Kondo Effect in Nanostructures

Transcription

1 Kondo Effect in Nanostructures Argonne National Laboratory May 7th 7 Enrico Rossi University of Illinois at Chicago Collaborators: Dirk K. Morr Argonne National Laboratory, May 7

2 The Kondo-effect R Metal with magnetic impurities Resistance minimum Metal with normal impurities Metal with superconducting transition T K T 93 s: Discovered experimentally; 964: Using perturbation theory Kondo explains existence of minimum, but his calculaion gives R diverging for T ; 97 s: Renormalization approach by Anderson and Wilson provides adequate theoretical framwork for understanding of the Kondo-effect. Argonne National Laboratory, May 7

3 From Anderson-model to Kondo-model ε d ε d + U ε f ε f ε d ε d + U ε f ε f For Low E nergies U degeneracy two fold U H = kσ ɛkc kσ c kσ + σ ɛdndσ+ Und nd + V kσ ( c dσ c kσ + c kσ c dσ ) H = kσ ɛkc kσ c kσ+ J σσ S c R,σ τ σσ c R,σ J > Antifferomagnetic coupling between mangetic impurity and host electrons Argonne National Laboratory, May 7 3

4 Kondo resonance The antiferromagnetic interaction J can form a bound state of energy E B between the lo- cal spin and one made up from the conduction electron states. Kondo cloud For T < T K E B T K N d For the impurity we have a resonant state N host ε d ε d + U ε for ɛ = ; For the host electrons the density of states at the Fermi energy is suppressed = increase of resistivity. ε F ε Argonne National Laboratory, May 7 4

5 Nanostructures: Quatum Dots GaAs Quantum Dot. S.M. Cronenwett et. al Science (998) (b) Γ N, ground state -> N+ excited ε N -> N+ tune ɛ d ; U tune width of energy level, V ds N- -> N I Γ; tune U ; V g D. Golhaber Gordon et al. Nature (998) Argonne National Laboratory, May 7 5

6 Nanostructures: Quantum Corrals + Scanning Tunneling Microscopy (STM) Atomic control of impurity position; By changing the size and/or shape of the corral, we can control the LDOS of conduction electrons; D. Eigler IBM Direct observation of the LDOS. By combining advances in nanofabrication and new probes like STM we can now: Control the parameter governing the Kondo-effect; Observe directly the Kondo-effect. Argonne National Laboratory, May 7 6

7 Creation of a quantum candle: Kondo resonance Manhoran et al. Nature 43, 5 () V STM Tip I di dv Vacuum = N(r,eV) Material Kondo resonance in occupied focus Quantum Image in unoccupied focus Argonne National Laboratory, May 7 7

8 Corral eigenmodes and quantum images Quantum images are projected through corral eigenmodes Fiete et al., PRL (); Agam and Schiller, PRL (); Porras et al. PRB (); Aligia, PRB (). Ψ Ψ E n = 3 E 3 n = E n = E Manoharan et al. Nature 43, 5 () Important unsolved questions: How does the Kondo effect emerge inside a quantum corral? How does the Kondo effect depend on the position of the magnetic impurity? What is the form of the Kondo resonance in space and energy in a quantum corral? Argonne National Laboratory, May 7 8

9 Theoretical approach We model the system with the following Hamiltonian: H = i,j,σ t ij c i,σ c j,σ + U c I,σ c I,σ + JS c R,α σ αβc R,β I=..N c,σ corral scatterers free electrons magnetic impurity non-magnetic We consider a two-dimensional host metal on a square lattice with dispersion ɛ k = k /m µ, where µ is the chemical potential. In the following we set the lattice constant a to unity and use E /ma as our unit of energy. We solve the problem in two steps: Step : Compute the eigenmodes of quantum corral using Generalized scattering theory D.K. Morr and N. Stavropoulos, PRL (4). Step : Calculate effect of magnetic impurity using large-n expansion N. Read and D. M. Newns, J. Phys. C (983). Argonne National Laboratory, May 7 9

10 Surface States and Bulk States Surface states: D E S Bulk states: 3D DOS µ The presence of the bulk states might complicate the analysis, because we can have: Coupling surface-bulk states; Tip tunneling in part to bulk states; Coupling of the Kondo impurity to the bulk states ( Knorr et al. PRL ()) However we can reduce these effect can minimized: Use ultrathin film (few atomic layers) grown on insulating or semiconducting substrates (S.J. Tang et al. PRL (6)); Presence of corral should enhance relative importance of surface states with respect to bulk states. The mirage experiment shows D character of the states on Cu() surface inside a corral. Argonne National Laboratory, May 7

11 Generalized Scattering theory The host electrons undergo multiple scattering with the atoms forming the quantum corral. e G c, the Green s function for the conduction electrons in presence of the corral only is given by: G c (r, r, iω n ) = G (r r, iω n ) + j,l G (r r j, iω n ) T jl (iω n )G (r l r, iω n ) Where the T -matrix satises the Bethe-Salpeter equation: T ij (iω n ) = Uδ ij + U l G (r i r l, iω n )T li (iω n ). x x G c = G + + x x x x N c (r, ω) = π Im[G c(r, r, ω + iδ)] Argonne National Laboratory, May 7

12 LDOS for corral with No Kondo impurities r = (,) r = (5,) DOS [/E ] µ ω+µ [E ] Argonne National Laboratory, May 7

13 Large-N expansion We know that the perturbation analysis in the Kondo coupling, J, breaks down. In the large-n expansion we express the spin S of the magnetic impurity in terms of fermionic operators, f m, f m : S = N N µ B f mf m m= with the constraint: S = N µ B = n f N f mf m =. m= In our case S = / and then N =. We can then rewrite the Hamiltonian in the form: H = i,j,σ t ij c i,σ c j,σ + U I=..N c,σ c I,σ c I,σ + J αβ c R,β f αf β c R,α We then find an effective action, S eff, function of two fields: Decoupled using HS transformation introduce HS field s s: The hybridization of f electrons with host electrons; ɛ f : Lagrange multiplier to impose the constraint. Argonne National Laboratory, May 7 3

14 Critical Kondo coupling: Jcr For the fields s and ɛf we then take the mean fields values, obtained by minimizing Sef f on the saddle point. Approximation is exact in the limit N. Solve the saddle point equations for different values of T and J. In the large-n approach for fixed T there is a minimum value of J, Jcr, for which the saddle point equations admit a solution T [E ] J cr [E ] u u u u no corral r = (5,) K () J [E ] s [E ] T =. [E ]; r K = (;) T =. [E ]; r K = (;) T =. [E ]; r K = (5;) T =. [E ]; r K = (5;) U=. U=.5 () K r = (,) Kondo effct No Kondo effect T K J T [E ] J cr [E ] u u u u no corral r = (5,) K () J [E ] s [E ] T =. [E ]; r K = (;) T =. [E ]; r K = (;) T =. [E ]; r K = (5;) T =. [E ]; r K = (5;) U=. U=.5 () K r = (,) J [m] T [m] (5,) (,) Argonne National Laboratory, May 7 4

15 Spatially Dependent Kondo-effect Tk [E] Jcr [E] Nc [/E] (a) (b) (c) J=.3 E no corral J=.3 E J=. E J=.5 E J=.5 E x along R=(x,) For fixed T we can see the dependence of J cr on the position of the magnetic impurity inside the corral. In particular we see that J cr is minimum where the conduction electron LDOS is maximum. For f ixed J we can tune T K by moving the impurity inside the corral. In particular T K is maximum where the conduction electron LDOS is maximum. Using the spatial dependence of J cr, T K we can: Turn on and off the Kondo-effect by simply moving the magnetic impurity inside the corral; Increase or decrease T K with respect to the case with no corral. E.R. and Dirk. K. Morr PRL (6). Argonne National Laboratory, May 7 5

16 LDOS with Kondo Impurity For a given J and T < T K solving the saddle point equations we nd the values of s and ɛ f. Once we know these values we can calculate the LDOS of the f electrons and of the host electrons taking into account the Kondo coupling. For the f electrons we have the Green s function: F (R, iω n ) = iω n ɛ f s G c (R, R, iω n ) and for the host electrons: and then: G(r, r, iω n ) = G c (r, r, iω n ) + s G c (r, R, iω n )F (R, iω n )G c (R, r, iω n ) N f (R, ω) = N π Im[F (R, ω + iδ)]; N(r, ω)tot = Im[G(r, r, ω + iδ)]; π Argonne National Laboratory, May 7 6

17 No corral..5 With Impurity Clean.4. LDOS c electrons..5 LDOS(ω=) c electrons.6..8 With Impurity Clean ω r 5 LDOS f electrons 5 Simple dip for c electrons Peak for f electrons Simple oscillations in real space ω Argonne National Laboratory, May 7 7

18 Kondo Resonances [/E ] N c tot N f [/E ] tot (a) (b) ω [E ] no Kondo impurity r K = (5,) r = ( 5,) ω [E ] Original mode decreased in amplitude; suppression of LDOS at ω = Kondo resonaces also away from the magnetic impurity Additional resonances N modes N + Kondo resonances Argonne National Laboratory, May 7 8

19 Spatial structure of Kondo resonances (a) ω = (b) ω =.65 E tot Nc [/E] tot Nc [/E] (c) ω = -.9 E ω =.8 E (d) tot Nc [/E] tot Nc [/E] Argonne National Laboratory, May 7 9

20 Different geometries ω =.5.8 ω =.6 J cr J cr r 3 DOS no Kondo imp. DOS with Kondo imp. ω =.6 DOS (ω µ) Argonne National Laboratory, May 7

21 Temperature dependence of Kondo resonances..5 N.5 r=r=(5,) T =. E T =.5 E T =.5 E T =.75 E T =. E T =.5 E [E ]. ε f s DOS T [E ] (ω µ) [E ] N f DOS f T =. E T =.5 E T =.5 E T =.75 E T =. E T =.5 E N DOS.5.5 T =. E T =.5 E T =.5 E T =.75 E T =. E T =.5 E r = R (ω µ) [E ] (ω µ) [E ] Argonne National Laboratory, May 7

22 Future Directions There are many possible extensions of the work presented. Motivated by recent experiments RAPID COMMUNICATIONS R853 W. CHEN, T. JAMNEALA, V. MADHAVAN, AND M. F. CROMMIE PRB 6 (a) 3 (b) 6 width (mv) 4 FIG.. Process of building a cobalt dimer from two individual cobalt atoms on the surface of Au~! ~constant current STM imaging parameters: I5539 A, V5. V. ~a! The atoms initially are 5 Å apart ~center-to-center!. ~b! The atoms after being repositioned with the STM tip to a separation of only 9 Å. ~c! The atoms after being positioned into the final dimer configuration ~approximate separation is 4 Å!. The disappearance of the Kondo resonance is a signature of the interaction between two cobalt atoms at the gold surface. In order to understand this interaction, we must address the different possible interaction mechanisms. The three most likely mechanisms are as follows: ~! quenching of the dimer magnetic moment, ~! antiferromagnetic coupling between the cobalt atoms, ~3! reduction of the exchange coupling between the dimer magnetic moment and surrounding conduction electrons. Here we discuss these mechanisms and argue why we believe that the third mechanism is the one that accurately describes the behavior of a cobalt dimer on Au~!. 6 4 AF FM W. Chen et al. PRB, 6 R859 (999) less as the energy is swept through E F (V5). The second curve in Fig. 3 shows the di/dv spectrum measured while holding the STM tip stationary over the center of an isolated cobalt atom on the gold surface. Here a narrow, asymmetric resonance can be seen near E F. This is the Kondo resonance for a single cobalt atom.6 ~This feature has also been interpreted as a bare d resonance,9 but we believe the width is too narrow for this to be the case.6! The width of the resonance yields T K 57 K, and the energy asymmetry is explained by the fact that the Kondo resonance is expressed via a Fano line shape.6,7 Spectra measured on different cobalt atoms did not deviate significantly from this curve for intercobalt separations ranging from hundreds of Å down to 6 Å. At separations less than 6 Å, however, cobalt atoms merge into a dimer and their electronic properties change dramatically. The third curve in Fig. 3 shows the di/dv spectrum measured with the STM tip held stationary over an atomically fabricated cobalt dimer. The Kondo resonance has disappeared. This abrupt disappearance of the Kondo resonance was seen consistently in spectra obtained from cobalt dimers using different STM tips ~different tips were obtained through field emission and controlled collision with the surface!. (c) EFM-EAF (mev) FIG.. Constant current STM topograph of a fabricated cobalt dimer ~upper left! positioned near a single cobalt atom. Height gradient is. Å per contour. Imaging parameters: I5539 A, V 5. V. di/dv (a.u.) Bias (mv) 5 3 P.Wahl et al PRL (6) we are now considering the problem of two or more FIG. 3. di/dv spectra obtained with the STM tip held over a single cobalt atom, an atomically fabricated cobalt dimer, and the clean gold surface ~curves have been shifted vertically!. The Kondo resonance can be seen for the individual cobalt atom, but is no longer present for the cobalt dimer. di/dv5538 V for all three spectra at V5. V ~the amplitude of the Kondo resonance reflects a 6% change in di/dv for the single-atom spectrum!. Argonne National Laboratory, May 7 FIG. : Kondo resonance of cobalt dimers on Cu() measured by STS at 6 K. A ends of the dimers are shown (green and black dots) to be equivalent. (a) Mod bottom) a compact dimer (.56 A ), a dimer at 5. A, at 5.7 A, at 7.4 A, at 7 infinite distance (> the A )corral. are depicted. The spectra are shown together with fit impurities inside dimer at 5. A also a simulated curve according to Eq. with J = 5 mev and the dimer at 5. A, a linear background had to be taken into account to obtain a shifted vertically for clarity) (b) shows the width of the resonance as a function o exchange interaction between cobalt adatoms on Cu() [5]. The three distinct different greys. from.56 A to 8. A together with the corresponding STS spectra. For the compact dimer the interaction between I in Fig. (c)] a perature T dimer

23 Conclusions We showed that the spatial structure of the corral s low energy eigenmode leads to: Spatial variations in the critical coupling J cr, in particular J cr is minimum where LDOS is maximum; Spatial variations of the Kondo temperature T K, in particular T K is maximum where LDOS is maximum Spatial dependence of the relation between J cr and T Quantum Corrals: a new probe for the Kondo effect! Argonne National Laboratory, May 7 3