Optimizing quantum transport in multi-barrier graphene systems using differential evolution

Abstract

Potential and mass barriers in graphene introduce electron scattering, modulating transmission probabilities. Complex multi-barrier setups allow electron transmission to be controlled with high precision, but have a huge design space of possible barrier geometries. This work presents a framework to optimize electronic transport in such systems using differential evolution algorithms. First, transfer matrix methods are employed to efficiently compute quantum transport through multi-barrier structures, before optimization is applied to find barrier configurations whose transmission profiles closely match a predefined target profile. To address the trade-off between the accuracy and complexity of resulting barrier configurations, regularization techniques are incorporated into the optimization process. Our approach demonstrates the potential for highly tunable electronic transport in graphene-based systems by exploiting evolution-inspired optimization techniques.

Figures (5)

• Figure 1: (a) Schematic of an arbitrary barrier configuration which can be considered using the transfer matrix method (b) Coefficients of right- $(a_n)$ and left-going $(b_n)$ waves on either side of an interface. Single angle transmissions through multi-barrier (c) potential and (d) mass systems where $\theta_\mathrm{inc.}=0.3$ rads and $\theta_\mathrm{inc.}=0$ rads, respectively. The blue curves in these panels correspond to simple double barrier systems, whereas the red curves show results for systems with 50 random barriers. The shaded regions highlight the domains spanned by 2000 such random configurations. The maximum barrier strength is set at $V_P=200$ meV for potential and $\Delta_M=240$ meV for mass barriers.
• Figure 2: Optimized mass barriers at normal incidence ($\theta_\mathrm{inc.}=0$) (a) An arbitrary target transmission profile (gray shaded area), with colored lines showing optimized transmissions for the different mutation ($\beta$) and crossover ($C$) parameter strategies in panels (b) and (c). (d) The evolution of the mean absolute error (MAE) over 2000 generations for 10 runs of each strategy, with the best solution for each case shown in bold. The purple lines correspond to a purely random strategy. The shaded boxes show the range of results obtained for each strategy. (e) Optimized barrier configurations for each strategy.
• Figure 3: Effect of changing barrier number ($N$) and population size ($P$) for mass barriers at normal incidence (a) An arbitrary target transmission profile (gray shaded area), with colored lines showing optimized transmissions for different values of $N$. (b) Accuracy achieved after 2,000 generations using different numbers of barriers ($N$) with a fixed population size ($P=100$). Colored dots correspond to the cases shown in panel (a). (c) Accuracy achieved for different population sizes $P$, with a fixed number of barriers $N=100$. (d) The evolution of the MAE over 2000 generations for different cases in (b) and (c), with the inset showing a zoom near the final convergence.
• Figure 4: Effect of regularization for mass barriers with $\theta_\mathrm{inc.}=0$ (a) An arbitrary target transmission profile (gray shaded area), with colored lines showing optimized transmissions for different regularization strengths ($\alpha$). (b) The MAE component of the optimized loss found for systems of various $\alpha$ values. (c) The barrier complexity associated with each optimized configuration. (d) The evolution of the MAE over 2000 generations for $\alpha=1$. (e) The best-fit barrier configurations for different $\alpha$ values.
• Figure 5: Optimization for different target behaviors. In all cases, a regularization of $\alpha=1$ is applied and both short ($N=20$, dashed) and long ($N=50$, solid) barrier systems are considered. The top subpanel for each case shows the target (shaded) and optimized transmissions, and the bottom subpanel shows both optimized structures. (a) and (b) have targets relevant for (multi) band pass or stop filters, using potential (blue) or mass (red) barriers and incident angles $\theta_\mathrm{inc.}=0.3$ and $\theta_\mathrm{inc.}=0$ rads, respectively. (c) shows the optimization of an angular-dependent target, corresponding to beam collimation, for $E=60$ meV. (d) considers incident electrons across a range of angles using an angle-averaged transmission in the target.