1 / 13

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

tameka
Download Presentation

Yoram Alhassid (Yale)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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).]

More Related