Kondo effect

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.{{Cite journal |first=Alex C |last=Hewson |author2=Jun Kondo |date=2009 |title=Kondo effect |journal=Scholarpedia |volume=4 |number=3 |page=7529 |doi=10.4249/scholarpedia.7529 |bibcode=2009SchpJ...4.7529H |author2-link=Jun Kondo |doi-access=free }}

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.{{cite journal|doi=10.1143/PTP.32.37|title=Resistance Minimum in Dilute Magnetic Alloys|date=1964|last1=Kondo|first1=Jun|journal=Progress of Theoretical Physics|volume=32|issue=1|pages=37–49|bibcode = 1964PThPh..32...37K |doi-access=free}} 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.

Theory

The dependence of the resistivity $\rho$ on temperature $T$ , including the Kondo effect, is written as

: $\rho(T) = \rho_0 + aT^2 + c_m \ln\frac{\mu}{T} + bT^5,$

where $\rho_0$ is the residual resistivity, the term $aT^2$ shows the contribution from the Fermi liquid properties, and the term $bT^5$ is from the lattice vibrations: $a$ , $b$ , $c_m$ and $\mu$ 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{{Cite journal |last1=Meissner |first1=W. |last2=Voigt |first2=B. |date=January 1930 |title=Messungen mit Hilfe von flüssigem Helium XI Widerstand der reinen Metalle in tiefen Temperaturen |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19303990702 |journal=Annalen der Physik |language=en |volume=399 |issue=7 |pages=761–797 |doi=10.1002/andp.19303990702 |bibcode=1930AnP...399..761M |issn=0003-3804|url-access=subscription }}{{Cite journal |last1=Meissner |first1=W. |last2=Voigt |first2=B. |date=January 1930 |title=Messungen mit Hilfe von flüssigem Helium XI Widerstand der reinen Metalle in tiefen Temperaturen |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19303990803 |journal=Annalen der Physik |language=en |volume=399 |issue=8 |pages=892–936 |doi=10.1002/andp.19303990803 |bibcode=1930AnP...399..892M |issn=0003-3804|url-access=subscription }} 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.{{Cite journal |last1=de Haas |first1=W. J. |last2=de Boer |first2=J. |last3=van dën Berg |first3=G. J. |date=1934-05-01 |title=The electrical resistance of gold, copper and lead at low temperatures |url=https://www.sciencedirect.com/science/article/pii/S0031891434803102 |journal=Physica |volume=1 |issue=7 |pages=1115–1124 |doi=10.1016/S0031-8914(34)80310-2 |bibcode=1934Phy.....1.1115D |issn=0031-8914|url-access=subscription }} Kondo described the three puzzling aspects that frustrated previous researchers who tried to explain the effect:{{Citation |last=Kondo |first=J. |title=Theory of Dilute Magnetic Alloys |date=1970-01-01 |series=Solid State Physics |volume=23 |pages=183–281 |editor-last=Seitz |editor-first=Frederick |url=https://www.sciencedirect.com/science/article/pii/S0081194708606165 |access-date=2024-06-01 |publisher=Academic Press |doi=10.1016/S0081-1947(08)60616-5 |isbn=978-0-12-607723-0 |editor2-last=Turnbull |editor2-first=David |editor3-last=Ehrenreich |editor3-first=Henry|url-access=subscription }}{{Cite journal |last=Kondo |first=Jun |date=January 2005 |title=Sticking to My Bush |url=https://journals.jps.jp/doi/10.1143/JPSJ.74.1 |journal=Journal of the Physical Society of Japan |language=en |volume=74 |issue=1 |pages=1–3 |doi=10.1143/JPSJ.74.1 |bibcode=2005JPSJ...74....1K |issn=0031-9015|url-access=subscription }}

The resistivity of a truly pure metal is expected to decrease monotonically, because with lower temperature, the probability of electron-phonon scattering decreases.
The resistivity should rapidly plateau when the temperature drops below the Debye temperature of the phonons, corresponding with the highest allowed mode of vibration of the metal. However, in the AuFe alloy, the resistivity continues to rise sharply below 0.01 K, yet there seemed to be no energy gap in AuFe alloy that small.
The phenomenon is universal, so any explanation should apply in general.

Experiments in the 1960s by Myriam Sarachik at Bell Laboratories showed that phenomenon was caused by magnetic impurity in nominally pure metals.{{Cite journal |last1=Sarachik |first1=M. P. |last2=Corenzwit |first2=E. |last3=Longinotti |first3=L. D. |date=1964-08-17 |title=Resistivity of Mo-Nb and Mo-Re Alloys Containing 1% Fe |url=https://link.aps.org/doi/10.1103/PhysRev.135.A1041 |journal=Physical Review |volume=135 |issue=4A |pages=A1041–A1045 |doi=10.1103/PhysRev.135.A1041|bibcode=1964PhRv..135.1041S |url-access=subscription }} When Kondo sent a preview of his paper to Sarachik, Sarachik confirmed the data fit the theory.{{Cite news |last=Chang |first=Kenneth |date=August 31, 2020 |title=Myriam Sarachik Never Gave Up on Physics |url=https://www.nytimes.com/2020/08/31/science/myriam-sarachik-physics.html |url-status=live |archive-url=https://web.archive.org/web/20200831164630/https://www.nytimes.com/2020/08/31/science/myriam-sarachik-physics.html |archive-date=August 31, 2020 |access-date=October 13, 2021 |work=The New York Times |language=en-US |issn=0362-4331}}

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.{{cite journal|doi=10.1103/PhysRev.124.41|url=http://cos.cumt.edu.cn/jpkc/dxwl/zl/zl1/Physical%20Review%20Classics/condensed/061.pdf|title=Localized Magnetic States in Metals|date=1961|last1=Anderson|first1=P.|journal=Physical Review|volume=124|issue=1|pages=41–53|bibcode=1961PhRv..124...41A}}{{Dead link|date=May 2023 |bot=InternetArchiveBot |fix-attempted=yes }} 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{{cite journal|last=Schrieffer, J.R.|author2=Wolff, P.A. |title=Relation between the Anderson and Kondo Hamiltonians|journal=Physical Review|date=September 1966|volume=149|issue=2|pages=491–492|doi=10.1103/PhysRev.149.491|bibcode = 1966PhRv..149..491S |s2cid=55838235 }} projects out the high energy charge excitations in the Anderson impurity model, obtaining the Kondo model as an effective Hamiltonian.

Image:kscheme.jpg

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|date=April 2020}}

The Kondo effect has been observed in quantum dot systems.{{cite journal|doi=10.1126/science.281.5376.540|pmid=9677192|first=Sara M.|last=Cronenwett|date=1998 |title=A Tunable Kondo Effect in Quantum Dots|journal=Science|volume=281|number=5376|pages=540–544|arxiv = cond-mat/9804211 |bibcode = 1998Sci...281..540C |s2cid=5139144}}{{Cite journal|url=http://kouwenhovenlab.tudelft.nl/wp-content/uploads/2011/10/97-revival-of-the-kondo.pdf|title=Revival of the Kondo|access-date=2016-08-19|archive-date=2017-05-17|archive-url=https://web.archive.org/web/20170517124645/http://kouwenhovenlab.tudelft.nl/wp-content/uploads/2011/10/97-revival-of-the-kondo.pdf|url-status=dead}} 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.{{Cite journal|last1=Neupane|first1=Madhab|last2=Alidoust|first2=Nasser|last3=Belopolski|first3=Ilya|last4=Bian|first4=Guang|last5=Xu|first5=Su-Yang|last6=Kim|first6=Dae-Jeong|last7=Shibayev|first7=Pavel P.|last8=Sanchez|first8=Daniel S.|last9=Zheng|first9=Hao|last10=Chang|first10=Tay-Rong|last11=Jeng|first11=Horng-Tay|display-authors=etal|date=2015-09-18|title=Fermi surface topology and hot spot distribution in the Kondo lattice system CeB₆|journal=Physical Review B|volume=92|issue=10|pages=104420|doi=10.1103/PhysRevB.92.104420|arxiv=1411.0302|bibcode=2015PhRvB..92j4420N|doi-access=free}}{{Cite journal|last1=Neupane|first1=M.|last2=Alidoust|first2=N.|last3=Xu|first3=S.-Y.|last4=Kondo|first4=T.|last5=Ishida|first5=Y.|last6=Kim|first6=D. J.|last7=Liu|first7=Chang|last8=Belopolski|first8=I.|last9=Jo|first9=Y. J.|last10=Chang|first10=T.-R.|last11=Jeng|first11=H.-T.|date=2013|title=Surface electronic structure of the topological Kondo-insulator candidate correlated electron system SmB₆|journal=Nature Communications|language=en|volume=4|issue=1|page=2991|doi=10.1038/ncomms3991|pmid=24346502|arxiv=1312.1979|bibcode=2013NatCo...4.2991N|issn=2041-1723|doi-access=free}}{{Citation|last1=Hasan|first1=M. Zahid|title=Topological Insulators, Topological Dirac semimetals, Topological Crystalline Insulators, and Topological Kondo Insulators|date=2015|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/9783527681594.ch4|work=Topological Insulators|pages=55–100|publisher=John Wiley & Sons, Ltd.|language=en|doi=10.1002/9783527681594.ch4|isbn=978-3-527-68159-4|access-date=2020-04-26|last2=Xu|first2=Su-Yang|last3=Neupane|first3=Madhab|url-access=subscription}}

In 2012, Beri and Cooper proposed a topological Kondo effect could be found with Majorana fermions,{{Cite journal|last1=Béri|first1=B.|last2=Cooper|first2=N. R.|year=2012|title=Topological Kondo Effect with Majorana Fermions|journal=Physical Review Letters|volume=109|issue=15|pages=156803|doi=10.1103/PhysRevLett.109.156803|pmid=23102351|arxiv=1206.2224|bibcode=2012PhRvL.109o6803B|s2cid=45712589}} while it has been shown that quantum simulations with ultracold atoms may also demonstrate the effect.{{Cite journal|last1=Buccheri|first1=F.|last2=Bruce|first2=G. D.|last3=Trombettoni|first3=A.|last4=Cassettari|first4=D.|last5=Babujian|first5=H.|last6=Korepin|first6=V. E.|last7=Sodano|first7=P.|date=2016-01-01|title=Holographic optical traps for atom-based topological Kondo devices|journal=New Journal of Physics|language=en|volume=18|issue=7|pages=075012|doi=10.1088/1367-2630/18/7/075012|issn=1367-2630|arxiv=1511.06574|bibcode=2016NJPh...18g5012B|s2cid=118610269}}

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{{Cite journal|last1=Dzsaber|first1=S.|last2=Prochaska|first2=L.|last3=Sidorenko|first3=A.|last4=Eguchi|first4=G.|last5=Svagera|first5=R.|last6=Waas|first6=M.|last7=Prokofiev|first7=A.|last8=Si|first8=Q.|last9=Paschen|first9=S.|date=2017-06-16|title=Kondo Insulator to Semimetal Transformation Tuned by Spin-Orbit Coupling|journal=Physical Review Letters|language=en|volume=118|issue=24|pages=246601|doi=10.1103/PhysRevLett.118.246601|pmid=28665644|arxiv=1612.03972|bibcode=2017PhRvL.118x6601D|issn=0031-9007|doi-access=free}} and, together with the theoretical work,{{Cite journal|last1=Lai|first1=H.H.|last2=Grefe|first2=S.E.|last3=Paschen|first3=S.|last4=Si|first4=Q.|year=2012|title=Weyl–Kondo semimetal in heavy-fermion systems|journal=Proceedings of the National Academy of Sciences of the United States of America |volume=115|issue=1|pages=93–97|doi=10.1073/pnas.1715851115|pmid=29255021|pmc=5776817|arxiv=1206.2224|bibcode=2018PNAS..115...93L|doi-access=free}} lead to the discovery of a new state,Gabbatiss, J. (2017) [https://www.independent.co.uk/news/science/quantum-material-physics-weyl-kondo-semimetal-vienna-university-technology-a8121141.html "Scientists discover entirely new material that cannot be explained by classical physics"], The Independent a correlation-driven Weyl semimetal. The team dubbed this new quantum material Weyl-Kondo semimetal.

Kondo effect

Theory

History

Examples

References

Further reading