The Kondo effect [1], named after the Japanese physicist Jun Kondo,  is one of the best-studied many-body phenomena in condensed matter physics. In metals with magnetic impurities it gives rise to a resistance minimum at low temperatures, beyond which the resistance increases again, something that does not happen for a clean metal. It is due to scattering of the electrons from the metal conduction electrons at the localized spin of the magnetic impurity. For lower and lower temperatures resonant spin-flip scattering of the conduction electrons become increasingly important for the resistance of the material. Ultimately the spin gets completely screened and a many-body Kondo resonance forms at the Fermi level, visible in electron spectroscopy as a sharp peak. The growth of the Kondo resonance with decreasing temperature is the smoking gun signature of these resonant spin flips and the resulting perfect screening of the spin by the surrounding electrons giving rise to a many-body singlet ground state. This process is closely related to how quarks become confined in hadrons, resulting there in colorless singlet states. Wilsons numerical renormalization group [2] gives a quantitative description of the Kondo screening process. And yet, recently some of the pioneering works on magnetic atoms on noble metal surfaces [3] – one of the archetypical Kondo systems – have been called into question [4]. This is all the more surprising, as the Kondo problem can be solved exactly using numerical methods such as the numerical renormalization group. This discrepancy between experiment and theory lays bare a fundamental issue in most experimental realizations of the Kondo effect: they can only see a small part of it.

The Kondo system consists of the sharp peak at the Fermi level, the Kondo resonance, along with the singly and doubly occupied impurity levels of the interacting magnetic impurity. While scanning tunneling microscopy investigations of atoms on surfaces often measured a feature at the Fermi level and attributed it to the Kondo resonance, they have never been able to simultaneously access the impurity states themselves, as they are strongly hybridized with the metallic substrate. The discussion around the spectroscopic signature in these experiments shows that a feature at the Fermi level alone is not sufficient to fully understand the underlying physics. To solve this ambiguity in the data, one needs a Kondo system where the magnetic impurity is kept somewhat isolated from the environment, though still allowing some interaction with the electron and being accessible to the spatial resolution granted by the scanning tunneling microscope. This we found in the mirror twin boundaries of MoS₂.

Mirror twin boundaries are line defects separating two mirror-symmetric areas of the pristine material, see Figure 1 (left). These boundaries are fascinating in many respects. They are partly filled by electrons to compensate for the polar charge that builds up in discontinuous regions of MoS₂ [5]. In our setup, these charged wires float above the graphene substrate, to which MoS₂ is only weakly bound by van der Waals forces. This isolation from the environment causes the electrons within the wire to exhibit one of the weirder effects in condensed matter physics: the electrons ‘split’ into separate charge and spin excitations, which can be measured with the scanning tunneling microscope [6].  And as a final twist: the electrons can only be found in discrete energy levels, like a particle in a box, due to the finite length of the boundaries.

Based on the work of Yang et al. [7], we reasoned that locally applied voltage pulses could switch the boundaries into a correlated magnetic state, where a degenerate energy level was split by the Coulomb energy into a singly occupied state below the Fermi level, making the boundary magnetic. One of the consequences of having these one-dimensional magnets on our sample would be the presence of a Kondo resonance at the Fermi level. This resonance would arise via spin-screening, where the electron bath in graphene interacts with the uncompensated spin in the mirror twin boundary, see Figure 1 (right).

At the time of measurement, the Kondo idea was admittedly not the one we thought most likely. While our system was in principle analogous to an atom on a metallic surface, there was a rather fundamental problem: while the Kondo effect was based on strong hybridization between the magnetic atom and its substrate, our grain boundaries were precisely so interesting because they were almost insensitive to their surroundings.

So we spent the first weeks of our measurement time trying other ideas, which we considered to have higher chances of success. Only when they kept failing, the process of which can be imagined as staring intently at datasets taken at different magnetic fields and seeing absolutely no difference between them, did we turn our attention to the small energy range around the Fermi level.

We enhanced our energy resolution and cooled down to 0.4 K. Immediately there appeared a small but unmistakable peak, right at the Fermi level, with a textbook Kondo shape. Finally putting the magnetic field to good use, we saw the peak split with increasing field, opening a box-like gap that grew at twice the Zeeman energy. We had our Kondo effect. Getting an extension for our allotted measurement time, we worked around the clock for the next month, subjecting as many grain boundaries as we could find to magnetic fields of all directions and magnitudes.

After that initial flurry of measuring, we took a step back. While it was great that we had in fact made the boundaries magnetic and that we could see the Kondo resonance, was there anything more to it? Here, the lack of interaction with its environment proved again to be decisive. The isolated nature of the boundaries had first seemed a drawback: the resonance was quite tiny. We realized later that this isolation simultaneously preserves the intrinsic elements out of which a Kondo system is composed.

To describe the Kondo effect, one needs the energetic locations and widths of the singly- and doubly-occupied peaks in the many-body spectral function of the magnetic impurity. All the properties of the resonance at the Fermi level are determined by these two peaks, using a microscopic model of the Kondo effect that stems from Anderson [8]. Measuring these two impurity level peaks with high precision was generally impossible in experiment, since they were always strongly hybridized with the substrate or the conductive measurement leads. But in our case, hybridization of the impurity levels with the substrate is absent, so the freestanding boundary states have kept the shape of the pristine spectral function, which we could resolve directly with the scanning tunneling microscope, see Figure 2.

Working as experimentalists together with Theo Costi, an expert on the numerical renormalization group theory approach to the Kondo problem, our experimental results could be reproduced accurately from first principles, using our knowledge of the impurity peaks to predict the Kondo resonance. A few feverish weeks of sending results back and forth resulted in the simulations in paper, where we learned to discount inelastic excitations broadening the peaks to arrive at the correct width. In this way, we were able to provide an almost perfect experimental realization of the Anderson model, see Figure 2. As a bonus, we could also spatially resolve the relation between the resonance and the spectral function of the magnetic impurity for the first time.

With this fully resolved Kondo system, we will soon delve into some of the unsolved problems surrounding magnetic interactions on surfaces. One immediate idea is to place our mirror twin boundaries on a superconducting substrate and map the Yu-Shiba-Rusinov that arise from the interaction between the spin in the boundary and the Cooper pairs. Since mirror twin boundaries are a feature of many semiconducting transition metal dichalcogenides, we are confident that these model magnets will also find uses in the wider community of solid state physicists beyond what we can currently imagine.

[1] Resistance Minimum in Dilute Magnetic Alloys, J. Kondo, Progress of Theoretical Physics 32, 37–49 (1964)

[2] The renormalization group: Critical phenomena and the Kondo problem, K. G. Wilson, Rev. Mod. Phys. 47, 773-840 (1975)

[3] Tunneling into a Single Magnetic Atom: Spectroscopic Evidence of the Kondo Resonance, V. Madhavan et al., Science 280, 567-569 (1998)

[4] A new view on the origin of zero-bias anomalies of Co atoms atop noble metal surfaces, J. Bouaziz et al., Nature Communications 11, 6112 (2020)

[5] Band Bending and Valence Band Quantization at Line Defects in MoS2, C. Murray et al., ACS Nano 14, 7, 9176–9187 (2020)

[6] Tomonaga-Luttinger Liquid in a Box: Electrons Confined within MoS2 Mirror-Twin Boundaries, W. Jolie et al., Phys. Rev. X 9, 011055 (2019)

[7] Manipulating Hubbard-type Coulomb blockade effect of metallic wires embedded in an insulator, X. Yang et al., National Science Review 10, 3 (2022)

[8] Anderson, P. W. A poor man's derivation of scaling laws for the Kondo problem. J. Phys. C: Solid State Phys. 3, 2436–2441 (1970)