Nanotechnology

New spin on correlated electrons

In the Kondo effect, the flow of electrons in a solid is modulated by magnetic impurities. Nanostructures such as carbon nanotubes can be designed to obtain even more complex versions of this intriguing effect.

The behaviour of electrons is difficult to predict. Undergraduates routinely calculate the quantum states of a hydrogen atom, but even a helium atom with two electrons displays complex dynamics. How much more complex are solids, with electrons in the billions. Nanotechnology now enables researchers to create controlled, or tunable, models of electronic behaviour in solids. On page 484 of this issue, Jarillo-Herrero et al.1 present perhaps the most sophisticated example of this approach to date, studying the flow of electrons through a carbon nanotube that is made to act like an exotic magnetic atom.

In attempting to explain the electronic properties of materials, theorists often construct ‘microscopic’ models, comprising electrons that can move freely (become delocalized), sit on localized sites and make transitions between these states. Such models offer powerful insights but also have drawbacks when applied to conventional bulk materials. In particular, electrons on many sites can interact with each other, making complete calculations impractical; and parameters such as an electron's tunnelling rate and interaction strength are not tunable, and can be difficult to measure precisely. Nanotechnology can remedy these limitations: by building submicrometre-sized structures in which one or a few electrons are isolated, researchers can know everything about the relevant electronic states, and can study electrons at a single site. Perhaps most importantly, nearly all of the significant parameters can be designed, measured and often tuned in situ.

The interaction of mobile electrons with magnetic impurities — atoms or ions with a non-zero magnetic moment — in a host metal has been a theme in solid-state physics for decades. In 1961, Anderson proposed that a magnetic impurity could be modelled as an electronic quantum state localized at a single site within a crystal. According to the Pauli exclusion principle, the site can be occupied by up to two electrons (with opposite spins), but Coulomb repulsion can make it energetically favourable for only one electron with arbitrary spin to reside there. If its spin opposes that of a nearby delocalized electron, the localized electron can lower its energy by tunnelling off the site and back on again. At low temperature, the electron ensures that it can tunnel by pairing with a single delocalized electron of opposite spin to form a state of zero total spin (a ‘singlet’) — a phenomenon known as the Kondo effect. In Anderson's microscopic model, the stability of the singlet ground state (and hence its temperature-dependent effect on electrical resistance) can be calculated in terms of basic parameters such as the binding energy of the localized electron and its rate of tunnelling between localized and delocalized states2.

Now, however, experimental physicists can design and fabricate single artificial magnetic impurities, such as semiconductor quantum dots3, molecules4,5 and nanotubes6, with substantial control over the basic parameters of Anderson's model. In each of these systems, conducting leads attached to the artificial impurity play the same role as the host metal of a traditional magnetic impurity. Electron flow from one lead to the other through the artificial impurity serves as a powerful probe of local electronic states. The tunability of such nanostructures, and their amenability to electrical measurement, have allowed highly quantitative studies of the Kondo effect7, and have spurred the construction of new Kondo systems not described by the minimal Anderson or Kondo model (see the Supplementary Information to ref. 1 for a review).

The essence of the Anderson model is a localized state with a twofold degeneracy (two different quantum states of equal energy), coupled to a sea of electrons with the same degeneracy. Normally this degeneracy comes from spin, but recently it has been verified that other twofold degeneracies can support the Kondo effect without spin8. This work forms the foundation for the achievements of Jarillo-Herrero et al.1.

The first non-spin Kondo implementation to be devised linked two sites capacitively, so that a charge could exist on one or the other, but not on both simultaneously8 (Fig. 1a). Since then, Sasaki et al.9 with a quantum dot, and now Jarillo-Herrero et al.1 with a nanotube, have used a single site at which two degenerate orbital states act as the localized ‘magnetic impurity’. Here the degeneracy stems from a spatial symmetry in the device, either associated with the structure of the material (Fig. 1c) or created by nanoscale patterning (Fig. 1b). In these experiments, a Kondo effect can occur because the local degree of freedom is also reflected in the nearby leads. With a traditional magnetic impurity, if the spin of the localized electron is pointed up, a nearby, delocalized electron will direct its spin down in response. Similarly, if there is an extra charge on the top site of Figure 1a, an electron in the nearby leads will move towards the bottom.

Figure 1: Degeneracy and the Kondo effect.
figure1

The basis of the Kondo effect is the screening of a localized twofold degeneracy by surrounding delocalized electrons. The degeneracy normally comes from spin, but with the advent of nanotechnology, researchers can create and study other types of degeneracy; a–c give examples. a, An electron can reside on either of two sites, but not both simultaneously because of Coulomb repulsion8. b, c, A spatial symmetry produces degenerate orbital levels11. b, The symmetry is the rotational symmetry of a circular quantum dot9. This is analogous to the rotational symmetry of a hydrogen atom that produces degenerate p orbitals. c, The lattice symmetry of a nanotube gives rise to a twofold orbital degeneracy1. Non-spin degeneracy may also give rise to the Kondo effect in bulk systems14.

Orbital degeneracy does not preclude spin degeneracy. The same surrounding delocalized electrons that electrostatically screen the local orbital state also magnetically screen the localized spin, so that the two degrees of freedom become intimately connected, together forming a new fourfold degenerate Kondo effect10,11. Jarillo-Herrero et al. have demonstrated that the spin and orbital states of a carbon nanotube can be finely controlled so as to observe this effect.

In an ideal nanotube, sets of four electronic states can be grouped together into a shell, much like those of an atom. All four states are degenerate, with two choices for spin and two for orbital state. Electrons can be added one at a time to the shell by fine control of the voltage on a nearby electrode. The first electron can reside in any of four states, with strong coupling to the metal leads producing a fourfold Kondo system. If a second electron is added to the nanotube, the Kondo effect vanishes, suggesting that the two electrons form a unique, non-degenerate ground state. The fourfold Kondo effect returns with the addition of a third electron, because now any one of the four states can be left empty. A fourth electron completes the shell, removing all degeneracy and thus eliminating the Kondo effect.

In experiments, the Kondo effect is generally identified by a distinctive logarithmic temperature dependence of electrical resistance. Fourfold Kondo is predicted (and now observed) to have the same temperature dependence as conventional twofold Kondo, but the two systems can be distinguished by their very different responses to a magnetic field. Applying a magnetic field creates and breaks a variety of degeneracies (Fig. 2), allowing Jarillo-Herrero et al. to observe not only fourfold Kondo but also three other types of Kondo effect associated with different degeneracies in a single nanotube.

Figure 2: Many kinds of Kondo.
figure2

For any number of electrons in a shell of a nanotube, the low-energy states can be identified. At zero magnetic field, a shell with either one or three electrons displays a fourfold degeneracy, giving rise to a novel fourfold Kondo effect (red). Applying a magnetic field first breaks orbital degeneracy, reducing fourfold Kondo to conventional twofold spin Kondo (blue), and then destroys even the twofold spin Kondo. Higher magnetic fields can bring together electronic states that were not degenerate at zero magnetic field, producing a variety of Kondo effects for both even and odd electron occupancy. Although all these Kondo effects are twofold (blue), none emerges from conventional spin degeneracy. Their various origins are explained in the Supplementary Information to ref. 1.

Researchers can unambiguously identify the states of several electrons in nanostructures, and even manipulate those states. Electrons in nanostructures can then be mapped onto simple microscopic models, which successfully predict a variety of novel correlated-electron behaviour. The coming years should see researchers building more complex models, and addressing phenomena that are at the centre of modern condensed-matter theory, such as quantum phase transitions12,13 and decoherence.

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Potok, R., Goldhaber-Gordon, D. New spin on correlated electrons. Nature 434, 451–452 (2005). https://doi.org/10.1038/434451a

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