Imaging quantum materials

Specialized imaging methods are now available to measure the quantum properties of materials with high sensitivity and resolution. These techniques are key to the design, synthesis and understanding of materials with exotic functionalities.

Electrons seem deceptively simple: elementary fermions with charge e and spin 1/2. Put them in a quantum material, and things get interesting. Their mass is small enough for their wave-like nature to be noticeable at energies that are accessible in many laboratories. They may be strongly correlated, meaning that what each electron does will strongly affect other electrons. They move through a material environment that may have lattice, spin, charge and orbital degrees of freedom — properties that depend on the atomic composition and crystal structure of the material itself. By appropriate optimization of these parameters, we can create a variety of astonishing phenomena and functionalities1 — from high-temperature superconductivity, to dissipationless conductivity at the boundaries of topological insulators, to emergent particles with fractional charge and statistics. In the environment of a real material, many factors contribute to the formation of spatial variation: quantum mechanical behaviour, strong interactions, complicated lattices, competing phases and susceptibility to disorder. With traditional bulk measurements, we can infer the existence of spatial variation; but with modern imaging methods, we can see it directly, quantify it and even manipulate it.

One might think that if a material has a uniform structure, any electronic phase that lives in it will also be uniform. However, the collective behaviour of a large number of electrons can lead to texture on length scales that are mesoscopic, meaning smaller than macroscopic but larger than atomic. Figure 1 showcases several examples of collective phenomena, visualized with various techniques. In a charge-density wave (Fig. 1a), the electrons or charge carriers form a standing wave pattern, leading to a superlattice distortion of the underlying lattice2. The current in such a charge-density wave can flow with stops and starts caused by pinning and electrostatic interactions. In uniform magnets with certain spin interactions that favour a chiral twist in the orientation of their spins, the smallest magnetization reversal must have a finite size because it must be surrounded by a twisting spin texture. The entire perturbation, called a skyrmion3,4 (Fig. 1b,c), is considered an attractive candidate for information technology in the field of spintronics5. In a superconductor, the boundary between superconducting and normal regions can be energetically favourable, such that the system maximizes this boundary by having many small cylindrical regions of normal material. The size of the smallest allowable region of normal material, called a vortex6,7 (Fig. 1d,e), is limited by the single-valued phase of the order parameter, which cannot wind by less than 2π around a vortex core. Vortices cause dissipation that limits the performance of superconducting cables and circuits, but are themselves candidates for information processing, and the structures of vortices give clues about the nature of the superconducting state. All of these mesoscopic structures can be 'pinned' when their energy depends on their location, but pinning is only one of the many ways in which variations in the material structure can imprint their pattern on functional electron phases.

Figure 1: Collective phenomena in quantum materials.
Figure 1

a, Charge-density waves in NbSe2, as seen in scanning tunnelling microscope topography2. b,c, Skyrmion lattice in Fe0.5Co0.5Si, as seen in (b) the lateral magnetization distribution obtained by Lorentz transmission electron microscopy3 and (c) the magnetic fields outside a sample imaged by magnetic force microscopy4. d,e, Quantized vortices in superconductors, appearing as (d) quasiparticle states in the vortex cores in Bi2Sr2CaCu2O8+δ seen in scanning tunnelling spectroscopy6 and (e) magnetic fields outside a sample of YBa2Cu3O6.991 seen by magnetic force microscopy7. The vortices in e are being simultaneously imaged and pulled through the sample. A strongly pinned vortex at lower temperatures (inset) does not move. Adapted from ref. 2, PNAS (a); ref. 3, Macmillan Publishers Ltd (b); ref. 4, AAAS (c); ref. 6, AAAS (d); and ref. 7, Macmillan Publishers Ltd (e).

Examples of the effects of variation in the underlying material structure are shown in Fig. 2. Naturally occurring structures such as local composition8 (Fig. 2a), point defects, twin domains9 (Fig. 2b), grains10, dislocations11, strain landscapes and potential landscapes12 (Fig. 2c) all break symmetry and change the environment of the electronic phase in ways that both inspire and test theoretical models of the underlying phases. These variations exist in all materials, but quantum materials can be either vastly more or vastly less responsive to disorder.

Figure 2: Interplay between electronic phases and spatial variation present in the underlying material structure or realized with patterning techniques.
Figure 2

a, Chromium dopant atoms in the magnetic topological insulator Cr0.08(Bi0.1Sb0.9)1.92Te3 as seen in scanning tunnelling microscope topographs. This dopant landscape is associated with nanoscale disorder in the Dirac mass seen by spectroscopy (not shown)8. b, Enhanced conductivity due to twin domains in a LaAlO3/SrTiO3 heterointerface, as seen by scanning SQUID microscopy imaging of the magnetic field produced by a transport current9. c, Carrier density (n2D) in graphene with hole-rich (blue) and electron-rich (red) regions, as seen by a scanning single-electron transistor12. d, Coexistence of metal and correlated Mott insulator phases through the insulator–metal transition in V2O3, as seen by near-field infrared microscopy14. The electronic transition seen here appears to be decoupled from a structural transition. e, Branched electron flow out of a quantum point contact in GaAs/AlGaAs two-dimensional electron gas, as seen by scanning gate microscopy15. f, Scanning tunnelling spectroscopy of a chain of magnetic (iron) atoms arranged on a superconductor (lead). This configuration is theoretically predicted to produce a one-dimensional topological superconductor with a localized Majorana mode, evidence for which appears in the localized mode seen at zero voltage16. Adapted from ref. 8, PNAS (a); ref. 9, Macmillan Publishers Ltd (b); ref. 12, Macmillan Publishers Ltd (c); ref. 14, Macmillan Publishers Ltd (d); ref. 15, Macmillan Publishers Ltd (e); and ref. 16, AAAS (f).

Multiple phases, such as spin-density wave and superconducting phases13, or metal and correlated Mott insulators14 (Fig. 2d), can phase-separate into distinct physical regions. When more than one phase is close to stability, phases compete with each other, greatly enhancing any underlying variation in the material and even creating spatial variation in an otherwise uniform environment. The electronic phase separation shown in Fig. 2d, for example, is decoupled from a structural phase transition14.

Patterning the sample offers additional ways to influence and probe electronic states15 (Fig. 2e) or even to design them atom by atom16 (Fig. 2f).

Bulk measurements such as electrical transport are usually the first way to study functionality in new samples. Transport measurements determine electrical and thermal conductivity and their dependence on external parameters such as temperature, magnetic field, electric field or strain. Magnetic measurements reveal ferromagnetism, paramagnetism and diamagnetism. Scattering measurements show how materials interact with incoming particles, such as optical electromagnetic radiation, X-rays or neutrons. Bulk measurements average the response in some way over the entire sample, so they can often reveal inhomogeneity. For example, at a phase transition, most measured quantities will change over a finite range of the tuning parameter used. If the sample has any inhomogeneity, the measured range will be larger than the intrinsic thermodynamic width. But the way in which different parts of the sample contribute to the average response strongly depends on the bulk parameter measured. For instance, phase transitions are often first identified through resistance measurements, which are dominated by the path of highest electrical conduction; in contrast, heat capacity measurements average the contributions from the entire sample. Spatially resolved imaging can effectively complement bulk measurements, revealing local patterns and helping to distinguish different types of spatial variation.

Over the past three decades, scientists have developed many specialized probes for imaging functional properties. Table 1 lists a few of the most ubiquitous and highest-impact scanning probes in the field of quantum materials. This list of techniques is certainly not exhaustive: others include scanning Kelvin probe microscopy, magnetic resonance force microscopy, scanning Josephson microscopy, scanning capacitance microscopy, scanning gate microscopy and many more. In fact, most interesting properties have at least one technique devoted to their measurement. Moreover, imaging does not need to be done by scanned probes: photons, electrons and other particles are also used as probes, as in photoemission17 and photoemission electron microscopy.

Table 1: Selected scanning probe techniques for quantum materials.

The spatial resolutions reported in Table 1 should be considered as approximate values; determining the resolution is often subtle. A point spread function, or impulse response, describes the response of the imaging system to a point source. The width of the point spread function is a good measure of the spatial resolution if it can be measured directly. Algorithms deconvolving the detected signal can improve the spatial resolution in some cases, although such deconvolution can often be difficult. For example, magnetic force microscopy measures the total force on a magnetized tip whose spatial details can only be known through considerable effort, such as by applying electron holographic methods to the tip itself18. Fortunately, because the tip is small and close to the surface, magnetic force microscopy offers relatively high spatial resolution without deconvolution. Hall probes and superconducting quantum interference devices (SQUIDs) both measure the magnetic field with seemingly simple point spread functions, but under real operating conditions their spatial resolution is limited by the distance from the sensor to the sample. Because magnetic fields spread out rapidly in free space, features that are small compared with this distance cannot be resolved. Nitrogen–vacancy centres19 can be said to have atomic spatial resolution, as listed in Table 1, but for measuring static magnetic fields, their spatial resolution is also limited by their distance from the source. Resonant phenomena can provide better spatial resolution20.

When planning an imaging experiment or interpreting its results, there are caveats and cautions to consider. Spatial resolution is highly desirable, but sensitivity to the property of interest is often more important. Indeed, it may be necessary to make trade-offs between spatial resolution, sensitivity, temporal resolution, the operating range of the experiment and the interpretability of the measurement. Many imaging techniques are two-dimensional and are sensitive only to the surface or near-surface region. Local measurements can be susceptible to noise simply because they measure a small amount of material, which can have less signal and more fluctuations than a large volume. The response is even more complicated if either the sensor or the sample is changed by the interaction. Small amounts of material and delicate states of the sample can be easily perturbed by the probe.

When the interaction between probe and sample is well understood, however, it is possible to take advantage of this effect to combine local measurement with local manipulation. For instance, the inset in Fig. 1e shows an image of a vortex taken with a magnetic force microscope at a temperature at which the vortex is strongly pinned. The main panel of Fig. 1e shows, at higher temperature, how such vortices respond to being pulled through the material by the force exerted by the probe: each vortex acts like an elastic string moving in an anisotropic pinning landscape. In Fig. 2f, a scanning tunnelling microscope was used to assemble a wire of magnetic atoms on a superconductor, deliberately designing the structure to create a Majorana bound state. The same scanning tunnelling microscope was used afterwards to perform tunnelling spectroscopy on the wire and look for the Majorana state.

Just as imaging provides information that is resolved in real space, other types of measurements provide information as a function of other parameters. For example, dynamic measurements provide information resolved in time. Scattering measurements such as angle-resolved photoemission spectroscopy or neutron scattering provide resolution in momentum space — hence these techniques are often called k-space microscopy or k-space imaging. A single experiment can also probe samples along multiple dimensions; for instance, a 'seven-dimensional' imaging experiment may provide time-resolved images in three spatial dimensions as a function of magnetic field, electric field and temperature. The term spectroscopy broadly describes any measurement with energy resolution. Many types of spectroscopy such as neutron spectroscopy, X-ray spectroscopy, optical conductivity or various kinds of tuned-voltage transport experiments show how materials interact with probes of specific energies. Measurements such as scanning tunnelling spectroscopy that combine energy resolution with spatial resolution have great impact in the characterization of quantum materials.

The widespread diffusion of imaging techniques has been enabled by a number of technological trends. Enhanced software and hardware capabilities have allowed automated control of experiments as well as storage and analysis of large datasets. Budding researchers, who once might have learned numerical methods during their PhD thesis if at all, now find it both desirable and possible to acquire numerical sophistication as early as they can. Nanofabrication facilities are widely available and can be used to make exotic scanning probes, while government-run beamlines provide access to high-quality beams of accelerated particles of many types. Both scanning technology and vibration control technology have improved and can be purchased from scientific instrumentation companies. Under the ongoing influence of these trends and the strong desire to make local measurements, imaging technology will continue to make huge improvements. Researchers will widely adopt techniques that can measure local properties with high spatial resolution: every material property that can be measured will be measured locally. Combined measurement modalities will show the relationship between different functions and between structure and function — for instance, by registering transport and diamagnetism, grain structure and ferroelectricity, chemical composition and any other property. The overall imaging time for large samples — which is usually rather long compared with bulk measurements — may be reduced by replacing single scanning probes with arrays of sensors or scanning beams, enabling imaging technologies to be used for high-resolution screening with fast turnaround.

As in all fields of science, clarity, transparency and reproducibility can accelerate the impact of experiments that image functionality. As imaging becomes more widespread, journal editors and reviewers could consider establishing community standards for the presentation of experimental results. Such standards may include the expectation of scale bars in all images, and colour maps in real units when false-colour plots are used; the original datasets used to perform all analyses should be reported or made available. The Methods section of manuscripts reporting imaging results should ideally answer several questions. What is the spatial resolution? What is the resolution in time, energy, momentum or other dimensions? How are these resolutions defined and determined? What is the contrast mechanism — that is, what property or properties does the imaging probe measure? How sensitive is the measurement to that property? What are the possible interactions between sample and probe, and what could be their effect on the measurements?

Looking ahead, precision imaging measurements with high resolution in space, time, momentum and/or energy will continue to produce textbook experimental results that demonstrate important concepts in a single image. Widespread, well-interpreted, fast-turnaround imaging will allow great improvements in sample growth and selection. With the support of imaging tools able to visualize exotic quantum states in any material structure, scientists around the world will improve their ability to grow, characterize and measure quantum materials, ultimately seeking to build a theoretical framework that correlates structural characteristics and quantum behaviour. We hope to predict the properties of those materials so well that we can specify a desirable functionality and then design materials optimized for that functionality. The more quantitative, interpretable and accessible our imaging capabilities become, the more they will accelerate the modelling–growth–experiment feedback loop, and hence the discovery and design of exotic functionality and phenomena.


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  1. Kathryn Ann Moler is at the Departments of Applied Physics and Physics of Stanford University, Stanford, California 94305-4045, USA

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Correspondence to Kathryn Ann Moler.