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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
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
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
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
19
Exciton creation operator
!
",Q = ke,kh kh ,ke ",Q#
!
B"Q+0
!
ake
+bkh
+0
!
B"Q+
= ake+bkh
+kh,ke ",Q#
20
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
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
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
$(%N)
close to equilibrium
!
"J#10
10N /N
th
41
Mott dissociation
!
" = N(aX
L)D
Excitons dissociate into an electron-hole plasma for
!
" #1
Exciton gas Electron-hole plasma
!
aX
42
!
" = N(aX
L)D
dilute exciton gas
dense electron-hole plasma!
EN
= NRX"1+ (...)# + ...[ ] = N$
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
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)