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

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

3. 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

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

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

6. 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!

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

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

9. 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

10. 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

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

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

13. 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

14. Cooper pairing in metals
BCS model:
retarded interaction

15. 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 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!

16. !

17. 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.

18. 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. 19

20. 300 K GaN microcavities: a polariton condensate at room temperature!
Below threshold
Above threshold
J.J. Baumberg, A. Kavokin et al., PRL 101, (2008)

21. 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??!

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

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

24. 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:

25. Boglyubov transformation:
Concentration of exciton-polaritons

26. Electron – electron interaction potential:
exciton mediated interaction
Coulomb repulsion

27. Results for a model GaN microcavity

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

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

30. 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= cm-1
nex=1011cm-1
electrons
holes
L=12 nm
2DEG l L

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

32. 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