Hybrid Bose-Fermi systems Alexey Kavokin University of Southampton, UK

Bosons Fermions Integer spin half-integer spin Bosonic stimulation Pauli exclusion principle BEC BCS Superfluidity Superconductivity And if they are coupled?

The previous lecture was about fermions In this lecture: quick reminder about Bose-Einstein condensation composite bosons: excitons superfluidity: Bogolyubov dispersion excitons + electrons: Fermi see + Bose gas exciton induced superconductivity interaction induced roton minimum, suppression of superfluidity All original results obtained in collaboration with Ivan Shelykh, Fabrice Laussy, Tom Taylor

Bose-Einstein condensation The distribution function: How many bosons do we have? Their concentration dimensionality of the system What happens if

Critical concentration: All extra bosons go to the condensate: depends on the mass, because and m3 BEC m2 m1<m2<m3 m1 T

Bose-Einstein condensation Superfluidity Superconductivity Bose-Einstein condensation All this happens at very low temperatures ... Condensation of cold atoms Exciton-polaritons: very light effective mass very high critical temperature for BEC!

Excitons: composite bosons EXCITON: an artificial ATOM Hole Atom Electron EXCITON + PHOTON = EXCITON-POLARITON Exciton polaritons are also composite bosons 7

POLARITON LASER what is it ? It is a coherent light source based on the Bose-condensat of exciton-polaritons in a microcavity

Concept of polariton lasing: Photon mode dispersion Extremely light effective mass Optically or electronically excited exciton-polaritons relax towards the ground state and Bose-condense there. Their relaxation is stimulated by final state population. The condensate emits spontaneously a coherent light 9

SUPERFLUIDITY In 1937 Kapitsa, Allen and Miserer discovered the superfluidity of He4 Lev Landau has proposed a phenomenological model of superfluidity E Nikolay Bogolyubov has created a theory of superfluidity of interacting bosons k Linear dispersion “sound” roton

Bogolyubov spectrum and superfluidity Gross-Pitaevskii equation for a conensate of interacting bosons substitution yields therefore

Resolving the linear system We obtain Bogolyubov spectrum responsible for superfluidity! k

LIGHT-INDUCED SUPERCONDUCTIVITY (Exciton mechanism of superconductivity revisited) Motivation: recent discovery of BEC of exciton polaritons Mechanism: exciton condensate instead of phonons Structure: metal-semiconductor sandwich or more complex heterostructures (microcavities) Starting point: Bose condensate of exciton polaritons put in contact to the Fermi see of electrons Electron –electron attraction: increases with increase of optical pumping! Result: light mediated BCS superconductivity: possibly very high Tc

Cooper pairing in metals BCS model: retarded interaction

Bardeen-Cooper-Schrieffer (BCS): Critical temperature: BCS: “weak coupling” regime Debye temperature Coupling constant Density of electronic states at the Fermi level Debye temperatures: Aluminium 428 K Cadmium 209 K Chromium 630 K Copper 343.5 K Gold 165 K Iron 470 K Lead 105 K Manganese 410 K Nickel 450 K Platinum 240 K Silicon 645 K Silver 225 K Tantalum 240 K Tin (white) 200 K Titanium 420 K Tungsten 400 K Zinc 327 K Carbon 2230 K Ice 192 K in conventional superconductors, which is why the critical temperature is very low!

!

An exciton mechanism may be realised in 2D metal-dielectric sandwiches (higher ). Non-equilibrium superconductivity has a great future BUT IT NEVER WORKED ! WHY ? Exciton-electron interaction still weak; Excitons are too fast (reduced retardation effect), consequently: 3) Coulomb repulsion becomes important.

Bose-Einstein condensation of exciton polaritons (2006-2010) In semiconductor microcavities excitons may be strongly coupled to photon modes An exciton is an electron-hole pair bound by Coulomb attraction photon exciton resonance Exciton-polaritons 193 articles in Physical Review Letters with « microcavity » in the title or abstract (compare to 368 with « graphene ») 18

19

300 K GaN microcavities: a polariton condensate at room temperature! Below threshold Above threshold J.J. Baumberg, A. Kavokin et al., PRL 101, 136409 (2008)

Our idea: Superconductivity mediated by a Bose-Einstein condensate of exciton-polaritons The condensate is created by resonant optical excitation BEC can exist at 300 K, why not superconductivity??!

We consider the following model structure: a heavily n-doped layer embedded between two neutral QWs in a microcavity

Electrons + exciton-polariton BEC: interaction Hamiltonian Coulomb repulsion Electron-polariton interactions Polariton-polariton interactions

Interactions: Electron-exciton interaction: L is the distance between exciton BEC and 2DEG l is the distance between electron and hole centers of mass in normal to QW plane direction Electron-electron interaction:

Boglyubov transformation: Concentration of exciton-polaritons

Electron – electron interaction potential: exciton mediated interaction Coulomb repulsion

Results for a model GaN microcavity

Our potential Comparison with BCS BCS potential We have: Much stronger attraction; Similar Debye temperature Peculiar shape of the potential Energy W

Solving the gap equation by iterations... we obtain the superconducting gap which vanishes at the crictical temperature

2DEG Now we know what may happen to fermions, But what will happen to bosons?? nex=109cm-1 L=55 nm L=25 nm nex=5 1010 cm-1 nex=1011cm-1 electrons holes L=12 nm 2DEG l L

Suppression of the Bose-Einstein condensation and superfluidity real space condensation classical fluid superfluid BEC

Conclusions: In Bose-Fermi systems with direct repulsive interaction of bosons and fermions, due to Froelich-like indirect interactions: Fermions attract fermions which results in Cooper pairing Bosons attract bosons which results in formation of the roton minimum and suppression of BEC