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Random number (bit) generators are crucial to secure communications, data transfer and storage, and electronic transactions, to carry out stochastic simulations and to many other applications. As software generated random sequences are not truly random, fast entropy sources such as quantum systems or classically chaotic systems can be viable alternatives provided they generate high-quality random sequences sufficiently fast. The discovery of spontaneous chaos in semiconductor superlattices at room temperature has produced a valuable nanotechnology option. Here we explain a mathematical model to describe spontaneous chaos in semiconductor superlattices at room temperature, solve it numerically to reveal the origin and characteristics of chaotic oscillations, and discuss the limitations of the model in view of known experiments. We also explain how to extract verified random bits from the analog chaotic signal produced by the superlattice.
In this paper, we comment the possible use of spontaneously chaotic semiconductor superlattices (SLs) as true random number generators. In Section 2, we discuss the mathematical model for a single SL under voltage bias. The model consists of a number of coupled stochastic differential equations together with algebraic boundary and voltage bias conditions. In Section 3.1, numerical solutions of the model equations show that the thermal and shot noises existing in the SL enhance stable spontaneous chaos in voltage intervals where the corresponding deterministic model exhibits chaos. The noises also induce chaos in nearby voltage intervals where the deterministic system had periodic oscillations. We also discuss the relation of our results to experiments and which features of the model need to be revised in order to optimize the chaotic oscillations. In Section 3.2 and following Ref. [15], we explain how to obtain a high-speed true random bit generator by processing the chaotic current oscillations provided by the device. Section 4 summarizes our findings and perspectives for fast random bit generators based on semiconductor superlattices. Two Appendices provide details on the derivation of the model equations.
Modeling electron transport in a SL is a bit more complicated than modeling mass transport in a fluid. One first thought could be following the route from Boltzmann equation to Navier-Stokes equations, as done in the kinetic theory of gases [21], and advocated in the mathematical literature on semiconductors [22, 23]. This has achieved some success in the case of strongly coupled SLs [24, 25], but none so far for weakly coupled SLs, as those displaying spontaneous chaos in experiments [15, 20]. The main reason for this failure is that Boltzmann-type equations for SLs are based on electrons populating minibands at zero electric field, and such a picture is far from reality in the presence of electric fields that are sufficiently strong: \(eFl>\Delta\), where \(-e
Spontaneous chaos at room temperature has been observed in quite recent experiments with voltage biased, doped, weakly coupled SLs [15, 20]. Before the 2012 experiments, spontaneous chaos was observed only at very low temperatures (from 4 to 77 K) [42]. The new key modification that allows observing oscillations of the current at room temperature is adding 55% of gallium in the barriers. The technical reasons are discussed in [20] and references cited therein. Early theoretical explanations of spontaneous chaotic oscillations at low temperature are based on complex dynamics of wave front solutions [43] when applied to discrete model equations such as those of Section 2; see Ref. [44]. At room temperature, wave fronts are not sharp, and spontaneous chaos arises due to other reasons, as explained in Ref. [27] and in what follows.
In the second voltage interval, small-amplitude current oscillations appear as a supercritical Hopf bifurcation from the stationary state. These oscillations correspond to the periodic creation of field pulses (charge dipoles) that die before arriving at the collector. An abrupt drop of the mean current in Figure 2(a) and a transition to an oscillation of richer harmonic content in the spectrum of Figure 2(b) mark the voltage beyond which the field pulses move throughout the whole SL as in Figure 1(c), and produce a large-amplitude current oscillation. The extra frequency appearing at the transition in Figure 2(b) suggests that the first oscillation becomes unstable because two complex conjugate Floquet multipliers thereof leave the unit circle, which would suggest a scenario of a direct route from a two-frequency quasiperiodic attractor to chaos [45]. That two oscillatory modes are present in the observed spontaneous chaotic oscillations is commented in Ref. [31], whose authors identify them as the dipole motion mode and the well-to-well hopping mode. In our simulations, they should correspond to the fully developed dipole motion of Figure 1(c) and to the confined dipole motion, respectively. Just after the mean current peak in Figure 2(a), there is a voltage interval of positive LLE that indicates sensitivity to initial conditions characteristic of a chaotic attractor.
Mean current, largest Lyapunov exponent and Fourier spectrum in terms of voltage. (a) Mean current and largest Lyapunov exponent vs voltage, and (b) Fourier spectrum vs voltage for the second oscillatory interval. The inset in panel (a) shows the mean current for a larger voltage interval and the vertical arrows mark supercritical Hopf bifurcation points bounding the second oscillatory voltage interval. Without noise, the voltage interval of spontaneous chaos is very narrow (3 mV width) and the LLE is only 0.25. Internal noise increases the LLE up to 0.76 and widens to 30 mV the voltage range for spontaneous chaos.
Experiments [15, 20, 31] show that oscillations appear for voltages on the first plateau and that they have frequencies about 7.5 times larger than those predicted by simulations of our mathematical model [27]. The current spikes observed in experiments are more irregular than those appearing in simulations. These features of oscillations observed in experiments point to the presence of imperfections not taken into account in the model. In earlier work on the role of imperfections [37], numerical simulations of a related discrete model showed that a 3% fluctuation in doping density could increment by a factor of 5 the oscillation frequency. Obvious imperfections that should be taken into account in our model include: (i) fluctuations of the doping density, (ii) fluctuations in \(d_B\) and \(d_W\), (iii) fluctuations in \(V_B\). Once imperfections are included in the mathematical model, we can pose the objective of optimizing chaos, i.e., introducing intentional imperfections so as to widen the voltage intervals for which there are chaotic oscillations, and increase the LLE and the complexity of attractors. These features would increase the usefulness of the device as a true random number generator.
The discovery of fast spontaneous chaotic oscillations of the current through semiconductor superlattices at room temperature brings to light their possible applications as true random bit generators [15]. Fast true random bit generators coming from tiny submicron all-electronic devices could be invaluable in secure communications and data storage. In this paper, we have discussed a mathematical model to describe spontaneous chaos in idealized superlattices with identical wells and barriers. Our numerical simulations show that spontaneous chaos possibly may appear directly from a two-frequency quasiperiodic attractor. We have also shown that the unavoidable shot and thermal noises existing in the nanostructure both enhance existing deterministic chaos (increasing its fractal dimension and largest Lyapunov exponent) and induce chaos in nearby voltage intervals. We have discussed that the differences between numerical and experimental results may be due to imperfections in the doping density, the gallium content in the barriers, and the size thereof. A better model needs to be developed to discuss the imperfections and their effect in the chaotic oscillations: ideally we could tune chaos via the introduction of controlled imperfections. We also explain how to extract verified random bit generators from a chaotic signal by digitalization and extraction of least significant bits from high order numerical derivatives, or by combining several chaotic signals coming either several superlattices or from far apart segments of the same long chaotic signal. 041b061a72