Kondo effect

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Kondo effect: How gold with a small amount of what were probably iron impurities behaves at low temperatures Classickondo.png
Kondo effect: How gold with a small amount of what were probably iron impurities behaves at low temperatures
Jun Kondo Jun Kondo.jpg
Jun Kondo

In physics, the Kondo effect describes the scattering of conduction electrons in a metal due to magnetic impurities, resulting in a characteristic change i.e. a minimum in electrical resistivity with temperature. [1] The cause of the effect was first explained by Jun Kondo, who applied third-order perturbation theory to the problem to account for scattering of s-orbital conduction electrons off d-orbital electrons localized at impurities (Kondo model). Kondo's calculation predicted that the scattering rate and the resulting part of the resistivity should increase logarithmically as the temperature approaches 0 K. [2] Extended to a lattice of magnetic impurities, the Kondo effect likely explains the formation of heavy fermions and Kondo insulators in intermetallic compounds, especially those involving rare earth elements such as cerium, praseodymium, and ytterbium, and actinide elements such as uranium. The Kondo effect has also been observed in quantum dot systems.

Contents

Theory

The dependence of the resistivity on temperature , including the Kondo effect, is written as

where is the residual resistivity, the term shows the contribution from the Fermi liquid properties, and the term is from the lattice vibrations: , , and are constants independent of temperature. Jun Kondo derived the third term with logarithmic dependence on temperature and the experimentally observed concentration dependence.

History

In 1930, Walther Meissner and B. Voigt [3] [4] observed that the resistivity of nominally pure gold reaches a minimum at 10 K, and similarly for nominally pure Cu at 2 K. Similar results were discovered in other metals. [5] Kondo described the three puzzling aspects that frustrated previous researchers who tried to explain the effect: [6] [7]

Experiments in the 1960s by Myriam Sarachik at Bell Laboratories showed that phenomenon was caused by magnetic impurity in nominally pure metals. [8] When Kondo sent a preview of his paper to Sarachik, Sarachik confirmed the data fit the theory. [9]

Kondo's solution was derived using perturbation theory resulting in a divergence as the temperature approaches 0 K, but later methods used non-perturbative techniques to refine his result. These improvements produced a finite resistivity but retained the feature of a resistance minimum at a non-zero temperature. One defines the Kondo temperature as the energy scale limiting the validity of the Kondo results. The Anderson impurity model and accompanying Wilsonian renormalization theory were an important contribution to understanding the underlying physics of the problem. [10] Based on the Schrieffer–Wolff transformation, it was shown that the Kondo model lies in the strong coupling regime of the Anderson impurity model. The Schrieffer–Wolff transformation [11] projects out the high energy charge excitations in the Anderson impurity model, obtaining the Kondo model as an effective Hamiltonian.

Schematic of the weakly coupled high temperature situation in which the magnetic moments of conduction electrons in the metal host pass by the impurity magnetic moment at speeds of vF, the Fermi velocity, experiencing only a mild antiferromagnetic correlation in the vicinity of the impurity. In contrast, as the temperature tends to zero the impurity magnetic moment and one conduction electron moment bind very strongly to form an overall non-magnetic state. Kscheme.jpg
Schematic of the weakly coupled high temperature situation in which the magnetic moments of conduction electrons in the metal host pass by the impurity magnetic moment at speeds of vF, the Fermi velocity, experiencing only a mild antiferromagnetic correlation in the vicinity of the impurity. In contrast, as the temperature tends to zero the impurity magnetic moment and one conduction electron moment bind very strongly to form an overall non-magnetic state.

The Kondo effect can be considered as an example of asymptotic freedom, i.e. a situation where the coupling becomes non-perturbatively strong at low temperatures and low energies. In the Kondo problem, the coupling refers to the interaction between the localized magnetic impurities and the itinerant electrons.

Examples

Extended to a lattice of magnetic ions, the Kondo effect likely explains the formation of heavy fermions and Kondo insulators in intermetallic compounds, especially those involving rare earth elements such as cerium, praseodymium, and ytterbium, and actinide elements such as uranium. In heavy fermion materials, the non-perturbative growth of the interaction leads to quasi-electrons with masses up to thousands of times the free electron mass, i.e., the electrons are dramatically slowed by the interactions. In a number of instances they are superconductors. It is believed that a manifestation of the Kondo effect is necessary for understanding the unusual metallic delta-phase of plutonium.[ citation needed ]

The Kondo effect has been observed in quantum dot systems. [12] [13] In such systems, a quantum dot with at least one unpaired electron behaves as a magnetic impurity, and when the dot is coupled to a metallic conduction band, the conduction electrons can scatter off the dot. This is completely analogous to the more traditional case of a magnetic impurity in a metal.

Band-structure hybridization and flat band topology in Kondo insulators have been imaged in angle-resolved photoemission spectroscopy experiments. [14] [15] [16]

In 2012, Beri and Cooper proposed a topological Kondo effect could be found with Majorana fermions, [17] while it has been shown that quantum simulations with ultracold atoms may also demonstrate the effect. [18]

In 2017, teams from the Vienna University of Technology and Rice University conducted experiments into the development of new materials made from the metals cerium, bismuth and palladium in specific combinations and theoretical work experimenting with models of such structures, respectively. The results of the experiments were published in December 2017 [19] and, together with the theoretical work, [20] lead to the discovery of a new state, [21] a correlation-driven Weyl semimetal. The team dubbed this new quantum material Weyl-Kondo semimetal.

Related Research Articles

The quantum Hall effect is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance Rxy exhibits steps that take on the quantized values

<span class="mw-page-title-main">Fermi liquid theory</span> Theoretical model in physics

Fermi liquid theory is a theoretical model of interacting fermions that describes the normal state of the conduction electrons in most metals at sufficiently low temperatures. The theory describes the behavior of many-body systems of particles in which the interactions between particles may be strong. The phenomenological theory of Fermi liquids was introduced by the Soviet physicist Lev Davidovich Landau in 1956, and later developed by Alexei Abrikosov and Isaak Khalatnikov using diagrammatic perturbation theory. The theory explains why some of the properties of an interacting fermion system are very similar to those of the ideal Fermi gas, and why other properties differ.

The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of , where e is the electron charge and h is the Planck constant. It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations have a fractional elementary charge and possibly also fractional statistics. The 1998 Nobel Prize in Physics was awarded to Robert Laughlin, Horst Störmer, and Daniel Tsui "for their discovery of a new form of quantum fluid with fractionally charged excitations". The microscopic origin of the FQHE is a major research topic in condensed matter physics.

A two-dimensional electron gas (2DEG) is a scientific model in solid-state physics. It is an electron gas that is free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels for motion in the third direction, which can then be ignored for most problems. Thus the electrons appear to be a 2D sheet embedded in a 3D world. The analogous construct of holes is called a two-dimensional hole gas (2DHG), and such systems have many useful and interesting properties.

<span class="mw-page-title-main">Majorana fermion</span> Fermion that is its own antiparticle

A Majorana fermion, also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesised by Ettore Majorana in 1937. The term is sometimes used in opposition to a Dirac fermion, which describes fermions that are not their own antiparticles.

In materials science, heavy fermion materials are a specific type of intermetallic compound, containing elements with 4f or 5f electrons in unfilled electron bands. Electrons are one type of fermion, and when they are found in such materials, they are sometimes referred to as heavy electrons. Heavy fermion materials have a low-temperature specific heat whose linear term is up to 1000 times larger than the value expected from the free electron model. The properties of the heavy fermion compounds often derive from the partly filled f-orbitals of rare-earth or actinide ions, which behave like localized magnetic moments.

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A composite fermion is the topological bound state of an electron and an even number of quantized vortices, sometimes visually pictured as the bound state of an electron and, attached, an even number of magnetic flux quanta. Composite fermions were originally envisioned in the context of the fractional quantum Hall effect, but subsequently took on a life of their own, exhibiting many other consequences and phenomena.

<span class="mw-page-title-main">Topological insulator</span> State of matter with insulating bulk but conductive boundary

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<span class="mw-page-title-main">Luttinger's theorem</span>

In condensed matter physics, Luttinger's theorem is a result derived by J. M. Luttinger and J. C. Ward in 1960 that has broad implications in the field of electron transport. It arises frequently in theoretical models of correlated electrons, such as the high-temperature superconductors, and in photoemission, where a metal's Fermi surface can be directly observed.

<span class="mw-page-title-main">Kondo insulator</span>

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Piers Coleman is a British-born theoretical physicist, working in the field of theoretical condensed matter physics. Coleman is professor of physics at Rutgers University in New Jersey and at Royal Holloway, University of London.

In condensed matter physics, a quantum spin liquid is a phase of matter that can be formed by interacting quantum spins in certain magnetic materials. Quantum spin liquids (QSL) are generally characterized by their long-range quantum entanglement, fractionalized excitations, and absence of ordinary magnetic order.

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<span class="mw-page-title-main">Dirac cone</span> Quantum effect in some non-metals

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Fractional Chern insulators (FCIs) are lattice generalizations of the fractional quantum Hall effect that have been studied theoretically since early 2010. They were first predicted to exist in topological flat bands carrying Chern numbers. They can appear in topologically non-trivial band structures even in the absence of the large magnetic fields needed for the fractional quantum Hall effect. They promise physical realizations at lower magnetic fields, higher temperatures, and with shorter characteristic length scales compared to their continuum counterparts. FCIs were initially studied by adding electron-electron interactions to a fractionally filled Chern insulator, in one-body models where the Chern band is quasi-flat, at zero magnetic field. The FCIs exhibit a fractional quantized Hall conductance.

References

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Further reading