Kondo Effect

Introduction to Contemporary Quantum Matter Physics Lecture 2: Kondo effect

Lectures by: Prof. Marc Janoschek & Johan Chang Course Outline: Feb. 10-14 Monday Tuesday Wednesday Thursday Friday Room Y16J33 Y16J33 Y16J33 Y16J33 Y16J33 10-10h45 Lecture 1 Lecture 4 Lecture 7 Lecture 10 Lecture 13 Johan Marc Johan Marc Johan Fermi-liquids Quantum Supercond. Magnetism Anomalous Phase Hall effect Transitions 11-11h45 Lecture 2 Lecture 5 Lecture 8 Lecture 11 Lecture 14 Marc Johan Marc Marc Johan Kondo-physics Non-Fermi Supercond. Skyrmions Charge Order liquids Lunch – Lunch - Lunch - Lunch - Lunch - Mensa Mensa Mensa Mensa Mensa

13h30- Lecture 3 Lecture 6 Lecture 9 Lecture 12 Lecture 15 14h15 Marc Johan Johan Marc Johan Heavy Fermions Supercond. Nematicity Skyrmions Charge Order Exercise Class Exercise Class 14h30-16 14h30-16 Further Reading

• A. C. Hewson, “The Kondo problem to heavy fermions,” Cambridge Studies in Magnetism, Cambridge University Press, Cambridge ; New York, (1993). • P. Schlottman. Physics Reports 181, 1-119 (1989). • B. A. Jones, “The Kondo Effect”, in Handbook of Magnetism and Advanced Magnetic Materials (Vol I), J. Wiley and Sons (2007). • P. Coleman, “Heavy Fermions: electrons at the edge of magnetism, ” in Handbook of Magnetism and Advanced Magnetic Materials (Vol I), J. Wiley and Sons (2007). • D. I. Khomskii, “Basic Aspects of the Quantum Theory of Solids,” Cambridge University Press, (2010). • Z. Fisk, H. R. Ott, T. M. Rice & J. L. Smith, “Heavy-electron metals,“ Nature 320, 124–129 (1986). • J. D. Thompson and Z. Fisk, J. Phys. Soc. Jpn. 81, 011002 (2012). • C. Pfleiderer, Rev. Mod. Phys. 81, 1551 (2009). • Timeline Low-Temperature Resistivity of a Typical Metal: Cu

→ Low-temperature residual resistivity is determined by material purity.

→ It basically describes temperature independent scattering from impurities.

→ RRR = residual resistivity ratio. Low-Temperature Resistivity of Gold

W. J. de Haas et al., Physica 1, 1115-1124 (1934) → Should impurities not lead to a temperature independent contribution to the resistivity?

→ What is going on? This is something new! Position of Resistivity Minimum

Resistivity ρ (μΩ-cm) H. H. Hill, et al. et Temperature Temperature , CNRS Colloq., 541 (1970) Colloq.,, CNRS 541 T (K) Specific Heat

LaAl2 with Ce impurities

S. Bader et al., Solid State Communications 16, 1263 (1975)

→ Low-temperature specific heat suggests presence of new cross-over temperature/energy scale TK. Magnetic Susceptibility

Pauli–like Susceptibility

Type equation here. Curie-Weiss behavior

1 / = 2 − 4

→ The susceptibility suggests that if T > TK local magnetic moment behavior is observed.

→ For T < TK non-magnetic behavior is observed. Summary of Temperature Dependence of Bulk Properties

→ Clearly the phenomena at hand sets a new energy (temperature) scale kBTK with dramatic consequences on material properties.

→ From the magnetic susceptibility we can see that above TK the system behaves like a paramagnetic with local magnetic moments.

→ Below TK a temperature-independent Pauli susceptibility is # observed. The large value of the Pauli susceptibility ! = $% implies a large electronic density of states!

→ This suggest that this phenomena involves magnetism (from magnetic impurities) and the conduction electrons (resistivity changes…) Kondo Effect: Magnetic Impurities in a Metallic Host

J. Kondo Prog. Theo. Phys. 28 772 (1962), Prog. Theo. Phys. 32, 37 (1964).

Figure from MJ et al., Science Advances e1500188 (2015)

Localized Impurity Moment Screened Impurity Moment

Further reading: P. Schlottman. Physics Reports 181, 1-119 (1989). B. A. Jones, “The Kondo Effect”, in Handbook of Magnetism and Advanced Magnetic Materials (Vol I), J. Wiley and Sons, 2007 A. C. Hewson, The Kondo problem to heavy fermions . Cambridge studies in magnetism (Cambridge University Press, Cambridge ; New York, 1993). Logarithmic Resistivity

J. Kondo Prog. Theo. Phys. 28 772 (1962), Prog. Theo. Phys. 32, 37 (1964).

→ Using this s-d interaction Hamiltonian with the exchange J Kondo calculated the scattering rate of conduction electrons off a magnetic impurity up to one order higher than Born approximation (D is the bandwidth): Bandwidth

D = bandwidth of electron band ! = electronic DOS

→ This explains the data well. → But what is the microscopic? Anderson Impurity Hamiltonian

Atomic Conduction States Electrons Ce: Xe (4f26s2)

P. W. Anderson, Phys. Rev. 124, 41 (1961). Further reading: P. Schlottman. Physics Reports 181, 1-119 (1989). B. A. Jones, “The Kondo Effect”, in Handbook of Magnetism and Advanced Magnetic Materials (Vol I), J. Wiley and Sons, 2007 A. C. Hewson, The Kondo problem to heavy fermions . Cambridge studies in magnetism (Cambridge University Press, Cambridge ; New York, 1993). Local Moment Formation

Atomic Conduction States Electrons Ce: Xe (4f26s2)

J. Kondo Formation of Kondo Resonance Prog. Theo. Phys. 28 772 (1962), (increased scattering!) Prog. Theo. Phys. 32, 37 (1964).

2 kBTK ~ !V

Atomic Conduction States Electrons Ce: Xe (4f26s2)

J. Kondo Formation of Kondo Resonance Prog. Theo. Phys. 28 772 (1962), (increased scattering!) Prog. Theo. Phys. 32, 37 (1964).

2 kBTK ~ !V

Jim Allen et al., Phys. Rev. 28, 5347 (1983). Asymptotic Freedom

J. Kondo Prog. Theo. Phys. 28 772 (1962), Prog. Theo. Phys. 32, 37 (1964).

→ Below local moments become confined by forming a singlet state with conduction electrons.

→ Above Kondo temperature local moments are asymptotically free.

→ This is similar to quark phyiscs. Probing Magnetic Interactions Via Inelastic Neutron Scattering

Detector Shielding Cryostat kf(t) Sample Q(t) Detector φ bank

Detector Tubes ki(t) Neutron Beam Chopper Neutron Source Kondo Effect in Quantum Dots

Takeshi Inoshita, Science 281, 526-527 (1998) Sara M. Cronenwett, Tjerk H. Oosterkamp, Leo P. Kouwenhoven, Science 281, 540-544 (1998)

→ Allows for temperature controlled tunneling rate through quantum dot.

Extra Slides Probing the Kondo Interactions With Neutron Scattering

CePd3 with Kondo temperature TK ≈ 600 K (~55 meV)

T < T K TK

T > TK

Fanelli, Lawrence et al, J Phys: Condens. Matter 26 (2014) 225602 The Magnetic Conundrum in δ-Pu

For extensive review see: J. C. Lashley et al., Phys. Rev. B 72, 054416 (2005)

P. Söderlind et al., Phys. Rev. B 50, 7291 (1994); S. Y. Savrasov and G. Kotliar, Phys. Rev. Lett. 84, 3670 (2000); J. Bouchet, et al., J. Phys.: Condens. Matter 12, 1723 (2000); O. Eriksson, et al., J. Alloys Compd. 287, 1 (1999); B. R. Cooper, et al., Philos. Mag. B 79, 683 (1999); S. Y. Savrasov, et al., Nature 10, 793 (2001); A. V. Postnikov and V. P. Antropov, Comput. Mater. Sci. 17, 438 (2000); Y. Wang and T. Sun, J. Phys.: Condens. Matter 12, L311 (2000); P. Söderlind, Europhys. Lett. 55, 525 (2001) & more

→ Conventional band structure theory reproduces the correct unit cell volumes, but predict the presence of a magnetic moment for α-Pu and δ-Pu → This is because the correct densities are achieved by the localization of f- electrons that give rise to magnetic moments.

→ Predicted magnetic moments range from 0.25-5 μB. → This issue has persisted since decades. LDA+DMFT dynamical spin susceptibility of δ-Pu

MJ, Pinaki Das, B. Chakrabarti, D. L. Abernathy, M. D. Lumsden, J. M. Lawrence, J. D. Thompson, G. H. Lander, J. N. Mitchell, S. Richmond, M. Ramos, F. Trouw, J.-X. Zhu, K. Haule, G. Kotliar, E. D. Bauer. Science Advances e1500188 (2015) Kondo-Valence Fluctuation in Plutonium

MJ et al., Science Advances e1500188 (2015) MJ et al., PNAS 114, E268 (2017)

→ Pu shows six allotropic phases with large volume changes between these phases (25%). → Our finding of Kondo-valence fluctuations gives a natural explanation for the instability towards volume changes. → Shows relevance of quantum phenomena for real materials (Here: nuclear stockpile) The Elephant in The Room: What is Quantum Matter?

→ It turns out there is no clear definition!!!

→ In a way this is deeply philosophical: All things are quantum at a level (i.e. the elephant) because the underlying interactions are “quantum”.

→ ‘Quantum matter’ exhibits macroscopic properties driven by dominant quantum interactions. Levels of ‘Quantumness’

A. Schofield, Contemporary Physics 40, 95 (1999) Ideal Gas Electron Gas (simple metal) Magnet • Non-interacting particles • Electrons follow Fermi-Dirac Statistics! • Classical physics cannot account for • Entirely described by • Electronic quasiparticle excitations, magnetism (Bohr-van Leeuwen theorem)! classical statistics however, do behave like non-interacting • Magnetic order requires electron spin & particles! Pauli exclusion principle • Magnetism is a macroscopic quantum Cooling Down Cooling Down state with broken rotational symmetry. Temperature

Superconductor Superfluid • Electronic quasiparticles form • Bose-Einstein condensation Cooper pair (spin-singlet) in k-space. of He atoms into ground state • A finite energy quantum is required • Macroscopic quantum state with to break pair (superconducting gap). zero viscosity. • Macroscopic quantum state exhibiting the Meissner effect.

‘Quantumness’