Topological materials by molecular beam epitaxy

During the past 40 years, topology has permeated condensed matter physics and has since risen to become one of the major tenets of the field. Topology is a very general concept used to classify objects based on their broad properties, which is quantified by a single (or multiple) number(s) called the topological invariant. In mathematics, this captures the similarities among soccer balls and oranges, which are distinguished from objects with holes such as coffee cups and doughnuts. In the context of condensed matter physics, however, the attributes of geometrical objects, such as surfaces, curvatures, etc., are analogously replaced by aspects of the electronic wavefunction, which can be used to define a wide variety of topological invariants. This has enabled understanding and predicting a wide variety of electronic, magnetic, and superconducting phases. Historically, one of the first and most celebrated applications of topology in condensed matter physics was the observation of quantized Hall resistance with vanishing longitudinal resistance at ultraclean heterointerfaces.¹ This has become known as the quantized Hall effect, which can be distinguished by the topological invariant called the Chern number or the Thouless–Kohmoto–Nightingale–den Nijs (TKNN) invariant.² The topological description gave a full quantum mechanical explanation of the chiral edge state responsible for the quantized transport: the bandgap created by the two-dimensional (2D) quantized cyclotron orbits gives rise to the non-zero Chern number and is spanned by gapless states that are bound to the one-dimensional (1D) edge. Such metallic states localized on the boundary are a general feature of topological materials.^3,4 After this initial success, important advances of topology quietly evolved until the early 2000s, when the quantum spin-Hall insulator was predicted in graphene⁵ and later in HgTe/CdTe quantum wells.⁶ Quantum spin-Hall insulators, which exhibit two spin-polarized, counter-circulating quantum Hall edge states, are described by a different topological invariant in the Z₂ class.⁷ After experimental confirmation of quantized transport emanating from the 1D edge states,⁸ the Z₂ classification scheme was generalized to three-dimensions⁹ and gained the name topological insulators (TIs).¹⁰ Experimental confirmation came rapidly through the observation of novel surface states within the bulk bandgap of Bi_1−xSb_x using angle-resolved photo-emission spectroscopy (ARPES) to map the surface band structure.¹¹ There are a variety of well-known electronic states that exist on surfaces of bulk materials,^12,13 which occur with a Z₂ topological number that is trivially zero. The topological surface states (TSSs), in contrast, that emerge on the surface of a so-called strong TI exhibit an odd number of linearly dispersing Dirac states, which are singly degenerate with the direction of the spin locked to the direction of the momentum.^3,4 This demonstrated that a large expanse of novel electronic states existed in materials both new, as well as those long studied in other contexts. This started off a near exponential rise in theoretical and experimental studies of topological materials.

Since the genesis of this field, many distinct states have emerged with a zoo of confirmed and candidate topological materials.¹⁴ Including TIs mentioned above, select examples that are at the forefront of research today include topological crystalline insulators, which possess spin–orbit induced band inversion; but unlike TIs, this crossing is protected by the mirror symmetry of the crystal lattice and exhibits an even number of 2D Dirac cones.^15,16 Nodal systems are a very rich and diverse class of materials, where there is a three-dimensional (3D) band crossing that is protected by a symmetry, either crystalline or time-reversal.¹⁷ These crossings, for example, can either occur at points or along lines of the Brillouin zone. When the crossing occurs at a point, this gives rise to Dirac and Weyl semimetals, with the latter exhibiting novel Fermi arc surface states.¹⁸ The crossing can also occur along a line, which are referred to as nodal line semimetals. Superconductors with well-formed bulk gaps also can exhibit topological structures with gapless states, which are manifestations of Majorana fermions as electronic quasiparticle excitations.¹⁹ Higher order topological states are 3D materials, where the topological states emerge on the lower dimensional edges called hinge states.²⁰ Furthermore, topology extends outside the realm of atomic-scale materials to systems where the periodicity ranges from the micrometer-scale to even the millimeter-scale in the areas of topological photonics²¹ and phononics,²² respectively. The insight given by topological phases spans the fundamental physics of condensed matter but also pushes into the realm of high energy physics including, for example, Majorana fermions, Weyl fermions, and axionic phenomena.

As with nearly all discoveries in condensed matter physics, a slew of possible applications emerge. Topological materials are no exception. The novel character of the topological states can be applied across a range of electronic, magnetic, optical and photonic, and phononic devices. The potentially useful applications of topological materials include efficient spin generators and spin-charge converters in spintronic devices,²³ basic elements to build qubits for fault-tolerant quantum computing,^24,25 energy-efficient microelectronic components,²⁶ and novel information storage media based on topological spin-textures such as skyrmions.²⁷ Among these, applications in spintronics and quantum computing have gained much attention and have motivated studies of magnetism and superconductivity in topological material-based systems.

Addressing both fundamental questions as well as pursuing practical applications necessitates creating high-quality materials as thin films. Advanced synthesis techniques, particularly, molecular beam epitaxy (MBE), can easily control growth at the submonolayer level with chemistries ranging across the periodic table. MBE is used to create systems where materials with dissimilar properties are joined at an atomically sharp interface. Furthermore, this level of precision and accuracy during synthesis opens many additional routes to control and manipulate topological systems, for instance, tuning dimensionality through thickness scaling or modifying symmetry at interfaces. Key examples are as follows: At an interface with a superconductor, Cooper pairs can leak across the interface, which represents a novel route to realize topological superconductivity through combining a superconductor with a topological material or a semiconductor that has strong spin–orbit coupling. Furthermore, when the thickness of a material is reduced below the scale of the wavelength of the electrons, the effective dimensionality is reduced, and the band structure is changed. This, as discussed above, enabled the first realization of a 2D TI state in a HgTe/CdTe quantum wells since band inversion in that system is driven by finite thickness. Also, in the TI Bi₂Se₃, finite thickness effects arise due to the wavefunctions on the top and bottom surfaces hybridizing when the thickness of the films is than five monolayers or less,²⁸ which is amplified by mixing In on Bi.²⁹ Additionally, 3D Dirac nodal metals have a Dirac point that is fourfold degenerate. When a symmetry, either time-reversal or spatial inversion, is broken, the single Dirac point splits into two Weyl points with twofold symmetry.^10,17,18,30 This can be achieved in bulk crystals which lack either symmetry, but thin film systems can be constructed to controllably break these as well. For example, interfaces generally break inversion symmetry, and carefully choosing materials with strong charge transfer can enhance such an effect. Combining topological materials as heterostructures with magnetic materials can break time-reversal symmetry. Furthermore, considering the rich array of magnetic phases and their domain structures, this enables spatially tuning topological band structures in non-trivial ways. Finally, the goal of new electronic motifs based on these exotic phases necessitates wafer-scale (lateral scale > cm) thin films to facilitate the micro- and nano-scale fabrication necessary for creating functional electronic devices.

MBE is the gold-standard for the growth of epitaxial materials. This is due to the fact that MBE enables a high-level of control over the growth process, which, as discussed here, is a key to understanding many aspects of topological materials. As shown in the schematic of a typical MBE growth system in Fig. 1(a), this level of control is rooted in how the constituent elements are delivered to the growing surface of the thin film. Each element is individually heated to a temperature where evaporation/sublimation takes place, which creates a beam of atoms (or molecules). These beams converge onto a substrate crystal with an atomically flat surface. There, the elements adsorb, undergo diffusion on the surface, and finally chemically bond. The temperature of the substrate can be tuned to an optimum value which, in a simplistic view, maximizes surface diffusion to confine nucleation and growth at the proper atomic sites. Furthermore, MBE growth takes place under ultrahigh vacuum (UHV) <1 × 10⁻⁹Torr (for comparison, atmosphere is 760 Torr), where the mean free path is orders-of-magnitude larger than the chamber. This serves several purposes. First, the atomic beams suffer no scattering in route to the substrate, which enables highly uniform growth while minimizing both thermal leakage among the cells, and elemental cross-contamination.³¹ Under UHV conditions, the incorporation of contaminants from the background gas is minimized. As a typical example, at 1 × 10⁻⁹ Torr, it will take well over an hour to adsorb a monolayer of contamination (for growth of ultraclean materials vacuum conditions can be as low as 10⁻¹¹–10⁻¹² Torr³²). Finally, systems are typically equipped with in situ diagnostics such as reflection high-energy electron diffraction (RHEED), which probes both the crystal quality as well as the morphology of the surface. This enables rapid feedback to optimize growth conditions.

To grow the best quality topological materials requires precise characterization of the surfaces both from a structural perspective but also from an electronic point of view. This has spurred the development of powerful platforms for growing materials and characterizing all the surface properties in a single system without breaking vacuum (in fact, such systems have a long history in semiconductors and other areas, but topological materials have brought them to the forefront of research). An example is shown in the schematic in Figs. 1(b)–1(d), which is modeled after Oak Ridge National Laboratory's systems for growth and characterization of topological and correlated matter based on the designs of ScientaOmicron. Such a system includes a load lock that serves to bring samples from atmosphere to UHV without compromising the vacuum conditions. This is attached to an intermediate chamber, where samples are distributed to the various growth or characterization stations; this also serves as a buffer between the high pressure of the load lock and the UHV conditions of the various stations. Multiple chambers are not only for growing topological materials, but also for growing metals, intermetallics, or novel complex oxide materials to serve as functional layers to be interfaced with topological materials, for example, superconductors or magnets. Furthermore, there is a station dedicated to surface preparation for heating, sputtering, or simple depositions. In terms of characterization, ARPES is an indispensable tool for probing the surface band structure.³³ Typical systems can be equipped with He-lamp sources (primary line is He-I at 21.2 eV), laser sources (polarization control with energies ranging across 6–11 eV), or even stationed at a synchrotron. Scanning tunneling microscopy (STM) enables direct real space probing of surface morphology, atomically resolved crystal structure, as well as direct energy-resolved imaging of the surface wavefunction that enables full maps of the quasiparticle interference spectrum, which has been a valuable probe for understanding the novel scattering process of TSS in TIs³⁴ as well as novel effects in 3D Dirac metals.³⁵ Furthermore, chemical information using x-ray photoemission spectroscopy (XPS) is indispensable to elucidate how valence states are modified at heterointerfaces. Since many topological materials can rapidly degrade or change in air, in situ transport has recently been very critical at separating properties endowed during growth from atmosphere effects,^36–38 as well as probing highly unstable materials such as Na₃Bi^39,40 and the monolayer superconductor FeSe.^41,42 However, ex situ characterization systems are still critical, and scientists have performed profound work on stand-alone MBE systems. For example, state-of-the-art STM instruments may be too sensitive to be practical when attached to such systems. Moreover, ARPES measurements performed on synchrotron beamlines with full control over energy, polarization, and spot size, are necessary to understand complexities of the matrix-element effects associated with the photoemission process.⁴³ In such situations, in situ capping layers have proven very effective,⁴⁴ as well as instruments called vacuum suitcases that maintain UHV conditions for weeks during transportation. Finally, transfer out of the MBE system is critical for a myriad of other probes such as high-magnetic-field magneto-transport, magnetometry, diffraction (x-ray, neutron, etc.), as well as device fabrication.

It has been over a decade since the discovery of TIs, which reinvigorated the field of topological materials. In this time frame, many exciting things were revealed and problems have been solved. The goal of this Perspective article is to frame future problems in the field of topological materials that concern MBE growth. As such, we discuss the role that MBE has played in many key discoveries, while highlighting the scientific aspects and challenges that led to these successes. This is a key to emphasizing open problems that will expand scientific understanding as well as push toward applications of topological materials. This work is organized as follows. Section II highlights how a detailed understanding of the synthesis is required to realize intrinsic properties of topological materials. Since, by far, the tetradymite TIs are the best studied, these materials are the key examples, and it is emphasized that most known and candidate topological materials have not been studied by MBE. Section III details current challenges and opportunities faced in the field of realizing topological superconductors and the push toward Majorana fermions for use in quantum computing. Section IV concerns the direction of integrating magnetism and topology, which enables control over time-reversal symmetry and is key for many possible applications in topological spintronics.

The key to understanding and ultimately utilizing topological phenomena in real materials is isolating the unusual topological states such that they dominate the electronic properties. This fundamentally requires precise control of the materials, which comes down to positioning the Fermi level at a particular location of the band structure, with examples schematically shown in Fig. 2(a). In TIs and topological crystalline insulators, the novel character emerges only when the Fermi level is located near the 2D Dirac point; similarly, in 3D Dirac and Weyl semimetals, the Fermi level should be close to the 3D Dirac or Weyl point for the electronic properties to not be obscured by trivial states. Furthermore, accessing the topological superconducting phases that host Majorana fermions requires positioning the Fermi level of the topological superconductors sufficiently close to the middle of the bulk energy gap, as discussed in Sec. III. Finally, as discussed in Sec. IV, to observe the quantum anomalous Hall effect requires controllably breaking time-reversal symmetry which opens a gap at the 2D Dirac point and results in the emergence of a 1D edge mode; yet if the Fermi level is far away from this location, quantization of the transport properties cannot be observed. To achieve this level of control, considerable effort needs to be expended to optimize the growth conditions for each individual material, which ultimately comes down to understanding and minimizing defects. Such progress is often slow and painstaking, but when mastered, history shows that the results can be revolutionary.

The band structure of a material fundamentally determines its topological class.^3,4 In most topological materials, the bands that give rise to the non-trivial topological invariant are not necessarily close to the Fermi energy. This ultimately makes them impractical for device applications since the topological states are located far (>eV) from the Fermi level, making it impossible for those states to dominate the low-energy electronic properties. However, for a subset of these materials, the topological states are located within, or near to, the bandgap formed by the valence band and conduction band. Key examples are the tetradymite TIs Bi₂Se₃, Bi₂Te₃, and Sb₂Te₃,⁵² topological crystalline insulators SnTe,¹⁶ and 3D Dirac semimetals such as Na₃Bi and Cd₃As₂,¹⁸ many of which are semiconductors with narrow bandgaps. Since all these materials are charge balanced, the Fermi level of an ideal material, E_F,ideal, should be intrinsically located within the bulk bandgap. Yet, defects in the form of vacancies, interstitial atoms, or impurities incorporated during the growth often dope the materials, thus shifting the Fermi level from this ideal location toward the conduction or valence bands, as shown in Fig. 2(a) as E_F,nonideal. This creates parallel conduction paths effectively shorting out the electronic character of the topological states. For most topological materials, this has been known prior to the initial studies and has been found to be the typical situation. Therefore, the challenge in this field is first to understand and address the defects incorporated during the synthesis.

The formation of defects is not unique to topological materials, in that every material forms an array of defect states. Control and mitigation of defects have been major drivers toward understanding basic phenomena in condensed matter physics. To demonstrate how charged defects lead to the occupation of the bulk conduction or valence bands, consider an isolated charge embedded in an insulator. This local charge is screened by the surrounding electronic states. This is described by a hydrogen-like potential that is rescaled by the dielectric constant of the material, which has an effective Bohr radius given by a = ε(m_e/m*)a_B, where ε is the dielectric constant, m_e is the electron mass, m* is the effective mass in the material, and a_B ≈ 0.5 Å is the Bohr radius in free space. The transition to a metallic phase (occupation of the conduction band or valence band) occurs when the density of such defects, N_3D, is sufficiently large that these local charge states overlap. This occurs at the so-called Mott criterion when N_D ≳ (0.26/a)³ with N_D being the critical density.⁵³ For semiconductors like Si, the critical density is of the order of 10¹⁷–10¹⁸ cm⁻³.⁵³ In stark contrast, the critical densities for TIs are two orders of magnitude lower in the range of 10¹⁴–10¹⁵ cm⁻³.^52,54 This results in the Fermi levels being well away from the Dirac point in nearly all samples.

The highly sensitive nature of topological materials to defects is fundamentally intertwined with the properties that make them topological.⁵⁵ First, for band inversion, a necessary condition for most topological materials, the anion and cation electronegativity need to be similar,⁵⁶ which implies that the top of the valence band is in close proximity to the bottom of the conduction band; this creates a situation where spin–orbit coupling, if sufficiently strong, can invert the bands. However, this is also the same criteria that makes the formation energy of defects relatively low, meaning that such materials will readily accommodate defects. To highlight the challenges of growing and controlling topological materials by MBE we examine the tetradymite TIs Bi₂Se₃, Bi₂Te₃, and Sb₂Te₃, which are the most widely studied topological materials and the formation of defects is the most well understood from both the fields of TIs as well as thermoelectrics.^52,56 (The growth of II-VI HgTe/CdTe and narrow gap III-V topological systems have long been studied in the context of semiconductor devices, see Ref. 57.) The most common defects in these materials are vacancies on the anion site, for example, Se vacancies in Bi₂Se₃, which are electron donors, and antisite defects where the cation (anion) can occupy the anion (cation) site which are, in most cases, acceptors (donors).^58,59 Since both cations and anions can adopt different formal valence states and have similar electronegativity, predicting the defect chemistry is not straightforward. For example, depending on the details of the synthesis, Bi₂Te₃ readily forms all types of defects, which enable it to exhibit either n-type or p-type character. In contrast, Sb₂Te₃ is solely p-type due to the predominance of antisite defects. In Bi₂Se₃, the large size mismatch among Bi and Se reduces antisite formation and favors Se vacancies giving a ubiquitous n-type character. Finally, the heavy atoms in these materials yield a high polarizability, and, thus, high dielectric constants: For example, Bi₂Se₃ ε ≈ 113, Bi₂Te₃ ε ≈ 290, and Sb₂Te₃ ε ≈ 168.⁵² Through the Mott criterion, the high dielectric constant effectively lowers N_D. Together, these attributes create a scenario which raises the defect density, while lowering the critical density N_D. For bulk materials, this has been overcome by careful mixing schemes among the anion and cation sites along with resonant doping. For example, careful synthesis of both Sn-doped (Bi_1−xSb_x)₂(Te_1−ySe_y)₃ and (Bi_1−xSb_x)₂Te₂S, and Bi₂Se₃ under high Ar pressure have exhibited bulk carrier freeze-out at low temperatures and resistivities in the range of 10–100 Ω cm, which is consistent with the Fermi level being inside the bulk bandgap.^60,61

In general, bulk crystals are grown under quite different conditions compared to thin films, which manifests as the formation of different types of defects with different concentrations (see Ref. 62 for additional information concerning bulk synthesis). Compared to bulk crystals of the tetradymites which are typically synthesized at temperatures in the range of 700–800 °C, thin films are grown at much lower temperatures, typically in the range of 200–300 °C. Due to the reduced kinetics, this can limit the formation of certain defects. A key example to illustrate this is the formation of the high energy Se vacancies in bulk crystals of Bi₂Se₃ that form at the 3A locations (middle of the quintuple layer unit). In bulk crystals, the higher growth temperatures enable the formation of these defects, and the quick cooling process tends to trap them. For thin films, in contrast, the low temperature growth prevents their formation and only allows the formation of the lower energy Se vacancies with typically reduced concentrations.⁶³ 

In contrast to bulk crystals, however, interfaces are a major source of defects for films. In particular, these occur in the form of chemical reactions, interdiffusion, and dislocations related to the processes of heteroepitaxy. This highlights the challenge of understanding the structural and chemical relations between the film and substrate, and since many of the topological materials are layered van der Waals structures, these properties are particularly complex. In so-called van der Waals epitaxy,^64,65 the lack of chemical bonds orthogonal to the growth direction reduces the ability for the individual layers to chemically adhere in-plane, which, in turn, reduces the effect of epitaxial strain. This generally results in strain-relaxation, where the film relaxes to the bulk lattice parameters, which typically occurs in the first one or two monolayers. This prevents utilizing the epitaxial mismatch between the substrate and the film to distort the bonding environment, and, thereby, modifying electronic properties. There are advantages, however, for van der Waals epitaxy. The reduced sensitivity to lattice mismatch enables films to be grown on substrates with either large strains or even amorphous surfaces, as well as other novel post-growth processing such as mechanical exfoliation of wafer-scale films⁶⁶ or post-growth crystallization.⁶⁷ A major challenge arises, however, during the formation of the first monolayer: Growth occurs from random nucleation points from which single monolayer islands grow laterally. These islands then merge to form the first monolayer. Since the film and substrate have different lattice parameters, the boundaries where the mergers occur form defects, such as rotation and tilting between adjacent grains and secondary nanostructures.^68,69 Furthermore, for van der Waals epitaxy, understanding interfacial bonding and how to best mitigate dangling bonds prior to deposition is critical.⁶⁵ Key examples of this are the formation of Si–Se as an initial step in growing Bi₂Se₃ on Si (111)^70,71 and SiO_2,⁷² and Ga–Se at the interface among Bi₂Se₃ on GaAs.^73,74 These aspects, combined with any interfacial chemical reaction related possibly to unpassivated bonds, lead to a high number of defects confined to the interface, as can be seen in Fig. 2(b); subsequent layers which are nearly commensurate with the first monolayer typically have far fewer defects. Transport measurements for Bi₂Se₃ films, for example, show that these interfacial defects are electron donors.^45,75 Through band-bending effects, these additional electrons occupy both the TSSs at the substrate/film interface as well as moving the Fermi energy well into the bulk conduction band.^45,54,76 The knowledge gained regarding the key signatures of these defects and how they are mitigated is important for future studies across topological materials, where it is critical to control the Fermi energy and place it sufficiently close to the Dirac point.

Interfacial defect control has been achieved by choosing substrates with either a specific lattice match or chemical compatibility. In particular, substrates with a matching hexagonal surface structure that have been used are α-Al₂O₃,^45,77 Si (111),^71,75,78 InP (111),^79–81 SrTiO₃ (111),^82,83 GaAs (111),^84,85 graphene-terminated SiC(0001),⁸⁶ CdS (0001),⁸⁷ as well as Si with amorphous SiO₂ termination.⁸⁸ For Bi₂Se₃ epitaxial strain on these substrates ranges from 0.25% mismatch on InP, to values as large as 15% on α-Al₂O₃. Despite this, the electrical properties are not found to vary significantly, in that minimum values of the sheet carrier densities are typically found to be in the range of 1–3 × 10¹³ cm⁻² and mobilities in the range of 1000–2000 cm² V⁻¹ s⁻¹; based on the separation of the Dirac point from the conduction band minimum in Bi₂Se₃, a sheet carrier density below 1 × 10¹³ cm⁻² is necessary, but not sufficient, for the Fermi level to be below the conduction band minimum.^54,76 Furthermore, with cracked Se (or Te), similar results are found.^89–91 Furthermore, thickness independence of the sheet carrier density highlights that the source of defects are mainly the interface. In fact, this can be further seen by comparing epitaxial growth of Bi₂Se₃ on substrates with the same in-plane symmetry to films grown on Si with amorphous SiO₂. Despite the lack of in-plane crystal structure, the large crystalline anisotropy of layered Bi₂Se₃ enables it to grow with grains that are all oriented with their c axes perpendicular to the substrate surface, yet with random in-plane orientations. Surprisingly, transport properties are nominally the same as epitaxial films grown on other substrates with sheet carrier densities around 1 × 10¹³ cm⁻² and mobilities around 2000 cm²V⁻¹s⁻¹.⁸⁸ This again highlights the high density of defects imparted at the interfaces. We next discuss two key strategies that have proven successful for Fermi-level control to realize intrinsic topological properties.

Understanding the limitation that the majority of defects are confined to the interface motivated the development of an MBE-grown virtual substrate that is both epitaxially as well as chemically matched to the layered tetradymite TIs.⁴⁹ The key to isolating transport through the TI was the material In₂Se₃, which has a polymorph that is nearly identical to the structure of the tetradymites, and is a trivial band insulator with a bandgap of about 1.4 eV.^92,93 There were, however, several key challenges, which necessitated a careful understanding of the materials science involved in the growth, which is detailed schematically in Fig. 2(c). The first was that the In₂Se₃ system has many polymorphs; stabilizing the correct α-phase is challenging, since growth directly on Al₂O₃ nucleates multiple phases of In₂Se₃. Stabilizing the correct phase required first growing a thin layer of Bi₂Se₃, which then favors the nucleation of α-In₂Se₃. The problem is that when a subsequent layer of Bi₂Se₃ is grown on top of this Bi₂Se₃/In₂Se₃ bilayer, the bottom Bi₂Se₃ will create an electrical short. Interestingly, the high volatility of Bi₂Se₃ relative to In₂Se₃ combined with the large bulk diffusion coefficient of layered materials enable a novel solution: Bi₂Se₃ in the Bi₂Se₃/In₂Se₃ bilayer can be sublimated out from underneath In₂Se₃ using a high temperature anneal step following the growth of In₂Se₃. This leaves a very high-quality In₂Se₃ virtual substrate for the subsequent growth of Bi₂Se₃. However, a secondary challenge emerged: The high bulk diffusion caused small amounts of In₂Se₃ to diffuse into Bi₂Se₃. This is problematic since low concentrations of In in Bi₂Se₃ can drive it into a topologically trivial state.⁹⁴ This was subsequently solved by slightly lowering the growth temperature and growing a second layer composed of (Bi_0.5In_0.5)₂Se₃, which together lowered the In diffusion well-below the critical value, where the topological phase transition occurs. Growing Bi₂Se₃ on this virtual substrate lowered the sheet carrier density by an order of magnitude to the range of 10¹² cm⁻² and raised the mobility of the TSS to the highest reported value, 16 000 cm²V⁻¹s⁻¹. The carrier density was sufficiently low, and combined with the high mobility enabled observing the quantum Hall effect in the high-magnetic field limit, as well as novel quantized Faraday and Kerr rotation angles,⁹⁵ which is a direct signature of axion electrodynamics.^96,97 Recently, the virtual substrate was improved by moving to In₂Se₃/(In_0.35Sb_0.65)₂Te₃ with a subsequently grown Ti-doped Sb₂Te₃. This lowered the defect density to the range of 10¹¹ cm⁻², but the mobility was lower at around 3000 cm²V⁻¹s⁻¹; this ultra-low carrier density enabled reaching the extreme quantum limit, where a novel insulating phase emerged when the external magnetic field drove a splitting of the zeroth Landau level associated with the TSSs.⁹⁸ 

As a means of electrically controlling defects, compensation doping schemes have also seen success, yet have faced key challenges relative to how dopants are incorporated into the layered tetradymite structure. One of the first schemes successfully used, which has long been understood and utilized in the context of thermoelectrics, was balancing acceptor and donor defects through alloying.⁵² As discussed above, based on the ability to readily form both antisite defects as well as Te vacancies Bi₂Te₃ can be made either n-type or p-type depending on the growth conditions. Sb₂Te₃ is only p-type due to the propensity to form antisite defects. Therefore, creating the mixture (Bi_1−xSb_x)₂Te₃ is a direct route to controllably tune between n-type and p-type. This saw initial success in lowering the carrier density relative to pure materials down to a value of 2 × 10¹² cm⁻² for x ≈ 0.95, yet the mobility remained relatively low initially at around 500 cm²V⁻¹s⁻¹.⁹⁹ Key improvements were achieved in transport by gate-control of the Fermi level,¹⁰⁰ which enabled the observation of the quantum Hall effect.^101,102 As detailed in Sec. IV, this ability to carefully balance defect types was a critical driver of the success in observing the quantum anomalous Hall effect in MBE-grown (Bi_1−xSb_x)₂Te₃ films with magnetic doping.

To close this section, we wish to point to open questions and emerging directions. As the field of topological materials has expanded, so has the number of candidate or confirmed systems. As such, a key direction is to diversify the topological materials grown as high-quality thin films. Beyond the tetradymites, a few materials have been thoroughly explored via MBE growth. Prime material classes where significant efforts have taken place are the rocksalt class of topological crystalline insulators, particularly SnTe,^103–109 and the 3D class of Dirac and Weyl semimetals, particularly Cd₃As₂^110–114 and Na₃Bi.^115–118 Interestingly, the only MBE-grown topological material that was not studied prior to confirmation of its topological states was Na₃Bi; the other materials have extensive histories prior to the genesis of the field of topological materials. This highlights the challenge of developing new materials by MBE and shows that researchers typically only attempt growth of materials that are reasonably likely to succeed.

In comparison to bulk crystal synthesis, where a scientist can pursue multiple growths in parallel across many different chemistries during a timeframe of a few weeks,⁶² growth and refinement of a particular class of MBE-grown materials takes place across many months to years. This is largely limited by the throughput of the MBE process as well as elemental cross-contamination, where each chamber is dedicated to one specific chemistry, e.g., a single chamber must be dedicated to, for example, either arsenic, phosphorous, or selenium/tellurium. Coupling this with the facts that (1) MBE chambers are only opened several times or less per year for maintenance or to exchange or recharge elements and that (2) the cost of each chamber ranges from hundreds of thousands to millions of dollars, naturally makes this a methodical area of research. As such, choices regarding material selection and experimental design need to be made carefully, which highlights the benefit of close collaboration with researchers performing bulk synthesis as well as theorists. To clarify the thought process regarding what might make a good candidate, we list some key questions an MBE researcher asks themselves before pursuing new materials:

1. Safety: The first question should always be are the elements hazardous? Adding toxic elements to an MBE system means that the researcher will have to come into close contact with them during any maintenance for the foreseeable life of the MBE chamber, hence toxic elements—Hg, Tl, As, Pb, Os, Be, etc., anything radioactive, anything pyrophoric—Na, K, Rb, white-P, etc., should be avoided if possible. Elements such as Cl, F, and S are also corrosive and may damage pumps and seals within the MBE system, which adds an additional layer of complexity. Here, we stress that a researcher must take the proper precautions when considering adding toxic elements to their MBE systems. Carefully read safety data sheets on individual elements and possibly reacted compounds; as discussed by May et al.,⁶² Hawley's Condensed Chemical Dictionary is a great resource for understanding hazards.¹¹⁹ Consider that the source material may form flakes, dust, or possibly vapor, and carefully plan how to mitigate exposure. Finally, when considering using a toxic element seek advice from colleagues in the MBE community who have experience dealing with such elements, as well as people in related fields and especially the safety staff at your institutions.

2. Has it been synthesized as a bulk crystal or as a thin film, and is there a thermodynamic phase diagram? These together hedge that attempts at synthesis will be favorable, while simultaneously providing important information about possible growth conditions, possible impurity phases, and defect structures. As discussed above, knowledge of the bulk phase diagram and synthesis guided the MBE growth of the tetradymites as well as motivating the charge balancing of (Bi_1−xSb_x)₂Te₃.

3. What elements are involved, is there a chamber compatible with that particular chemistry (in a sense rhetorical here), and are those elements amenable to MBE? Regarding the latter, the question comes down to how the atoms can be delivered to the substrate in a controllable way that imparts a low kinetic energy. For the use in an MBE process, the elements should have sufficiently high vapor pressure, 10⁻⁵–10⁻⁴ Torr, below a temperature of about 2000 °C. Accessible elements are most alkaline metals, alkaline earths, most of the d-block excluding the 4d Zr-Ru, and 5d Hf-Pt, which are refractory and require an e-beam evaporator or some other mechanism,¹²⁰ rare earths, and most p-block elements other than some of the halides.

4. Is there a commercially available substrate where a low-energy surface is a good match to the surface symmetry and lattice parameter of the desired material? Is it possible to achieve atomically clean surfaces in situ, and is it available with high-quality atomically flat surfaces, all while being cost effective?

5. What is the volatility of each element as well as their compounds? The fewer the elements the easier, since the deposition rate of each element has to be carefully calibrated and slight deviations can result in incorrect stoichiometries. In specific situations for some binaries (for example, the tetradymites fit this, but also the archetype-MBE material GaAs) where one element is highly volatile (Se or Te), at a given growth temperature, the growth can be entirely controlled by the flux of the other atoms (Bi or Sb), where the excess unbonded Se or Te simply evaporates away resulting in an adsorption controlled growth window. This is the best-case scenario, and many materials do not follow this, with common examples being the ternary perovskite oxides, e.g., SrTiO₃, where the Sr to Ti ratio has to be manually controlled, with a typical error being >1%. Furthermore, as shown in the x-ray diffraction 2θ–θ scans in Fig. 3, attempts at MBE growth of Co₃Sn₂Se₂ at 400 °C from the ternary chalcogenide family of magnetic Weyl phases^121,122 showed that highly volatile SnSe₂ forms, then evaporates, resulting in Co₇Se₈ at typical vapor pressures of Se; this highlights a specific unforeseen difficulty that can arise when exploring new topological materials by MBE.

6. Is the resulting material air stable? Since thin films both have a large surface to volume ratio and the fact that the novel states of topological materials are often on the surfaces, chemical reactions with the air can be extraordinarily detrimental. Whereas much understanding can be gained using in situ probes, as highlighted in Fig. 1, most high-magnetic-field magnetometers and transport systems, x-ray diffraction systems, and clean room processes, for example, are ex situ. Much effort has to be expended discovering and understanding air-stability problems as well as searching for and discerning the effects of capping layers, as has been demonstrated for the tetradymites, which are fairly air stable compared to, for example, Na₃Bi.

Based on these guidelines, there are many important areas in science and technology that can be addressed by undertaking and refining the synthesis of new topological materials by MBE. The tetradymite TIs are by far the most well explored family of topological materials, yet there are still many open questions, particularly regarding heterostructures with superconductors and magnets, and the ternary compounds such as MnBi₂Te₄. These are discussed in the following sections. Of particular interest are the nodal semimetals for which there is very little studied outside of the excellent work done on thin films of 3D Dirac semimetals Na₃Bi and Cd₃As₂. For example, many intermetallic compounds, particularly the Heuslers are widely unexplored,^123,124 as well as square-net compounds,¹²⁵ and work is just emerging regarding candidate materials and understanding the challenges related to the growth. Furthermore, higher-order topological phases²⁰ represent an exciting area where the precise control of surface morphology and island formation available in MBE can be employed to realize novel states that emerge at 1D and 0D boundaries of materials. Beyond these specific examples, novel large scale numerical efforts have presented a near endless number of compounds available to explore,¹²⁶ many of which are great candidates for MBE growth.

Interplay of topology and superconductivity results in an emerging class of quantum matter aptly called topological superconductivity. Topological superconducting phases relate the topologically non-trivial structure of the gapless boundary modes (Majorana modes) to the gapped bulk with a non-zero topological invariant.^3,4,19 One of the most intriguing features is the emergence of Majorana fermions—particles that are their own antiparticles—which emerge as quasiparticles in these condensed matter systems.^127,128 Majorana fermions come in different forms. In contrast to most known particles and quasiparticles, Majorana fermions exhibit novel non-Abelian exchange statistics. When two indistinguishable particles are exchanged, fermions and Bosons accumulate a phase of −1 and 1, respectively, with their quantum states unchanged, and anyons which belong to the Abelian group, accumulate a non-integer phase between −1 and 1. Upon exchange of multiple Majorana fermions, which are non-Abelian anyons, the quantum state evolves to a unique state that depends only on their exchange (or braiding) trajectories. Confirmation of the existence of Majorana fermions and proof of non-Abelian statistics have been vigorously pursued over the past decade. Achieving these milestones is critical for potential applications in topological quantum computing since it directly enables an extremely long decoherence time, which has long been a key challenge in this field.^24,25

Topological superconductivity can be realized in a variety of MBE-grown materials systems. This can be either single layers of intrinsic topological superconductors or engineered heterointerfaces. The latter includes non-superconducting topological materials or semiconductors, both with strong spin–orbit coupling, which are interfaced with s-wave superconductors. Synthesis of high-quality topological superconductors (intrinsic or engineered) as well as conclusive experimental evidence of key features have both been major challenges. Topological superconducting phases naturally emerge in spinless p-wave superconductors with triplet pairing symmetry (p-wave pairing in 1D and p_x ± p_y pairing in 2D). As first proposed by Kitaev in a 1D tight-binding chain with p-wave pairing, discrete states at zero energy (E = 0) are bound to the ends of the chain under certain conditions,¹²⁹ as illustrated in Fig. 4(a). Such states are Majorana zero modes and can be realized in 1D intrinsic topological superconductors as well as in 1D superconductor/TI¹³⁰ and 1D superconductor/semiconductor hybrid systems.¹³¹ Creating interfaces that are transparent to superconductivity is one of the key challenges, and, by improving the interface in hybrid systems, enhanced superconducting and Majorana-zero-mode features have been observed. In 2D topological superconductors, gapless chiral Majorana edge modes can propagate along the edges of 2D topological superconductors and emerge on the boundaries of vortices with Majorana zero modes at their cores as shown in Fig. 4(b).^127,132 Theory proposed that propagating chiral Majorana edge modes can arise in integer quantum Hall or quantum anomalous Hall insulators when coupled with an s-wave superconductor.^133,134 However, experimental evidence of the resulting half-quantized conductance plateaus in such systems has been under debate.^135,136 In the vortex cores of 2D topological superconductors, Majorana zero modes give rise to zero-bias conductance peaks in, for example, scanning tunneling spectroscopy. 3D (time-reversal invariant) topological superconductors are the superconducting counterparts of 3D TIs. They host 2D gapless Majorana fermion surface states (Majorana cones) residing within the superconducting gap [Fig. 4(c)].¹³⁷ The use of interconnected MBE-ARPES and MBE-STM systems is critical to reveal the signatures of 2D and 3D topological superconductivity. For example, gapless surface states and zero-bias conductance peaks have been observed within magnetic vortices on the surface of β-PdBi₂ thin films grown by MBE.^138,139 In the remainder of this section, we will discuss intrinsic and engineered topological superconductors systems enabled by MBE and further challenges for advanced studies.

Superconductivity can be induced in the tetradymite TIs by doping or intercalation, resulting in possible topological superconductivity with strong spin–orbit coupling. This has been observed in bulk crystals of Bi₂Se₃ where dopant atoms were intercalated between van der Waals layers: Cu_xBi₂Se₃ with T_c ≈ 3.5 K,¹⁴⁰ Sr_xBi₂Se₃ with T_c ≈ 3.0 K,¹⁴¹ Nb_xBi₂Se₃ with T_c ≈ 3.4 K,¹⁴² and Tl_xBi₂Te₃ with T_c ≈ 2.3 K.¹⁴³ Among these, Cu_xBi₂Se₃ has been grown by MBE and extensively studied. In bulk crystals, an intercalated Cu atom is a strong electron donor, which raises the carrier density and thereby stabilizes the superconducting phase. In contrast, when grown by MBE, Cu is an electron acceptor and strongly lowers the carrier density, which highlights the amphoteric behavior of Cu in Bi₂Se₃.⁵² When the films are thin, this, in fact, lowers the Fermi level sufficiently to isolate transport through the TSS, yet precludes superconductivity.⁷⁶ Cu-doped Bi₂Se₃ shows the complexity that emerges for defects and dopants in the extremely different growth regimes of bulk crystals and MBE. This highlights critical challenges that need to be understood to integrate superconductivity in known topological materials via doping.

Sr₂RuO₄ is one of the promising candidates for intrinsic topological superconductivity. This material has the same crystal structure as the layered copper oxide (cuprates) high-temperature superconductor (La,Sr)₂CuO₄. However, the superconducting transition temperature (⁠ $T c$⁠), near 1 K, in Sr₂RuO₄ is nearly two orders of magnitude lower than that of the cuprates.¹⁴⁴ Furthermore, in contrast to the antiferromagnetic ordering in the cuprates, Sr₂RuO₄ has ferromagnetic ordering that led to the exciting theoretical proposal of spin-triplet (odd-parity) superconductivity when the spins align and break time-reversal symmetry.¹⁴⁵ There has been experimental evidence of possible chiral spin-triplet pairing, which is also called chiral p-wave pairing. However, there are still unsolved issues and consensus on the nature of the chiral p-wave superconductivity in Sr₂RuO₄ has not been reached.^146–149 Nevertheless, there are many interesting aspects and open questions regarding Sr₂RuO₄ which can be addressed from the perspective of MBE growth. For example, $T c$ of bulk Sr₂RuO₄ can be as high as 1.5 K and is highly sensitive to defects, stoichiometry, and strain; MBE has good control over these aspects which has led to the growth of pure crystalline thin films by oxide-MBE.¹⁵⁰ Yet, not all Sr₂RuO₄ thin films exhibit superconductivity due to the highly sensitive nature to defects. High-quality films can be synthesized by MBE with an adsorption-controlled growth window, where the Sr to Ru ratio is controlled by the formation and desorption of excess volatile Ru-O molecules. This results in either SrRuO₃ or Sr₂RuO₄ depending on the temperature and flux ratio. These adsorption-controlled MBE films have shown superconductivity^151–153 with enhanced T_c up to 1.8 K when strained precisely on NdGaO₃ (110).¹⁵³ Furthermore, epitaxial strain in Sr₂RuO₄ can manipulate the Fermi surface topology.¹⁵⁴ Finally, Sr₂RuO₄ is proposed to be classified as a topological crystalline superconductor where symmetry-protected Majorana fermions reside.¹⁵⁵ Experimentally, half-quantum vortices that are supported by spin-triplet paring was observed in Sr₂RuO₄,¹⁵⁶ which can host Majorana zero modes in the vortex cores. However, direct evidence of Majorana bound states is lacking, which highlights that there are many open questions regarding this novel topological material.

Advances in material synthesis and fabrication methods have allowed rapid development of highly crystalline 2D superconductors prepared by van der Waals epitaxy and mechanical exfoliation.¹⁵⁷ Among the 2D superconductors, only a few turned out to host intrinsically non-trivial topological band structures, including Fe-based high- $T c$ superconductors, transition metal dichalcogenide PdTe₂, and β-Bi₂Pd.

The superconducting transition temperature of bulk FeSe is 8 K. Astonishingly, however, this can be significantly enhanced by reducing film thickness down to a monolayer on the SrTiO₃ surface⁴¹ as well as by electron doping the FeSe layer.⁴² Interfacial electron–phonon interactions between the FeSe electrons and SrTiO₃ phonons are thought to be a key ingredient to stabilizing the high-T_c superconductivity.¹⁵⁸ Furthermore, the interfacial atomic structure has to be carefully engineered by MBE growth through precise control of substrate treatment, stoichiometry, growth mode, and post-annealing—all of which strongly affect reproducibility.⁴¹ Although, T_c of a monolayer FeSe film on SrTiO₃ is reported as high as 100 K,¹⁵⁹ it has been very controversial because it strongly depends on the quality of the interface between FeSe and SrTiO₃ and on the specific probes used to characterize the superconductivity. Monolayer FeSe films are highly sensitive to air exposure so that a protective capping layer is required for ex-situ transport. However, the capping layer affects the FeSe film properties with T_c reduced by 10 s of Kelvin in comparison to T_c measured by in situ surface-sensitive techniques such as STM and ARPES. The challenge of probing highly unstable interfacial superconductivity by a variety of techniques highlights the importance of multi-connected UHV systems equipped with in situ transport. Theory proposed topologically non-trivial states in band structures of FeSe and Fe(Te,Se) thin films.^160,161 Integration of high T_c superconductivity and non-trivial topology makes these materials unique and attractive. Signatures of Majorana bound states localized in the vortex cores have been demonstrated in Fe(Te,Se) on SrTiO₃, by using interconnected MBE and low-temperature STM systems as shown in Figs. 5(a) and 5(b).^162,163 Due to the high transition temperature, Fe-based high-T_c superconductors may serve as material platforms for a practical topological quantum computer operating near liquid nitrogen temperature. Challenges still remain in clarifying mechanisms of the high T_c superconductivity in Fe-based superconductors and in designing routes to use the Majorana bound states for a qubit.

The 2D material PdTe₂ is classified as a type-II Dirac semimetal.^164,165 Interplay between the non-trivial topological nature of the electronic bands and superconductivity makes PdTe₂ a promising candidate for a 2D topological superconductor. In contrast to interface-enhanced superconductivity in monolayer FeSe/SrTiO₃, monolayer PdTe₂ is a narrow bandgap semiconductor and is not superconducting, whereas multilayer PdTe₂ is metallic and superconducting with T_c ≈ 1.6 K.^166,167 A careful study of the thickness-dependent electronic properties of PdTe₂ requires precise layer-by-layer growth control by MBE and in situ characterization techniques such as STM and ARPES. Superconductivity in PdTe₂ is found to be insensitive to the substrate, which reflects the van der Waals epitaxy. Thin films of PdTe₂ have been epitaxially grown on various substrates such as SrTiO₃ (001), graphene-SiC (0001), and Bi₂Te₃.^166–168 Experimental studies revealed both type I and type II superconductivity in PdTe₂.^166,169 However, experimental demonstration of signatures of topological superconductivity in PdTe₂ remains an immediate challenge.

The layered material β-Bi₂Pd is another 2D superconductor with T_c ≈ 5.4 K. β-Bi₂Pd exhibits several TSSs that have been experimentally confirmed by spin-resolved ARPES measurements on bulk crystals, and theoretically categorized to belong to the Z₂ topological class, analogous to strong TIs such as Bi₂Se₃.¹⁷⁰ Thin films of β-Bi₂Pd can be grown by MBE with different growth modes at different growth temperatures. The sticking coefficient of Bi is small at a substrate temperature above 200 °C, at which synthesis of β-Bi₂Pd required Bi-rich conditions (Bi/Pd flux ratio > 3). MBE growth of β-Bi₂Pd films on SrTiO₃ (001) at temperatures between 300 and 350 °C proceeds in the Volmer–Weber growth mode, where epitaxial islands with their thickness down to a single unit cell form on the substrate with various lateral sizes controlled by the Bi/Pd flux ratio.¹³⁹ In contrast, the sticking coefficient of Bi becomes comparable to that of Pd at room temperature, and β-Bi₂Pd thin film growth with the Bi to Pd flux ratio tuned to 2:1 proceeds in layer-by-layer fashion.¹⁷¹ In addition to the typical bulk superconducting features, zero-bias conductance peaks, possibly from Majorana zero modes, were observed in vortices of a β-Bi₂Pd film by in situ scanning tunneling spectroscopy as shown in Figs. 5(c)–5(e).¹³⁹ Furthermore, unique features of the superconducting state in β-Bi₂Pd have been observed: First, the TSS superconducting gap is anomalously larger than the bulk superconducting gap,¹³⁸ which is unlike most topological superconductors where the bulk superconducting gap is larger than the topological superconducting gap. The large superconducting gap observed in TSSs may result in more stable Majorana zero modes that persist to higher temperatures. Second, possible transport evidence for time-reversal-invariant topological superconductivity and Majorana surface states on a 3D topological superconductor have been recently demonstrated in bulk crystals doped by K.¹⁷² Finally, Little–Parks devices showed the magnetic flux was half-quantized, which further indicated the unconventional nature of the superconductivity.¹⁷³ The study of superconductivity in pristine and doped β-Bi₂Pd films may enable the realization of the trivial-to-topological quantum phase transition in superconducting states.

Topological superconducting states can be artificially engineered by proximity-induced superconductivity in topological materials. When a 3D TI is interfaced with an s-wave superconductor, effective spinless p-wave superconductivity emerges in the 2D TSSs, which can exhibit Majorana bound states in artificial vortices.¹⁷⁴ For 2D TIs, spin-filtered edge states proximitized by an s-wave superconductor can become 1D topological superconductors that resemble Kitaev's model for a 1D spinless p-wave superconductor.¹²⁹ 1D nanowires of 3D TIs and topological crystalline insulators can host topological superconducting states when in contact with an s-wave superconductor.¹³⁰ These engineered topological superconducting states based on TIs and topological crystalline insulators can be extended to other topological materials as well.

Superconductors can be ex situ processed by e-beam evaporation, sputtering, and focused ion beam induced deposition to build superconductor-topological material hybrid systems. Ex situ superconductors are critical to devise such hybrid systems using topological nanostructures¹⁷⁵ in the forms of flakes, plates, ribbons, belts, and wires, either by mechanical exfoliation from bulk crystals, or by direct synthesis by vapor-liquid-solid (VLS) or chemical vapor deposition (CVD). The VLS growth methods typically use a metal (Au) particle catalyst that becomes liquid and collect source material in vapor phase. Above the solubility limit, the supersaturated source material starts to form a crystalline solid in the layer-by-layer fashion at the liquid/substrate interface. In the CVD growth method, the source material in the vapor phase directly deposits on the substrate to form nanostructures or a film. To achieve a clean interface between a superconductor and a topological material processed ex situ, proper native-oxide removal is required via ion milling or chemical wet etching before the superconductor deposition. However, precise control over the native-oxide removal is very challenging, and the resulting interface typically contains defects and disorder as a result of the damage to the surface of the topological material as well as from the amorphous or the multi-grained nature of the polycrystalline superconductor. The defective interface can cause unwanted subgap states and suppress features of topological superconductivity. With the atomic precision of MBE layer-by-layer growth, thin-film-based superconductor-topological material hybrid systems can be synthesized, which is advantageous for further device development. Transparent interfaces with minimal defects can be obtained either by in situ MBE growth of both the topological material then the superconductor or by MBE growth of topological materials on a properly surface-treated/desorbed ex situ superconductor, such as Bi₂Se₃ on NbSe₂ (Fig. 6).^176,177 In contrast to the case of deposition and lift-off of patterned superconducting electrodes by ex situ fabrication, it is challenging to pattern the superconductor layer in MBE-grown superconductor-topological material hybrid systems. For example, when patterning a superconductor on top of a topological material, selective etching of the superconductor is hard to achieve, and the remaining bare surface of the topological material is likely to be damaged by the etching process. One solution to form clean junctions is to add a stencil/shadow mask for in situ superconductor deposition. A recent study demonstrated full in situ lithography of superconductor-TI hybrid devices via MBE by using two monolithically integrated hardmasks for selective-area growth (SAG) of (Bi_1−xSb_x)₂Te₃ TI thin films and stencil lithography of a superconductor.¹⁷⁸ 

Superconducting features induced in the topological states have been experimentally demonstrated in various superconductor-topological material systems. Josephson effects in superconductor-TI hybrid Josephson junctions have shown induced superconducting features linked to the TSSs.^179–182 Furthermore, signatures of Majorana bound states, such as the fractional Josephson effect characterized by 4π periodic current-phase relation and suppression of odd Shapiro steps mediated by the 2D TI HgTe,¹⁸³ tetradymite 3D TIs,^178,184,185 and Dirac semimetals Bi_1−xSb_x¹⁸⁶ and Ca₃As₂¹⁸⁷ as well as zero-bias conductance peaks in vortices on Bi₂Te₃ grown on NbSe₂ [Figs. 6(c) and 6(d)],¹⁸⁸ have been reported.

Similarly, topological superconducting states can be artificially engineered by proximity-induced superconductivity in semiconductors with strong spin–orbit coupling. The spin-degenerate parabolic band structure of a 1D semiconductor splits its up-spin and down-spin bands due to spin–orbit coupling. When an external magnetic field is applied, the associated Zeeman splitting opens an energy gap at E = 0. At small energies within the gap, the spin-textured band structure can be considered effectively spinless. Interfacing an s-wave superconductor induces effective spinless p-wave superconductivity in the semiconductor, and Majorana zero modes emerge at the ends of the 1D superconductor-semiconductor wire.^131,189 Rashba spin–orbit-coupled semiconductors of InAs and InSb nanowires are a natural choice to realize the engineered 1D topological superconductor. These 1D nanowire-based superconductor-semiconductor systems have been extensively studied and have demonstrated zero-bias conductance peaks as possible evidence of Majorana zero modes.^190,191 MBE is viable to grow InAs and InSb nanowires by the VLS growth method. Among these, MBE-grown InAs nanowires have been widely used. However, MBE growth of InSb nanowires is challenging due to the narrow growth window. In contrast, InSb nanowires grown by metal-organic vapor phase epitaxy (MOVPE) are more accessible and have been widely used for Majorana physics studies. As in most of the other topological material systems, advances in 1D topological superconductors have been made possible by improvements in materials. Development of an epitaxial superconductor (Al) by MBE on InAs and InSb nanowires^192,193 has resulted in a highly transparent interface, which exhibited features of induced superconductivity and topological superconductivity, as shown in Fig. 7(a).^194–196 To achieve clean Al-InSb junctions, MOVPE InSb nanowires were ex situ transferred to a multi-chamber MBE system, cleaned by atomic-hydrogen in UHV, and coated by epitaxial Al in situ by MBE with shadowing by adjacent nanowires.¹⁹³ 

Toward the application of quantum computing based on superconductor-semiconductor systems, it is crucial to develop scalable platforms that can construct networks of 1D channels in a controllable manner. A 2D electron gas (2DEG) made of semiconductor heterostructures grown by MBE can provide high-mobility semiconductor channels and can be used, when interfaced with an s-wave superconductor, as a scalable platform for topological qubits consisting of multiple Majorana zero-modes. InAs and InSb quantum wells and InAs/GaSb double quantum wells have been investigated for this purpose.^197–205 For a transparent superconductor–2DEG interface, a 2DEG needs to be placed close to the surface so that the wavefunctions of the superconductor and 2DEG overlap. If the quantum well top barrier between the superconductor and 2DEG is very thin (none or just a few monolayers), strong coupling between the two can be achieved, and, in return, the electron mobility of the 2DEG is reduced due to a significant contribution of remote ionized impurity scattering at the interface. With thicker top barriers, the electron mobility of the 2DEG improves, but the coupling between the superconductor and 2DEG is similarly reduced. An optimum thickness for the top barrier, which maximizes the mobility of the 2DEG, while maintaining a strong coupling with a superconductor is found to be around 10 nm in epitaxial Al-InAs 2DEG systems. In addition to the remote ionized impurity scattering at the surface, there are other scattering mechanisms that contribute to the electrical properties of a 2DEG. In comparison to InAs 2DEGs grown on InP and GaAs substrates, InAs 2DEGs grown on GaSb have a smaller lattice mismatch, and, thus, significantly reduced misfit dislocations and the related scattering. Nearly lattice-matched barriers of Al_1−xGa_xSb can be used for the InAs 2DEGs grown on GaSb substrates, and smoother surface morphology and higher electron mobility were observed.²⁰⁶ The material choice for the quantum well barrier and capping layer determines the conduction band profile, types of defects forming in the barrier and at the interfaces and surface, and Fermi level pinning at the surface. For example, a near-surface InAs 2DEG with 10 nm-thick AlGaSb top barrier and InGaAs capping (3 nm), grown on the GaSb substrate, showed an electron mobility more than an order of magnitude higher than that of InAs 2DEGs with a 10-nm-thick InGaAs barrier, grown on InP.²⁰⁷ By etching Al into a narrow wire geometry and applying a global top gate to deplete electrons in the outer areas, a 1D superconductor–semiconductor quantum wire can be effectively formed. Tunneling spectroscopy at the end of the quantum wire has revealed signatures of Majorana zero modes,²⁰⁸ as shown in Fig. 7(b).

With a motivation for advanced quantum devices and a scalable quantum computer, SAG of in-plane 1D wires of InAs and InSb^50,209–214 by MBE, MOVPE, and chemical beam epitaxy has been investigated with a subsequent in situ epitaxial superconductor. Advantages of SAG are scalability and flexibility in designing nanowire networks as well as reduced post-fabrication steps through pre-fabrication of nanowire patterns on SiO_x or SiN_x dielectrics before growth, which could minimize defects introduced during device processing. There are certain devices that can be cleanly prepared by SAG. For example, three-terminal superconductor-semiconductor hybrid devices consisting of two normal leads at the ends of a nanowire and one in situ superconductor lead in the middle of the nanowire can be prepared to investigate the correlation of end-to-end subgap states and the splitting of zero-bias conductance peaks, as shown in Fig. 7(c).^215,216 In comparison to InAs and InSb nanowires by the VLS method, SAG wires contain more sources of defects and likely have lower electron mobility. SAG wires are directly grown on substrates with a significant, in most cases, lattice mismatch, resulting in misfit dislocations on the substrate surface and stacking faults along the {111} planes in the InAs or InSb wires.⁵⁰ In addition, pre-fabrication processing including surface cleaning on the substrate can damage the surface and introduce disorder when growing SAG wires. Incomplete etching of nanowire patterns on SiO_x or SiN_x may leave residues that inhibit semiconductor growth, and, thus, cause voids in thin nanowires or formation of grain boundaries in thick nanowires. An immediate challenge in SAG is to reduce the defects and disorder for better electron transport properties.

The pursuit of experimental confirmation of Majorana fermions and non-Abelian statistics will continue through using intrinsic and engineered topological superconductors. Demonstration of non-Abelian statistics based on multiple Majorana bound states is directly related to the realization of a topological qubit. Due to the proposed fault-tolerant nature of topological qubits, confirming these experimental milestones will represent a great step forward in the fields of condensed matter physics as well as quantum information science. Toward this goal, foundational theoretical and experimental work has begun: creation/manipulation of Majorana zero modes at the ends of 1D topological superconductors as well as in vortex cores in 2D and 3D topological superconductors, fractional Josephson effects mediated by topological materials, and chiral Majorana edge modes in quantum anomalous Hall insulators proximitized by a superconductor. Up to the present time, robust signatures of Majorana zero modes in superconductor–semiconductor hybrid systems have been evidenced by tunneling conductance measurements and Coulomb blockade measurements. Based on these achievements, a topological qubit may be realized in the near future by using semiconductor nanowires grown by the VLS and SAG methods and semiconductor 2DEGs. To obtain the theoretically proposed benefits of the topological protection and extremely long decoherence time in a qubit, key challenges remain regarding improving materials, such as reducing defects in the semiconductor platforms and creating superconducting islands with clean superconductor-semiconductor junctions. Once a topological qubit is achieved, overcoming scalability and reproducibility across multiple devices will be critical to the development of multi-qubit systems. However, as with other industrial scale devices enabled by MBE,^217–219 platforms such as semiconductor SAG wires and 2DEGs will be at the forefront of quantum devices over the coming decade.

Ferromagnetism, where all spins in a material coherently align along the same direction, fundamentally breaks time-reversal symmetry.^220,221 Since degeneracies among energy levels are always protected by a symmetry, breaking these is a means for controlling the band structure of a material. This naturally leads to merging magnetism with topological materials as a route to control the electronic character of the bulk and the boundary states. This is most clearly demonstrated for the quantum anomalous Hall effect.^222,223 For the case of TIs, the bulk conduction and valence bands are inverted by strong spin–orbit coupling, which obey time-reversal symmetry; the corresponding degeneracy at the 2D Dirac point on the surfaces for a 3D TI and edges for a 2D TI are protected by this symmetry, in that they must exist so long as this symmetry is intact. Introducing time-reversal symmetry breaking in the form of ferromagnetism aligned perpendicular to the surface can break this degeneracy in a topologically non-trivial way, as shown in Fig. 2(a). Take the case of the 3D TI: the resulting energy gap in the surface state enables the system to be described by another topological invariant, the Chern number. This places these materials in the same topological class as the quantum Hall states, and thus exhibiting chiral 1D quantized edge states, yet at a zero external magnetic field. This is called the quantum anomalous Hall effect, which was first envisioned in Haldane's graphene model with a magnetic field woven into the unit cell.²²⁴ It took around a quarter of a century for experimental verification in MBE-grown tetradymite films that were carefully charge balanced and doped by transition metals to introduce magnetism.²²⁵ Beyond the quantum anomalous Hall effect, a rich array of novel phenomena and possible technological applications emerge when magnetism and topological materials are merged. As highlighted here, understanding and utilizing MBE growth is the beginning step to achieve these goals.

As discussed in Sec. II, understanding the MBE growth from the perspective of defect formation and how it translates into changes of the topological properties was the key enabler for realizations of the quantized Hall effect. It was first predicted theoretically that Bi₂Se₃, Bi₂Te₃, and Sb₂Te₃ would become ferromagnetic when doped by the cations Cr or Fe.²²⁶ When made thin, this magnetic ground state would open a gap at the Dirac point which would give rise to a non-zero Chern number, thus hosting the quantum anomalous Hall state. Experimentally, this required doping the materials to stabilize a magnetic phase, while not reducing spin–orbit coupling such that the topological properties are not endangered, i.e., inadvertent reduction of spin–orbit coupling can drive the system into a topologically trivial phase. Furthermore, this required simultaneous Fermi level control such that it can be tuned into the magnetic exchange-gap at the Dirac point. Each of these aspects has proven a distinct challenge.

First, Bi₂Se₃ has the largest bulk bandgap, and the Dirac point is well-separated from the bulk bands. However, it is strongly n-type, and it was only recent that thin films could be made with sufficiently low carrier density to place the Fermi level close to the Dirac point. Furthermore, doping transition metals simultaneously failed to stabilize ferromagnetism, raising the Fermi level deep into the bulk conduction band, while weakening spin–orbit coupling to a point where the materials become non-topological.²²⁷ These aspects were overcome by understanding subtleties related to the origins of both ferromagnetism and the primary sources of charged defects in Cr-doped Bi₂Se₃.²²⁸ Doping Cr into Bi₂Se₃ raises the Fermi energy and simultaneously the Hall resistivity shows signatures of paramagnetism emerging due to the net moments of the Cr. However, ferromagnetism does not emerge, which is likely related to raising the Fermi energy and suppressing the spin–orbit coupling strength, and, thereby, reducing the net ferromagnetic interaction. These challenges were overcome by utilizing the virtual substrate method to lower interfacial defects (see Sec. II) combined with Ca-doping, and adapting an interfacial strategy where monolayers of (Cr_0.5Bi_0.5)₂Se₃ were grown at the interfaces of Bi₂Se₃.²²⁹ These remote layers utilized the long-range magnetic interaction to amplify the ferromagnetic coupling, while leveraging the short range nature of spin–orbit coupling to protect band-inversion in the Bi₂Se₃ layer. Combined with lowering the global Fermi level, this stabilized a ferromagnetic phase in Bi₂Se₃. This strategy highlights a multifaceted origin for magnetism in the tetradymites, which occurs when J_eff²-χ_L⁻¹χ_e⁻¹> 0 with J_eff being the effective exchange coupling and χ_L and χ_e being the magnetic susceptibilities of the local moments and free electrons, respectively. For topological materials χ_e should be large due to the strong spin–orbit coupling through the van Vleck mechanism.^226,230 However, interfacially induced magnetism indicates that ferromagnetism emerges due to a combination of enhancing J_eff through the long range Ruderman–Kittel–Kasuya–Yosida interaction of the conduction electrons as well as maintaining a large spin–orbit coupling by keeping the bulk of the film undoped. Finally, although the quantum anomalous Hall effect was not observed in these films, a novel aspect was observed for the first time. The sign of the anomalous Hall effect is reversed in Bi₂Se₃ relative to all other magnetic topological materials, which is an important aspect for engineering the Berry phase contribution to the Hall effect and ultimately the quantum anomalous Hall effect.^226,230

Given the challenges with Bi₂Se₃, early work toward realizing the quantum anomalous Hall effect focused on magnetically doping Bi₂Te₃ and Sb₂Te₃. The challenges with making Bi₂Se₃ magnetic are less obtrusive in the telluride system since spin–orbit coupling is stronger, and, therefore, more robust to the weakening associated with doping with transition metals necessary for magnetism. This has resulted in observing ferromagnetism in the pure compounds when doped by V, Cr, or Mn.^231–235 Furthermore, the intrinsic ambipolar defect chemistry in the tellurides, discussed in Sec. II, gives a chemical route to control the Fermi level. MBE growth of Cr_0.22(Bi_xSb_1−x)_1.78Te₃ showed that with magnetic doping, the carrier type can be tuned continuously from p-to-n type by increasing Bi content (x), which maintains the ferromagnetic state.²³³ Furthermore, as the charge compensation point was approached, the Hall resistivity increased and saturated near about 3 kΩ, about 1 tenth of the resistance quanta h/e²≈ 25 kΩ. This provided an early hint that accessing the quantum anomalous Hall effect was feasible. However, this was limited by the ability to control the defect chemistry of these tetradymite films, but the carrier density was sufficiently low to enable fine tuning to be done with an external gate voltage. SrTiO₃ is intrinsically close to a ferroelectric instability, thus possessing a huge dielectric constant at low temperatures, ε ≈ 20 000 at around 4 K.^236,237 Experimentally, this enables microfabrication to be skipped as the gate voltage can be applied directly through the SrTiO₃ substrates, even with millimeter-scale thickness. This enables controlling the number of carriers into the range of 10¹³ cm⁻²,²³⁸ which is well above the density of carriers in MBE-grown tetradymite films.⁸³ Switching to SrTiO₃ (111) substrates and growing a 5 QL Cr_0.15(Bi_0.1Sb_0.9)_1.85Te₃ film enabled the first observation of the quantum anomalous Hall effect, where the Hall resistance becomes quantized in units of e²/h, and the longitudinal resistance drops to zero, as shown in Fig. 8.²²⁵ Despite the Curie temperature of Cr_0.15(Bi_0.1Sb_0.9)_1.85Te₃ being about 10 K, the quantized phase only appeared when the temperature was reduced by two orders of magnitude to less than ∼100 mK. At 30 mK, the Hall conductivity was quantized at about 0.987e²/h at a maximum magnetic field of 18 T. Further work on V-doping showed improvements to the quantization at around 0.9998e²/h.²³⁹ This points to a complexity that arises in the band structure.²³⁹ For the (Bi_1−xSb_x)₂Te₃ system the Dirac point is near the top of the valence band, which has a local maximum away from the Brillouin zone center. Therefore, when the Fermi level is positioned within the exchange gap, there should necessarily be parallel bulk conduction which shorts the quantized edge transport. Ironically, the high levels of disorder may rescue the system from this non-ideality, in that they may localize the parallel bulk states at very low temperatures due to Anderson localization.

Moving away from homogeneous single-layer materials to heterostructures have enabled raising the transition temperatures substantially. Mogi et al started from a single-layer 8 QL Cr_0.10(Bi_0.22Sb_0.78)_1.90Te₃ which exhibited near-quantization (80%) at 0.5 K and full quantization at 50 mK.²²⁹ Like the Bi₂Se₃ discussed above which this work inspired, heavy Cr-doped layers [Cr_0.46(Bi_0.22Sb_0.78)_1.54Te₃] at the top and bottom layers of (Bi_0.22Sb_0.78)₂Te₃ substantially increased the onset of the quantum anomalous Hall phase to better than 90% quantization at 0.5 K. Extending this again to a pentalayer structure, consisting of a trilayer with the additional layers of (Bi_0.22Sb_0.78)₂Te₃ below and on top of the heavily Cr-doped layers further increased the quantization to near 100% for temperatures near 2 K. The rationale for this success of this layering scheme likely was the larger Cr-doping achievable with the spatial confinement in the pentalayer geometry, which could enhance the exchange gap, reduce sample inhomogeneity, or both.

Whereas early hints of the quantum anomalous Hall effect were observed in exfoliated bulk crystals,²⁴⁰ there are many examples of novel phenomena that highlight the importance and unique ability of MBE to engineer materials at the atomic level. First is the enhanced transition temperature using the pentalayer geometry. Beyond this, the atomic scale control of MBE can enable independently tuning the magnetism at the top and bottom surfaces. For example, in the trilayer Cr-(Bi_1−xSb_x)₂Te₃/(Bi_1−xSb_x)₂Te₃/V-(Bi_1−xSb_x)₂Te₃ heterostructure, the Cr-doped layer acts as a soft-ferromagnet and the V-doped layer a hard ferromagnet. As such, when the magnetic field is swept in a loop, the longitudinal resistance and Hall resistance are expected to exhibit a structure with multiple coercive fields where the top and bottom layers switch independently. When the magnetizations are antiparallel, the system formally obeys time-reversal symmetry. In this regime, a zero-resistance Hall plateau emerges, and the resistance exhibits a tremendous increase.^241,242 This behavior is indicative of an axion insulating state where the surface states are all gapped. In a separate system, when the magnetic layers are spatially symmetric, Cr-(Bi_1−xSb_x)₂Te₃/(Bi_1−xSb_x)₂Te₃/Cr-(Bi_1−xSb_x)₂Te₃ the Hall effect exhibits an additional feature. When the system is in the QAH state and the magnetic field is swept close to the coercive field an additional hump-like feature emerges.²⁴³ This can be associated with the topological Hall effect, which emerges due to chiral spin textures that arise as the system changes its magnetization. Achieving this in a topological heterostructure requires global breaking of time-reversal symmetry, careful control over the Fermi level such that its in the exchange gap, and significant Dzyaloshinskii–Moriya (DM) interaction resulting from broken inversion symmetry and strong spin–orbit coupling. Although the system is formally inversion symmetric, the topological Hall effect only emerges at large bottom-gate voltages. This likely points to the non-zero DM emerging when there is a net electric field that breaks inversion symmetry. These examples together highlight how the precise control of synthesis enabled by MBE can drive discovery as well as shed light on the fundamental origins of emergent phases of topological matter.

Going forward, many questions regarding magnetic topological materials remain open. First, what is the maximum temperature the quantum anomalous Hall effect can be observed? Can this be as high as room temperature? This latter question seems tantalizingly plausible given that the quantum Hall effect can be observed in graphene as high as room temperature.²⁴⁴ This was made possible by the large room temperature mobility and applying a sufficiently strong magnetic field (45 T) such that the cyclotron energy is of order the temperature.²⁴⁴ Analogously, the cyclotron energy should be replaced by the exchange gap energy, Δ, for the quantum anomalous Hall effect. As shown in Fig. 2(a), the exchange gap opened within the 2D surface band is spanned by the 1D chiral edge mode. Therefore, so long as the temperature is sufficiently small relative to Δ, then the quantized edge conduction should be measurable. Temperature bounds can arise for several reasons. The first is that, obviously, the temperature needs to be below the Curie temperature; for many materials that can be integrated with topological materials as heterostructures the Curie temperature can be much higher than room temperature. The second condition that determines the upper limit is the thermal activation of carriers across the exchange gap in the 2D topological surface bands (see Ref. 55). Considering a mobility around 2000 cm²V⁻¹s⁻¹ yields a resistance of 10 × h/e² for a carrier concentration of 1 × 10¹⁰ cm⁻², which corresponds to a deviation from perfect quantization by about 10%, and gives an upper bound for how much parallel conduction is acceptable to observe the quantum anomalous Hall effect. Quantitatively, this shows that an exchange gap of at least 0.3 eV is necessary for observing quantized transport near room temperature. Such a large gap, corresponding to a Curie temperature well over 1000 K, likely precludes this. However, considering an upper temperature near 77 K (liquid nitrogen temperature) corresponds to much smaller gaps, of order of 0.05 eV, which are well within theoretical predictions of heterointerfaces among TIs and magnets, which highlights open directions of MBE in the field.

Finally, the quantum anomalous Hall effect has recently been observed in intrinsic materials without any doping. The two examples are MnBi₂Te₄^245,246 and twisted bilayer graphene precisely aligned to hBN.²⁴⁷ The former is an intrinsically antiferromagnetic TI, which orders with a layered A-type structure.^248,249 The quantum anomalous Hall effect emerged in atomically thin flakes when exposed to an applied magnetic field, which resulted in the time-reversal symmetry breaking necessary to gap the surface band structure. This is a critical advance beyond the pioneering studies that first revealed the quantum anomalous Hall effect. In the graphene system, quantized transport was observed due to orbital magnetism that emerges in carbon systems and can be accessed when two layers of graphene are twisted to a precise angle of 1.15° and aligned to hBN. These two systems show the powerful tunability of atomically precise heterostructures and represent two areas, where key advancements can be made in MBE growth. Although at this moment it is not clear how to achieve precise alignment of the individual layers necessary for the bilayer/hBN system, high quality MnBi₂Te₄ films should be possible to synthesize by MBE. MnBi₂Te₄ is a derivative of the tetradymite structure with an additional layer of Mn-Te inserted into the quintuple layer structure.⁵⁶ This forms a septuple layer composed of Te–Bi–Te–Mn–Te–Bi–Te, as shown in Fig. 9(a). MBE growth will enable many routes to precisely tune the magnetic ground state and interfacial properties. However, initial reports of the growth relied on alternating depositions of Bi₂Te₃ and MnTe layers, which subsequently intermixed to form the septuple layers.^250,251 Going beyond this initial work, codeposition approaches^252,253 have confirmed that an adsorption controlled growth window can be achieved for MnBi₂Te₄ as well as the next member of the series MnBi₄Te₇, as shown in Figs. 9(b)–9(c) with stoichiometry confirmed by Rutherford backscattering spectroscopy.²⁵⁴ With increasing Bi:Mn flux ratio, these films showed the evolution of the topological band structure as well as the magnetic ground state, which are key data that will help clarify some of the remaining mysteries in this material. This accomplishment further suggests that higher members MnBi₆Te₁₀, or MnBi₈Te₁₃^255–257 may also be possible to synthesize either through adsorption controlled growth or through a layer-by-layer approach. Fully understanding the synthesis of this material and how to integrate it with other functional materials as heterostructures will open many new routes to realize the quantum anomalous Hall effect as well as other exotic phases.

The unusual spin-textures found in topological materials make them attractive candidates for spintronic device applications, where the spin degrees of freedom are used to perform memory and logic operations rather than just the charge dynamics.²⁵⁸ Moving away from the paradigm of charge-based electronics can offer many advantages including greatly reduced energy dissipation, faster operation, as well as novel routes to data storage. However, the field faces key challenges regarding generation, manipulation, and detection of spin currents as well as performing switching and gate operations since spin currents are not conserved, unlike charge currents. The field of “topological spintronics” has emerged to address many of these challenges based on the novel character of the spin-polarized TSSs. In particular, the intrinsic locking of the direction of the spin to the direction of the wave vector, and thus the direction of the current flow, can mitigate many of the current challenges. For one, spin current generators can be realized in topological materials, where spin-polarized currents can be produced through the spin-momentum locking of the TSSs as well as via spin Hall effect in the strong-spin–orbit-coupled bulk states. Switching can be enabled by the intercoupling between ferromagnetic layers and the spin-polarized surfaces. Magnetization switching of ferromagnetic layers have been demonstrated via spin-transfer torque exerted by topological materials. Realizing many of these goals requires understanding the detailed physics of spin-polarized transport in topological materials as well as the strong physical and chemical interactions at the heterointerfaces among topological and magnetic materials, which are necessary for a topological spintronic device.

The current-induced spin polarization in TSSs can be electrically detected using a potentiometric geometry with a ferromagnet (FM) voltage probe on a TI channel,²⁵⁹ as illustrated in Fig. 10(a). In this configuration, the direction of the current flow in the TI determines the direction of the spin polarization on the TI surface, and an external magnetic field aligns the magnetization of the FM voltage probe parallel or anti-parallel to the direction of the TI spin polarization. Measured voltage from the FM voltage probe should switch its sign when the relative direction of the TI spin polarization and the FM magnetization changes. The amplitude of the voltage change when it switches is proportional to the projection of the spin polarization onto the FM. Experiments have revealed step-like hysteretic spin signals associated with the direction of spin polarization in the TI and the magnetization of the FM voltage probe, as shown in Fig. 10(b).^46,260–266 A high spin polarization ratio in the TSSs, due to the suppression of backscattering, has been experimentally confirmed by potentiometric measurements. Moreover, larger spin signals were observed when the chemical potential is tuned into the bulk bandgap and toward the Dirac point, due to reduced conduction through trivial bulk bands [Fig. 10(c)].⁴⁶ We note that the sign of the measured spin signal is inconsistent between research groups.²⁶⁷ Theoretical models to interpret such current-induced spin polarization as well as the experimental results need to be reconciled.

Spin-charge conversion is an essential component for spintronic technologies as it facilitates interfacing with current charge-based electronics. Currently, heavy metals are the materials most common in such devices. When a charge current flows through materials such as Pt, Ta, or W, for example, spin-transfer torque by spin-charge conversion in the heavy metal causes the magnetization of an interfaced FM to precess or reverse its orientation.²⁶⁸ In most materials, this is found to require a large current density, which causes deleterious heating. However, increased efficiency of spin-charge conversion can be achieved by basing such devices on TIs. Efficiency from experimental studies of both charge-to-spin conversion and spin-to-charge conversion has been studied using TIs. Schematics of representative device for charge-to-spin conversion and spin-to-charge conversion are illustrated in Fig. 11. The charge-current-induced spin-polarized electrons in TIs can exert a spin-transfer torque to an FM (or spin–orbit torque when the origin of the spin polarization is related to spin–orbit coupling, such as the spin Hall effect), which works as a means of charge-to-spin conversion [Fig. 11(a)]. The first demonstration of a TI-based (Bi₂Se₃/Pt bilayer system) spin-transfer torque was measured by ferromagnetic resonance [Fig. 12(a)]. This exhibited a record efficiency for this effect at room temperature.²³ The spin-charge-conversion efficiency can generally be quantified by the spin Hall angle (or spin torque ratio) $θ ‖≡( 2 e / ℏ) J S , ‖/J=( 2 e / ℏ) σ S , ‖/σ$⁠, where $J S , ‖$ and $σ S , ‖$ are the parallel components of the spin current density and spin conductivity, respectively, absorbed by the FM, and J and $σ$ are the charge current density and charge conductivity, respectively, in the spin current source material. The spin-transfer torque induced by the TI is strong enough to switch the magnetization of the adjacent FM metal layers with in-plane and perpendicular magnetic anisotropies at room temperature.^269,270 Current-induced spin-polarized electrons in TIs can exchange-couple to an adjacent FM layer. Such TI/FM bilayers can exhibit unidirectional magnetoresistance (UMR), R_UMR,²⁷¹ which is dependent on the TI current direction. This can be seen in Fig. 11(b) where high and low resistance states depend on the relative direction of the TI spin polarization and the FM magnetization. For the charge-to-spin conversion process associated with UMR, the figure-of-merit is expressed as R_UMR per current density (j) per total resistance (R): R_UMR/j/R. In the Bi₂Se₃/CoFeB bilayer system, R_UMR/j/R was found to be twice as large in comparison to the best reported Ta/Co bilayers.²⁷² 

Spin-to-charge conversion can be achieved in TI-based spin pumping devices where precessional magnetization dynamics of an FM injects a spin current into a TI, as shown in Fig. 11(c). The injected spin current can be converted into a charge current in the TI through the inverse Edelstein effect. This is facilitated by the 2D surface states in combination with the inverse spin Hall effect by the bulk states. In Bi₂Se₃/CoFeB spin pumping devices, the resulting spin Hall angle at room temperature turned out to be greater than that of heavy metal-based spin pumping devices.²⁷³ From experimental reports of spin-charge conversion, the measured spin Hall angle based on TI/FM systems widely varies from 0.001 to 2.²⁷⁴ Different mechanisms of spin-charge conversion with a single material may result in different values of the spin Hall angle. Materials quality and TI-FM interface quality can also play an important role in determining the spin-charge conversion efficiency, as discussed next.

Thin films of MBE-grown TIs are promising platforms for large-scale device fabrication. One of the major challenges in fabricating TI/FM bilayer devices is to make clean interfaces between the materials. Oxides and the subsequent surface damage associated with their removal, and defects introduced at the interface during fabrication significantly degrade the efficiency of spin-charge conversion. One solution to avoid the degradation is to grow in situ FM layers on TI films. TI/magnetic-TI bilayers of (Bi,Sb)₂Te₃/Cr-doped (Bi,Sb)₂Te₃ grown by MBE have been demonstrated to show spin–orbit torque with magnetization switching of the magnetic TI layers [Fig. 12(b)]^275,276 as well as large UMR.²⁷⁷ Although highly efficient spin-charge conversion was observed in TI/magnetic-TI bilayer devices, there are key limitations. The most prominent is the low Curie temperature of the ferromagnetic TI layer (below 40 K). This requires development of other in situ magnetic materials for room-temperature applications. An alternative solution to achieve a clean interface is in situ capping of TIs directly after MBE growth, using materials such as Se on Bi₂Se₃. This preserves air-sensitive surfaces of TIs⁸¹ and opens up the use of various magnetic materials prepared ex situ by other deposition techniques such as e-beam evaporation and sputtering. There are multiple routes to remove the Se capping layer from Bi₂Se₃ leaving atomically clean surfaces.⁴⁴ The Se capping layer can be removed by heating the substrate at 200–250 °C under Se flux with in situ monitoring such as RHEED in an MBE chamber; the recovery of the diffraction pattern of Bi₂Se₃ is a clear indicator of Se-desorption. The Se beam flux prevents further creation of Se vacancies and likely facilitates removal of Se-O.⁴⁴ Although Se beam flux and in situ monitoring are not available in many cases, a gentle pre-sputtering of the Se-capping layer combined with careful control of substrate temperature and desorption time could recover a clean Bi₂Se₃ surface. Subsequent in vacuo deposition of the FM layer will likely be a key in obtaining a clean interface between Bi₂Se₃ and the FM layer.

There are a few issues in spin-charge conversion using Bi₂Se₃/FM-metal bilayer systems. One major issue is the coexistence of bulk conduction and surface conduction in Bi₂Se₃ thin films. The Fermi level in Bi₂Se₃ films is typically located slightly above the conduction band edge. Because Bi₂Se₃ intrinsically has a strong spin–orbit coupling, spin accumulation at the surface through the spin Hall effect in the bulk, which, when combined with the spin polarization by TSSs, is not negligible. In fact, it could positively enhance the overall spin Hall angle since the sign of the spin polarization is the same as that of the TSSs. In contrast, if the Fermi level is located further up in the bulk bands, Rashba spin-split states may play an additional role. Particularly, these states reduce the overall spin polarization signal since its orientation is opposite relative to the TSSs. To avoid this negative contribution, the Fermi level needs to be near or below the conduction band edge, which can be achieved by minimizing defects at the interface, preferentially using TIs that are more tolerant to such defects, or by carefully choosing materials with optimum band alignment. The second major issue is the shunting of charge current via the FM metal. This occurs since the conductivity of the FM is orders of magnitude lower than Bi₂Se₃. Despite the efficient spin-charge conversion in Bi₂Se₃, the overall current density through the Bi₂Se₃/FM-metal bilayer structure is significantly higher than what is in the Bi₂Se₃ layer. To overcome the shunting issue, insulating (or highly resistive) FMs are desirable for Bi₂Se₃/FM bilayer systems.

TI/FM-insulator bilayers with a clean interface can be prepared by MBE. FM insulators have been found in the forms of oxides, nitrides, sulfides, and dilute magnetic semiconductors. However, there are key challenges related to the integration of these with topological materials. Typical growth temperatures of crystalline FM insulators are much higher than that of the Bi-chalcogenide TIs. In addition, lattice matching with substrates or epilayers is critical for the growth of FM insulators. Given these conditions, MBE growth of TIs on FM insulators is more applicable, in most cases, than the growth of FM insulators on TIs. FM insulators for TI/FM bilayer structures prepared by MBE include yttrium iron garnet (YIG) Y₃Fe₅O₁₂, M-type hexagonal ferrite BaFe₁₂O₁₉, the dilute magnetic semiconductor (Ga_1−xMn_x)As, and the Heisenberg FM insulator EuS.^278–282 

For MBE growth of TIs on crystalline magnetic oxides such as YIG and BaFe₁₂O₁₉ films, which can be prepared by sputtering or pulsed laser deposition, TIs can be grown by either one-step or two-step growths. In the case of large lattice mismatches between two layers, two-step growth may produce smoother crystalline TI films.⁷¹ Spin pumping into Bi₂Se₃ from YIG revealed that the spin-to-charge conversion efficiency drops rapidly when the Bi₂Se₃ thickness decreases below 6 QLs. The efficiency remains almost constant with Bi₂Se₃ thickness being greater than 6 QLs.²⁷⁸ This result correlates with a decrease in the spin polarization with a hybridization gap in Bi₂Se₃ under 6 QLs²⁸³ and implies that the TSSs play a dominant role in spin-to-charge conversion in Bi₂Se₃/YIG bilayer systems. Spin-to-charge conversion in this system was further investigated using Cr-(Bi_1−xSb_x)₂Te₃ on YIG, and the combination of composition control and electrostatic gating enabled tuning the surface and bulk contributions.²⁸⁴ Wang et al. found that the spin Hall conductivity did not change substantially when the Fermi level was tuned across the bulk bandgap. It was concluded that the spin-to-charge conversion originates from either the full spin–orbit coupled bulk states or the spin-momentum-locked TSSs. Strong interfacial exchange coupling revealed induced magnetism in the TI layer. In the Bi₂Se₃/BaFe₁₂O₁₉ bilayer system, hysteretic anomalous Hall effect as well as magnetization switching of the BaFe₁₂O₁₉ layer by spin–orbit torque from Bi₂Se₃ was observed even at 300 K.²⁸⁰ The spin–orbit torque switching efficiency is comparable with Pt/BaFe₁₂O₁₉ bilayer systems at 300 K, however, the efficiency becomes 30 times higher at 3 K most likely due to enhanced surface conduction at lower temperatures.

TI/FM-insulator bilayer systems are an ideal platform to study the breaking of time-reversal symmetry and opening of an energy gap at the Dirac point due to exchange coupling. This was shown using (Bi,Sb)₂(Te,Se)₃ films, with compositions tuned to reduce bulk conduction, grown on highly resistive FM (Ga,Mn)As. With the magnetization oriented normal to the interfaces an exchange gap was opened, which was indicated by a modification of a quantum correction to the magnetoresistivity called the weak antilocalization effect.²⁸¹ In 3D TIs, the strong spin-momentum locking causes a coherent cancellation of backscattering due to the accumulation of phase factors of ±π for loops of opposite orientation, which effectively reduces the probability for resistive backscattering. The emergence of an energy gap at the Dirac point modifies the Berry phase to be $π( 1 − Δ / ( 2 E F ))$ with Δ being the gap size and $E F$ being the Fermi energy, and weak localization is expected to arise as the Fermi level approaches the energy gap.²⁸⁵ In the (Bi,Sb)₂(Te,Se)₃/GaMnAs bilayer system, the crossover between weak antilocalization and weak localization was observed and attributed to the gap opening by breaking time-reversal symmetry.²⁸¹ 

High spin-charge-conversion efficiency in the tetradymite TIs is promising for spintronic applications in computation, logic, and memory devices. However, high efficiency alone may not guarantee that TI-based devices can immediately be embedded in the current technology and replace the metal-based spintronic devices. One of the challenges is the high resistivity of tetradymites, which could cause additional power consumption, compared to the metal-based spintronic devices. More experimental effort is required to understand and utilize exotic new topological materials in current spintronic device motifs, as well as in new approaches such as topological antiferromagnetic spintronics.²⁸⁵ For example, in the topological material α-Sn, resonant spin pumping from Fe through Ag into MBE-grown α-Sn films induced lateral charge current due to the inverse Edelstein effect.²⁸⁶ The resulting spin Hall angle was well above those of heavy metals and comparable with that from spin-pumping in Bi₂Se₃/FM-metal systems. Furthermore, the topological Kondo insulator SmB₆ shows TSSs with insulating bulk at low temperatures. Similarly, spin injection experiments on SmB₆ revealed spin-to-charge conversion by the inverse Edelstein effect linked to the surface states.²⁸⁷ These studies highlight many functional aspects of the spin-polarized band structure across the family of topological materials which can be unlocked by mastering MBE growth.

To conclude, the goal of this Perspective article is to highlight the strong interrelation of MBE growth and future advances in the field of topological matter. This is demonstrated by many historical examples of how MBE growth has led to serendipitous discoveries of topological phases. These subsequently pushed the development of theory for both understanding the physics of materials and predicting new phases of matter. Over the past decade and a half, this strong interrelation among theory and experiment has blossomed. This synergy has led to rapid progress in predicting topological phases in certain materials as well as baseline confirmation, particularly related to novel band structures. As such, there are many open questions that can be addressed using thin films grown by MBE, and certainly many surprises awaiting discovery. This predicts an exciting decade to come that is full of new discoveries which will push closer to practical applications of topological materials.

Particularly, attempting and refining the growth of new materials is an open area that is very broad in scope. This is historically a well-proven route to new discoveries. Currently, only a handful of known and predicted topological materials have been synthesized by MBE, with the majority of the work focused on the tetradymite family including Bi₂Se₃, Bi₂Te₃, and Sb₂Te₃. These materials have been at the basis of most of the exciting discoveries, and certainly will continue to play an important role. This is especially clear with the recent exciting discoveries made in the ternary tetradymites series, particularly the intrinsic magnetic TI, MnBi₂Te₄. Refinement of the growth of this material and higher order compounds,²⁵⁴ as well as integration with strong magnetic materials will open the door to answering fundamental questions, particularly, what is the maximum temperature the quantum anomalous Hall effect can be observed. Beyond TIs, many new topological phases have emerged and represent a totally unexplored space for MBE growth. The broad class of nodal metals is the biggest example of these, with preliminary work being confined to the 3D Dirac semimetals, but leaving very little work done on the Weyl semimetals and the myriad of other nodal metals. Therefore, implementing MBE's ability to control dimensionality, symmetry, and proximity-coupling across interfaces will no doubt lead to many interesting observations.

As the diverse character of the topological states gives many new routes to perform computation or data storage, topological devices represent an area of critical importance. Implementation of any such device, either a prototype or mass production, will ultimately require a high-quality thin film either as a single slab of material or as a heterostructure of multiple materials with dissimilar properties. Achieving this can lead to new technologies ranging from spintronics to quantum computation. Current barriers are the materials, which are fundamentally difficult to control since the materials-character that makes them topological often lowers the barrier for the formation of defects while amplifying their deleterious effects.⁵⁵ Therefore, a new generation of materials is needed, which are robust during synthesis, air stable, and stable during necessary processing and integration with other materials. Furthermore, the typical scales for the bandgap (<0.3 eV), dielectric constants (>100), and effective masses (∼0.1 m_e) have been shown to be barriers to intrinsic properties dominated by the topological states, and realizing room temperature operation of a topological device will require optimizing these key parameters.⁵⁵ The timeframe to overcome such challenges can be given by considering other fields: The field of quantum computation has only recently advanced to the level of integrating multiple quantum devices into a single processor that can realistically compete with a classical computer.^288,289 This triumph took nearly 30 years²⁹⁰ of dedicated effort to understand and master relatively simple materials such as Al/Al₂O₃ and Si. In contrast, the subfield of topological quantum computation has a relatively short history and has not found its silicon, let alone achieved the experimental milestone of a topological qubit. But rapid theoretical and experimental progress has been made to determine motifs that can serve as the basis of a topological qubit as well as identifying several materials systems with promising properties. As such, the future will be filled with excitement as MBE growth is refined, the unusual topological states are better understood, and new applications and devices emerge.

We would like to thank Roman Engel-Herbert, Seongshik Oh, Brian Sales for insightful comments, as well as Anjali Rathore for assistance with literature review. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (manuscript preparation and MBE growth) and by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy (structural characterization).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.