Hybrid Bose-Fermi systems Alexey Kavokin University of Southampton, UK.
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Transcript of Hybrid Bose-Fermi systems Alexey Kavokin University of Southampton, UK.
Hybrid Bose-Fermi systems
Alexey KavokinUniversity of Southampton, UK
Bosons Fermions
Integer spin half-integer spin
BCSBEC
Pauli exclusion principleBosonic stimulation
And if they are coupled?
Superfluidity Superconductivity
1
, ,
exp 1
BE
B
f k TE k
k T
1, ,
exp 1
FD
B
f k TE k
k T
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
1
, , ,
exp 1
B
B
f k TE k
k T
The distribution function:
How many bosons do we have? k
B TkfµTN
,,),(
kdkfn
R
TNµTn d
BddR
0
0 ),(2
1),(lim),(
Their concentration
dimensionality of the system
1exp
11lim),(0
TkR
µTn
B
dR
What happens if
0,
0?
2 2
2
kE k
m
kdkfTn d
Bdµc
0
0 ),(2
1lim)(
Critical concentration:
1exp
1,,
Tk
kETkf
B
B
)()()(0 TnTnTn c
All extra bosons go to the condensate:
( )cn T depends on the mass, because
2 2
2
kE k
m
and
T
( )cn T
BEC
m1
m2
m3
m1<m2<m3
Bose-Einstein condensation
Superconductivity Superfluidity
Condensation of cold atoms
All this happens at very low temperatures ...
Exciton-polaritons: very light effective mass very high critical temperature for BEC!
EXCITON: an artificial ATOM
Hole
Electron
m810
Atom
m1010
Excitons: composite bosons
EXCITON + PHOTON = EXCITON-POLARITON
Exciton polaritons are also composite bosons
POLARITON LASER
what is it ?
0,0 0,3 0,6 0,9 1,2 1,5
1,4
2,1
2,8
Ref
ract
ive
inde
xMicrometers
3 /2
Field intensity
QW's
/4
It is a coherent light source based on the Bose-condensat of exciton-polaritons in a microcavity
3.58 3.59 3.60 3.61 3.62 3.63 3.64 3.65
-15-10
-5051015202530
3.58 3.59 3.60 3.61 3.62 3.63 3.64 3.65
-15-10
-5051015202530
Energy (eV)
An
gle
(d
egre
e)
Concept of polariton lasing:
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
Extremely light effective mass 5 4
010 10 m
Photon mode dispersion
22
2k
Ln
c
In 1937 Kapitsa, Allen and Miserer discovered the superfluidity of He4
Lev Landau has proposed a phenomenological model of superfluidity
Nikolay Bogolyubov has created a theory of superfluidity of interacting bosons
SUPERFLUIDITY
k
E
kEkEkEb 22
Linear dispersion “sound”roton
Bogolyubov spectrum and superfluidity
*ViTt
i
Gross-Pitaevskii equation for a conensate of interacting bosons
trkitrki eCAen
*
substitution
* * *
* *2 ,
i kr t i kr t i kr t i kr t i kr t i kr t
i kr t i kr t i kr t i kr t
A e C e E k Ae C e Ae C e
Vn Ae C e Vn A e Ce o A C
yields
A AE k A C
* * * *C C E k A C
therefore
nV
Resolving the linear system
det 0E k
E k
We obtain
2 2 2 2E k E k
Bogolyubov spectrum responsible for superfluidity!
kEkEkEb 22
0A E k C
0A C E k
bE k
k
Mechanism: exciton condensate instead of phonons
Result: light mediated BCS superconductivity: possibly very high Tc
Starting point: Bose condensate of exciton polaritons put in contact to the Fermi see of electrons
Structure: metal-semiconductor sandwich or more complex heterostructures (microcavities)
Electron –electron attraction: increases with increase of optical pumping!
Motivation: recent discovery of BEC of exciton polaritons
(Exciton mechanism of superconductivity revisited)
LIGHT-INDUCED SUPERCONDUCTIVITY
Cooper pairing in metals
retarded interaction
BCS model:
Bardeen-Cooper-Schrieffer (BCS): Critical temperature:
Density of electronic states at the Fermi level
Coupling constant
Debye temperature
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
1 in conventional superconductors,
which is why the critical temperature is very low!
BCS: “weak coupling” regime
!
•An exciton mechanism may be realised in 2D metal-dielectric sandwiches (higher ).
•Non-equilibrium superconductivity has a great future
BUT IT NEVER WORKED ! WHY ?
1) Exciton-electron interaction still weak;
2) Excitons are too fast (reduced retardation effect), consequently:
3) Coulomb repulsion becomes important.
In semiconductor microcavities excitons may be strongly coupled to photon modes
Exciton-polaritons
exciton
photon
An exciton is an electron-hole pair bound by Coulomb attraction
193 articles in Physical Review Letters with « microcavity » in the title or abstract (compare to 368 with « graphene »)
Bose-Einstein condensation of exciton polaritons (2006-2010)
resonance
GaN microcavities: a polariton condensate at room temperature!
Below threshold Above threshold
J.J. Baumberg, A. Kavokin et al., PRL 101, 136409 (2008)
300 K
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??!
a heavily n-doped layer embedded between two neutral QWs in a microcavity
We consider the following model structure:
Electrons + exciton-polariton BEC: interaction Hamiltonian
Electron-polariton interactions
Polariton-polariton interactions
Coulomb repulsion
Interactions:
Electron-exciton interaction:
Electron-electron 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
Boglyubov transformation:
Concentration of exciton-polaritons
Electron – electron interaction potential:
exciton mediated interaction
Coulomb repulsion
Results for a model GaN microcavity
Comparison with BCS
Energy1
W
BCS potential
Our potential
We have:
1) Much stronger attraction;
2) Similar Debye temperature
3) Peculiar shape of the potential
Solving the gap equation by iterations...
we obtain the superconducting gap which vanishes at the crictical temperature
Now we know what may happen to fermions,
But what will happen to bosons??
2D
EG
electrons
holes
l
L
L=12 nm
L=25 nm
L=55 nm nex=109cm-1
nex=5 1010 cm-1
nex=1011cm-1
BEC
Suppression of the Bose-Einstein condensation and superfluidity
real space condensation
superfluid
classical fluid
Conclusions:
In Bose-Fermi systems with direct repulsive interaction of bosons and fermions, due to Froelich-like indirect interactions:
1. Fermions attract fermions which results in Cooper pairing
2. Bosons attract bosons which results in formation of the roton minimum and suppression of BEC