ISSN 10637761, Journal of Experimental and Theoretical Physics, 2014, Vol. 119, No. 1, pp. 115123. Pleiades Publishing, Inc., 2014.Original Russian Text A.V. Gorbunov, V.B. Timofeev, 2014, published in Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, 2014, Vol. 146, No. 1, pp. 133143.
Recently, we have found  that Zeeman splittingcompensation is observed in a potential trap near thewindow in a Schottky gate upon the accumulation of2D spatially indirect dipolar excitons (a 25nmwideGaAs/AlGaAs quantum well) at magnetic fields belowsome critical value, Bcr 2 T. The spin splitting suppression effect was predicted theoretically for a thermodynamically equilibrium Bose condensate of 2Dexciton polaritons in an optical microcavity at zerotemperature . This effect is related to the spinornature of intracavity polaritons. They have the spinS = 1 with two allowed spin projections onto the structure growth axis, Sz = 1, corresponding to two opposite directions of the circular polarization, . Theexchange interaction in a spinor Bose condensate isassumed to be so arranged that the coparallel spinsare repelled, while the antiparallel ones are attractedor repelled, but more weakly. As a result, in theabsence of a magnetic field, the situation where thenumber of spins with the projections Sz = +1 and Sz =1 is the same is energetically favorable. This corresponds to a linear polarization of the emitted luminescence light. A sharp increase in the degree of linearpolarization is actually observed experimentally whenthe threshold of Bose condensation is exceeded in particle concentration (see, e.g., ). Obviously, all spinsin a sufficiently strong perpendicular magnetic field(Faraday geometry) will be aligned with the field andwill fill the lower Zeeman sublevel. However, in weakmagnetic fields, as was shown in , no spin splitting(Zeeman effect) must be observed, because the redshift, a decrease in energy due to the filling of the
lower spin sublevel, is exactly compensated by theblue shift, an increase in energy due to the mutualrepulsion between the parallel spins filling the lowersublevel. Compensation takes place until the magneticfield exceeds its critical value, Bcr = 2nU1/gB, where nis the particle concentration, g is the gfactor, B is theBohr magneton, and U1 is a phenomenological coefficient that describes the linearcircular dichroism ofthe Bose condensate and that is proportional to thedifference of the interaction coefficients of the paralleland antiparallel spins . In the range 0 < B < Bcr, thepolarization is elliptical and gradually approaches thecircular one with growing field. At B Bcr, the polarization is purely circular and Zeeman splitting proportional to (B Bcr) is observed.
Despite the fact that the system of exciton polaritons in an optical microcavity is obviously not an equilibrium one (the lifetime of an intracavity polariton istypically a few picoseconds), the phenomenon of spinsplitting compensation has been detected experimentally  for an excitonpolariton condensate in aGaAs microcavity under quasistationary nonresonant photoexcitation conditions at magnetic fields B 1.7 T. There was no Zeeman splitting with an accuracyof 5 eV and an elliptical polarization was actuallyobserved. However, the sign of the circular polarization was negative up to 3 T, i.e., there was condensation not to the lower Zeeman sublevel but to the upperone, confirming that this system is a nonequilibriumone. In this situation, the applicability of the equilibrium model  raises serious doubts.
Zeeman splitting suppression for intracavitypolaritons has also been observed under resonant pho
ELECTRONIC PROPERTIESOF SOLID
Dipolar Excitons in a Potential Trap in a Magnetic FieldA. V. Gorbunov* and V. B. Timofeev
Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia*email: firstname.lastname@example.org
Received December 18, 2013
AbstractThe conditions for observing the Zeeman spin splitting compensation in an exciton Bose gas havebeen investigated. The magnetoluminescence of spatially indirect, dipolar excitons in a 25nmwideGaAs/AlGaAs quantum well upon their accumulation in a lateral electrostatic trap has been studied in theFaraday geometry. The critical magnetic field Bcr below which the spin (paramagnetic) splitting of the luminescence line for a heavyhole exciton at the trap center is almost completely compensated due to theexchange interaction in a dense Bose gas has been found to increase linearly with exciton concentration inqualitative agreement with the theory. Using a potential trap is fundamentally important. Incomplete compensation is observed in a homogeneous photoexcitation spot for dipolar excitons: the splitting is considerably smaller than that for a spatially direct exciton but differs noticeably from zero. The spin splitting compensation effect is observed only under neutral charge balance conditionsthere is no Zeeman splitting suppression in a charged quantum well.
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toexcitation conditions (the regime of an optical parametric oscillator), in a definitely nonequilibriumpolariton condensate . However, to describe thisphenomenon, the authors had to develop a completelydifferent physical model suggesting the simultaneousexistence of two nonequilibrium condensates withopposite directions of the circular polarization coherently coupled through the spinflip polaritonpolariton scattering processes.
Subsequently, the experiments with a nonequilibrium polariton condensate  were explained by themechanism of optical orientation of polariton spins ina magnetic field . In this case, in addition to theapplied external field, the following was taken intoaccount: (a) the internal field arising from the anisotropy of the electronhole exchange interaction andleading to the original linear polarization and (b) theeffective internal field associated with the excitonexciton exchange interaction. The precession of thepolariton spins around the direction of the total magnetic field leads to their optical orientation along thisdirection. As a result, the dependences of the linearand circular polarizations as well as the Zeeman splitting on external magnetic field turn out to be qualitatively similar to those derived in .
In contrast to an exciton polariton, the lowest (inenergy) heavyhole (hh) exciton in GaAs is not aspinor: in addition to the optically active brightexciton with the spin projections Sz = 1, there is adark exciton uncoupled to light for which Sz = 2.According to some theoretical calculations (see, e.g.,), the groundstate energy for the dark exciton mustbe slightly lower than that for the bright one. Therefore, precisely the dark excitons must primarily condense with decreasing temperature. The properties ofa 4component exciton Bose condensate in a magnetic field have recently been analyzed theoretically in, where, in particular, the possibility of phase transitions in a magnetic field between the states of a condensate with different numbers of components waspredicted. Thus, it could be expected a priori that theexperimentally derived behavior of the spin polarization for an exciton condensate in a magnetic fieldwould be more complex than that for exciton polaritons. Therefore, the experimental detection of theZeeman splitting compensation effect for dipolarexcitons in  turned out to be quite an unexpectedevent. It should be noted that, despite its much longerlifetime (~1 ns), the investigated system of dipolarexcitons also turned out to be far from equilibrium inspin degrees of freedom: just as for exciton polaritons, the upper Zeeman sublevel exhibited a higherpopulation than the lower one . Obviously, the phenomenon of spin splitting compensation can hardly bedescribed by a simple model for an equilibrium system in the case of dipolar excitons as well.
This paper is devoted to clarifying the conditionsunder which there is spin splitting compensation for aBose gas of dipolar excitons in a wide single quantum
well. For this purpose, we investigated, in particular,the dependences on exciton concentration and oncharge balance in the quantum well. We found that thecritical magnetic field Bcr actually increases linearlywith exciton density. At the same time, the breakdownof electrical neutrality in the quantum well affects radically the Zeeman splitting: its behavior in a magneticfield becomes more complex, while the compensationeffect vanishes. We made a comparison with thebehavior of the spin polarization in a magnetic field forboth a spatially direct exciton in the absence of anelectric field and a spatially indirect exciton without alateral potential trap. The spin splitting of the directexciton line behaves in complete agreement with theavailable data for wide GaAs/AlGaAs quantum wells. Incomplete compensation is observed fordipolar excitons without a potential trap: the splittingis considerably smaller than that for a direct excitonbut differs noticeably from zero. The previouslydetected giant blue shift of the dipolarexcitonenergy in weak magnetic fields  is due to the presence of uncompensated charges in the potentialtrapthe shift increases with their concentration.
2. THE EXPERIMENTAL TECHNIQUE
We investigated spatially indirect dipolar excitonsin a wide (25 nm) single GaAs/AlGaAs quantum wellplaced in an electric field transverse to the heterolayers. An external voltage U was applied to the upperelectrode, a Schottky gate at the heterostructure surface, relative to the lower builtin electrode, a conducting electron channel in a doped quantum wellinside the structure. The distance between the electrodes was d 265 nm, and the builtin negativepotential due to the Schottky barrier was determinedfrom the Schottky diode opening voltage (the flatband regime): U0 +500 mV. Thus, the electric fieldstrength inside the available parallelplate capacitorwas estimated from the relation F = (U U0)/d andreached 10 kV cm1, for example, at voltage U =+235 mV. Owing to the applied electric field, thedipolar excitons have a large dipole moment in theground state (over 100 D). In the system under study,such excitons do not bind into molecules or othermultiparticle complexes due to the dipoledipolerepulsion.
The photoexcitation of excitons and the observation of their luminescence were done through a circular window 7 m in diameter in the opaque metal layerof the Schottky gate (100nmthick Au/Cr film). Inthe absence of a magnetic field, dipolar excitons wereaccumulated in a ringshaped lateral trap thatappeared along the window perimeter because of thehighly nonuniform electric field [12, 13]. As wasshown in , an appreciable number of indirect excitons in a perpendicular magnetic field (Faraday geometry) are concentrated near the window center. Excitons are accumulated in such a magnetoelectrostatic
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DIPOLAR EXCITONS IN A POTENTIAL TRAP IN A MAGNETIC FIELD 117
trap in crossed radial electric and perpendicular magnetic fieldssuch a geometry of the experiment wasproposed previously in  to realize exciton Bosecondensation. Precisely these excitons at the windowcenter exhibit the spin splitting compensation effect inweak magnetic fields  and the properties of preciselythese excitons were studied in this paper.
Dipolar excitons were excited by the simultaneousaction of two continuouswave lasers with wavelengthssb = 782 nm (photoexcitation with a photon energybelow the energy gap in the AlGaAs barrier, subbarrier excitation) and ob = 659 nm (overbarrier photoexcitation). Their emission was focused on the sample into a spot ~20 m in diameter. By combining theemission from these lasers and experimentally choosing the ratio of their powers, we achieved maximumcompensation of extra charges in the trap and maintained the exciton system itself as close to neutrality aspossible. Previously, it was established that the neutrality of a photoexcited electronhole system isextremely important for the realization of excitonBose condensation [12, 13]. Therefore, special attention was given to the question of charge balance whenthe influence of the exciton density on the spin splitting compensation effect was studied. We controlledneutrality based on two spectral peculiarities: first,based on the emergence of the line of a bound(charged) exciton (trion) lower in energy approximately by 1 meV from the line of a free (neutral) hhexciton, which looks like a shoulder on the red slope ofthe main line in the case of weak deviation from neutrality; second, based on the presence of the line of alighthole (lh) exciton in the luminescence spectrum,which is weaker than the main hhexciton line by tensof times and lies higher in energy by a few millielectronvolts. As the concentration of uncompensatedcharge carriers increases in the quantum well, the lhexciton line is redshifted, closer to the hhexciton line,its intensity decreases, and, in the long run, it completely vanishes . This is because, first, the relaxation of hot lighthole excitons to the lowest statethrough their scattering by uncompensated charges ismuch faster than that on neutral excitons. Second, thepresence of free carriers leads to a screening of theCoulomb interaction and to a decrease in the excitonbinding energy .
The sample was immersed directly in liquid 4Heinside a superconducting solenoid in an optical cryostat in which experiments could be carried out in therange of magnetic fields 0 < B < 6 T at a temperatureT 1.7 K. The luminescence light was collected bymeans of a fusedsilica lens with a focal length of12.5 mm and a numerical aperture NA 0.4 located infront of the sample inside the solenoid. The magnifiedimage of the window in the Schottky gate throughwhich the photoexcitation and the luminescenceobservation was done was projected onto the entranceslit of a spectrometer (focal length 500 mm) equippedwith a cooled silicon CCD camera at the exit. In this
paper, most of the spectral measurements were madewithout a spatial resolution. The photoluminescencepolarization was analyzed by means of a Glan prismand a quarterwave phase plate.
3. EXPERIMENTAL RESULTSAND DISCUSSION
We were primarily interested in how the system ofdipolar excitons in a magnetic field behaved as theBose gas density changed. When preparing our experiments on varying the exciton concentration, we foundthe intensity of the luminescence line at the windowcenter Im directly related to this concentration todepend nontrivially on the overbarrier photoexcitationpower Pob at fixed subbarrier photoexcitation powerPsb and external voltage U applied to the sample. It canbe seen from Fig. 1 that a nonmonotonic dependenceof the line intensity Im(Pob) with its maximum near10 W (the points in this region are in...