PG5295 Muitos Corpos 1 Electronic Transport in Quantum dots 2 Kondo effect: Intro/theory. 3 Kondo effect in nanostructures

Transcription

1 PG5295 Muitos Corpos 1 Electronic Transport in Quantum dots 2 Kondo effect: Intro/theory. 3 Kondo effect in nanostructures Prof. Luis Gregório Dias DFMT

2 PG5295 Muitos Corpos 1 Electronic Transport in Quantum dots 2 Kondo effect: Intro/theory. 3 Kondo effect in nanostructures Prof. Luis Gregório Dias DFMT Basic references for today s lecture: A.C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge Press, R. Bulla, T. Costi, Prushcke, Rev. Mod. Phys (in press) arxiv K.G. Wilson, Rev. Mod. Phys (1975).

3 Lecture 2: Outline Kondo effect: Intro. Kondo s original idea: Perturbation theory. Numerical Renormalization Group (NRG). s-d and Anderson models. NRG results for the local density of states.

4 Coulomb Blockade in Quantum Dots Coulomb Blockade in Quantum Dots: dot spectroscopy Y. Alhassid Rev. Mod. Phys (2000).

5 Coulomb Diamonds (Stability Diagram) ev sd ev gate Coulomb Blockade in Quantum Dots L. P. Kouwenhoven et al. Science (1996).

6 Carbon nanotube Quantum dots. Makarovski, Zhukov, Liu, Filkenstein PRB R (2007). Carbon nanotubes depsited on top of mettalic electrodes. Quantum dots defined within the carbon nanotubes. More structure than in quantum dots: shell structure due to orbital degeneracy. Gleb Filkenstein s webpage:

7 Lecture 2 (coming up )

8 More is Different The behavior of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of simple extrapolation of the properties of a few particles. Instead, at each level of complexity entirely new properties appear and the understanding of the new behaviors requires research which I think is as fundamental in its nature as any other. Phillip W. Anderson, More is Different, Science (1972)

9 Can you make atoms out of atoms? Energy Ga As 4p 4p 4s 3d 3d 4s [Ar] 3d 10 4s 2 4p 1 [Ar] 4s 2 3d 10 4p 3 Many Atoms! GaAs crystal E conduction Atomic Energy levels E F Band gap Many Atoms! Band structure M. Rohlfing et al. PRB (1993) valence k

10 From atoms to metals, plus atoms E Conduction band E Many Atoms! filled (few) ATOM E F Metal (non magnetic)? Magnetic impurities (e.g., transition atoms, with unfilled d-levels, f-levels (REarths )) Is the resulting compound still a metal?

11 Kondo effect µ Fe /µ B Magnetic impurity in a metal. 30 s - Resisivity measurements: minimum in ρ(t); T min depends on c imp. 60 s - Correlation between the existence of a Curie-Weiss component in the susceptibility (magnetic moment) and resistance minimum. ρ/ρ 4.2K 1% Fe Mo.9 Nb.1 Mo.2 Nb.8 Mo.8 Nb.2 Mo.7 Nb.3 Top: A.M. Clogston et al Phys. Rev (1962). Bottom: M.P. Sarachik et al Phys. Rev. 135 A1041 (1964). T ( o K)

12 Kondo effect M.P. Sarachik et al Phys. Rev. 135 A1041 (1964). ρ/ρ 4.2K Mo.2 Nb.8 1% Fe Mo.7 Nb.3 ξ K ~ v F /k B T K Mo.8 Nb.2 Mo.9 Nb.1 ρ(t) T ( o K) Characteristic Resistivity decreases increases energy with scale: the decreasing Kondo temperature T (Kondo (usual) T K effect)

13 Kondo problem: s-d Hamiltonian Kondo problem: s-wave coupling with spin impurity (s-d model): Conduction band E filled E F Metal (non magnetic, s-band) Magnetic impurity (unfilled d-level)

14 Kondo s explanation for T min (1964) H = J S c c + S c c + s-d k k k k k,k Spin: J>0 AFM k z ( c c c c ) + S + e k k k k c c k kσ kσ Metal: Free waves Many-body effect: virtual bound state near the Fermi energy. AFM coupling (J>0) spin-flip scattering Kondo problem: s-wave coupling with spin impurity (s-d model): Metal DOS -D ε F D ε

15 Kondo s explanation for T min (1964) Perturbation theory in J 3 : Kondo calculated the conductivity in the linear response regime spin 2 kbt Rimp J 1 4J ρ0 log D 5 kbt Rtot ( T ) = at cimp Rimp log D T min R D = 1/5 c imp 1/5 imp 5ak B Only one free paramenter: the Kondo temperature T K Temperature at which the perturbative expansion diverges. 1 2J ρ0 k T ~ De B K

16 Kondo s explanation for T min (1964) ( ) 5 log kbt Rtot T = at cimp Rimp D Theory diverges logarithmically for T 0 or D. What is (T<T K perturbation expasion no longer holds) going on?{ Experiments show finite R as T 0 or D. ρ(ε) -D ε F D ε

17 Kondo Impurity and Lattice models Concentrated Diluted case (Kondo case: impurity Kondo Lattice model) (e.g., some heavy-fermion materials) ξ K ~ v F /k B T K Kondo impurity model suitable for diluted impurities in metals. Some rare-earth compounds (localized 4f or 5f shells) can be described as Kondo lattices. This includes so called heavy fermion materials (e.g. Cerium and Uraniumbased compounds: CeCu 2 Si 2 ; UBe 13 ; etc).

18 Kondo Lattice models Concentrated case: Kondo Lattice (e.g., some heavy-fermion materials) Kondo impurity model suitable for diluted impurities in metals. Some rare-earth compounds (localized 4f or 5f shells) can be described as Kondo lattices. This includes so called heavy fermion materials (e.g. Cerium and Uraniumbased compounds CeCu 2 Si 2, UBe 13 ).

19 A little bit of Kondo history: Early 30s : Resistance minimum in some metals Early 50s : theoretical work on impurities in metals Virtual Bound States (Friedel) 1961: Anderson model for magnetic impurities in metals 1964: s-d model and Kondo solution (PT) 1970: Anderson Poor s man scaling : Wilson s Numerical Renormalization Group (non PT) 1980 : Andrei and Wiegmann s exact solution

20 A little bit of Kondo history: Early 30s : Resistance minimum in some metals Early 50s : theoretical work on impurities in metals Virtual Bound States (Friedel) Kenneth G. Wilson Physics Nobel Prize in : Anderson model for magnetic impurities in "for his theory for critical phenomena in connection metals with phase transitions" 1964: s-d model and Kondo solution (PT) 1970: Anderson Poor s man scaling : 75: Wilson s s Numerical Renormalization Group (non PT) 1980 : Andrei and Wiegmann s exact solution

21 Kondo s explanation for T min (1964) What is going on?{ 1 k T / D B ( ) 5 log kbt Rtot T = at cimp Rimp D Diverges logarithmically for T 0 or D. (T<T K perturbation expasion no longer holds) Experiments show finite R as T 0 or D. The log comes from something like: dε kbt = log ε D All energy scales contribute! -D ρ(ε) ε F D ε

22 Perturbative Discretization of CB ε = (E-E F )/D = ( E)/D

23 Perturbative Discretization of CB = ( E)/D A n = log 1 1 n Want to keep all contributions for D? ε cut-off = 1 n max = -1 Not a good approach! A 7 > A 6 > A 5 > A 4 > A 3 > A 2 > A 1

24 Wilson s CB Logarithmic Discretization n =Λ -n (Λ=2) ε = (E-E F )/D

25 Wilson s CB Logarithmic Discretization A = log Λ= const. n n =Λ -n -n ε cut-off no n max = Λ Now you re ok! A 3 = A 2 = A 1 (Λ=2)

26 Kondo problem: s-d Hamiltonian Kondo problem: s-wave coupling with spin impurity (s-d model): ρ(ε) Conduction band E filled E F Metal (non magnetic, s-band) Magnetic impurity (unfilled d-level)

27 Kondo s-d Hamiltonian H = J S c c + S c c + s-d k k k k k,k k z ( c c c c ) + S + e k k k k c c k kσ kσ ρ(ε) From continuum k to a discretized band. Transform H s-d into a linear chain form (exact, as long as the chain is infinite):

28 New Hamiltonian (Wilson s RG method) Logarithmic CB discretization is the key to avoid divergences! Map: conduction band Linear Chain Lanczos algorithm. Site n new energy scale: DΛ -(n+1) < ε k - ε F < DΛ -n Iterative numerical solution J γ 1 ρ(ε) γ 2 γ 3... γ n ~Λ -n/2

29 Logarithmic Discretization. Steps: 1. Slice the conduction band in intervals in a log scale (parameter Λ) 2. Continuum spectrum approximated by a single state 3. Mapping into a tight binding chain: sites correspond to different energy scales. t n ~Λ -n/2

30 Wilson s CB Logarithmic Discretization Logarithmic Discretization (in space): Λ>1 ρ(ε)

31 Wilson s CB Logarithmic Discretization Logarithmic Discretization (in energy): Λ>1

32 New Hamiltonian (Wilson) Recurrence relation (Renormalization procedure). ρ(ε) J γ 1 γ 2 γ 3... γ n ~Λ -n/2

33 New Hamiltonian (Wilson) Suppose you diagonalize H N getting E k and k> and you want to diagonalize H N+1 using this basis. First, you expand your basis: k 0... Then you calculate <k,a f + N k,a >, <k,a f N k,a >and you have the matrix elements for H N+1 (sounds easy, right?)

34 Intrinsic Difficulty You ran into problems when N~5. The basis is too large! (grows as 2 (2N+1) ) 0 N=0; (just the impurity); 2 states (up and down) N=1; 8 states N=2; 32 states N=5; 2048 states ( ) N=20; 2.199x10 12 states: 1 byte per state 20 HDs just to store the basis. And we might go up to N=180; 1.88x states. Can we store this basis? (Hint: The number of atoms in the universe is ~ ) Cut-off the basis lowest ~1500 or so in the next round (Even then, you end up having to diagonalize a 4000x4000 matrix )....

35 Renormalization Procedure Iterative numerical solution. Renormalize by Λ 1/2. Keep low energy states. J γ 1 ξ N γ 2 γ γ n ~ ξ n Λ -n/2 H N H N+1

36 Renormalization Group Transformation Renormalization Group transformation: (Rescale energy by Λ 1/2 ). ξ N... H N Fixed point H*: indicates scale invariance. H N+1 Fixed points

37 Spectral function calculation Local Density of states: Lehmann representation. At each NRG step, you define

38 Spectral function At each NRG step:

39 Spectral function calculation (Costi) To get a continuos curve, need to broaden deltas. Best choice: log gaussian

40 NRG on Anderson model: LDOS ε 1 +U 1 t γ 1 γ 2 γ 3... ε 1 γ n ~Λ -n/2 Single-particle peaks at ε d and ε d +U. Many-body peak at the Fermi energy: Kondo resonance (width ~T K ). NRG: good resolution at low ω (log discretization). Γ ε d ~T K Γ ε d + U

41 Numerical Renormalization Group What can you do? Describe the physics at different energy scales for arbitrary J. Probe the parameter phase diagram. Crossing between the free and screened magnetic moment regimes. Energy scale of the transition is of order T k ~T k

42 Recent Developments: TD-NRG Application: time dependent impurity problems

43 Summary: NRG overview NRG method: designed to handle quantum impurity problems All energy scales treated on the same footing. Non-perturbative: can access transitions between fixed points in the parameter space Calculation of physical properties

44 Anderson Model e d +U ε F e d t D with Quantum dot language e d : energy level U: Coulomb repulsion e F : Fermi energy in the metal t: Hybridization D: bandwidth e d : position of the level (V g ) U: Charging energy e F : Fermi energy in the leads t: dot-lead tunneling D: bandwidth

45 Schrieffer- Wolff Transformation Anderson Model Existence of localized moment V kd << U Schrieffer-Wolff transformation ε E d +U s-d Model ε F E d

46 Schrieffer- Wolff Transformation From: Anderson Model (single occupation) with To: s-d (Kondo) Model

47 History of Kondo Phenomena Observed in the 30s Explained in the 60s Numerically Calculated in the 70s (NRG) Exactly solved in the 80s (Bethe-Ansatz) So, what s new about it? Kondo correlations observed in many different set ups: Transport in quantum dots, quantum wires, etc STM measurements of magnetic structures on metallic surfaces (e.g., single atoms, molecules. Quantum mirage )...