Uncovering spatiotemporal patterns in semiconductor superlattices by efficient data processing tools

Time periodic patterns in a semiconductor superlattice, relevant to microwave generation, are obtained upon numerical integration of a known set of drift-diffusion equations. The associated spatiotemporal transport mechanisms are uncovered by applying (to the computed data) two recent data processing tools, known as the higher order dynamic mode decomposition and the spatiotemporal Koopman decomposition. Outcomes include a clear identification of the asymptotic self-sustained oscillations of the current density (isolated from the transient dynamics) and an accurate description of the electric field traveling pulse in terms of its dispersion diagram. In addition, a preliminary version of a data-driven reduced order model is constructed, which allows for extremely fast online simulations of the system response over a range of different configurations.

Figure 1

Temporal evolution of the current density in the considered timespan.

Figure 2

Spatiotemporal color diagram for the evolution of the electric field in the considered timespan. The intensity of $F$ varies from blue to red as $F$ increases. The two horizontal white lines indicate the values $x={\tilde{x}}_1$ and ${\tilde{x}}_2$, as defined in Eq. (12).

Figure 3

Plots of $F({\tilde{x}}_1,t)$ and $F({\tilde{x}}_2,t)$ vs $t$, with ${\tilde{x}}_1$ and ${\tilde{x}}_2$ as defined in Eq. (12). Since ${\tilde{x}}_1<{\tilde{x}}_2$, the two curves can be distinguished because, consistently with Fig. 2, $F({\tilde{x}}_2,t)$ shows higher peaks than $F({\tilde{x}}_1,t)$.

Figure 4

Counterparts of Figs. 1 and 3 for the HODMD method application with tunable parameters as in Eq. (20) and $K=3000$ snapshots in $0\le t\le 300$. The original data are plotted with thin solid lines, while the reconstructed data are plotted with thick dashed lines.

Figure 5

Growth rates (a) and amplitudes (b) vs the frequencies for the retained modes, using the HODMD method with tunable parameters as in Eq. (20) and $K=3000$ snapshots in $0\le t\le 300$. In plot (a), positive and negative growth rates are displayed in red and black, respectively.

Figure 6

Counterpart of Fig. 5, considering only the lower group of points in both plots.

Figure 7

Counterpart of Fig. 4 for the periodic attractor.

Figure 8

Counterpart of Fig. 7 for the purely decaying dynamics.

Figure 9

Spatial RMS norm of the electric field for the strictly decaying dynamics.

Figure 10

Counterpart of Fig. 5 for the application of HODMD to snapshots contained in the transient stage (24), with the tunable parameters given in Eq. (25).

Figure 11

Counterpart of Fig. 4 when applying extrapolated HODMD using snapshots in the transient stage (24), with the tunable parameters in Eq. (25). In both plots, the original data are shown with thin solid lines, the reconstructed data in the transient stage are shown with thick dashed black lines, and the extrapolated data are shown with thick dashed blue lines.

Figure 12

Counterpart of Fig. 10 for the application of HODMD to snapshots contained in the timespan (26), with the tunable parameters given in Eq. (27).

Figure 13

Counterpart of Fig. 11 when applying extrapolated HODMD using snapshots in the timespan (26), with the tunable parameters given in Eq. (27).

Figure 14

Fourier amplitudes vs the frequencies obtained via FFT applied to data of the current density in the temporal intervals $0\le t\le 300$ (a) and $200\le t\le 300$ (b).

Figure 15

Counterpart of Fig. 2 for the electric field in the periodic attractor (a) and the scaled electric field defined in Eq. (30) (b). In both plots, the dashed white line indicates the trajectory of a particle moving with the overall propagation velocity $c$ defined in Eq. (29). Thus, the slope of this line coincides with $c$.

Figure 16

The scale defined in Eq. (31).

Figure 17

Dispersion diagram resulting from the application of the STKD method to the scaled electric field in Eq. (30) in the periodic attractor, using the tunable parameters in Eq. (32).

Figure 18

Temporal evolution of the current density (a) and the electric field at $x={\tilde{x}}_1$ and ${\tilde{x}}_2$ as defined in Eq. (12) (b), for the synchronized HODMD reconstruction of the periodic attractor corresponding to $\sigma ={\sigma }_5=0.7$, in the time interval (44).

Figure 19

Counterpart of Fig. 18 for the periodic attractor corresponding to $\sigma ={\sigma }_1=0.3$.

Figure 20

Synchronized attractor for ${\sigma }^{\text{new}}=0.65$, in the time interval (44), for the current density (a) and the electric field at $x={\tilde{x}}_1$ and ${\tilde{x}}_2$ as defined in Eq. (12) (b). In these plots, the exact evolution is displayed in thin solid lines, while the outcome of the HODMD-based data-driven ROM is depicted with thick dashed lines.

Figure 21

Counterpart of Fig. 20 for ${\sigma }^{\text{new}}=0.35$.