Topological Quantum Materials for Realizing Majorana Quasiparticles

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Review

Topological quantum materials for realizing Majorana quasiparticles Stephen R. Lee, Peter A. Sharma, Ana L. Lima-Sharma, Wei Pan, and Tina M. Nenoff Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.8b04383 • Publication Date (Web): 30 Nov 2018 Downloaded from http://pubs.acs.org on December 2, 2018

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Chemistry of Materials

Topological quantum materials for realizing Majorana quasiparticles Stephen R. Lee, † Peter A. Sharma, † Ana L. Lima-Sharma, † Wei Pan, ‡ and Tina M. Nenoff *, † †Sandia

National Laboratories, Albuquerque, NM 87185, United States

‡Sandia National Laboratories, Livermore, CA 94551-0969, United States ABSTRACT: In the last decade, basic physics, chemistry, and materials-science research on topological quantum materials – and their potential use to implement reliable quantum computers – has rapidly expanded to become a major endeavor. A pivotal goal of this research has been to realize materials hosting Majorana quasiparticles, thereby making topological quantum computing a technological reality. While this goal remains elusive, recent data-mining studies, performed using topological quantum chemistry methodologies, have identified some thousands of potential topological materials – some, perhaps many, with potential for hosting Majoranas. We write this review for advanced-materials researchers who are interested in joining this expanding search, but who are not currently specialists in topology. The first half of the review addresses, in readily understood terms, three main areas associated with topological sciences: (1) a description of topological quantum materials and how they enable quantum computing (2) an explanation of Majorana quasiparticles, the important topologically-endowed properties, and how it arises quantum mechanically, and (3) a description of the basic classes of topological materials where Majoranas might be found. The second half of the review details selected materials systems where intense research efforts are underway to demonstrate nontrivial topological phenomena in the search for Majoranas. Specific materials reviewed include the group II-V semiconductors (Cd3As2), the layered chalcogenides (MX2, ZrTe5), and the rare-earth pyrochlore iridates (A2Ir2O7 , A= Eu, Pr). In each case, we describe crystallographic structures, bulk phase diagrams, materials synthesis methods (bulk, thin film, and/or nanowire forms), methods used to characterize topological phenomena, and potential evidence for existence of Majorana quasiparticles.

1. INTRODUCTION Majorana zero modes – elusive quasiparticles often described as their own antiparticles – have generated broad research interest because of their position at the juncture of two ongoing science and technology revolutions, one in quantum computing 1, 2, 3 and the other in topological quantum materials. 4, 5, 6, 7, 8, 9,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29 As recounted in recent reviews, quantum computing offers the possibility of greatly increased computing power, 1, 3 whereas topological materials offer not only new pathways towards quantum computing, 30, 31, 32, 33 but also the promise of highperformance optical devices, 26 thermoelectric devices, 23, 34 spintronic devices, 23 and catalysts. 26 Success in these technological endeavors requires a more comprehensive understanding and mastery of the physical and chemical behavior of topological quantum materials than has been achieved to date.

In Section 5 we present the fourth theme for the chemist or materials scientist seeking to synthesize, characterize, process, or otherwise advance topological quantum materials (TQMs). We review selected materials classes currently pursued as some of the most-likely hosts of topologically non-trivial materials phases with properties suitable for realizing Majoranas. These materials include: (i) the group II-V semiconductors (Cd3As2); (ii) the layered chalcogenides (MX2, ZrTe5); and (iii) the rare-earth pyrochlore iridates (A2Ir2O7 , where A= rare-earth elements such as Eu and Pr). Using these materials as exemplars, we highlight both traditional materials-science aspects and currently published experimental evidence supporting existence of the elusive Majorana zero modes within the solid state.

2. THE TECHNOLOGICAL DRIVER – TOPOLOGICAL QUANTUM COMPUTING In the words of Stearne, 2 the goal of quantum computing Our review considers four interconnected themes of is “to realize a new type of computation essentially different interest to chemists and materials scientists seeking a from that of classical computer science.” He goes on to note straightforward introduction to topological quantum that when fully realized, quantum computers will be able to materials for quantum computing. The first three themes solve new classes of complex problems so hard that classical span from topological quantum computing (Section 2); to computers could never feasibly solve them. Among these, the topological science of Majorana quasiparticles, herein at we find numerous truly hard many-body problems of longtimes, simply Majoranas (Section 3); to the realization of standing interest to chemists, physicists, and materials these quasiparticles in the solid state using a variety scientists. The promise to efficiently solve such problems topological-materials approaches (Section 4). We aim to stems from the massive superposition of quantum states provide brief, descriptive overviews of selected topologicaluniquely possible through the coherent interference of the science areas. quantum bits, the qubits, that quantum computers operate ACS Paragon Plus Environment 1

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upon. Because the number of superposed states increases exponentially with the number of qubits, even a ~100-qubit quantum computer may one-day outperform even the largest of classical computers – at least for certain key problems. Beyond solving the previously unsolvable, a more enduring reason we pursue quantum computing arises because the endeavor itself fundamentally advances the ways we think about the fundamental laws of chemistry and physics. 2

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2.2 The Quantum Statistics of Particle Exchange. To demystify the topologically realized quasiparticles found in such states, and to begin seeing how that they might be used to process information, we again appeal to the familiar – in this case, the quantum statistics of fermions and bosons. 30 In contrast to the fermions and bosons typically found in the ordinary three-dimensional (3D) space of bulk materials, the topology giving rise to our quasiparticles requires them to exist in a reduced-dimensional space, for instance, as states inhabiting the 2D planar surface or interface of a bulk crystal, or as states found in a 1D nanowire. Enabled by the reduced dimensionality, our quasiparticles turn out to be anyons, particles that obey quantum statistics bounded by those of fermions and bosons. Whereas the spatial exchange of two identical particles yields wavefunctions that are antisymmetric for fermions and symmetric for bosons – the exchange within a 2D space inhabited by anyons can introduce wavefunctions differing by an any arbitrary phase shift (hence the name, “anyons”). For anyons, a second exchange of the same type (consider the exchanges both to occur via a counter-clockwise rotation of the two particles) no longer must yield back the original state, and may instead cause a nontrivial phase shift:

2.1 Noise-Induced Decoherence: A Role for Topology. The qubits that make up a quantum computer are never fully isolated from their environment, such that environmental effects inevitably introduce noise into the qubits. Because the qubits rely on coherent interference of often-delicate internal quantum states to perform quantum computations, the noise-induced decoherence of these states presents a major hurdle to error-free computation. While errorcorrecting quantum-information theories, algorithms, and devices are being developed to cope with this challenge, 2 qubits based on physical processes that are intrinsically isolated from environmental interaction would greatly enhance the prospects of quantum computing. In recent years, the rapidly growing field of topological materials has merged with quantum computing, forming yet another new discipline, topological quantum computing, that offers prospects for implementing qubits that are intrinsically more immune to decoherence errors.

𝜓(𝑟1,𝑟2 )

𝑒2𝑖𝜃𝜓(𝑟1,𝑟2) ,

(1)

where  is the arbitrary change of phase for a single exchange. 30, 33 This phase shift, imbued by the topology of the material’s bandstructure, is not only topologically protected from the environment, but also provides a topological means for quantum-mechanically propagating information through the material in a manner facilitating quantum computation.

Those of us not steeped in these topics may well ask: What are topological materials, and how may they deliver a qubit intrinsically well-isolated from its environment? The answer rests in part with the mathematical science of topology, which involves the study of shapes (or more broadly, manifolds or sets), often with emphasis on global properties intrinsic to a particular class of shapes. In the case of topological materials phases – the topological insulators, topological semimetals, and topological superconductors now receiving such intense research interest – the focus shifts to the topology, shape, and symmetry of the underlying bandstructure and wavefunctions of these new materials phases.

In a given topological materials phase, one of the conventional forms of matter most familiar to us – the metals, semiconductors, insulators, and superconductors – can coexist alongside the exotic topology-induced behavior of the phase, with both aspects of the phase arising from the Figure 1. Schematic diagram showing the braiding (or exchange) bandstructure topology. Topological scientists often of two indistinguishable particles moving on a 2D plane versus describe the conventional-phase behaviors as arising from time for various types of quantum statistics. trivial topology, and any exotic behaviors as arising from 2.3 Particle Braiding. To further glimpse how such non-trivial topology. This terminology reflects the central computations become possible, we note that Eqn. (1) enables importance of topology in this new field; it is not a comment a quantum-topological process called braiding that involves on the transformative influences that the so-called trivial a series of physical particle exchanges over time within a phases have brought to humanity. The nontrivial, intrinsic discrete ensemble of anyons. 35 A hypothetical example of physical properties of states enabled by bandstructure braiding, with the braiding path occurring in time as the topology can be thought of as arising directly from nonanyons exchange, appears schematically in Figure 1 for the local, global features of the materials’ phase. As such, these evolution of two indistinguishable particle states with time. properties and states are robust to destruction by local Within topological science, a pair of like, clockwise, perturbations within the phase. Because the properties and anyonic exchanges occurring within a braiding process are states tend to persist so long as the materials phase exists, often termed a winding. One also sees in Figure 1 that a onesuch properties and states are often described as being winding braid of bosons or fermions produces no net change topologically protected. And we perhaps begin to see how in the phase, and thus, the braiding carries no information. we might topologically realize a more powerful isolation from nearby fermions or bosons found in the environment. ACS Paragon Plus Environment 2

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Chemistry of Materials (blue spheres) or holes (white spheres) present in the topological material must be interacting with superconducting Cooper pairs (grey ellipses). Additionally, the Cooper pairs and any unpaired electronic states must all be of one spin. Such a topological superconductor has oddparity, triplet-based Cooper pairs, and materials of this type are typically called p-wave superconductors after the porbital-like distribution of their superconducting electronic states. In these peculiar circumstances, we see in Figure 2(a) that an electron becomes nearly equivalent to a hole plus a Cooper pair. As we will see in Kitaev’s quantum mechanical theory below, it takes two Majorana quasiparticle states to represent a single fermionic state. Thus in Figure 2, we are writing “pictorial equations” where two Majorana quasiparticles appear on each side of the equivalence signs.

As thoroughly described in the review of Nayak et al., 30 not just any anyonic particles will do for implementing quantum computation. Indeed, the anyons must have quantum statistics that give rise to non-trivial braiding statistics, which in oversimplified terms is to say that the braided particles must not give rise to only the phases =0 (or π). Such a result gives rise to a trivial boson (or fermion), which effectively vanishes from the system of topological quasiparticle states under consideration. Anyons resulting in only trivial braiding statistics are termed Abelian, whereas anyons resulting in non-trivial braiding statistics are termed non-Abelian. Again, bringing forward the familiar, we note that to say the braiding transformations of the wavefunctions are non-Abelian is simply to say they are non-commutative. We come to an important point: To implement faulttolerant topological quantum computing, we seek very specific types of topological materials – those giving rise to non-Abelian anyons capable of directly transmitting quantum information via braided particle-exchange interactions. 30, 36 Once such materials are clearly realized, devices utilizing braiding concepts to implement qubits become possible. 30, 37

3. MAJORANA QUASIPARTICLES AND KITAEV THEORY Because Majoranas are thought to obey the just-described non-Abelian anyonic quantum statistics, these quasiparticles may be the particles necessary for topological quantum computing. 32 The increasingly common technical phrase “Majorana fermion” recognizes original theoretical developments by Ettore Majorana, who worked in 1937 to extend Dirac’s 1928 equations for spin 1/2 fields, recasting them in a form involving only real numbers. 38 Majorana’s refashioned theory made respectable the idea that spin 1/2 particles could be their own antiparticles. 39 Early applications of his work were in the realm of high-energy physics, where prior to any experimental studies, neutrinos were thought to be possible examples of these unusual Majorana fermions. In the eight decades elapsed since Majorana’s time, no fundamental particles have been identified that possess the hallmarks of the originally posed Majorana fermions, though indeed, the search continues within elementary-particle physics. 39

Figure 2. Illustrations of paired Majorana-quasiparticle excitations suggesting how they might arise within a ‘Fermi-like sea’ of isospin Cooper pairs within a p-wave superconductor. Adapted by permission from Springer Nature: Wilczek, F. Majorana Returns, Nature Physics, 2009, 5, 614. 39

Keeping the above discussion in mind, we consider how Majorana quasiparticles could acquire some of the seemingly exotic properties often attributed to them. First, because of the required uniformity of spin states, all quasiparticle states become functionally spinless. Second, the implied equivalence of an electron and a hole+Cooper pair – whenever these fermions are well-separated within the isospin sea of Cooper pairs – suggests that these fermions’ underlying Majorana quasiparticles are effectively neutral. Third, further imagine that we take each of the particle ensembles that form Figure 2(b); we then place them into a sea of Cooper pairs; finally, we bring the two ensembles together to intimately overlap in space. Because the resulting annihilation of the two pseudo-equivalent ensembles will return a Cooper pair to the surrounding sea of p-wave states, one may see how these quasiparticles might be viewed as their own antiparticles.

3.1 Majorana Quasiparticles. The early quantum theory of Majorana provides historical context for today’s topological quantum materials in that the unique quasiparticles enabled by bandstructure topology can also be thought of as being their own antiparticles. Because of this similarity to Majorana’s 1937 theory, these proposed quasiparticle excitations realized in the solid state are called Majorana quasiparticles. 39 As noted by Das Sarma et al., 32 however, there are significant departures from Majorana’s original work. Most notably, the original Majorana fermions actually do obey ordinary fermi-Dirac statistics, 32 whereas solid-state Majorana quasiparticles obey the distinctly different, non-Abelian anyonic statistics, noted earlier. 32

3.2 Kitaev’s 1D Chain Model of a Superconducting Nanowire. To move from our qualitative picture into the realm of quantum mechanics, we turn to founding quantum mechanical theory for Majoranas developed by Kitaev. 40 The idea of this section is to describe how quantum mechanics gives rise to a Majorana zero mode. Kitaev’s original, 2001 superconducting-nanowireinspired conceptual model is shown schematically in Figure 3. The model consists of a linear, 1D chain of isospin sites, with each fermion site capable of holding a single electron. Following Kitaev’s original notation, fermionic creation and annihilation operators, aj and aj†, can be defined for each site. To write down the Hamiltonian for the chain of L sites

To picture how Majorana quasiparticles might come to exist within real materials comprised of fermions or bosons, we appeal to Figure 2. The figure shows a key prerequisite for the existence of Majoranas: the underlying electrons ACS Paragon Plus Environment 3


(L >> 1), Kitaev defines w as the site-hopping amplitude,  as the chemical potential, and =||ei as the superconducting gap. Using these definitions, the ordinary Hamiltonian describing the total energy of the quantummechanical model system formed by Kitaev’s chain of fermions may be written:

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operators do not appear in the Hamiltonian (Eqn. 6), they represent two additional distinct topological phases embedded within even this simple chain model. He further concludes that only one of the two phases has nontrivial, non-bulk character with a pair of occupied Majoranaquasiparticle states at the ends of the chain, as pictured in Figure 3(b).

The two end-states each have zero energy ( = 0) making them 2-fold degenerate. Taken together, the end states can 1 ∑𝑗 ―𝑤(𝑎𝑗† 𝑎𝑗 + 1 + 𝑎𝑗†+ 1𝑎𝑗) ― 𝜇 𝑎𝑗† 𝑎𝑗 ― 2 + ∆𝑎𝑗𝑎𝑗 + 1 + ∆ ∗ 𝑎𝑗†+ 1be 𝑎𝑗†further viewed as a fermion, albeit a very unusual one, that is highly delocalized across the entire length of the . (2) chain, even as the separate Majoranas localize at each end of the chain. 6 These paired, degenerate, zero-energy, To reveal the presence of a possible topological phase individually neutral, localized states are a highly simplified hosting Majoranas, Kitaev rewrites the ordinary theoretical example of the famous Majorana zero modes that Hamiltonian in terms of Majorana quasiparticles. This are now sought after by experimental researchers worldrewrite requires Kitaev to define a pair of special Majorana wide. operators in terms of the underlying fermions: 𝐻1 =

[

(

]

)

c2j-1 = ei2aj + e-i/2aj†, c2j = -i ei/2aj + i e-i/2aj†,

(3)

where j=1,..., L. These operators serve two important purposes. First, they formally define two Majorana quasiparticle operators (and sites) for each fermion. Second, they impose mathematical constraints onto the underlying fermions such that quantum mechanical expressions written in terms of c2j-1 and c2j (instead of the more general aj and aj†) reflect some of the unusual quantum-mechanical properties first proposed by Majorana. By using Eqn. (3), Kitaev recasts Eqn. (2) to obtain the corresponding Majorana Hamiltonian: i

𝐻1 = 2 ∑𝑗[ ―𝜇𝑐2𝑗 ― 1𝑐2𝑗 + (|Δ| + 𝑤)𝑐2𝑗𝑐2𝑗 + 1 + (|Δ| ― 𝑤)𝑐2𝑗 ― 1𝑐2𝑗 + 1 ] . (4) Using this constrained representation, we can now look for Majorana states hidden within the more general fermionic system represented by Eqn. (2). To see if Majoranas emerge, we will simplify the Majorana Hamiltonian for two special cases originally presented by Kitaev. Consider case 1:  < 0 with || = w = 0. The two Hamiltonians, Eqns. (2) and (4), become

Figure 3. Two types of Majorana-site pairings occur along a linear chain of fermion sites in Kitaev’s simplified quantum-mechanical model of a superconducting nanowire: (a) Case 1 ( < 0 with || = w = 0) where each pair of Majorana operators (c2j-1 , c2j) align with a corresponding fermion site j (j=1,...,L); (b) Case 2 ( = 0 with || = w > 0) where each pair of Majorana operators (c2j , c2j+1) now bridge two adjacent-fermion sites, j and j+1. For Case 2, only the two end states in panel (b) will be occupied. The insets at lower left and lower right notionally tie the end-of-chain states back to the qualitative picture sketched in Figure 2, as do the sketched-in indications of bulk-phase Cooper pairing. (See text for further explanation of the variables and labels used in Kitaev’s chain model.)

We close the Kitaev-model discussion with one further essential concept. These zero modes cannot exist without the topological defects represented by the truncated ends of the chain. Importantly, in real bulk materials, these defects 1 i 𝐻1 = ―𝜇∑𝑗 𝑎𝑗† 𝑎𝑗 ― 2 = ― 2𝜇∑𝑗𝑐2𝑗 ― 1𝑐2𝑗 . (5) are not always our typical materials-science crystal-lattice defects; rather, these defects exist more generally as Here, the fermionic operators, aj and aj†, and the Majorana materials-intrinsic edge states arising around the periphery operators, c2j-1 and c2j, are aligned to the corresponding of a 2D plane comprised of topologically induced surface fermion sites, j, at all points along the chain, as shown in states, where the near-edge local surface potential is Figure 3(a). The occupation number for the ground-state necessarily varying due to lateral truncation of the material. Majorana-operators paired together for each site, j, is 0. In Alternatively, the needed defects can be intentionally this case, no Majoranas exist and the chain is a trivial introduced and engineered by applying magnetic or electric superconductor. fields to topological materials, with the defects then arising either as quantized, magnetic-flux-induced vortices, or as Next, consider instead case 2:  = 0 with || = w > 0. The electric-field inhomogeneities or discontinuities. Some Majorana Hamiltonian becomes conventional crystal defects may also serve, particularly 𝐻1 = ― i𝑤∑𝑗𝑐2𝑗𝑐2𝑗 + 1 . (6) those associated with either crystallographic boundaries or symmetry breaking; dislocations, grain boundaries, and Now, the Majorana operators, c2j and c2j+1, bridge adjacent domain walls are examples. Indeed, edge dislocations in fermion sites, j, as shown in Figure 3(b). More importantly, ruthenates (Sr2RuO4), perovskite oxides (SrTiO3, KTaO3), the two Majorana operators, c1 and c2L , appear unpaired at and other topological materials have been quantum each end of the chain. Kitaev goes on to argue from more mechanically analyzed as proposed defects hosting complex lattice theory 39 that because these unpaired Majoranas. 41, 42, 43, 44 ACS Paragon Plus Environment

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Since the time of Kitaev’s original chain model, both the number and sophistication of theoretical approaches for realizing Majorana zero modes has flourished. And as each new theoretical innovation has appeared, a corresponding wave of experimental materials investigations seeking to reveal actual Majoranas has quickly appeared as well, with the field of topological quantum materials seeming to grow in exponential fashion – conceptually resembling the earliernoted, but arguably still-aspirational, exponential growth of superposed quantum-states and computing power with the number of qubits. In the next Section, we review this burgeoning body of research by providing a directory to some of the most important materials approaches, foundational theories, and previous literature reviews that may prove valuable to chemists and materials scientists seeking to break into the field and/or further expand the search for materials harboring Majorana zero modes.

complex, high-mobility, semiconductor heterostructures containing a 2D electron gas (2DEG). At very low temperatures the 2DEG can be viewed as a BCS (BardeenCooper-Schaffer) state comprised of composite fermions. These states are essentially triplet-based topological superconducting states as needed for Majoranas. Das Sarma et al. touch on this possibility when presenting an early overview where they make pedagogical use of the then already-known physics of the FQHE to explain topological states and their use in quantum computing. 52 Avron et al. also offer a brief perspective look at topological aspects of the QHE. 53 Das Sarma et al. note that while some FQHE states found in GaAs are extremely fragile, being barely observable even at ~ 20 mK in the best materials, these states nonetheless have suitable statistics for hosting Majoranas suitable for quantum computing. Because many more robust materials candidates with less-demanding temperature requirements have since emerged, the FQHE in high-mobility GaAs has receded from its early preeminent position in the still-expanding race to demonstrate Majoranas and non-Abelian statistics. 33

4. MATERIALS APPROACHES FOR REALIZING MAJORANAS To produce experimental evidence for the existence of Majorana quasiparticles, widely differing materials and measurement approaches have been increasingly pursued. In this section, we outline the previous and ongoing materials that have generated wide interest among TQM researchers. The section first touches upon materials of historical interest, and then goes on to describe the now wide-ranging topological materials classes – superconductors, insulators, semiconductors, and semimetals – that are being increasingly explored in the search for Majorana quasiparticles. Given our previous statement that Majoranas require p-wave topological superconductors, we will also describe how proximity-based s-wave superconductors are deployed in conjunction with the non-superconducting topological materials to broadly realize p-wave-like superconductive behavior in the surface region of these otherwise non-superconducting TQMs.

Superfluid He3. While fundamental studies of the exotic low-temperature phases of He3 went forward long ago, 1959-1975, the recognition of the topological importance of superfluid He3 came much later. The non-Abelian anyonic character of He3 noted in 2000 by Read and Green, in their work on the FQHE. 50 The extremely low temperatures and high pressures, ~ 2-3 mK and ~30 atm, needed to realize the superfluid phase may limit interest in He3 as a practical host for Majoranas. 4.2 Topological Superconductors. The p-wave (1D) and p+ip (2D) topological superconductors are the materials classes most closely embodying the linear-chain and honeycomb-lattice models used by Kitaev to introduce topology into quantum computing. As such, the essence of how they may enable Majorana zero modes rests with the chain model already discussed in Section 3.2. Notably, these topological superconductor materials are thought to rarely occur in nature.

4.1 Historical Materials Implementations. The materials first proposed as possible hosts for Majorana quasiparticles are GaAs/AlAs heterostructures. They became famous through revealing studies of integer and fractional quantum hall effects (IQHE and FQHE), and superfluid He3, which enjoyed a much earlier period of intense research interest through studies of then-new fermi liquids. 45, 46, 47, 48

Materials Studies. Specific materials where topological superconductivity has been investigated include Sr2RuO4 and UPt3. These two materials are described in detail by Kallin and Berlinsky in their focused review of chiral superconductors. 13 The ruthenate, Sr2RuO4, becomes superconducting below ~1K, 54 and half-flux-quantum vortices have been detected in magnetometry of Sr2RuO4 micro-flakes. 55 The heavy fermion metal, UPt3, shows theoretical promise but remains relatively uninvestigated from a topological perspective. Interestingly, it does become superconducting below ~0.5K. 56 The scarcity of candidate topological superconductors and their limited progress towards Majoranas have driven pursuit of the topological alternatives to be discussed below.

GaAs/AlGaAs and the v=5/2 Fractional Quantum HallEffect. The importance of materials exhibiting the fractional quantum hall effect (FQHE) arises in part because they motivated some of the earliest consideration of topological effects occurring within the solid state at low temperature. The theoretical works of Moore and Read, 49 and of Read and Green 50 are especially notable because of their early recognition of the v=5/2 quantum Hall effect as a topological state of matter possessing the quantum statistics of non-Abelian anyons. (They coined these quasiparticles, “nonabelions,” a name which has not stuck, though it seems apt.). The earlier-yet Nobel-prize-winning work of Thouless et al. 51 on the integer QHE introduced the mathematics of topology into condensed-matter physics, paving the way for present-day topological quantum materials science as a distinct field of endeavor.

Foundational Theory. In addition to the founding theory developed to describe the IQHE and FQHE, 51, 49, 50 much of the topological and quantum mechanical theory of Majoranas in the topological superconductors was developed by Kitaev, as already noted. 30, 40, 57 In addition, Sengupta et al. 58 developed a related theory where they Notably, the v=5/2 FQHE studies have long utilized considered the possibility of Majoranas in 1D organic ACS Paragon Plus Environment 5


superconductors. Notably, their paper provides an early theoretical analysis of how to detect Majoranas using measurements of zero-bias conductance peaks to detect tunneling of midgap Andreev bound states, and using measurements of magnetic susceptibility to detect spininduced paramagnetic-moment signatures. Gurarie et al. 59 theoretically analyze the topological states realized for pwave superfluidity in a spin-polarized fermi gas, where they uncover topological superconducting phases closely related to those in the Kitaev models. They too point to superfluid He3 as a candidate host for Majoranas. Shortly thereafter, Das Sarma et al., conceptually connected the then-known chiral p+ip symmetry of superfluid He3 to that suspected for Sr2RuO4, proposing it as capable of supporting non-Abelian braiding statistics. 60

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This finding suggests significant new research opportunities for chemists and materials scientists, especially those with expertise in these existing-materials candidates, and with a desire to broaden their research into topological-materials science. Even prior to these data-mining exercises, the exploration of proximity-based effects in topological insulators has become quite active, with the intense research interest driven by their intriguing bandstructure. In a topological insulator, a large energy gap separates the highest occupied electronic band from the lowest empty band (just like a “regular” insulator); in addition, these bulk insulating states coexist with gapless surface or edges (depending on the dimensionality) that lead to corresponding conducting surface states. The topological states can appear in both 2D or 3D crystals and require the presence of spin-orbit coupling and time-reversal symmetry. Figure 4 schematically contrasts the nontrivial and trivial bandstructure possibilities within a topological insulator. 66,

Related Reviews. Qi and Zhang, 4 Alicea, 6 Beenakker, 6 Sato and Fujimota, 12 Kallin and Berlinsky, Chiu et al., 15 Sato and Ando, 15 and Trebst 23 present major reviews of the topological superconductors. The reviews of Alicea, 6 and of Sato and Ando, 15 offer clear discussions of the quantum mechanics and topological science showing how Majorana quasiparticles become possible in idealized 1D p-wave and 2D p+ip superconductors, offering significantly more depth than our own Section 3.2. For those seeking terseness, Beenakker and Kouwenhoven offer a contrastingly brief, but articulate, introductory overview. 61 The review of Trebst 23 may appeal to readers because of its discussion of bonddirectional interactions within the Kitaev formalism, and because of its materials-oriented discussions of ruthenates and iridates. In addition, Trebst briefly cites further theoretical developments beyond the original Kitaev models, where effects arising from phenomena such as vacancies, impurities, disorder, strain, and topological-phase nucleation have been theoretically pursued. 23

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Figure 4. Schematic bandstructure of (a) a nontrivial topologicalinsulator phase, and (b) a trivial topological-insulator phase. Grey represents the normal bulk-insulator band states, and black lines labeled with “” represent various topological surface states. Note that the surface states connect the bulk bands in case (a), but do not in case (b). Reprinted figures with permission from American Physical Society as follows: Fu, et al., Phys. Rev. Lett., 2007, 98, 106803. Copyright 2007 by the American Physical Society. 66

4.3 Topological Insulators and the Rise of the Proximity Effect. In contrast to the p-wave examples noted above, conventional s-wave superconductors are topologically trivial; additionally, their Cooper pairs comprise electrons with opposite spin projections (singlets). As a result, they cannot directly host spinless Majorana quasiparticles. Yet, s-wave superconductors have taken on a central materials role within TQMs. This role stems from the well-known proximity effect, whereby a superconducting phase induces superconductivity in normal materials through intimate interfacial contact of the two materials. Proximity-induced superconductivity within the surface region of a topological insulator, because of contact with an s-wave superconductor, produces the functional equivalent of a topological superconductor. Similar functionality obtains for proximitybased approaches utilizing either semiconductors or topological semimetals, as well.

Materials Studies. As described by the Tian et al. review, 17 several generations of topological-insulator materials have

already been subject to extensive study. These generations (i) began with HgTe implemented as thin heterostructural quantum wells within CdTe, which are found to exhibit 2D topological-insulator states; 68 (ii) these were followed by the discovery of Bi1-xSbx alloys as the first generation of topological insulators exhibiting 3D states; (iii) the related 3D, stoichiometric compounds followed, including Bi2Se3, Bi2Te3, and Sb2Te3; (iv) more recently, these give way to further materials with 3D topological states, including stoichiometric SnTe, PbxSn1-xSe(Te) alloys, and yet other compounds and alloys in this region of the periodic table. Other relevant types of materials are certain iridates, 24 and the more unusual compound, -RuCl3. 69, 70 All of these topological insulators are detailed in the Related Reviews of topological insulators catalogued below.

Because s-wave superconductors are ubiquitous and well known, and because many non-superconducting topological Foundational Theory. In early 2007, theory specifically materials candidates continue to emerge, proximity-based treating topological insulators appeared in Fu, Kane, and approaches greatly expand the set of materials wherein Mele, and in Fu and Kane. 66, 67 In 2008, the latter authors Majorana quasiparticle might ultimately be realized. 6 developed theoretical extensions describing a topological Indeed, the news pages in Nature 62 recently cite preprint insulator with its surface proximal to a patterned s-wave papers reporting that systematic examinations of extensive superconducting layer [Figure 5(a)], where the induced 63, 64, 65 stoichiometric-compound-materials databases now superconductivity within the patterned line-junction is found predict the existence of some ~ 6000 to 8000 materials to open a small gap, 20, into the junction’s topological wherein significant topological states are likely to be found. ACS Paragon Plus Environment 6

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surface states [compare Figure 5(b) to the individual  crossings in Figure 4(a), where no gaps are opened]. 71 In 2009, they then further developed an interferometric scheme utilizing topological insulators to search for Majorana quasiparticles. 72 Additional foundational theory was unveiled by Cook et al., 73 who examined the formation of Majorana quasiparticles in a topological insulator nanowire proximate to an s-wave superconductor. Since simple 1D wire-like geometries preclude braiding within a 2D plane, true-nanowire or nanowire-like patterned tri-junctions [Figure 5(c)], and larger nanowire-based networks have been conceptualized to facilitate braiding and computational operations within nanowire-based TQMs. 71, 74 Akhmerov et al. describe a theoretical scheme wherein a proximate topological insulator is used to implement a combined qubit and Fabry-Perot interferometer. The qubit utilizes Majorana zero modes at vortex pairs, and the interferometer reads-out the qubit to fermionic states. 75 Cook et al. further advance the theory of Majoranas within proximate topological insulators by examining their stability subject to variations in the magnetic flux or the on-site potential. Their results indicate that unpaired solid-state Majoranas should persist under experimentally realizable conditions. Notably, some of the above-described innovations are previewed by Kitaev.

remarkably clear introduction to spin-orbit effects in both topological insulators and semiconductors, including a terse discussion of their connection to Majorana quasiparticles. 11 Quite recently, Uchida and Kawasaki reviewed the topological properties of oxide thin films and interfaces, emphasizing both the emergence of iridate/oxide-based topological insulators 22 and the increasing importance of materials growth and interface quality.

(b)

(a)

30, 40, 57

(c) Figure 6. (a) 1D-nanowire semiconductor proximate to an s-wave superconductor; 74 (b) schematic diagram of the semiconductor’s original and proximity-induced spinless bandstructure regimes; 74 (c) an alternative implementation utilizing a 2D semiconducting thin-film and magnetic insulator. 81 Figures 6(a) and 6(b) are reprinted by permission from Springer Nature: Nature Phys., NonAbelian statistics and topological quantum information processing in 1D wire networks. Alicea, et. al., 2011. Reprinted Figure 6(c) with permission from American Physical Society as follows: Sau, et al., Phys. Rev. Lett., 104, 040502, 2010. Copyright 2010 by the American Physical Society.

4.4 Semiconducting nanowires and thin films. As shown in Figure 6(a), the semiconductor-nanowire approach seeks to implement Majorana zero modes using an engineered topological structure comprised of a semiconducting wire with high spin-orbit coupling and a proximate s-wave superconductor, with the structure placed in an external magnetic field, B. The proximate superconductor serves the same role as in the topological insulator schemes, but now providing Cooper pairs to a semiconductor host. The high spin-orbit coupling and the applied field laterally splits a degenerate band in the semiconductor [Figure 6(b)], with mixing of spin states occurring near k=0, and with a gap introduced by the superconductivity. The chemical potential is tuned to the gap to induce the topological phase. Implementation of braiding can be provisioned using trijunction-wiring schemes, just as in the proximate topological insulators. Implementations utilizing semiconductor thin films draw upon similar physics, with a magnetic insulator layer providing the needed field [Figure 6(c)].

Figure 5. (a) An s-wave superconducting layer (S) deposited on top a topological insulator (TI), where S has a long, patterned opening of width W, forming a wire-like, superconducting line junction at the topological insulator surface; (b) spectrum of the line junction when  = W = 0; (c) a trijunction extension of the linejunction concept; (d) phase diagram of the trijunction’s Majorana bound states (MBS), where shading indicates MBS should exist (see Ref. [71] for details). Reprinted figures with permission from American Physical Society as follows: Fu, et al., Phys. Rev. Lett., 100, 096407, 2008. Copyright 2008 by the American Physical Society. 71

Related Reviews. Qi and Zhang, 4 Alicea, 6 Cava et al., 12 Ando, 23 and Sato and Fujimota, 12 present major reviews of topological insulators. Additionally, Sasaki and Mizushima review the doping of topological insulators to achieve topological superconductivity. 9 The review of Cava et al., 12 is of interest because of its focus on the structural and chemical aspects of topological insulator materials, especially those related to Bi2Te3, the related stoichiometric tetradymites, and their solid solutions. For an introduction to topological insulators, readers may refer to an earlier pedagogical overview by Qi and Zhang. 76 Within a review Materials Studies. As detailed by Lutchyn et al., 18 the focused on spin-orbit coupling, Manchon et al. give a ACS Paragon Plus Environment 7


semiconductors most used to implement this approach include InAs and InSb, grown either by molecular beam epitaxy (MBE) for 2D quantum-well devices, or by metalcatalyzed vapor-liquid-solid (VLS) growth for 1D quantumwire devices. The proximate superconductors used in these studies are NbTiN or Al. Because the semiconductorsuperconductor interface quality is critical, the metal layer (Al) is deposited in-situ in the case of MBE. 15 Because we will focus on only semimetal materials exemplars in Section 5, we note here that experiments on hybrid-superconducting semiconductor nanowires have led to a variety of evidence for Majoranas. Both the fractional a.c. Josephson effect 77 and zero-bias conductance peaks have been reported. 78, 79 Recent work includes detailed Coulomb-blockade spectroscopy of the zero-bias conductance while varying the nanowire length and the applied magnetic field. 80 We separately discuss the significance of these types of measurements as evidence for Majoranas below in Section 4.6.

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distinct characteristic bandstructures [Figure 7], which we will briefly discuss before directing the reader to the more in-depth literature. One of two salient differences between the topological insulators and their topological semimetal counterparts is simply that the bulk states of the topological semimetals have valence and conduction bands that touch, whereas topological insulators do not. A second difference arises because the nontrivial topological states of a semimetal rest with the nodes positioned at the point where the Dirac cones (extrema of the bulk bands) touch, such that the bulk bands themselves give rise to the topology. In contrast, the topological states that bridge a TIs wellseparated bulk bands might be viewed as less fundamentally connected to the nontrivial topology giving the conducting topological insulator surface states. 21, 26, 86 Because the topological semimetals have 3D topology, the formation of edge states for Majoranas may be more dependent on reduced dimensionalities explicitly achieved through use of nanowires, patterned thin films, supported thin platelets, and/or heterostructural interfaces.

Foundational Theory. The use of a semiconductors in combination with an s-wave superconductor was suggested in multiple theoretical papers. Sau et al. propose a threelayer embodiment comprised of a patterned s-wave superconductor, a semiconductor layer, and a supporting magnetic insulator, as in Figure 6(c). 81, 82 Alicea propose an alternative geometry where the magnetic insulator is omitted, and a magnetic field is applied parallel to the semiconductor layer (not shown). 83 Lutchyn et al. propose nanowire use, in a manner similar to Figure 6(a), but with the underlying s-wave superconductor physically gapped near the mid-length portion of the wire, and with the wire embedded in a superconducting-quantum-interferencedevice (SQUID) circuit. 84 Notably, experimental research to detect Majoranas intensified soon after the papers by Sau et al. 82 and Lutchyn et al. 84 were published. This increased experimental focus was enabled by their proposal to improve detection of Majorana zero modes via the observation of a quantized zero-bias conductance peak within a semiconductor-nanowire-superconductor heterostructure.

(a) Weyl

(b) Dirac

(c) Nodal line

Figure 7. Schematic diagrams of the characteristic near-band-edge bandstructure for the three main types of topological semimetals. 88 Reprinted figure with permission from Koshino, et al., Phys. Rev. B, 2018, 93, 045201. Copyright 2018 by the America Physical Society.

Turning first to the Weyl semimetals, we again find a name that has historical roots in the Dirac equation. As recounted by Ciudad, 87 Hermann Weyl, working in 1929, examined the Dirac equation for the case of massless Related Reviews. Alicea, 6 as well as Lutchyn et al. 1518 fermions, and reported that two independent equations could present reviews of the use of semiconductor materials be written, whose solutions were chirally related. proximate to s-wave superconductors in order to realize Quasiparticle states in topological semimetals that exhibit Majorana quasiparticles. While Alicea’s review clearly analogous chiral behavior are thus called the Weyl summarizes relevant theory, relating it back to the simplified semimetals. Because of the two-solution chirality, a Kitaev models of topological superconductors, the review of minimum of two overlapping sets of Dirac cones arise to Lutchyn et al. has a strong, complementary focus on produce semimetal character [Figure 7(a)]. In this case, a materials-science aspects including semiconductor growth, pair of Weyl nodes form at the conic intersection bridging superconductor-semiconductor interface quality, and the bulk bands. The Weyl nodes are connected by a materials characterization. Lutchyn et al.’s further topologically protected fermi arc [Figure 8] that gives rise discussions of experimental signatures of Majorana to spin-polarized surface-conducting states. Existence of quasiparticles, and of proposed experiments towards these unremovable Weyl nodes requires that a material quantum-computing applications, also prove interesting. possess either broken time-reversal symmetry (magnetic Yet other reviews touch upon use of proximate materials) or broken inversion symmetry semiconductors. 5, 13 Finally, Winkler extensively details the (noncentrosymmetric crystals). Moreover, the nodes require physics of spin-orbit coupling in semiconductors, 85 which three-spatial dimensions (hence, 3D or bulk materials) may be of value for those considering alternative within which at least one pair of accidental band touchings semiconductor materials. occurs (these accidents of nature are sometimes colorfully called diabolical points). The geometry in which the bands 4.5 Topological Semimetals. The topological semimetals, touch leads to two types of Weyls, type I and type II. Type comparative newcomers to the science of TQMs, may be I indicates that the band extrema are touching, as in Figure classified into three broad groups, termed the Weyl, Dirac, 7(a), and type II indicates that the band extrema are not and nodal-line semimetals. These groups are defined by ACS Paragon Plus Environment 8

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touching, such that electron and hole pockets exist at the band extrema, which now rest adjacent to the Weyl nodes (not shown). 21, 26, 86

science theme yet to blossom 101 and the extensive use topological multilayers to band-engineer topology. Building on Murakami, many first-principles analyses have gone forward to find relevant semimetals. Early efforts towards Weyl semimetals include Xu et. al. (HgCr2Se4), 102 Witczak-Krempa and Kim (pyrochlore iridates), 103 Halász and Balents (HgTe/CdTe), 104 Weng et al. (transition-metal monophosphides and monoarsenides), 105 and Soluyanov et al. (WTe2). 106 Early efforts towards Dirac semimetals include Young et al. -cristobalite BiO2), 107 Wang et al. (A3Bi, A=Na, K, Rb), 108 Wang et al. (Cd3As2), 109 and Cao et al. (group-IV allotropes). 110 Some of these materials remain topologically uninvestigated in experiments. More recently, device theory has begun to appear, just as in the previous TQMs cycles, with Chen et al. theoretically treating Josephson junction behavior in Cd3As2. 111

Second, the Dirac semimetals have a 2-fold higher degree of band degeneracy, which eliminates the doublet structure seen for Weyl semimetals [Figure 7 (b) vs Figure 7 (a)]. These semimetals may arise two different ways: (i) either because of a special nonsymmetrical space-group symmetry, involving both a rotation and a translation, or alternatively, (ii) because two Dirac points arise on an axis of rotation, with rotational symmetry acting to provide topological protection. Dirac semimetals of the latter type predominate in studies to date. Lastly, the nodal-line semimetals have Dirac cones that partially overlap near their tips, forming a loop or line structure in the region of intersection [Figure 7 (c)]. 88 Both the Dirac and nodal-line semimetals possess four-fold degeneracy, whereas the Weyl semimetals possess two-fold degeneracy because of the split bandstructure. Multiple band touchings may occur in a given semimetal, along with corresponding sets of Dirac cones within the Brillouin zone. 21, 26, 86 As a notable example of the rich topological structure that arises, Xu et al. find 24 bulk Weyl cones in semimetal Cd3As2. 89

Related Reviews. Wang et al., 15 Yan and Felser, 19 Armitage et al., 21 Schoop et al., 26 and Klemenz et al. 112 have presented reviews dealing with various aspects of topological semimetals. The review of Wang et al. focuses specifically on quantum transport in Dirac and Weyl semimetals, making it of most interest to readers seeking experimental evidence for Majorana quasiparticles in these materials. 15 The review of Armitage et al. is especially complete, featuring a detailed discussion of the topological properties of the semimetals, theoretical consideration of how such materials may be realized, an overview of materials considerations allowing identification of realsemimetal candidates, and an overview of experiments to identify topology in semimetal systems and to study their quantum transport behavior. 21 In the course of their review, they provide an exhaustive survey of the semimetal literature. The subsequent review of Schoop et al., 26 lends a strong narrative focus on chemistry and materials science. Two further aspects will attract some readers: first, the properties of the topological semimetals are explained in terms of the better-known, simpler 2D semimetal material, graphene; second, the authors discuss applications of topological semimetals beyond quantum computing, with emphasis on catalysis. The very recent review of Klemenz et al. features a tighter focus on square-net topological semimetals and their structure-property relationships. 112 Finally, for readers seeking a brief, but technically thorough overview, Burkov clearly recounts the topological semimetals in a commentary article. 86

Figure 8. Schematic diagram of a fermi arc connecting two Weyl nodes formed in a Weyl semimetal (The corresponding Dirac cones are omitted for simplicity). 89 From Xu, S.-Y., et al., Science, 2015, 349(6248), 613-617. Reprinted with permission from AAAS.

Materials Studies. Guided by recent theoretical developments, several candidate topological semimetals have been experimentally investigated. Initial studies of Weyl semimetals include pyrochlore iridates, HgTe/CdTeheterostructures and alloys, 90 and the monopnictides TaAs, TaP, NbAs, and NbP. 89, 91, 92, 93 Initial studies of Dirac semimetals include the Na3Bi-class materials, 94, 95, 96 and stoichiometric Cd3As2, as well as the related (Cd1-xZnx)3As2 alloys, 97, 98 and the chalcogenide ZrTe5. 99 The literature for the topological semimetals has already grown far larger than these brief examples; below, we direct readers to several recent, more-extensive reviews. We also provide moredetailed reviews of Cd3As2, ZrTe5 , and the pyrochlore iridates in Section 5.

4.6 Experimental Methods for Probing Topological Materials. Several of the previously noted reviews discuss the experimental methods used to characterize the properties of topological materials and to probe for the existence of Majorana quasiparticles within them. Here, we briefly sketch the most important methods that have been applied or proposed within the literature. In Section 5, we will Foundational Theory. Theory for the topological highlight which methods have been used for the specific semimetals stems from Murakami, 100 who in 2007 analyzed materials under review. However, it is important to phase transitions between the quantum-spin Hall (QSH) and distinguish between evidence that supports the topological insulator phases in 3D materials. This showed that a novel properties of the band structure – which would indirectly gapless semimetal phase originates between the QSH and suggest that Majorana quasiparticles could be realized – and topological insulator phases because of the topological evidence that shows the existence of Majorana states. nature of the 3D-semimetal nodes. In 2011, Burkov and Evidence for the right topological band structure is easier to Balents theoretically examined realization of a 3D Weyl obtain, covers more experimental techniques, and so is more semimetal phase in a TI multilayer, presaging a materialswidely reported in the literature. Experiments for Majorana ACS Paragon Plus Environment 9


quasiparticles are more controversial in interpretation and generally much more difficult to perform as complicated device geometries and lower temperatures are needed.

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providing a visualization of the fermi surfaces in the bulk and at the surface. ARPES has thus been used extensively for topological properties of band structures in quantum materials. 117 ARPES can also be used to determine the degree of spin polarization of energy bands. 118 However, ARPES experiments can be irreproducible due to their high surface sensitivity and the consequent need for very clean samples. A frequent criticism of ARPES is the uncertainty about whether the measured surface band structure is representative of the bulk band structure. However, if successful, these measurements can be directly compared to theoretical predictions for topological band structure.

Direct evidence for the existence of Majorana quasiparticles in the solid state involves observing the nonAbelian statistics through a braiding experiment, as discussed previously. Braiding in this context means exchanging the positions of a Majorana pair and observing a difference in phase. This type of experiment has not been done yet for any candidate system. In addition, there are numerous proposals for braiding, but it is not yet clear how to accomplish this experiment. Because this is a rapidly evolving field and not directly a materials problem, we do not discuss braiding experiments that have been attempted or proposed to date.

Scanning-Tunneling Microscopy. Combined scanning tunneling microscopy and spectroscopy (STM and STS) are another tool for understanding topological quantum materials. 119 This is a spatially resolved surfacespectroscopic technique in that the tunneling current between a small metal tip and a metallic surface is proportional to the local density of states. This technique is also capable of measuring the fermi surface using quasiparticle interference resulting from the scattering of surface defects. 120 STS is considered complementary to ARPES because it is a real space probe at the cost of momentum space resolution.

To support the claim that a given material is a candidate for supporting Majorana quasiparticles, several proximate experiments are often done. Topologically non-trivial electronic states in proximity to an s-wave superconductor are needed to form Majorana quasiparticles. These electronic states must then be shown to be topologically nontrivial; selected experimental methods for doing this demonstration follow. Magnetoresistance. The simplest measurement to perform is longitudinal magnetoresistance. In Weyl and Dirac semimetals, there is a negative quadratic magnetoresistance when the electric field direction is parallel to the applied magnetic field, at sufficiently low magnetic fields, 113 which is sometimes referred to as the chiral anomaly. This effect is predicted to be observed sufficiently close to the Dirac or Weyl nodes. A large negative magnetoresistance is not unique to exact Dirac or Weyl nodes. 114 A negative longitudinal magnetoresistance can also be caused by artifacts from non-uniform current flow in irregularly shaped samples. 115 Magnetoresistance is not considered conclusive proof of a topologically nontrivial band structure that could support Majorana quasiparticles because it is difficult to interpret theoretically and due to the presence of extrinsic artifacts.

4.7 Experimental Methods for Probing the Existence of Majorana Quasiparticles. Braiding is considered the ultimate proof of the existence of Majorana states. However, two important proximate experiments have emerged that indirectly support the existence of Majorana quasiparticles. Zero-Bias Conductance. One difficulty with detecting Majorana quasiparticles is that they are charge neutral. However, the tunneling conductance between a topologically non-trivial non-superconducting region and an s-wave superconductor can be modified due to the presence of a Majorana state at the interface of a normalsuperconducting junction. 121, 122, 123, 124 Sau et al. proposed that electron tunneling at a normal-superconductor junction should exhibit a peak in the conductance at zero voltage. 82, 18 This zero-bias conductance (ZBC) peak would be the simplest proof of the existence of Majorana states at the end of such a wire. However, several authors have discussed how other effects, unrelated to the presence of Majorana end states, can lead to a ZBC peak. Some of the discussed artifacts for a Majorana-induced ZBC peak include disorder 125, 126 and conventional Andreev bound states, 127, 128 and so, this measurement is not conclusive. Nonetheless, this experiment is a necessary step for realizing a system for controlling Majorana end states and is a precursor to any braiding experiment. When analyzing ZBC peaks, one should be aware of these artifacts and perform numerous control experiments; a good example can be found in Nichele et al. for the InAs nanowire-aluminum system. 129

Quantum Oscillations. Traditional quantum oscillation measurements, where a transport or thermodynamic property is measured as a function of magnetic field, allow you to measure the fermi surface and effective mass of charge carriers. Thus, one is also to gain insight into whether a material has the right topology in the band structure to allow Majorana quasiparticles. For example, the presence of a Dirac cone and associated Berry’s phase in the band structure can be determined. 116 Quantum oscillation measurements are often confusing due to the presence of multiple bands, whose contribution to the signal are weighted differently depending on a complicated mixture of carrier mobility and density of states. Furthermore, surface states must be distinguished from bulk states, and bandstructure simulations are usually needed to inform the interpretation of measurements.

Josephson Tunneling. The ZBC peak is fundamentally a single electron tunneling effect. When considering the tunneling of Cooper pairs from the superconducting electrode, more exotic effects are predicted to occur that can Photoemission. Topological properties of the band provide further evidence for Majorana end states. 77, 123, 130, structure are in principle accessible directly using 131 We summarize one example experiment to highlight this spectroscopic methods. Angle resolved photoemission technique. In a superconductor-normal metalspectroscopy (ARPES) allows the determination of energysuperconductor junction, the current or voltage across the momentum relations of electron states in solids, thereby ACS Paragon Plus Environment 10

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Chemistry of Materials topological behavior. For example, few-monolayer-thick superconducting layers have been predicted to become topological superconductors. 135 There is thus an intense materials search for 2D materials that can be easily formed into layers of few monolayer thickness.

junction is periodic in the magnetic flux through a closed loop terminated by the junction, considering only the tunneling of Cooper pairs. The presence of states within the superconducting gap disturbs this periodicity because there is a contribution to the current that is not due to Cooper pairs. For a superconductor-metal-superconductor junction, where the “metal” has a topologically non-trivial band structure as described earlier, Majorana end states are expected to form, and the periodicity in the current-phase relation changes from 2 to 4. This 4 periodicity can be examined under microwave irradiation, and is referred to as the ac fractional Josephson effect. 132 This fractional Josephson tunneling effect is considered a stronger form of evidence for Majorana quasiparticles, in fact one can view this type of experiment as an indirect form of non-Abelian braiding. 133 Nevertheless, non-topological alternative explanations have been suggested for this behavior. 134

In addition, as already pointed out above for the case of topological insulators and semiconductors, a onedimensional nanowire with strong spin-orbit coupling proximitized to a conventional s-wave superconductor 81 has been proposed to create and detect Majorana zero modes. Notably, this approach may also be applied to the topological semimetals reviewed here as exemplars. Similarly, electron- or hole-doping of topological insulators has been pursued to realize intrinsic bulk topological superconductors in 2D and 3D (e.g., copper-doped Bi2Se3). 136 This doping approach has also been proposed to apply to the topological semimetals, again yielding topological superconductors that may also host Majorana quasiparticles. 137 Provided that such strategies prove useful for modifying the behavior of not only the topological insulators -- but also the topological semimetals -- the semimetals, too, take on broad importance in the growing search for Majoranas.

5. MATERIALS EXEMPLARS In an effort to bring physical evidence to the theory predicting both topological phenomena and the existence of Majorana quasiparticles, we next present three materials exemplars whose structural characteristics have recently led to relevant topological or Majorana-related behaviors. These exemplars are: (i) Cd3As2, a II-V compound which is predicted to be either a Dirac or a Weyl semimetal, depending on the symmetry of the low-temperature  phase; (ii) layered dichalcogenides (MX2), and ZrTe5 where it is predicted to be a Dirac semimetal; and (iii) the rare-earth pyrochlore iridates, which may be either topological insulators or topological semimetals. We endeavor to describe in detail the crystallographic structures, bulk phase diagrams, materials-synthesis methods (for bulk, thin film, and/or nanowire forms), and recent evidence that either establishes topological behavior, or suggests existence of Majorana quasiparticles.

5.1 Cd3As2 and Related Group II-V Materials. The II-V compound Cd3As2 is predicted as either a Dirac or a Weyl semimetal. It is synthesized either as bulk crystals or thin films by solid state methods, as described below. The crystal chemistry of this type of cadmium arsenide is described as a distorted superstructure of an anti-fluorite (M2X) structure type, and is considered the low temperature crystal structure phase. The anti-fluorite arrangement consists of an FCC sublattice of As atoms, each surrounded by both 6 Cd atoms and two vacancies (at diagonals from each other) all sitting at the cube vertexes. Ali, et al., 138 have shown the phase to have a tetragonal unit cell of a = 12.633(3) Å and c = 25.427(7) Å in the centrosymmetric I41/acd space group. There are three other phases; they include a high temperature phase in the Fm-3m space group, and an intermediate temperature phase in the P42/nmc space group, and a low temperature phase crystallizing in a noncentrosymmetric I41cd space group. The two low temperature phases are a result of the different ordering of the CdAs tetrahedral units inside the structure. 139 Three of the four main polymorph structures of Cd3As2 are shown in Figure 9.

The I41/acd (No. 142) centrosymmetric phase has a needle crystal morphology along the direction and is generally grown in the 575 °C region. By contrast, the alternative low-temperature phase not shown in the figure, 138 I41cd, has a plate like morphology with a large (112) area, Figure 9. The structure of Cd3As2. Reprinted with permission from Inorg. Chem. 53, 8, 4062-4067. Copyright 2014 American and is a noncentrosymmetric phase. It has been reported by Chemical Society. Ali, et al., that the low temperature phase of the I41/acd space group is the phase which enables Cd3As2 to be a Dimensionality and symmetry requirements have a great symmetry constrained 3D Dirac semimetal, as described in influence on the realization of Majorana quasiparticles in Section 4.5, and the electronic 3D analog of graphene. 138 materials. The original Kitaev model met such requirements The powder x-ray diffraction patterns allowing (in part) through used of an idealized 1D chain geometry. differentiation of the two of the low temperature polymorphs Real materials prove more complex, because inversion must are shown in Figure 10. 139 AFM measurements confirmed be broken to obtain the correct band structure to realize Weyl the effects of crystal structure defects at the atomic scale as or Dirac fermions, and time reversal symmetry must also be affecting the valence band and enabling the mobility of the broken for Weyl fermions. One can break inversion carriers in the conduction band. 140 symmetry by making extremely thin layers to realize ACS Paragon Plus Environment 11


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Reprinted by permission from Springer Nature, J. Phase Equilibria, The As-Cd (Arsenic-Cadmium) System, H. Okamoto, 1992.

The growth of crystals of the I41cd space group by Sankar, et al. 139 have utilized a multizone single-crystal growth method by self-selecting vapor growth (SSVG). Stoichiometric amounts of 99.9999% pure elements of Cd and As were added to double quartz ampoules, which were evacuated and sealed. The ampoule was then heated to 50 °C above the melting point and held for 4 hours to enable a homogenous mixture of elements to form. This mixture was then quenched. The resulting ingot was isothermally annealed at 700 °C for 3 days and then air cooled. It is then crushed and loaded into another growth ampoule and placed within a 3-zone furnace at a temperature range of 675-575500 °C for 10 days. The resultant crystals are usually (112) oriented. Figure 10. Synchrotron powder X-ray diffraction pattern for Cd3As2 and the corresponding Rietveld refinement. 139 A comparison of the I41/acd and I41cd phases is presented.

The Cd/As thermodynamic phase diagram in Figure 11 indicates a number of Cd As2 phases [ (stable), ’, ” Figure S4. Synchrotron x-ray3 diffraction patterns of Cd As with I41cd and I41/acd symmetries (metastable),  with specific temperature dependencies. 3In 2 general, the Cd3As2 phase transition temperatures are  to are compared. The diffraction peaks are more pronounced for I41cd than I41/acd, indicating that ’ ~227 °C, ’ to  ~469 °C, and ” to  at ~595 °C, with the Cd3As2 melting at ~716 °C. For purposes of this review, Cd3As2 of I41cd has a higher crystallinity. we are focusing on the low temperature  phase. There are a variety of methods of growing single crystals of this phase with different preferred orientations. Next, we summarize Figure 12. A single crystal of Cd3As2. some of the successfully used methods. 141

138 Reprinted with permission from Inorg. Chem. 2014, 53, 8, 4062-4067. Copyright 2014 American Chemical Society.

Cd3As2 Single Crystals. Interest in single crystals of this phase for topological applications are generally with the focus for eventual exfoliation to a thin film. The growth of single crystals of the I41/acd space group is carried out under flux, with a Cd-rich melt of Cd5-Cd3As2 being required. The metals are added to a quartz glass ampoule, evacuated, and sealed. The mixture is heated to 825 °C, and held for 48 hours. To ensure single crystal growth, a slow cool rate of 6 °C/hour is performed to a temperature of 425 °C. The sample is subsequently centrifuged. This preparation ensured 100% Cd3As2 crystal growth from a 1:5 flux. The recovered single crystals are silvery-metallic in appearance, as seen in Figure 12. 138

Cd3As2 nanostructures. Nanostructured phases have been grown on Si substrates by the simple variation of Ar gas temperature and pressures during straightforward chemical vapor deposition (CVD). The resulting morphologies include nanowires, nanobelts, nanoplates, and nano-octahedra. 142 To grow and vary these nanostructures, the authors utilized the following CVD-based methodologies. A mixture of Cd and As elemental metals were mixed inside a test tube that is placed inside a quartz tube that is then sealed. The experiments were conducted in a single temperature tube furnace. Therein, the metals are vaporized and transported toward the cooler end of the tube where it is believed that Cd nanodrops form nucleation sites Figure S5. STM topography measurements performed on the cleaved surface of I41which /acd the CdAs nanocrystals then on the(112) Si substrates, upon condense. When growth conditions are at an Ar flow rate of 50 sccm, and at pressures ranging from 100 to 120 Pa, respectively, nanocrystal rods of Cd3As2 grow in progressively larger sizes with pressure. When the pressure is increased further to 140 Pa, a mixture of rods and plates is grown. At approximately 150 Pa, plate-only phases are grown. Finally, at over 200 Pa, individual nanoplates and unique nanooctogons are grown. Though Cd3As2 has generally been grown as a single crystal and then mechanically cleaved or exfoliated to enable a thin crystal for device fabrication and testing, there are examples in the literature of Cd3As2 thin-film growth. Methods pursued in recent TQMs studies include molecularbeam epitaxy (MBE) and pulsed-laser deposition (PLD). Figure 11. Phase diagram for the Cd-As materials system; four Two examples of Cd3As2 growth by these methods appear distinct phases are seen for Cd3As2 at the 60 atm% Cd position. 141 next. ACS Paragon Plus Environment 12

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Molecular Beam Epitaxy. Thin film growth of Cd3As2 has been successfully accomplished by MBE, as shown in Figure 13, 143 resulting in a vacancy-ordered, tetragonalstructure Dirac-semimetal phase at low growth temperatures. The Cd3As2 was evaporated from solid pieces by using an effusion cell, with the BEP fluxes varying between 2x10-7 and 5x10-6 Torr. Typical film-growth times were 60 min, during which 100–300 nm thick films were grown at substrate temperatures between 110 °C and 220 °C. The films were successfully grown on (111) B oriented GaAs substrates, onto which 140-nm-thick (111) GaSb buffer layers were grown prior to Cd3As2 growth. The orientation relationship is described by (112) Cd3As2 || (111) GaSb, and [1¯10] Cd3As2 || [¯101] GaSb. 143 Figure 14. Cd3As2 film grown on a substrate of SrTiO3 by combing low-temperature PLD and elevated-temperature solid-phase epitaxy. The cap of Si3N4/TiO2 is deposited in-situ during PLD and serves to prevent Cd3As2 desorption during the post-PLD regrowth anneal (10 nm scale bar). 144

Realized Topological Behavior in Cd3As2. Among the many known Dirac semimetals, Cd3As2 features uniquely useful properties: it has a high electron mobility, 145 it can be epitaxially grown as a planar heterostructure, 146 and it is air stable, 91 which is ideal for real-world applications and research experiments. These properties combined make Cd3As2 an unusually promising Dirac-semimetal material for topological quantum-computation applications. A detailed accounting as to the history of predictions and experimental evidence for fermi arc surface states, quantum oscillations, and deduced non-trivial Berry phase in Cd3As2, is found in Yu, et al. 147

Figure 13. Low-magnification HAADF-STEM image of a Cd3As2 single-heterolayer grown by MBE on a GaAs-GaSb substrate and buffer. 143

Since the discovery of Cd3As2 as a stable Dirac semimetal, multiple approaches have been used to realize topological superconductivity in this material. In initial efforts about three years ago, two groups using a pointcontact technique have reported tentative evidence of unconventional superconductivity in Cd3As2. 151, 152 However, in these studies, the hallmark of superconductivity, a zero-resistance state, was not observed. Later, by using a high-pressure technique, superconductivity and the zero-resistance state were achieved. 153 However, further X-ray diffraction (XRD) analysis showed that the pressure-induced superconducting transition was accompanied by a structural phase transition. 153, 154 Thus, it is not clear whether the induced superconductivity was topological or not. In contrast, topological superconductivity has clearly been achieved through the proximity effect in topological insulators. 71, 155 Therefore, it is expected that the presence of the s-wave pairing in the fermi arc states in Cd3As2 can lead to the formation of surface topological superconductivity, and topologically protected gapless Andreev bound states (or Majorana zero modes). 91, 148, 149, 150

Pulsed Laser Deposition. Cd3As2 film growth by PLD involves use of a KrF excimer laser, which ablates a Cd3As2 polycrystalline target to create a growth flux. The stoichiometric targets are prepared by mixing 6N5 Cd and 7N5 As shots at a ratio of 3:2, followed by annealing of the mixture at 950 °C for 48 hours in a vacuum-sealed silica tube. The anneal step is followed by grinding and pelletizing the compound, and then re-sintering at 250 °C for 30 hours. Using PLD of these and other targets, Cd3As2 thin films and TiO2 / Si3N4 capping layers were successively deposited on (001) SrTiO3 single-crystalline substrates, with the growths conducted at room temperature with a base pressure of

Topological Quantum Materials for Realizing Majorana Quasiparticles

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