Effet Kondo dans les nanostructures: Morceaux choisis

Transcription

1 Effet Kondo dans les nanostructures: Morceaux choisis Pascal SIMON Rencontre du GDR Méso: Aussois du 05 au 08 Octobre 2009

2 OUTLINE I. The traditional (old-fashioned?) Kondo effect II. Direct access to single magnetic impurity physics III. Kondo effect in artificial magnetic impurities (quantum dots) IV. Quantum phase transitions at the nano scale V. Kondo effect in various environments VI. Conclusion and some perspectives

3 OUTLINE I. The traditional (old-fashioned?) Kondo effect II. Direct access to single magnetic impurity physics III. Kondo effect in artificial magnetic impurities (quantum dots) IV. Quantum phase transitions at the nano scale V. Kondo effect in various environments VI. Conclusion and some perspectives

4 Introduction to the Kondo effect De Haas et al., 1934

5 Introduction to the Kondo effect De Haas et al., 1934

6 Introduction to the Kondo effect De Haas et al., 1934 Local spin exchange interaction (known as the s-d model) Kondo interaction

7 Explanation of the resistivity minimum.1 J. Kondo J. Kondo J. J

8 Explanation of the resistivity minimum.1 Second order calculation in J Like non-magnetic scatterers! J. Kondo J. Kondo J. J

9 Explanation of the resistivity minimum.1 Second order calculation in J Like non-magnetic scatterers! J. Kondo J. Kondo J. Third order calculation in J J

10 Explanation of the resistivity minimum.1 Second order calculation in J Like non-magnetic scatterers! J. Kondo J. Kondo J. Third order calculation in J New feature J

11 Explanation of the resistivity minimum.1 Second order calculation in J Like non-magnetic scatterers! J. Kondo J. Kondo J. Third order calculation in J New feature Total resistivity J

12 Explanation of the resistivity minimum.1 Second order calculation in J Like non-magnetic scatterers! J. Kondo J. Kondo J. Third order calculation in J New feature Total resistivity This expression has a minimum J

13 Explanation of the resistivity minimum.2 Jun Kondo, Prog. Theor. Phys. (1964)

14 Where the logs are coming from? Second order diagrams for the T matrix Virtual processes with spin flip Spin flip scattering leads to a logarithmically divergent inverse lifetime

15 A Kondo Temperature A A However, this expression is perturbative in J and is valid provided

16 A Kondo Temperature A A However, this expression is perturbative in J and is valid provided A new energy scale appears in the problem. This is a way to define the Kondo temperature

17 A Kondo Temperature A A However, this expression is perturbative in J and is valid provided A new energy scale appears in the problem. This is a way to define the Kondo temperature Question: what happens at lower temperature? The so called Kondo problem

18 A Kondo Temperature A A However, this expression is perturbative in J and is valid provided A new energy scale appears in the problem. This is a way to define the Kondo temperature Question: what happens at lower temperature? The so called Kondo problem All orders? Susceptibility

19 A Kondo Temperature A A However, this expression is perturbative in J and is valid provided A new energy scale appears in the problem. This is a way to define the Kondo temperature Question: what happens at lower temperature? The so called Kondo problem All orders? Susceptibility All quantities seem to diverge at the Kondo temperature!

20 The scaling hypothesis Since all quantities diverge at the same energy scale This suggest some scaling hypothesis where F, G are universal scaling functions - High temperature - Lower and intermediate temperature???

21 All temperature behaviour - Low temperature This corresponds to the Fermi liquid regime: the impurity spin forms a singlet with an electron of the Fermi sea Nozières, All temperature: Numerical renormalization group (Major conceptual step) Wilson, 1975 (Nobel prize 1982) T. Costi, A. Hewson, V Zlatic, J. Phys C Mat., 1994

22 Main signature A sharp peak anomaly grows at the Fermi level in the impurity density of states T. Costi, A. Hewson, V Zlatic, J. Phys C Mat., 1994 How to experimentally measure such local property?

23 History of Kondo phenomena - Resistance minimum observed in the `30s -... And explained in the `60s by Kondo - Log divergence problem solved by Wilson NRG: `70s - Bethe Ansatz solution (essentially exact): `80s - Spectroscopy analysis by conformal field theory: `90s

24 History of Kondo phenomena - Resistance minimum observed in the `30s -... And explained in the `60s by Kondo - Log divergence problem solved by Wilson NRG: `70s - Bethe Ansatz solution (essentially exact): `80s - Spectroscopy analysis by conformal field theory: `90s And so what s new about it?

25 History of Kondo phenomena - Resistance minimum observed in the `30s -... And explained in the `60s by Kondo - Log divergence problem solved by Wilson NRG: `70s - Bethe Ansatz solution (essentially exact): `80s - Spectroscopy analysis by conformal field theory: `90s And so what s new about it? Kondo signatures in electronic transport observed in many different nanoscopic set ups: `00s - STM measurements of magnetic structures on metallic surfaces - Quantum dots (experimental control of parameters) - New insights: multi-impurity systems, correlated environment,...

26 OUTLINE I. The traditional (old-fashioned?) Kondo effect II. Direct access to single magnetic impurity physics III. Kondo effect in artificial magnetic impurities (quantum dots) IV. Quantum phase transitions at the nano scale V. Kondo effect in various environments VI. Conclusion and some perspectives

27 Scanning Tunneling Microscopy

28 Tunneling Spectroscopy tunnel of a single magnetic impurity H.C. Manoharan et al., Nature 403, 512 (2000)

29 Different spectrum shapes q = Fano factor depends on the surface & tunneling matrix elements

30 Toward the contact regime Contact regime Tunneling regime N. Néel et al., Phys. Rev. Lett. (2007)

31 OUTLINE I. The traditional (old-fashioned?) Kondo effect II. Direct access to single magnetic impurity physics III. Kondo effect in artificial magnetic impurities (quantum dots) IV. Quantum phase transitions at the nano scale V. Kondo effect in various environments VI. Conclusion and some perspectives

32 What about quantum dots Goldhaber-Gordon et al., Nature (1998) From Kouwenhoven s group From Tarucha s group Cobden et al., Nature (2001) Bockath et al., Nature (2002) Roch et al., Nature (2008) And many others... - Quantum dots can hold a few hundreds electrons - A current can be driven through them THIS IS THE NANO SCALE

33 Coulomb blockade Energy scales: Coulomb energy Level spacing Level width Temperature

34 Model hamiltonian for low T Ground state for N odd

35 Model hamiltonian for low T Ground state for N odd Free spin!

36 Model hamiltonian for low T Ground state for N odd Free spin! Quantum fluctuations g N odd N-1 N N+1

37 Model hamiltonian for low T Ground state for N odd Free spin! Quantum fluctuations g N odd N-1 N N+1 In this temperature regime, we can use a model based on the Anderson Hamiltonian with

38 Kondo effect in quantum dots Cotunneling regime Virtual charge transitions can be accompanied with a spin flip inside the dot! Like an ordinary magnetic impurity in a metal We can indeed map in this regime the Anderson model to the Kondo model The Kondo effect is expected to occur at low T (zero bias anomaly)

39 Conductance in quantums dots Van der Wiel et al., Science (2000)

40 Kondo ridges

41 Scaling of the conductance Van der Wiel et al., Science (2000)

42 Comparision between the scaling functions

43 Spectroscopic analysis of the Kondo resonance Van der Wiel et al., Science 289, 2105 (2000)

44 Recover the original Kondo measurements? Kondo ``antiresonance Sato et al., PRL, 95, (2005)

45 Perspectives offered by quantum dots

46 Perspectives offered by quantum dots Ability to fully control: 1) The caracteristics of one or several artificial magnetic impurities 2) Their various couplings and positions 3) The nature of the electronic environment

47 Perspectives offered by quantum dots Ability to fully control: 1) The caracteristics of one or several artificial magnetic impurities 2) Their various couplings and positions 3) The nature of the electronic environment The leads can be either metallic, ferromagnetic superconducting, of finite size, etc.

48 Perspectives offered by quantum dots Ability to fully control: 1) The caracteristics of one or several artificial magnetic impurities 2) Their various couplings and positions 3) The nature of the electronic environment The leads can be either metallic, ferromagnetic superconducting, of finite size, etc. This offers a new possibility to scrutinize electronic correlations at the nano scale (probe = transport) New physics to be discovered

49 OUTLINE I. The traditional (old-fashioned?) Kondo effect II. Direct access to single magnetic impurity physics III. Kondo effect in artificial magnetic impurities (quantum dots) IV. Quantum ``phase transitions at the nano scale V. Kondo effect in various environments VI. Conclusion and some perspectives

50 Road toward quantum criticality Phase 1 Phase 2 Control parameter Quantum critical point Can we realize such physics around quantum dots?

51 Examples in impurity problems e

52 Examples in impurity problems The two-channel Kondo effect: e

53 Examples in impurity problems The two-channel Kondo effect: This fixed point is unstable and demands e

54 Examples in impurity problems The two-channel Kondo effect: This fixed point is unstable and demands The control parameter is therefore e

55 Examples in impurity problems The two-channel Kondo effect: This fixed point is unstable and demands The control parameter is therefore The two-impurity Kondo effect: e The control parameter is

56 Mesoscopic device for 2-channel Kondo From Goldhaber-Gordon et al., Nature 2007.

57 The two channels [Oreg & Goldhaber-Gordon, Phys. Rev. Lett. 90, 13 (2003)] 2 L 1 R Singular transition Cox & Zawadowski, Adv. in Phys. (1998) Pustilnik, Borda, Glazman, von Delft, PRB 69, (2004)

58 Scaling analysis of 1-channel Kondo physics

59

60

61 2 channel Kondo scaling

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63 Consider one electron per dot Double small dots Integrate out charge fluctuations: Direct spin exchange interaction Kondo couplings Describe the 2-impurity Kondo model

64 Leading processes Symmetrical version has been investigated a lot (~ heavy fermions): 2 A special electron-hole symmetry is needed to get a quantum phase transition Consider:

65 Existence of a quantum phase transition Statement: ALWAYS displays a true QPT! PARITY is not needed electron-hole symmetry can be broken 1 2 [G.Zarand, C.H. Chung, PS, and M. Vojta, PRL 2006]

66 Dangerous processes What does destroy QPT? Charge transfer! Most dangerous process: These are renormalized, and grow up Scaling analysis: Cross-over scale [G.Zarand,C.H. Chung, PS, and M. Vojta, PRL 2006]

67 Conductance [G.Z.,C.H. Chung, P. Simon, and M. Vojta, to appear in PRL] [G.Zarand,C.H. Chung, PS, and M. Vojta, PRL 2006]

68 Simpler geometry The transport properties can be computed exactly near the QCP provided E. Sela and I Affleck, PRL 2009

69 Other Quantum Phase Transitions - Singlet-Triplet transition See the talk by F. Balestro - From screened to a local moment regime In Aharonov-Bohm interferometer like devices - In triple quantum dot? Possible existence of a fully stable non Fermi liquid phase? See L. Dias da Silva, PS, N. Sandler, K. Ingersent, S. Ulloa, PRL 2009 See Lazarovits, PS, G. Zarand, L. Szunyoth, PRL 2005 K. Ingersent, A. W.W. Ludwig, I. Affleck, PRL 2005

70 OUTLINE I. The traditional (old-fashioned?) Kondo effect II. Direct access to single magnetic impurity physics III. Kondo effect in artificial magnetic impurities (quantum dots) IV. Quantum ``phase transitions at the nano scale V. Kondo effect in various environments VI. Conclusion and some perspectives

71 What type of electronic environment? SC DOT SC SC OR See the talks by R. Deblock and V. Meden DOT OR... See the talk by J. Hauptmann New physics to be envisioned and explored compared to the bulk situation

72 Kondo effect in a ferromagnetic environment.1 Main effect of the polarized electrodes They generate an effective magnetic exchange field Martinek et al., 2004

73 Kondo effect in a ferromagnetic environment.2 NRG results by Martinek et al., PRL 2004 Splitting of the Kondo resonance

74 Kondo effect in a ferromagnetic environment.3 Possibility of compensating this effective exchange field by a genuine magnetic field Restoration of the Kondo effect Martinek et al., PRL 2004

75 Kondo effect in a ferromagnetic environment.4 Possibility of compensating this effective exchange field With the dot plunger gate voltage Restoration of the Kondo effect Experiments: see talk by J. Hauptmann Martinek et al., PRL 2004

76 Kondo effect in a superconducting environment Problem complex because of - Coulomb interaction on the dot - Superconducting correlations in the leads However, each phenomenon is characterized by a single energy scale: the Kondo temperature the superconducting gap

77 Some recent experimental realizations with nanotubes Eichler et al., 2008 See talk by R. Deblock

78 0 Junction

79 0 Junction

80 0 Junction

81 0 Junction

82 0 Junction 0 Junction

83 Junction What happens when there is a localized spin on the dot?

84 Junction

85 Junction

86 Junction

87 Junction The sign of the Josephson current is sensitive to spin

88 Kondo effect in a superconducting environment Phase diagram to be expected ``0 Junction Junction Possibility of controlling the sign of the Josephson current with the gate voltage See talk by R. Deblock For a quantitative analysis of the phenomenon See talk by V. Meden For a spectroscopic analysis See T. Meng, PS, S Florens, PRB 2009

89 OUTLINE I. The traditional (old-fashioned?) Kondo effect II. Direct access to single magnetic impurity physics III. Kondo effect in artificial magnetic impurities (quantum dots) IV. Quantum ``phase transitions at the nano scale V. Kondo effect in various environments VI. Conclusion and some perspectives

90 properties Summary & Perspectives Resurgence of the Kondo effect due to - A control of the geometry - A full control of the set of parameter - A control of the electronic environment - Direct access to transport (thererefore spectral) Rich and new exotic physics: - Evidences of Quantum phase transitions - Non Fermi liquid physics in quantum dots Tremendous progress in numerical impurity solvers - NRG, DMRG, FRG, etc. - Application to DMFT (Dynamical Mean Field Theory) and calculation of genuine material properties

91 Summary & Perspectives Ideal playground to analyze non-equilibrium properties in correlated (nano) systems - Look at the non-equilibrium (ac) conductance - Noise see the talk by Takis Kontos - Extend RG out of equilibrium - Dynamical properties in the vicinity of a QPT Very nice from a fundamental point of view BUT can we do something useful out of this? - YES the Kondo effect is robust Use it for quantum Spintronics D. Feinberg, PS, APL 2004, PRL 2006 see the poster by Denis Feinberg