(ii) Transverse spatiotemporal dynamics in a resonant tunneling diode

Figure 4: Schematic energy band structure of the resonant tunneling diode (DBRT). See [25]

Figure 5: Chaotic spiking (a) and breathing (b) of the DBRT current density patterns. For each, the spatiotemporal pattern of the electron density, the projection of the phase portraits on the global current-voltage plane and the time series of the voltage $U$ are shown. Parameter: $\epsilon =16.5$ (a) and $\epsilon = 9.1$ (b). See [37].

The schematic energy band structure of the resonant tunneling diode (DBRT) is shown in Fig. 2. The electrons tunnel from the emitter contact through the left barrier into the quantum well and from there through the right barrier to the collector. The dynamical variables in this case are the space-dependent electron density $a(x,t)$ in the quantum well (activator), as well as the voltage applied to the tunneling diode $u(t)$ (inhibitor) (each in dimensionless units), where $x$ is the transverse spatial coordinate vertical to the current transport direction. Based on our previous work on transverse dynamics [6,8], the next task was the microscopic calculation of the tunneling currents $J_{ew}$ and $J_{ec}$ [16]. Expanding the model by adding control forces $F_a$ , $F_u$ , one obtains in appropriate units a system of equations of the form

$$\frac{\partial a}{\partial t} = \frac{\partial }{\partial x} \left(D(a)\frac{\partial a}{\partial x}\right) + f(a,u) - KF_a(x,t),$$ (8)

$$\frac{du}{dt} = \frac{1}{\varepsilon} \left(U_0 - u - r \langle j \rangle\right)- KF_u(t).$$ (9)

Here, the nonlinear function $f(a,u)$ characterizes the difference between the inflow and the ouflow of the tunneling currents $J_{ew}$ , $J_{ec}$ and $D(a)$ is an effective diffusion coefficient. Equation (9) describes the global coupling of the system through an applied circuit with a resistance $r$ at an external voltage $U_0$ . $\epsilon$ is a time scale parameter and $\langle j \rangle$ is the spatially averaged current density. This reaction-diffusion system is structurally like the one examined in (1), but in contrast to that one, leads to a $Z$ -shaped current-voltage characteristic rather than to an $S$ -shaped one.

Without control, $K=0$ , we found transverse trigger fronts in the bistable regime [26], stochastic pulse trains in the excitable regime [25] and breathing current filaments and spatiotemporal spiking in the oscillatory regime [16].

Our research in collaboration with P. Rodin (St. Petersburg) showed that the dynamical behaviour of the DBRT can be chaotic when an electric circuit acts on the device [37]. Formally this can be achieved by choosing a negative $r$ in Eq. (9). Thus we could prove both breathing and spiking chaotic behaviour as shown in Fig. 2.5. The complete bifurcation diagram in Fig. 2.6 shows a complex bifurcation scenario, which was further examined in [25].

Figure 6: Bifurcation diagram of maxima and minima of the voltage $U$ vs the timescale parameter $\epsilon$ . Thick dotted lines: spatially homogeneous solution, thick dashed lines: periodic breathing, thick solid lines: periodic spiking. See [25]

Figure 7: Control of an unstable periodic orbit using a local control scheme without voltage feedback for $\epsilon = 9.1$ . (a) Control regime in the $K$ -$R$ plane. $\bullet$ means successful control, $\cdot$ no control, solid lines: analytical results. See (5). (b) The largest real part $\Lambda$ of the Floquet spectrum vs $K$ ($R=-0.55$ ). Dotted lines mean complex conjugate pairs of eigenvalues. See [25]

Now we switch on the control, i.e. $K\neq 0$ [25]. The aim here was to compare the effectiveness of various control methods. Our starting point is the theoretically well understood diagonal control $F_u=F_{\text{vf}}$ , $F_a=F_{\text{loc}}$ , where $F_{\text{vf}}$ and $F_{\text{loc}}$ are calculated similarly to (3) and (4). As already done for the generic model in (i), we can numerically reproduce analytical conditions for successful control also for the DBRT model (5) [25].

Figure 8: Like in Fig. 7, but with a pure voltage control. See [25].

For local control without voltage feedback, i.e. $F_u=0$ , $F_a=F_{\text{loc}}$ , the control regime deforms as in Fig. 7(a). Floquet diagrams are essential for the bifurcation analysis as shown in Fig. 7(b). In this case it follows from the Floquet diagram, that the left boundary of the control regime is associated with a flip bifurcation whereas the lower and right boundary are each associated with Hopf bifurcations.

The most suitable control scheme for practical applications is a pure voltage feedback, $F_u=F_{\text{vf}}$ , $F_a=0$ , where the physically easily accessible voltage variable is used in the control. We managed with this simple method to stabilize an unstable periodic spatiotemporal orbit for the DBRT. [25]. The control regime (cf. Fig.8(a)) is is in comparison to diagonal control obviously smaller, which can also be seen in the corresponding Floquet diagram (Fig.8). Further interesting control schemes arise by choosing a spatially averaged control force $F_a(x,t) = \langle F_{\text{loc}}(x',t) \rangle_{x'}$ [25].