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