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We report and quantify a large orbital-Hall torque generated by Nb and Ru, which we identify from a strong dependence of torques on the ferromagnets. This is manifested as a sign reversal and strong enhancement in the damping-like torques measured in Nb (or Ru)/Ni bilayers as compared to Nb (or Ru)/FeCoB bilayers. The long-range nature of orbital transport in the ferromagnet is revealed by the thickness dependences of Ni in Nb (or Ru)/Ni bilayers which are markedly different from the regular spin absorption in the ferromagnet that takes place within a few angstroms and thus it uniquely distinguishes the orbital Hall torque from the spin Hall torque.

The recent emergence of magnetic van der Waals materials allows for the investigation of current induced magnetization manipulation in two dimensional materials. Uniquely, Fe3GeTe2 has a crystalline structure that allows for the presence of bulk spin-orbit torques (SOTs), that we quantify in a Fe3GeTe2 flake. From the symmetry of the measured torques, we identify the current induced effective fields using harmonic analysis and find dominant bulk SOTs, which arise from the symmetry in the crystal structure. Our results show that Fe3GeTe2 uniquely can exhibit bulk SOTs in addition to the conventional interfacial SOTs enabling magnetization manipulation even in thick single layers without the need for complex multilayer engineering., Comment: 19 pages 4 figures

Here, we report first-principles calculations of laser-induced photocurrents at the surface of a prototype Rashba system. By referring to Keldysh nonequilibrium formalism combined with the Wannier interpolation scheme, we perform first-principles electronic structure calculations of a prototype BiAg2 surface alloy, which is a well-known material realization of the Rashba model. In addition to the nonmagnetic ground state situation, we also study the case of in-plane magnetized BiAg2. We calculate the laser-induced charge photocurrents for the ferromagnetic case and the laser-induced spin photocurrents for both the nonmagnetic and the ferromagnetic cases. Our results confirm the emergence of very large in-plane photocurrents as predicted by the Rashba model and are in agreement with previous experimental measurements of THz emission generated at Bi/Ag interfaces. The resulting photocurrents satisfy all the symmetry restrictions with respect to the light helicity and the magnetization direction. We provide microscopic insights into the symmetry and magnitude of the computed currents based on the ab initio multiband electronic structure of the system, and scrutinize the importance of resonant two-band and three-band transitions for driven currents, thereby establishing a benchmark picture of photocurrents at Rashba-like surfaces and interfaces. Our work contributes to establishing the interfacial Rashba spin-orbit interaction as a major mechanism for the generation of in-plane photocurrents, which are of great interest in the field of ultrafast and terahertz spintronics.

Among two-dimensional materials, Fe3GeTe2 has come to occupy a very important place owing to its ferromagnetic nature with one of the highest Curie temperatures among known van der Waals materials and the potential for hosting skyrmions. In this combined experimental and theoretical work, we investigate the magnetic bubble domains as well as the microscopic domain wall profile using spin-polarized scanning tunneling microscopy in combination with atomistic spin-dynamics simulations performed with parameters from density functional theory calculations. We find a weak magneto-electric effect influencing the domain wall width by the electric field in the tunneling junction and determine the critical magnetic field for the collapse of the bubble domains. Our findings shed light on the origins of complex magnetism that Fe3GeTe2 exhibits.

The emergence of spin-orbit torques as a promising approach to energy-efficient magnetic switching has generated large interest in material systems with easily and fully tunable spin-orbit torques. Here, current-induced spin-orbit torques in VO$_2$/NiFe heterostructures were investigated using spin-torque ferromagnetic resonance, where the VO$_2$ layer undergoes a prominent insulator-metal transition. A roughly two-fold increase in the Gilbert damping parameter, $\alpha$, with temperature was attributed to the change in the VO$_2$/NiFe interface spin absorption across the VO$_2$ phase transition. More remarkably, a large modulation ($\pm$100%) and a sign change of the current-induced spin-orbit torque across the VO$_2$ phase transition suggest two competing spin-orbit torque generating mechanisms. The bulk spin Hall effect in metallic VO$_2$, corroborated by our first-principles calculation of spin Hall conductivity, $\sigma_{SH} \approx 10^4 \frac{\hbar}{e} \Omega^{-1} m^{-1}$, is verified as the main source of the spin-orbit torque in the metallic phase. The self-induced/anomalous torque in NiFe, of the opposite sign and a similar magnitude to the bulk spin Hall effect in metallic VO$_2$, could be the other competing mechanism that dominates as temperature decreases. For applications, the strong tunability of the torque strength and direction opens a new route to tailor spin-orbit torques of materials which undergo phase transitions for new device functionalities., Comment: 16 pages, 5 figures

Motivated by recent progress in two-dimensional (2D) spintronics, we present a monolayer of Eu deposited on 1H−WSe2 as a promising platform for engineering spin-orbit effects and Berry curvature. By first-principles calculations based on density functional theory, we show that Eu/WSe2 exhibits intriguing properties such as high magnetic anisotropy, valley-dependent polarization of spin and orbital angular momenta, and Rashba textures. These originate from magnetic and spin-orbit proximity effects at the interface and the interplay between localized 4f magnetic moments of Eu and mobile charge carriers of Eu and WSe2. The analysis of the magnetic properties reveal a ferromagnetic configuration with an out-of-plane easy axis of the magnetization, which favor a pronounced anomalous Hall effect in the proposed system. Thus, we promote 4f rare-earth metals deposited on top of a transition-metal dichalcogenides as a promising platform for 2D spintronics.

We propose a concept of non-collinear spin current, whose spin polarization varies in space even in non-magnetic crystals. While it is commonly assumed that the spin polarization of the spin Hall current is uniform, asymmetric local crystal potential generally allows the spin polarization to be non-collinear in space. Based on microscopic considerations we demonstrate that such non-collinear spin Hall currents can be observed for example in layered Kagome Mn$_3$X (X = Ge, Sn) compounds. Moreover, by referring to atomistic spin dynamics simulations we show that non-collinear spin currents can be used to switch the chiral spin texture of Mn$_3$X in a deterministic way even in the absence of an external magnetic field. Our theoretical prediction can be readily tested in experiments, which will open a novel route toward electric control of complex spin structures in non-collinear antiferromagnets.

We explore spin dynamics in Cu(1,3-bdc), a quasi-2D topological magnon insulator. The results show that the thermal evolution of Land\'e $g$-factor ($g$) is anisotropic: $g_\textrm{in-plane}$ reduces while $g_\textrm{out-plane}$ increases with increasing temperature $T$. Moreover, the anisotropy of the $g$-factor ($\Delta g$) and the anisotropy of saturation magnetization ($\Delta M_\textrm{s}$) are correlated below 4 K, but they diverge above 4 K. We show that the electronic orbital moment contributes to the $g$ anisotropy at lower $T$, while the topological orbital moment induced by thermally excited spin chirality dictates the $g$ anisotropy at higher $T$. Our work suggests an interplay among topology, spin chirality, and orbital magnetism in Cu(1,3-bdc)., Comment: 6 pages, 4 figures

The discovery of topological states of matter has led to a revolution in materials research. When external or intrinsic parameters break certain symmetries, global properties of topological materials change drastically. A paramount example is the emergence of Weyl nodes under broken inversion symmetry, acting like magnetic monopoles in momentum space. However, while a rich variety of non-trivial quantum phases could in principle also originate from broken time-reversal symmetry, realizing systems that combine magnetism with complex topological properties is remarkably elusive due to both considerable experimental and theoretical challenges. Here, we demonstrate that giant open Fermi arcs are created at the surface of ultrathin hybrid magnets. The Fermi-surface topology of an atomically thin ferromagnet is substantially modified by the hybridization with a heavy-metal substrate, giving rise to Fermi-surface discontinuities that are bridged by the Fermi arcs. Due to the interplay between magnetism and topology, we can control both the shape and the location of the Fermi arcs by tuning the magnetization direction. The hybridization points in the Fermi surface can be attributed to a non-trivial "mixed" topology and induce hot spots in the Berry curvature, dominating spin and charge transport as well as magneto-electric coupling effects., 14 pages, 10 figures

Recently discovered high-quality nodal chain spin-gapless semimetals $M$F$_3$ ($M$ = Pd, Mn) feature an ultra-clean nodal chain in the spin up channel residing right at the Fermi level and displaying a large spin gap leading to a 100\% spin-polarization of transport properties. Here, we investigate both intrinsic and extrinsic contributions to anomalous and spin transport in this class of materials. The dominant intrinsic origin is found to originate entirely from the gapped nodal chains without the entanglement of any other trivial bands. The side-jump mechanism is predicted to be negligibly small, but intrinsic skew-scattering enhances the intrinsic Hall and Nernst signals significantly, leading to large values of respective conductivities. Our findings open a new material platform for exploring strong anomalous and spin transport properties in magnetic topological semimetals., 7 pages, 4 figures

The bosonic analogs of topological insulators have been proposed in numerous theoretical works, but their experimental realization is still very rare, especially for spin systems. Recently, two-dimensional (2D) honeycomb van der Waals ferromagnets have emerged as a new platform for topological spin excitations. Here, via a comprehensive inelastic neutron scattering study and theoretical analysis of the spin-wave excitations, we report the realization of topological magnon insulators in CrXTe

Spin-orbit torque (SOT) is an emerging technology that enables the efficient manipulation of spintronic devices. The initial processes of interest in SOTs involved electric fields, spin-orbit coupling, conduction electron spins and magnetization. More recently interest has grown to include a variety of other processes that include phonons, magnons, or heat. Over the past decade, many materials have been explored to achieve a larger SOT efficiency. Recently, holistic design to maximize the performance of SOT devices has extended material research from a nonmagnetic layer to a magnetic layer. The rapid development of SOT has spurred a variety of SOT-based applications. In this Roadmap paper, we first review the theories of SOTs by introducing the various mechanisms thought to generate or control SOTs, such as the spin Hall effect, the Rashba-Edelstein effect, the orbital Hall effect, thermal gradients, magnons, and strain effects. Then, we discuss the materials that enable these effects, including metals, metallic alloys, topological insulators, two-dimensional materials, and complex oxides. We also discuss the important roles in SOT devices of different types of magnetic layers. Afterward, we discuss device applications utilizing SOTs. We discuss and compare three-terminal and two-terminal SOT-magnetoresistive random-access memories (MRAMs); we mention various schemes to eliminate the need for an external field. We provide technological application considerations for SOT-MRAM and give perspectives on SOT-based neuromorphic devices and circuits. In addition to SOT-MRAM, we present SOT-based spintronic terahertz generators, nano-oscillators, and domain wall and skyrmion racetrack memories. This paper aims to achieve a comprehensive review of SOT theory, materials, and applications, guiding future SOT development in both the academic and industrial sectors., an invited paper in the "Advances in Magnetics" series

The exchange interaction governs static and dynamic magnetism. This fundamental interaction comes in two flavours-symmetric and antisymmetric. The symmetric interaction leads to ferro- and antiferromagnetism, and the antisymmetric interaction has attracted significant interest owing to its major role in promoting topologically non-trivial spin textures that promise fast, energy-efficient devices. So far, the antisymmetric exchange interaction has been found to be rather short ranged and limited to a single magnetic layer. Here we report a long-range antisymmetric interlayer exchange interaction in perpendicularly magnetized synthetic antiferromagnets with parallel and antiparallel magnetization alignments. Asymmetric hysteresis loops under an in-plane field reveal a unidirectional and chiral nature of this interaction, which results in canted magnetic structures. We explain our results by considering spin-orbit coupling combined with reduced symmetry in multilayers. Our discovery of a long-range chiral interaction provides an additional handle to engineer magnetic structures and could enable three-dimensional topological structures.

Topological semimetals in ferromagnetic materials have attracted enormous attention due to the potential applications in spintronics. Using the first-principles density functional theory together with an effective lattice model, here we present a new family of topological semimetals with a fully spin-polarized nodal loop in alkaline-metal monochalcogenide $MX$ ($M$ = Li, Na, K, Rb, Cs; $X$ = S, Se, Te) monolayers. The half-metallic ferromagnetism can be established in $MX$ monolayers, in which one nodal loop formed by two crossing bands with the same spin components is found at the Fermi energy. This nodal loop half-metal survives even when considering the spin-orbit coupling owing to the symmetry protection provided by the $\mathcal{M}_{z}$ mirror plane. The quantum anomalous Hall state and Weyl-like semimetal in this system can be also achieved by rotating the spin from the out-of-plane to the in-plane direction. The $MX$ monolayers hosting rich topological phases thus offer an excellent materials platform for realizing the advanced spintronics concepts.

Recent experimental observation of an unexpectedly large current-induced spin-orbit torque in surface oxidized Cu on top of a ferromagnet pointed to a possibly prominent role of the orbital Rashba effect (ORE) in this system. Here, we use first principles methods to investigate the ORE in a system of oxygen monolayer deposited on top of a Cu(111) film. We show that surface oxidization of the Cu film leads to a gigantic enhancement of the ORE near the Fermi energy. The resulting chiral orbital texture in the momentum space is exceptionally strong, reaching as much as $\ensuremath{\sim}0.5\ensuremath{\hbar}$ in magnitude. We find that resonant hybridization between O $p$ states and Cu $d$ states is responsible for the emergence of the ORE at the interface. We also present a minimal model that captures the emergence of the ORE through the $pd$ hybridization mechanism. We demonstrate that an application of an external electric field to the system generates colossal orbital Hall currents which are an order of magnitude larger than the spin Hall currents found in heavy metals. This implies that the ``orbital torque'' mechanism may be significant in surface oxidized Cu/ferromagnet structures. Our results encourage an experimental verification of the rich orbital physics in surface oxidized Cu films through optical measurements such as angle-resolved photoemission spectroscopy and momentum microscopy.

Motivated by the importance of understanding competing mechanisms to current-induced spin-orbit torque in complex magnets, we develop a unified theory of current-induced spin-orbital coupled dynamics. The theory describes angular momentum transfer between different degrees of freedom in solids, e.g., the electron orbital and spin, the crystal lattice, and the magnetic order parameter. Based on the continuity equations for the spin and orbital angular momenta, we derive equations of motion that relate spin and orbital current fluxes and torques describing the transfer of angular momentum between different degrees of freedom. We then propose a classification scheme for the mechanisms of the current-induced torque in magnetic bilayers. Based on our first-principles implementation, we apply our formalism to two different magnetic bilayers, Fe/W(110) and Ni/W(110), which are chosen such that the orbital and spin Hall effects in W have opposite sign and the resulting spin- and orbital-mediated torques can compete with each other. We find that while the spin torque arising from the spin Hall effect of W is the dominant mechanism of the current-induced torque in Fe/W(110), the dominant mechanism in Ni/W(110) is the orbital torque originating in the orbital Hall effect of W. It leads to negative and positive effective spin Hall angles, respectively, which can be directly identified in experiments. This clearly demonstrates that our formalism is ideal for studying the angular momentum transfer dynamics in spin-orbit coupled systems as it goes beyond the "spin current picture" by naturally incorporating the spin and orbital degrees of freedom on an equal footing. Our calculations reveal that, in addition to the spin and orbital torque, other contributions such as the interfacial torque and self-induced anomalous torque within the ferromagnet are not negligible in both material systems., Comment: 26 pages, 13 figures

Motivated by the recently observed unconventional Hall effect in ultra-thin films of ferromagnetic SrRuO$_3$ (SRO) we investigate the effect of strain-induced oxygen octahedral distortion in the electronic structure and anomalous Hall response of the SRO ultra-thin films by virtue of density functional theory calculations. Our findings reveal that the ferromagnetic SRO films grown on SrTiO$_3$ (in-plane strain of $-$0.47$\%$) have an orthorhombic (both tilting and rotation) distorted structure and with an increasing amount of substrate-induced compressive strain the octahedral tilting angle is found to be suppressed gradually, with SRO films grown on NdGaO$_3$ (in-plane strain of $-$1.7$\%$) stabilized in the tetragonal distorted structure (with zero tilting). Our Berry curvature calculations predict a positive value of the anomalous Hall conductivity of $+$76\,S/cm at $-$1.7$\%$ strain, whereas it is found to be negative ($-$156\,S/cm) at $-$0.47$\%$ strain. We attribute the found behavior of the anomalous Hall effect to the nodal point dynamics in the electronic structure arising in response to tailoring the oxygen octahedral distortion driven by the substrate-induced strain. We also calculate strain-mediated anomalous Hall conductivity as a function of reduced magnetization obtained by scaling down the magnitude of the exchange field inside Ru atoms finding good qualitative agreement with experimental observations, which indicates a strong impact of longitudinal thermal fluctuations of Ru spin moments on the anomalous Hall effect in this system., 8 pages, 4 figures. arXiv admin note: text overlap with arXiv:2002.05393

We investigate the influence of magnons on the temperature-dependence and the anisotropy of the spin-orbit torque (SOT). For this purpose we use 3rd order perturbation theory in the framework of the Keldysh formalism in order to derive suitable equations to compute the magnonic SOT. We find several contributions to the magnonic SOT, which depend differently on the spin-wave stiffness $\mathcal{A}$ and on the temperature $T$, with the dominating contribution scaling like $T^{2}/\mathcal{A}^{2}$. Based on this formalism we compute the magnonic SOT in the ferromagnetic Rashba model. For large Rashba parameters the magnonic SOT is strongly anisotropic and for small quasiparticle broadening it may become larger than the non-magnonic SOT.

Merging the fields of topology and magnetism expands the scope of fundamental quantum phenomena with novel functionalities for topological spintronics enormously. Here, we theoretically demonstrate that ferromagnetism provides an efficient means to achieve a topological switching between crystalline and higher-order topological insulator phases in two dimensions. Using a tight-binding model and first-principles calculations, we identify layered NpSb as a long-awaited two-dimensional topological crystalline insulator with intrinsic ferromagnetic order with a band gap which is as large as 220 meV. We show that when ${\mathcal{M}}_{z}$ symmetry is preserved for the out of plane magnetization of this material, it exhibits a pair of gapless edge states along all boundaries and carries a nonzero mirror Chern number ${\mathcal{C}}_{\mathcal{M}}=1$. Remarkably, when rotating the magnetization into the plane a higher-order topological insulator phase with a parity-based invariant ${\ensuremath{\nu}}_{2\text{D}}=1$ is achieved, and in-gap topological corner states emerge. Our results pave the way to understanding and engineering topological insulating states in two-dimensional ferromagnets.

We investigate laser-induced torques in magnetically non-collinear ferromagnets with a spin-spiral magnetic structure using \textit{ab-initio} calculations. Since spin-spirals may be used to approximate the magnetization gradients locally in domain walls and skyrmions, our method may be used to obtain the laser-induced torques in such objects from a multiscale approach. Employing the generalized Bloch-theorem we obtain the electronic structure computationally efficiently. We employ our method to assess the laser-induced torques in bcc Fe, hcp Co, and L$_{1}0$ FePt when a spin-spiral magnetic structure is imposed. We find that the laser-induced torques in these magnetically noncollinear systems may be orders of magnitude larger than those in the corresponding magnetically collinear systems and that they exist both for linearly and circularly polarized light. This result suggests that laser-induced torques driven by noncollinear magnetic order or by magnetic fluctuations may contribute significantly to processes in ultrafast magnetism.

In metallic noncentrosymmetric crystals and at surfaces the response of spin currents and charge currents to applied electric fields contains contributions that are second order in the electric field, which are forbidden by symmetry in centrosymmetric systems. Thereby, photocurrents and spin photocurrents can be generated in inversion asymmetric metals by the application of femtosecond laser pulses. We study the laser-induced charge current in the ferromagnetic Rashba model with in-plane magnetization and find that this \textit{magnetic photogalvanic effect} can be tuned to be comparable in size to the laser-induced photocurrents measured experimentally in magnetic bilayer systems such as Co/Pt. Additionally, we show that femtosecond laser pulses excite strong spin currents in the nonmagnetic Rashba model when the Rashba parameter is large.

In solids, electronic Bloch states are formed by atomic orbitals. While it is natural to expect that orbital composition and information about Bloch states can be manipulated and transported, in analogy to the spin degree of freedom extensively studied in past decades, it has been assumed that orbital quenching by the crystal field prevents significant dynamics of orbital degrees of freedom. However, recent studies reveal that an orbital current, given by the flow of electrons with a finite orbital angular momentum, can be electrically generated and transported in wide classes of materials despite the effect of orbital quenching in the ground state. Orbital currents also play a fundamental role in the mechanisms of other transport phenomena such as spin Hall effect and valley Hall effect. Most importantly, it has been proposed that orbital currents can be used to induce magnetization dynamics, which is one of the most pivotal and explored aspects of magnetism. Here, we give an overview of recent progress and the current status of research on orbital currents. We review proposed physical mechanisms for generating orbital currents and discuss candidate materials where orbital currents are manifest. We review recent experiments on orbital current generation and transport and discuss various experimental methods to quantify this elusive object at the heart of $orbitronics$ $-$ an area which exploits the orbital degree of freedom as an information carrier in solid-state devices., Comment: Invited article from EPL Perspective

Interfaces are ubiquitous in materials science, and in devices in particular. As device dimensions are constantly shrinking, understanding the physical properties emerging at interfaces is crucial to exploit them for applications, here for spintronics. Using first-principles techniques and Monte Carlo simulations, we investigate the mutual magnetic interaction at the interface between graphene and an antiferromagnetic semiconductor BaMnO3. We find that graphene deeply affects the magnetic state of the substrate, down to several layers below the interface, by inducing an overall magnetic softening, and switching the in-plane magnetic ordering from antiferromagnetic to ferromagnetic. The graphene- BaMnO3 system presents a Rashba gap 300 times larger than in pristine graphene, leading to a flavor of quantum anomalous Hall effect (QAHE), a hybrid QAHE, characterized by the coexistence of metallic and topological insulating states. These findings could be exploited to fabricate devices that use graphene to control the magnetic configuration of a substrate., Z.Z. acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) Grant No. ZA 780/3-1, the Ramon y Cajal MINECO program (Grant No. RYC-2016-19344), the EC H2020-EINFRA-5-2015 MaX Center of Excellence (Grant No. 676598), the Spanish MINECO (Grant No. FIS2015-64886-C5-3-P), the CERCA programme of the Generalitat de Catalunya (Grant No. 2017SGR1506), and by the Severo Ochoa programme (MINECO, Grant No. SEV-2013-0295). M.J.V. acknowledges funding from the Communauté française de Belgique ARC grant (Grant No. AIMED 15/19-09). Y.M. acknowledges funding from the German Research Foundation (Deutsche Forschungsgemeinschaft), Grant No. MO 1731/5-1. This work has been also financially supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Center SPP 1666. The authors acknowledge computer time from the PRACE-3IP and 4IP on resources Lindgren, Archer, and Salomon (EU Grants No. RI-312763 and No. 653838), CECI, SEGI-ULg and Zenobe hosted by CENAERO (Grant No. GA 1117545), the JARA-HPC projects (No. jara0088, No. JIAS16, and No. JHPC39), the JARA-HPC Vergabegremium and VSR commission on the supercomputer JURECA at Forschungszentrum Jülich.

Controlling magnetism by electric fields offers a highly attractive perspective for designing future generations of energy-efficient information technologies. Here, we demonstrate that the magnitude of current-induced spin-orbit torques in thin perpendicularly magnetized CoFeB films can be tuned and even increased by electric-field generated piezoelectric strain. Using theoretical calculations, we uncover that the subtle interplay of spin-orbit coupling, crystal symmetry, and orbital polarization is at the core of the observed strain dependence of spin-orbit torques. Our results open a path to integrating two energy efficient spin manipulation approaches, the electric-field-induced strain and the current-induced magnetization switching, thereby enabling novel device concepts.

The recent demonstration of topological states in antiferromagnets (AFMs) provides an exciting platform for exploring prominent physical phenomena and applications of antiferromagnetic spintronics. A famous example is the AFM topological insulator (TI) state, which, however, was still not observed in two dimensions. Using a tight-binding model and first-principles calculations, we show that, in contrast to previously observed AFM topological insulators in three dimensions, an AFM TI can emerge in two dimensions as a result of a nonsymmorphic symmetry that combines the twofold rotation symmetry and half-lattice translation. Based on the spin Chern number, Wannier charge centers, and gapless edge states analysis, we identify intrinsic AFM $X\mathrm{Mn}Y$ ($X=\mathrm{Sr}$ and Ba, $Y=\mathrm{Sn}$ and Pb) quintuple layers as experimentally feasible examples of predicted topological states with a stable crystal structure and giant magnitude of the nontrivial band gaps, reaching as much as 186 meV for SrMnPb, thereby promoting these systems as promising candidates for innovative spintronics applications.

Physical review research 2(1), 013063 (2020). doi:10.1103/PhysRevResearch.2.013063, Published by APS, College Park, MD