1. BEC-BCS cross-over in the exciton gas
Monique Combescot
Institut des NanoSciences de Paris, CNRS
Université Pierre et Marie Curie
Taipei march 2012

2. 2) Under a density increase 3) Optical traps for dark excitons
Very dilute limit
2) Under a density increase
3) Optical traps for dark excitons

3. Part 1
very dilute limit

4. Semiconductor conduction band photon valence band  photon
conduction electron
valence electron

5. 
electron
photon 
hole
photon
hole
electron

6. hole: full valence band minus one electron
formidable reduction of the many-body problem !

7. valence electron
hole

8. Naive view of an exciton
electron
hole
exciton similar to Hydrogen atom
light effective mass
dielectric constant

9. The true story is
far more complex

10. Coulomb interaction in a periodic lattice
Bloch states

11. repulsive scattering between conduction and valence electrons

12. Intraband Coulomb processes
must be dressed by a dielectric constant
results from many-body effects induced by a
« boiling valence band »

13. 

14. 

15.  

17. Direct scatterings between valence and/or conduction electrons:
repulsive as for free electrons
reduced by the semiconductor dielectric constant

18. repulsion between conduction and valence electrons
attraction between electrons and holes
We turn from conduction/valence electrons to electrons/holes

19. Attraction between one electron and one hole
reduced by dielectric constant

20. One Wannier exciton
set of ladder diagrams between one electron and one hole
conserved through Coulomb scatterings
Wannier exciton
energy
wave function

21. Exciton creation operator

22. Excitons are boson-like particles
as bosons
but not exactly

23. « Pauli scattering » m j i n
just what you read

24. Being boson-like particles, excitons, as cold atoms, must have
a Bose-Einstein condensation
Yet, the precise nature of composite boson condensate
is not known !
close to
but surely not exactly

25. Exciton BEC searched for decades …. but never evidenced !
Reason: the condensate must be dark …
and search has been done through
photoluminescence experiments

26. Although small, interband Coulomb interaction also exists
exists for pairs
J= (+1, 0, -1) only
emission and reabsorption
of a virtual photon

27. conduction band made of S atomic level
valence band made of P atomic level

28. valence-conduction Coulomb processes exist for bright excitons only 
Electron keeps its spin 

29. conduction band
valence band
pair
bright
coupled to photons
uncoupled to photons
dark
no valence-conduction
Coulomb processes

30. (repulsive) interband Coulomb processes push
Coulomb intraband
exists for all
Coulomb interband
exists for bright only
(repulsive) interband Coulomb processes push
bright exciton above dark exciton

31. Exciton BEC must be dark …
Dark excitons have the lowest energy
just because they are dark !
Exciton BEC must be dark …
No hope to see it (directly) with
photoluminescence experiments
Better to know where condensation takes place !

32. Pairs are created in bright states by photon absorption
However, dark / bright excitons are linked by carrier exchange
1/2 +2 -1
-3/2
3/2 -2 +1
-1/2
dark
bright

33. Dark condensate must have a linear polarization
minimum for
linear polarization

34. So,
in the dilute limit,
electrons and holes
form excitons.
At low T, we must have
a dark exciton
Bose-Einstein condensate

35. How can we see a dark condensate ?
parabolic trap
dark
lowest state
lowest trap level
we predict the appearance of a dark spot
at the center of the trap
when BEC takes place

37. Under a density increase
Part 2
Under a density increase

38. appears in the condensate
a bright component
appears in the condensate

39. In quantum wells
electrons
excitons
heavy holes
We first forget degeneracy
dark excitons
bright excitons
lowest state
dark condensate

40. When the density increases, we must include interactions
the total Hamiltonian then is
ground state
standard mean field treatment
minimum

41. minimum
threshold
for
lowest state
condensate has a bright component

42. Josephson oscillations
and
are conjugate variables
Hamilton equations
close to equilibrium
for

43. an electron-hole plasma
(B)
a phase separation
takes place between
an exciton gas
and
an electron-hole plasma

44. Excitons dissociate into an electron-hole plasma for
Mott dissociation
Electron-hole plasma
Exciton gas
Excitons dissociate into an electron-hole plasma for

45. dense electron-hole plasma
dilute exciton gas
positive to avoid collapse
dense electron-hole plasma
in 3D
the average pair energy has a minimum in the dense limit
in 2D

46. « electron-hole droplets » in Ge and Si
Phase separation

47. for
Phase separation along double tangente

48. Phase separation

49. dense electron-hole plasma
a BCS condensation
occurs in the
dense electron-hole plasma

50. Even if strongly screened by Fermi seas
of electron-hole free carriers
electrons and holes still attract each other
Formation of « electron-hole Cooper pairs »
neutral, so not superconducting
but still superfluid

51. Again, dark pairs have the lowest energy
exchange coupling must however bring a bright component
in the condensate since it is dense
Degeneracy between the two dark and the two bright pairs
must lead to a linear polarisation
as optimum state

52. dark condensate
« Gray » condensate
Phase separation
exciton BEC
exciton BCS

53. So,
under a density increase,
-a bright component appears in the condensate.
-a phase separation appears between exciton gas and e-h plasma
-a BCS condensation of excitonic Cooper pairs takes place in the e-h plasma

54. Optical traps for dark excitons
Part 3
Optical traps for dark excitons

55. Exciton-photon interaction

56. 1) Momentum conservation
electron k+Qp Qp
photon
hole -k
exciton Qp Qp
photon

57. Spin conservation
photon
3/2
-1/2
S=1
photon
S=1
-1/2
3/2
exciton S=1

58. Optical traps
with
standing waves

59. In a standing wave (+Q, -Q)K
exciton K
photon +Q +Q
exciton K
K + 2Q
photon +Q
- Q
an exciton K becomes a superposition of (K, K +2Q, K- 2Q)
so, it is trapped Potential depth: a fraction of meV

60. with linearly polarized photons
+3/2 or -3/2
photon +Q
- Q
exciton K
K + 2Q
-1/2
-1/2
with linearly polarized photons
dark and bright excitons are equally trapped
by making exchange with different parts of the light
Exciton condensation
should appear as the trap centers turning dark
when the temperature decreases

61. The electron-hole system is really fascinating !
Conclusion
The electron-hole system is really fascinating !
1) At low density, excitons are formed.
They have a BEC condensation into a dark state
with linear polarisation
2) Under a density increase,
- a bright component appears in the condensate
- a phase separation takes place with coexistence of
an exiton gas and an electron-hole plasma
- a BEC condensation of excitonic Cooper pairs takes
place in the dense plasma
3) Dark excitons can be optically trapped by standing waves
through carrier exchange with virtual exciton
coupled to photon

62. Monique Combescot, Odile Betbeder-Matibet, Roland Combescot
References
Bose-Einstein condensation of excitons: the key role of dark excitons
Monique Combescot, Odile Betbeder-Matibet, Roland Combescot
Phys. Rev. Lett. 99, (2007)
Optical traps for dark excitons
Monique Combescot, Michael Moore, Carlo Piermarocchi
Phys. Rev. Lett. 106, (2011)
« Gray » BCS condensate of excitons and internal Josephson oscillations
Roland Combescot, Monique Combescot
Arxiv