Hybrid Bose-Fermi systems
Alexey Kavokin
University of Southampton, UK
Bosons
Fermions
Integer spin


f BE k , T ,  
half-integer spin
1
 
E k 
 1
exp 
 kBT 



Bosonic stimulation
1
 
E k 
 1
exp 
 kBT 


Pauli exclusion principle
BEC
Superfluidity

f FD k , T ,  
BCS
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:
1
fB k ,T ,  
,
E k 
 1
exp 
 kBT 



How many bosons do we have?

2
  0,
 


N (T , µ)   f B k , T , 
k2
E k  
2m

k
Their concentration
n(T , µ)  lim R  
n0 (T , µ)  lim R  
What happens if
  0?
N (T ,  )
Rd



d
 n0 
f (k ,  )d k
d  B
2  0
1
dimensionality of the system
1
1
Rd
  
exp
 1
 k BT 
Critical concentration:



d
nc (T )  lim µ  0
f (k ,  )d k
d  B
2  0
1
All extra bosons go to the condensate:
n0 (T )  n(T )  nc (T )
nc (T )

depends on the mass, because


f B k ,T ,  
nc (T )
BEC
2
1

E k  
exp
 1
k
T
 B


k2
E k  
2m
and
m3
m2
m1<m2<m3
m1
T
Superfluidity
Superconductivity
Bose-Einstein
condensation
Condensation of
cold atoms
All this happens
at very low
temperatures ...
Exciton-polaritons: very light effective mass
very high critical temperature for BEC!

Excitons: composite bosons
EXCITON: an artificial ATOM
8
10 m
Hole
Atom
Electron
EXCITON + PHOTON = EXCITON-POLARITON
Exciton polaritons are also composite bosons
1010 m
POLARITON LASER
QW's
what is it ?
/4
Field intensity
Refractive index
2,8
2,1
3 /2
1,4
0,0
0,3
0,6
0,9
Micrometers
It is a coherent light source based on the
Bose-condensat of exciton-polaritons in
a microcavity
Ang
l
e (d
e gr
ee)
30
25
20
15
10
5
0
-5
-10
-15
3.58
3.59
3.60
3.61
3.62
Energy (eV)
3.63
3.64
3.65
1,2
1,5
Concept of polariton lasing:
Photon mode
dispersion

 2 
2
n 
 k
c
 L 
Extremely light
effective mass
 10 5  10 4  m0
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
2
SUPERFLUIDITY
In 1937 Kapitsa, Allen and
Miserer discovered the
superfluidity of He4
Lev Landau has proposed a
phenomenological model of
superfluidity
E
Eb k   E 2 k   2E k 
Nikolay Bogolyubov has created
a theory of superfluidity of
interacting bosons
Linear dispersion “sound”
roton
k
Bogolyubov spectrum and superfluidity
Gross-Pitaevskii equation for a conensate of interacting bosons

i
 T  i     V    * 
t
  nV
substitution
  n  Ae 

i kr t
  C*ei krt 

yields
A e

i kr t

2Vn Ae


 C * e
i kr t


 i kr t
 C *e


 E  k  Ae

 

 i kr t
 Vn A*e




i kr t
 i kr t
 C *e
 Ce

 i kr t

i kr t
therefore
A   AE  k     A  C 
C *   C * E  k     A*  C * 


 
  Ae

i kr t
  o  A, C 

 C *e

 i kr t


Resolving the linear system
A E  k        C  0
 A  C  E  k        0

  E  k   

det 
0

  E  k    

We obtain
2  E2  k   2 E  k 
2
Eb  k 
Bogolyubov spectrum responsible for superfluidity!
Eb k   E 2 k   2E k 
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:
  1
Aluminium 428 K
Platinum 240 K
Cadmium 209 K
Silicon 645 K
Chromium 630 K
Silver 225 K
in conventional superconductors,
Copper 343.5 K
Tantalum 240 K
Gold 165 K
Tin (white) 200 K
which is why the critical temperature is
very low!
Iron 470 K
Titanium 420 K
Lead 105 K
Tungsten 400 K
Manganese 410 K
Zinc 327 K
Nickel 450 K
Carbon 2230 K
Ice 192 K
!
•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.
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 »)
GaN microcavities: a polariton condensate at room temperature!
300 K
Below threshold
J.J. Baumberg, A. Kavokin et al.,
PRL 101, 136409 (2008)
Above threshold
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
Electron-electron interaction:
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
Our potential
Comparison with BCS
BCS potential
We have:
  1
Energy
1) Much stronger attraction;
2) Similar Debye temperature
3) Peculiar shape of the potential
W
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??
holes
electrons
2DEG
l
L
L=55 nm
nex=109cm-1
L=25 nm
nex=5 1010 cm-1
L=12 nm
nex=1011cm-1
Suppression of the Bose-Einstein condensation and superfluidity
real space condensation
classical fluid
BEC
superfluid
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