Resonant Pairing of Excitons in Semiconductor Heterostructures

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by  Sergey V. Andreev

Resonant Pairing of Excitons in Semiconductor Heterostructures
S. V. Andreev1,2,∗
1CNRS, LPTMS, Universit´e Paris Sud, UMR8626, 91405 Orsay, France
2ITMO University, St. Petersburg 197101, Russia
(Dated: 1er f´evrier 2016)
Wesuggestindirectexcitonsin2Dsemiconductorheterostructuresasaplatformforrealizationof
a bosonic analog of the Bardeen-Cooper-Schrieffer superconductor. The quantum phase transition
to a biexcitonic gapped state can be controlled in situ by tuning the electric field applied to the
structure in the growth direction. The proposed playground should allow one to go to strongly
6 correlated and high-temperature regimes, unattainable with Feshbach resonant atomic gases.
1
0
PACSnumbers:71.35.Lk,34.50.Cx,67.10.Ba,74.10.+v
2

n
a Thephenomenonofresonantpairingliesattheheartof nant interaction may result in formation of a fragmented
J superconductivityinmetals.HereCooperpairsoffermio- biexcitonic supersolid.Thisnewstronglycorrelatedstate

9 nic particles – electrons, can Bose-Einstein condense to of bosonic matter would be robust to fluctuations of all
2 carry electric charge without dissipation. In-depth study kindwhichusuallyspoilsuperconductivityinlowdimen-

of this scenario, commonly known as Bardeen-Cooper- sions.Intransitionmetaldichalcogenidesexcitonicsuper-
]
s Schrieffer (BCS) theory, has been performed by using solidity could be used to realize dissipationless transport
a
the technique of Feshbach resonances (FR’s) in ultra- of electrons and holes at record high temperatures.
g
- coldatomicgases.InFermigasesthistechniquehasallo- In order to better present our idea we first recall the
t wed for observation of a crossover from a BCS-like state basicphenomenologyoftheFRinatomicsystems[Fig.1
n
a made of spatially overlapping pairs of atoms to a Bose- (a)].Atlowenergiesinteractionoftwoatomsintheopen
u Einstein condensate (BEC) of tightly bound diatomic channel (OC) can be modeled as scattering via an ersatz
q
molecules [1–5]. This so-called BCS-BEC crossover has two-bodypotentialschematicallyshowninFig.1(b).At
.
t become a paradigm of the many-body physics, sharing short distances this potential has a minimum separated
a
m importantanalogieswithhigh-temperaturesuperconduc- from the continuum by a large barrier. A (quasi-)bound
tivity [6] and neutron stars [7–9]. state inside the well corresponds to the closed molecu-
-
d A natural idea expounded in a series of papers [10] lar channel (CC), the outer continuum of states to the
n has been to apply the same FR technique to degenerate OC and the barrier models coupling of the two channels
o
Bose gases. It has been shown that in the case of bo- due to hyperfine interaction [1]. The energy ε of the dis-
c
[ sons the smooth crossover is replaced by a thermodyna- crete level is proportional to the magnetic-field detuning
micallysharpphasetransitionfromacoherentmixtureof oftheOCwithrespecttotheCC[15].Forastrictlytwo-
1
atoms and molecules to a pure molecular superfluid [11]. dimensional (2D) collision (relevant for our system) the
v
The latter is distinguished by the absence of atomic off- scattering length would be given by
8
6 diagonallong-rangeorderandgappedatomicexcitations.
a=r eα, (1)
9 Though being of great fundamental interest on its own ∗
7 right,untilnowthisresearchhasnotmetitsapplication- whereα=ε/β,theparameterβ characterizesthebarrier
0
orientedcounterpartinthephysicsofsolidstate.Moreo- transmission (for ε (cid:29) β it gives the lifetime of a quasi-
.
1 ver, experimental attempts to realize a unitary Bose gas bound state inside the well according to τ = (cid:126)/πβ) and
0
of atoms did not succeed. This is due to coalescence of r is the microscopic rangeof thepotential. Bychanging
6 ∗
three and more atoms (few-body recombination) [12, 13] ε from negative to positive values one could realize the
1
: and mechanical instability when approaching the reso- scattering regimes where a (cid:28) r∗ and a (cid:29) r∗, respecti-
v
nance on the attractive side [14]. vely.
i
X In this Letter we propose a new setting for study and Our proposal of an excitonic FR is based on the follo-
r manipulation of resonantly paired bosonic superfluids. wingobservation.Considerexcitonsintheirgroundstate
a
Bosonic quasiparticles we consider are indirect excitons inawidezinc-blendesemiconductorquantumwell(QW).
in biased semiconductor heterostructures. Our excitonic These are bosons composed of an electron with the spin
analog of BCS is expected to be stable across the whole ±1/2 and a heavy-hole with the spin ±3/2. Depending
range of scattering lengths. The scattering length of the onthemutualorientationofthefermionicspins,thespin
excitons can be conveniently tuned by the bias electric of an exciton can take four possible values : ±1 (the so-
field. A distinct feature of an indirect exciton is a large called ”bright” excitons) and ±2 (”dark” excitons) [16].
dipole moment oriented perpendicularly to the structure Interactionoftwobright(dark)excitonshavingthesame
plane. In the system under consideration an interplay spin,aswellasinteractionofabrightexcitonwithadark
between the long-range dipolar repulsion and the reso- one, is repulsive. At short distances such excitons avoid
2
and the generic potential introduced to model the FR in
atomic systems [Fig. 1 (b)]. In particular, the result (1)
withthesubstitution(2)appliesdirectlytogivethelow-
energyexcitonicscatteringlength.Thelatterthuscanbe
controlled by tuning the effective distance in the vicinity
of d . The narrow interval
c
|d−d |(cid:28)∆d∝β, (4)
c
corresponds to the regime of vanishing interaction. Here
not only the proportionality law (3) does not hold, but
the very meaning of the parameter ε as the energy of a
(quasi-)boundstateisnolongeradequate.Ford(cid:54)d this
c
energy is given by ε¯ = −(cid:126)2/ma2, where the scattering
Figure 1. Basic idea of the excitonic Feshbach reso-
length a diverges according to the exponential law (1)
nance (FR). In the atomic FR (a), scattering of two atoms
with the power
along the dashed potential curve, called open channel (OC),
can be modified by coupling to the closed molecular channel
α=α ∝∆d/(d −d). (5)
(CC)(solidcurve).TheeffectofCConOCcanbetakeninto 0 c
accountbyreplacingtheactualOCpotentialbytheonesche- Full evolution of the shape of the exciton interaction
matically shown in (b). Remarkably, the potential of exactly
potential as a function of d has been calculated numeri-
thesametypedescribesinteractionoftwoindirectexcitonsin
cally for GaAs coupled quantum wells (CQW’s) [17, 18].
coupledsemiconductorlayers(c).Theenergyεofthe(quasi-
ThestructureconsistsoftwoGaAslayersseparatedbya
)biexciton is proportional to the difference d−d , where d is
c
the distance between the layers and d is the critical separa- thin AlGaAs barrier [Fig. 1 (c)]. From these studies one
c
tion at which the true bound state disappears. can deduce d ≈ 7 nm. A straightforward dimensional
c
analysis [18] indicates that d should scale as the effec-
c
tive electron Bohr radius a =(cid:126)2κ/m e2 when changing
e e
each other due to the Pauli exclusion of the constituent
the compound. Clearly, these arguments can be adopted
electronsand(or)holes.Ontheotherhand,theexhange
to a wide single QW as well. The advantage of the single
of fermions in a pair of the bright (dark) excitons with
QW with respect to the CQW configuration is that it
the opposite spins can result in binding of these excitons
offers a possibility to explore excitonic interaction over a
into molecules (biexcitons) [19].
wider range of d, including the limit d→0.
Suppose now that we apply an electric field in the
Crucially, the Fermi statistics of electrons and holes
direction perpendicular to the QW plane. The excitons
prohibitsboundstatesofmorethantwoexcitons.Invir-
would become polarized in the same direction. The pair-
tueofthePauliprinciple,interactionofthethirdexciton
wiseinteractioninallchannelswouldacquirepronounced
with at least one exciton in the pair is always repulsive
repulsivecharacteratthedistancesoftheorderofthedi-
[20]. The absence of trimers and larger excitonic com-
polar length
plexes in quantum wells has been confirmed experimen-
tally [21]. Hence, one may think of using the proposed
r =me2d2/κ(cid:126)2, (2)
∗ playground for realization of a stable bosonic analog of
where κ is the dielectric constant of the semiconductor, BCS. A distinct property of resonantly paired excitons
m is the exciton mass and d is the effective distance bet- would be long-range dipolar repulsion. In what follows
weenanelectronandaholelayerinabiasedQW.Inthe we shall discuss how it could manifest in collective beha-
channel where the excitons have opposite spins, the di- vior of the system.
polar repulsion would introduce a potential barrier bet- We start with the dilute regime, where one can ap-
ween the outer continuum of states and the biexciton. proach the problem perturbatively. For simplicity we
Withincreaseofdthebiexcitonbindingenergy|ε|would shall consider a binary mixture of bright (or dark) ex-
decrease, until, at some ctitical value d (to be specified citonsonlyandassumeequalpopulationofup(”↑”)and
c
below)thetrueboundstatewoulddisappearandbecome down(”↓”)spinbranches.Spin-polarizedconfigurations,
replaced by a resonance (the state with ε>0). Close to relevantforpossibleexperimentsinmagneticfield,willbe
d studied elsewhere. The Hamiltonian of the system reads
c
ε∝d−dc, (3) Hˆ =(cid:90) (cid:88) Ψˆ†σ(ρ)(cid:18)−2(cid:126)m2 ∆+Vext(ρ)(cid:19)Ψˆσ(ρ)dρ+
σ=↑,↓
which holds both for d > d and d < d , providing that
ε(cid:29)β. c c 1(cid:90) (cid:88)Ψˆ†(ρ)Ψˆ† (ρ(cid:48))V (ρ−ρ(cid:48))Ψˆ (ρ)Ψˆ (ρ(cid:48))dρ(cid:48)dρ
2 σ σ(cid:48) σσ(cid:48) σ σ(cid:48)
Onecansee,thatthereisaone-to-onecorrespondence σ,σ(cid:48)
between the interaction of excitons with opposite spins (6)
3
where integration is taken over the structure area, ρ =
(x,y). In the ultracold limit the microscopic two-body
interaction V (ρ − ρ(cid:48)) can be substituted by effec-
σσ(cid:48)
tive k-dependent pseudo-potentials, V2D(k,k(cid:48))=g −
σσ(cid:48) σσ(cid:48)
2π(cid:126)2/m|k−k(cid:48)|r for a pure 2D (V (ρ)≡0) [23] and
∗ ext
g
V1D(k ,k(cid:48))= √σσ(cid:48)
σσ(cid:48) x x
2πa
y
(7)
(cid:126)2
+ (|k −k(cid:48)|r )2ln(|k −k(cid:48)|r )
mr x x ∗ x x ∗
∗
for a quasi-1D geometry [24]. The latter is realized by
introducing the external potential V (y) = mω2y2/2
ext y
tightly confining the system in one direction, and mo-
delsawave-guideofthehalf-widtha =(cid:112)(cid:126)/mω inthe
y y
structure plane. The momentum-dependent terms in the
Figure 2. Fragmented biexcitonic supersolid. Fine tu-
aboveformulaedescribethelong-rangedipolarrepulsion
ningofthecontactpartoftheexcitoninteractionbymeansof
(commonfeatureforallchannels).Thecontactpartswill
theelectricfieldyieldsarotoninstability(upperspectrumon
be taken as positive constants for interaction of excitons the right) of a uniform density distribution (light gray color
having the same spin, g = g ≡ g > 0, and of the on the left). The instability drives the condensate to a su-
↑↑ ↓↓ bg
resonant type persolid state, characterized by periodical modulation of the
density(gray,ontheleft).Stabilityofthisstateisguaranteed
(cid:126)2 2π bythree-bodyrepulsiveinteractionofexcitons(singlearrows)
g =g + (8)
↑↓ bg mln(1/ka)+(cid:126)2k2/mβ withtheirbiexcitonicmolecules(pairedarrows).Uponaden-
sity increase the supersolid fragments into a periodical chain
for the channel where a biexciton can be formed. Here a ofmolecularcondensates(darkgray),characterizedbystrong
is the 2D scattering length given by Eq. (1) and in the repulsionandagappedelementaryexcitationspectrum(bot-
condensateoneshouldlet(cid:126)2k2/m=2µfortheenergyof tom, on the right). We take the parameters typical for the
experiments on GaAs CQW’s [22] (see methods).
colliding excitons, with µ being the chemical potential.
The formula (8) is only meaningful if ε (cid:29) β. In the in-
terval (4), where the bound state disappears, one should
[25]. The interactions manifest in the second branch. At
substractg from(8)andusetheresult(5)forthepower
bg small k it has the typical linear form with the slope
√
of the exponent in (1). c = (cid:112)ng/m(cid:126)2 (n ≡ n / 2πa ). Away from the reso-
Having in mind possible application of our theory to 1 y
nance where g ≈ g the linear dispersion law monoto-
investigation of exciton superconductivity in 2D wires, bg
nously turns to a quadratic one at higher momenta. By
we shall focus on the quasi-1D geometry. Main conclu-
rampingεdowntoµ,however,onecanmakegtobeano-
sions drawn here hold for the 2D case as well. Let us
malously small [see Eq.(9) below], so that ε (k) develops
assume ε (cid:29) µ > 0, so that the situation schematically r
a roton-maxon structure.
illustrated in the bottom of Fig. 1 (b) is realized, with
Rotonization of the spectrum implies a dynamical in-
theenergyoftheresonancegreatlyexceedingtheexciton
stability [23, 26, 27]. In the frame of the model (6)
energy. In this case the ground state (GS) of (6) corres-
the system would collapse [28]. Such pathological beha-
ponds to the true kinetic equilibrium of the system with
vior can be regularized by introducing three-body repul-
respect to the binary collisions. The GS wave-function
sive forces [29]. In our case these can enter the game
is the excitonic order parameter with the components
√ on the attractive side of the resonance due to forma-
Ψ↑,↓(x,y) = (n1/2 πay)1/2e−y2/2a2y, having Gaussian tion of weakly bound excitonic pairs. The pair effec-
profiles across the wave-guide and uniform 1D densities
tively behaves as a single body in collisions with the
n (x) ≡ n in the longitudinal direction. The chemical
1 1 √ third particle, which can give rise to the three-body
potential reads µ = n1g/ 2πay +(cid:126)ωy/2. The effective term g/2(cid:82)(Ψˆ†Ψˆ†Ψˆ†Ψˆ Ψˆ Ψˆ + Ψˆ†Ψˆ†Ψˆ†Ψˆ Ψˆ Ψˆ )dρ at
coupling constant g ≡ (g + g )/2 is governed by ε ↑ ↑ ↓ ↑ ↑ ↓ ↑ ↓ ↓ ↑ ↓ ↓
↑↑ ↑↓ the two-body approximation level [30]. The three-body
according to Eq.(8). The standard Bogoliubov approach
repulsion prevents the collapse. Instead, at the point
yields the elementary excitation spectrum of the form
(cid:114)πa (cid:126)2 (cid:114) 3 n g
εm(k)=Ek ≡(cid:126)2k2/2m g =gc ≡ 2 ry me−1/2n1r∗−1− 32π a1 (9)
(cid:113) √ ∗ y
ε (k)= E2+2n E [g/ 2πa +r ln(kr )(cid:126)2k2/m].
r k 1 k y ∗ ∗
theGSundergoesafirst-orderquantumphasetransition
The first branch, having the form of a free particle dis- to a supersolid – a condensate with a periodical modula-
persion, describes excitation of magnons (spin waves) tation of density [26, 29, 31].
4
Physically, the onset of the roton instability reflects configuration [33, 35], akin to the fragmented BEC in
tendency of the system to crystallize. It is well known opticallattices[36].Inthisregimethetunnelingbetween
that for particles interacting via long-range repulsive theadjacentlatticesitesisfrozenandthecondensatesdo
forces it may be profitable to arrange into a periodic not talk with each other. The system resembles more a
structure at sufficiently high pressure and low tempera- crystal than aquantum liquid. However, as we have seen
ture [32]. The effect of resonant attraction in a dipolar above, by going deep into the molecular regime (µ (cid:29) ε)
BEC consists in possibility of building up a lattice po- the cells of this crystal can be made superfluid. A re-
tential already in the dilute limit. The lattice constant markable quality of such ”superfluid train” would be its
is of the order of the healing length ξ = (cid:112)(cid:126)2/mng, i. e. robustnesstothermalandquantumfluctuations.Indeed,
in contrast to usual crystals it spans a macroscopically thelong-wavefluctuationsofthephase,whichareknown
large amount of particles. to preclude BEC in low dimensions [37], become sup-
By increasing the pressure one can bring the system pressed as soon as the links between the cells are bro-
to the dense and, generally speaking, strongly correla- ken.Sodothephase-slipevents,whichdestroysuperflui-
ted regime. The perturbative theory fails to predict pro- dity in 1D [38] : within each trapped condensate these
perties of the GS there. Instead, some phenomenologi- are energetically forbidden. The only topological defects
cal arguments can be applied. Thus, it can be postula- whichcanproliferateinanisolatedcellarevortices,that
ted [24, 33, 34], that the equilibrium state of a strongly raises the temperature at which the system becomes su-
coupled excitonic BCS still manifests macroscopic long- perfluid up to the Kosterlitz-Thouless (KT) transition
range order, though the healing length is now much less point kT ∼(cid:126)2n/m.
KT
thanthelatticeconstant.Eachunitcellofthisstatemay
The upper limit on the density n is imposed by quan-
be regarded as a trapped 2D BEC in the Thomas-Fermi
tum dissociation of excitons that occurs when the mean
limit. Possible quantum phases and transition between
inter-exciton distance becomes comparable to the exci-
them within a condesate can be examined by using the
ton size. The latter canbe significantly reduced by using
Hamiltonian [24]
the so-called van der Waals heterostructures based on
Hˆ(cid:48) =(cid:90) (cid:88) Ψˆ†(ρ)(cid:18)− (cid:126)2 ∆−µ¯ (cid:19)Ψˆ (ρ)dρ+ transition metal dichalcogenides (TMD’s) [39]. A gene-
σ 2mσ σ σ ral strategy for achieving record-high values of TKT with
σ=↑,↓,B TMD’s has been worked out in [40]. Implementation of
1(cid:90) (cid:88)Ψˆ†(ρ)Ψˆ† (ρ(cid:48))V (ρ−ρ(cid:48))Ψˆ (ρ)Ψˆ (ρ(cid:48))dρ(cid:48)dρ the fragmented biexcitonic supersolid in these structures
2 σ σ(cid:48) σσ(cid:48) σ σ(cid:48) thuscouldpavethewaytohigh-T counterflow supercon-
σ,σ(cid:48) c
(cid:90) (cid:114) (cid:126)2β (cid:90) ductivity of electrons and holes [41]. Inconvenience rela-
+ε Ψˆ† Ψˆ dρ− (Ψˆ†Ψˆ†Ψˆ +Ψˆ Ψˆ Ψˆ† )dρ, tedtotheelectron-holeradiativerecombinationcouldbe
B B 2πm ↑ ↓ B ↑ ↓ B
avoided by using the dark excitonic states [42].
(10)
In conclusion, we have shown that dipolar excitons in
withµ¯ =µ¯ =µ¯ /2≡µ¯beingthelocal chemicalpoten-
↑ ↓ B coupled 2D films can be used for realization of a stable
tials and m = m = m /2 ≡ m. The two-body poten-
↑ ↓ B bosonic analog of the BCS superconductor. The state of
tialscanbetakenintheformV (ρ−ρ(cid:48))=g δ(ρ−ρ(cid:48))
σσ(cid:48) σσ(cid:48) a resonantly paired excitonic gas can be controlled by
with some parameters g > 0 to be defined from the
σσ(cid:48) theelectricfieldappliedperpendicularlytothestructure
experiment.Theresonantinteractionin(10)appearsex-
plane. The proposed setting should allow one to create
plicitlyasthelasttermwhichconvertstwoexcitonswith
novel dense and strongly-correlated quantum phases of
oppositespinstoabiexciton(thecorrespodingfieldope-
bosons.Asanillustartion,wepredictafragmentedbiex-
rator is labeled by ”B”) and vice versa.
citonic supersolid. We expect this new collective state of
Byadjustingtheexternalbiasvoltagesuchthatµ(cid:29)ε
mattertoberemarkablyrobusttofluctuations.Thispro-
one can completely eliminate the excitonic component
pertycanbeofparticularinterestforrealizationofroom-
and obtain a purely molecular (biexcitonic) BEC. The
temperaturecounterflowsuperurrentsintransitionmetal
elementaryexcitationspectrumofthisphaseisshownin
dichalcogenides,wheremethodsforachievingrecord-high
Fig. (2). In addition to the usual sound mode, it has a
degeneracy temperatures have been recently suggested
gapped branch corresponding to the pair-breaking exci-
[40].
tations. The gap can be controlled by the applied elec-
tricfield(viatheparameterε).Bothbranchessatisfythe TheauthoraknowledgesD.S.Petrov,A.A.Varlamov
Landau criterion for superfluidity. For µ(cid:29)ε the gapped and G. V. Shlyapnikov for clarifying discussions. The
mode lies above the phonon one and the critical velocity research leading to these results received funding from
is given by the velocity of sound c =(cid:112)ng /m(cid:126)2. the European Research Council (FR7/2007-2013 Grant
B BB
For sufficiently large values of g ’s quantum fluc- Agreement No. 341197) and from the Government of
σσ(cid:48)
tuations arising from depleted regions in between the the Russian Federation (Grant 074-U01) through ITMO
condensates drive the supersolid to a number-squeezed Postdoctoral Fellowship scheme.
5
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