## Abstract

The synthesis of bulk crystals, thin films and nanostructures plays a seminal role in expanding the frontiers of quantum materials. Crystal growers accomplish this by creating materials aimed at harnessing the complex interplay between quantum wavefunctions and various factors such as dimensionality, topology, Coulomb interactions and symmetry. This Review provides a synthesis perspective on how this discovery of quantum materials takes place. After introducing the general paradigms that arise in this context, we provide a few examples to illustrate how thin-film growers in particular exploit quantum confinement, topology, disorder and interfacial heterogeneity to realize new quantum materials.

## Main

Although the properties of all materials are fundamentally explained by quantum mechanics, the classification 'quantum materials' refers specifically to materials whose defining behaviour is rooted in the quantum world, with no classical analogue. A recent US Department of Energy workshop^{1} resulted in a consensus definition of quantum materials as being “solids with exotic physical properties, arising from the quantum mechanical properties of their constituent electrons.” Despite the contemporary sheen suggested by the name itself and this definition, the properties of quantum materials have of course been known to mankind for a long time: the quintessential manifestations are ferromagnetism, reputedly discovered in lodestone by the Greeks around 2,500 years ago, and superconductivity, first discovered by Heike Onnes a little over a century ago. However, it is only during the materials revolution of the 'age of silicon' that physicists have developed a sophisticated-enough understanding of the properties of quantum materials to explore them through systematic synthesis. The science and art of crystal growth have played a seminal role in leading us through this voyage of discovery, revealing a vast panorama populated by materials whose quantum properties stem from a complex interplay between factors such as reduced dimensionality, quantum confinement, quantum coherence, quantum fluctuations, topology of wavefunctions, electron–electron interactions, relativistic spin–orbit interactions and fundamental symmetries. Examples of these exotic quantum properties include the integer and fractional quantum Hall effects in two-dimensional electron gases (2DEGs)^{2,3}, high-temperature superconductivity in the cuprates^{4}, triplet superconductivity in the ruthenates^{5}, the quantum spin Hall and quantum anomalous Hall effects in topological insulators^{6,7}, emergent electronic behaviour, superconductivity, magnetism at complex heterointerfaces^{8,9,10,11,12,13,14}, metal–insulator transitions^{15} and quantum spin liquids^{16}.

Discovering new quantum behaviour has commonly been the outcome of a well-established tradition in condensed matter physics: experimentalists often venture into uncharted territory by synthesizing a new crystal, guided by some combination of (often incorrect) received wisdom and (sometimes correct) intuition about the characteristics needed to unearth unprecedented quantum properties. Examples of this approach are abundant, but probably best exemplified by two of the major discoveries in contemporary physics: high-temperature superconductivity in the cuprates and the fractional quantum Hall effect. The former discovery occurred by hunting for materials in which Jahn–Teller polarons might lead to a strong electron–phonon coupling^{17}, whereas the latter was the result of an elegantly simple scheme ('modulation doping') for reducing scattering from donor impurities in a 2DEG^{18}. In both cases, the actual discoveries are best categorized as the outcome of ingenious serendipity: the genius lies in deciding where to look even with incomplete knowledge and then capitalizing on the unexpected surprise. But there is another highly effective paradigm that has emerged in recent years for the discovery of quantum materials. This relies on advances in contemporary condensed matter theory that allow quantum materials to be conceived on a theorist's sketchpad, brought to life by the crystal grower, then exhibited on the stage of physics and technology by sophisticated experimental measurements. This paradigm began in the 1980s with the design of quantum semiconductor structures (quantum wells, quantum wires, quantum dots, heterostructures and superlattices) that allowed the precise definition of envelope wavefunctions for electrons in modulated crystals^{19}. Over the past decade, this approach has advanced well beyond the mere design of wavefunction amplitudes towards the tailoring of wavefunction topology^{20,21}. This has led to the synthesis of topological insulators and topological semimetals in which the ordinary electron can mimic the behaviour of exotic particles hitherto imagined to be relevant only for particle physics. Examples include massless helical Dirac fermions, Weyl fermions, axions and Majorana fermions. Topological quantum materials now serve as platforms for staging fundamentally new physical phenomena, such as the quantum spin Hall effect, the quantum anomalous Hall effect, the chiral anomaly, and the topological magneto-electric effect, wherein these quasiparticles play key roles.

A number of existing articles have addressed quantum materials from the viewpoint of theory and measurements^{22,23,24,25,26}. This Review, in contrast, discusses the discovery of quantum materials from the perspective of a crystal grower. We try to address the following questions: how do crystal growers contribute to the process of discovering a quantum material? When do we take a path of emergent discovery? And when do we “listen to theorists,” ignoring the infamous admonition by Bernd T. Matthias to do otherwise? Is crystalline perfection a necessity? Or can the messy reality of actual crystals, with all their blemishes, actually contribute to interesting physics? What are the future opportunities and strategies for making the next big discovery in quantum materials? These are questions that affect crystal growers in different ways across every specialized area of condensed matter physics. Figure 1 summarizes key aspects of various modern approaches to the synthesis of quantum materials. The science of contemporary crystal growth has many facets and, in a single Review, it is impossible to provide a generalized picture that covers the enormous advances made in bulk crystal growth, thin-film growth and nanostructure synthesis. Here we offer a few vignettes, admittedly limited glimpses centred on thin films that leave out many aspects of a vast endeavour, especially the rich world of bulk single-crystal growth. Nonetheless, we hope that these descriptions serve well to illustrate how crystal growers contribute to the discovery of quantum materials.

We begin by discussing the first generation of design-to-order quantum materials — low-dimensional semiconductor quantum confined structures — for which crystal growers have developed exquisite control over crystalline purity and new phenomena have emerged directly as a consequence of this perfection. We also briefly address in this context the recent rapid development of crystal growth of single-atomic-layer materials, where both quantum confinement and magnetism are taken to their true 2D limit. Next, we discuss the second generation of design-to-order quantum materials, wherein crystal growers have been able to follow theoretical predictions to design the topology of crystalline band structures that are impervious to disorder. Nonetheless, crystalline imperfections must be overcome in order to reveal clearly the topological states in conventional transport experiments. The next section addresses the role of disorder in emergent quantum behaviour: it is an essential component of phenomena such as metal–insulator transitions but seems to also lurk in the background of the recently discovered quantum anomalous Hall effect. Finally, we provide an overview of emergence created by interfacing with heterogeneous materials, in particular addressing the surprising appearance of superconductivity and ferromagnetism at oxide interfaces.

## Quantum confinement

Our ability to quantum engineer the physical properties of materials by using parameters such as quantum confinement, strain and material composition to tailor wavefunctions has been well-honed since the 1980s, when synthesis techniques such as molecular beam epitaxy (MBE), metal–organic chemical vapour deposition and variants thereof enabled the creation of semiconductor quantum wells and superlattices. By controlling the thickness and composition of semiconductor thin films along the growth axis, crystal growers learned how to tailor the potential seen by electrons and holes in the bands of semiconductors at length scales comparable to their de Broglie wavelength, thus modulating the envelope wavefunction through quantum confinement in one spatial direction. The crystal grower's challenge in this context was to make materials with the least amount of defects possible, characterized by parameters such as narrow X-ray rocking curve widths, high electron mobility and narrow inhomogeneous spectral linewidths. The development of ultrahigh-mobility 2DEGs is a quintessential example in this context. Once crystal growers figured out how to optimally separate 2DEGs from scattering centres through careful design of the modulation doping heterostructure, a systematic programme of technological refinement in MBE techniques was carried out by the leading practitioners in the field, with attention paid to the purity of source materials, the design of MBE chamber components and the detailed sample growth protocols^{27}. Figure 2a is an historical summary of the improvements in 2DEG mobility, showing an enhancement of four orders of magnitude obtained over several decades^{28}, with Fig. 2b showing the structure of a characteristic sample used for generating 2DEG samples with record mobility (about 35 × 10^{6} cm^{2} V^{−1} s^{−1} at 4.2 K). Achieving these exceptional values of electron mobility requires a concerted sequence of sample growths to remove residual impurities from the source materials (Fig. 2c). The highly reduced disorder allows the study of 2DEGs with very low carrier densities, in which electron–electron interactions dominate over the kinetic energy without suffering the deleterious effects of reduced screening. When such 2DEGs are subject to a strong enough magnetic field and a low enough temperature to freeze out thermal excitations, stable new quasiparticles (composite fermions) emerge^{29}, leading to the fractional quantum Hall effect as well as a cornucopia of emergent properties at ultrahigh mobility, including bubble/stripe phases and new fractional states due to interactions between composite fermions themselves^{28}. Further developments and refinements in both epitaxial growth and chemical synthesis, as well as post-synthetic processing, naturally extended this type of wavefunction engineering to the other two spatial dimensions, resulting in quasi-1D quantum wires^{30} and 0D quantum dots^{31}. Parallel developments in the growth of magnetic heterostructures, both epitaxial and polycrystalline, have allowed design and control over magnetic properties such as magnetic anisotropy, interlayer coupling and spin-dependent tunnelling^{32}. At the interface of these fields, magnetic semiconductor quantum structures have allowed researchers to tailor the interaction between the extended band states of conventional semiconductors and the localized spins associated with magnetic dopants^{33,34}.

Although the epitaxial growth of semiconductors with near-atomic layer control provides a powerful approach for achieving effectively 2D electron systems, the actual physical dimensionality of the active layers is usually 3D. All we need is a confining layer for electronic states thin enough to energetically separate the ground state from the first excited state so as to freeze out the associated spatial degree of freedom. Similarly, effectively 2D magnetic materials can be realized in 3D crystals wherein the exchange coupling between localized spins in a given plane is much stronger than that between spins in different planes^{35}. The advent of graphene and other single- (or few-) atomic layer materials dramatically changed this situation by providing the first truly 2D quantum materials^{36}. In contrast with the expensive and technologically elaborate thin-film deposition methods used for epitaxial growth of 2D semiconductor and magnetic thin films, samples of these 2D materials can be fabricated from van der Waals bonded crystals using the low-cost method of exfoliation. This methodology has rapidly led to the discovery of a wide variety of 2D quantum phenomena, including exceptionally robust excitons with binding energies of up to 500 meV (ref. 37), exotic 2D superconductivity^{38,39} and 2D ferromagnetism^{40,41}. The manufacturing demands of technology have of course encouraged the development of synthesis beyond simple exfoliation to wafer-scale growth^{42,43} and the development of complex device heterostructures^{44}

## Topology by design

When the integer quantum Hall effect was first discovered, one of the deep mysteries it brought to the fore was the robust universality of the phenomenon: regardless of the material that hosts a 2DEG, the Hall conductance is precisely quantized in units of *e*^{2}/*h* as long as the electrons can make enough cyclotron orbits in a magnetic field before being scattered. This universality is now understood to be a consequence of the distinct topological nature of the quantum Hall insulator^{45}, which is characterized by a non-zero Chern number that makes it fundamentally distinct from the surrounding vacuum, where the Chern number is zero. The boundary between these fundamentally distinct topological states of matter carries current in protected 'edge states' that do not dissipate energy. This perspective on seeking materials whose electronic spectrum is characterized by different topological invariants has led to an explosion of theoretical activity that predicts a variety of topological phases of matter: a few prominent examples include topological insulators^{46}, topological Kondo insulators^{47}, topological crystalline insulators^{48}, Dirac semimetals^{49} and Weyl semimetals^{50,51,52}. In each of these cases, the topological invariant of interest arises because of some fundamental symmetry, such as time-reversal or symmetry related to crystal structure. Theorists are in luck with these predictions because the underlying physics, although sophisticated in methodology, involves single-particle band structure calculations that can be carried out with reasonable accuracy using analytical and first-principles methods. Experimentalists have been quick to capitalize on this aspect, synthesizing the materials of interest in bulk, thin film and nanostructure form, then using methods such as angle-resolved photoemission spectroscopy (ARPES) and electrical transport to demonstrate the existence of the predicted electronic states^{53,54,55,56,57,58}

The different challenges and opportunities presented by topological materials provide an excellent example of the contrasting approaches of crystal growers to bulk and thin-film samples. Bulk crystal growers are able to pounce on a prediction, rapidly pursuing the exploration of a potentially interesting material. For instance, experimental evidence for the helical Dirac surface states of 3D topological insulators was shown in the tetradymite family of crystals^{59} using ARPES with spin-resolved detection soon after their theoretical prediction. However, these measurements also indicated that such crystals — although long-known to be excellent thermoelectrics — were far from being ideal topological insulators: since they are narrow-bandgap semiconductors, their chemical potential of as-grown crystals was typically located within the bulk states due to extrinsic doping by point defects such as chalcogen vacancies. Similarly, thin-film growers, despite years of experience in perfecting other semiconductors such as the group IV, III–V and II–VI semiconductors, also found themselves struggling to synthesize tetradymite thin films in their intrinsic state. In MBE growth, we can ideally grow pure crystals under an ultrahigh vacuum, set the ratios of incident elemental fluxes at will to control stoichiometry, exploit non-equilibrium growth to overcome the thermodynamic constraints of bulk crystal growth, and use substrates to stabilize stubborn phases. The MBE growth of tetradymite thin films (such as Bi_{2}Se_{3}, Bi_{2}Te_{3}, Sb_{2}Te_{3} and derivative compounds) can indeed be readily carried out on a variety of substrates, yielding quintuple-layer by quintuple-layer growth and helical Dirac states detected in ARPES (Fig. 3). Since these are van der Waals-bonded crystals, the usual constraint imposed by lattice-matching is relaxed. However, thin films of these materials are still far from the structural perfection attained in other MBE-grown semiconductors, often displaying textured surfaces, twins and antiphase domain boundaries, as well as mosaic twists and tilts^{60}. The density of these structural defects can be reduced by careful choice of the substrate^{61}, resulting in very narrow X-ray rocking curve widths in Bi_{2}Se_{3} grown on (111) InP, for instance. However, even these films never become monocrystalline over length scales greater than a few micrometres. More importantly, they remain extrinsic, with carrier densities that place the chemical potential in the bulk bands.

One solution to producing more insulating samples of chalcogenide topological insulators is to exploit the defects themselves: in some chalcogenides, such as Bi_{2}Se_{3} and Bi_{2}Te_{3}, the easily formed point defects are donors (for instance, Se vacancies), whereas in others, such as Sb_{2}Te_{3}, they are acceptors. Synthesizing compound crystals that combine such materials (Fig. 3d) leads to self-compensation and a low carrier density with the chemical potential located in the bulk bandgap, or at least low enough to allow further tuning using electrostatic gates. This strategy has been effectively exploited in both bulk crystal growth^{62,63,64,65} and thin-film growth (Fig. 3e)^{66}. This does, however, come at a price: even though the topological surface states become dominant in electrical transport, they are not completely exempt from scattering even in the presence of time-reversal symmetry, and the alloy disorder in these compound crystals does not improve the situation. Ideally, crystal growers would like to stick with a single topological insulator such as Bi_{2}Se_{3} and figure out how to synthesize defect-free, intrinsic crystals. Nature, however, makes the defect chemistry in this material challenging: Se vacancies (donors) form at low energy cost, and attempts to suppress them using a high Se flux result in the formation of Se antisites, which are also donors. Non-equilibrium approaches to thin-film growth offer promising opportunities to address this problem. A successful example is the MBE growth of high-mobility, low-carrier-density Bi_{2}Se_{3} thin films on a substrate created by an elaborate deposition and annealing sequence, followed by an electron depleting overlayer^{67}. The realization of these thin films has been directly responsible for the experimental discovery of the topological magnetoelectric effect and axion electrodynamics using the quantized Faraday/Kerr effect in the terahertz domain^{68}.

## The role of disorder

Although theorists are often loath to take into account imperfections and disorder in crystals because of the serious challenges they present for calculations, crystal growers must invariably confront the presence of defects. As elegantly phrased in ref. 1, “embracing disorder” in a controlled manner may indeed be a viable route for crystal growers to not only discover new quantum materials but also understand existing ones. Indeed, in hindsight it has been realized that many characteristic phenomena in quantum materials depend on the presence of disorder. For instance, the integer quantum Hall effect requires a delicate balance between disorder and perfection. It is well-recognized that disorder is also an important ingredient in the behaviour of strongly correlated electron systems: it may be responsible for stabilizing correlated behaviour, such as nematic phases in iron pnictides and charge density waves in NbSe_{2} (ref. 69). It is also an inevitable ingredient in the physics of the metal–insulator transition in strongly correlated electron systems; it is therefore important in experiments to disentangle the extrinsic effects of disorder from the intrinsic influence of bandwidth^{70,71}. As an illustration of the unexpected role disorder can play in quantum materials, we now discuss the recently discovered quantum anomalous Hall insulator.

As discussed earlier, the band structure and wavefunctions of some quantum materials are characterized by a topological invariant derived from the presence of a fundamental symmetry. When this symmetry is broken, it often leads to the emergence of exotic states. For example, the massless nature of the helical Dirac surface states in 3D topological insulators is destroyed if time-reversal symmetry is broken when these states are coupled to ferromagnetic order with a magnetization perpendicular to the surface. This results in the opening of a gap in the Dirac cone of 3D topological insulators^{72}. In principle, this can be realized if a 3D topological insulator is doped with appropriate magnetic ions. Although the presence of a gap reaching tens of meV can be seen in magnetically doped topological insulators by ARPES^{73,74}, it is now being recognized that this may originate from impurity-induced resonances, rather than the breaking of time-reversal symmetry^{75}. More robust and quantitative evidence for the opening of a broken time-reversal symmetry gap arises from quantum transport measurements carried out with the chemical potential (or Fermi level) adjusted to be within this gap. This gives access to 1D chiral edge states of the quantum anomalous Hall insulator that are truly free of dissipation (σ_{xx} = 0) and ensures the Hall conductivity is quantized (σ_{xy} = *e*^{2}/*h*), even at zero magnetic field^{76}. This phenomenon is a variant of Haldane's concept of a Chern insulator^{77} and is qualitatively distinct from the quantum Hall insulator, which requires the formation of Landau levels in the presence of an external magnetic field. The first predictions of the quantum anomalous Hall effect directed experimentalists to pursue ultrathin Cr-doped (Bi,Sb)_{2}Te_{3} thin films, and crystal growers immediately set about searching for this predicted phenomenon. Remarkably, the first example of a quantum anomalous Hall insulator was realized in exactly the type of sample envisaged by theory^{7}. But, as always, the real world of materials invariably holds surprises.

The first surprise was the precise measurement of the quantum anomalous Hall effect^{78,79} in a material system that theory explicitly advised against — namely V-doping — in which calculations predicted a metallic rather than insulating ferromagnet^{76}. The quantum anomalous Hall effect has been confirmed by many other groups in both Cr- and V-doped (Bi,Sb)_{2}Te_{3} thin films^{80,81,82,83,84}. We note that even though transmission electron microscopy shows the formation of homogeneous phase pure films, magnetically doped topological insulator thin films are quite disordered and cannot be described as high-quality crystals. Structural defects abound, including vacancies, twins and mosaicity (Fig. 4a–c). The structural disorder is also accompanied by magnetic disorder: scanning superconducting quantum interference device (SQUID) magnetometry reveals that, at least at temperatures down to 250 mK, magnetization reversal occurs locally because strongly pinned domains change orientation without propagation of domain walls (Fig. 4e–f)^{85}. This is the second surprise, namely that ballistic edge transport occurs in samples whose Drude mobility is limited to a few hundred cm^{2} V^{−1} s^{−1}. Nonetheless, a precisely quantized anomalous Hall effect, at the level of 4 parts in 10,000, is observed at temperatures below 100 mK, even in millimetre-scale patterned Hall bar devices (Fig. 4d). This is perhaps the clearest evidence for topologically protected transport in topological insulator crystals and it ironically occurs when time-reversal symmetry is broken. Indeed, recent studies seem to suggest that this disorder may even be necessary for attaining good quantization since it localizes other dissipative conduction paths that are not edge states^{86}.

The magnetic gap opened by broken time-reversal symmetry can be measured directly by employing the temperature dependence of the activated longitudinal conductivity of the quantum anomalous Hall edge channels, *σ _{xx}* ≈ exp(−

*Δ*/

*kT*). These measurements present a third surprise. The activation gap (150–200 μeV) is much smaller than the exchange energy implied by the Curie temperature (about 2 meV)

^{84}. This discrepancy does not seem to be currently understood, but we speculate that it arises from the presence of significant structural, electronic and magnetic disorder, which likely leads to spatial variations in the chemical potential and resulting exchange interactions between itinerant electrons and local moments.

An interesting challenge is to raise the temperature for observing the quantum anomalous Hall effect. Crystal growers are already making progress by using intuitive analogies inspired by modulation doping^{87} and trying to exploit the interaction between the topological surface states of an undoped topological insulator interfaced with ferrimagnetic insulators^{88}. If these strategies succeed, we can envision potential technological concepts that could exploit these robust dissipation-free edge states for chiral spintronics, a natural extension of the rapidly developing area of topological spintronics that aims to exploit helical Dirac states for spin–charge conversion^{89,90}.

## Heterogeneous interfaces

It has long been recognized that juxtaposing a normal metal with a superconductor induces superconductivity through a proximity effect^{91,92}. Extensive studies of this phenomenon have led to further discoveries of proximity-induced superconductivity in more exotic geometries such as ferromagnets^{93} and topological insulators^{94,95}. Similarly, proximity to a ferromagnetic material can also induce magnetic order in conventional metals^{96}, graphene^{97} and topological insulators^{88}. Even more interesting is the case of emergent quantum behaviour that seemingly appears out of nowhere when two ordinary materials are interfaced with each other. A classic example of such behaviour is the case of LaAlO_{3} and SrTiO_{3}, a heterointerface whose study was motivated by a desire to artificially engineer polar discontinuities and exploit the resulting charge transfer to create a low-dimensional electron system. Indeed, oxide heterointerfaces in general provide a rich playground for a variety of emergent quantum phenomena^{98}

In the simplest picture of the ionic limit, the LaAlO_{3} lattice in the [001] direction may be viewed as alternating oppositely charged sheets, whereas SrTiO_{3} is charge-neutral: the heterointerface between these two wide-bandgap materials then creates an opportunity for intrinsically forming a 2DEG. This could also happen for extrinsic reasons, due to imperfections such as oxygen vacancies. The epitaxial growth of LaAlO_{3} on (001) SrTiO_{3} substrates, carried out with precise control over the nature of the heterointerface, showed that although the (AlO_{2})^{−}/(SrO)^{0} interface is highly insulating, the (LaO)^{+}/(SrO)^{0} interface yields a high-density, high-mobility 2DEG^{8}. An immediate question that arose then was the nature of the ground state of this 2DEG. Experiments soon discovered a rich range of behaviour: the 2DEG could be superconducting with *T*_{C} around 200 mK (ref. 9), it could be ferromagnetic^{10} and, most fascinating of all, superconductivity and ferromagnetism could coexist in phase separation^{11,12}. Well-controlled epitaxial growth of thin films played a crucial role in these discoveries; the key lies in systematic manipulation of the heterointerface through parameters such as the nature of the termination of the SrTiO_{3} substrate and the oxygen partial pressure during growth of LaAlO_{3}. Crystal growers continue to extend the lessons learned from the study of the LaAlO_{3}/SrTiO_{3} heterointerface to a variety of other oxide heterointerfaces, such as SmTiO_{3}/SrTiO_{3} (ref. 99). There are likely to be more surprises in store as materials such as these are explored further.

The heterointerface between SrTiO_{3} and other non-oxide materials also continues to serve as a treasure trove for discovering emergent quantum phenomena. For instance, a particularly exciting opportunity arises from the optical gating of topological insulator thin films^{100} and local optical control of ferromagnetic order in magnetic topological insulators grown on SrTiO_{3} substrates^{101}. Perhaps the most intriguing recent discovery is the observation of high-temperature interfacial superconductivity in single-unit-cell β-phase FeSe grown by MBE on (001) SrTiO_{3}. We note that bulk FeSe is a low-temperature superconductor with *T*_{C} = 8 K under ambient conditions. The first hint of an enhanced *T*_{C} in single-unit-cell FeSe films arose from *in vacuo* scanning tunnelling spectroscopy studies that showed a large superconducting gap of *Δ* ∼ 20 meV, thus suggesting an estimated superconducting transition temperature *T*_{C} of around 80 K (ref. 13). High-resolution ARPES measurements of similarly grown samples also find a gap of *Δ* ∼ 13 meV and *T*_{C} of around 60 K (ref. 102). Importantly, replica bands in these ARPES spectra are attributed to the removal of quanta from oxygen-related bosonic modes in the SrTiO_{3} substrate, which suggests that the interfacial coupling between electrons in FeSe and optical phonons in SrTiO_{3} probably plays an important role in enhancing superconductivity. Further credence to the interfacial nature of the observed superconductivity arises from the observation that scanning tunnelling spectroscopy and ARPES, both surface-sensitive probes, only see superconductivity in single-unit-cell samples, and not in thicker samples. This is consistent with *ex situ* transport measurements on surface-passivated films, which typically yield a maximum *T*_{C} of only around 30 K since it is only the first unit cell that supports superconductivity. The most tantalizing results reported on single-unit-cell FeSe/SrTiO_{3} stem from *in vacuo* 'deeply invasive' four-probe resistance measurements that indicate a remarkably high *T*_{C} of around 100 K (Fig. 5)^{14}. Whether or not this astonishing result will stand the test of time remains to be seen, but it is a credible extension of the promise shown in earlier scanning tunnelling spectroscopy and ARPES measurements.

## Outlook

The rise of quantum materials in the contemporary sense of the term is a direct result of crystal growers effectively navigating toward materials that enable an emergent interplay between quantum confinement, topology, interactions, quantum fluctuations and quantum coherence. Perfection in materials synthesis has been important for discovering exciting new physics in this endeavour, but disorder has also often unexpectedly played a key role. As crystal growers keep advancing the state of their art, interesting new physics will undoubtedly be discovered. For instance, 2DEGs with mobilities exceeding 35 × 10^{6} cm^{2} V^{−1} s^{−1} might result in new quantum states of composite fermions; better controlled oxide interfaces might provide an alternative route from the cuprates towards high-temperature superconductivity; and quantum anomalous Hall insulators with reduced disorder might allow the realization of emergent topological states in zero field. But is this really where quantum materials can have a long-lasting scientific impact? This crystal grower sees an invaluable lesson in the unprecedented success of semiconductor quantum structures and thin-film magnetism, where the engineering of wavefunction amplitudes (including orbital and spin degrees of freedom) was motivated by both fundamental questions and a clear-eyed vision of broad technological impact in information technologies. In an analogous manner, we now have the opportunity to develop new generations of quantum materials aimed at creating, initializing and manipulating the full quantum mechanical wavefunction, both in terms of amplitude and phase. This exciting brave new world has already begun to be populated, using well-established quantum materials such as superconductors^{103}, single-atom transistors^{104} and semiconductor quantum dots^{105} for quantum computing. The engineering of hybrid quantum materials promises to make even broader technological impact on quantum information technologies. One vision is to create solid-state platforms for scalable quantum entangled systems wherein distant qubits on a chip are coupled via a quantum bus created by a macroscopic quantum state. Such schemes have so far been pursued by using superconducting qubits that communicate with each other via virtual photons in a microwave cavity^{106}. Another path forward is the pursuit of hybrid superconductor/semiconductor nanostructures that could host braided Majorana states for topological quantum computing^{107}. Can clever hybrid materials synthesis provide a way to make heterogeneous solid-state systems wherein such functionality is seamlessly integrated? The nexus of macroscopic magnetism with single spins presents equally exciting opportunities. For instance, the coherent interfacing of macroscopic spin states (magnons in a ferrimagnet) with localized ones (single spin defects in nanodiamonds) holds promise for on-chip quantum conduits^{108}. Multiferroic materials^{109} that entangle ferroelectric and magnetic order parameters could provide an even more powerful quantum platform in this context. Recent studies of a quantum anomalous Hall insulator have also shown possible evidence for the quantum tunnelling of macroscopic magnetic states out of a 'false vacuum'^{84}. Could this present a pathway for coherently manipulating a quantum memory using an edge current? The opportunities for the synthesis community to develop complex quantum material configurations for quantum information processing are indeed very exciting. The challenges are immense, but we might anticipate that the rapid progress that took place in the development of quantum materials in recent decades will be repeated in the years to come.

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## Acknowledgements

The author thanks A. Richardella and J. Kally for the unpublished atomic force microscopy and transmission electron microscopy data shown in this Review. This work is supported by the Pennsylvania State University 2D Crystal Consortium—Materials Innovation Platform (2DCC-MIP), funded through National Science Foundation Cooperative Agreement DMR-1539916. The author also acknowledges support from the National Science Foundation (DMR-1306510), the Office of Naval Research (N00014-15-1-2370, N00014-15-1-2675), the Army Research Office Multidisciplinary University Research Initiative (W911NF-12-1-0461) and C-SPIN, one of six centres of STARnet, a Semiconductor Research Corporation programme, sponsored by MARCO and DARPA.

## Author information

## Affiliations

### Department of Physics and Materials Research Institute, Pennsylvania State University, University Park, Pennsylvania 16802, USA

- Nitin Samarth

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### Competing interests

The author declares no competing financial interests.

## Corresponding author

Correspondence to Nitin Samarth.

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