Transcript Slide 1
The Kondo effect in multiple quantum dot systems and deformable molecules A. Ramšak* J. Mravlje T. Rejec* R. Žitko J. Bonča* * Department of Physics Faculty of Mathematics and Physics University of Ljubljana Outline (1) Conductance (2) Kondo in a single quantum dot (3) Methods (4) Double quantum dots (5) Triple quantum dots (6) Deformable molecules (7) Center-of-mass motion (8) Summary I A Conductance Vsd ~ I G G(Vgate ) Vsd Vgate ~ n 2 1 Vg I A Conductance Vsd ~ I G G(Vgate ) Vsd Vgate ~ n 2 1 Vg Non-interacting systems: U=0 n 2 1 dot () 1 0.8 0.6 0.4 0.2 0 0.5 1 1.5 2 2.5 3 Non-interacting systems: U=0 n 2 1 G( ) dot () 2e2 G( ) [ ] h 1 0.8 0.6 0.4 0.2 0 0.5 1 1.5 2 2.5 3 The Anderson model: U > 0 d U d U 0 2e2 G( ) [ ] h 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 U 0 U ( ) 1 0.8 0.6 0.4 0.2 0 00 0.5 1 or 1.5 22 2.5 2.5 33 T U 0 T TK U 2e2 G( ) [ ] h 1 0.8 0.6 0.4 0.2 0 00 0.5 1 or 1.5 22 2.5 2.5 33 The Kondo effect in a quantum dot d U () d 0 d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () d d U The Kondo effect in a quantum dot d U d () 1 TK U e 2 d d ( d U ) d U U The Kondo effect in a quantum dot The Kondo effect in a quantum dot The Kondo effect in a quantum dot The Kondo effect in a quantum dot A( d U / 2, ) d U / 2 “Ring system” 1 “Open system” “Open system” IF open system is Fermi liquid the GS energy of a large ring system is an universal function of flux T. Rejec and A. Ramšak, PRB 68, 033306 (2003); 68 035342 (2003) N 1 N 4n 1(3); Nel even N 1 N 4n 1(3); Nel odd g G / G0 Linear conductance from the ground-state energy T 0 See also: J. Favand and F. Mila (Phys. J. 1998); O. Sushkov (PRB 2001); R. Molina et al. (PRB 2003) Linear conductance from the ground-state energy T 0 Linear conductance from the ground-state energy T 0 Aharonov – Bohm rings Broken time-reversal symmetry T. Rejec and A. Ramšak, PRB 68, 033306 (2003) The Kondo effect in a quantum dot numerical tests… U=0 The Kondo effect in a quantum dot: finite temperature T low T U=0 high T The Kondo effect in a quantum dot: finite temperature T low T high T The Kondo effect in a quantum dot: finite temperature T low T high T Fingerprints of Kondo… A( d U / 2, ) d U / 2 t1 / t 2 M loc S / Smax Fingerprints of Kondo… A( d U / 2, ) d U / 2 t1 / t 2 M loc S / Smax Fingerprints of Kondo… A( d U / 2, ) d U / 2 t1 / t 2 Fingerprints of Kondo… A( d U / 2, ) d U / 2 t1 / t 2 M loc S / Smax Multiple quantum dot systems Chan et al, Nanotechnology 15, 609 (2004) Elzerman et al, PRB 67, 16308 (2003) QD QD Electrostatic gates Vidan et al, Applied Phys. Lett. 85, 3602 (2004) Double quantum dot t2 t12 t'' Double quantum dot (a) t2 U (b) t2 d t2 J RKKY 2TK U V 0 2 t12 (c) J 4 2TK U t2 t12 t'' Double quantum dot (a) t2 U (b) t2 d t2 U V 0 2 t12 (c) J 4 TK U (d) t2 0 & V ~ U : SU (4) Kondo (a) t2 U (b) t2 d t2 U V 0 2 t12 (c) J 4 TK U (d) t2 0 & V ~ U : SU (4) Kondo half filling: n 2 Also for finite hibridization: TK [SU (4)] ~ U TK [SU (2)] J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 73, 241305(R) (2006) hibridization Double quantum dot S1 S2 1 Kondo U / t12 3 AFM U / t12 50 2 Kondo U / t12 60 t t Double quantum dot S1 S2 1 Kondo U / t12 3 AFM U / t12 50 2 Kondo U / t12 60 t t Double quantum dot S1 S2 1 Kondo U / t12 3 AFM U / t12 50 2 Kondo U / t12 60 t t Other topologies: local singlet vs the Kondo effect A. Ramšak, J. Mravlje, R. Žitko, and J. Bonča, quant-ph/0608065. Thermal equilibrium: A-B spin corelations T/J 2 4t J U S ASB B/J Zero magnetic field, thermal equilibrium B 0, T const. 2 4t A J U B Zero magnetic field, temperature A B J /U 4t 2 J U Zero magnetic field, temperature A B J /U 4t 2 J U Zero magnetic field, temperature A B Jc2 J /U Zero magnetic field, temperature A B J /U J c3 ~ Triple quantum dot d 2t2 d d 2t2 Triple quantum dot d 2t2 d U d d 2t2 Triple quantum dot d Triple quantum dot d Triple quantum dot d Triple quantum dot U / t2 1 2 5 10 20 Triple quantum dot t ~ T1K Triple quantum dot t ~ T1K Triple quantum dot t ~ J t ~ T1K Triple quantum dot t ~ J J ~ 2T1K t ~ T1K Triple quantum dot Deformable molecules Deformable molecules e Deformable molecules e Change in: • local energy • hopping matrix elements H. Park et al. Nature 407 (2000) … Modeling Old knowledge … Isolated molecule: H d n Unn a a M (n 1)(a a ) Ueff E(2) 2E(1) U 2 M 2 M , d molecule attached to the leads: Lang & Firsov transformation: ~ 1 M ( a a )( n1) H U HU, U e the result: M , d Reduction of U and narrowing of the level-width A.C. Hewson and D.M. News J.Phys C 13 (1980) K. Schönhammer and O. Gunnarsson PRB 30 (1984) decrease U negative U: A. Taraphder and P. Coleman, PRL 66, 2814 (1991). M J. Mravlje, A. Ramšak, and T. Rejec, PRB 72, 121403(R) (2005); See also: P.S. Cornaglia, D.R. Grempel, and H. Ness, Phys. Rev. B 71, 075320 (2005), A. Mitra, I. Aleiner, and A.J. Millis, Phys. Rev. B 79, 245302 (2004). Molecules with a center of mass motion J. Mravlje, A. Ramšak, and T. Rejec, submitted to PRB Molecules with a center of mass motion Molecules with a center of mass motion Molecules with a center of mass motion Molecules with a center of mass motion Friedel sum rule: Molecules with a center of mass motion Friedel sum rule: Molecules with a center of mass motion A B Molecules with a center of mass motion Molecules with a center of mass motion Summary • Linear conductance at T=0 can then be extracted from the GS energy: • The Kondo effect: Low temperature destiny of quantum dots ad summary… U t d U d t g G / G0 Formulae are exact IF the system is Fermi liquid note: • linear conductance • zero temperature • non-interacting single-channel leads Conductance formalisms U≠0 U=0 non-equilibrium transport: T ≠ 0, V ≠ 0 Landauer – Büttiker formula Meir – Wingreen formula linear response regime: T ≠ 0, V ~ 0 Kubo formula zero-temperature linear response: T = 0, V ~ 0 In Fermi liquid systems Fisher – Lee relation … Proof of the method Step 1. Conductance of a Fermi liquid system at T=0 Kubo T=0 define (n.i.: Fisher-Lee) ‘Landauer’ Step 2. Quasiparticle hamiltonian (Landau Fermi liquid) Step 3. Quasiparticles in a finite system N Step 4. Validity of the conductance formulas