Driven by topological properties, two-dimensional valleytronics is pivotal for the design of low-power electronic devices. Consequently, incorporating simulation experiments of valleytronic devices into solid-state physics curricula significantly enhances the practicality and innovativeness of a course.

Leveraging band theory and the tight-binding model, this experiment presents a simulation design for two-dimensional graphene-based valleytronic devices. First, taking graphene as a prototype, the physical foundations of valleytronics within a hexagonal honeycomb structure are reviewed, and the fundamental characteristics of the valley degree of freedom are introduced. Second, a two-dimensional electronic device model is constructed based on monolayer graphene to simulate the band structure and spatial electron distribution of topologically confined electronic states in a single-channel configuration. Third, the electronic properties of two-dimensional topological valley devices with three and four channels are investigated. The channel number parity significantly alters the band structure at the Fermi level, indicating that interchannel coupling of electronic states is decisive for the electronic properties of the system. Specifically, in the three-channel configuration, each valley hosts a single linear topological conducting near-zero-energy state, with these states having opposite propagation directions for the two valleys. Notably, these states are not localized in a single channel but are uniformly distributed near the first and third channels. In contrast, in the four-channel system, interchannel coupling opens a global band gap in the band structure, and consequently, the system exhibits insulating behavior at zero energy. At finite low energies, the wavefunctions of the electronic states display strong interchannel coupling. Furthermore, the mechanisms influencing these multichannel two-dimensional topological transport devices are explored by varying sublattice onsite energy ratios and external magnetic fields.

The results demonstrate that modifying the sublattice onsite energy ratio between adjacent regions effectively regulates the distribution proportion of electrons across real-space channels. Specifically, in the three-channel configuration, an increase in U₂/U₁ causes the zero-energy states that are originally uniformly distributed near the first and third channels to progressively localize in the first channel. This process culminates in the exclusive localization of zero-energy states in the first channel owing to the decoupling of electronic wavefunctions across channels. In contrast, decreasing U₂/U₁ reverses this trend, leading to a gradual accumulation of zero-energy states near the third channel. A similar modulation phenomenon is also observed in the four-channel system. Moreover, applying an external magnetic field breaks the time-reversal symmetry of the system, substantially modifying the band structure and spatially separating electrons from different valleys. Additionally, the transmission direction and distribution channels of electrons in real space can be manipulated by adjusting the magnetic field direction or the carrier type.

Given the physical equivalence between modulating sublattice onsite energy in a monolayer system and applying a perpendicular external electric field in a multilayer system, these results indicate that external electric or magnetic fields exert critical regulatory effects on topological conducting states. This influence enables the selective control of transmission channels and directions for electrons in different valleys within the device. This simulation experiment not only demonstrates an intuitive application of band theory in two-dimensional topological electronic devices but also provides an ideal platform for teaching practices and research training in solid-state physics–related courses.

Topology optimization is an innovative computational technique that determines the optimal material distribution or structural configuration to meet the specific design criteria. This technique is crucial in the intelligent microwave device design due to its high degree of freedom compared with shape or size optimization. Microwave engineering is an important fundamental course for physics, electronic engineering, and communication engineering students. Incorporating device topology optimization experiments into the curriculum can make the abstract concepts in the microwave theory and network theory more easy to comprehend, this not only bridges classroom knowledge with advanced industrial design techniques but also cultivates students’ analytical, modeling, and engineering skills.

Using concrete examples of antenna matching network optimization, the experiment reported in this study explores the method and implementation techniques of topology optimization. The investigated network topology optimization experiment involves two interconnected phases: theoretical modeling and numerical optimization. Initially, this study uses the impedance matrix approach for the accurate and fast modeling of a grid-like antenna matching network. Once the connection relationship in the matching network is determined, a closed-form expression of the network scattering parameters can be obtained explicitly, eliminating the need for computationally expensive full-wave simulations and thus accelerating the forward problem solving during each iteration of topology optimization. Subsequently, a genetic algorithm, which is a type of powerful stochastic optimization algorithm, is leveraged to optimize the connection relationship in the grid-like network, achieving an intelligent on-demand design of the antenna operating frequency band. The genetic algorithm can handle the ill-posed optimization problem using mechanisms such as selection, crossover, and mutation to yield solutions that approximate the optimal solution of the search process.

This study demonstrates the resultant structures and performances of topology-optimized narrow-bandwidth monopole antennas operating at center frequencies of 1.00, 1.20, and 1.40 GHz, as well as a wideband antenna operating in the range of 1.099–1.395 GHz. The narrow-bandwidth design is achieved by introducing a single resonance of the matching-network-loaded antenna that closely matches the prescribed center frequency, while the wideband design emerges from creating two resonances with an appropriate frequency separation. Two important figures of merit—the reflection coefficient and antenna’s total efficiency—are introduced in these experiments to assess the antenna performance. These metrics measure the power reflection from the antenna feeding port and the ratio of the radiated power to the incident power, respectively. Following the topology optimization of the grid-like antenna feeding network, the values of the port reflection coefficients of the monopole antennas reduce notably from nearly 0 dB to below −10 dB over the specified frequency band, and the antenna efficiencies are generally higher than 90% over the operating band, with peak efficiency values exceeding 95%.

This study systematically elucidates the theory and workflow of network topology optimization, emphasizing the combined use of impedance matrix modeling and genetic algorithms. Numerical optimization experiments are conducted to obtain an on-demand intelligent design of the antenna matching network. These experiments reveal a substantial improvement in the antenna radiation performance after incorporating the topology-optimized matching network. This experiment serves as a comprehensive and advanced training opportunity for microwave- and antenna-related courses, significantly enhancing undergraduate students’ ability to tackle complex engineering problems.