Chaos and interactions in nano-size metallic grains: the competition between superconductivity and ferromagnetism Yoram Alhassid (Yale) Introduction Universal.

Chaos and interactions in nano-size metallic grains: the competition between superconductivity and

ferromagnetismYoram Alhassid (Yale)

• 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

Sebastian Schmidt (Yale, ETH Zurich)

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 ).

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)]

2† † 2 2 †( )

2 s

eH a a a a N J S G P P

C

• A ferromagnetic exchange interaction with exchange constant ( is the total spin of the grain).S11111111111111

• Discrete single-particle levels (spin degenerate) and wave functions that follow random matrix theory (RMT).

† † †P a a

• BCS-like pairing interaction with coupling ( creates pairs of spin up/down electrons).

0G

0sJ

• Charging energy term describing a grain with capacitance andelectrons : constant interaction (CI) model.

NC

Competition: Pairing correlations and one-body term favor minimal ground-state spin, while spin exchange interaction favors maximal spin polarization.

A derivation from symmetry principles[Y. Alhassid, H.A. Weidemuller, A. Wobst, PRB 72, 045318 (2005)]

† † †;

1v

2H a a a a a a

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:

; 0v v sJ G

2 2 †0 0

1 ˆˆ ˆ ˆ ˆv / 2 v / 22 S S SV J N J N J S G P P

The eigenstates factorizes into two parts:| ; B SU M

(i) are zero-spin eigenstates of the reduced BCS Hamiltonian in a subset of doubly-occupied and empty levels.U

|U

| B SM 2S

Eigenstates of the universal Hamiltonian:

(ii) are eigenstates of , obtained by coupling spin-1/2 singly-occupied levels in to total spin and spin projection .

Example of8 electronsin 7 levels:3 pairs plus 2 singles

= blue levels = red levels

B S M

BU

Quantum phase diagram (ground-state spin)

Exact solution: there is a coexistence regime of superconductingand ferromagnetic correlations ( ). 0S

Mean field (S-dependent BCS): lowest solutions with do not have pairing correlations (gap is zero).

S. Schmidt, Y.A., K. van Houcke, Europhys. Lett. 80, 47004 (2007)

0S

[ Ying et al, PRB 74, 012503 (2006)]

Ground-state spin in the plane (for an equally-spaced single-particle spectrum)

/ /sJ

• A Zeeman field broadens the coexistence regime and makes itaccessible to typical values of

For a fixed the spin increases bydiscrete steps as a function of

//sJ

1S

sJ

Controling the coexistence regime: a Zeeman field

Stoner staircase

• Spin jumps: the first step can have

(Ground-state spin versus ) /sJ

Experiments: it is difficult to measure the ground-state spin.

Quantum dots: CI model

• Conductance peak height (for <<T<< )

is the partial width of the single-electron resonance to decay into the left (right) lead: rc2 where rc is the point contact.

follows RMT wave function statistics.

Exp: Chang et al. PRL 76 1695 (1996)

Exp: Folk et al., PRL 76 1699 (1996)

2 1 l r

l r

eG

kT

( )l r

[R. Jalabert, A.D. Stone, Y. Alhassid, PRL 68, 3468 (1992)].

Transport: Coulomb blockade conductance

( )cr

Peak height distributions

Quantum dots: charging + exchange correlations[[Y. Alhassid and T. Rupp, PRL 91, 056801 (2003)]

Excellent agreement of theory and experiment for the peak spacing width (2)

Conductance Conductance Peak spacingsPeak spacings 22

Conductance peak heightsConductance peak heights ggmaxmax

Better quantitative agreement for the ratioat

Excellent agreementfor peak height distribution at 0.1kT

0.6kT

Experiments: C.M. Marcus et al. (1998)

max max( ) /g g

(Exchange constant = )0.3

Nano-size metallic grains: charging, exchange + pairing correlations

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)].

,T

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.

, S ', 'S

,' ';

l rS S

, , † 2' '; 0 ,| ( ', ' || ( )|| , )|l r l rS S l rS r S

Only a single level contributes:

(i) The electron tunnels into an empty level and blocks it: (ii) The electron tunnels into a singly-occupied level :

' { }B B ' { }U U

k

† †, ,( ) ( )l r k l r k

k

r r a

S. Schmidt and Y. Alhassid, arXive: 0802.0901, PRL, in press (2008)

Mesoscopic fluctuations of the conductance peaks

(i) Peak-spacing statistics ( )

Peak-spacing distributions

• Exchange suppresses bimodality while pairing enhances it.

Average peak spacing

Single-particle energies and wave functions described by random matrixstatistics (GOE).

0.1T

(ii) Peak-height statistics ( )

Peak-height distributions

• Exchange interaction suppresses the peak-height fluctuations.

Mesoscopic signatures of coexistence of pairing and exchange correlations for and : bimodality of peak spacing distribution andsuppression of peak height fluctuations.

/ 0.5 / 0.6sJ

Peak height fluctuation width

0.1T

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.

Effects of spin-orbit scattering in the presence of pairing and exchangecorrelations: g-factor statistics,… (time-reversal remains a good symmetry).

[Spin-orbit + exchange: D. Gorokhov and P. Brouwer, PRB 69, 155417 (2004).]

Open problems

Experimental candidates: platinum ( ), vanadium ( ).

/ 0.6 0.7sJ

/ 0.6sJ