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] Experiments in the 1960s by Myriam Sarachik at Bell Laboratories provided the first data that confirmed the Kondo effect. [3] 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.

Background

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. [4] 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 [5] 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. [6] [7] 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. [8] [9] [10]

In 2012, Beri and Cooper proposed a topological Kondo effect could be found with Majorana fermions, [11] while it has been shown that quantum simulations with ultracold atoms may also demonstrate the effect. [12]

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 [13] and, together with the theoretical work, [14] lead to the discovery of a new state, [15] 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.

<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.

The Anderson impurity model, named after Philip Warren Anderson, is a Hamiltonian that is used to describe magnetic impurities embedded in metals. It is often applied to the description of Kondo effect-type problems, such as heavy fermion systems and Kondo insulators. In its simplest form, the model contains a term describing the kinetic energy of the conduction electrons, a two-level term with an on-site Coulomb repulsion that models the impurity energy levels, and a hybridization term that couples conduction and impurity orbitals. For a single impurity, the Hamiltonian takes the form

The quantum spin Hall state is a state of matter proposed to exist in special, two-dimensional semiconductors that have a quantized spin-Hall conductance and a vanishing charge-Hall conductance. The quantum spin Hall state of matter is the cousin of the integer quantum Hall state, and that does not require the application of a large magnetic field. The quantum spin Hall state does not break charge conservation symmetry and spin- conservation symmetry.

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

A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material.

<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>

In solid-state physics, Kondo insulators are understood as materials with strongly correlated electrons, that open up a narrow band gap at low temperatures with the chemical potential lying in the gap, whereas in heavy fermion materials the chemical potential is located in the conduction band.

<span class="mw-page-title-main">Piers Coleman</span> British-American physicist

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.

The Kondo model is a model for a single localized quantum impurity coupled to a large reservoir of delocalized and noninteracting electrons. The quantum impurity is represented by a spin-1/2 particle, and is coupled to a continuous band of noninteracting electrons by an antiferromagnetic exchange coupling . The Kondo model is used as a model for metals containing magnetic impurities, as well as quantum dot systems.

<span class="mw-page-title-main">Samarium hexaboride</span> Chemical compound

Samarium hexaboride (SmB6) is an intermediate-valence compound where samarium is present both as Sm2+ and Sm3+ ions at the ratio 3:7. It is a Kondo insulator having a metallic surface state.

Weyl semimetals are semimetals or metals whose quasiparticle excitation is the Weyl fermion, a particle that played a crucial role in quantum field theory but has not been observed as a fundamental particle in vacuum. In these materials, electrons have a linear dispersion relation, making them a solid-state analogue of relativistic massless particles.

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

In physics, Dirac cones are features that occur in some electronic band structures that describe unusual electron transport properties of materials like graphene and topological insulators. In these materials, at energies near the Fermi level, the valence band and conduction band take the shape of the upper and lower halves of a conical surface, meeting at what are called Dirac points.

<span class="mw-page-title-main">Electronic properties of graphene</span>

Graphene is a semimetal whose conduction and valence bands meet at the Dirac points, which are six locations in momentum space, the vertices of its hexagonal Brillouin zone, divided into two non-equivalent sets of three points. The two sets are labeled K and K'. The sets give graphene a valley degeneracy of gv = 2. By contrast, for traditional semiconductors the primary point of interest is generally Γ, where momentum is zero. Four electronic properties separate it from other condensed matter systems.

The term Dirac matter refers to a class of condensed matter systems which can be effectively described by the Dirac equation. Even though the Dirac equation itself was formulated for fermions, the quasi-particles present within Dirac matter can be of any statistics. As a consequence, Dirac matter can be distinguished in fermionic, bosonic or anyonic Dirac matter. Prominent examples of Dirac matter are graphene and other Dirac semimetals, topological insulators, Weyl semimetals, various high-temperature superconductors with -wave pairing and liquid helium-3. The effective theory of such systems is classified by a specific choice of the Dirac mass, the Dirac velocity, the gamma matrices and the space-time curvature. The universal treatment of the class of Dirac matter in terms of an effective theory leads to a common features with respect to the density of states, the heat capacity and impurity scattering.

Magnetic topological insulators are three dimensional magnetic materials with a non-trivial topological index protected by a symmetry other than time-reversal. In contrast with a non-magnetic topological insulator, a magnetic topological insulator can have naturally gapped surface states as long as the quantizing symmetry is broken at the surface. These gapped surfaces exhibit a topologically protected half-quantized surface anomalous Hall conductivity perpendicular to the surface. The sign of the half-quantized surface anomalous Hall conductivity depends on the specific surface termination.

References

  1. ↑ Hewson, Alex C; Jun Kondo (2009). "Kondo effect". Scholarpedia. 4 (3): 7529. Bibcode:2009SchpJ...4.7529H. doi: 10.4249/scholarpedia.7529 .
  2. ↑ Kondo, Jun (1964). "Resistance Minimum in Dilute Magnetic Alloys". Progress of Theoretical Physics. 32 (1): 37–49. Bibcode:1964PThPh..32...37K. doi: 10.1143/PTP.32.37 .
  3. ↑ Chang, Kenneth (August 31, 2020). "Myriam Sarachik Never Gave Up on Physics". The New York Times. ISSN   0362-4331. Archived from the original on August 31, 2020. Retrieved October 13, 2021.
  4. ↑ Anderson, P. (1961). "Localized Magnetic States in Metals" (PDF). Physical Review. 124 (1): 41–53. Bibcode:1961PhRv..124...41A. doi:10.1103/PhysRev.124.41.[ permanent dead link ]
  5. ↑ Schrieffer, J.R.; Wolff, P.A. (September 1966). "Relation between the Anderson and Kondo Hamiltonians". Physical Review. 149 (2): 491–492. Bibcode:1966PhRv..149..491S. doi:10.1103/PhysRev.149.491. S2CID   55838235.
  6. ↑ Cronenwett, Sara M. (1998). "A Tunable Kondo Effect in Quantum Dots". Science. 281 (5376): 540–544. arXiv: cond-mat/9804211 . Bibcode:1998Sci...281..540C. doi:10.1126/science.281.5376.540. PMID   9677192. S2CID   5139144.
  7. ↑ "Revival of the Kondo" (PDF).{{cite journal}}: Cite journal requires |journal= (help)
  8. ↑ Neupane, Madhab; Alidoust, Nasser; Belopolski, Ilya; Bian, Guang; Xu, Su-Yang; Kim, Dae-Jeong; Shibayev, Pavel P.; Sanchez, Daniel S.; Zheng, Hao; Chang, Tay-Rong; Jeng, Horng-Tay; et al. (2015-09-18). "Fermi surface topology and hot spot distribution in the Kondo lattice system CeB6". Physical Review B. 92 (10): 104420. arXiv: 1411.0302 . Bibcode:2015PhRvB..92j4420N. doi: 10.1103/PhysRevB.92.104420 .
  9. ↑ Neupane, M.; Alidoust, N.; Xu, S.-Y.; Kondo, T.; Ishida, Y.; Kim, D. J.; Liu, Chang; Belopolski, I.; Jo, Y. J.; Chang, T.-R.; Jeng, H.-T. (2013). "Surface electronic structure of the topological Kondo-insulator candidate correlated electron system SmB6". Nature Communications. 4 (1): 2991. arXiv: 1312.1979 . Bibcode:2013NatCo...4.2991N. doi: 10.1038/ncomms3991 . ISSN   2041-1723. PMID   24346502.
  10. ↑ Hasan, M. Zahid; Xu, Su-Yang; Neupane, Madhab (2015), "Topological Insulators, Topological Dirac semimetals, Topological Crystalline Insulators, and Topological Kondo Insulators", Topological Insulators, John Wiley & Sons, Ltd., pp. 55–100, doi:10.1002/9783527681594.ch4, ISBN   978-3-527-68159-4 , retrieved 2020-04-26
  11. ↑ Béri, B.; Cooper, N. R. (2012). "Topological Kondo Effect with Majorana Fermions". Physical Review Letters. 109 (15): 156803. arXiv: 1206.2224 . Bibcode:2012PhRvL.109o6803B. doi:10.1103/PhysRevLett.109.156803. PMID   23102351. S2CID   45712589.
  12. ↑ Buccheri, F.; Bruce, G. D.; Trombettoni, A.; Cassettari, D.; Babujian, H.; Korepin, V. E.; Sodano, P. (2016-01-01). "Holographic optical traps for atom-based topological Kondo devices". New Journal of Physics. 18 (7): 075012. arXiv: 1511.06574 . Bibcode:2016NJPh...18g5012B. doi:10.1088/1367-2630/18/7/075012. ISSN   1367-2630. S2CID   118610269.
  13. ↑ Dzsaber, S.; Prochaska, L.; Sidorenko, A.; Eguchi, G.; Svagera, R.; Waas, M.; Prokofiev, A.; Si, Q.; Paschen, S. (2017-06-16). "Kondo Insulator to Semimetal Transformation Tuned by Spin-Orbit Coupling". Physical Review Letters. 118 (24): 246601. arXiv: 1612.03972 . Bibcode:2017PhRvL.118x6601D. doi: 10.1103/PhysRevLett.118.246601 . ISSN   0031-9007. PMID   28665644.
  14. ↑ Lai, H.H.; Grefe, S.E.; Paschen, S.; Si, Q. (2012). "Weyl–Kondo semimetal in heavy-fermion systems". Proceedings of the National Academy of Sciences of the United States of America. 115 (1): 93–97. arXiv: 1206.2224 . Bibcode:2018PNAS..115...93L. doi: 10.1073/pnas.1715851115 . PMC   5776817 . PMID   29255021.
  15. ↑ Gabbatiss, J. (2017) "Scientists discover entirely new material that cannot be explained by classical physics", The Independent
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