Rules for Phase Shifts of Quantum Oscillations in Topological Nodal-Line Semimetals

Recent progress on topological semimetal IrO2: electronic structures, synthesis, and transport properties

5dtransition metal oxides, such as iridates, have attracted significant interest in condensed matter physics throughout the past decade owing to their fascinating physical properties that arise from intrinsically strong spin-orbit coupling (SOC) and its interplay with other interactions of comparable energy scales. Among the rich family of iridates, iridium dioxide (IrO2), a simple binary compound long known as a promising catalyst for water splitting, has recently been demonstrated to possess novel topological states and exotic transport properties. The strong SOC and the nonsymmorphic symmetry that IrO2possesses introduce symmetry-protected Dirac nodal lines (DNLs) within its band structure as well as a large spin Hall effect in the transport. Here, we review recent advances pertaining to the study of this unique SOC oxide, with an emphasis on the understanding of the topological electronic structures, syntheses of high crystalline quality nanostructures, and experimental measurements of its fundamental transport properties. In particular, the theoretical origin of the presence of the fourfold degenerate DNLs in band structure and its implications in the angle-resolved photoemission spectroscopy measurement and in the spin Hall effect are discussed. We further introduce a variety of synthesis techniques to achieve IrO2nanostructures, such as epitaxial thin films and single crystalline nanowires, with the goal of understanding the roles that each key parameter plays in the growth process. Finally, we review the electrical, spin, and thermal transport studies. The transport properties under variable temperatures and magnetic fields reveal themselves to be uniquely sensitive and modifiable by strain, dimensionality (bulk, thin film, nanowire), quantum confinement, film texture, and disorder. The sensitivity, stemming from the competing energy scales of SOC, disorder, and other interactions, enables the creation of a variety of intriguing quantum states of matter.

Evidence of unconventional superconductivity on the surface of the nodal semimetal CaAg1-xPdxP

Surface states of topological materials provide extreme electronic states for unconventional superconducting states. CaAg1-xPdxP is an ideal candidate for a nodal-line Dirac semimetal with drumhead surface states and no additional bulk bands. Here, we report that CaAg1-xPdxP has surface states that exhibit unconventional superconductivity (SC) around 1.5 K. Extremely sharp magnetoresistance, tuned by surface-sensitive gating, determines the surface origin of the ultrahigh-mobility "electrons." The Pd-doping elevates the Fermi level towards the surface states, and as a result, the critical temperature (Tc) is increased up to 1.7 K from 1.2 K for undoped CaAgP. Furthermore, a soft point-contact study at the surface of Pd-doped CaAgP proved the emergence of unconventional SC on the surface. We observed the bell-shaped conductance spectra, a hallmark of the unconventional SC. Ultrahigh mobility carriers derived from the surface flat bands generate a new class of unconventional SC.

Evidence for unconventional superconductivity and nontrivial topology in PdTe

PdTe is a superconductor with T_c ~ 4.25 K. Recently, evidence for bulk-nodal and surface-nodeless gap features has been reported in PdTe. Here, we investigate the physical properties of PdTe in both the normal and superconducting states via specific heat and magnetic torque measurements and first-principles calculations. Below T_c, the electronic specific heat initially decreases in T³ behavior (1.5 K < T < T_c) then exponentially decays. Using the two-band model, the superconducting specific heat can be well described with two energy gaps: one is 0.372 meV and another 1.93 meV. The calculated bulk band structure consists of two electron bands (α and β) and two hole bands (γ and η) at the Fermi level. Experimental detection of the de Haas-van Alphen (dHvA) oscillations allows us to identify four frequencies (F_α = 65 T, F_β = 658 T, F_γ = 1154 T, and F_η = 1867 T for H // a), consistent with theoretical predictions. Nontrivial α and β bands are further identified via both calculations and the angle dependence of the dHvA oscillations. Our results suggest that PdTe is a candidate for unconventional superconductivity.

Magnetic Breakdown and Topology in the Kagome Superconductor CsV_{3}Sb_{5} under High Magnetic Field

The recently discovered layered kagome metals of composition AV_{3}Sb_{5} (A=K, Rb, Cs) exhibit a complex interplay among superconductivity, charge density wave order, topologically nontrivial electronic band structure and geometrical frustration. Here, we probe the electronic band structure underlying these exotic correlated electronic states in CsV_{3}Sb_{5} with quantum oscillation measurements in pulsed fields up to 86 T. The high-field data reveal a sequence of magnetic breakdown orbits that allows the construction of a model for the folded Fermi surface of CsV_{3}Sb_{5}. The dominant features are large triangular Fermi surface sheets that cover almost half the folded Brillouin zone. These sheets have not yet been detected in angle resolved photoemission spectroscopy and display pronounced nesting. The Berry phases of the electron orbits have been deduced from Landau level fan diagrams near the quantum limit without the need for extrapolations, thereby unambiguously establishing the nontrivial topological character of several electron bands in this kagome lattice superconductor.

Negative transverse magnetoresistance due to the negative off-diagonal mass in linear dispersion materials

This study calculated the magnetoresistance (MR) in the Dirac electron system, Dresselhaus-Kip-Kittel (DKK) model, and nodal-line semimetals based on the semiclassical Boltzmann theory, with particular focus on the detailed energy dispersion structure. The negative off-diagonal effective-mass was found to induce negative transverse MR owing to the energy dispersion effect. The impact of the off-diagonal mass was more prominent in case of a linear energy dispersion. Further, Dirac electron systems could realize negative MR even if the Fermi surface was perfectly spherical. The obtained negative MR in the DKK model may explain the long-standing mystery in p-type Si.

Two -dimensional semimetal AlSb monolayer with multiple nodal-loops and extraordinary transport properties under uniaxial strain

Two-dimensional (2D) nodal-loop semimetal (NLSM) materials have attracted much attention for their high-speed and low-consumption transporting properties as well as their fantastic symmetry protection mechanisms. In this paper, using systematic first-principles calculations, we present an excellent NLSM candidate, a 2D AlSb monolayer, in which the conduction and valence bands cross with each other forming fascinating multiple nodal-loop (NL) states. The NLSM properties of the AlSb monolayer are protected by its glide mirror symmetry, which was confirmed using a symmetry-constrained six-band tight-binding model. The transport properties of the AlSb monolayer under in-plane uniaxial strains are also studied, based on a non-equilibrium Green's function method. It is found that both compressive and tensile strains from -10% to 10% improve the transporting properties of AlSb, and it is interesting to see that flexure configurations are energetically favored when compressive uniaxial strains are applied. Our studies not only provide a novel 2D NLSM candidate with a new symmetry protection mechanism, but also raise the novel possibility for the detection of out-of-plane flexure in 2D semimetal materials.

Quantum transport evidence of isolated topological nodal-line fermions

Anomalous transport responses, dictated by the nontrivial band topology, are the key for application of topological materials to advanced electronics and spintronics. One promising platform is topological nodal-line semimetals due to their rich topology and exotic physical properties. However, their transport signatures have often been masked by the complexity in band crossings or the coexisting topologically trivial states. Here we show that, in slightly hole-doped SrAs₃, the single-loop nodal-line states are well-isolated from the trivial states and entirely determine the transport responses. The characteristic torus-shaped Fermi surface and the associated encircling Berry flux of nodal-line fermions are clearly manifested by quantum oscillations of the magnetotransport properties and the quantum interference effect resulting in the two-dimensional behaviors of weak antilocalization. These unique quantum transport signatures make the isolated nodal-line fermions in SrAs₃ desirable for novel devices based on their topological charge and spin transport.

Fermi level fluctuations, reduced effective masses and Zeeman effect during quantum oscillations in nodal line semimetals

We probe quantum oscillations in nodal line semimetals (NLSM) by considering an NLSM continuum model under strong magnetic field and report the characteristics of the Landau level (LL) spectra and the fluctuations in the Fermi level as the field in a direction perpendicular to the nodal plane is varied through. Based on the results on parallel magnetization, we demonstrate the growth of quantum oscillation with field strength as well as its constancy in period when plotted against 1/B. We find that the density of states (DOS) which show series of peaks in succession, witness bifurcation of those peaks due to Zeeman effect. For field normal to nodal plane, such bifurcations are discernible only if the electron effective mass is considerably smaller than its free value, which usually happens in these systems. Though a reduced effective massm* causes the Zeeman splitting to become small compared to LL spacings, experimental results indicate a manifold increase in the Landegfactor which again amplifies the Zeeman contribution. We also consider magnetic field in the nodal plane for which the DOS peaks do not repeat periodically with energy anymore. The spectra become more spread out and the Zeeman splittings become less prominent. We find the low energy topological regime, that appears with such in-plane field set up, to shrink further with reducedm* values. However, such topological regime can be stretched out in case there are smaller Fermi velocities for electrons in the direction normal to the nodal plane.

Magneto-Optical Transport Properties of Type-II Nodal Line Semimetals

We investigate the magneto-optical transport properties and Landau levels of type-II nodal line semimetals. The tilted liner dispersion in type-II nodal line semimetals makes the conduction band and valence band asymmetric, and Landau levels are coupling in the presence of a magnetic field. We find the background of absorption peaks is curved. The oscillation peaks are tailless with the change of magnetic field. Through tuning tilt term, we find the absorption peaks of optical conductivity change from incomplete degenerate structure to splitting double peaks structure. We also find interband absorption peaks is no longer zero in the imaginary part of Hall conductivity. With the change of the tilt term, the contribution of the absorption peak has two forms, one is that the negative peak only appears at high frequencies, and the other is two adjacent peaks with opposite signs. In addition, the resistivity, circularly polarized light and magnetic oscillation of Hall conductivity are studied.

Increasing the number of topological nodal lines in semimetals via uniaxial pressure

The application of pressure has been demonstrated to induce intriguing phase transitions in topological nodal-line semimetals. In this work we analyze how uniaxial pressure affects the topological character of BaSn[Formula: see text], a Dirac nodal-line semimetal in the absence of spin-orbit coupling. Using calculations based on the density functional theory and a model tight-binding Hamiltonian, we find the emergence of a second nodal line for pressures higher than 4 GPa. We examine the topological features of both phases demonstrating that a nontrivial character is present in both of them. Thus, providing evidence of a topological-to-topological phase transition in which the number of topological nodal lines increases. The orbital overlap increase between Ba [Formula: see text] and [Formula: see text] orbitals and Sn [Formula: see text] orbitals and the preservation of crystal symmetries are found to be responsible for the advent of this transition. Furthermore, we pave the way to experimentally test this kind of transition by obtaining a topological relation between the zero-energy modes that arise in each phase when a magnetic field is applied.

Quantum oscillation and Landau-Zener transition in untilted nodal line semimetals under a time-periodic magnetic field

Nodal line semimetals (NLSM) exhibit interesting quantum oscillation (QO) characteristics when acted upon by a strong magnetic field. We study the combined effect of strong direct and alternating magnetic field, perpendicular to the nodal plane in an untilted NLSM in order to probe the behavior of the low lying Landau level states that can periodically become gapless for suitably chosen field parameters. The oscillatory field variation, as opposed to a steady one, has interesting impact on the QO phenomena with the Landau tubes crossing the Fermi surface extremally two times per cycle. Furthermore, the low energy modes can witness Landau-Zener like transitions between valence and conduction band providing further routes to conduction. We discuss such transition phenomena following the framework of adiabatic-impulse approximation for slow quenches. Next we also investigate the effect of oscillating magnetic field acting parallel to the nodal loop where topologically nontrivial magnetic oscillations at low energies can be witnessed. Therefore, with proper parameters chosen, one can engineer topological transitions to occur periodically in such systems as the oscillating field is swept through its cycles.

Topological Quantum Materials from the Viewpoint of Chemistry

Topology, a mathematical concept, has recently become a popular and truly transdisciplinary topic encompassing condensed matter physics, solid state chemistry, and materials science. Since there is a direct connection between real space, namely atoms, valence electrons, bonds, and orbitals, and reciprocal space, namely bands and Fermi surfaces, via symmetry and topology, classifying topological materials within a single-particle picture is possible. Currently, most materials are classified as trivial insulators, semimetals, and metals or as topological insulators, Dirac and Weyl nodal-line semimetals, and topological metals. The key ingredients for topology are certain symmetries, the inert pair effect of the outer electrons leading to inversion of the conduction and valence bands, and spin-orbit coupling. This review presents the topological concepts related to solids from the viewpoint of a solid-state chemist, summarizes techniques for growing single crystals, and describes basic physical property measurement techniques to characterize topological materials beyond their structure and provide examples of such materials. Finally, a brief outlook on the impact of topology in other areas of chemistry is provided at the end of the article.

Quantum oscillations with angular dependence in PdTe2single crystals

The layered transition-metal dichalcogenide PdTe₂has been discovered to possess bulk Dirac points as well as topological surface states. By measuring the magnetization (up to 7 T) and magnetic torque (up to 35 T) in single crystalline PdTe₂, we observe distinct de Haas-van Alphen (dHvA) oscillations. Eight frequencies are identified withH||c, with two low frequencies (F_α= 8 T andF_β= 117 T) dominating the spectrum. The effective masses obtained by fitting the Lifshitz-Kosevich (LK) equation to the data aremα*=0.059m0andmβ*=0.067m0wherem₀is the free electron mass. The corresponding Landau fan diagrams allow the determination of the Berry phase for these oscillations resulting in values of ∼0.67πfor the 3D α band (hole-type) (down to the 1st Landau level) and ∼0.23π-0.73πfor the 3D β band (electron-type) (down to the 3rd Landau level). By investigating the angular dependence of the dHvA oscillations, we find that the frequencies and the corresponding Berry phase (Φ_B) vary with the field direction, with a Φ_B∼ 0 whenHis 10°-30° away from theabplane for both α and β bands. The multiple band nature of PdTe₂is further confirmed from Hall effect measurements.

Quantum Transport Signatures of a Close Candidate for a Type II Nodal-Line Semimetal

The nodal-line semimetal is a new type of topological state of matter in which the crossing of two energy bands forms a nodal loop. In the absence of spin-orbit coupling, Mg₃Bi₂ is predicted as a type II nodal-line semimetal, which may evolve to a topological insulator with a small energy gap of ∼35 meV in the presence of spin-orbit coupling. However, the transport evidence is still lacking. Here, we measure the magneto-transport in Mg₃Bi₂. At low temperatures, the magnetoconductivity exhibits a weak antilocalization behavior. We fit the experimental data with a magnetoconductivity formula for the weak antilocalization effect of three-dimensional nodal-line semimetals as well as the well-known Hikami-Larkin-Nagaoka formula for two-dimensional weak (anti)localization effects. By comparing the fitting results of these two theories, we demonstrate that the weak antilocalization in Mg₃Bi₂ is better described by the theory for nodal-line semimetals. Our work will inspire more explorations to use the new weak localization theory to identify a large spectrum of nodal-line semimetals.

Weak Localization and Antilocalization in Nodal-Line Semimetals: Dimensionality and Topological Effects

New materials such as nodal-line semimetals offer a unique setting for novel transport phenomena. Here, we calculate the quantum correction to conductivity in a disordered nodal-line semimetal. The torus-shaped Fermi surface and encircled π Berry flux carried by the nodal loop result in a fascinating interplay between the effective dimensionality of electron diffusion and band topology, which depends on the scattering range of the impurity potential relative to the size of the nodal loop. For a short-range impurity potential, backscattering is dominated by the interference paths that do not encircle the nodal loop, yielding a 3D weak localization effect. In contrast, for a long-range impurity potential, the electrons effectively diffuse in various 2D planes and the backscattering is dominated by the interference paths that encircle the nodal loop. The latter leads to weak antilocalization with a 2D scaling law. Our results are consistent with symmetry consideration, where the two regimes correspond to the orthogonal and symplectic classes, respectively. Furthermore, we present weak-field magnetoconductivity calculations at low temperatures for realistic experimental parameters and predict that clear scaling signatures ∝sqrt[B] and ∝-lnB, respectively. The crossover between the 3D weak localization and 2D weak antilocalization can be probed by tuning the Fermi energy, giving a unique transport signature of the nodal-line semimetal.

Magnetic impurity in topological nodal loop semimetals

We investigate the Kondo effect of a spin-1/2 magnetic impurity in a topological nodal loop semimetal, in which band touchings form a nodal loop. The Fermi surface of a nodal loop semimetal is a torus or a drum-like structure, which is determined by chemical potential. When the chemical potential μ lies at the nodal loop ([Formula: see text]), the magnetic impurity and the conduction electrons form bound states only if their coupling exceeds a critical value. As the chemical potential is tuned away from the nodal loop, the Fermi surface becomes a torus or drum-like structure and the impurity and the host material always favor a bound state due to the finite density of state. Due to the anisotropic dispersion relationship in the energy band, the spatial spin-spin correlations [Formula: see text]([Formula: see text]) are of power-law decay with the decay rates proportional to [Formula: see text] and [Formula: see text] in different directions, respectively. The product [Formula: see text] and [Formula: see text] oscillates in coordinate space and the period is enhanced gradually as the Fermi surface evolves from a torus surface into a drum-like structure.

Galvanomagnetic properties of the putative type-II Dirac semimetal PtTe2

Platinum ditelluride has recently been characterized, based on angle-resolved photoemission spectroscopy data and electronic band structure calculations, as a possible representative of type-II Dirac semimetals. Here, we report on the magnetotransport behavior (electrical resistivity, Hall effect) in this compound, investigated on high-quality single-crystalline specimens. The magnetoresistance (MR) of PtTe₂ is large (over 3000% at T = 1.8 K in B = 9 T) and unsaturated in strong fields in the entire temperature range studied. The MR isotherms obey a Kohler’s type scaling with the exponent m = 1.69, different from the case of ideal electron-hole compensation. In applied magnetic fields, the resistivity shows a low-temperature plateau, characteristic of topological semimetals. In strong fields, well-resolved Shubnikov – de Haas (SdH) oscillations with two principle frequencies were found, and their analysis yielded charge mobilities of the order of 10³ cm² V⁻¹ s⁻¹ and rather small effective masses of charge carriers, 0.11 m_e and 0.21 m_e. However, the extracted Berry phases point to trivial character of the electronic bands involved in the SdH oscillations. The Hall effect data corroborated a multi-band character of the electrical conductivity in PtTe₂, with moderate charge compensation.

Realistic Floquet Semimetal with Exotic Topological Linkages between Arbitrarily Many Nodal Loops

Valence and conduction bands in nodal loop semimetals (NLSMs) touch along closed loops in momentum space. If such loops can proliferate and link intricately, NLSMs become exotic topological phases, which require nonlocal hopping and are therefore unrealistic in conventional quantum materials or cold atom systems alike. In this Letter, we show how this hurdle can be surmounted through an experimentally feasible periodic driving scheme. In particular, by tuning the period of a two-step periodic driving or certain experimentally accessible parameters, we can generate arbitrarily many nodal loops that are linked with various levels of complexity. Furthermore, we propose to use both a Berry-phase related winding number and the Alexander polynomial topological invariant to characterize the fascinating linkages among the nodal loops. This Letter thus presents a class of exotic Floquet topological phases that has hitherto not been proposed in any realistic setup. Possible experimental confirmation of such exotic topological phases is also discussed.