Spatial Spreading of the Condensate of Magnetoexcitons in a Quantum Hall Insulator

The spatial spreading of a dense ensemble of spin cyclotron magnetoexcitons in a quantum Hall insulator at the filling factor ν = 2 is visualized using an optical system with a high aperture ratio. It is found that nondiffusive propagation over macroscopic distances is characteristic not only of excitons with a momentum on the order of the reciprocal magnetic length, which form a coherent condensate of magnetoexcitons, but also of excitons with very low momenta. The nondiffusive propagation of magnetoexciton condensates in real space is accompanied by a huge threshold increase in the amplitude of light reflection from excitations. The possible explanations of the observed behavior are discussed.

The fundamental possibility of the Bose-Einstein condensation of excitons first attracted the attention of theorists in the early 1960s [1][2][3] and became one of the most actively discussed after the pioneering works of L.V. Keldysh and A.N. Kozlov [4,5]. To attain condensation, one needs to create a dense long-lived ensemble of excitons and cool this ensemble to the temperature of the phase transition to a condensed state, as later done for systems of atoms in magnetic traps [6,7]. Obviously, this program cannot be implemented in direct-gap semiconductors because of the short radiative recombination time of excitons. For this reason, the greatest progress toward the creation of dense exciton ensembles was achieved in ultrapure indirect-gap semiconductors like Ge and Si. However, owing to the valley degeneracy in the conduction band, the phase transition to another condensed state of the exciton matter, i.e., the electron-hole liquid [8], was observed in these materials instead of Bose-Einstein condensation. All attempts to alter the situation with valley degeneracy by applying uniaxial stress or magnetic field led to only partial lifting of the degeneracy and the observation of non-Boltzmann statistics of excitons [9]. Therefore, efforts to create an exciton Bose-Einstein condensate in three-dimensional solid-state systems remained unsuccessful. With the advent of high-mobility two-dimensional (2D) systems in quantum wells (QWs) based on III-V and II-VI semiconductor materials, it became widely anticipated that novel condensed states of excitons would be found in these systems. Indeed, nonequilibrium exciton-polariton condensates were observed and investigated in QWs embedded in Bragg cavities [10], which unquestionably became a major achieve-ment in the physics of 2D excitons. The properties of exciton-polariton condensates are similar to those of laser systems owing to the significant mixing of excitons with light, and the question about the possibility of the condensation of 2D excitons themselves remains open to date. We develop a new approach to the problem of exciton condensation, proposing to condense magnetoexcitons, i.e., excitations in twodimensional electron systems (2DESs) subjected to a quantizing external magnetic field.
Magneto-optical studies of 2DESs have been actively pursued for almost half a century and the corresponding bibliography is very extensive (see, e.g., reviews [11,12]). Theoretically, the problem of a 2D Wannier-Mott exciton in a transverse magnetic field was considered quite long ago [13]. The prospects for the condensation of magnetoexcitons looked very promising: the properties of a 2D gas of such excitations turn out to be close to those of an ideal Bose gas [14]. However, the application of a magnetic field to "conventional" excitons, formed by a valence-band hole and a conduction-band electron, did not lead to condensation in experiments, even in the case of longlived spatially indirect excitons in double GaAs/AlGaAs QWs [15]. The idea of the condensation of cyclotron magnetoexcitons, in which the electron and the hole are at different Landau levels in the conduction band, was more recently proposed in [16]. The most promising in this regard turned out triplet cyclotron magnetoexcitons (TCME, or spin-flip excitons) in a quantum Hall insulator (electron filling factor ν = 2), fairly well studied both experimentally and theoretically [17,18]. They are formed by an electron

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vacancy (Fermi hole) in the completely filled zeroth electron Landau level and an excited electron with the opposite spin in the empty first Landau level. The TCME is the lowest energy excitation in the system. Furthermore, TCMEs are long-lived composite bosons with the spin S = 1 whose lifetime reaches milliseconds [19]. At temperatures T ≲ 1 K and concentrations n ex ~ 1-10% of the density of magnetic flux quanta, a magnetoexciton phase named the magnetofermionic condensate is formed in the quantum Hall insulator, which is a Fermi system [19]. This state is an experimental example of the condensation of composite bosons in the space of generalized momenta q, which are quantities that depend on both spatial coordinates and their gradients [20]. The new phase is macroscopically coherent because its response to an external electromagnetic field is at least an order of magnitude stronger than that in a rarefied exciton gas [19]. Recent interferometric studies have shown that the coherence length in this condensate is at least 10 μm [21]. A distinctive feature of this condensate is its ability to spread out of the photoexcitation region over macroscopic distances into the bulk of the quantum Hall insulator. The spread of the condensate at distances of ≈2 mm, i.e., in fact, over the entire size of the sample under study, was demonstrated in [19]. The first experiments on the visualization of this spreading [22] showed that the transport mechanism is nondiffusive. First, the range of TCME transport in the condensed state is at least three orders of magnitude longer than the diffusion length of magnetoexcitons in the gas phase. Second, the spatial profile of the condensate density in no way matches the Gaussian distribution. The velocity of the exciton motion within a wide pump spot in experiments is estimated at ≳10 3 cm/s [23]. Such a rapid spin transfer over macroscopic distances is certainly of much interest, but the physical nature of this unusual phenomenon is unclear and requires further research.
Here, we investigate the propagation of a magnetofermionic (magnetoexciton) condensate in a quantum Hall insulator using the optical visualization of the spreading pattern with high spatial resolution.
The sample under study was a high-quality heterostructure containing a single symmetrically doped GaAs/AlGaAs QW with a width of 31 nm, an electron density in the 2D channel of n e = 2 × 10 11 cm -2 , and a dark mobility of μ e = 1.5 × 10 7 cm 2 /(V s). The sample with a size of ~3 × 3 mm was mounted in an insert with liquid 3 He equipped with an optical window, which, in turn, was placed in a 4 He cryostat with a superconducting solenoid. The experiments were carried out in the temperature range from 0.55 to 1.5 K in magnetic fields up to 6 T perpendicular to the QW plane.
A single-mode laser diode with a wavelength of about 780 nm was used as an optical source for the formation of an ensemble of nonequilibrium cyclotron magnetoexcitons and the excitation of photoluminescence (PL), and Toptica DL Pro tunable CW semiconductor laser with a linewidth of 1 MHz was used to monitor resonant reflection. The high spatial coherence of the latter makes it difficult to observe the image of the sample in reflected light because of parasitic interference and a speckle structure. To reduce the degree of coherence, the beam of the probe laser was focused on a rotating frosted-glass plate in a spot that was then imaged into the cryostat. A high-aperture-ratio two-lens imaging system mounted inside the 3 He insert was used to focus light from the pump and probe lasers on the sample surface. A detailed description of the system is given in [22,24]. The size of the pump spot on the sample could be as small as ≈5 μm. The size of the probe spot was selected to uniformly fill the field of view of the optical system (≈190 μm). The same pair of lenses was used to collect both resonantly reflected laser radiation and PL radiation and to guide it out of the cryostat as a collimated beam. A ×30 magnified image of the sample was formed by a long-focal-length objective at the entrance slit of a grating spectrometer equipped with a cooled CCD camera. The signal caused by reflection from the sample surface was suppressed using a pair of crossed linear polarizers placed outside the cryostat, one at the input in the probe laser beam and the other at the output in the reflected beam. Radiation exiting the cryostat was passed through an interference filter with a 10-nm passband centered at 820 nm to cut off pump laser light. It should be stressed that the measurement setup is very sensitive to the accuracy of focusing at the surface of the sample immersed in liquid 3 He. A special positioning unit was designed to ensure smooth translation of the stage with the sample along the optical axis. With the optimum adjustment, it was possible to approach the theoretically expected spatial resolution of ≈1 μm (see [21]).
The spin-flip exciton in a quantum Hall insulator (ν = 2) was observed for the first time using inelastic light scattering [17,25]. Perhaps it would also be possible to identify its presence in the absorption spectrum by fabricating a structure with a Bragg mirror located behind the QW, as done in [26]. Both methods are technically challenging. In our experiments, the main method of detecting TCMEs is photoinduced resonant reflection (PRR), where the reflection coefficient of light with a wavelength corresponding to the 0-0 optical transition between the zeroth Landau levels of heavy holes in the valence band and electrons in the conduction band changes when the pump is turned on [27]. This method detects the presence of photoexcited Fermi holes that are constituents of cyclotron magnetoexcitons (triplet magnetoexcitons themselves are "dark" quasiparticles that, in the dipole approximation, do not interact with an electromagnetic field). At the same time, PRR offers no information about the momentum q of magnetoexcitons containing these Fermi holes. Meanwhile, the minimum of the TCME dispersion curve appears at the inverse magnetic length q min ≈ 1/l B rather than at q = 0 [28]. In a field of 4 T, the magnetic length ≈ 10 -6 cm, so that a fairly high momentum has to be given away upon relaxation. It turns out that, in addition to PRR, one needs to simultaneously record the PL spectra of the 2DES: these measurements provide information on the momentum distribution function of magnetoexcitons. When nonequilibrium excitations appear in the system, features associated with translationally invariant three-particle complexes constructed from a dark triplet magnetoexciton and an extra Fermi hole are seen in the PL spectra [29,30]. There are two types of such complexes. If two Fermi holes have the same spin projections along the magnetic field, they form a spin triplet; if the spin projections of the holes are opposite, a spin singlet is formed. The hole-spin-triplet three-particle state is a trion. An electron bound in a trion cannot be involved in plasma oscillations. The trion energy carries no information about the momentum of the constituent magnetoexciton. In fact, the intensity of the trion line reflects the total density of magnetoexcitons. The hole-spin-singlet state is a plasmaron, since the photoexcited electron from a triplet magnetoexciton can recombine with one of the Fermi holes, transferring the energy and momentum to a new electron-hole pair (plasma oscillation). A plasmaron may be considered as a magnetoplasmon bound to an extra Fermi hole. In contrast to trions, the plasmaron PL spectrum carries information about both the total number of magnetoexcitons and the energy distribution function of plasmarons. From the latter, in turn, we can find the distribution function of magnetoexcitons over the momenta q they had at the time of the formation of the plasmaron containing them [30]. It was shown that complete thermalization does not occur in a rarefied gas of triplet magnetoexcitons because the conditions of energy and momentum conservation cannot be satisfied simultaneously [31]. Optical pumping generates nonequilibrium TCMEs with momentum q . 0, and relaxation to the lowest energy state becomes possible only via exciton-exciton scattering upon reaching a certain critical exciton density [32]. As a result, the TCME ensemble consists of magnetoexcitons with nearly-zero momenta and magnetoexcitons at the energy minimum with momenta on the order of the inverse magnetic length. It is proved experimentally [31] that the rapid transfer of the exciton density over long distances is carried out only by TCMEs occupying the energy minimum near momenta q min ~ 1/l B . Thus, an intense plasmaron band in the PL spectrum with a pronounced maximum in the region of q min is a signature of the formation of a magnetoexciton condensate.
The use of a high-resolution imaging optical system with precise sharpness adjustment in our experiments made it possible to observe qualitatively new = / B l c eB features in the spreading of the magnetoexciton condensate. The profile of the laser spot was well described by a Gaussian distribution with a width of 10 μm at half-maximum and 30 μm at the base. The size of the photoexcitation region coincides with the PL spot, and the distribution of the PL intensity corresponds to that of the laser intensity. The usual procedure for tuning to the resonance with the 0-0 optical transition in PRR measurements is to find the probe laser wavelength λ max where the spectrally recorded signal of the intensity of the laser line in the beam reflected from the sample is the largest. Figures 1a-1d demonstrate how the spatial distribution of the PRR intensity changes in this case with the optical pump power P pump . At the lowest powers, a round inhomogeneous ring with a diameter of about 30 μm is observed. The brightness of the ring increases with the power; then, it turns into a disk whose diameter increases gradually by a factor of 1.5-2. The spot itself does not look completely homogeneous, with a structure of thin dark lines discernible within the spot. The spatial distribution of the PRR signal changes dramatically when the probe laser is detuned to a wavelength λ 2 that is 0.2-0.3 nm shorter than λ max (Figs. 1e-1h).
While there are no qualitative differences at the weakest pumps, a much wider region flares up with an increase in P pump , the bright area being crossed by a grid of thin lines perpendicular to each other. The size of this region increases rapidly with the pump until it fills the entire field of view.
The TCME spreading patterns displayed in Fig. 1 can be explained by recalling the results of [31]. It was shown that the TCME ensemble consists of magnetoexcitons with nearly zero momenta and magnetoexcitons at the energy minimum with momenta on the order of the inverse magnetic length, and the rapid transfer of exciton density over long distances is carried out only by the latter. Evidently, PRR measurements at the wavelength λ 2 probe TCMEs with momenta q min ~ 1/l B , whereas measurements at wavelength λ max probe TCMEs with momenta q . 0. The former demonstrate the ability to spread over a hundred microns or more in accordance with the earlier results [19,22]. However, the PRR signal distributions shown in Fig. 1 indicate that TCMEs with low momenta also propagate in the sample in a quite nontrivial fashion. The spatial distribution of TCMEs with low momenta has nothing to do with diffusion. In fact, the TCME density is described by a step function, and the diameter of the plateau is two orders of magnitude greater than the mean free path of a single TCME in a rarefied exciton gas [19]. Thus, it can be assumed that the spreading of excitons with low momenta in a dense ensemble of TCMEs has a collective character as well. This observation qualitatively confirms the theoretical statement made in [33] that two types of condensed states should exist for ν = 2: one is the state formed by TCMEs with q = 0 and the other is the condensate of magnetoexcitons with q = q min ~ 1/l B .   Figures 2a and 2b show the pump-power dependences of the spectrally recorded PRR signal and the area of the PRR region determined by visualization when the probe laser is tuned to λ max and λ 2 , respectively. All these dependences are qualitatively similar and demonstrate a threshold behavior. At the wavelength λ max , the threshold is about 1 μW. Below the threshold, the spectral intensity increases slowly (sublinearly); in the range from 0.6 to 1.5 μW, the signal increases approximately sevenfold, and then remains unchanged. The area of the PRR spot is almost constant both below and above the threshold, and the increase at the threshold is approximately a factor of 3. At the wavelength λ 2 , the threshold occurs at 0.4 μW, and both the spectral signal and the area increase by a factor of almost 20. Evidently, the area of the PRR spot at the wavelength λ 2 begins to exceed the size of the field of view at a low pump power of 1.5 μW, and, for this reason, both the intensity of the PRR signal and the spreading area level off at higher powers.
The dependences shown in Fig. 2b demonstrate that magnetoexcitons condensed at the energy minimum near q min ~ 1/l B begin to spread over macroscopic distances right above the condensation threshold and flee still further with a further increase in the pump power (i.e., with an increase in the exciton density in the photoexcitation spot). The PRR spectral signal increases concurrently with the area of the spreading spot; i.e., in the first approximation, the concentration of these excitations remains constant. The condensation of TCMEs with momenta q ≈ 0 occurs at about two times higher pump power. The PRR spot increases stepwise by a factor of 1.5-2 and ceases to grow further. The intensity of the PRR signal behaves in a similar way, but increases significantly stronger at the threshold. It follows that the density of magnetoexcitons inside the spot increases in this case by a factor of 2-3.
To gain additional information, we measured PL spectra. In these measurements, the entrance slit of the spectrometer defines a narrow vertical strip passing through the center of the photoexcitation region in the images shown in Fig. 1. Figure 3 shows a series of PL spectra recorded for pump powers increasing from 0.1 to 10 μW. At the lowest powers, the spectrum features the usual single-particle doublet in σ + and σcircular polarizations. However, beginning with a low pump power of 0.3 μW, a shoulder associated with the formation of trions (T) appears in the σ + polarization, and a weak plasmaron band (Pln) shifted to lower energies by 2 meV from the single-particle transition line is observed in the σpolarization. The intensity of the plasmaron band becomes comparable to that of the σsingle-particle line at pump powers of about 1 μW, and the plasmaron becomes more intense at 1.5 μW. A further increase in P pump leads to the gradual weakening and broadening of the plasmaron band. It almost vanishes at 10 μW, whereas the σsingleparticle line reappears.
Therefore, the condensate of magnetoexcitons that is formed near q min ~ 1/l B begins to spread to tens of microns at low photoexcitation power densities ≲0.5 W/cm 2 . As the pump increases, the spreading spot grows rapidly and reaches the size of the field of view for 1.5 μW. The accompanying increase in the spectral intensity of the PRR signal is directly related to the increase in the area of the spreading spot. The PRR signal corresponding to TCMEs with low momenta also exhibits a threshold behavior versus the pump power. The onset of the plateau in the PRR signal at pump powers above ~1.5 μW correlates with the maximum plasmaron band intensity in the PL spectrum. Beginning with this photoexcitation power, only a certain fixed fraction of TCMEs remains in the region of q . 0 in the momentum space and occupies a spot with a diameter of about 50 μm in real space, and all other triplet magnetoexcitons fall to the energy minimum, where they are condensed and, therefore, rapidly flow out of the field of view. Now, the main question is why an increase in the pump power by a factor of 2-3 in the threshold region leads to an increase in the reflection signal by a factor of 7 or even 20. Figure 2c shows the dependence of the integrated PL intensity on the pump power. This dependence is very close to linear and no jumps are observed. This means that the concentration of photoexcited TCMEs should also increase linearly with the pump power. One of the explanations, initially suggested in [19] and later developed in theoretical papers [33,34], is that, in the presence of a random potential in the system, it is the transition from an incoherent magnetoexciton gas to a coherent condensate phase that should lead to a significant increase in optical absorption. It may also be suggested that the observed strong stepwise increase in the PRR signal is caused by the angular redistribution of reflected light. For example, reflection may become concentrated near the normal to the surface of the sample. Such effects are usually associated with the emergence of spatial coherence in the system; see, e.g., theoretical and experimental results in [35,36]. Recently, the occurrence of high spatial coherence in the magnetoexciton condensate under study was confirmed experimentally [21], but additional studies are evidently required to verify this assumption.