Efekt Kondo i kwantowe zjawiska krytyczne w układach nanoskopowcyh. Ireneusz Weymann Wydział Fizyki, Uniwersytet im. Adama Mickiewicza w Poznaniu

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1 Efekt Kondo i kwantowe zjawiska krytyczne w układach nanoskopowcyh Ireneusz Weymann Wydział Fizyki, Uniwersytet im. Adama Mickiewicza w Poznaniu

2 Introduction: The Kondo effect in metals de Haas, de Boer and van den Berg, Physica 1, 1115 (1934) I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

3 Introduction: The Kondo effect in metals Abrikosov, Physics 2, 5 (1965) Kondo problem T=0 behavior by FL theory Nozieres, J. Low Temp. Phys. 17, 31 (1974) Numerical renormalization group (NRG) Wilson, Rev. Mod. Phys. 47, 773 (1975) I. Weymann, The Kondo Kondo effect effect and in quantum criticality dots in QDs Konferencja KDM, Poznań,

4 Introduction: The Kondo effect in quantum dots Single spin trapped in a quantum dot acts as magnetic impurity. At low temperatures, T<T K, It can be screened by conduction electrons. Theoretical predictions: source drain Instead of resistance increase now one observes enhancement of conductance I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

5 Introduction: The Kondo effect in quantum dots metals quantum dots I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

6 Introduction: The Kondo effect in quantum dots I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

7 Introduction: The Kondo effect in quantum dots Pustilnik & Glazman, JPCM 2004 I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

8 Model: Single-impurity Anderson model source G L e G R drain e +U Anderson Hamiltonian I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

9 Method: Numerical Renormalization Group (NRG) Crossover from high to low temperature requires nonperturbative method: Wilson s NRG Wilson, RMP 1975 see also Bulla, Costi, Pruschke, RMP 2008 Main ideas: Logarithmic discretization of conduction band, with discretization parameter Λ Mapping the conduction band onto a Wilson chain with exponentially decaying hoppings, t n ~ Λ -N/2 Iteratively diagonalize the Wilson chain Developments: chain geometry calculation of transport properties [Costi et al. JPCM 1994] density matrix NRG [Hofstetter, PRL 2000] QD t t 0 t 1 complete basis set [Anders, Schiller, PRL 2005] H full density matrix NRG [Weichselbaum, 0 H von Delft, 1 H PRL ] flexible full density-matrix NRG [Budapest Wilson NRG, chain 2008] t 2 K. G. Wilson I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

10 Method: NRG Iterative diagonalization of Wilson chain QD ~ L -n / 2 s 0 s 1 dimension of Hilbert space grows as 2 n for spinless fermions s 2 s 3 s 4 Truncation needed! I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

11 Method: NRG Energy scale separation QD Truncation based on energy scale separation E 1s E 2s s 0 s 1 kept states (K) s 2 s 3 s 4 E 3s E 4 s discarded states (D) I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

12 Method: NRG Complete eigenbasis QD Anders&Schiller [2005]: use discarded states to construct approximate complete many-body basis of NRG Hamiltonian e - environmental state E 1s E 2s s 0 s 1 kept states (K) s 2 s 3 s 4 E 3s E 4 s discarded states (D) I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

13 Method: Budapest Flexible DM-NRG We use the Budapest Flexible DM-NRG code O. Legeza, C. P. Moca, A. I. Toth, I. Weymann, G. Zarand, arxiv: (2008). The code is avaiable at The code can calculate arbitrary correlation functions of local operators using arbitrary number of both Abelian and non-abelian symmetries Linear response conductance: I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

14 Single-impurity Anderson model: Spectral function e e +U I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

15 Single-impurity Anderson model: Spectral function Development of Kondo resonance at low temperatures e e +U I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

16 Numerical issues Depends greatly on considered model Typical parameters (for single run): Number of Wilson sites: N~60 Number of states per Wilson site: >10 4 Numerical requirements (for single run): RAM: >20GB Fast HDD: >500GB Number of runs: >10 4 I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

17 Types of the Kondo effect Different types of the Kondo effect fully-screened Kondo effect (SIAM, SU(4) Kondo) underscreened Kondo effect (S>1/2, SMMs) overscreened Kondo effect (S=1/2 coupled to 2 channels) 2S= n 2S> n 2S< n n number of screening channels I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

18 SU(4) Kondo effect in DQDs: Emergent SU(4) Kondo physics in a spincharge-entangled double quantum dot Collaborators: Experiment: Stanford: A.J. Keller S. Amasha I.G. Rau D. Goldhaber-Gordon San Jose: J.A. Katine Weizmann: Hadas Shtrikman Theory: Budapest: C.P. Moca, G. Zarand Poznań: I.W.

19 SU(4) Kondo effect in CNT I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

20 SU(4) Kondo effect in DQDs 200nm 2DEG (GaAs/AlGaAs) Double quantum dot with strong inter-dot capacitive coupling: Each dot has its own drain and source electrodes The P gates enable tuning the orbital levels of the dots The W gates enable tuning the dotlead tunnel rates The interdot tunnel rates are tuned to be negligible using the C gates I. Weymann, The Kondo Kondo effect effect and in quantum criticality dots in QDs Konferencja KDM, Poznań,

21 SU(4) Kondo effect in DQDs 2DEG (GaAs/AlGaAs) 200nm Triple points, where three charge states are degenerate. Along the line between triple points (LBTP), the charge states (N 1 +1,N 2 ) and (N 1,N 2 +1) are degenerate. This constitutes a pseudospin. The SU(4) Kondo effect expected along LBTP I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

22 SU(4) Kondo effect in DQDs: Theoretical modeling Hamiltonian Linear conductance through dot j: NRG calculations with 4 symmetries: SU(2) spin SU(2) spin U(1) charge U(1) charge I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

23 SU(4) Kondo effect in DQDs Experiment NRG A.J. Keller, S. Amasha, IW, C.P. Moca, I.G. Rau, J.A. Katine, H. Shtrikman, G. Zarand, D. Goldhaber-Gordon, Nat. Phys. 10, 145 (2014) I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

24 SU(4) Kondo effect in DQDs A.J. Keller, S. Amasha, IW, C.P. Moca, I.G. Rau, J.A. Katine, H. Shtrikman, G. Zarand, D. Goldhaber-Gordon, Nat. Phys. 10, 145 (2014) I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

25 SU(4) Kondo effect in DQDs: temperature dependence of conductance ε 1 = ε 2 = -0.03meV = -0.3U A.J. Keller, S. Amasha, IW, C.P. Moca, I.G. Rau, J.A. Katine, H. Shtrikman, G. Zarand, D. Goldhaber-Gordon, Nat. Phys. 10, 145 (2014) I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

26 The 2-channel Kondo effect: Universal Fermi liquid crossover and quantum criticality in a mesoscopic system Collaborators: Experiment: Stanford: A.J. Keller L. Peters D. Goldhaber-Gordon Weizmann: D. Mahalu V. Umansky Theory: Budapest: C.P. Moca, G. Zarand Poznań: I.W.

27 The 2-channel Kondo effect Multi-channel Kondo model introduced by Nozieres and Blandin: n channels couple to a spin-s impurity Nozieres and Blandin, Journal de Physique 41 (3), 193 (1980) For n=2 and S=1/2 one gets the two-channel Kondo (2CK) model: no charge transfer channel 1 channel 2 J 1 J 2 Competition between channels 1 and 2 leads to QPT I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

28 The 2-channel Kondo phase diagram Fermi liquid scale T* Fermi liquid 2 (FL) non-fermi liquid (NFL) quantum critical Fermi liquid 1 (FL) 2-channel Kondo quantum critical point (QCP) Fermi liquid to non-fermi liquid cross-over at T* I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

29 The 2-channel Kondo realization in quantum dots grain source QD drain charge transfer to grain forbidden when leads and grain s electrons compete to screen the spin in the dot I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

30 The 2-channel Kondo realization in quantum dots grain source QD drain I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

31 2CK: Experimental setup and theoretical model Hamiltonian: I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

32 2CK: Linear conductance and 2CK lines a G (e 2 /h) b G (e 2 /h) CK - ( ( )) / E C / E C c / U G (e 2 /h) CK V BWT (mv) / U V LP (mv) -210 /2 d 20 mk ) 1/2 e 20 mk I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

33 2CK: Universal 2CK scaling of conductance channel Kondo scaling 0 - / U V LP (mv) -210 V (m (A(0,T)-A(,T)) / T 1/2 d mk (G(0,T)-G(V SD,T)) / (kt) 1/2 e mk (- / T) 1/ (ev SD / kt) 1/2 2CK scaling: I. Affleck & A.W.W. Ludwig, PRB 48, 7297 (1993) I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

34 The 2-channel Kondo phase diagram Fermi liquid scale T* Fermi liquid 2 (FL) non-fermi liquid (NFL) quantum critical Fermi liquid 1 (FL) 2-channel Kondo quantum critical point (QCP) Extract FL scale T* I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

35 2CK: Extracting Fermi liquid scale T* Detune the couplings and look for deviations from 2CK scaling Fit to CFT description of universal crossover between FL and NFL by E. Sela, A.K. Mitchell and L. Fritz, PRL 106, (2011) A.K. Mitchell and E. Sela, PRB 85, (2012) a (G(0,T)-G(V SD,T)) / (kt) 1/ CK scaling 20 mk 40 mk 52 mk 80 mk 109 mk 130 mk 149 mk b (G(0,T)-G(V SD,T)) / (kt) 1/ Crossover 20 mk Crossover 40 mk c (ev SD / kt) 1/2-3 0 (ev SD / kt) 1/ I. Weymann, d Kondo effect G (e 2 and /h) quantum criticality in QDs Experiment e Konferencja G (ekdm, 2 /h) Poznań,

36 (ev 2CK: Universal Fermi SD / kt) 1/2 liquid cross-over and FL scale T* -3 0 (ev SD / kt) 1/2 3 d G (e 2 /h) Experiment e G (e 2 /h) NRG calculations V SD ( V) ( ev) 0-20 T* ( ev) QCP QCP QCP QCP P / P / T* ( ev) V BWT (mv) / E C Measurement of Fermi liquid scale T* Fermi liquid scale T* vanishes at QCP Quadratic dependence of T* on gate voltage around QCP I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

37 2CK: Universal Fermi liquid cross-over and FL scale T* I. Weymann, Kondo effect and quantum criticality in QDs Konferencja KDM, Poznań,

38 Summary Kondo effect in single quantum dots Observation of the SU(4) Kondo effect in double quantum dots The 2-channel Kondo effect and universal Fermi liquid crossover in a dot-grain device

39 Thank you for your attention Calculations performed at