Explosive Bifurcation to Chaos in Weakly Coupled Semiconductor Superlattices

K. J. Luo, H. T. Grahn, K. H. Ploog, L. L. Bonilla
1998 Physical Review Letters  
The bifurcation scenario to chaos has been studied for vertical transport in an incommensurately driven superlattice system. With increasing driving amplitude, quasiperiodic, frequency-locked, and chaotic oscillations are identified by using Poincaré maps, which show a variety of attractors in the chaotic regime. The dimension of the attractor is abruptly increased in the transition process, i.e., the bifurcation between frequency locking and chaos is explosive. However, the bifurcation pattern
more » ... bifurcation pattern depends strongly on the applied dc bias, providing clear evidence that the system is spatially inhomogeneous in the vertical direction. [S0031-9007(98)06837-9] PACS numbers: 73.50.Fq, 73.20.Dx, 73.40.Gk Spatiotemporal chaos has been studied both experimentally and theoretically in liquid and chemical systems [1], in coupled map lattices [2], as well as in solid-state systems [3] [4] [5] [6] [7] . When the investigated system is driven by an incommensurate frequency, i.e., the ratio of the natural frequency and the driving frequency is an irrational number, the transition to chaos has been predicted theoretically to occur via the following routes: either quasiperiodicity ! frequency locking ! chaos or directly from quasiperiodicity to chaos [2] [3] [4] [5] . Both routes to chaos have been observed in a number of experiments [3] [4] [5] . Vertical transport in weakly coupled semiconductor superlattices (SL's) is known to exhibit nonlinear phenomena such as domain formation [8-10], multistability [11] , and self-sustained current oscillations [12] [13] [14] . It has been shown that the self-sustained current oscillations are due to a quasi-one-dimensional motion of the domain boundary in the SL direction, i.e., vertical to the SL layers [8, [12] [13] [14] [15] . Therefore, the temporal behavior of the current oscillations is directly related to the vertical spatiotemporal motion of the domain boundary inside the SL. Theoretical studies predict the appearance of chaos in such SL systems accompanied by the breakdown of spatiotemporal coherence of the motion of the domain boundary [15] . Although driven and undriven chaos of these current oscillations have been recently observed in the frequency power spectra [16] , little is known about the transition process between synchronization (frequency locking) and chaos as well as the actual type of chaotic behavior in the experimentally investigated SL system. This type of information can be obtained only from real-time measurements. In this paper, we present Poincaré maps in the presence of an external driving voltage applied parallel to the growth direction of a weakly coupled semiconductor SL. These Poincaré maps clearly indicate that the transition from frequency locking to chaos is accompanied by a loss of spatiotemporal coherence. Furthermore, the Poincaré maps reveal that a number of attractors with varying complexity exist in the chaotic regime. However, for different dc biases, completely different routes to chaos are observed. At the same time, the observed bifurcation patterns are more complicated than the ones predicted by the theoretical investigation reported in Ref. [15] . The experimental discrepancy with the theory as well as the different observed routes to chaos may be due to the spatial inhomogeneity of the SL in the growth direction. The investigated sample consists of a doped, 40-period, weakly coupled SL with 9.0 nm GaAs wells and 4.0 nm AlAs barriers. For a more detailed description of the structure and contact layers, see Refs. [16, 17] . The dc and ac electric fields are applied in parallel to the SL direction. The experimental data are recorded in a He-flow cryostat at 5 K using high-frequency coaxial cables with a bandwidth of 20 GHz. The driving voltage in the form of V ac sin͑2pf d t͒ is generated with a Wavetek 50 MHz pulse/function generator (model 81), where f d denotes the driving frequency. The power spectra of the current oscillations are detected with an Advantest R3361 spectrum analyzer. The real-time current traces are recorded with a Hewlett-Packard 54720A digital oscilloscope, which is triggered by the synchronization signal from the pulse/ function generator, using a sampling rate of 1 GSa͞s and 32 768 points/snapshot. The resulting time resolution was about 20-50 points per period t d of the driving frequency, which corresponds to about 600-1600 t d per snapshot. The dc bias V dc is fixed at two different voltages in the second plateau of the time-averaged current-voltage characteristic, where current oscillations due to a recycling motion of a charge monopole have been observed [14] . Without any ac driving voltage, the system exhibits periodic self-sustained current oscillations with an intrinsic fundamental frequency f 0 f i ͑V ac 0͒ of 30.5 MHz at V dc 6.574 V and 11.4 MHz at 7.080 V. In the following, we will fix f d at f 0 3 ͑1 1 p 5 ͒͞2 (the golden mean) and vary V ac to study the bifurcation scenario to chaos. Figure 1 shows the bifurcation diagrams of the 1290 0031-9007͞98͞81(6)͞1290(4)$15.00
doi:10.1103/physrevlett.81.1290 fatcat:lq2ih3b7ojf37c6zca4udw32a4