SERC School on Magnetism and Superconductivity ’06 The Kondo Effect Vijay B. Shenoy ([email protected]) Centre for Condensed Matter Theory Indian Institute of Science VBS The Kondo Effect – 0 SERC School on Magnetism and Superconductivity ’06 Overview What is Kondo effect? Motivation – Why do this? Plan of the Lectures VBS The Kondo Effect – 1 SERC School on Magnetism and Superconductivity ’06 Kondo... This is Jun Kondo...but, whats his effect? Something that happens in metals... VBS The Kondo Effect – 2 SERC School on Magnetism and Superconductivity ’06 Metals, Wonder Materials! What is a metal? Many electron system with a Fermi Surface Key: Gapless excitations Linear T specific heat Temperature independent magnetic susceptibility ... Key: “Protected” from repulsive interactions by Pauli – Fermi liquid VBS The Kondo Effect – 3 SERC School on Magnetism and Superconductivity ’06 Resistivity in Metals Almost constant at “low” temperatures...all way to linear at high temperatures VBS The Kondo Effect – 4 SERC School on Magnetism and Superconductivity ’06 Resistivity in Metals...There’s More! Increases with impurity content Has some “universal” features... VBS The Kondo Effect – 5 SERC School on Magnetism and Superconductivity ’06 The Kondo Effect! There is a “resistance minimum” in some cases! VBS The Kondo Effect – 6 SERC School on Magnetism and Superconductivity ’06 When does this happen? Impurity susceptibility: Fe in Cu Happens when impurity is “magnetic”! Whats the big deal? Why bother? VBS The Kondo Effect – 7 SERC School on Magnetism and Superconductivity ’06 Why bother? “Universal” relation between dimensionless “magnetic part” of resistivity and temperature!! “Universal” relations observed for other properties VBS (susceptibility etc.) The Kondo Effect – 8 SERC School on Magnetism and Superconductivity ’06 Not Impressed? How about Quantum Dots? Regions that can hold a few hundred electrons! Can drive a current through these! This is Nano! VBS The Kondo Effect – 9 SERC School on Magnetism and Superconductivity ’06 Universal Stuff in QDs too! Universal relation between dimensionless conductance and temperature! VBS The Kondo Effect – 10 SERC School on Magnetism and Superconductivity ’06 There’s More! Dimensionless resistivity vs. temperature in “Kondo metals” and dimensionless conductance vs. temperature show the same universal function!!!! VBS The Kondo Effect – 11 SERC School on Magnetism and Superconductivity ’06 Why Bother? These results cannot just be happenstance! ... The Kondo problem is a cornerstone in physics...it gave rise to a lot of physics! ... Two Nobel Prizes! ... Dynamical Mean Field Theory (DMFT): Revival of interest in quantum impurity problems VBS The Kondo Effect – 12 SERC School on Magnetism and Superconductivity ’06 Plan of the Lectures When is an impurity “magnetic”? Anderson Model The sd model: “Low energy” physics Whence resistivity minimum? (“High” T ) The Kondo Problem! Variational approach to the Kondo ground state (“Low” T ) “Interpolation” – Scaling/RG ideas Local Fermi Liquid ideas VBS The Kondo Effect – 13 SERC School on Magnetism and Superconductivity ’06 Prerequisites Must know Band theory of metals (One e physics) Essential Stat. Mech. No harm if you know Second quantization Greens function Great if you know Diagrammatic perturbation theory! VBS The Kondo Effect – 14 SERC School on Magnetism and Superconductivity ’06 How do we model an impurity in a metal? Host metal A set of Bloch states kσ (σ - spin index) | i † Hamiltonian Hm = kσ ε(k)ckσckσ Bandwidth 2D, i. e.,P D ε(k) D − ≤ ≤ Density of states ρ(ε) Impurity with “d-levels” Impurity “d-level” dσ with energy ε | i d † Hamiltonian Hd = σ εdcdσcdσ Host-impurity “Hybridization”P † ∗ † Hybridization : Hh = kσ Vkckσcdσ + Vk cdσckσ P Hamiltonian H = Hm + Hd + Hh specified when ε(k), εd and Vk are given VBS The Kondo Effect – 15 SERC School on Magnetism and Superconductivity ’06 The Impurity Question Given that we have the host metal at some filling, say the chemical potential is “mid-band” (µ =0), what “happens” to the impurity d-level? Is the impurity level occupied? Is the impurity “magnetic”, i. e., does it retain a “spin”? Recap: What happens in a free ion? Answer the question in a simple model: Host ρ(ε) = ρ0 constant...”flat band” Vk = V , a constant Now, a three parameter model (ρ0, εd and V ) And, µ is given to be zero VBS The Kondo Effect – 16 SERC School on Magnetism and Superconductivity ’06 The Impure Answer: Physics Arguments From physics arguments Impurity level will be occupied if εd < µ (Of course!) But an electron cannot stay at the impurity for ever...due to hybridization...the impurity state picks up a “lifetime” Estimate of lifetime τ (~ = 1): ∆ = 1 V 2ρ τ ∼ | | 0 The impurity level behaves like a “virtual state” with “energy eigenvalue” = ε i Ed d − τ This implies a “Lorentizan” density of state for the impurity 1 ∆ ρ (ε) = d π (ε ε )2 + ∆2 − d VBS The Kondo Effect – 17 SERC School on Magnetism and Superconductivity ’06 echnical Digression: The Resolvent Green’s Function Hamiltonian: H The Green’s operator (function) ∂ (i H)G(t t′) = δ(t t′)1 ∂t − − − In frequency language G+(ω) = (ω+ H)−1 − Recall: 1 = 1 = P 1 iπδ(x) x+ x+iη x − Why bother? Well,` ´ 1 1 ρ(ε) = Im(TrG(ω)), ρ (ε) = ImG (ω) −π a −π aa The Greens function: route to calculate density of states! VBS The Kondo Effect – 18 SERC School on Magnetism and Superconductivity ’06 Back to Impurity Density of States... Hamiltonian: H = Hm + Hd +Hh (Perturbation Hh) H0 | {z −1} 1 Notation, G0 = (ω H0) , G0dσ = − ω−εd The impurity Green’s function (after a bit of algebra) G0dσ 1 Gdσ(ω) = 2 = 2 1 G |Vk| ω ε |Vk| − 0dσ k ω−ε(k) − d − k ω−ε(k) 2 1 P PVk , Ed = εd + P | | , ≈ ω (Ed i∆) εd ε(k) − − Xk − ∆ = π V 2δ(ε ε(k)) | k| d − Xk Physics: Impurity level forms a “virtual state” VBS The Kondo Effect – 19 SERC School on Magnetism and Superconductivity ’06 Impurity Density of States...and...!! In the simple three parameter model 2 εd D 2 Ed = εd + ρ0 V ln − , ∆= πρ0 V | | ε + D | | ˛ d ˛ ˛ ˛ ˛ ˛ This implies a “Lorentizan”˛ density˛ of state for the impurity 1 1 ∆ ρdσ(ε)= ImGdσ(ε)= − π π (ε E )2 + ∆2 − d The occupancy of the virtual level µ 1 1 −1 Ed µ ndσ = dε nF (ε) ρdσ(ε) tan − h i −∞ ≈ 2 − π ∆ Z T =0 „ « |{z} But.. S n n =0! This model says impurity h zi ∼ h d↑ − d↓i can never have a moment...Whats the missing physics? VBS The Kondo Effect – 20 SERC School on Magnetism and Superconductivity ’06 The Anderson Model Anderson, PR 124, 41 (1961), realized the moment formation has origins in strong correlations! (contribution from Wolff/Fridel) Introduced local Coulomb correlations on the impurity † Hd = εdcdσcdσ + Und↑nd↓ σ X U is the “d orbital” Coulomb repulsion energy The Anderson Model † † H = ε(k)ckσckσ + εdcdσcdσ + Und↑nd↓ + σ Xkσ X † ∗ † Vkckσcdσ + Vk cdσckσ Xkσ Can we see that this is likely to produce a moment? VBS The Kondo Effect – 21 SERC School on Magnetism and Superconductivity ’06 Take a Moment...“See” the Moment S 1 † Impurity spin operator = 2 σσ′ cdσ~τσσ′ cdσ′ Now, S S = 5 (n + n ) 3Pn n · 4 d↑ d↓ − 4 d↑ d↓ The impurity Hamiltonian can now be written as 5 3 H = (ε + U) c† c US S d d 4 dσ dσ − 4 · Xσ ′ εd | {z } Thus, if U is “large enough”, and if we can arrange things such that nd =0, the impurity will maximize S S...will pick uph ai moment! 6 · So, what competes with this?...Kinetic energy (hybridization)! VBS The Kondo Effect – 22 SERC School on Magnetism and Superconductivity ’06 The Hartree-Fock Theory Hartree-Fock (Mean Field) treatment of Coulomb HHF = ε c† c + U( n n + n n n n ) d d dσ dσ h d↓i d↑ h d↑i d↓ − h d↑ih d↓i Xσ Note: n are unknown (variational) parameters h dσi HF Anderson Hamiltonian HHF = ε(k)c† c + (ε + U n ) c† c U n n + kσ kσ d h dσ¯ i dσ dσ − h d↑ih d↓i kσ σ X X ε˜dσ † ∗ † | {z } Vkckσcdσ + Vk cdσckσ Xkσ This is a one particle Hamiltonian that we have already solved...apply our results to “each spin channel”... VBS The Kondo Effect – 23 SERC School on Magnetism and Superconductivity ’06 The Hartree-Fock Solution Four parameter Anderson model (εd, ρ0,V,U) at chemical potential µ Green’s function 1 Gdσ(ω)= ω (E i∆) − dσ − 2 εd D 2 Edσ = εd + ρ0 V ln − +U ndσ¯ , ∆= πρ0 V | | ε + D h i | | ˛ d ˛ ˛ ˛ ≡ε ˛ ˛ d ˛ ˛ | {z } DOS etc. 1 ∆ ρdσ(ε) = π (ε E )2 + ∆2 − dσ 1 1 −1 Edσ µ ndσ = tan − h i 2 − π ∆ „ « VBS The Kondo Effect – 24 SERC School on Magnetism and Superconductivity ’06 The Hartree-Fock Solution The occupancy condition enables us to determine n h dσi 1 1 −1 εd + U nd↓ µ nd↑ = tan h i− h i 2 − π ∆ „ « 1 1 −1 εd + U nd↑ µ nd↓ = tan h i− h i 2 − π ∆ „ « Question: When is it possible to have n = n ? h d↑i 6 h d↑i Need to find a condition on εd, ρ0,V,U! This will tell us when an impurity can be magnetic... The condition will depend only on two parameters...(ε µ)/U and ∆/U d − VBS The Kondo Effect – 25 SERC School on Magnetism and Superconductivity ’06 The Hartree-Fock Solution If the impurity develops a moment, then the DOS picture will look like Ed Ed ↑ ↓ µ ρd (ε) ρd (ε) ↑ ↓ ∆ ∆ ε So moment definitely forms when ε < µ with ∆ ε µ , AND ε + U > µ with ∆ ε + U µ d ≪ | d − | d ≪ | d − | Physics: (i) Impurity level must be occupied by only “one spin species” (ii) The hybridization should be small enough to “hold this spin species at the impurity” VBS The Kondo Effect – 26 SERC School on Magnetism and Superconductivity ’06 The Hartree-Fock Solution Can work out the exact condition for a special case..