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Yoram Alhassid (Yale)

Chaos and interactions in nano-size metallic grains: the competition between superconductivity and ferromagnetism. Yoram Alhassid (Yale). Sebastian Schmidt (Yale, ETH Zurich). Introduction Universal Hamiltonian for a chaotic grain: the competition between

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Yoram Alhassid (Yale)

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  1. Chaos and interactions in nano-size metallic grains: the competition between superconductivity and ferromagnetism Yoram Alhassid (Yale) Sebastian Schmidt (Yale, ETH Zurich) • Introduction • Universal Hamiltonian for a chaotic grain: the competition between • superconductivity (pairing correlations) and ferromagnetism (exchange • correlations). • Quantum phase diagram (ground-state spin). • Transport: mesoscopic fluctuations of Coulomb blockade conductance • Conclusion

  2. Introduction: metallic grains (nanoparticles) • Discrete energy levels extracted from non-linear conductance measurements. • Superconducting at low temperatures [von Delft and Ralph, Phys. Rep. 345, 61 (2001)] • A pairing gap was observed in spectra of size ~ 10 nm grains. • Explained by BCS theory: • valid in the bulk limit = single-particle level spacing = pairing gap However, in grains smaller than ~ 3 nm, , the fluctuations dominate and “superconductivity would no longer be possible” ( Anderson).

  3. Universal Hamiltonian for a chaotic grain An isolated chaotic grain with a large number of electrons is described by the universal Hamiltonian[Kurland, Aleiner, Altshuler, PRB 62, 14886 (2000)] • Discrete single-particle levels (spin degenerate) and wave functions that follow random matrix theory (RMT). • Charging energy term describing a grain with capacitance and • electrons : constant interaction (CI) model. • BCS-like pairing interaction with coupling ( creates pairs of spin up/down electrons). • A ferromagnetic exchange interaction with exchange constant ( is the total spin of the grain). Competition: Pairing correlations and one-body term favor minimal ground-state spin, while spin exchange interaction favors maximal spin polarization.

  4. A derivation from symmetry principles[Y. Alhassid, H.A. Weidemuller, A. Wobst, PRB 72, 045318 (2005)] Hamiltonian of interacting electrons in a dot: • The randomness of the single-particle wave functions induces randomness in the two-body interaction matrix elements. • Cumulants of the interaction matrix elements are determined by requiring invariance under a change of the single-particle basis (single-particle dynamics are chaotic). Averages: There are three (two) invariants in the orthogonal (unitary) symmetry:

  5. Eigenstates of the universal Hamiltonian: The eigenstates factorizes into two parts: (i) are zero-spin eigenstates of the reduced BCS Hamiltonian in a subset of doubly-occupied and empty levels. Example of 8 electrons in 7 levels: 3 pairs plus 2 singles = blue levels = red levels (ii) are eigenstates of , obtained by coupling spin-1/2 singly-occupied levels in to total spin and spin projection .

  6. Quantum phase diagram (ground-state spin) S. Schmidt, Y.A., K. van Houcke, Europhys. Lett. 80, 47004 (2007) [ Ying et al, PRB 74, 012503 (2006)] Ground-state spin in the plane (for an equally-spaced single-particle spectrum) Exact solution: there is a coexistence regime of superconducting and ferromagnetic correlations ( ). Mean field (S-dependent BCS): lowest solutions with do not have pairing correlations (gap is zero).

  7. Controling the coexistence regime: a Zeeman field • A Zeeman field broadens the • coexistence regime and makes it • accessible to typical values of Stoner staircase (Ground-state spin versus ) For a fixed the spin increases by discrete steps as a function of • Spin jumps: the first step can have Experiments: it is difficult to measure the ground-state spin.

  8. Transport: Coulomb blockade conductance Quantum dots: CI model [R. Jalabert, A.D. Stone, Y. Alhassid, PRL 68, 3468 (1992)]. follows RMT wave function statistics. • Conductance peak height (forG <<T<<D) is the partial width of the single-electron resonance to decay into the left (right) lead: Gla ½fl (rc)½2where rcis the point contact. Peak height distributions Exp: Folk et al., PRL 76 1699 (1996) Exp: Chang et al. PRL 76 1695 (1996)

  9. Quantum dots: charging + exchange correlations [Y. Alhassid and T. Rupp, PRL 91, 056801 (2003)] Conductance peak heightsgmax Conductance Peak spacingsD2 (Exchange constant = ) Excellent agreement of theory and experiment for the peak spacing widths(D2) Better quantitative agreement for the ratio at Excellent agreement for peak height distribution at Experiments: C.M. Marcus et al. (1998)

  10. Nano-size metallic grains: charging, exchange + pairing correlations S. Schmidt and Y. Alhassid, arXive: 0802.0901, PRL, in press (2008) For a grain weakly-coupled to leads we can use the rate equation formalism plus linear response in the presence of interactions [Alhassid, Rupp, Kaminski, Glazman, PRB 69, 115331 (2004)]. The linear conductance is calculated from the many-body energies of the dot and the lead-grain tunneling rates between many-body eigenstates of the N-electron grain and of the (N+1)-electron grain. • Only a single level contributes: • The electron tunnels into an empty level and blocks it: • The electron tunnels into a singly-occupied level :

  11. Mesoscopic fluctuations of the conductance peaks Single-particle energies and wave functions described by random matrix statistics (GOE). (i) Peak-spacing statistics ( ) Peak-spacing distributions Average peak spacing • Exchange suppresses bimodality while pairing enhances it.

  12. (ii) Peak-height statistics ( ) Peak-height distributions Peak height fluctuation width • Exchange interaction suppresses the peak-height fluctuations. Mesoscopic signatures of coexistence of pairing and exchange correlations for and : bimodality of peak spacing distribution and suppression of peak height fluctuations.

  13. Conclusion • A nano-size chaotic metallic grain is described by the universal Hamiltonian • a competition between superconductivity and ferromagnetism in a finite-size system. • Quantum phase diagram (ground-state spin): coexistence regime of superconductivity and ferromagnetism. • Transport: signatures of coexistence between pairing and exchange correlations in the mesoscopic conductance fluctuations. Experimental candidates: platinum ( ), vanadium ( ). Open problems Effects of spin-orbit scattering in the presence of pairing and exchange correlations: g-factor statistics,… (time-reversal remains a good symmetry). [Spin-orbit + exchange: D. Gorokhov and P. Brouwer, PRB 69, 155417 (2004).]

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