Abstract
A two-dimensional (2D) electron system in a perpendicular magnetic field acts like a macroscopic strongly correlated source of radiation with inherent temporal and spatial correlations. The formation of the correlated macroscopic state of 2D electrons passes through uniform and nonuniform states of the electron subsystem. These are reflected in two types of noise of photoresponse in the 2D system. The strange attractor characterized by small-sized dynamics in the phase space of a photoluminescence of 2D electrons is revealed.
Keywords
- giant optical fluctuations
- highly ordered correlated state of electrons
- nonlinear dynamics
- quantum Hall effect
- strange attractor
1. Introduction
The formation of a highly ordered correlated state in low-dimensional electron systems is well known to produce fluctuations of their physical parameters (a potential fluctuation, spin and charge fluctuations, phase fluctuations of wave functions, etc.) [1]. The fluctuation character of various kinds of phenomena can be implicitly indicative of their fundamental nature. In particular, the studies of quantum shot noise made it possible to reveal the fractional electron effective charge [2,3]. The strong correlation effects may occur in a quasi-two-dimensional (2D) electron system in a transverse magnetic field, such as the fractional quantum Hall effect (FQHE) [4] and the pinned Wigner solid [5]. However, to date, quantum-sized electron systems have not been investigated in terms of the fluctuations in their optical response. In that context, the optical methods could provide information that is beyond the reach of the magnetoresistance studies. These are the single-particle density of states immediately below the Fermi level, single-particle energy gaps, some information on a random potential (the amplitude and characteristic length scales), interference-enhanced effects, polarization correlations, coherence time, and so on. It is also important to emphasize here that quantum localization effects are not the strong restriction for the optical technique in the QHE regime [6].
In a series of articles [7–17], we reported that the giant optical fluctuations (GOFs) occur in a photoexcited 2D electron system (GaAs/AlGaAs quantum wells) in the QHE regime. Initially, it was revealed that the rated value of a variance of an intensity of radiative recombination of 2D electrons with photoexcited holes increases by several orders of magnitude for certain values of the Landau-level filling factor
Fluctuations of optical signals are considered traditionally in the theory of optical coherence [18]. It is thus considered that the source of radiation is in a stationary state close to thermodynamic equilibrium and only field fluctuations are taken into account. However, besides optical coherence, intrinsic times of quantum dynamics of an electron subsystem can be perceptible in an optical signal. It seems that the consideration of source fluctuations under thermodynamically nonequilibrium conditions is a reasonable generalization of the existing theory. In particular, the source can be in a deterministic chaos regime, as it must occur at critical points and phase transitions. QHE is the macroscopic quantum effect that is manifested in the quantization of the Hall resistance
2. GOF detection under QHE conditions: sample structure and initial experimental conditions
High-quality samples containing a 250-Å-wide GaAs/Al0.3Ga0.7As quantum well were used as a standard object where the effect of GOF is observed [7]. Figure 1 shows the sample structure in which the radiative recombination of 2D electrons with photoexcited holes was studied.
The samples were grown by molecular beam epitaxy on a GaAs substrate by the following scheme: GaAs buffer layer 3000 Å thick, undoped GaAs/AlGaAs (30/100 Å) superlattice 13,000 Å in total thickness, GaAs quantum well 250 Å thick, AlGaAs spacer 400 Å thick, doped AlGaAs:Si layer (doping level, 1018 cm–3) 650 Å thick, and GaAs cap layer 100 Å thick. The characteristic mobility of 2D electrons and the concentration in these structures were
Figure 2a illustrates a simple method to find out the location of the required filling factor
Thus, this distribution of photon counts differs substantially from the Poisson distribution and the magnitude of fluctuations has a maximum in the vicinity of the filling factor
It should be noted here that such a regime occurs also for
3. Macroscopic character of giant fluctuations of 2D electrons: a multifiber scheme technique
We applied the correlation analysis to the study of formation and characteristic aspects of the revealed macroscopic state in a 2D electron system. Correlation functions are quite often used in the analysis of noise due to the ease of interpretation. Correlation spectroscopy was for the first time applied to study of intensity fluctuations by Hanbury Brown and Tviss [20]. Subsequently, it was gradually resulting in a situation where the correlation spectroscopy became the traditional tool of quantum optics. However, as mentioned above, this method had rarely been used for studying fluctuations near critical points or phase transitions. Meanwhile, the correlation spectroscopy could be used successfully for spatial and temporal analysis of optical fluctuations of complex systems. In a critical point, a characteristic range of correlations (correlation radius) significantly exceeds the interparticle distance and a system is becoming susceptible to changes of local concentrations [21]. If the photoexcited 2D electron system is in the quasi-equilibrium under IQHE conditions, then the intensity of recombination radiation consists of a convolution of 2D electron and photoexcited hole distribution functions [6]:
The width of the experimentally measured hole distribution function
In steady-state conditions, time dependence of the radiation intensity represents a realization of a stationary random process. The intensity correlation functions depend only on the time difference. The information on the concentration redistribution is contained in correlation functions of intensity fluctuations. In analyzing the autocorrelation function, it is possible to study periodic processes to find the characteristic time of a relaxation and dynamics of attractors. Cross-correlation function can provide information on spatial correlations in the system of interacting carriers.
We used special multifiber optical scheme to record and analyze the spatial correlation between the radiation intensities of PL signals from different points of a large sample (
where
where |
With further increase in magnetic field (
The range of an intensity noise covers a much wider range of a magnetic field (~2 T; see Figure 6) than it occurs in the case of a small spot of photoexcitation (0.005–0.01 T). We decided to analyze the noise region in terms of possible regularities in its dynamics and turned to the analysis of autocorrelation and cross-correlation functions of 2D PL signals in this region.
It was revealed that time dependences of the PL autocorrelation
Here,
Here, it must be taken into consideration that damped autocorrelation function is one of the criteria for the strange attractor of a dynamical system (see the next section). Figure 7a shows these correlations. Practically complete coincidence of the functions
where ∆
Thus, cross-correlation functions are sensitive to this difference in phase of stationary waves in corresponding points of the sample (Figure 7b). A phase difference indicates that the phase is not a random variable, but it has a fixed value, unalterable by the switching. It appears that a 2D PL intensity is modulated by “a standing wave”, whose phase is defined by experimental conditions. An example of 2D electron systems in which a standing wave can be generated is plasmon excitation in the microwave (MW) field [ [24]
4. Phase space portrait of the GOFs: beginning of the instability in the 2D system in a vicinity of ν = 2; overview of the GOF effect and its possible mechanisms
It is well known that the comprehensive analysis of the time-series data of fluctuating signals may give useful information on the evolution of many kinds of dynamical systems. The study of GOF time dependences, using this idea, showed that there are complex regimes of a motion in our 2D electron system in the vicinity of
In particular, it was natural to expect a specific dynamics of PL intensity fluctuations with the presence of a modulating standing wave in a point of uniformity of the electron density. The phase portrait of a possible strange attractor of dissipative dynamical systems can be received using the methods described in the works [25–27].
It was shown in [26] that a time series of measurements of a single observable
Sufficiently long (1–3 h) time sequences
Here,
As an autocorrelation function of a periodic function is a function with the same period, the characteristic time
This time may be called an electronic “coherence” time. It should be noted that optical coherence time in the system studied is very short and it is defined by a spectral line width (
Thus, there is a mechanism that makes 2D electron concentration uniformly distributed across the all illuminated surface (
One possible scenario of this specific ordered 2D electron state formation under QHE conditions is the phase transition in the 2D electron system. A phenomenological model, describing the correlation functions of 2D PL signals, has been developed based on experimental data [17].
It gives the explanation for various phase conditions of a 2D electron system in a vicinity of the filling factor
Under QHE conditions, small electron-density fluctuations can give rise to giant fluctuations of the conductivity of the 2D electron system and lead to fluctuation metal-insulator transitions. Another possibility involves the occurrence of a new coherent macroscopic state of the electron system described by a common wave function with a unified phase similar to superfluidity. Such coherent ordered electron states could occur due to the quantum leakage, such as the steady-state Josephson effect.
5. Summary
Thus, a system of 2D electrons in a perpendicular magnetic field in a vicinity of the filling factor
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