Half- and quarter-metals in rhombohedral trilayer graphene.

Ferromagnetism is most common in transition metal compounds where electrons occupy highly localized d orbitals. However, ferromagnetic order may also arise in low-density two-dimensional electron systems1-5. Here we show that gate-tuned van Hove singularities in rhombohedral trilayer graphene6 drive spontaneous ferromagnetic polarization of the electron system into one or more spin and valley flavours. Using capacitance and transport measurements, we observe a cascade of transitions tuned to the density and electronic displacement field between phases in which quantum oscillations have fourfold, twofold or onefold degeneracy, associated with a spin- and valley-degenerate normal metal, spin-polarized 'half-metal', and spin- and valley-polarized 'quarter-metal', respectively. For electron doping, the salient features of the data are well captured by a phenomenological Stoner model7 that includes valley-anisotropic interactions. For hole filling, we observe a richer phase diagram featuring a delicate interplay of broken symmetries and transitions in the Fermi surface topology. Finally, we introduce a moiré superlattice using a rotationally aligned hexagonal boron nitride substrate5,8. Remarkably, we find that the isospin order is only weakly perturbed, with the moiré potential catalysing the formation of topologically nontrivial gapped states whenever itinerant half- or quarter-metal states occur at half- or quarter-superlattice band filling. Our results show that rhombohedral graphene is an ideal platform for well-controlled tests of many-body theory, and reveal magnetism in moiré materials4,5,9,10 to be fundamentally itinerant in nature.

Imaging currents in HgTe quantum wells in the quantum spin Hall regime.

The quantum spin Hall (QSH) state is a state of matter characterized by a non-trivial topology of its band structure, and associated conducting edge channels. The QSH state was predicted and experimentally demonstrated to be realized in HgTe quantum wells. The existence of the edge channels has been inferred from local and non-local transport measurements in sufficiently small devices. Here we directly confirm the existence of the edge channels by imaging the magnetic fields produced by current flowing in large Hall bars made from HgTe quantum wells. These images distinguish between current that passes through each edge and the bulk. On tuning the bulk conductivity by gating or raising the temperature, we observe a regime in which the edge channels clearly coexist with the conducting bulk, providing input to the question of how ballistic transport may be limited in the edge channels. Our results represent a versatile method for characterization of new QSH materials systems.

Locally enhanced conductivity due to the tetragonal domain structure in LaAlO3/SrTiO3 heterointerfaces.

The ability to control materials properties through interface engineering is demonstrated by the appearance of conductivity at the interface of certain insulators, most famously the {001} interface of the band insulators LaAlO3 and TiO2-terminated SrTiO3 (STO; refs 1, 2). Transport and other measurements in this system show a plethora of diverse physical phenomena. To better understand the interface conductivity, we used scanning superconducting quantum interference device microscopy to image the magnetic field locally generated by current in an interface. At low temperature, we found that the current flowed in conductive narrow paths oriented along the crystallographic axes, embedded in a less conductive background. The configuration of these paths changed on thermal cycling above the STO cubic-to-tetragonal structural transition temperature, implying that the local conductivity is strongly modified by the STO tetragonal domain structure. The interplay between substrate domains and the interface provides an additional mechanism for understanding and controlling the behaviour of heterostructures.

Images of edge current in InAs/GaSb quantum wells.

Quantum spin Hall devices with edges much longer than several microns do not display ballistic transport; that is, their measured conductances are much less than e(2)/h per edge. We imaged edge currents in InAs/GaSb quantum wells with long edges and determined an effective edge resistance. Surprisingly, although the effective edge resistance is much greater than h/e(2), it is independent of temperature up to 30 K within experimental resolution. Known candidate scattering mechanisms do not explain our observation of an effective edge resistance that is large yet temperature independent.

Observation of fractional Chern insulators in a van der Waals heterostructure.

Topologically ordered phases are characterized by long-range quantum entanglement and fractional statistics rather than by symmetry breaking. First observed in a fractionally filled continuum Landau level, topological order has since been proposed to arise more generally at fractional fillings of topologically nontrivial Chern bands. Here we report the observation of gapped states at fractional fillings of Harper-Hofstadter bands arising from the interplay of a magnetic field and a superlattice potential in a bilayer graphene-hexagonal boron nitride heterostructure. We observed phases at fractional filling of bands with Chern indices [Formula: see text] Some of these phases, in [Formula: see text] and [Formula: see text] bands, are characterized by fractional Hall conductance-that is, they are known as fractional Chern insulators and constitute an example of topological order beyond Landau levels.

Scanning SQUID susceptometers with sub-micron spatial resolution.

Superconducting QUantum Interference Device (SQUID) microscopy has excellent magnetic field sensitivity, but suffers from modest spatial resolution when compared with other scanning probes. This spatial resolution is determined by both the size of the field sensitive area and the spacing between this area and the sample surface. In this paper we describe scanning SQUID susceptometers that achieve sub-micron spatial resolution while retaining a white noise floor flux sensitivity of ≈2µΦ0/Hz1/2. This high spatial resolution is accomplished by deep sub-micron feature sizes, well shielded pickup loops fabricated using a planarized process, and a deep etch step that minimizes the spacing between the sample surface and the SQUID pickup loop. We describe the design, modeling, fabrication, and testing of these sensors. Although sub-micron spatial resolution has been achieved previously in scanning SQUID sensors, our sensors not only achieve high spatial resolution but also have integrated modulation coils for flux feedback, integrated field coils for susceptibility measurements, and batch processing. They are therefore a generally applicable tool for imaging sample magnetization, currents, and susceptibilities with higher spatial resolution than previous susceptometers.