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 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 complicat...

Static domain formation in doped semiconductor superlattices results in many branches in the currentvoltage characteristic separated by a discontinuity in the current. The transition process from one branch to the next has been studied experimentally by adding an ac bias with different amplitudes to a dc bias close to a discontinuity and recording the time-resolved current. The relocation time of the domain boundary depends exponentially on the difference between the final static current and the maximum or minimum current value of the corresponding branch, which is reached before the relocation of the domain boundary takes place. A universal relationship between the relocation time and the current difference has been observed. ͓S0163-1829͑98͒50812-0͔The current-voltage (I-V) characteristic of doped, weakly coupled superlattices ͑SL's͒ exhibits-under formation of static electric-field domains-many sharp branches, which are separated by a discontinuity in the current. Under domain formation, the electric field in the SL's breaks up into two regions of constant field, which are separated by a domain boundary. The domain boundary is formed by a charge accumulation layer, which is confined to one or several SL periods, i.e., one or several quantum wells of the SL. When the applied bias voltage is swept from one current branch to the next across a discontinuity, the domain boundary moves exactly by one SL period. 1 Although there has been previously a large amount of investigations on static domain formation, 2-7 dynamical processes such as the domain formation time have only been studied recently. 8,9 However, the actual motion of the domain boundary, which occurs during a current jump from one branch to the next, remains unclear.In this paper, we determine the relocation time of the domain boundary in a weakly coupled SL by fixing the dc bias (V dc ) near a discontinuity of the I-V characteristic and adding an ac square pulse voltage with different amplitudes (V ac ). The transient behavior of the current is measured as a function of V ac . When the total applied bias sweeps across a current jump, e.g., from one branch to the next, the current response exhibits a delay, which becomes faster with increasing V ac . The delay time depends exponentially on the difference between the final stabilized current and the maximum or minimum current of the initial current branch. A universal relationship between the decay time and the current difference has been observed for all cases.The investigated sample consists of a 40-period, weakly coupled SL with 9.0 nm GaAs wells and 4.0 nm AlAs barriers grown by molecular beam epitaxy on a ͑100͒ n ϩ -GaAs substrate. The central 5 nm of each well are n doped with Si at 3.0ϫ10 17 cm Ϫ3 . The SL is sandwiched between two highly Si-doped Al x Ga 1Ϫx As contact layers forming an n ϩ -n-n ϩ diode. The sample is etched to yield mesas with a diameter of 120 m. 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 ti...

Topological nodal-line semimetals are predicted to exhibit unique drumhead-like surface states (DSS). Yet, a direct detection of such states remains a challenge. Here, we propose spin-resolved transport in a junction between a normal metal and a spin-orbit coupled nodal-line semimetal as the mechanism for their detection. Specifically, we find that in such an interface, the DSS induce resonant spin-flipped reflection. This effect can be probed by both vertical spin transport and lateral charge transport between anti-parallel magnetic terminals. In the tunneling limit of the junction, both spin and charge conductances exhibit a resonant peak around zero energy, providing a unique evidence of the DSS. This signature is robust to both dispersive-DSS and interface disorder. Based on numerical calculations, we show that the scheme can be implemented in the topological semimetal HgCr2Se4.

The multifractal dimension of chaotic attractors has been studied in a weakly coupled superlattice driven by an incommensurate sinusoidal voltage as a function of the driving voltage amplitude. The derived multifractal dimension for the observed bifurcation sequence shows different characteristics for chaotic, quasiperiodic, and frequency-locked attractors. In the chaotic regime, strange attractors are observed. Even in the quasiperiodic regime, attractors with a certain degree of strangeness may exist. From the observed multifractal dimensions, the deterministic nature of the chaotic oscillations is clearly identified. ͓S0163-1829͑99͒03032-5͔

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