Kondo effect in a quantum dot without spin Hyun-Woo Lee (Postech) & Sejoong Kim (Postech  MIT) References: H.-W. Lee & S. Kim, cond-mat/ P. Silvestrov & Y. Imry, cond-mat/ V. Kashcheyevs, A. Schiller, A. Aharony, & O. Entin-Wohlman, cond-mat/

Kondo effect  Temperature dependence of resistance  Resistance minimum

Kondo effect (continued) [J. Kondo, Prog. Theor. Phys. ’64]  Scattering by magnetic impurities Before After s - d model

Scattering amplitude for a channel of where N : number of d-electrons, N(0) : density of state at E F D : width of conduction electron distribution around E F This lnT dependence combined with the phonon contribution ( T 5 dependence) makes a resistance minimum in R(T). LogT dependence in R(T) J k,q = J where –D <  k,  q < D = 0 otherwise (*) 엄종화

Kondo effect (continued)  High T vs. low T Kondo singlet Cf. Asymptotic freedom

T K ~ T 일 때, Hamann expression (Phys. Rev. 1967) For T K > T, take (-) in the equation T K < T, take (+) in the equation T K << T 일 때, When T << T K,  ~  0 - cT 2 : unitary limit Kondo effect (*) 엄종화

Kondo effect in AuFe(26ppm) wire Hamann expression (Phys. Rev. 1967) From fitting the Hamann expression to  (T), we obtain S = 0.12, T K = 0.99 K. Concentration of AuFe is estimated by the slope of  => 26 ppm in the above figure Slope of Kondo resistivity = 0.11 n  cm / (ppm decade K) (*) 엄종화

Kondo effect in quantum dot

Quantum dot (QD)  “Metallic” limit ~ e 2 /2C >> kT n VgVg

Transport through a QD  Orthodox theory of Coulomb blockade Transport due to charge fluctuations Transport due to charge fluctuations

Quantum confinement  Single particle energy quantization  E >> kT

Even-odd effect  Spin singlet (S=0) vs doublet (S=1/2)  QD with odd n = magnetic impurity ??? n VgVg /2 0 S=1/2

Kondo effect in QD ?  Hamiltonian  Spin flip via second order processes c.f. n VgVg Before After

Kondo effect in QD w/ odd n  Theories T. K. Ng and P. A. Lee T. K. Ng and P. A. Lee Phys. Rev. Lett. 61, 1768 (1988)Phys. Rev. Lett. 61, 1768 (1988) L. I. Glazman and M. E. Raikh L. I. Glazman and M. E. Raikh JETP Lett. 47, 452 (1988)JETP Lett. 47, 452 (1988)  Experiments D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abush-Magder, U. Meirav, and M. A. Kaster D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abush-Magder, U. Meirav, and M. A. Kaster Nature 391, 156 (1998); Phys. Rev. Lett. 81, 5225 (1998)Nature 391, 156 (1998); Phys. Rev. Lett. 81, 5225 (1998) S. M. Cronenwett, T. H. Oostercamp, and L. P. Kouwenhoven S. M. Cronenwett, T. H. Oostercamp, and L. P. Kouwenhoven Science 281, 540 (1998)Science 281, 540 (1998)  Kondo suppression of  Kondo suppression of R

Unitary limit of the Kondo effect in SET [W. G. van der Wiel et al., Science ’00] G at V gl = -413mV shows logarithmic T dependence (inset), and saturates below 90mK (unitary limit) This experiment shows a unitary limit = 2 e 2 / h (  R =  L 의 경우 ) V gl was fixed at -413mV. V SD was biased between S and D. Kondo resonance peak G(T)G(T) 온도대역 : 15 mK – 800 mK FWHM (*) 엄종화

Kondo temperature: T K ; Costi et al., J. Phys.: Condense. Matter 6, 2519 (1994) In Anderson model, An empirical function ; Goldhaber-Gordon et al., PRL 81, 5225 (1998) T K in Log scale : universal functional form of T / T K s is a fit parameter, but is almost constant ~0.2 in the Kondo regime. (*) 엄종화

Kondo effect in QD w/o spin?

Beyond orthodox theory  Anomalous transmission phase Yacoby et al., PRL 74, 4047 (1995); Schuster et al., Nature 385, 417 (1997); Avinun-Khalish et al., Nature 436, 529 (2005) Yacoby et al., PRL 74, 4047 (1995); Schuster et al., Nature 385, 417 (1997); Avinun-Khalish et al., Nature 436, 529 (2005) Bruder et al., PRL 76, 114 (1995); Lee, PRL 82, 2358 (1999); Silvestrov et al., PRL 85, 2565 (2000) Bruder et al., PRL 76, 114 (1995); Lee, PRL 82, 2358 (1999); Silvestrov et al., PRL 85, 2565 (2000)  Population switching Silvestrov et al., PRL 85, 2565 (2000); Konig et al., PRB 71, R (2005); Sindel et al., PRB 72, (2005) Silvestrov et al., PRL 85, 2565 (2000); Konig et al., PRB 71, R (2005); Sindel et al., PRB 72, (2005)  Correlation-induced resonace Meden et al., PRL 96, (2006); PRB 73, (2006) Meden et al., PRL 96, (2006); PRB 73, (2006)

Two level QD  QD w/ two single-particle level  Source & Drain  Tunneling  “Spin” ? t1Lt1L t1Rt1R t2Lt2L t1Rt1R tt

Pseudospin for  1 = 2 (= )  Unitary transformations Pseudospin Pseudospin up Pseudospin down

Schrieffer-Wolf transformation: QD system (Anderson model)  s - d model  Fock space decomposition  Full Hamiltonian  Projection to n =1 Fock space 012

Effective Hamiltonian H s-d  Total Hamiltonian Anisotropic antiferro-exchange Anisotropic antiferro-exchange U(1) instead of SU(2)U(1) instead of SU(2) Pseudomagnetic field B z eff Pseudomagnetic field B z eff (*) For  =  (*) For  =  SU(2): J z =J + =J -SU(2): J z =J + =J - B z eff =0B z eff =0

Pseudomagnetic field B z eff  Expectation value For   >   For   >    Population switching from  level to  level with decreasing   Population switching from  level to  level with decreasing  0-U/2  =+U/2 012 hzhz

Population switching (PS) [Silvestrov & Imry, PRL’00]  Energy renormalization  eff =  bare +  (hopping)  eff =  bare +  (hopping)  : gate voltage dependent  : gate voltage dependent Charge 1  0 U U Charge 1  2 0-U/2  =+U/2 012

Poor man’s scaling  [1] Fock space decomposition  [2] Full Hamiltonian  [3] Projection to “g” sector of Fock space New Hamiltonian w/ reduced D New Hamiltonian w/ reduced D  [4] Back to [1] D DD DD

Scaling equations  Exchange J ’s Scaling invariant Scaling invariant Integration: Characteristic energy scale (Kondo temperature) Integration: Characteristic energy scale (Kondo temperature)  Pseudomagnetic field B z eff Integration: Integration:

Anisotropic s - d model  Approximation   Anisotropic s - d model Exact solution (via Bethe ansatz) available !!! Exact solution (via Bethe ansatz) available !!! Tsvelick & Wiegmann, Adv. Phys. 32, 453 (1983)Tsvelick & Wiegmann, Adv. Phys. 32, 453 (1983)

Conductance at T=0   and  scattering states  Friedel sum rule  Landauer-Buttiker formula

Anisotropic s - d model [Tsvelick and Wiegmann, Adv. Phys. (1983)]   S z  =(  n   -  n   )/2 vs. h z  G vs. 

Cf. Conventional spin Kondo  Conventional spin Kondo  Kondo w/o spin Correlation-induced resonance Correlation-induced resonance [Meden & Marquardt, PRL 96, (2006)] 0-U/2  =+U/2 012

w/o degeneracy  1 - 2  0  Same Unitary transformation  Additional pseudomagnetic field  h Parallel to z Parallel to z Shift of CIRsShift of CIRs Perpendicular to z Perpendicular to z Asymmetry in CIRs (Fano-like)Asymmetry in CIRs (Fano-like)

Summary  Kondo effect in QD w/o spin Distinct conductance pattern (cf. spin Kondo in QD) Distinct conductance pattern (cf. spin Kondo in QD)  Future directions w/o degeneracy w/o degeneracy Temperature dependence Temperature dependence Pseudospin & real spin Pseudospin & real spin Real Spin [SU(2)]Real Spin [SU(2)] Pseudospin [Not SU(2) invariant]Pseudospin [Not SU(2) invariant] Connection w/ anomalous transmission phase problem ?Connection w/ anomalous transmission phase problem ?