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At least 199 records · Page 11

Topological and magnetic properties of the interacting Bernevig-Hughes-Zhang model

We investigate the effects of electronic correlations on the Bernevig-Hughes-Zhang model using the real-space density matrix renormalization group (DMRG) algorithm. We introduce a method to probe topological phase transitions in systems with strong correlations using DMRG, substantiated by an unsupervised machine learning methodology that analyzes the orbital structure of the real-space edges. Including the full multi-orbital Hubbard interaction term, we construct a phase diagram as a function of a gap parameter (m) and the Hubbard interaction strength (U) via exact DMRG simulations on N×4 cylinders. Our analysis confirms that the topological phase persists in the presence of interactions, consistent with previous studies, but it also reveals an intriguing phase transition from a paramagnetic to a stripey antiferromagnetic topological insulator. The combination of the magnetic structure factor, strength of magnetic moments, and the orbitally resolved density, provides real-space information on both topology and magnetism in a strongly correlated system.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Local spin structure in the layered van der Waals materials MnPS 𝑥 ⁢Se 3−𝑥

Two-dimensional (2D) layered materials, whether in bulk form or reduced to just a single layer, have potential applications in spintronics and capacity for advanced quantum phenomena. A prerequisite for harnessing these opportunities lies in gaining a comprehensive understanding of the spin behavior in 2D materials. The low dimensionality motivates an understanding of the spin correlations over a wide length scale, from local to long-range order. In this context, we focus on the magnetism in bulk MnPSe 3 and MnPS 3 , 2D layered van der Waals antiferromagnetic semiconductors. These materials have similar honeycomb Mn layers and magnetic ordering temperatures, but distinct spin orientations and exchange interactions. Here, we utilize neutron scattering to gain deeper insights into the local magnetic structures and spin correlations in the paramagnetic and ordered phases by systematically investigating a MnPS 𝑥 ⁢Se 3−𝑥 (𝑥 = 0, 1, 1.5, 2, 3) series of powder samples using total neutron scattering measurements. By employing magnetic pair distribution function analysis, we unraveled the short-range magnetic correlations in these materials and explored how the nonmagnetic anion S/Se mixing impacts the magnetic correlations. The results reveal that the magnetism can be gradually tuned through alteration of the nonmagnetic S/Se content, which tunes the atomic structure. The change in magnetic structure is also accompanied by a control of the magnetic correlation length within the 2D honeycomb layers. Complimentary inelastic neutron scattering measurements allowed a quantification of the change in the magnetic exchange interactions for the series and further highlighted the gradual evolution of spin interactions in the series MnPS 𝑥 ⁢Se 3−𝑥 .

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Inhomogeneous magnetic ordered state and evolution of magnetic fluctuations in Sr⁢(Co 1–x ⁢Ni x ) 2 ⁢P 2 revealed by 31 P NMR

SrCo 2 ⁢P 2 with a tetragonal structure is known to be a Stoner-enhanced Pauli paramagnetic metal being nearly ferromagnetic. Recently J. Schmidt et al. [Phys. Rev. B 108, 174415 (2023)] reported that a ferromagnetic ordered state is actually induced by a small Ni substitution for Co of x = 0.02 in Sr⁢(Co 1–x ⁢Ni x ) 2 ⁢P 2 where an antiferromagnetic ordered phase also appears by further Ni substitution with x = 0.06–0.35. Here, in this work, using nuclear magnetic resonance (NMR) measurements on 31 P nuclei, we have investigated how the magnetic properties change by the Ni substitution in Sr⁢(Co 1–x ⁢Ni x ) 2 ⁢P 2 from a microscopic point of view, especially focusing on the evolution of magnetic fluctuations with the Ni substitution and the characterization of the magnetically ordered states. The temperature dependencies of the 31 P spin-lattice relaxation rate divided by temperature (1/T 1 ⁢T) and Knight shift (K) for SrCo 2 ⁢P 2 are reasonably explained by a model where a double-peak structure for the density of states near the Fermi energy is assumed. Based on a Korringa ratio analysis using the T 1 and K data, ferromagnetic spin fluctuations are found to dominate in the ferromagnetic Sr⁢(Co 1–x ⁢Ni x ) 2 ⁢P 2 as well as the antiferromagnets where no clear antiferromagnetic fluctuations are observed. We also found the distribution of the ordered Co moments in the magnetically ordered states from the analysis of the 31 P-NMR spectra exhibiting a characteristic rectangular-like shape.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Ferromagnetism and structural phase transition in rhombohedral PrI r 3

The synthesis, structural, magnetic, thermal, and transport properties are reported for polycrystalline PrI⁢r 3 . At room temperature PrI⁢r 3 displays the rhombohedral space group R–3m and a PuNi 3 -type structure. At around 70 K a phase transition to a monoclinic C⁢2/m structure is observed and continued cooling reveals temperature independent behavior of the unit cell volume. Further, PrI⁢r 3 undergoes a paramagnetic to ferromagnetic transition with T c =7.5K. The temperature dependent magnetic susceptibility follows the Curie-Weiss law with a positive Curie-Weiss temperature, and an effective moment that is close to the theoretical effective moment for a free P⁢r +3 ion. Finally, the structural transition introduces further complexity into the behavior of PuNi 3 -type materials and highlights the importance of temperature-dependent structural studies to complement physical property measurements in intermetallic compounds.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Out-of-plane magnetic phase diagram of the Kitaev quantum magnet Na 2 Co 2 TeO 6

We have investigated the magnetic properties and mapped out the phase diagram of the honeycomb magnet Na 2⁢ Co 2 ⁢TeO 6 with Co 3d 7 in out-of-plane magnetic fields. This material has previously been proposed to show nearest-neighbor Kitaev interactions between Co spins and maybe even Kitaev quantum spin liquid behavior in high fields. At 0.01 T, we observe a thermal phase transition at T N = 27K, transitioning from a paramagnetic state to a canonical ferrimagnetic state. Upon increasing the magnetic field, a spin-floplike phase transition occurred before saturation of J = 1/2 between 10 K and T N . Below 10 K, a peak-dip-peak structure emerges between 10 and 17 T in the magnetic susceptibility (d⁢M/⁢dH) before the magnetic saturation, reminiscent of magnetization plateau behavior. The measurement of the magnetocaloric effect also shows dip-peak-dip behavior in this field range. Our data can be explained by an XXZ model with a single ion anisotropy and possibly small Kitaev and Γ exchange interactions. Finally, we also determined the magnetization saturation field that helps constrain the energy scale of the exchange interactions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Weak-coupling theory of magic-angle twisted bilayer graphene

Strong correlations occur in magic-angle twisted bilayer graphene (MATBG) when the octet of flat moiré minibands centered on charge neutrality (CN) is partially occupied. The octet consists of a single valence band and a single conduction band for each of four degenerate spin-valley flavors. Motivated by the importance of Hartree electrostatic interactions in determining the filling-factor-dependent band structure, we use a time-dependent Hartree approximation to gain insight into electronic correlations. We find that the electronic compressibility is dominated by Hartree interactions, that paramagnetic states are stable over a range of density near CN, and that the dependence of energy on flavor polarization is strongly overestimated by mean-field theory. Published by the American Physical Society 2024

Materials Science↗

Magnetic and quadrupole coupling emerging in NdB 4 with geometrically frustrated orthogonal antiferromagnetic dimer

Unusual successive order in NdB 4 with the frustrated Shastry-Sutherland lattice was investigated by means of inelastic neutron scattering. Here, we found that the pseudoquartet with |±5/2⟩ and |±7/2⟩ in the paramagnetic state splits into four levels in the magnetically ordered states. The observed spectra are in good agreement with the 𝑓-electron states obtained from the on-site Hamiltonian with the magnetic interaction ⟨𝐽 𝑥 ⟩ ⁢𝐽 𝑥 , the coupling of 𝐽 𝑥 with quadrupole 𝑂 𝑧⁢𝑥 , and the crystalline electric field potential. The 𝑓-electron states reproduce the temperature dependences of the in-plane moment 𝑚 ⊥ = ⟨𝐽 𝑥 ⟩, which is the primary order parameter, and a weakly induced 𝑚 𝑐 = ⟨𝐽 𝑧 ⟩ as the secondary order parameter. The magnetic-quadrupole coupling emerges and plays an essential role under the geometrical frustration, where the Heisenberg interactions are canceled out.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Coarsening dynamics of Ising-nematic order in a frustrated Heisenberg antiferromagnet

We study the phase ordering dynamics of the classical antiferromagnetic 𝐽 1 −𝐽 2 (nearest-neighbor and next-nearest-neighbor couplings) Heisenberg model on the square lattice in the strong frustration regime (𝐽 2 /𝐽 1 > 1/2). While thermal fluctuations preclude any long-range magnetic order at finite temperatures, the system exhibits a long-range spin-driven nematic phase at low temperatures. The transition into the nematic phase is further shown to belong to the two-dimensional Ising universality class based on the critical exponents near the phase transition. Our large-scale stochastic Landau-Lifshitz-Gilbert simulations find a two-stage phase ordering when the system is quenched from a high-temperature paramagnetic state into the nematic phase. In the early stage, collinear alignments of spins lead to a locally saturated Ising-nematic order. Once domains of well-defined Ising order are developed, the late-stage relaxation is dominated by curvature-driven domain coarsening, as described by the Allen-Cahn equation. The characteristic size of Ising-nematic domains scales as the square root of time, similar to the kinetic Ising model described by the time-dependent Ginzburg-Landau theory. Our results confirm that the late-stage ordering kinetics of the spin-driven nematic, which is a vestigial order of the frustrated Heisenberg model, belongs to the dynamical universality class of a nonconserved Ising order. Interestingly, the system shows no violation of the superuniversality hypothesis under weak bond disorder. The dynamic scaling invariance is preserved in the presence of weak bond disorder. Here, we also discuss possible applications of our results to materials for which vestigial Ising-nematic order is realized.

Antiferromagnets↗

Role of magnetic and structural symmetry breaking in forming the Mott insulating gap in Nb 3 ⁢Cl 8

The α-phase of the gapped insulator Nb 3 Cl 8 has recently emerged as the long-sought critical testing bed for examining the importance of strong interelectronic correlation vs symmetry breaking in understanding insulation of such Mott compounds. Structural symmetry breaking detected by density functional theory (DFT) energy lowering (such as dimer formation, disproportionation, or Jahn-Teller distortions) explains insulation in both d-electron Mott-like systems and in non-d-electron cases without recourse to strong correlation. Yet, in Nb 3 Cl 8 , structural symmetry breaking alone (viz. formation of Nb trimers) fails to explain insulation, leading instead to a partially occupied metallic flat band, in contrast with experimental observations. We examine the role of magnetic symmetry breaking, noting that Nb 3 Cl 8 is an observed paramagnet (not an antiferromagnet), thus potentially carrying also short-range ordered magnetic moments. Describing the latter as a polymorphous distribution of nonzero local moments with total zero net magnetization is demonstrated to lower the DFT total energy, while gapping the system without recourse to strong correlation or long-range magnetic order. This suggests that degeneracy removal by symmetry breaking in mean-field-like approaches—either structural, or magnetic, or both—can reduce or eliminate the need for strong correlation, allowing the use of DFT for such Mott systems.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Large critical fields in superconducting Ti 4 ⁢Ir 2⁢ O from spin-orbit coupling

The recently synthesized 𝜂-carbide-type superconductors exhibit large critical fields. A notable example is Ti 4 ⁢Ir2⁢ O, for which the upper critical field strongly violates the Pauli paramagnetic limit, a behavior that is unusual for cubic materials that preserve inversion symmetry. Here, by combining density functional theory (DFT) and analytic modeling, we provide an explanation for this enhanced Pauli limiting field. We show that the nonsymmorphic Fd⁢-3 ⁢m symmetry implies that the electronic states near the X points exhibit strong spin-orbit coupling (SOC), which leads to a vanishing effective 𝑔 factor and enables the enhanced Pauli limiting field. Furthermore, our DFT results reveal a Van Hove singularity peak near the X points, accounting for ∼65% of the total density of states (DOS), occurring near the chemical potential. We propose that the strong SOC and enhanced DOS in the vicinity of the X points provide the origin of the observed enhancement of the critical field. This leads to a prediction that the magnetic field will lead to a strongly momentum-dependent gap suppression. As a result, the gap due to electronic states away from (near to) the X points will be rapidly (slowly) suppressed by fields.

Critical field↗

Light-induced anomalous Hall conductivity in the massive three-dimensional Dirac semimetal Co 3 ⁢Sn 2 ⁢S 2

Weyl semimetals can emerge from Dirac semimetals when the time-reversal or spatial-inversion symmetries are broken. Recently, it has been proposed based on Floquet theory that Dirac semimetals can be converted into Weyl semimetals even by shining circularly polarized light (CPL). Here we have investigated the possibility of such a Dirac-Weyl conversion by measuring the CPL-induced anomalous Hall conductivity (AHC) in a massive 3D Dirac semimetal Co 3 ⁢Sn 2 ⁢S 2 in the paramagnetic phase using ultrafast midinfrared pump-terahertz Faraday rotation probe spectroscopy. We find that the field-strength and driving frequency dependence of the observed AHC is well accounted for by CPL-induced nonzero Berry curvature associated with the splitting of the Dirac bands as predicted by the Floquet theory. As a result, the estimated splitting of the Dirac bands reaches about 60% of the mass gap and the calculated CPL-induced AHC quantitatively reproduces the experimental observation, demonstrating a promising route toward the realization of Floquet-Weyl states from massive Dirac semimetals.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Complex fermiology and electronic structure of antiferromagnet EuSnP

Throughout this work, we studied the electronic structure of a layered antiferromagnetic metal, EuSnP, in the paramagnetic and in the antiferromagnetic phase using angle-resolved photoemission spectroscopy (ARPES) alongside density-functional theory (DFT)-based first-principles calculations. The temperature dependence of the magnetic susceptibility measurements exhibits an antiferromagnetic transition at a Néel temperature of 21 K. Employing high-resolution ARPES, the valence-band structure was measured at several temperatures above and below the Néel temperature, which produced identical spectra independent of temperature. Through analysis of the ARPES results presented here, we attribute the temperature-independent spectra to the weak coupling of the Sn and P conduction electrons with Eu 4⁢ƒ states.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Exploring the impact of Cr-doping on the crystallographic and magnetic structure of Mn 5 Si 3 antiferromagnetic alloy

Here, the role of Cr-doping on the structural and magnetic ground states of Mn 5 Si 3 alloy has been investigated through temperature-dependent neutron powder diffraction (NPD) and X-ray absorption fine structure (XAFS) techniques. All the Cr-doped alloys of nominal composition Mn 5-x Cr x Si 3 (for x = 0.05, 0.1, and 0.2) undergo two first-order magneto-structural phase transitions from hexagonal (space group: P6 3 /mcm) paramagnetic → orthorhombic (space group: Ccmm) collinear antiferromagnetic → orthorhombic (space group: Cc2m) non-collinear antiferromagnetic phase on cooling from room temperature. NPD studies at different constant temperatures indicate that both anti ferromagnetic phases are commensurate in nature and can be represented by q = (0,1,0) magnetic propagation vector for all the Cr-doped alloys. Such doping at the Mn-site results in a significant modification of the non-collinear antiferromagnetic structure (both moment size and orientation) and hence affects the unusual magnetic properties, like inverted hysteresis loop, thermomagnetic irreversibility, etc. The XAFS measurements were performed to interpret the local environment of doped Cr atoms in detail, which is critical for a microscopic understanding of the unusual properties of this class of Cr-doped Mn 5 Si 3 alloys. The analysis confirms the elemental state of Cr in the doped alloys and indicates a high degree of preservation of local crystallographic structure with varying Cr concentration and sample temperature. Doping induces intriguing changes in XAFS patterns, elucidated through different types of scattering mechanisms associated with the central absorbing Cr atom.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Influence of Nb alloying on Nb recrystallization and the upper critical field of Nb 3 ⁢Sn

Nb 3 Sn conductors are important candidates for high-field magnets for particle accelerators, and they continue to be widely used for many laboratory and NMR magnets. However, the critical current density, J c , of present Nb 3 Sn conductors declines swiftly above 12-15 T. State-of-the-art Ta- and Ti-doped strands exhibit upper critical field, H c2 , values of ~ 24-26.5 T (4.2 K) and do not reach the FCC target J c , which serves as the present stretch target for Nb 3 Sn development. As recently demonstrated, to meet this goal requires enhanced vortex pinning but an independent and supplementary approach is to significantly enhance H c2 . In this study, we have arc-melted multiple Nb alloys with added Hf, Zr, Ta and Ti and drawn them successfully into monofilament wires to investigate the possibilities of H c2 enhancement through alloying. H c2 (T) was measured for all samples in fields up to 16 T and some up to 31 T. We have found that all alloys show good agreement with the standard Werthamer, Helfand, and Hohenberg (WHH) fitting procedure without the need to adjust the paramagnetic limitation parameter (α) and spin-orbit scattering parameter (λ so ). The evaluation of dH c2 /dT near T c , which is proportional to the electronic specific heat coefficient γ and the normal state resistivity ρn, allows a better understanding of the induced disorder introduced by alloying in the A15 phase. So far, we have observed that Hf alloying of pure Nb can enhance H c2 (0) by 3-4 T to ~28 T, while adding just 1 at. %Hf or Zr into a Nb4Ta base alloy can raise H c2 (0) to ~31 T. Very importantly we find that Hf and Zr raise the alloy recrystallization temperature above the usual A15 reaction temperature range of 650°C – 750°C, thus ensuring denser A15 phase nucleation in the Nb alloy grain boundaries, possibly leading to a more homogeneous A15 phase Sn content and refined A15 grain size. Furthermore, the potential for further advancements in Nb 3 Sn properties is explored in relation to the recrystallization of the Nb alloy and the factors controlling the upper critical field.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Tuning the interlayer coupling in La 0.7 Sr 0.3 Mn 0.95 Ru 0.05 O 3 / LaNiO 3 multilayers with perpendicular magnetic anisotropy

In ferromagnetic oxide epitaxial multilayers, magnetic properties and interlayer coupling are determined by a variety of factors. Beyond the contribution of interlayer exchange coupling, strain and interfacial effects, such as structural reconstructions or charge transfer, play significant roles, resulting in complex magnetic behavior. In this study, the interlayer coupling of ferromagnetic La 0.7 ⁢Sr 0.3⁢ Mn 0.95 ⁢Ru 0.05 ⁢O 3 (LSMRO) layers (8 nm thick) was investigated, when separated by epitaxial spacers of paramagnetic metallic LaNiO 3 (LNO), in stacks exhibiting perpendicular magnetic anisotropy. By varying the thickness of the spacer, it was found that the coupling between two LSMRO layers changes from antiferromagnetic (with a 4-unit-cell-thick LNO spacer) to ferromagnetic (with a 6-unit-cell-thick LNO spacer). For multilayers comprising five LSMRO layers and a 4-unit-cell-thick LNO spacer, the antiferromagnetic coupling was preserved. However, the effective magnetic anisotropy changed, causing the magnetization to cant more towards the in-plane direction. Further, this behavior was corroborated by X-ray magnetic circular dichroism (XMCD) investigations at the Mn and Ni L 3 edges. The XMCD results indicated that the 4-unit-cell-thick LNO spacer in the multilayer become magnetically ordered, closely following the magnetization of adjacent LSMRO layers.

36 MATERIALS SCIENCE↗

Tuning the spin dynamics and magnetic phase transitions of the Cantor alloy via composition and sample processing protocols: A muon spin relaxation study

CrMnFeCoNi, also called the Cantor alloy, is a well-known high-entropy alloy whose magnetic properties have recently become a focus of attention. Here, we present a detailed muon spin relaxation study of the influence of chemical composition and sample processing protocols on the magnetic phase transitions and spin dynamics of several different Cantor alloy samples. Specific samples studied include a pristine equiatomic sample, samples with deficient and excess Mn content, and equiatomic samples magnetized in a field of 9 T or plastically deformed in pressures up to 0.5 GPa. The results confirm the sensitive dependence of the transition temperature on composition and demonstrate that post-synthesis pressure treatments cause the transition to become significantly less homogeneous throughout the sample volume. In addition, we observe critical spin dynamics in the vicinity of the transition in all samples, reminiscent of canonical spin glasses and magnetic materials with ideal continuous phase transitions. Application of an external magnetic field suppresses the critical dynamics in the Mn-deficient sample, while the equiatomic and Mn-rich samples show more robust critical dynamics. The spin-flip thermal activation energy in the paramagnetic phase increases with Mn content, ranging from 3.1⁢(3) × 10 -21 J for 0% Mn to 1.2⁢(2) × 10 -20 J for 30% Mn content. These results shed light on critical magnetic behavior in environments of extreme chemical disorder and demonstrate the tunability of spin dynamics in the Cantor alloy via chemical composition and sample processing.

36 MATERIALS SCIENCE↗

Synthesis and characterization of the novel breathing pyrochlore compound Ba 3 ⁢Tm 2⁢ Zn 5⁢ O 11

In this study, a novel material from the rare-earth based breathing pyrochlore family, Ba 3 ⁢Tm 2 ⁢Zn 5 ⁢O 11 , was successfully synthesized. Powder x-ray diffraction and high-resolution powder neutron diffraction confirmed phase purity and the $F\bar4$3⁢𝑚 breathing pyrochlore crystal structure, while thermogravimetric analysis revealed incongruent melting behavior compared to its counterpart, Ba 3 ⁢Yb 2⁢ Zn 5⁢ O 11 . High-quality single crystals of Ba 3 ⁢Tm 2 ⁢Zn 5 ⁢O 11 were grown using the traveling solvent floating zone technique and assessed using Laue x-ray diffraction and single crystal x-ray diffraction. Further, thermodynamic characterization indicated paramagnetic behavior down to 0.05 K, and inelastic neutron scattering measurements identified distinct dispersionless crystal electric field energy bands, with the fitted crystal electric field model predicting a single-ion singlet ground state and an energy gap of ∼9 meV separating it from the first excited (singlet) state. Additional low-energy excitation studies on single crystals revealed dispersionless bands at 0.8 and 1 meV. Computed phonon dispersions from first-principles calculations ruled out phonons as the origin of these modes, further illustrating the puzzling and unique properties of Ba 3 ⁢Tm⁢ 2 Zn 5 ⁢O 11 .

36 MATERIALS SCIENCE↗

Atomic dynamics in 𝑀⁢Cr⁢𝑋 2 (𝑀=Ag, Cu; 𝑋 = S, Se) across magnetic and superionic transitions

Here, a systematic study of atomic dynamics and thermal properties of the family of layered chalcogenide compounds 𝑀⁢Cr⁢𝑋 2 (𝑀= Ag, Cu; 𝑋 = S, Se) was performed, including neutron and x-ray scattering, thermal characterization, and first-principles simulations. In all compounds, we observe a breakdown of specific phonon modes across the superionic phase transition, for phonons whose eigenvectors exhibit large contributions of mobile ions. In particular, the nondispersive portions of transverse acoustic (TA) branches at short-wavelengths and the low-energy optical phonons with large contributions from Ag + or Cu + become severely damped in the superionic phase. However, well-defined quasiparticles persist in the superionic state for long-wavelength TA phonons. In the case of AgCrS 2 , the coupling of lattice dynamics with its antiferromagnetic transition was also investigated. The magnetic ordering couples with the monoclinic–rhombohedral structural transition, and the Cr 3+ spin arrangement strongly affects the phonon dispersions. We qualitatively reproduce the magnetic and nuclear components of the INS measurement for antiferromagnetic AgCrS 2 by combining models of spin-waves and spin-polarized first-principles phonon simulations. Quasielastic magnetic fluctuations persist in the paramagnetic phase up to high temperature, but are clearly distinguished from the nuclear component through their momentum dependence. Finally, we report measurements of the thermal properties of the selenide compounds and find good agreement with our DFT simulations.

36 MATERIALS SCIENCE↗