Transition from spin-orbit to hyperfine dominated spin relaxation in a coldfluid of dipolar excitons

Ran Finkelstein,1, ∗ Kobi Cohen,1 Benoit Jouault,2 Ken West,3

Loren N. Pfeiffer,3 Masha Vladimirova,2 and Ronen Rapaport1, 4

1Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel2Laboratoire Charles Coulomb, UMR 5221 CNRS-Universite de Montpellier, Montpellier, F-34095, France

3Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA.4Applied Physics Department, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel

We measure the spin-resolved transport of dipolar excitons in a biased GaAs double quantum wellstructure. From these measurements we extract both spin lifetime and mobility of the excitons. Wefind that below a temperature of 4.8K, there is a sharp increase in the spin lifetime of the excitons,together with a sharp reduction in their mobility. At T = 1.5K, where the excitons have the lowestmobility, we observe an anomalous non-monotonous dependence of the exciton spin lifetime on themobility. Below a critical power the spin lifetime increases with increasing mobility and density,while above the critical power the opposite trend is observed. We interpret this transition as anevidence of the interplay between two different spin dephasing mechanisms: at low mobility thedephasing is dominated by the hyperfine interaction with the lattice nuclei spins, while at highermobility the spin-orbit interaction dominates, and a Dyakonov-Perel spin relaxation takes over. Theexcitation power and temperature regime where the hyperfine interaction induced spin dephasing isobserved correlates with the regime where a dark dipolar quantum liquid was reported recently ona similar sample.

I. INTRODUCTION

The spin degree of freedom plays a crucial rolein electronic, magnetic, optical and transport prop-erties of semiconductor nanostructures, and spin isalso one of the most promising candidates for encod-ing and transporting quantum information in suchstructures. The state of an electron spin is verysensitive to interactions with its environment. Twomain interactions are known to affect electron spin:spin-orbit interaction, that couples the spin of elec-tron to its motion, and hyperfine interaction thatcouples electron spin to the spin of the lattice nu-clei. They give rise to striking phenomena such asthe spin Hall effect1, persistent spin helix2 and self-sustained polarization oscillations3, but on the sametime they are the two main sources of spin relax-ation and dephasing4,5. Relative importance of thesetwo types of interaction is related to the electronmobility6,7.

This fact is particularly interesting for an elec-tron bound in a dipolar, or indirect exciton (IX),a Coulomb-bound pair made of an electron and ahole, where each constituent is confined in a sepa-rate GaAs quantum well separated from the otherby a thin (Al, Ga)As barrier. Such IXs form a longliving two-dimensional fluid of interacting boson-likeparticles, with a complex, four-fold internal spinstructure. Two of the IX spin states can coupleto the light (bright states), they have spin projec-

tion along the growth axis Sz = ±1, while twoother spin states (Sz = ±2) are dark. It was pre-dicted that Bose-Einstein condensation of the IXgas into the dark spin state should occur below acritical temperature8, and that the condensed phaseshould be strongly correlated due to the strong dipo-lar interactions between the particles, similarly to aquantum liquid9,10. Recent experiments on dipolarIXs have reported a condensation to a dark, corre-lated quantum liquid-like phase11–13, with a stronglyquenched IX motion12–14, as well as reports on a con-densate with an extended spatial coherence15–17 anda ballistic spin motion18.

There are many open questions on the properties,the microscopic nature, and the formation dynamicsof the newly observed collective phase. For example,the darkening of the IX fluid is suggested to be at-tributed to a spontaneous condensation of IX to thelow energy dark spin states8,11,13,17. In this context,the spin of the electron inside the IX could in princi-ple be such a complementary, sensitive probe of IXdynamics. Indeed, long spin lifetimes (up to tens ofnanoseconds) have been reported for IX19–24. Thisis due to dramatic reduction of the electron-holeexchange interaction within IX, which is responsi-ble for the fast spin relaxation of traditional, directexcitons (DXs) in semiconductor quantum wells25.Because hole spin relaxation is usually much fasterthan spin relaxation of an electron, IX spin lifetimeis determined by the constituent electron spin life-

arX

iv:1

706.

0086

1v1

[co

nd-m

at.q

uant

-gas

] 2

Jun

201

7

time. Thus, for IX spin relaxation we can expect aphenomenology similar to that of a two-dimensionalelectron gas. When IXs are localized (e. g. as ob-served in low mobility correlated liquid of IXs, orin the case of strong disorder), the spin dephasingcould be dominated by the hyperfine interaction, viathe fluctuations of the Overhauser field BN

3–5. Thismechanism is schematically represented in Fig. 1 (d):electron spin relaxation rate is determined by theelectron correlation time τc (during which the Over-hauser field experienced by an electron can be con-sidered as constant) and the squared average valueof BN . In contrast, in a regime where IXs are highlymobile, fluctuations of the Overhauser field are av-eraged out. In this case electron spin dephasing ismainly due to spin-orbit interaction, via Dyakonov-Perel (DP) mechanism (Fig. 1 (c))26. Here the rele-vant time is the momentum scattering time τp.

A natural way to probe these two regimes is tostudy the correlation between IX spin lifetime andmobility, by changing the lattice temperature andthe exciton density. To get access to both spin life-time and mobility, spatial extension of the IX cloudupon point-like excitation, and its polarization de-gree should be analyzed in the same experiment.Previous works have reported a significant decreaseof the IX spin lifetime with increasing IX diffusivity,a behavior consistent with DP relaxation mechanismof IX20. However, the role of the Overhauser field inthe exciton spin dynamics has not been evidencedso far. Indeed, when hyperfine mechanism of spinrelaxation dominates, one should expect an increaseof the IX spin lifetime with increasing IX diffusivityand density.

In this work we report, for the first time in anyexcitonic system to our knowledge, a temperatureand density regime where a transition from a de-creasing spin lifetime to an increasing spin lifetimewith increasing mobility takes place. This suggeststhat hyperfine interaction does play a role in the IXspin relaxation when a transition from high to lowmobility state of the IX fluid occurs. Remarkably,the transition temperature between the two regimesis found to closely match the previously reportedcritical temperature for the condensation of an IXliquid.

The paper is organized as follows: The sample andexperimental setup are presented in Section II. It isfollowed by the description of experimental results(Section III), as well as the drift-diffusion model andits assumptions (Section IV). Section V contains thediscussion of the observed power and temperaturedependences and their interpretation in terms of in-terplay between spin-orbit and hyperfine interaction.

DX IX

-

+

E (b) (a)

𝐕

𝝈+ 𝐬

𝒌

𝑩𝒆𝒇𝒇

(c) 𝐵𝑁(𝑟1)

𝑠

𝑁

𝜏𝑐 𝜏𝑐

Δ𝑡 = 𝜏𝑝

𝐵(𝑘′)

𝑠

𝑠 (d)

𝑘′

FIG. 1. (a) Sketch of the sample and experimental setup.Coupled quantum wells subjected to an electric field inthe z direction are excited with circularly polarized lightresonant with the DX energy.(b) Band diagram of the bi-ased GaAs/AlGaAs coupled quantum wells. A resonantexcitation of the DX transition is followed by an electrontunneling across the barrier, and the formation of IXs.(c) Hyperfine interaction-induced spin relaxation for lo-calized excitons: relaxation rate is inversely proportionalto the electron spin correlation time τc. This mechanismis quenched for mobile excitons; (d) Dyakonov-Perel spinrelaxation mechanism for moving excitons via fluctua-tions of the effective magnetic field induced by spin-orbitinteraction. Relaxation rate is proportional to the mo-mentum scattering time τp

The conclusions are given in Section VI.

II. SAMPLE AND EXPERIMENTAL SETUP

The sample that we use in this work has alreadybeen studied in Refs. 11 and 13. It contains a dou-ble quantum well (DQW) structure with two 12 nm-wide GaAs quantum wells separated by a 4 nm-wideAl0.45Ga0.55As barrier. An electric field is appliedalong the growth axis between a semi-transparent10 nm-thick Ti top circular gate with a diameter of300 µm and n+-doped GaAs substrate, as depictedin Fig. 1(a). Fig. 1(b) shows the band diagram ofthe biased DQW structure with direct exciton (DX)and IX transitions. A tunable Titanium-Sapphirelaser is focused through a ×20 objective (numericalaperture NA=0.42) to a beam of 3 µm diame-ter at the sample surface. The sample is mountedinside an immersion He4 optical cryostat. The pho-toluminescence (PL) is collected through the sameobjective, and split into two beams with orthogo-nal polarizations (σ+ and σ−), simultaneously im-

2

<+

6.9K-5 0 5

<-

339 nW-5 0 5

<+

3.5K

<-

338 nW

<+

1.5K

<-

418 nW

<+

6.9K-5 0 5

1540

1545

1550 <-

71 nW-5 0 5

<+

6.9K-5 0 5

<-

4929 nW-5 0 5

<+

3.5K

<-

4714 nW

<+

1.5K

<-

4971 nW

<+

3.5K1540

1545

1550 <-

69 nW

<+

1.5K1540

1545

1550 <-

63 nW(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Position(µm)

Energy(m

eV)

FIG. 2. Color-encoded spatially resolved PL spectra of IX in the two orthogonal circular polarizations. Themeasurements at three different excitation powers and three temperatures are shown. Each data set is normalized toits maximum in σ+ polarization.

-5 0 50

0.5

1

-5 0 5 -5 0 5

-5 0 50

0.5

1

-5 0 5 -5 0 5

-5 0 50

0.5

1

-5 0 5 -5 0 5

70nW 5µW400nW1.5K 1.5K 1.5K

(a) (b) (c)

70nW 5µW400nW3.5K 3.5K 3.5K

(d) (f)(e)

70nW 5µW400nW6.9K 6.9K 6.9K

(i)(h)(g)

Intensity

(arb.units)

Position(µm)

FIG. 3. Spatial distribution of IX emission intensity at several temperatures and excitation powers. σ+ (blue circles)and σ− (red squares) intensity profiles are compared to the excitation spot profile (black dashed line). The solid linesare a fit to the model presented in Section IV.

3

aged onto a spectrometer with magnification of 40.The spectrometer includes a CCD with a pixel sizeof 26 µm, such that the spatial imaging resolutionis 0.65 µm. We use photoluminescence excitation(PLE) experiments to determine the energy of theDX state in our sample, EDX = 1.558 eV (see Ap-pendix). To optimize the spin pumping efficiency,we tune a σ+-polarized excitation exactly to thatenergy. The excitation power is varied, resulting invariation of the resulting IX densities. The appliedbias is fixed at Vg = 3 V, corresponding to the IXemission energy E = 1.5408 eV at the lowest mea-sured temperature and excitation power. To deter-mine the radiative lifetime τr of the IXs, we directlymeasured τ0r for a specific DX-IX energy separation∆0

DX−IX = 26 meV, using a time-resolved PL experi-

ment. A radiative lifetime of τ0r = 460 ns was found.To extract τr for all other experiments, we use thequadratic analytic model which was recently shownto be a good approximation for the dependence ofτr on the DX-IX energy splitting ∆DX−IX:11,27

τr = τ0r

(∆DX−IX

∆0DX−IX

)2

. (1)

III. RESULTS

Fig. 2 shows typical color maps of a spatial cross-section of PL spectra as a function of the distancefrom the excitation spot for two circular polariza-tions. These measurements are taken at three differ-ent excitation powers and three temperatures. Foreach pair of measurements σ+/σ− the intensity isnormalized to its maximum in σ+ polarization. Sev-eral observations could be made from these measure-ments: (i) The IX emission energy increases withincreasing excitation power for each temperature.This is a signature of the increasing IX density. Thisenergy shift results in the reduction of the IX life-time τr (see Eq. 1 and Fig. 4 (a)); (ii) the IXemission is partially polarized, which means that thetransfer between DX and IX states partially con-serves the polarization, in agreement with previousreports19–22,28; (iii) there are variations in the spa-tial extent, polarization and shape of the IX cloudfor different excitation powers. This is a signature ofthe dipole-dipole interactions between IXs. There isa non-trivial dependence of the diffusion coefficienton the IX density and temperature, due to an inter-

play between drift and diffusion currents of IX29. Toget a closer look on the shape of the IX emission in-tensity profile and its polarization we plot in Fig. 3the spectrally integrated intensity of the IX emissionin σ+ (blue symbols) and σ− (red symbols) polar-izations for the same set of measurements as thoseshown in Fig. 2. The excitation spot size is shown bya black dashed line. To extract the IX spin lifetimeand diffusivity from these data sets, we fit them tothe drift-diffusion model presented in the next Sec-tion.

IV. MODEL

We consider the simplest model that accounts forradial drift, diffusion, and spin relaxation of IXs cre-ated by circularly polarized pump beam with a R0-radius Gaussian spot. The same model was used inRef. 20.

Let n = [n+2n+1n−1n−2] be the vector of IX den-sities in each of four IX spin states, D is the IX diffu-sion coefficient. IX mobility µ is related to the diffu-sion coefficient via Einstein relation (linear transportregime30), µ = D/kBT , where kB is the Boltzmannconstant and T is the lattice temperature. The po-tential u0 responsible for the drift of IXs out fromthe excitation spot is due to the dipole-dipole inter-action between IXs29–32. In the simplest case u0 isgiven by the plain capacitor formula u0 = 4π2d/ε,where d is the distance between the quantum wellcenters, and ε is the dielectric constant9,31,33,34.For the samples studied in this work, this yieldsu0 = 1.6 × 1010 meV cm2. The exciton generationrate under σ+1 excitation in the sample plane canbe written as Λ+1 = 2(Np/R

20) exp

(−r2/R2

0

), where

r is the in-plane distance from the excitation spotcenter, and Np is the number of excitons generatedper second.

The drift-diffusion equation including spin dy-namics of the four exciton states reads

dn

dt= ∇(D∇n) + µn∇(u0(nb + nd)) +wn+ Λ, (2)

where nb+nd = (n+2+n+1+n−1+n−2) is the total(bright and dark) exciton density, and Λ = [0Λ+100]is the generation rate vector. The exciton recom-bination and spin relaxation (including individualelectron and hole spin flips, as well as the excitonspin flips) is represented by the matrix w35:

4

w =

−(w+

e + w+h ) w−

e w−h 0

w+e −( 1

τr+ w−

e + w−h + wex) wex w+

h

w+h wex −( 1

τr+ w−

e + w−h + wex) w+

e

0 w−h w−

e −(w+e + w+

h )

(3)

Here wex = 1/2τex is exciton spin flip rate betweenSz = ±1 states, w±

e(h) = τ−1e(h)/(1 + exp(±∆/kBT ))

is electron (hole) spin flip rate, and ∆ is the split-ting between Sz = ±1 (bright) and Sz = ±2 (dark)states. As ∆ kBT in the range of temperatures ofinterest we assume that electron(hole) spin flip rateis simply given by we(h) = τ−1

e(h)/2.

Assuming (i) fast hole relaxation τh τe, τex and(ii) very slow exciton relaxation τex τe, Eq. 3 canbe simplified. For the bright exciton densities weget:

2dn+1

dt= 2∇(D∇n+1 + 2µn+1∇(u0nb))

− 1

τrn+1 −

1

τe(n+1 − n−1) + Λ,

(4)

2dn−1

dt= 2∇(D∇n−1 + 2µn−1∇(u0nb))

− 1

τrn−1 −

1

τe(n−1 − n+1).

(5)

Note that these equations do explicitly account forthe dark excitons dynamics in the structure, butthey assume that bright-dark conversion is infinitelyfast, so that n+2 = n−1 and n−2 = n+1. One cansee from Eqs. 4, 5 that within the assumptions madehere, the polarization lifetime (the time correspond-ing to the transition from one bright IX state to an-other) is simply equal to the electron spin-flip timeτs = τe. In the following we will refer to this timeτs as the IX spin lifetime.

In order to find the steady state solutions for theexciton density, we numerically solve this set of equa-tions. The full width at half maximum (FWHM)of the Gaussian excitation spot radius is extractedfrom the fit to the Gaussian function of the laserspot profile (dashed lines in Fig. 3).

When fitting the model to the experimental data,we assumed that the number of excitons generatedper second Np increases linearly with the excitationpower. At lowest power P0 = 70 nW we have chosenN0p = 3×108 s−1. The value of N0

p is the first fittingparameter (actually, this value depends on the choiceof u0, the true fitting parameter is N0

pu0). The twoother fitting parameters are the electron spin relax-ation time τe (or, equivalently, IX spin lifetime τs)

(a)

(b)

(c)

(d)

= r (ns)

0

50

100

150

200

250

= s (ns)

0

5

10

15

20

25

301.5K2.5K3.5K4.8K6.9K

Power (7W)10-1 100 101

D (c

m²/s

)

10-3

10-2

10-1

100

Power (7W)10-1 100 101

FWH

M (µ

m)

4

5

6

7

8

FIG. 4. (a) IX radiative lifetime τr used in the model-ing, see Section II and Eq. 1; (b) IX cloud width at halfmaximum of intensity profile as a function of excitationpower for several temperatures. IX spin lifetime τs (c)and exciton diffusion coefficient (d) are extracted fromthe drift-diffusion modeling of the PL spatial profiles, asthose shown in Fig. 3.

and the exciton diffusion constant D. The recom-bination time τr is determined for each set of dataaccording to Eq. 1. The resulting values are shownin Fig. 4 (a).

The comparison between the data and the calcu-lated PL intensity (assumed to be proportional tothe density of bright excitons) is shown in Fig. 3 forthree different temperatures and powers. One cansee in Fig. 4 (c),(d) that both τs and D obtainedwithin the fitting procedure appear to be tempera-ture and power dependent. In the next Section wediscuss this dependence and the underlying physicalmechanisms.

5

10-3 10-2 10-1 100

D (cm2/s)

0

10

20

30

40

= s (ns

)1.5 K2.5 K3.5 K4.8 K6.9 K

FIG. 5. IX spin lifetime as a function of the diffusionconstant as extracted from the drift-diffusion modeling.Black line is the theoretical estimation based on Eq. 8.

V. DISCUSSION

To better understand the spin dynamics of IXs,it is instructive to compare their spin lifetime to thediffusion constant, that determines the transport dy-namics of the IX cloud. Both the spin and the trans-port show three distinct temperature regimes, as canbe seen in Fig. 4.

(i) At high temperature, T ≥ 4.8 K, the spin life-time is rather short (τs < 5 ns), and almost does notchange with excitation power. At the same time, Dalso remains approximately constant, and the sizeof the cloud exhibits a weak, non-monotonous be-havior with power. The fact that D is indepen-dent of excitation power is typical for a thermallyactivated cloud30. The non-monotonous cloud sizeis due to the interplay between diffusion and driftof the excitons. It can be shown analytically thatin the case of diffusion-dominated transport, theFWHM of the cloud does not depend on Np, thenumber of particles injected in the system per unittime, but only on (Dτr)

1/2. By contrast, in thecase of drift-dominated transport, the FWHM of the

cloud increases as (DNp/kBT )1/4

τ1/2r . The non-

monotonous behavior of the cloud size when exci-tation power increases could thus be the signatureof transition between these two regimes29,32,36.

(ii) A very different behavior is seen in the lowertemperature regime, 1.5 K < T < 4.8 K. Thespin lifetime sharply rises as the temperature de-creases, reaching an almost a 5-fold increase at 2.5 K.Spin lifetime also exhibits a strong power depen-dence, with a clear threshold power of ' 0.5 µW,above which it strongly decreases. This trend of

τs is clearly anti-correlated with both the size ofthe cloud and the diffusion constant obtained fromthe modeling. Indeed, the diffusion constant Dsharply decreases with decreasing temperature inthis regime. As for the power dependence, D isapproximately constant with increasing power upto the same threshold of ' 0.5 µW, and increasesonly above this threshold. The IX cloud does notshow any expansion up to a similar excitation power,and then experience a strong expansion. Such anti-correlation of τs and D is well explained within theDP mechanism, where the IX spin relaxation timeτs reads:

τ−1s,SO = Ω2

SOτp, (6)

where τp is the momentum scattering time, ΩSO =2βk is the electron spin precession frequency inthe presence of an effective magnetic field due tospin-orbit interaction, β is the spin-orbit interac-

tion constant, k =√

2mexckBT/h2me/mexc is the

thermal wave vector of an electron within exciton,me(mexc) is the electron (exciton) mass. Becauseτp = Dmexc/kBT , an increase of the diffusion con-stant leads to a decrease of the spin relaxation time,as has been reported by Leonard et al.20. Note how-ever, that the persistent localization of the IX cloudbelow T = 4.8 K over a very large range of the ex-citation powers is consistent with previous report ofa transition into a spatially localized dark liquid ofIX12,13, which was interpreted as a condensation intoa dark quantum liquid of dark spin states11,13,14,17.In particular, Cohen et al. observed a similar powerthreshold, over which there is a strong increase inIX mobility and the IX cloud expansion13,14.

(iii) Focusing on the lowest measured temperatureof 1.5 K (which also corresponds to the temperaturewhere the transition to the liquid phase in Ref. 13was found to be fully established), a very distinct be-havior of τs with power is observed. In this regimea clear increase of τs when the power increases isseen, together with a slow increase in D. This cor-relation between D and τs continues up to ' 0.5µW, after which a strong increase of D is observedtogether with a strong decrease of τs, where the pre-viously discussed anti-correlation between τs and Dis established again. This low power regime of corre-lation can not be understood within DP mechanismand suggests that the hyperfine interaction with thenuclear spins of the GaAs lattice is involved in thespin relaxation process4,6,7. Indeed, the fluctuatingOverhauser field BN created by the nuclear spinsinduces a spin relaxation given by a formula verysimilar to the one corresponding to the DP mecha-

6

nism:

τ−1s,N =

1

3Ω2Nτc, (7)

where ΩN = gµBBN is the electron spin precessionfrequency due to the fluctuating Overhauser field,τc is the correlation time of this field and g is theelectron g-factor. The correlation time τc is relatedto the exciton diffusion coefficient and can be es-timated as the time over which an exciton movesby a distance corresponding to its own radius aexc:τc = a2exc/D. In contrast to the spin relaxation in-duced by spin-orbit interaction, hyperfine inducedspin-relaxation is enhanced when exciton diffusionslows down. Therefore, this relaxation mechanismcould explain the initial increase of τs with power,as observed at T = 1.5 K.

This is the first time, to our knowledge, that suchspin relaxation mechanism is observed in an exci-tonic system. It stems from the very low mobilityregime achieved for the IX cloud, allowing for thehyperfine interaction induced dephasing to becomedominant. As the power increases, there is a tran-sition from the hyperfine interaction to the DP de-phasing mechanism, corresponding to the sharp in-crease in the IX diffusivity. While here we did notcarefully map the state of the IX fluid as a functionof power and temperature as was done in Ref. 13,the temperature and power regimes where strong IXfluid localization and hyperfine-induced spin relax-ation is observed in this work, seems to overlap withthe regime where the IX liquid formation was ob-served. We thus can speculate that the reason forthis strong localization of the IX cloud could be theformation of a dark IX liquid, similar to the previ-ous reports11–14,17. If this is indeed the case, it showsthat measuring spin polarization can be a comple-mentary, sensitive tool for mapping the collectivestates of IX fluids.

In order to get a unified picture of the observedphenomena, we write the total spin relaxation time,due to both hyperfine and spin-orbit relaxations, as:

τ−1s = τ−1

s,N + τ−1s,SO, (8)

where τs clearly has a non-monotonous dependenceon the diffusion constant. Fig. 5 shows τs(D), ascalculated from Eq. 8 (solid line) or extracted frommeasurements fitted by the drift-diffusion model ofEqs. 4,5 (symbols). We assume BN = bN/

√NN ,

where bN = 5.3 T is the nuclear field at saturation3

and NN ' 105 is the number of nuclei within the ex-citon volume Vexc = πa2excL, L is the quantum wellwidth. The g-factor of the L = 12 nm GaAs quan-tum well is g ' 0.355 and the spin-orbit constant

β = 25 meV·A20. The theoretical curve τs(D) hasa non-monotonous dependence and reaches a max-imum τmaxs =

√3/8vth/(ΩSOΩNaexc) at Dmax =

(ΩN/ΩSO)aexcvth/√

6, where vth =√

2kBT/mexc

is the thermal velocity of excitons. Remarkably,this curve does not depend on temperature, becauseΩSO and vth have the same temperature depen-dence. Numerically one finds τmaxs ' 35 ns andDmax ' 3.7×10−3 cm2/s, close to the experimentalobservation. The theoretical curve being universal,it can be directly compared to the experimental dataextracted from the numerical modeling at any tem-perature. Since there are no fitting parameters inthe theory, the agreement can be considered as sat-isfactory.

VI. CONCLUSIONS

To conclude, our analysis suggests that IX spinlifetime is limited not only by the spin-orbit inter-action via DP mechanism, but also by the hyperfineinteraction with the spins of the lattice nuclei. Thecrossover between these two different regimes corre-sponds to a transition from a high mobility state ofthe fluid to a very low mobility state, and may becorrelated with the power and temperature regimewhere a condensation of a dipolar IX fluid was pre-viously reported. These results suggest that the IXspin polarization could be a sensitive supplementaryprobe for the onset of the collective states in dipo-lar fluids of excitons. Further work, both theoreticaland experimental, is required to confirm the rela-tion between spin dynamics and collective proper-ties of IX liquids. In particular, it would be in-structive to combine the constant energy scheme(EDX − EIX = const) used in Ref. 13 with res-onant spin polarized excitation used in this workto achieve better control over bright and dark ex-citon densities. Theoretically, the simplified modelthat we used must be extended to the case of non-thermal distribution of the IX population betweenbright and dark states. The assumption of the in-finitely fast hole spin relaxation must be lifted in thiscase. The comparison between such model and thefull set of data obtained within the constant energyscheme will settle definitively the origin of excitonlocalization, and the onset of the hyperfine-inducedspin relaxation.

7

ACKNOWLEDGMENTS

We would like to acknowledge financial supportfrom the German Deutsche Forschungsgemeinschaft(Grant No. SA-598/9), from the German IsraeliFoundation (Grant No. GIF I-1277-303.10/2014),from the Israeli Science Foundation (Grant No.1319/12) and French National Research Agency(Grant OBELIX, No. ANR-15-CE30-0020-02). Thework at Princeton University was funded by theGordon and Betty Moore Foundation through theEmergent Phenomena in Quantum Systems initia-tive Grant No. GBMF4420, and by the NationalScience Foundation Materials Research Science andEngineering Center Grant No. DMR-1420541.

APPENDIX

Dependence on excitation energy

In order to determine the resonant energy levelsof the DQW structure, a photo-luminescence exci-tation (PLE) experiment was conducted in the fol-lowing way: a tunable CW Titanium-Sapphire laserwas scanned through a wavelength range coveringthe expected optical direct exciton resonances, whilemonitoring the (e1:hh1) 1S IX PL energy, intensityand polarization.

Fig. 6 shows the results of a polarization-resolvedPLE experiment. Fig. 6(a) and (b) show the in-tegrated intensity and the peak energy of IX as afunction of the excitation energy of the laser. Thereare two distinct peaks, separated by about 14 meVin energy. We attribute the high energy peak to the(e1:lh1) 1S DX absorption line and the lower one tothe (e1:hh1) 1S DX. A large blue shift of the IX lineis observed when the excitation is at resonance withone of the DX absorption lines. This is not surpris-ing as it is expected that the DX resonances wouldhave a large absorption cross-section and thereforean efficient conversion of photons to IX population,by carrier tunneling to adjacent quantum wells.

Fig. 6(c) presents the DOP of IX as a functionof the excitation energy. It can be seen that theDOP of the IX is maximal for laser excitation res-onant with the DX transitions. While for excita-tion at the (e1:hh1) DX transition, IX are polar-ized in parallel to the exciting laser (positive DOP),an opposite polarization appears when the laser isresonant with the (e1:lh1) DX transition (negative

DOP). This could be explained by the followingmechanism: a resonant excitation with σ+ polarized

FIG. 6. Polarization-resolved PLE of IX. (a) The inte-grated PL intensity, monitored at the IX line for bothpolarizations. (b) The PL energy, and (c) The degreeof circular Polarization (DOP). A negative DOP meansthat the PL of the IX has the opposite circular polar-ization with respect to the circularly polarized excitinglaser. The laser power here is 1.3 µW and is circularlypolarized with σ+. The energies of (e1:hh1) DX and(e1:lh1) DX are pointed in (c) as a guide to the eye.

light at the (e1:lh1) line creates the following excitonstate:

∣∣ 12 ,

12

⟩. The light-hole in the exciton rapidly

relaxes to the heavy-hole sub-band where it couldbe either at

∣∣ 12 ,

32

⟩state, which is dark or at

∣∣ 12 ,−

32

⟩state, which is bright and has Sz = −1, . Thus aninitial population of IX with polarization oppositeto the laser is created. From here on, the IX popu-lation follows the spin relaxation mechanism. Such”opposite” initial polarization was observed in singleGaAs QW by time resolved experiments but not tothis extent37. Another interesting feature is that thedensity and the DOP following a (e1:lh1) DX excita-tion seem to be larger than at a (e1:hh1) DX excita-tion. The increase in density is in contrast with theknown oscillator strength ratios in bulk GaAs, whichis three times larger for the (e1:hh1) transition thanthe (e1:lh1) transition38.

8

∗ Current Address: Department of Physics of Com-plex Systems, Weizmann Institute of Science, Re-hovot 76100, Israel

1 J. Wunderlich, B. Kaestner, J. Sinova, and T. Jung-wirth, Phys. Rev. Lett. 94, 047204 (2005).

2 J. D. Koralek, C. P. Weber, J. Orenstein, B. A.Bernevig, S.-C. Zhang, S. Mack, and D. D.Awschalom, Nature 458, 610 (2009).

3 F. Meier and B. Zakharchenya, eds., Optical orienta-tion, Modern Problems in Condensed Matter ScienceSeries, Vol. 8 (North-Holland, Amsterdam, 1984).

4 K. V. Kavokin, Semicond. Sci. Technol. 23, 114009(2008).

5 M. I. Dyakonov, ed., Spin Physics in Semiconduc-tors, Springer Series in Solid-State Sciences (Springer,Berlin, 2008).

6 R. I. Dzhioev, K. V. Kavokin, V. L. Korenev, M. V.Lazarev, B. Y. Meltser, M. N. Stepanova, B. P. Za-kharchenya, D. Gammon, and D. S. Katzer, Phys.Rev. B 66, 245204 (2002).

7 Z. Chen, S. G. Carter, R. Bratschitsch, P. Dawson,and S. T. Cundiff, Nat. Phys. 3, 265 (2007).

8 M. Combescot, O. Betbeder-Matibet, andR. Combescot, Phys. Rev. Lett. 99, 176403 (2007).

9 B. Laikhtman and R. Rapaport, Europhys. Lett. 87,27010 (2009).

10 Y. E. Lozovik, I. L. Kurbakov, and I. V. Ovchinnikov,Solid State Commun. 126, 269 (2003).

11 Y. Shilo, K. Cohen, B. Laikhtman, K. West, L. Pfeif-fer, and R. Rapaport, Nat. Commun. 4, 2335 (2013).

12 M. Stern, V. Umansky, and I. Bar-Joseph, Science343, 55 (2014).

13 K. Cohen, Y. Shilo, K. West, L. Pfeiffer, and R. Ra-paport, Nano Lett. 16, 3726 (2016).

14 Y. Mazuz-Harpaz, K. Cohen, and R. Ra-paport, (2017), 10.1016/j.spmi.2017.01.022,arXiv:1701.02598.

15 A. A. High, J. R. Leonard, A. T. Hammack, M. M.Fogler, L. V. Butov, A. V. Kavokin, K. L. Campman,and A. C. Gossard, Nature 483, 584 (2012).

16 A. A. High, J. R. Leonard, M. Remeika, L. V. Butov,M. Hanson, and A. C. Gossard, Nano Lett. 12, 2605(2012).

17 M. Alloing, M. Beian, M. Lewenstein, D. Fuster,Y. Gonzalez, L. Gonzalez, R. Combescot,M. Combescot, and F. Dubin, Europhys. Lett.107 (2013), 10.1209/0295-5075/107/10012.

18 A. High, A. T. Hammack, J. R. Leonard, S. Yang,L. V. Butov, T. Ostatnicky, M. Vladimirova, A. V.Kavokin, T. C. H. Liew, K. L. Campman, andA. C. Gossard, Phys. Rev. Lett. 110, 1 (2013),arXiv:1302.3852.

19 A. Violante, R. Hey, and P. V. Santos, Phys. Rev. B

91, 125302 (2015).20 J. R. Leonard, Y. Y. Kuznetsova, S. Yang, L. V. Bu-

tov, T. Ostatnicky, A. Kavokin, and A. C. Gossard,Nano Lett. 9, 4204 (2009).

21 K. Kowalik-Seidl, X. P. Vogele, B. N. Rimpfl,S. Manus, J. P. Kotthaus, D. Schuh, W. Wegscheider,and A. W. Holleitner, Appl. Phys. Lett. 97, 011104(2010).

22 M. Beian, M. Alloing, E. Cambril, C. Gomez Car-bonell, J. Osmond, A. Lematre, and F. Dubin, Eu-rophys. Lett. 110 (2015).

23 P. Andreakou, S. Cronenberger, D. Scalbert, A. Nali-tov, N. A. Gippius, A. V. Kavokin, M. Nawrocki, J. R.Leonard, L. V. Butov, K. L. Campman, A. C. Gos-sard, and M. Vladimirova, Phys. Rev. B 91, 125437(2015).

24 P. Andreakou, A. V. Mikhailov, S. Cronenberger,D. Scalbert, A. Nalitov, A. V. Kavokin, M. Nawrocki,L. V. Butov, K. L. Campman, A. C. Gossard, andM. Vladimirova, Phys. Rev. B 93, 115410 (2016).

25 A. Vinattieri, J. Shah, T. C. Damen, K. W. Goossen,L. N. Pfeiffer, M. Z. Maialle, and L. J. Sham, Appl.Phys. Lett. 63, 3164 (1993).

26 M. I. D’yakonov and V. Perel, Zh. Eksp. Teor. Fiz 33,1954 (1971).

27 Y. Mazuz-Harpaz, K. Cohen, B. Laikhtman, R. Ra-paport, K. West, and L. N. Pfeiffer, Phys. Rev. B 95,155302 (2017).

28 M. Poggio, G. M. Steeves, R. C. Myers, N. P. Stern,A. C. Gossard, and D. D. Awschalom, Phys. Rev. B70, 121305 (2004).

29 R. Rapaport, G. Chen, and S. H. Simon, Phys. Rev.B 73, 033319 (2006).

30 R. Rapaport, G. Chen, S. Simon, O. Mitrofanov,L. Pfeiffer, and P. Platzman, Phys. Rev. B 72, 075428(2005).

31 A. L. Ivanov, Europhys. Lett. 59, 586 (2002).32 R. Rapaport and G. Chen, J. Phys. Condens. Matter

19, 295207 (2007).33 C. Schindler and R. Zimmermann, Phys. Rev. B 78,

045313 (2008).34 M. Remeika, J. R. Leonard, C. J. Dorow, M. M.

Fogler, L. V. Butov, M. Hanson, and A. C. Gossard,Phys. Rev. B 92, 115311 (2015).

35 M. Z. Maialle, E. A. de Andrada e Silva, and L. J.Sham, Phys. Rev. B 47, 15776 (1993).

36 Z. Voros, R. Balili, D. Snoke, L. Pfeiffer, andK. West, Phys. Rev. Lett. 94, 226401 (2005).

37 B. Dareys, X. Marie, T. Amand, J. Barrau,Y. Shekun, I. Razdobreev, and R. Planel, Superlat-tices Microstruct. 13, 353 (1993).

38 P. Y. Yu and M. Cardona, Fundamentals of Semicon-ductors, Graduate Texts in Physics (Springer BerlinHeidelberg, Berlin, Heidelberg, 2010).

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