Noise-Induced Front Motion: Signature of a Global Bifurcation

We show that front motion can be induced by noise in a spatially extended excitable system with a global constraint. Our model system is a semiconductor superlattice exhibiting complex dynamics of electron accumulation and depletion fronts. The presence of noise induces a global change in the dynamics of the system forcing stationary fronts to move through the entire device. We demonstrate the effect of coherence resonance in our model; i.e., there is an optimal level of noise at which the regularity of front motion is enhanced. Physical insight is provided by relating the space-time dynamics of the fronts with a phase-space analysis.

Figure 1

Noise-induced front motion: Space-time plots of the electron density for (a) $D=0$ (no noise), (b) $D=0.5\mathrm{A}{\mathrm{s}}^{1/2}/{\mathrm{m}}^2$, and (c) $D=2.0\mathrm{A}{\mathrm{s}}^{1/2}/{\mathrm{m}}^2$. Light and dark shading corresponds to electron accumulation and depletion fronts, respectively. The emitter is at the bottom. Parameters: $U=2.99\mathrm{V}$, $\sigma =2.0821012488{\Omega }^{-1}{\mathrm{m}}^{-1}$, $N_D={10}^{11}{\mathrm{cm}}^{-2}$, $T=20\mathrm{K}$, $N=100$ GaAs wells of width $w=8\mathrm{nm}$, and ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$ barriers of width $b=5\mathrm{nm}$, energies $E^a=41.5\mathrm{meV}$, $E^b=160\mathrm{meV}$, scattering width $\Gamma =8\mathrm{meV}$, transition matrix elements $H_{m,m+1}^{a,b}=-e{F}_m\times 0.0127\mathrm{m}$, $H_{m+1,m}^{a,a}=-0.688\mathrm{meV}$, $H_{m+1,m}^{b,b}=1.263\mathrm{meV}$, as in Ref. 9.

Figure 2

(a) Three noise realizations of the current density $J(t)$. From top to bottom, $D=0.8$, $D=2.0$, and $D=5.0\mathrm{A}{\mathrm{s}}^{1/2}/{\mathrm{m}}^2$. (b) Mean interspike interval (top panel) and its normalized fluctuations $R_T$ (bottom panel) versus noise intensity. Lines, constant $D$; diamonds, $D\sim J_{m-1\to m}^{1/2}$ [18]. The inset shows the peak frequency versus $D$.

Figure 3

Bifurcation diagram in the ($\sigma$, $U$) plane. Thick or hatched lines mark the transition from stationary to moving fronts via a Hopf or a saddle-node bifurcation on a limit cycle, respectively. The inset shows a blowup of a small part of the hatched line revealing its sawtoothlike structure. Dark (green) and light (yellow) regions correspond to stationary and moving fronts, respectively, where the numbers denote the positions of the stationary accumulation front in the superlattice. Upper inset shows the frequency $f$ of the limit cycle which is born above the critical point (marked by a cross in the lower inset) as function of $U$.

Figure 4

(a) Phase portrait in terms of electron densities $n_{65}$ and $n_{64}$, normalized to the donor density $N_D$, below the global bifurcation. (b) Space-time plot and (c) time series of $n_{65}$ for the trajectory shown in (a). The different parts of the trajectory are labeled by roman numerals I–VI in (a), (b), and (c). Parameters as in Fig. 1, $D=0$.