Topological LC circuit and its realization in microstrip transmission lines

To harness at will propagation of electromagnetic (EM) waves is important for valuable applications ranging from imaging and sensing well below the EM wavelength to realizing IoT, self-driving automobile and advanced information processing. We propose that topological propagations of EM waves can be achieved in microstrip transmission-line systems: a sandwich structure of grounded metal film, dielectric substrate and printed metallic wires. When the metallic wires form a honeycomb pattern, linear frequency-momentum dispersions appear in the normal frequency EM modes, much similar to the Dirac cones in the electronic energy-momentum dispersions seen in graphene. Implementing a periodic hexagonal pattern of wide/narrow microstrip lines opens a frequency band gap, and generates in a topological EM state mimicking the quantum spin Hall effect (QSHE). We reveal theoretically the physics by considering an LC circuit and paralleling it to a tight-binding model. The planar and open structure of this system provides a unique platform for resolving the pseudospin, pseudospin-momentum locking and p-d orbital hybridization in the topological edge EM propagation.