In the OSTI Collections: the Kondo Effect

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Article Acknowledgement:
Dr. William N. Watson, Physicist
DOE Office of Scientific and Technical Information

Improved Kondo-effect calculations

Kondo lattices; superconductivity

Related phenomena

References

Reports available through OSTI's SciTech Connect

Additional references

If the positive and negative charges of an atom’s quarks and electrons balance each other out, the atom as a whole is electrically neutral and on average neither attracts nor repels other charged particles. An atom’s quarks and electrons also have individual magnetic fields of their own, which are associated with their variously oriented spins^[^Wikipedia^] and orbital motions^[^Wikipedia^] in the atom. The different orientations generally mean that the magnetic fields largely cancel each other out and make the atom nonmagnetic, but if the spins and orbital motions don’t cancel out completely, the atom as a whole will have a magnetic field and affect other magnetic particles nearby.

Neighboring atoms may bind together by sharing some of their outermost electrons. Shared electrons can be confined by electromagnetic forces to pairs of atoms, as in a molecule, or move rather freely among large numbers of atoms, as in a metal. A metal’s shared electrons can move randomly, but they will drift in a common general direction to form an electric current if appropriate electromagnetic forces are applied to the metal. The more uniform the arrangement of the metal’s atoms, the less the metal will resist the motion of the current.

Uniformity in the atoms’ arrangement is lessened by the atoms’ own random thermal motions, which disturb the flow of the metal’s shared electrons. Resistance to the flow generally decreases as a metal gets colder and its atoms’ random vibrations lessen. So it was surprising when physicists found many decades ago that some metals, when cooled past a certain point, stop decreasing their resistance to current and actually become more resistive as they get colder. These metals consist mainly of nonmagnetic atoms with some magnetic ones among them—in effect, a solid solution of some magnetic atoms dissolved in a nonmagnetic mass.

A 1964 paper^[^PTP^] by physicist Jun Kondo^[^AIST^] presented his mathematical analysis of the forces at work in these metals, which showed that the magnetic field of each magnetic atom induces nearby shared electrons to orient their own magnetic fields in the opposite direction. When the metal is cooled below a certain temperature, these reorientations largely cancel the overall magnetic field at large distances from the magnetic atoms and also disturb the flow of the shared electrons, increasingly so as the metal is cooled further. This Kondo effect^[^Wikipedia^], and phenomena associated with it, are important for electronics and in other ways. Some of the research reports on the subject that are available through SciTech Connect provide new variations on standard mathematical analyses of the Kondo effect. Other reports deal with a somewhat different situation, in which the magnetic atoms constitute a dense, regularly spaced lattice throughout the material. In this case the shared electrons mediate significant interactions among the magnetic atoms, a situation that doesn’t appear in the dilute solid solutions that Kondo analyzed. Still other reports describe phenomena that are related to the Kondo effect in quite different ways.

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Improved Kondo-effect calculations

While the Kondo effect was first found and explored in otherwise nonmagnetic bulk metals with some magnetic atoms included, the effect has more recently been observed in two-dimensional atomic arrays at metal surfaces by examining them with a scanning tunneling microscope (or STM)^[^Wikipedia^], whose probe’s electric charge “tunnel” differently between the probe and different array atoms^[^Wikipedia^] as the probe approaches or recedes from them. The Kondo effect should also appear in the one-atom-thick sheets of graphene^[^Wikipedia^,^OSTI^], a semiconductor with less resistance to electric current than any known metal^[^Wikipedia^], if the graphene is similarly augmented with magnetic atoms.

Figure 1. Schematic diagram for a magnetic atom (red) adsorbed onto a sheet of graphene, probed by the tip of a scanning tunneling microscope (STM)^[^Wikipedia^]. (From “Scanning tunneling spectroscopy of a magnetic atom on graphene in the Kondo regime”^[^{SciTech Connect}^], p. 2.)

According to calculations presented in two different papers, the exact details of the Kondo effect should depend on what kind of magnetic atoms are added to the graphene. Magnetic atoms differ in their magnetism because of differences in how their constituent quarks and electrons are arranged, and in which magnetic fields get cancelled out by different particles’ spins and orbital motions. Different magnetic atoms thus interact differently with current-carrying electrons in a bulk metal or two-dimensional conductor.

Calculations described in “Scanning tunneling spectroscopy of a magnetic atom on graphene in the Kondo regime”^[^{SciTech Connect}^] show how the Kondo effect should be manifested when graphene is augmented by atoms whose uncancelled magnetic fields are entirely from the spin of a single electron in each atom that moves in what’s known as an s orbital^[^Wikipedia^]. The paper’s authors found that if the graphene is doped^[^Wikipedia^] with some other atoms that supply either more or fewer conduction electrons overall than carbon atoms alone would, and if the magnetic atom’s energy is below a critical value, then the graphene will exhibit an ordinary-looking Kondo effect as it becomes increasingly resistant to current the more it’s cooled below a certain temperature. On the other hand, if the magnetic atom’s energy is above the critical value, the graphene (whether it’s doped or not) will still exhibit a Kondo-like effect, but the way the graphene’s resistance depends on the temperature will be quite different, because in this situation, the forces within the material that the graphene’s conduction electrons are subject to makes them react to additional forces as if they had no rest mass.

According to the calculation described in “Theory of Fano Resonances in Graphene: The Kondo effect probed by STM”^[^{SciTech Connect}^], a similar type of Kondo effect should occur if the magnetic atom is like cobalt, whose uncancelled magnetic field is associated with an electron in one of its d orbitals^[^Wikipedia^]. This report’s authors found that the energies of the cobalt’s d-orbital electrons, which would be more nearly equal in an isolated cobalt atom, would be much more different from each other in graphene, and that electrons in the most energetic d-orbital state would produce a Kondo effect in the cobalt-graphene system. Also, this Kondo effect is related to the conduction electrons behaving as if they had no rest mass in this case either.

The calculations reported in these papers involved solving specific versions of a general equation that describes significant electron interactions in a conductor that contains magnetic atoms. A complete solution of the general equation itself would describe any material that might exhibit the Kondo effect, not just specific materials like a graphene sheet. But only for relatively few equations do we know general methods for solving them exactly, and the equation discussed in these two papers is not among them. In situations like this, researchers will often seek a solution for a simpler equation that will at least approximately describe the features they want to understand. While this often works out, sometimes the equation being solved turns out to oversimplify the phenomena of interest, so that its solution either describes them very inaccurately, or fails to indicate that they even exist. In that case a different simplification is called for.

“Validity of equation-of-motion approach to Kondo problem in the large N limit”^[^{SciTech Connect}^] describes a method of making just such an improved analysis of the Kondo effect in nonmagnetic conductors containing magnetic atoms. While the analysis its authors sought to improve upon had successfully described the materials’ resistivity and other features for higher temperatures, one symptom of its shortcomings was its inaccurate description of the ways that conduction electrons could move when the uncancelled electron in a magnetic atom had only two possible states of motion. In metals, the electrons that are shared among the atoms and can constitute electric currents can move only in certain definite ways, each way having a characteristic energy. It’s almost 100% certain that electrons will actually be executing the lowest-energy motions, and equally certain that no electrons will move with very high energies. Any electron motions that would be possible at one particular energy level (the Fermi level^[^Wikipedia^]) have a precisely 50% probability of being executed by an electron if the metal is at a uniform temperature^[^Wikipedia^]. In materials that exhibit the Kondo effect, the possible shared-electron motions whose energy levels are within a given small range are more numerous if that range includes the Fermi level than if that range is immediately above or below it. Expressed differently, the density of possible shared-electron states near a given energy is higher near the Fermi level than just above or below it. However, the earlier analysis yielded a distribution of electron-state energy levels that didn’t reflect this. The authors of the improved analysis succeeded by accounting for previously omitted details of the electron interactions that had seemed relatively insignificant, but turned out to be responsible for features like the electron-energy distribution.

Figure 2. (From “Validity of equation-of-motion approach to Kondo problem in the large N limit”^[^{SciTech Connect}^], p. 3.) Graph showing how many states of motion are calculated to be possible for shared electrons in a material that exhibits the Kondo effect when each magnetic atom has N possible states for the magnetic field of its uncancelled electron. The horizontal axis represents possible energies, centered on the material’s Fermi level^[^Wikipedia^]; the vertical axis represents the number of possible states of motion for shared electrons per unit energy range (“density of states”, or DOS). Results of the new calculations reported in “Validity of equation-of-motion approach to Kondo problem in the large N limit” are represented by solid lines and show peaks in the density of states near the Fermi level—an improvement on earlier calculations that failed to show this phenomenon for the case N=2, as the dashed line representing that result illustrates.

A progress report on another group’s work (“Scalable Methods for Electronic Excitations and Optical Responses of Nanostructures: Mathematics to Algorithms to Observables”^[^{SciTech Connect}^]) describes the status of a different effort to mathematically account for this phenomenon in realistic models of nanometer-scale physical systems. Recognizing that earlier analyses represented effects on the electrons by an average of the forces acting on them, these researchers added to these analyses mathematical descriptions of the electron states that accounted for the electrons’ correlations.

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Kondo lattices; superconductivity

The preceding reports dealt with materials in which the magnetic atoms had no significant direct or indirect interactions with each other. In those cases one may describe the material as though it consists of only one of its portions—one that contains all the nonmagnetic atoms in the neighborhood of a single magnetic atom. This is actually a reasonable approximation since the magnetic atoms in different neighborhoods barely affect each other: with respect to the interactions between shared and magnetic-atom electrons that produce the Kondo effect, the material as a whole acts much like a collection of separate neighborhoods. However, an explicit accounting of multiple magnetic atoms is needed to understand materials whose magnetic atoms are numerous and form a regularly-spaced lattice. These magnetic atoms can directly affect each other, particularly if their magnetism derives from electrons in atomic d or f orbitals^[^Wikipedia^].

Figure 3. Schematic diagrams comparing the neighborhood of a magnetic atom in a dilute magnetic solid that exhibits the Kondo effect with a similarly-sized neighborhood in a material with a higher density of magnetic atoms in a regularly-spaced lattice arrangement (a “Kondo lattice”). Red spheres represent the magnetic atoms, orange arrows indicate the direction of those atoms’ magnetic fields, the light blue area represents the “sea” of electrons that are shared among all the atoms in the material and are free to move among them and form electric currents, and the blue arrows represent the magnetic field directions of shared electrons close to the magnetic atoms. Nonmagnetic atoms’ locations aren’t represented in this diagram. The schematics illustrate a condition in which magnetic atoms induce shared electrons near them to orient their magnetic fields in the opposite direction. In this condition, these electrons inhibit the flow of other shared electrons, thus increasing the material’s resistance to electric current. (From “Kondo Physics and Unconventional Superconductivity in the U Intermetallic U₂PtC₂ Revealed by NMR”^[^{SciTech Connect}^], pp. 7 and 8/slides 5 and 6.)

In a project reported in “Synchrotron Studies of Quantum Emergence in Non-Low Dimensional Materials Final Report”^[^{SciTech Connect}^], the momenta and energies of electrons in this latter class of materials were measured from how the electrons responded to x-rays from synchrotrons^[^Wikipedia^], and mapped at exceptionally high resolution with a view to clarifying how such materials’ electron structure is related to effects of strong interactions among the electrons, including a variation of the Kondo effect. Among the materials examined were those having the general chemical formula^[^Wikipedia^] Ce_1-xYb_xCoIn₅, with x having various values between 0 and 1 inclusive. Comparison of electron momenta and energies in CeCoIn₅ (Ce_1-xYb_xCoIn₅ with x=0) at a temperature of 26 kelvins (-247.15 °C) with mathematical derivations of what they should be under different conditions showed that the most energetic electrons from the Ce atoms^[^Wikipedia^] apparently remained confined to those atoms rather than leaving them to be shared among all the atoms in the material.

Two other theoretical analyses lead to different conclusions about the temperatures at which two phenomena should occur: (1) the shared electrons’ moving coherently and responding to external forces as if their rest mass had greatly increased, and (2) a change in the range of possible electron motions whose energies are below the Fermi level. If the premises of one analysis are correct, both phenomena would be expected to occur together; if the other analysis’ different premises are correct instead, the first phenomenon should occur at a much higher temperature than the second. The first analysis indicates that the coherent shared-electron state (called a “Kondo liquid”) should have formed in CeCoIn₅ at 45 kelvins (almost twice the experimental temperature), but the range of possible electron energies below the Fermi level wasn’t altered. However, if the coherent Kondo liquid did indeed form before the CeCoIn₅ reached 26 kelvins, the lack of change in the sub-Fermi level electron motions could be consistent with the second theory. Other experiments to directly determine the temperature at which a Kondo liquid does form might help rule out at least one of the two theories.

The report’s author notes that strongly-interacting electron effects exemplify various types of phenomena of wider interest. For one thing, they are an example of collective behaviors, which occur in relatively simple systems like Kondo-lattice materials, and in systems of much greater complexity, such as human societies. Different as these systems are, the more easily analyzed behaviors of electrons in solids might provide some paradigms to serve as a starting point for understanding societal phenomena. For another thing, they exemplify collective electron behaviors that would not occur among weakly-interacting or noninteracting electrons. If different collective electron effects become sufficiently balanced with each other in a material, it reaches a “quantum critical point”^[^Wikipedia^] like the one attained when a material is poised between states like magnetism and superconductivity—both of which are technologically useful.

Researchers assumed the second theory considered in this study when they analyzed the results of experiments they performed and reported in “Magnetic excitations in Kondo liquid: superconductivity and hidden magnetic quantum critical fluctuations”^[^{SciTech Connect}^].

Like electrons, the particles in atomic nuclei have orbital motions and spins that are associated with magnetic fields; if any of the particles has a magnetic field that isn’t cancelled by the field of some other particle in the nucleus, the nucleus as a whole will act as a magnet. If the nucleus is aligned parallel or antiparallel to an external magnetic field, and then exposed to radio waves or microwaves of an appropriate frequency, it will either absorb energy from the waves or emit waves of the same frequency (a phenomenon known as nuclear magnetic resonance, or NMR^[^Wikipedia^]). The appropriate radiowave or microwave frequency depends on the external magnetic field strength, and also on the magnetic fields of the neighboring electrons—which means that, for a given metallic atom, this frequency will depend on whether the atom is among other atoms of the same metal or among nonmetallic atoms in a different material. This frequency difference is named the Knight shift^[^Wikipedia^] after its discoverer, Walter D. Knight^[^Wikipedia^].

Unusual variations with temperature occur in the Knight shift observed for both normally conducting and superconducting CeCoIn₅, which result from changes in the state of its electrons due to different causes. Assuming the aforementioned theory of how the shared electrons’ effective rest mass increases, the researchers deduced what it implied about different processes’ contributions to the Knight shift variations. These deductions, in turn, suggest among other things “a highly promising candidate for the physical origin” of CeCoIn₅’s superconductivity—an “unconventional” sort of superconductivity that apparently isn’t caused by the same mechanism^[^Wikipedia^] that operates in the earliest-known superconductors.

Besides the unconventionality of their superconductivity mechanism, another reason for the interest in superconductors with Kondo lattices is the fact that earlier experiments had suggested magnetic materials couldn’t superconduct. This is brought out in a slide presentation about nuclear magnetic resonance experiments with a different unconventional superconductor, entitled “Kondo Physics and Unconventional Superconductivity in the U Intermetallic U₂PtC₂ Revealed by NMR”^[^{SciTech Connect}^]. The evidence gained from these experiments suggests that electrons from the material’s platinum atoms bind into pairs that enter a single coherent state, as in conventional superconductors, but differ in that the electrons in each pair have their spins parallel instead of antiparallel.

Determining exactly how electrons in a superconductor bind together to form pairs would solve a significant problem. Unpaired electrons necessarily occupy different states of motion and spin—a quantum-mechanical consequence of their spins having a component along any direction of half a unit h of angular momentum^[^Wikipedia^]. Bound pairs of electrons, however, have a whole number of units h of angular momentum, and those pairs can move identically to form a supercurrent which can’t be deflected by conducting-material irregularities that resist currents of differently-moving electrons. But electrons can only bind into pairs if they overcome the mutual electrical repulsion of their identical negative charges. For a material to superconduct, it has to have some internal interactions that overcome the electrons’ mutual repulsion. A possible set of interactions for unconventional superconductors was examined in a technical report entitled “Topological confinement and superconductivity”^[^{SciTech Connect}^], which was later published in revised form as “Robust pairing mechanism from repulsive interactions” in the journal Physical Review B. This report followed up on the observation that unconventional superconductors become superconducting under conditions close to those under which the atoms’ magnetic fields align oppositely to those of their immediate neighbors^[^Wikipedia^]. The authors’ calculations demonstrate that the interplay between this alternating-alignment tendency of neighboring atoms and the sharing of electrons among atoms throughout the material, in materials having a particular type of Kondo lattice, can result in the binding of electrons into superconducting pairs.

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Related phenomena

As we’ve seen above, the strong interactions of the numerous, closely-spaced magnetic atoms in a Kondo lattice lead to electronic phenomena that don’t happen in the dilute magnetic-atom solutions originally studied by Kondo. However, the presence of a Kondo lattice alone doesn’t necessarily make all electronic phenomena different. “Q-dependence of the spin fluctuations in the intermediate valence compound CePd₃”^[^{SciTech Connect}^] describes experiments in which neutrons were found to scatter quite differently from a Kondo-lattice solid depending on the solid’s temperature. The neutrons scattered from CePd₃ at 7 kelvins (as well as from LaPd₃ at that temperature) lost energy in a pattern that suggested they were interacting with a regular, relatively dense array of magnetic cerium atoms whose most energetic f-orbital electrons were coherent . But from room-temperature CePd₃ and LaPd₃, the neutrons lost very little energy, and their scattering pattern was what would be expected if the materials had relatively few magnetic atoms that interacted little or not at all.

Many of the aforementioned studies deal with compounds of cerium^[^Wikipedia^], an element that “continues to excite both theoretical and experimental work based on its unique behavior among the elements”. Like some other solid materials, cerium compounds have more than one stable arrangement of their atoms. Which arrangement is most stable depends on conditions like temperature and pressure; change the conditions sufficiently and the atoms will rearrange. One rearrangement in cerium compounds is particularly striking, and is exemplified by pure cerium. At room temperature (about 300 kelvins) and atmospheric pressure, pure cerium has one atomic arrangement, but under a few hundred times atmospheric pressure, the atoms suddenly collapse into a different stable phase into a volume less than 86% of what it originally occupied. At higher temperatures, the volume change is less; at a critical temperature around 480 kelvins, the volume doesn’t suddenly collapse at any pressure.

Why this happens has been a subject of speculation, leading to different theoretical models of the phenomenon. One model links the collapse to the properties of the Kondo lattice of magnetic atoms in cerium alloys. (In pure cerium, all the atoms constitute a Kondo lattice.) According to this model, the volume collapse is caused by a change in how the shared electrons in cerium screen the magnetic fields of the atoms’ most energetic f-orbital electrons. Mathematical calculations show general corroboration of this concept, and indicate that cerium alloys should have a lower critical temperature below which the alloys’ volume shouldn’t suddenly change with pressure. Results of experiments to provide “the first truly quantitative test” of this model are described in “Thermal signatures of the Kondo volume collapse in cerium”^[^{SciTech Connect}^], and include data on the pressure-volume relationship for cerium from room temperature to 775 kelvins. The data suggests that effect of temperature variation on cerium’s atomic arrangement changes profoundly across the transition between cerium’s collapsed and uncollapsed phases, reflecting how greatly the interatomic forces between cerium atoms are affected by how well the atoms’ magnetic fields are screened.

While the Kondo effect may be directly related to phenomena like those just mentioned, other effects analogous to the Kondo effect have received attention in other studies.

One possible analog to the Kondo effect in solids is a similar effect in individual molecules. Calculations to examine the causes of unusual magnetic properties in certain organometallic molecules suggest their origin in “the intertwined nature of both a strong magnetic coupling strength and the electronic structure of the metal”, according to the report “Decamethylytterbocene Complexes of Bipyridines and Diazabutadienes: Multiconfigurational Ground States and Open-Shell Singlet Formation”^[^{SciTech Connect}^]. “The analogy of this molecular phenomenon to the Kondo effect in solid-state intermetallic magnetism,” it says, “continues to have predictive power, furthering the assertion that these organometallic molecules may be ideal systems with which to study the Kondo effect on the nanoscale.” The authors state that the results “have implications for understanding chemical bonding not only in organolanthanide complexes, but also for f-element chemistry in general, as well as understanding magnetic interactions in nanoparticles and devices.”

Figure 4. Schematic diagram showing bipyridine and diazabutadiene molecules discussed in “Decamethylytterbocene Complexes of Bipyridines and Diazabutadienes: Multiconfigurational Ground States and Open-Shell Singlet Formation”^[^{SciTech Connect}^]. The figures show general structural formulas in which the unlabeled vertices represent carbon atoms, the Ns represent nitrogen atoms, and the Rs and R´s represent various atoms or groups listed in the table beneath the structural formulas. Abbreviated names for the different molecules with the specified R and R´ groups are listed in the “abbreviation” columns. (From “Decamethylytterbocene Complexes of Bipyridines and Diazabutadienes: Multiconfigurational Ground States and Open-Shell Singlet Formation”^[^{SciTech Connect}^], p. 5.)

Experiments related to another type of resistance increase with cooling were conducted with a dilute solid solution whose atoms don’t differ in their possible magnetic-field orientations, but in the amount of electric charge they might retain. The report “Evidence for charge Kondo effect in superconducting Tl-doped PbTe”^[^{SciTech Connect}^] makes the case that this phenomenon occurs when lead telluride (PbTe) is doped with thallium (Tl), one of several elements that can lose 1 or 3 electrons when combining with other atoms but apparently doesn’t lose other numbers. Thallium-doped lead telluride is also a superconductor, and the report’s authors note that the tunneling of pairs of electrons on and off the thallium atoms, changing its charge from +3 to +1 and back, would provide a way to bond those electron pairs together to form a supercurrent. The authors measured single crystals’ attraction to magnetic fields, in which thallium atoms substituted for between 0.3% and 1.5% of the lead atoms, as well as the crystals’ electrical resistivity under different magnetic fields. They observed “an anomalous low-temperature upturn in the resistivity that scales in magnitude with the Tl concentration, with a temperature dependence that is consistent with the Kondo effect”, “demonstrated that this behavior does not arise from either magnetic impurities or from localization effects”, and noted that the observed charge Kondo behavior “directly attests to an electronic pairing mechanism for superconductivity … which potentially accounts for the anomalously high [maximum superconducting temperature] T_c of this material”.

Finally, a very different Kondo effect analog is described in “A Maximally Supersymmetric Kondo Model”^[^{SciTech Connect}^]. The “Kondo model” in this case resembles a material that exhibits the ordinary Kondo effect, but the system in question is a set of elementary particles, hypothesized to consist of strings^[^Wikipedia^] whose endpoints are constrained to lie on membranelike entities (“branes”^[^Wikipedia^]). Since the mathematical relationships between different features of brane theory are analogous to the relationships between a dilute solid solution of magnetic atoms and the solvent’s shared electrons, understanding of the Kondo effect informed the solution of analogous brane-theory problems considered in the report.

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References

Wikipedia