Solid State Communications 157 (2013) 6–10

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Solid State Communications

0038-10

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n Corr

E-m

timofee

journal homepage: www.elsevier.com/locate/ssc

Compensation of dipolar-exciton spin splitting in magnetic field

A.V. Gorbunov n, V.B. Timofeev

Institute of Solid State Physics RAS, Chernogolovka, Moscow Region 142432, Russian Federation

a r t i c l e i n f o

Article history:

Received 26 November 2012

Accepted 12 December 2012

by Y.E. Lozoviksplitting of the luminescence line of the heavy-hole excitons in the trap centre is completely

compensated at magnetic field below critical value E2 T. The effect of spin splitting compensation is

Available online 19 December 2012

Keywords:

A. Excitons

A. Quantum wells

D. Bose condensation

D. Spin splitting

98/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ssc.2012.12.008

esponding author. Tel.: þ7 496 5228314.

ail addresses: gorbunov@issp.ac.ru (A.V. Gorb

v@issp.ac.ru (V.B. Timofeev).

a b s t r a c t

Magnetoluminescence of spatially indirect dipolar excitons in 25 nm GaAs/AlGaAs single quantum well

collected within a lateral potential trap has been studied in Faraday geometry. The paramagnetic spin

caused by the exchange interaction in dense exciton Bose gas which is in qualitative agreement with

the existing theoretical concepts.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Spin degrees of freedom in excitonic Bose systems and theexchange interaction between the spin-oriented components ofthe exciton ensemble become apparent in polarization of emittedirradiation. Analysis of the latter enables to considerably enhanceunderstanding of the properties of the system in question.Specifically, in the case of microcavity exciton–polariton Bosecondensate the phenomenon of Zeeman spin splitting suppres-sion in magnetic field below some critical value, BoBc. has beenrecently predicted theoretically [1] and later confirmed experi-mentally [2,3]. The effect is related to the spinor nature of excitonpolaritons in microcavities: optically active heavy-hole excitonspossess two spin projections onto the growth axis of the struc-ture: Sz¼71. Accordingly, luminescence light propagating alongthe axis contains two components circularly polarized in oppositedirections: sþ and s� . In the absence of both magnetic field andinterparticle interaction the components are degenerate. Whentaking into account the exchange interaction the state of excitonBose condensate becomes energetically favourable with equalnumbers of bosons with up and down spins. It means that theluminescence light of exciton Bose condensate must be linearlypolarized. If the system has no preferential direction, spontaneoussymmetry breaking may take place which is manifested in arandom-time varying direction of linear polarization. But in mostexperiments structural anisotropy of the objects under study resultsin pinning of the linear polarization to a definite crystallographic

ll rights reserved.

unov),

direction. In particular, in GaAs/AlGaAs quantum wells (QW) grownin the (001) plane two mutually transverse directions, [110] and[110], have a priority. Linear polarization along /110S has beenobserved both in exciton–polariton Bose condensate [2,4] and incondensate of spatially indirect, dipolar excitons [5]. Two orthogo-nal linearly polarized components are split in energy (usually by asmall value of �10�5–10�4 eV) and the linear polarization resultsfrom the primary occupation in the Bose condensate of the lowest-energy state. At zero magnetic field, with increasing optical pump-ing, the linear polarization degree rlin grows stepwise near thethreshold of Bose–Einstein condensation (BEC). Magnetic fieldswitching results in appearance of circularly polarized sþ- ands�-components, each of them being connected with a state wherethe spins are co-parallel and aligned with the magnetic field oropposed to it. According to theory [1], at thermodynamic equili-brium and zero temperature Zeeman energy splitting of thecomponents is suppressed until the state with anti-parallel spinsremains energetically more favourable than the state withco-parallel spins: the polariton–polariton exchange interactionexactly compensates Zeeman splitting, whereas the luminescencelight remains elliptically polarized. With increasing magnetic field B

a critical value Bc is achieved, when the energy of one of the spin-oriented states in the magnetic field becomes lower than the energyof the state with anti-parallel spins: there appears (sþ�s�)-splitting proportional to the increment of magnetic field DB¼B�Bc.Indeed, suppression of Zeeman splitting at small magnetic fieldshas been observed experimentally in photoluminescence ofexciton–polariton condensate [2,3] in spite of the obvious non-equilibrium.

Unlike the exciton–polariton in a microcavity the exciton itselfin GaAs/AlGaAs heterostructures has a fourfold spin degeneracy.Besides an optically active exciton with spin Sz¼71, there exists

+

A.V. Gorbunov, V.B. Timofeev / Solid State Communications 157 (2013) 6–10 7

an optically non-active dark exciton with Sz¼72. For the latteroptical transitions are forbidden in the dipole approximation. Theproperties of the ‘‘four-component’’ exciton Bose condensate inmagnetic field have been recently analyzed theoretically [6]. Inparticular, predicted was the feasibility of phase transitions inmagnetic field between condensate states composed of differentnumbers of components. It means that the polarization behaviourof condensed excitons in magnetic field may noticeably differfrom that of cavity exciton–polariton condensate.

In this work we have experimentally studied the effect ofperpendicular magnetic field B? on spatially indirect dipolarexcitons in a 25-nm GaAs/AlGaAs QW collected in an electrostaticpotential trap nearby a round window in the top Schottky gate[7,8]. The spatially nonhomogeneous electric field of the trapcontains not only perpendicular component F?, normal to thesurface, but also lateral one FII, parallel to the plane of the QW. Incrossed magnetic field B and a lateral component of electric fieldFII excitons move along direction FII�B, i.e. the dispersion curvefor excitons E(k) shifts in the momentum space so that excitonswith finite momentum ka0 possess minimal energy. As a result,optical transition with photon emission becomes indirect in themomentum space and one should expect an increase of theexciton radiative lifetime and, hence, deeper exciton cooling. Forspatially-indirect interwell excitons subjected to homogeneousperpendicular electric F? and parallel, in-plane, magnetic field BII

the effect has been studied both theoretically [9] and experimen-tally [10,11]. The theory of direct intrawell exciton in homoge-neous parallel electric FII and perpendicular magnetic B? fieldshas been developed in [12,13].

In the present work indirect excitons are experimentallyexplored in crossed fields, homogeneous perpendicular magneticfield B? and inhomogeneous parallel electric field FII. Owing to thesymmetry, in the vicinity of the round window in the Schottkygate the in-plane component of static electric field FII has onlyradial constituent Fr, which is maximal near the window edge anddecays down to null exactly in the window centre. The geometryof the crossed fields with radially symmetric in-plane electricfield Fr and a normal to the plane magnetic field B? causes theexcitons to move along the ring trajectories around the windowcentre. In other words, a magnetoelectric trap for spatiallyindirect excitons can be realized. Essentially the same idea, withthe use of a point contact on top of a cylindrical sample insteadof a window in the metal film, has been previously proposed inRef. [14] as a fairly promising approach to experimental imple-mentation of exciton Bose condensation.

1,505 1,510 1,515 1,520

σ

σ-

0.86 meV

0.34meV

Inte

nsity

Energy, eV

0.81 meV

0.52 meV

Edge

Centre

Fig. 1. Circularly polarized photoluminescence spectra at the window edge (left

series) and in the middle of the window (right series) versus magnetic field B

increasing from 0 to 6 T with step 0.5 T from bottom to top. Some linewidths

(FWHM) are shown. Photoexcitation power is Pob¼11.5 mW for overbarrier laser

and Psb¼118 mW for sub-barrier laser. Temperature T¼1.6 K.

2. Experimental technique

Spatially indirect dipolar excitons were studied in a single wide(25 nm) GaAs QW at electric field normal to heterolayers, appliedbetween the metal Schottky gate on top of the AlGaAs/GaAs-heterostructure and the conducting electron layer inside thestructure. Both photoexcitation and photoluminescence detectionwas done through a round window ø7 mm in the opaque Schottkygate (100 nm thick Au/Cr film). Dipolar excitons were collected in acircular lateral potential trap that appeared along the windowperimeter owing to the strongly nonhomogeneous electric field[15,16]. The sample inside the superconducting magnet was placedin liquid 4He in an optical cryostat that enabled experiments in therange of magnetic fields 0oBo6 T at TE1.6 K.

Dipolar excitons were simultaneously excited with two con-tinuous wave lasers with wavelengths lsb¼782 nm (photonenergy below the energy gap in the AlGaAs barrier, ‘‘subbarrier’’excitation) and lob¼659 nm (‘‘overbarrier’’ excitation), the laserspot diameter on the sample surface being �30 mm. Combining

the two laser photoexcitations allowed to maintain the excitonsystem as close to neutrality as possible [15,17].

The magnified image of the window in the Schottky gate wasprojected on the entrance slit of the spectrometer equipped witha cooled silicon CCD-camera. The use of an imaging spectrometerenabled recording the luminescence spectra with spatial resolu-tion (r2 mm) along the direction of the spectral slit. As a resultluminescence spectra from the top/bottom edge of the windowand from its centre were obtained simultaneously. In the zerothorder of diffraction grating image of the sample in differentspectral ranges could be registered with the help of a narrowband (E0.7 nm) interference filter. The luminescence polariza-tion was analyzed by means of a Glan prism and a quarter-wavephase plate.

3. Experimental results and discussion

The circularly polarized photoluminescence spectra versusmagnetic field are presented in Fig. 1. The right (sþ) and left(s�) circularly polarized components are shown as solid anddashed lines, respectively. At the chosen value of the electric field,F?E7.6 kV/cm, the width of the spectral line of dipolar excitonswithout magnetic field (FWHM) amounts to DEZ0.8 meV both atthe edge and in the centre of the window. With switchedmagnetic field the lines narrow noticeably: at BZ1.5 T DE equalsto E0.5 meV at the edge and E0.35 meV in the centre. Besidesthe strong heavy-hole-exciton (hh) line, a much weaker(30C100-fold) line of light-hole exciton (lh) as well as very weaklines of the higher-energy excited states of both hh- and lh-excitons are observed at the window centre (not shown).

Fig. 2 shows the behaviour of exciton-line energy E as a functionof magnetic field. In the range of low fields, B?r1.3 T, anunusually large quadratic-in-field blue shift of the exciton lines isobserved: E2.2 meV/T2 for hh-exciton at the window edge. Thesevalues exceed, by at least an order of magnitude, the normalLangevin diamagnetic shift, which is determined by the in-wellarea of the dipolar-exciton and stays typically below 0.1 meV/T2 inGaAs QWs [18]. The ‘‘giant’’ shift is neither observed in the flat-band regime, at F?E0, nor seen in homogeneous perpendicularelectric field [17]. The effect may be reasonably connected with thepresence of the radial field component in the QW plane in thenonhomogeneous electric field inside the window. In crossed

1,505

1,510

1,515

1,520

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0 1 2 3 4 5 6

0.00

0.05

0.10

0.15

0.20

0.0

0.5

1.0

1.5

E, e

V

HH (edge)

HH (centre)

LH (centre) σ+

σ-

LH (centre)

HH (centre)

HH (edge)

ΔEZ,

meV

ΔEZ,

meV

Pol

ariz

atio

n de

gree

B, Tesla

Fig. 2. Energy (a), Zeeman splitting (b, c) and circular-polarization degree (d) vs.

magnetic field B for the luminescence line of dipolar heavy-hole exciton at the

edge (circles) and in the middle of the window (squares), as well as for the light-

hole exciton in the middle (triangles). Solid symbols in Fig. 2a correspond to

s� and open ones to sþ circularly polarized components, respectively.

A.V. Gorbunov, V.B. Timofeev / Solid State Communications 157 (2013) 6–108

magnetic and electric fields the central symmetry of the lowestexciton state assumes to get broken, higher states with largerangular moments being admixed. In pure electrostatics the magni-tude of radial component Fr(r) is maximal in the vicinity of thewindow edge, decaying monotonously down to zero in the windowcentre: Fr(0)¼0. However, the actual distribution Fr(r) is difficultto calculate: one should take into account field screening byphotogenerated charge carriers. At the window edge Fr is expectedto be of the same order of magnitude as F?. Since the value of theapplied external electric field is sufficiently large (E7.6 kV/cm), the

presence of the lateral component inside the window should be alsoappreciable.

At B?Z1.3 T dependence E(B) is close to linear (see Fig. 2a).The slope of the straight line for hh-exciton both at the edge andin the centre of the window amounts to E0.81 meV/T. Obviously,in the limit of strong magnetic field energy E(B) is determined bythe behaviour of the lowest Landau level for exciton: Eo¼:eB/2mc¼:oc/2, where reduced mass m¼memh/(meþmh), me and mh

are the electron and hole masses, respectively, and oc¼eB/mc isthe cyclotron frequency. But here, in the range of moderatemagnetic field, exciton binding energy Eb and cyclotron energy:oc are of the same order. That is why the value of the effectivereduced mass extracted from the slope of dependence E(B),meffE0.071mo, is definitely overestimated: it exceeds the electronmass in GaAs meE0.066mo, (here mo is the free electron mass).

Spin- (paramagnetic) splitting in the magnetic field of the lineof indirect hh-exciton into sþ- and s�- components occurs both inthe centre and at the edge of the window, but in quite a differentway. Fig. 2b shows the correspondent dependencies of Zeemansplitting DEZ¼Es� � Esþ as functions of magnetic field B?.

At the window edge the splitting grows linearly in the range ofmagnetic field 0rBr1.5 T, i.e. DEZ¼mBgB, where mB is the Bohrmagneton, and the effective g-factor of an exciton is gxEþ0.9(the energy of the s�-component is higher). With 1.5rBr4 Tthe splitting value remains more or less constant, DEZE0.07 meV,while with further field increase it diminishes and approacheszero. This spin splitting behaviour is fairly common in wide GaAs/AlGaAs QWs. It has been previously observed both in non-dopedQWs in the absence of electric field [19] and in a weakly dopedQW in a perpendicular electric field [20]. The dependence with asmall spin splitting value and a change of the g-factor sign inmoderate magnetic fields has been described theoretically inthe effective-mass approximation, with regard for valence bandmixing of hole states [21].

On the contrary, Zeeman splitting for hh-excitons in themiddle of the window is quite unusual: it appears to be compen-sated (with an accuracy of 720 meV) at low magnetic fields,below BE2 T. At B42 T DEZ grows linearly with field, whilegxE�1.5 (in this case the energy of the sþ-component is higher).Simultaneously, in the same place of the trap, the spin splittingfor the lh-excitons increases monotonically with BZ0 (seeFig. 2c). It is characterized by an effective g-factor, gxEþ7, justas it occurs in a 25-nm AlGaAs/GaAs QW under homogeneousphotoexcitation without any lateral trap [17].

Thus, in spite of the fact that Bose-condensation effects havebeen previously observed exactly for hh-excitons collected in thering electrostatic trap at the edge of the window in the Schottkygate [7,8], no indication of compensated Zeeman splitting forexcitons in magnetic field was found for such type of a trap. Itshould be noted that spectrally selective registration of theluminescence images shows a considerable change in the spatialdistribution of the dipolar-exciton luminescence at the windowedge when the magnetic field is turned on (see Fig. 3). In theabsence of magnetic field (Fig. 3a) a typical pattern of two pairs ofbright spots located symmetrically in vertices of a square withdiagonals along directions [110] and [110] [7,8,15] is observed,being close to a square with sides parallel to [100] n [100] due toinsufficient spatial resolution. In magnetic field (Fig. 3b) the spotsspread along the perimeter of the window and a square with sidesalong [110] and [110] is observed. Therefore, the exciton motionalong the circular trajectories in crossed perpendicular magneticfield B? and radial electric field Fr may be suggested to destroy theconditions for exciton accumulation in an annular lateral trap upto the critical density sufficient for BEC.

In the middle of the window, on the contrary, without magneticfield no sign of exciton Bose condensation has been previously

[110]

[110]

Fig. 3. Spatial distribution of dipolar-exciton luminescence in the circular electrostatic trap at the edge of the window in the Schottky gate: (a) in the absence of magnetic

field, B?¼0, and (b) in magnetic field B?¼3 T.

A.V. Gorbunov, V.B. Timofeev / Solid State Communications 157 (2013) 6–10 9

found [15]. With magnetic field, however, just in the windowcentre exciton spin splitting behaves exactly as it was predicted forspinor Bose condensate [1]: there is no splitting in low magneticfield, 0rBrBcE2 T, while at B4Bc it grows linearly with fieldincrement DB¼B�Bc. The observed 2.5-fold narrowing in themagnetic field of the hh-exciton luminescence line points to thepossibility of attaining a high exciton density that is sufficient toform degenerate gas of interacting Bose particles. Note that theexternal electric field used in the present work is almost by anorder of magnitude stronger than in the previous studies [7,8,15].Hence, the effective field strength F? in the middle of the windowis also notably higher. So, the excitons in this region are alsospatially indirect though with a smaller dipole moment than at theedge. Besides, just in the vicinity of the window centre a magneto-electric trap, similar to that proposed in Ref. [14], in which dipolarexcitons are collected by means of circular winding in crossedfields, might be realized. Clearly, at the same conditions theconcentration of lh-excitons is lower by around two orders ofmagnitude. As a result in this, much less dense, exciton system nodegeneracy is achieved, the collective effects are weak and nounexpected spin slitting is observed.

Circular-polarization degree, rcirc¼(Isþ� Is�)/(Isþþ Is�), whereIsþ and Is� are the intensities in the line maximum at sþ- and s�-polarization, respectively, behaves quite similarly for hh-excitons inthe middle and at the edge of the window (see Fig. 2d): rcirc

increases monotonously with magnetic field from 0 up to E0.2. Butin the middle of the window the stronger sþ-component alwayshas a higher energy at spin splitting—the exciton spin system at thewindow centre is out of thermodynamic equilibrium, the degree ofnonequilibrium increasing with magnetic field. At the window edgethe more intense sþ-component has a lower energy than the s�-component, i.e. the system is more close to equilibrium. It is notsurprising because the lifetime of the indirect exciton at the windowedge was found to exceed that at the centre by at least an orderof magnitude [22]. Nevertheless, in sufficiently strong magneticfield the spin equilibrium in this region is also destroyed, as rcirc

increases with magnetic field while the value of splitting DEZ

decreases.In the absence of a magnetic field the luminescence line of

indirect excitons at the window edge is linearly polarized alongdirection /110S in the plane of quantum well {001} with thelinear polarization degree rlinE0.2, the luminescence in thewindow middle being unpolarized. In the explored range ofmagnetic field the linear polarization characteristics do not change.

4. Conclusions

Upon accumulation of spatially-indirect dipolar excitons in awide GaAs/AlGaAs quantum well inside an electrostatic trapformed by the nonhomogeneous electric field nearby the windowin the Schottky gate suppression of spin splitting has been foundin a perpendicular to quantum well magnetic field (Faradaygeometry) in the field range below BcE2 T. The effect is relatedto Zeeman splitting compensation due to the exchange interac-tion in dense degenerate gas of Bose particles with nonzero spin.Besides, at low magnetic fields, Br1.3 T, an anomalously largequadratic-in-field high-energy shift is observed, up to E2.2 meV/T2, which is connected with the presence of a considerableparallel component of the electric field in the quantum wellplane. In crossed fields, perpendicular magnetic and axiallysymmetric radial electric fields, exciton motion along the circulartrajectories around the window centre is possible which mayresult in mixing of the centrally-symmetric lowest-energy state ofthe exciton with higher states possessing larger angular moments.

Acknowledgements

We thank A.I. Il’in (IPMT-HPM RAS) for electron lithography ofthe samples. The work was supported by the Russian Foundationfor Basic Research; the Presidium of the Russian Academy ofSciences (programme ‘‘Basic Research in Nanotechnologies andNanomaterials’’); and the Division of Physical Sciences, RussianAcademy of Sciences (programme ‘‘Strongly Correlated Electronsin Solids and Solid State Structures’’).

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