (Platform for the Accelerated Realization, Analysis, and Discovery of Interface Materials) under cooperative agreement nos. DMR-2039380 (sample and device fabrication) and DMR-1807810 (optical measurements). S.K. and D.C. were supported in part by an NSF CAREER grant (DMR-2237522) and a Sloan research fellowship (to D.C.). The growth of hBN crystals was supported by the Elemental Strategy Initiative of MEXT, Japan, and CREST (JPMJCR15F3), JST. We used the Cornell Center for Materials Research Shared Facilities supported through the NSF MRSEC programme (DMR-1719875) and the Cornell NanoScale Facility supported by NSF grant NNCI-2025233. We also acknowledge support from a David and Lucille Packard Fellowship (to K.F.M.), Kavli Postdoctoral Fellowship (to W.Z.) and Swiss Science Foundation Postdoc Fellowship (to P.K.).
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B.S., W.Z. and Z.H. fabricated the devices. W.Z. and B.S. performed the electrical transport measurements and analysed the data with help from Z.H. and Y.Z. W.Z., Z.T. and P.K. performed the optical measurements. S.K. and D.C. performed the theoretical analysis. K.W. and T.T. grew the bulk hBN crystals. J.S. and K.F.M. designed the scientific objectives and oversaw the project. All authors discussed the results and commented on the manuscript.
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Extended data
Extended Data Fig. 1 Characterization of the moiré density.
a, Quantum oscillations are observed in the longitudinal resistance as a function of itinerant hole density in the Kondo lattice region under an out-of-plane magnetic field of 13.6 T at 1.6 K. A constant has been subtracted from the resistance to bring out the oscillations. The first five Landau levels (LLs) are labeled. The density is calibrated by the LLs. b, Resistance as a function of hole density (upper axis). The two sharp resistance peaks are assigned to be at filling \(\nu\) = 1 and 2. The density separation between the two peaks is the moiré density.
Extended Data Fig. 2 Determination of the Kondo lattice region (device 1).
a,b, Longitudinal resistance \({R}_{{xx}}\) (a) and MCD (b) as a function of total filling factor, \(\nu ,\) in the bilayer and out-of-plane electric field, E, at temperature T=1.6 K. The out-of-plane magnetic field \(B\) is 13.5 T (a) and 0.1 T (b). In the Kondo lattice region (bounded by the red dashed lines), the quantum oscillations from the WSe2 layer are independent of E and shown as pronounced vertical stripes; the small-field MCD is also enhanced at filling factor slightly above 1. The slope of the red lines in a is about two times of the slope in b since the WSe2 band is spin-polarized for filing factor below about 0.4 at 13.5 T and spin degenerate at 0.1 T. c, Magnetic-field dependence of the Mott gap. The electric-field span of the Kondo lattice region, multiplied by the interlayer dipole moment (0.26 \(e\times\) nm), provides an estimate of the Mott gap (see Ref. 12). The error bars are the combined uncertainty of the interlayer dipole moment and the electric-field span of the region.
Extended Data Fig. 3 Results at other electric fields of the Kondo Lattice region (device 1).
a,b, Temperature dependence of \({R}_{{\rm{xx}}}\) (a) and magnetic-field dependence of \({R}_{{xy}}\) (b) at representative itinerant hole densities for E = 0.634 V/nm. The lattice temperature is 10 mK. c, Magnetic susceptibility (upper) and Curie-Weiss temperature (lower) from the analysis in d versus itinerant hole density at representative temperatures. The error bars in upper panel are one standard deviation of the zero-field slope of MCD. The error bars in lower panel are one standard deviation of the best-fit values for the Curie-Weiss analysis. d, Curie-Weiss analysis of the temperature dependence of inverse magnetic susceptibility for x = 0 (bottom), 0.05 (middle) and 0.11 (top). The electric field is fixed at E = 0.615 V/nm for c and d.
Extended Data Fig. 4 Analysis of the temperature dependent Rxx at E = 0.638 V/nm and ν=1+x (device 1).
a, Determination of the critical density of the metal-insulator transition: \({R}_{{xx}}\) (symbols) follows a power-law temperature dependence, \({R}_{{xx}} \sim {T}^{-0.48}\) (solid line), at the critical density of \({x}_{c}\approx\) 0.04. b, Determination of the Kondo temperature (\({T}^{* }\)) for \({x > x}_{c}\): temperature dependence of \({R}_{{xx}}\) in linear scale and its polynomial fit (solid line) (upper panel) and \(\left|\frac{d{R}_{{xx}}}{{dT}}\right|\) evaluated using the fit function (lower panel). \({T}^{* }\) is determined as the temperature corresponding to the first minimum. c, \({R}_{{xx}}\) as a function of \({T}^{2}\) at low temperatures for \({x > x}_{c}\). Coefficient \(A\) is determined from the slope of the linear fit (solid line). d, \({R}_{{xx}}\) (log scale) versus \({T}^{-1/2}\) at itinerant hole densities below \({x}_{c}\). Solid lines are linear fits to the Efros-Shklovskii variable range hopping model.
Extended Data Fig. 5 Reproducibility of the main results (device 2).
a, Temperature dependent \({R}_{{xx}}\) at varying itinerant hole densities. b, Dependence of the Kondo temperature \({T}^{* }\) and coefficient \({A}^{1/2}\) on itinerant hole density. Error bars denote the standard deviation from data fitting. c, At the critical density \({x}_{c}\approx 0.04\), \({R}_{{xx}}\) follows a power-law temperature dependence, \({R}_{{xx}} \sim {T}^{-0.23}\) (solid line). d, Magnetic-field dependence of \({R}_{{xy}}\) at varying itinerant hole densities at \(T\) = 10 mK. e, Magnetic-field dependence of MCD at representative itinerant hole densities at T=1.6 K. f, Itinerant hole density dependence of the Kondo temperature (\({T}^{* }\)) and Curie-Weiss temperature (\(\theta\)). Ferromagnetic correlations between the local moments persist up to \(x\approx 0.09\). The vertical dashed line denotes the critical density. The error bars are one standard deviation of the Curie-Weiss temperature from the fits.
Extended Data Fig. 6 Comparison to the case of ν=2+x (device 1).
a, Doping dependence of coefficient \({A}^{1/2}\) extracted from the temperature dependence of longitudinal resistance \({R}_{{xx}}\). At low temperatures, \({R}_{{xx}}\) follows the dependence of \(A{T}^{2}\) (inset, solid lines). The coefficient \({A}^{1/2}\) is over an order of magnitude smaller than for the case of \(\nu =1+x\) (Fig. 1e). Error bars denote the standard deviation from data fitting. b, Magnetic-field dependence of \({R}_{{xy}}\) at varying itinerant hole densities at 10 mK. The anomalous Hall response is absent for any densities.
Extended Data Fig. 7 Reproducibility of the magnetic hysteresis (device 3).
a, Optical image of Device 3. b, Magnetic-field dependence of Hall resistance at 0.6 K from two different pairs of Hall probes as shown in a. Both show a hysteretic behavior.
Extended Data Fig. 8 MCD spectrum (device 1).
a, MCD spectrum centered around the exciton resonance of MoTe2 as a function of total filling factor \(\nu\) at T=1.6 K, \(B\) = 0.05 T and E = 0.638 V/nm. b, MCD spectrum as a function of magnetic field at x = 0.03 and T=1.6 K. The absolute value of the MCD is integrated over the spectral window between the black dashed lines. c, Linecut of b at zero magnetic field for two different magnetic-field scan directions.
Extended Data Fig. 9 Mean-field phase diagram and inverse participation ratio (IPR).
a, MF phase diagram as a function of itinerant hole density \(x\) for a \(L\times L\) (distorted) honeycomb lattice system (\(L=12\)). The results are obtained by averaging over 50 uncorrelated disorder configurations. The Kondo-singlet MF parameter, \(\left\langle \left|\phi \right|\right\rangle\) (green), and the ferromagnetic component of the magnetization density of the Mo layer, \(\left\langle {M}_{f}\right\rangle\) (blue), and of the W layer, \({\rm{|}}{\rm{\langle }}{M}_{c}{\rm{\rangle }}{\rm{|}}\) (orange). The W layer magnetization is multiplied by 30 to match the overall scale of the plot. b, Disorder averaged IPR as a function of system size. Three representative values of \(x\) are chosen: \(x=0.04\) (FM phase), \(x=0.1\) (near the critical point), and \(x=0.175\) (heavy Fermi liquid phase). The error bars are one standard deviation of the results in 50 uncorrelated disorder configurations.
Extended Data Fig. 10 Real-space snapshots across the phase transition.
Local expectation values of \(|\phi |\) (left), \({N}_{f},{N}_{c}\) (middle), and \({M}_{f},{M}_{c}\) (right) for a disorder configuration at five representative values of \(x\) (ascending order from top to bottom). At \(x=0.03\), both itinerant holes and local moments show long-range ferromagnetic order. As \(x\) is increased, local moments begin to give way to the Kondo singlet formation in an inhomogeneous fashion. At such sites, the local number constraint is violated. As \(x\) exceeds the critical value, the Kondo singlets form globally and the ferromagnetism disappears.
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Zhao, W., Shen, B., Tao, Z. et al. Emergence of ferromagnetism at the onset of moiré Kondo breakdown. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02636-4
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DOI: https://doi.org/10.1038/s41567-024-02636-4
