Water in a river shows a variety of flow patterns and whirls. Any obstacle in the river, such as a bridge pillar or simply a rough bank, will lead to a distinctive flow pattern. It has been comparatively less obvious how electrons flow in a solid. But in a paper in Nature, Sulpizio et al.1 report an experiment in which the flow pattern of electrons in an electrical conductor is imaged.
The electrical resistance of a metal is caused by electrons being scattered from impurities in the material’s atomic lattice or from lattice vibrations called phonons. However, it is not affected by electron–electron scattering. When two electrons scatter off each other, their individual momenta are changed by the scattering event. But the total momentum of the two electrons is conserved, as is the total momentum of a sea of electrons in a metal. Therefore, simply measuring the resistance of a metal will not unveil the effects of electron–electron scattering.
To nail down these effects, materials need to be tuned to a regime in which electron–electron scattering is dominant and the electrons flow like a viscous liquid2,3. At low temperatures, electron–electron (as well as electron–phonon) scattering is suppressed and electron–impurity scattering dominates. Conversely, at high temperatures, electron–phonon scattering takes over. For graphene (a single layer of carbon atoms arranged in a honeycomb lattice), there is an intermediate temperature range4 (50–250 kelvin) for which the rate of electron–electron scattering is the highest among all scattering rates (Fig. 1). However, even in this case, the material’s resistance will not be modified by electron–electron scattering because of momentum conservation.
One way to investigate the viscous-flow regime has been to measure a local resistance, known as vicinity resistance4, on an extremely small scale. The value of this quantity changes sign in the case of viscous flow. Another option has been to observe an effect called superballistic resistance5 for electrons flowing through a narrow opening in a material. Here, the resistance is reduced below the value expected for a ballistic system, in which there is effectively no scattering. Such pioneering experiments were crucial for demonstrating that viscous electron flow can be important in electron transport. However, they provide only indirect evidence for the existence of such flow and do not give insights into the spatial arrangements of flow patterns.
Electrons passing through a sample of a conducting material are driven by an electric field. As a result, there is a voltage gradient along the direction of current flow. Unfortunately, this local voltage gradient is independent of the flow regime. But when a weak magnetic field is applied to the sample, another voltage, known as a Hall voltage, is produced perpendicular to the direction of current flow. The spatial profile of the Hall voltage does provide information about the flow characteristics.
Sulpizio and colleagues use a sensitive electric-field sensor that enables local probing of this Hall voltage. The sensor is an innovative technology developed by this research group6. It consists of an electronic device called a single-electron transistor, the conductance of which depends sensitively on its electrostatic environment.
In the present work, the sensor is made from ultraclean carbon nanotubes. Individual electrons are confined within these nanotubes by electrodes. Such an arrangement provides the required sensitivity for detecting weak electric fields or voltage gradients, such as those associated with the Hall voltage. The spatial resolution of the sensor is limited by its size and the distance of the sensor to the object to be probed.
Changing the temperature and the number of charge carriers per given area in the sample induces different flow regimes, which lead to different Hall-voltage profiles. Sulpizio et al. use this property to image local electric fields in a uniform layer of graphene, and investigate the transition between the regime in which electron–electron scattering dominates and those in which electron–phonon or electron–impurity scattering takes over.
The authors demonstrate experimentally how electron–electron scattering alters the Hall-voltage profile of a uniform conductor. Viscous flow in liquids leads to turbulence and whirls, depending on the viscosity of the liquid and on obstacles to the flow. However, the observation of such features in electron transport is beyond the scope of the present work and could require different experimental tools, such as sensitive magnetic-field sensors, or samples that have complex geometries.
What do Sulpizio and colleagues’ results mean for our understanding of electron transport in conductors? In the viscous regime, the flow of electrons is described by a universal hydrodynamic concept known as Poiseuille flow. The authors’ imaging of electronic Poiseuille flow is a breakthrough in the study of electron transport as well as a demonstration of a sophisticated imaging technique that combines high spatial resolution with extreme sensitivity. We now know that electron flow can be diffusive, ballistic or viscous, and that there are experimental tools for differentiating between these regimes.
For solid-state systems in general, electron–electron interactions are relevant for phenomena as diverse as ferromagnetism (the familiar type of magnetism found in iron bar magnets) and the fractional quantum Hall effect (whereby electrons in a strong magnetic field act together to behave like particles that have a fractional electric charge). The authors’ technique could also be used to investigate, on a local scale, the superconductivity that was discovered last year in a twisted bilayer of graphene7. The potential to extract local information about strongly interacting systems of electrons will have far-reaching consequences for this field. Further applications of the technique could enable local probing of electric fields as they arise in complex quantum circuits — which might one day lead to a quantum computer.
Nature 576, 45-46 (2019)