Enter oxides. Various complex oxides can exhibit all of these properties, and that has led to a concerted effort to develop materials growth techniques to create high quality oxide thin films, with an eye toward creating the same kind of atomically sharp heterointerfaces as in III-Vs. A foundational paper is this one by Ohtomo and Hwang, where they used pulsed laser deposition to produce a heterojunction between LaAlO3, an insulating transparent oxide, and SrTiO3, another insulating transparent oxide (though one known to be almost a ferroelectric). Despite the fact that both of those parent constituents are band insulators, the interface between the two was found to play host to a two-dimensional gas of electrons with remarkable properties. The wikipedia article linked above is pretty good, so you should read it if you're interested.
When you think about it, this is really remarkable. You take an insulator, and another insulator, and yet the interface between them acts like a metal. Where did the charge carriers come from? (It's complicated - charge transfer from LAO to STO, but the free surface of the LAO and its chemical termination is hugely important.) What is happening right at that interface? (It's complicated. There can be some lattice distortion from the growth process. There can be oxygen vacancies and other kinds of defects. Below about 105 K the STO substrate distorts "ferroelastically", further complicating matters.) Do the charge carriers live more on one side of the interface than the other, as in III-V interfaces, where the (conduction) band offset between the two layers can act like a potential barrier, and the same charge transfer that spills electrons onto one side leads to a self-consistent electrostatic potential that holds the charge layer right against that interface? (Yes.)
Even just looking at the LAO/STO system, there is a ton of exciting work being performed. Directly relevant to the meeting I just attended, Jeremy Levy's group at Pitt has been at the forefront of creating nanoscale electronic structures at the LAO/STO interface and examining their properties. It turns out (one of these fortunate things!) that you can use a conductive atomic force microscope tip to do (reversible) electrochemistry at the free LAO surface, and basically draw conductive structures with nm resolution at the buried LAO/STO interface right below. This is a very powerful technique, and it's enabled the study of the basic science of electronic transport at this interface at the nanoscale.
Beyond LAO/STO, over the same period there has been great progress in complex oxide materials growth by groups at a number of universities and at national labs. I will refrain from trying to list them since I don't know them all and don't want to offend with the sin of inadvertent omission. It is now possible to prepare a dizzying array of material types (ferromagnetic insulators like GdTiO3; antiferromagnetic insulators like SmTiO3; Mott insulators like LaTiO3; nickelates; superconducting cuprates; etc.) and complicated multilayers and superlattices of these systems. It's far too early to say where this is all going, but historically the ability to grow new material systems of high quality with excellent precision tends to pay big dividends in the long term, even if they're not the benefits originally envisioned.