Atomically thin materials such as graphene, are ideal systems to study electronic states in two dimension. Topological effects in 2D systems manifest itself at the bulk-vacuum interface, i.e., the edges. Microwave Impedance Microscopy, as a spatially resolved technique, can provide direct insights on the transition of electronic states across edges. The figure shows the MIM study of quantum Hall effects in monolayer graphene. The edge states are clearly observed as the bulk goes through a transition when Landau levels are completely filled. The carrier density in the graphene can be easily tuned by electrostatic gating so that we can use MIM to resolve the LL structure throughout the graphene region. MIM measurement in such system has a few unique aspects: 1) MIM can probe sub-surface regions. In this case, the graphene is sandwiched between two hBN layers, which is a key to make high quality device. 2) MIM and transport can be performed simultaneously on the same Hall bar structure. This offers a meaningful comparison of the two measurements, which provides a comprehensive understanding of the underlying physics. More details can be found at PRL 117, 186601 (2016).

    The interface between different materials, different phases in the same material, or different domains of the same phase, can be regarded as a discontinuity of certain order, and can often host electronic states with very different properties from rest of the material. MIM is a powerful technique to study such interface states: firstly being a spatially resolved technique MIM can easily access these states that often emerge in highly inhomogeneous environment; secondly MIM is very sensitive to detect conductive features therefore particularly good at identification of such states. The figure on the right shows an example of MIM study on metallic domain walls in Nd2Ir2O7. This material is an antiferromagnet with a special All-In-All-Out (AIAO) antiferromagnetic order which has two different types of domains, AIAO and its time reversal counterpart, AOAI. The domain walls between these two domains turn out to host metallic states as a result of unique synergy between spin-orbit coupling and electronic correlation. MIM is used to directly resolve such conductive domain wall states, and their spatial configurations further facilitate study of the dynamics of the AIAO order. More details can be found at Science 350, 538 (2015).