Feedback lock-in: A versatile multi-terminal measurement system for electrical transport devices.

We present the design and implementation of a measurement system that enables parallel drive and detection of small currents and voltages at numerous electrical contacts to a multi-terminal electrical device. This system, which we term a feedback lock-in, combines digital control-loop feedback with software-defined lock-in measurements to dynamically source currents and measure small, pre-amplified potentials. The effective input impedance of each current/voltage probe can be set via software, permitting any given contact to behave as an open-circuit voltage lead or as a virtually grounded current source/sink. This enables programmatic switching of measurement configurations and permits measurement of currents at multiple drain contacts without the use of current preamplifiers. Our 32-channel implementation relies on commercially available digital input/output boards, home-built voltage preamplifiers, and custom open-source software. With our feedback lock-in, we demonstrate differential measurement sensitivity comparable to a widely used commercially available lock-in amplifier and perform efficient multi-terminal electrical transport measurements on twisted bilayer graphene and SrTiO3 quantum point contacts. The feedback lock-in also enables a new style of measurement using multiple current probes, which we demonstrate on a ballistic graphene device.

Directional ballistic transport in the two-dimensional metal PdCoO2.

In an idealized infinite crystal, the material properties are constrained by the symmetries of the unit cell. The point-group symmetry is broken by the sample shape of any finite crystal, but this is commonly unobservable in macroscopic metals. To sense the shape-induced symmetry lowering in such metals, long-lived bulk states originating from an anisotropic Fermi surface are needed. Here we show how a strongly facetted Fermi surface and the long quasiparticle mean free path present in microstructures of PdCoO2 yield an in-plane resistivity anisotropy that is forbidden by symmetry on an infinite hexagonal lattice. We fabricate bar-shaped transport devices narrower than the mean free path from single crystals using focused ion beam milling, such that the ballistic charge carriers at low temperatures frequently collide with both of the side walls that define the channel. Two symmetry-forbidden transport signatures appear: the in-plane resistivity anisotropy exceeds a factor of 2, and a transverse voltage appears in zero magnetic field. Using ballistic Monte Carlo simulations and a numerical solution of the Boltzmann equation, we identify the orientation of the narrow channel as the source of symmetry breaking.

Microwave-Frequency Scanning Gate Microscopy of a Si/SiGe Double Quantum Dot.

Conventional transport methods provide quantitative information on spin, orbital, and valley states in quantum dots but lack spatial resolution. Scanning tunneling microscopy, on the other hand, provides exquisite spatial resolution at the expense of speed. Working to combine the spatial resolution and energy sensitivity of scanning probe microscopy with the speed of microwave measurements, we couple a metallic tip to a Si/SiGe double quantum dot (DQD) that is integrated with a charge detector. We first demonstrate that the dc-biased tip can be used to change the occupancy of the DQD. We then apply microwaves through the tip to drive photon-assisted tunneling (PAT). We infer the DQD level diagram from the frequency and detuning dependence of the tunneling resonances. These measurements allow the resolution of ∼65 µeV excited states, an energy consistent with valley splittings in Si/SiGe. This work demonstrates the feasibility of scanning gate experiments with Si/SiGe devices.

Evidence of Orbital Ferromagnetism in Twisted Bilayer Graphene Aligned to Hexagonal Boron Nitride.

We have previously reported ferromagnetism evinced by a large hysteretic anomalous Hall effect in twisted bilayer graphene (tBLG). Subsequent measurements of a quantized Hall resistance and small longitudinal resistance confirmed that this magnetic state is a Chern insulator. Here, we report that when tilting the sample in an external magnetic field, the ferromagnetism is highly anisotropic. Because spin-orbit coupling is weak in graphene, such anisotropy is unlikely to come from spin but rather favors theories in which the ferromagnetism is orbital. We know of no other case in which ferromagnetism has a purely orbital origin. For an applied in-plane field larger than 5 T, the out-of-plane magnetization is destroyed, suggesting a transition to a new phase.

Super-geometric electron focusing on the hexagonal Fermi surface of PdCoO2.

Geometric electron optics may be implemented in solids when electron transport is ballistic on the length scale of a device. Currently, this is realized mainly in 2D materials characterized by circular Fermi surfaces. Here we demonstrate that the nearly perfectly hexagonal Fermi surface of PdCoO2 gives rise to highly directional ballistic transport. We probe this directional ballistic regime in a single crystal of PdCoO2 by use of focused ion beam (FIB) micro-machining, defining crystalline ballistic circuits with features as small as 250 nm. The peculiar hexagonal Fermi surface naturally leads to enhanced electron self-focusing effects in a magnetic field compared to circular Fermi surfaces. This super-geometric focusing can be quantitatively predicted for arbitrary device geometry, based on the hexagonal cyclotron orbits appearing in this material. These results suggest a novel class of ballistic electronic devices exploiting the unique transport characteristics of strongly faceted Fermi surfaces.

Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene.

When two sheets of graphene are stacked at a small twist angle, the resulting flat superlattice minibands are expected to strongly enhance electron-electron interactions. Here, we present evidence that near three-quarters ([Formula: see text]) filling of the conduction miniband, these enhanced interactions drive the twisted bilayer graphene into a ferromagnetic state. In a narrow density range around an apparent insulating state at [Formula: see text], we observe emergent ferromagnetic hysteresis, with a giant anomalous Hall (AH) effect as large as 10.4 kilohms and indications of chiral edge states. Notably, the magnetization of the sample can be reversed by applying a small direct current. Although the AH resistance is not quantized, and dissipation is present, our measurements suggest that the system may be an incipient Chern insulator.

Real-time vibrations of a carbon nanotube.

The field of miniature mechanical oscillators is rapidly evolving, with emerging applications including signal processing, biological detection1 and fundamental tests of quantum mechanics2. As the dimensions of a mechanical oscillator shrink to the molecular scale, such as in a carbon nanotube resonator3-7, their vibrations become increasingly coupled and strongly interacting8,9 until even weak thermal fluctuations could make the oscillator nonlinear10-13. The mechanics at this scale possesses rich dynamics, unexplored because an efficient way of detecting the motion in real time is lacking. Here we directly measure the thermal vibrations of a carbon nanotube in real time using a high-finesse micrometre-scale silicon nitride optical cavity as a sensitive photonic microscope. With the high displacement sensitivity of 700 fm Hz-1/2 and the fine time resolution of this technique, we were able to discover a realm of dynamics undetected by previous time-averaged measurements and a room-temperature coherence that is nearly three orders of magnitude longer than previously reported. We find that the discrepancy in the coherence stems from long-time non-equilibrium dynamics, analogous to the Fermi-Pasta-Ulam-Tsingou recurrence seen in nonlinear systems14. Our data unveil the emergence of a weakly chaotic mechanical breather15, in which vibrational energy is recurrently shared among several resonance modes-dynamics that we are able to reproduce using a simple numerical model. These experiments open up the study of nonlinear mechanical systems in the Brownian limit (that is, when a system is driven solely by thermal fluctuations) and present an integrated, sensitive, high-bandwidth nanophotonic interface for carbon nanotube resonators.

Absorptive pinhole collimators for ballistic Dirac fermions in graphene.

Ballistic electrons in solids can have mean free paths far larger than the smallest features patterned by lithography. This has allowed development and study of solid-state electron-optical devices such as beam splitters and quantum point contacts, which have informed our understanding of electron flow and interactions. Recently, high-mobility graphene has emerged as an ideal two-dimensional semimetal that hosts unique chiral electron-optical effects due to its honeycomb crystalline lattice. However, this chiral transport prevents the simple use of electrostatic gates to define electron-optical devices in graphene. Here we present a method of creating highly collimated electron beams in graphene based on collinear pairs of slits, with absorptive sidewalls between the slits. By this method, we achieve beams with angular width 18° or narrower, and transmission matching classical ballistic predictions.

Graphene kirigami.

For centuries, practitioners of origami ('ori', fold; 'kami', paper) and kirigami ('kiru', cut) have fashioned sheets of paper into beautiful and complex three-dimensional structures. Both techniques are scalable, and scientists and engineers are adapting them to different two-dimensional starting materials to create structures from the macro- to the microscale. Here we show that graphene is well suited for kirigami, allowing us to build robust microscale structures with tunable mechanical properties. The material parameter crucial for kirigami is the Föppl-von Kármán number γ: an indication of the ratio between in-plane stiffness and out-of-plane bending stiffness, with high numbers corresponding to membranes that more easily bend and crumple than they stretch and shear. To determine γ, we measure the bending stiffness of graphene monolayers that are 10-100 micrometres in size and obtain a value that is thousands of times higher than the predicted atomic-scale bending stiffness. Interferometric imaging attributes this finding to ripples in the membrane that stiffen the graphene sheets considerably, to the extent that γ is comparable to that of a standard piece of paper. We may therefore apply ideas from kirigami to graphene sheets to build mechanical metamaterials such as stretchable electrodes, springs, and hinges. These results establish graphene kirigami as a simple yet powerful and customizable approach for fashioning one-atom-thick graphene sheets into resilient and movable parts with microscale dimensions.

Magnetically Actuated Single-Walled Carbon Nanotubes.

We couple magnetic tweezer techniques with standard lithography methods to make magnetically actuated single-walled carbon nanotube (SWNT) devices. Parallel arrays of 4-10 µm-long SWNT cantilevers are patterned with one end anchored to the substrate and the other end attached to a micron-scale iron magnetic tag that is free to move in solution. Thermal fluctuations of this tag provide a direct measurement of the spring constant of the SWNT cantilevers, yielding values of 10(-7)-10(-8) N/m. This tag is also a handle for applying forces and torques using externally applied magnetic field gradients. These techniques provide a platform on which interaction forces between SWNTs and other objects such as biomolecules and cells can be measured in situ.

Fluctuation broadening in carbon nanotube resonators.

We simulated the behavior of suspended carbon nanotube resonators over a broad range of temperatures to explore the physics of semiflexible polymers in underdamped environments. We find that thermal fluctuations induce strong coupling between resonance modes. This effect leads to spectral fluctuations that readily account for the experimentally observed quality factors Q ∼ 100 at 300 K. Using a mean-field approach to describe fluctuations, we analytically calculate Q and frequency shifts in tensioned and buckled carbon nanotubes and find excellent agreement with simulations.