1

BEC-BCS cross-over in theexciton gas

Monique Combescot

Institut des NanoSciences de Paris, CNRSUniversité Pierre et Marie Curie

Evora oct. 2012

2

1)Dilute limit: BEC condensate of linearly polarized dark excitons

2)Under a density increase: - mixture of dark and bright condensates - phase separation between BEC and BCS condensates

3

dilute limit

Part 1

4

In the dilute limit, electrons and holes form excitons. At low T, they undergo a Bose-Einstein condensation in a linearly polarized dark state

5

Semiconductor

photon

photon

valenceelectron

conduction electron

!

Qp

!

kv

!

kc = kv +Qp

!

kv

!

kc

valence band

conduction band

6

photon

photon

hole

electron

!

Qp

!

kh

= "kv

!

ke = kv +Qp!

kh

!

ke

hole

electron

7

hole: full valence band minus one electron

formidable reduction of the many-body problem !

8

Naive view of an exciton

exciton similar to Hydrogen atom

light effective mass!

+ e

!

"eelectron

hole

dielectric constant

!

me

*" 0.1m

e

!

"sc#10

!

RX

=me

4

2h2"sc

2#13.6eV

0.1

102

$

% &

'

( )

!

RX"10meV

!

aX

=h2"sc

me2# 0.53A

10

0.1

$

% &

'

( )

!

aX" 50A

9

The true story is

far more complex

10

Coulomb interaction in a periodic lattice

Bloch states

!

nk = anks

+v

!

r nk =eik.r

L3 2unk(r)

!

unk(r) = u

nk(r + a)

11

!

VCoul

= Ve"e

"V e"e

!

=1

2..............a " n

1k1

+q

+a " n

2k2#q

+an

2k2

an1k1

$

!

vq ( " n 1,n1)v#q ( " n

2,n

2)

!

vq (n,n) =4"e2

L3q2

!

vq ( " n # n) = O(q0)

!

n1k1

!

n2k2

!

" n 2k2# q

!

" n 1k1+ q

repulsive scattering between conduction and valence electrons

12

Intraband Coulomb processes

!

4"e2

L3q2

!

c

!

c

!

c

!

c

!

c

!

c

!

v

!

v

!

v

!

v

!

v

!

v

dressed by a dielectric constant

which results from many-body effects induced by « boiling valence band »

13

!

c

!

c

!

c

!

v

!

c

!

c

!

c

!

v

!

2

!

c

!

c

!

c

!

v

!

c

!

c

!

c

!

v

!

1

!

c

!

c!

c

!

c

!

c

!

v

!

1

!

1

!

2

!

2

14

!

c

!

c!

c

!

c

!

c

!

v

!

c

!

c

!

c

!

c

!

c

!

c

!

c

!

c!

c

!

c

!

v

!

v

!

4"e2

L3q2

!

1 q

!

1 q

!

1 q

!

1 q

!

q0

!

...

q2

!

...

q2

!

vq (n,n) =4"e2

L3q2

!

vq ( " n # n) = O(q0)

15

!

n1

!

4"e2

#scL3q2

!

n2

Direct scatterings between valence and/or conduction electrons:

repulsive as between free electrons reduced by the semiconductor dielectric constant

!

n1

!

n2

16

repulsion between conduction and valence electrons

attraction between electrons and holes

!

ack1

+q

+avk

2"q

+avk

2

ack1

!

"avk2

avk2"q

+

We turn from conduction/valence electrons to electrons/holes

!

ack

+= a

k

+

!

avk

= b"k+

!

Vvc ="4#e2

$scL3q2

q%0

&k1k2

& ak1

+q

+b"k

2

+b"k

2+qak

1

!

"k2

+ q = # k 2

17

!

k2!

k1

!

k2" q

!

k1

+ q

!

e

!

e

!

h

!

h

!

"4#e2

$scL3q2

Attraction between one electron and one hole reduced by dielectric constant

18

One Wannier exciton

!

e

!

e

!

h

!

h

!

e

!

e

!

h

!

h

!

e

!

e

!

h

!

h

!

ke + kh =Q conserved through Coulomb scatterings

Wannier exciton

!

",Q = ke,kh ........#

energy

wave function !

"# +Q2

2(me + mh )

!

re,rh ",Q =eiQ.R

L3 2#" (r)

!

R =m

ere

+ mhrh

me

+ mh

!

r = re" r

h

!

kh,ke ",Q

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Exciton creation operator

!

",Q = ke,kh kh ,ke ",Q#

!

B"Q+0

!

ake

+bkh

+0

!

B"Q+

= ake+bkh

+kh,ke ",Q#

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Excitons are boson-like particles

!

i = ("i,Q

i)

!

B j

+,Bi

+[ ]"

= B j

+Bi

+" Bi

+B j

+ = 0 as bosons

!

Bm,B

i

+[ ]"

= #mi"D

mi

!

Dmi,B j

+[ ]"

= { .....................n

# }Bn

+

!

+

!

i

!

j

!

n

!

m

!

i

!

j

!

n

!

m

but not exactly

21

Being boson-like particles, excitons, as cold atoms, must undergo Bose-Einstein condensation

Yet, the precise nature of composite boson condensate is not known !

close to

but surely not exactly!

(B0

+)N0 + ...

!

(B+)N0

22

Exciton BEC searched for decades …. but never evidenced !

Reason: the condensate must be dark … and search has been done through photoluminescence experiments

23

conduction band

valence band!

l = 0

!

l =1

!

lh

z= ±1,0

!

s =1/2

!

jez

= ±1

2

!

s =1/2

!

sh

z= ±

1

2

!

jhz

= ±3

2,±1

2

pair

!

Jehz

= jez

+ jhz

= ±2,±1,0

24

Although small, interband Coulomb interaction also exists

exists for bright pairs J= (+1, 0, -1) only

!

c1

!

c2

!

v1

!

v2

!

c1

!

v1

!

v2

!

c2

!

v2

!

c2

!

c1

!

v1

!

e1

!

h1

!

e2

!

h2

emission and reabsorption of a virtual photon

25

!

Jeh

z= ±1,0

!

Jeh

z= ±2

bright

dark

coupled to photons

uncoupled to photons

no valence-conduction Coulomb processes

!

" ±,#

26

Coulomb intraband exists for all

Coulomb interband exists for bright only

(repulsive) interband Coulomb processes push bright exciton above dark exciton

!

h

!

e

27

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 !

28

Pairs are created in bright states by photon absorption

-1

+1

+2

-2

-3/2

1/2

3/2

-1/2

brightdark

However, dark / bright excitons are linked by carrier exchange

29

Dark condensate must have a linear polarization

!

B+

= a2B2

++ a"2B"2

+

!

HN

=0 B

NHB

+N0

0 BNB

+N0

!

a2

"(+2) + a#2

"(#2)

!

a2(+2) + a"2("2)!

a2(+2) + a"2("2)

!

a"2("2)!

a2(+2)

!

a2(+2)

!

a2

"(+2)

!

a"2#("2)

!

a"2("2)

!

+

!

a2

2

= a"2

2

=1 2

linear polarization

minimum for

!

a2

"(+2) + a#2

"(#2)

!

a2

"(+2)

!

a"2#("2)

!

HN

30

So, in the dilute limit, electrons and holes form excitons. At low T, they undergo a Bose-Einstein condensation in a linearly polarized dark state

31

How can we see a dark condensate ?

parabolic trap

lowest state dark

lowest trap level

we predict the appearance of a dark spot at the center of the trap when BEC takes place

32

33

Optical trap: standing wave (+Q, -Q)

K exciton becomes superposition of (K, K +2Q, K- 2Q)

so, it is trapped. Potential depth: a fraction of meV

photon +Q+Q

exciton KK

photon +Q- Q

exciton KK + 2Q

34

Under a density increase

Part 2

35

a bright componentappears in the condensate

(A)

36

dark excitons

bright excitons

!

Dk

+

!

Bk

+

!

Hkin

=k2

2M" D

k

+Dk

+ (#0

+k2

2M" )B

k

+Bk

lowest state

dark condensate

!

(D0

+)N0

37

When the density increases, we must include interactions

!

2

the Hamiltonian then reads

!

H =k2

2M" D

k

+Dk

+ (#0

+k2

2M" )B

k

+Bk

+

!

D+D " N

d

!

1

!

"1!

"2

!

k1!

k2

!

k3

!

k4

!

g " Bk4

+Bk

3

+Dk

2

Dk1

+ h.c.

mean field treatment

!

B+B " N

b

!

D " Ndei# d

!

B " Nbei# b

!

H " #0Nb + 2g NbNd cos2($b %$d )

!

"gminimum

38

!

H " #0Nb $ 2 gNbNd

minimum

!

N " Nb

!

"H

"Nb

= 0

!

Nb

(0)=2 gN "#

0

4 g

!

Nth ="0

2 g

!

N > Nth

for

!

(D0

+)N"N

b

( 0)

(B0

+)Nb

( 0)

0the lowest state

condensate has a bright component

!

"0#10µeV

!

m " 0.1!

"#0

$ exch(00

00)

!

"#0

RX

(L

aX

)D

!

Nth

L2"10

9cm

#2

threshold

39

and

!

" ="b#"

d

!

h2"J

2= 32gbd

2Nb

(0)Nd

(0)= 2#

0

2(N2

Nth

2$1)

!

!

"0#10µeV

!

"N # "N (0)+ ....cos$

Jt

Josephson oscillations

!

"N = (Nb# N

d) /2 are conjugate variables

Hamilton equations

!

d("N)

dt=#H

#$

!

d"

dt= #

$H

$(%N)

close to equilibrium

!

"J#10

10N /N

th

40

a phase separation takes place between exciton gas and electron-hole plasma

(B)

41

Mott dissociation

!

" = N(aX

L)D

Excitons dissociate into an electron-hole plasma for

!

" #1

Exciton gas Electron-hole plasma

!

aX

42

!

" = N(aX

L)D

dilute exciton gas

dense electron-hole plasma!

EN

= NRX"1+ (...)# + ...[ ] = N$

X(#)

positive to avoid collapse

!

EN

= N (...)kF

2

2m" (...)e2k

F+ ...

#

$ %

&

' ( = N)eh (*)

!

N " (kFL)

D

!

"eh(#) $%n2 3 &'n1 3

!

"#n $%n1 2

in 3D

in 2D

the average pair energy has aminimum in the dense limit

43

!

" = N(aX

L)D!

"(#)

!

"RX

!

"0

« electron-hole droplets » in Ge and Si

!

"0(L

aX

)D

!

N

!

T

!

eh

!

eh

phase separation

!

0

!

x

44

!

"(#)

!

" = N(aX

L)D

!

"RX

!

"2

!

"1

!

EN

= N1"(#

1) + N

2"(#

2)

!

N = N1+ N

2

!

dN1

= "dN2

!

0 =dE

N

dN1

= "(#1) +#

1$ " (#

1) %"(#

2) %#

2$ " (#

2)

!

" # ($1) = " # ($

2)for

!

"(#2) $"(#

1)

#2$#

1

= % " (#1) = % " (#

2)

Phase separation along double tangente

45

!

"2(L

aX

)D

!

N

!

T

!

x!

eh

!

eh

!

"1(L

aX

)D

phase separation

!

0

46

a BCS condensation occurs in the dense electron-hole plasma

(C)

47

Even if strongly screened

electrons and holes still attract each other

!

......ak+b"k

+# = BCooper+

Formation of « electron-hole Cooper pairs »

neutral, so not superconducting but still superfluid

48

Again, dark pairs have the lowest energy

Exchange coupling must however bring a bright component to the condensate since it is dense

Moreover,degeneracy between the two dark and the two bright pairs must lead to a linear polarisation as optimum state

!

(B2

++ B"2

+)N"N

1

( 0)

(B1

++ B"1

+)N1

( 0)

0

49

!

"2(L

aX

)D

!

N

!

T

!

eh

!

"1(L

aX

)D

exciton BEC exciton BCS

phase separation

dark condensate « gray » condensate

!

x

50

So, under a density increase, -a bright component appears inthe condensate. -a phase separation occursbetween exciton gas and e-h plasma -a BCS condensation of excitonicCooper pairs takes place in the e-hplasma

51

ConclusionElectron-hole systems are quite rich !

1) At low density, excitons are formed. They suffer BEC condensation into a dark state with linear polarisation2) Under a density increase, - a bright component appears in the condensate - a phase separation occurs between exiton gas and electron-hole plasma - a BCS condensation of excitonic Cooper pairs takes place in the dense plasma

52

References

Bose-Einstein condensation of excitons: the key role of dark excitons

Monique Combescot, Odile Betbeder-Matibet, Roland Combescot

Phys. Rev. Lett. 99, 176403 (2007)

Optical traps for dark excitons

Monique Combescot, Michael Moore, Carlo Piermarocchi

Phys. Rev. Lett. 106, 206404 (2011)

« Gray » BCS condensate of excitons and internal Josephson oscillations

Roland Combescot, Monique Combescot

Phys. Rev. Lett. 109, 26401 (2012)