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We consider interaction energies E f [ L ] between a point O ∈ R d , d ≥ 2 , and a lattice L containing O, where the interaction potential f is assumed to be radially symmetric and decaying sufficiently fast at infinity. We investigate the conservation of optimality results for E f when integer sublattices kL are removed (periodic arrays of vacancies) or substituted (periodic arrays of substitutional defects). We consider separately the non-shifted ( O ∈ k L ) and shifted ( O ∉ k L ) cases and we derive several general conditions ensuring the (non-)optimality of a universal optimizer among lattices for the new energy including defects. Furthermore, in the case of inverse power laws and Lennard-Jones-type potentials, we give necessary and sufficient conditions on non-shifted periodic vacancies or substitutional defects for the conservation of minimality results at fixed density. Different examples of applications are presented, including optimality results for the Kagome lattice and energy comparisons of certain ionic-like structures. [ABSTRACT FROM AUTHOR]

In this study, we explored the electronic properties of a two-dimensional (2D) kagome lattice in the presence of an external magnetic field, spin-orbit coupling (SOC), and strain. Our focus was on investigating the energy-dependent behavior of the electronic heat capacity and paramagnetic spin susceptibility in response to changing factors. Employing the Green's function approach, we successfully determined the energy dependence of the electronic heat capacity and paramagnetic susceptibility using the tight-binding model. Notably, at low energies, the electronic heat capacity almost reaches the Schottky anomaly peak, while the paramagnetic susceptibility sharply decreases. Furthermore, we conducted a detailed investigation of the energy-dependent paramagnetic susceptibility and electronic heat capacity of a kagome lattice monolayer, considering the influences of SOC and magnetic field factors. Our investigation of the density of states (DOS) in the presence of SOC suggests that semimetal-to-insulator phase transitions occur. Additionally, throughout our study, we observed variations in the DOS spectrum, indicating the influence of both compressive and tensile strains. We found that these alterations, along with the splitting of levels under different conditions, led to changes in the material's electrical properties. [Display omitted] • Single orbital lattice hopping, tight-binding approach is employed. • By applying SOC, phase transfer from semimetal to insulator is created. • Applying an external magnetic field creates a band gap. • The paramagnetic susceptibility demonstrated a decreasing trend in low energy. • In low temperatures, an upward trend for the heat capacity is observed. [ABSTRACT FROM AUTHOR]

Interfacial structural transformations arising from solute segregation play a crucial role in the performance of materials. However, these micro-transitions are often difficult to reveal due to their buried nature. Here, in a typical Mg-rare-earth model system, we present the discovery of interfacial Kagome lattice transitions triggered by segregation of solutes coupled with bi-crystalline lattice interactions, in three types of partially coherent tilt grain boundaries (GBs), using atomic-resolution scanning transmission electron microscopy observations plus theoretical calculations. The results demonstrate that the minimization of GB elastic strain is responsible for the selective segregation of solutes at interfacial specific sites, resulting in a confined phase transition of the embedded triangular-quadrilateral (10 1 ¯ 1) Mg lattice to a topological hexagonal-triangular Kagome lattice via atomic shuffling. Furthermore, the pressure-temperature conditions that trigger such interfacial Kagome lattice transitions have also been determined. These findings not only enrich the transition pathways formed by the Kagome lattice but also provide a bottom-up approach for investigating interfacial structural transformations induced by solute/impurity segregation in polycrystalline materials. [Display omitted] [ABSTRACT FROM AUTHOR]

The frustrated magnets exhibiting kagome geometries and the investigation of their structure-property relationships have garnered substantial attention as potential candidates for quantum spin liquid (QSL). In this study, β -Mn 2 (OH) 3 Cl nanosheets with distorted kagome geometries were synthesized via a solid-state reaction, and low-temperature magnetic measurements revealed a spin-glass-like magnetic phase transition around 40 K in the frustrated β -Mn 2 (OH) 3 Cl. A pressure-induced structural phase transition was further realized by employing in situ synchrotron X-ray diffraction, leading to a hexagonal crystal structure with a nearly perfect kagome lattice of magnetic Mn2+ ions. High-pressure Raman and UV–vis absorption measurements indicates that the topological evolution can be attributed to the synergistic effects of pressure driven internal hydrogen bonds and the intrinsic Jahn-Teller effects. This study provides a strategy for synthesizing candidate compounds of QSL and sheds light on the mechanism underlying the topological evolution of kagome lattices under high pressure. [Display omitted] • β -Mn 2 (OH) 3 Cl nanosheets were synthesized via a solid-state reaction and exhibited spin glass behavior at low temperatures. • β -Mn 2 (OH) 3 Cl undergoes an irreversible phase transition under high pressure to form a nearly perfect kagome lattice. • The hydrogen bonding and the Jahn-Teller effect are key factors in its high-pressure structural transition of β -Mn 2 (OH) 3 Cl. [ABSTRACT FROM AUTHOR]

We investigate the anisotropic magnetic response in a kagome antiferromagnetic compound Fe 1- x Co x Sn (x = 0, 0.04, 0.06, 0.14). The magnetic behaviors of three distinct spin orderings, namely the planar, tilted, and axial spin states, are examined under various conditions, including doping rate, temperature, and magnetic fields. We confirm that the magnetic susceptibility of all three spin states has significant anisotropy, which depends on both the magnitude and direction of the applied magnetic field. The planar and axial states show scaling behavior in magnetic susceptibility, while the tilted state does not. Furthermore, employing the differential magnetic susceptibility, we observe that the axial spins display discontinuous spin-flop transitions, while the planar and tilted spins exhibit a continuous spin reorientation. The magnetic phase diagrams, summarizing our observations, highlight the anisotropic nature of magnetic behaviors in the Fe 1- x Co x Sn system. These diagrams clearly show that the magnetic responses are not overlapping and follow distinct mechanisms in their interaction with magnetic fields. We suggest that the axial state may arise due to the dopants as an obstacle within the kagome lattice, as evidenced by similar magnetic behaviors in Fe 0.96 Ni 0.06 Sn. [Display omitted] ● Magnetic phase diagrams of AFM Fe 1- x Co x Sn (x < 0.14) single crystals are proposed. ● The influence of magnetic fields on planar, tilted, and axial orderings is examined. ● Magnetic susceptibility can be scaled only in planar and axial spin states. ● Axial spins undergo spin-flops, while planar and tilted spins continuously change. ● The axial state may arise due to the dopants as an obstacle within the kagome lattice. [ABSTRACT FROM AUTHOR]

We investigate the magnetisation and internal energy of the David star in the Kagome lattice by the effective field theory developed by Kaneyoshi. First-order, Second-order and successive phase transitions are observed for different spin-down orientations. Spin-down orientation of identical atoms causes a second-order phase transition, spin-down orientation of different atoms causes a first-order and successive phase transitions. Minimum internal energy is obtained as UDavid star =−5 for all spin-up orientations, whereas it increases (UDavid star > −5) for at least one spin-down orientation. For spin-down orientation of A↑–A↓, internal energy becomes zero at a certain temperature and we call this temperature 'zero internal energy temperature (Tzie)'. At T < Tzie=1.99, internal energy is positive, whereas it is negative for other spin states. Therefore, spin-induced minimum internal energy broken is observed except for all spin-up (A↑, B↑ and C↑) cases. [ABSTRACT FROM AUTHOR]

Single crystals of U 2 Mn 3 Ge and U 2 Fe 3 Ge with a Kagome lattice structure were synthesized using a high-temperature self-flux crystal growth method. The physical properties of these crystals were characterized through measurements of resistivity, magnetism, and specific heat. U 2 Fe 3 Ge exhibits ferromagnetic ground state and anomalous Hall effect, and U 2 Mn 3 Ge demonstrates a complex magnetic structure. Both compounds exhibit large Sommerfeld coefficient, indicating coexistence of heavy Fermion behaviour with magnetism. Our results suggest that this U 2 TM 3 Ge (TM = Mn, Fe, Co) family is a promising platform to investigate the interplay of magnetism, Kondo physics and the Kagome lattice., (Creative Commons Attribution license.)

Interesting behaviors may emerge in the magnetic frustrated materials with significant site-mixing disorder. We present the results of the structural, magnetic susceptibility, and specific heat measurements of Dy 3 Sb 3 Zn 2 O 14 with ∼20%Dy/Zn site-mixing disorder, which results in either a diluted 2D triangular lattice, or an intermediate structure between the kagome and pyrochlore lattice. In addition to the sharp anomaly of the temperature dependence of specific heat atT∼0.35 K, which was attributed to the emergent charge order state for the sample with less disorder, a broad peak atT∼1.5 K, and a small hump belowT∼0.1 K are observed. The measured temperature dependence of specific heat and the Monte Carlo simulation suggest that the magnetic frustration persists despite of a strong site-mixing disorder., (© 2024 IOP Publishing Ltd.)

In this work, we demonstrate the formation and electronic influence of lateral heterointerfaces in FeSn containing Kagome and honeycomb layers. Lateral heterostructures offer spatially resolved property control, enabling the integration of dissimilar materials and promoting phenomena not typically observed in vertical heterostructures. Using the molecular beam epitaxy technique, we achieve a controllable synthesis of lateral heterostructures in the Kagome metal FeSn. With scanning tunneling microscopy/spectroscopy in conjunction with first-principles calculations, we provide a comprehensive understanding of the bonding motif connecting the Fe 3 Sn-terminated Kagome and Sn 2 -terminated honeycomb surfaces. More importantly, we reveal a distance-dependent evolution of the electronic states in the vicinity of the heterointerfaces. This evolution is significantly influenced by the orbital character of the flat bands. Our findings suggest an approach to modulate the electronic properties of the Kagome lattice, which should be beneficial for the development of future quantum devices.

We develop a multi-band semiclassical trajectory (MBSCT) method for studying the high harmonic generation (HHG) from solids, which is fundamentally similar to the Boltzmann equation but describe the electron density distribution in a different way and can simulate the electron transitions between bands, thereby depicting a richer array of physical processes. Compared to other theoretical methods, such as the time-dependent Schrödinger equation, the semiconductor Bloch equation, and time-dependent density functional theory, our MBSCT method avoids issues like massive consumption of computational resources and the need for wave function phase correction. Moreover, we focus on Kagome-type materials to justify the MBSCT method and investigate the influence of flat band on HHG in strong laser fields. The simulated results show that the intensity of certain harmonic orders is suppressed by the flat band, implying harmonic spectroscopy as a potential all-optical approach for characterising nonequilibrium physics of flat-band quantum materials., (© 2024 IOP Publishing Ltd.)

This paper details the nonlinear design of adaptive lattices by determination and enhancement of compliant modes and optimizing the designed structure for delivering high amplitude actuation. The particular focus is the kagome lattice geometry—a pattern with some unique and useful actuation properties. Developing a novel design tool, the stiffness matrix of the beam assembly is calculated using a developed second-order geometrically nonlinear beam finite element formulation allowing large rotations. Based on this formulation in conjunction with singular value decomposition of the stiffness matrix, the modal optimization technique reduces the continuous structure with many degrees of freedom to a small number of low energy modes, which form the basis of designing the adaptive structure. For delivering high-amplitude actuation, the designed structure needs to be re-optimized due to changes in the nonlinear stiffness matrix under large deformation. This is performed via Bayesian optimization and by removing some internal members of the lattice. The integrity and feasibility of the optimum design is guaranteed via defining some constraints on removed members. [ABSTRACT FROM AUTHOR]

The magnon band topology due to the Dzyaloshinskii–Moriya interaction (DMI) and its relevant topological thermal Hall effect has been extensively studied in kagome lattice magnets. In this theoretical investigation, we report a new mechanism for phase transitions between topological phases of magnons in kagome ferromagnets by tuning the anisotropic nearest-neighbor ferromagnetic interaction and DMI. Using the linear spin-wave theory, we calculate the Chern number and thermal Hall conductivity of magnons in low temperature regime. We show the magnon band structures and magnonic edge states in each topological phase. From the topological phase diagram, we find a sign reversal of the thermal Hall conductivity upon tuning the modulation factors. We explicitly demonstrate the correspondence of thermal Hall conductivity with the propagation direction of the magnonic edge states. Finally, we discuss candidate materials as experimental realizations of our theoretical model.

The high-field regime of the spin-s XXZ antiferromagnet on the kagome lattice gives rise to macroscopically degenerate ground states thanks to a completely flat lowest single-magnon band. The corresponding excitations can be localized on loops in real space and have been coined "localized magnons". Thus, the description of the many-body ground states amounts to characterizing the allowed classical loop configurations and eliminating the quantum mechanical linear relations between them. Here, we investigate this loop-gas description on finite kagome lattices with open boundary conditions and compare the results with exact diagonalization for the spin-1/2 XY model on the same lattice. We find that the loop gas provides an exact account of the degenerate ground-state manifold while a hard-hexagon description misses contributions from nested loop configurations. The densest packing of the loops corresponds to a magnon crystal that according to the zero-temperature magnetization curve is a stable ground state of the spin-1/2 XY model in a window of magnetic fields of about 4% of the saturation field just below this saturation field. We also present numerical results for the specific heat obtained by the related methods of thermal pure quantum (TPQ) states and the finite-temperature Lanczos method (FTLM). For a field in the stability range of the magnon crystal, one finds a low-temperature maximum of the specific heat that corresponds to a finite-temperature phase transition into the magnon crystal at low temperatures. [ABSTRACT FROM AUTHOR]

Magnetic two-dimensional (2D) topological insulators with spontaneous magnetization have been predicted to host quantum anomalous Hall effects (QAHEs). For organic topological insulators, the QAHE only exists in honeycomb or Kagome organometallic lattices based on theoretical calculations. Recently, coloring-triangle (CT) lattice has been found to be mathematically equivalent to a Kagome lattice, suggesting a potential 2D lattice to realize QAHE. Here, based on first-principles calculations, we predict an organometallic CT lattice, Cu-dicyanobenzene (DCB), to be a stable QAH insulator. It exhibits ferromagnetic (FM) properties as a result of the charge transfer from metal atoms to DCB molecules. Moreover, based on the Ising model, the Curie temperature of the FM ordering is calculated to be around 100 K. Both the Chern numbers and the chiral edge states of the semi-infinite Cu-DCB edge structure, which occur inside the spin-orbit coupling band gap, confirm its nontrivial topological properties. These make the Cu-DCB CT lattice an ideal candidate to enrich the family of QAH insulators. [ABSTRACT FROM AUTHOR]

• Tunable frequency ranges of topological edge states based on inter- and intra-cell couplings. • Frequency selecting function can be realized by the specific inter- and intra-cell couplings with two different types of domains. • Two functional waveguides based on the frequency selecting have been designed to demonstrate the concept, which are beneficial to the on-chip transmission. Topological photonic insulators provide a novel approach to achieve high-efficiency terahertz transmissions due to the presence of the topological edge states. In this work, the frequency ranges of the topological edge states can be controlled by varying the radius of rods in the kagome lattice and the distance between them. In addition, a gap can be generated between two different frequency ranges and the frequency selecting functions can be realized by constructing different types of domains. Similarly, sandwich a non-trivial photonic crystal between two trivial ones also capable of frequency selection for its topological edge states. Our findings suggest that this work has significant potential for applications in photonic integrated devices. [ABSTRACT FROM AUTHOR]

Kagome lattices have attracted much attention because of their potential in superconductivity and topological applications. In this work, we proposed three two-dimensional Janus breathing-Kagome lattices, Nb 3 TeCl 7 , Nb 3 TeBr 7 and Nb 3 TeI 7 , and further investigated their stability, electronic and thermoelectric (TE) transport performance. We illustrated that the three materials all possess high stability, and low cleavage energy of 0.21–0.22 J/m2. Besides, they exhibit common features of Kagome lattices, such as fat bands and van Hove singularities near band-edges, which lead to high Seebeck coefficient (0.37–0.57 mV/K) and TE power factor of (2.79 − 3.71 mW/K2m). Further, owing to their asymmetric Janus structures, the strong coupling between acoustic and optical phonons, as well as low frequency for lying-phonons, the three lattices also possess low lattice thermal conductivity of 1.06–2.76 W/mK. Ultimately, they deliver promising TE figure of merit of {0.28, 0.39, 0.43} at 300 K, and increase further to {1.09, 1.39, 1.40} at 700 K, demonstrating breathing-Kagome lattice Nb 3 TeX 7 are suitable candidates in TE application. [ABSTRACT FROM AUTHOR]

Thermal Hall conductivity κ x y and Hall spin conductivity σ x y in the nearest-neighbors antiferromagnetic Heisenberg model on kagome lattice with out-of-plane Dzyaloshinskii–Moriya interaction (DMI), which has been an active area of study in condensed matter physics, are studied using the linear spin-wave theory. We analyze the effect of this coupling on the behavior of κ x y as well as σ x y as well. For all cases analyzed, the behavior of κ x y and σ x y as a function of temperature T is analyzed where the behavior obtained for T finite, being due to the magnons thermally excited. We get a strong effect of temperature and DMI interaction on the behavior of both conductivities. • The thermal Hall conductivity and spin Hall conductivity are calculated. • The behavior of conductivities as a function of temperature is analyzed. • The effect of Dzyaloshinskii–Moriya interaction and temperature on conductivities are analyzed. [ABSTRACT FROM AUTHOR]

We report an ultra-high vacuum low-temperature scanning tunneling microscopy (STM) study of the C60 monolayer grown on Cd(0001). Individual C60 molecules adsorbed on Cd(0001) may exhibit a bright or dim contrast in STM images. When deposited at low temperatures close to 100 K, C60 thin films present a curved structure to release strain due to dominant molecule–substrate interactions. Moreover, edge dislocation appears when two different wavy structures encounter each other, which has seldomly been observed in molecular self-assembly. When growth temperature rose, we found two forms of symmetric kagome lattice superstructures, 2 × 2 and 4 × 4, at room temperature (RT) and 310 K, respectively. The results provide new insight into the growth behavior of C60 films.

Mineral francisites Cu 3 Bi(SeO 3 ) 2 O 2 Cl are unique compounds with interesting quasi two-dimensional structure along with fascinating magnetic properties. The magnetic properties can be fine-tuned when non-magnetic Bi is replaced by a suitable rare earth (RE) metal. It is because of the inclusion of additional magnetic sub-centre RE apart from Cu. Temperature dependent Raman spectroscopy measurements in RE based francisites [Cu 3 RE(SeO 3 ) 2 O 2 Cl, shortly RECufr] were performed in the range of 11 K-295 K. Among the three studied RECufr (LaCufr, NdCufr, and DyCufr) compounds, the properties of phonon vibration vary from moderate (in DyCufr) to weak (in LaCufr) spin phonon coupled and the absence of spin phonon coupling (SPC) (i.e. strictly anharmonic in nature) was observed in NdCufr and the reason for this observation has been provided. More specifically, two Raman-active phonons soften below the antiferromagnetic ordering temperature of T N ≈ 39 K in DyCufr compound, indicating the existence of moderate SPC. This trend of phonon vibration is correlated with magnetic properties, particularly field induced metamagnetic transition (MMT). Strong MMT enabled DyCufr develops SPC, while weak MMT enabled NdCufr is unable to develop SPC., (© 2024 IOP Publishing Ltd.)

Two-dimensional (2D) magnets exhibit unique physical properties for potential applications in spintronics. To date, most 2D ferromagnets are obtained by mechanical exfoliation of bulk materials with van der Waals interlayer interactions, and the synthesis of single- or few-layer 2D ferromagnets with strong interlayer coupling remains experimentally challenging. Here, we report the epitaxial growth of 2D non-van der Waals ferromagnetic bilayer FeSb on SrTiO 3 (001) substrates stabilized by strong coupling to the substrate, which exhibits in-plane magnetic anisotropy and a Curie temperature above 390 K. In situ low-temperature scanning tunneling microscopy/spectroscopy and density-functional theory calculations further reveal that an Fe Kagome layer terminates the bilayer FeSb. Our results open a new avenue for further exploring emergent quantum phenomena from the interplay of ferromagnetism and topology for application in spintronics.

Averievite-type compounds with the general formula (MX)[Cu5O2(TO4)], where M = alkali metal, X = halogen and T = P, V, have been synthesized by crystallization from gases and structurally characterized for six different compositions: 1 (M = Cs; X = Cl; T = P), 2 (M = Cs; X = Cl; T = V), 3 (M = Rb; X = Cl; T = P), 4 (M = K; X = Br; T = P), 5 (M = K; X = Cl; T = P) and 6 (M = Cu; X = Cl; T = V). The crystal structures of the compounds are based upon the same structural unit, the layer consisting of a kagome lattice of Cu2+ ions and are composed from corner-sharing (OCu4) anion-centered tetrahedra. Each tetrahedron shares common corners with three neighboring tetrahedra, forming hexagonal rings, linked into the two-dimensional [O2Cu5]6+ sheets parallel to (001). The layers are interlinked by (T5+O4) tetrahedra (T5+ = V, P) attached to the bases of the oxocentered tetrahedra in a “face-to-face” manner. The resulting electroneutral 3D framework {[O2Cu5](T5+O4)2}0 possesses channels occupied by monovalent metal cations M+ and halide ions X−. The halide ions are located at the centers of the hexagonal rings of the kagome nets, whereas the metal cations are in the interlayer space. There are at least four different structure types of the averievite-type compounds: the P-3m1 archetype, the 2 × 2 × 1 superstructure with the P-3 space group, the monoclinically distorted 1 × 1 × 2 superstructure with the C2/c symmetry and the low-temperature P21/c superstructure with a doubled unit cell relative to the high-temperature archetype. The formation of a particular structure type is controlled by the interplay of the chemical composition and temperature. Changing the chemical composition may lead to modification of the structure type, which opens up the possibility to tune the geometrical parameters of the kagome net of Cu2+ ions.

In order to improve the large anomalous Hall effect (AHE) in Mn3Sn thin films, we eliminated the co-existing Mn2Sn phase in the films by changing the composition; 50 nm thick polycrystalline Mn3+xSn thin films were fabricated on Si/SiO2 substrates by the sputtering method followed by a thermal annealing process in vacuum. The film compositions were Mn70Sn30(sample-A), Mn75Sn25(sample-B), and Mn80Sn20(sample-C) in as-deposited state and were slightly changed to be Mn75Sn25(sample-A), Mn77Sn23(sample-B), and Mn78Sn22(sample-C), respectively, after the annealing at 500 °C. From a structural analysis by X-ray diffractometry, the sample-C was considered to crystallize to Mn3Sn phase without passing the crystallization of Mn2Sn phase at 300 °C, differently from the sample-A. The saturation magnetization, $M_{\mathrm {S}}$ , of the sample-A significantly increased below 250 K, corresponding with the Curie temperature of Mn2Sn. On the other hand, $M_{\mathrm {S}}$ did not show significant changes with cooling temperature in the samples-B and -C. An AHE was observed at the room temperature in all the samples. The anomalous Hall conductivity, $\sigma _{\mathrm {AH}}$ , at the room temperature increased in magnitude, as the content of Mn increased. The sign of $\sigma _{\mathrm {AH}}$ changed from negative to positive in the sample-A with cooling temperature. On the other hand, the sign remained negative in the sample-C. These differences might be due to the elimination of co-existing Mn2Sn phase in the Mn3Sn thin films with enlarging the Mn content from the stoichiometry. Consequently, we successfully improved the large AHE in polycrystalline antiferromagnetic Mn3Sn thin films. [ABSTRACT FROM AUTHOR]

Highlights • Excessive dipole interaction radius limitation leads to losing phase transition. • At least 3 coordination spheres must be taken into account in kagome ASI model. • Phase transition disappears at dilution by more than 0.35 dilution concentration. Abstract The investigations carried out in this paper show the influence of cutoff dipole interaction radius and dilution on the basic thermodynamic characteristics of the kagome artificial spin ice with free boundary conditions. The phase transition disappearance at the dipole interaction radius limitation was shown in our previous study by Shevchenko et al. (2017). This work continues this study and answers the question of the coordination spheres numbers that must be taken into account, in order not to lose the basic collective phenomena. The answer is at least three coordination spheres with 14 nearest neighbors must be taken into account for kagome artificial spin ice. Restriction to a smaller cutoff radius leads to significant changes in the thermodynamic behavior of the main characteristics of the system. An increase of the interaction radius shifts the phase transition temperature to the low-temperature area. The effects of dilution on kagome artificial spin ice were also investigated. It is shown that the phase transition occurs up to p c ≈ 0.35 dilution concentrations, that well coincides with the Sykes and Essam theory. Further dilution leads to the phase transition disappearance. [ABSTRACT FROM AUTHOR]

Abstract We study diluted antiferromagnetic Ising models on triangular and kagome lattices in a magnetic field, using the replica-exchange Monte Carlo method. We observe seven and five plateaus in the magnetization curve of the diluted antiferromagnetic Ising model on the triangular and kagome lattices, respectively, when a magnetic field is applied. These observations contrast with the two plateaus observed in the pure model. The origin of multiple plateaus is investigated by considering the spin configuration of triangles in the diluted models. We compare these results with those of a diluted antiferromagnetic Ising model on the three-dimensional pyrochlore lattice in a magnetic field pointing in the [111] direction, sometimes referred to as the "kagome-ice" problem. We discuss the similarity and dissimilarity of the magnetization curves of the "kagome-ice" state and the two-dimensional kagome lattice. Highlights • Calculations of diluted antiferromagnetic Ising models on triangular and kagome lattices were performed. • The origin of multiple magnetization plateaus is investigated in diluted models. • The similarity and dissimilarity of the magnetization curves of the "kagome-ice" state were discussed. [ABSTRACT FROM AUTHOR]

Highlights • Zero temperature phase diagrams for Ising-Heisenberg expanded Kagomé lattice. • The origin of Ising Kagomé lattice remains in Ising-Heisenberg expanded Kagomé lattice. • Spontaneous magnetization and critical temperature for Ising-Heisenberg expanded Kagomé lattice. Abstract Here we consider the Ising-Heisenberg model in the expanded Kagomé lattice, also known as triangle-dodecagon (3–12) or star lattice. This model can still be understood as a decorated honeycomb lattice. Assuming that the Heisenberg spins are at the vertices of the triangle while other spins are of the Ising type. Thus, this model is equivalent to an effective Kagomé Ising lattice, through the decoration transformation technique. Thus this means that the model is exactly solvable so we can study the most relevant properties of this model. Like the phase diagram at zero temperature, exhibiting a frustrated phase, a ferromagnetic phase, a classical ferrimagnetic phase and a quantum ferrimagnetic phase. We observed that Heisenberg spin exchange interaction influences the frustrated phase, but we rigorously verify that the magnitude and origin of the frustration emerge in a similar way to antiferromagnetic Ising Kagomé lattice. Likewise, the thermodynamic properties of the model can also be obtained, such as the critical temperature as a dependence of the Hamiltonian parameters and the spontaneous magnetization of the model. Besides, we investigated the entropy of the model, identifying its residual entropy in the frustrated region. Even we analyze the specific heat behavior as a temperature dependence, to deal with the phase transition. [ABSTRACT FROM AUTHOR]

Highlights • Based on the kagome lattice, a photonic topological insulator is formed. • Berry curvature and Chern number of the band structure are calculated. • Unidirectional propagation and topological protection are demonstrated. • Edge states may propagate along the same boundary but in opposite directions. Abstract We investigate topological insulator properties of the kagome helical waveguide array, by utilizing theoretical and numerical means. The time-reversal symmetry of the kagome lattice is broken by helicity, and a photonic topological insulator is formed. The Berry curvature and Chern number of the band structure are calculated, to verify the topological phase transition, caused by the helicity. Unidirectional propagation and topological protection properties of the edge states at certain momentum values are demonstrated. Interestingly, the same kagome helical waveguide array, may support topological edge states with different momenta propagating along the same boundary but in opposite directions. [ABSTRACT FROM AUTHOR]

Abstract We investigate the antiferromagnetic spin- 1 ∕ 2 Ising model in the presence of the external magnetic field on the geometrically frustrated kagome lattice using the effective field theory cluster approximation up to the size of the cluster consisting of 12 connected sites which form typical basic geometrical structure of the kagome lattice. The magnetization properties, systems of the ground states, as well as the specific heat capacity behavior of the model are investigated depending on the cluster size. It is shown that properties of the model related to the frustration, such as the formation of discrete system of ground states, strongly depend on the used cluster approximation. It is shown that besides physically relevant ground states the model also exhibits the existence of ground states which are related to the used effective field theory technique and should disappear in the limit n → ∞ of the n -site cluster approximation with simultaneous formation of physically relevant ground states of the Ising model on the kagome lattice. The unphysical origin of these ground states is also demonstrated by the low temperature behavior of the specific heat capacity which exhibits even inverse Schottky peaks with negative values of the specific heat capacity which are reduced with increasing of the cluster size approximation towards physically acceptable behavior. For completeness, the dependence of the position of the critical temperature and the behavior of the spontaneous magnetization and specific heat capacity of the ferromagnetic model in the zero external magnetic field on the size of the cluster approximation is also briefly discussed. Highlights • Dependence of magnetization properties of the model on cluster size is studied. • The formation of the magnetization plateaus for low temperatures is shown. • The existence of the single-point ground states is also shown. • Anomalies of the specific heat capacity are investigated. • Dependence of the critical temperature of the ferromagnetic model on the size of cluster approximation is studied. [ABSTRACT FROM AUTHOR]

Highlights • At coverage less than 1 ML, Bi atoms self-assemble into the cluster array with (5 × 5) periodicity, (37 × 37) R 25. 3 ∘ in a Kagome lattice and (p × 3) stripes, sequentially. • The (5 × 5) periodicity is a low temperature (< 100 K) phase and behavior as gas at room temperature. • The results reveal the Van der Waals interaction between Au(111) and Bi adatoms. • We have optimized the small window for a stable 2D Bi in Kagome and honeycomb lattice on Au(111). Abstract Atomically thin Bismuth (Bi) films are predicated to be candidates for high temperature quantum spin hall materials. Here, we reported the structural evolutions and the corresponding electronic properties of Bi ultrathin films on Au(111) at initial stage as a function of coverage using low-energy electron diffraction (LEED), scanning tunneling microscopy (STM) and photoemission spectroscopy (PES). The small window for a stable two dimensional (2D) Bi in Kagome and honeycomb lattice on Au(111) has been optimized. LEED and STM results show that Bi atoms self-assemble into dimered (5 × 5) superdots, (37 × 37) R 25. 3 ∘ in a Kagome lattice and (p × √3) stripes sequentially in the first monolayer regime and into bi-layered Bi (110) islands afterwards. UPS results show the disappearance of Au(111) surface states at coverage of 0.18 ML, revealing the Van der Waals interactions between Au(111) and Bi adatoms. The repulsive interactions between Bi atoms prevent the 2D Bi structure from islands forming at initial stage and benefit large domains in (37 × 37) R 25. 3 ∘ Kagome lattice. Such 2D Bi may be interesting because of its relevance in terms of novel physical properties. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]

Magnetic competition in topological kagome magnets is studied by incorporating the spin–orbit coupling, anisotropic Hund coupling and spin exchange into a tight-binding electron dynamics in the kagome lattice. Using the Bogoliubov variational principle we find the stable phases at zero and finite temperatures. At zero temperature and in the strong Ising-Hund coupling regime, a magnetic tunability from the out-of-plane ferromagnetism to the in-plane antiferromagnetism is achieved through a universal property of the critical in-plane Hund coupling. At low temperature the out-of-plane ferromagnetism is stable until a finite crossing temperature. Above the crossing temperature the in-plane antiferromagnetism is stable, but the magnetization of the out-of-plane ferromagnetism still survives. This suggests a metastable coexistence of these magnetic phases in a finite temperature range. A large anomalous Hall conductance is observed in the Ising-Hund coupling limit.

Spin and charge are two interrelated properties of electrons. Most of previous works on topological phases of matter tend to study the topological electronic bands and topological magnon excitations separately. In this work, by combining density functional theory and the Schwinger boson method, we determine the topological electronic band structure and topological magnon spectrum simultaneously in the ferromagnetic phase of CoCu _3 (OH) _6 Cl _2 . This Herbertsmithite material has an ABC-stacking Kagome-lattice structure with fractional occupancies on the Cu sites. We find that the hopping parameters of the electronic and magnonic bands can mutually influence each other and predict in further model studies the possibility of magnetically driven topological phase transitions in the electronic bands. Our study also provides a useful method to deal with correlated topological magnetic systems with narrow band gaps self-consistently.

The high-field regime of the spin-s XXZ antiferromagnet on the kagome lattice gives rise to macroscopically degenerate ground states thanks to a completely flat lowest single-magnon band. The corresponding excitations can be localized on loops in real space and have been coined "localized magnons". Thus, the description of the many-body ground states amounts to characterizing the allowed classical loop configurations and eliminating the quantum mechanical linear relations between them. Here, we investigate this loop-gas description on finite kagome lattices with open boundary conditions and compare the results with exact diagonalization for the spin-1/2 XY model on the same lattice. We find that the loop gas provides an exact account of the degenerate ground-state manifold while a hard-hexagon description misses contributions from nested loop configurations. The densest packing of the loops corresponds to a magnon crystal that according to the zero-temperature magnetization curve is a stable ground state of the spin-1/2 XY model in a window of magnetic fields of about 4% of the saturation field just below this saturation field. We also present numerical results for the specific heat obtained by the related methods of thermal pure quantum (TPQ) states and the finite-temperature Lanczos method (FTLM). For a field in the stability range of the magnon crystal, one finds a low-temperature maximum of the specific heat that corresponds to a finite-temperature phase transition into the magnon crystal at low temperatures.

A new mineral, dioskouriite, CaCu4Cl6(OH)4∙4H2O, represented by two polytypes, monoclinic (2M) and orthorhombic (2O), which occur together, was found in moderately hot zones of two active fumaroles, Glavnaya Tenoritovaya and Arsenatnaya, at the Second scoria cone of the Northern Breakthrough of the Great Tolbachik Fissure Eruption, Tolbachik volcano, Kamchatka, Russia. Dioskouriite seems to be a product of the interactions involving high-temperature sublimate minerals, fumarolic gas and atmospheric water vapor at temperatures not higher than 150 °C. It is associated with avdoninite, belloite, chlorothionite, eriochalcite, sylvite, halite, carnallite, mitscherlichite, chrysothallite, sanguite, romanorlovite, feodosiyite, mellizinkalite, flinteite, kainite, gypsum, sellaite and earlier hematite, tenorite and chalcocyanite in Glavnaya Tenoritovaya and with avdoninite and earlier hematite, tenorite, fluorophlogopite, diopside, clinoenstatite, sanidine, halite, aphthitalite-group sulfates, anhydrite, pseudobrookite, powellite and baryte in Arsenatnaya. Dioskouriite forms tabular, lamellar or flattened prismatic, typically sword-like crystals up to 0.01 mm × 0.04 mm × 0.1 mm combined in groups or crusts up to 1 × 2 mm2 in area. The mineral is transparent, bright green with vitreous luster. It is brittle; cleavage is distinct. The Mohs hardness is ca. 3. Dmeas is 2.75(1) and Dcalc is 2.765 for dioskouriite-2O and 2.820 g cm−3 for dioskouriite-2M. Dioskouriite-2O is optically biaxial (+), α = 1.695(4), β = 1.715(8), γ = 1.750(6) and 2Vmeas. = 70(10)°. The Raman spectrum is reported. The chemical composition (wt%, electron microprobe data, H2O calculated by total difference; dioskouriite-2O/dioskouriite-2M) is: K2O 0.03/0.21; MgO 0.08/0.47; CaO 8.99/8.60; CuO 49.24/49.06; Cl 32.53/32.66; H2O(calc.) 16.48/16.38; -O=Cl −7.35/−7.38; total 100/100. The empirical formulae based on 14 O + Cl apfu are: dioskouriite-2O: Ca1.04(Cu4.02Mg0.01)Σ4.03[Cl5.96(OH)3.90O0.14]Σ10∙4H2O; dioskouriite-2M: (Ca1.00K0.03)Σ4.03(Cu4.01Mg0.08)Σ4.09[Cl5.99(OH)3.83O0.18]Σ10∙4H2O. Dioskouriite-2M has the space group P21/c, a = 7.2792(8), b = 10.3000(7), c = 20.758(2) Å, β = 100.238(11)°, V = 1531.6(2) Å3 and Z = 4; dioskouriite-2O: P212121, a = 7.3193(7), b = 10.3710(10), c = 20.560(3) Å, V = 1560.6(3) Å3 and Z = 4. The crystal structure (solved from single-crystal XRD data, R = 0.104 and 0.081 for dioskouriite-2M and -2O, respectively) is unique. The structures of both polytypes are based upon identical BAB layers parallel to (001) and composed from Cu2+-centered polyhedra. The core of each layer is formed by a sheet A of edge-sharing mixed-ligand octahedra centered by Cu(1), Cu(2), Cu(3), Cu(5) and Cu(6) atoms, whereas distorted Cu(4)(OH)2Cl3 tetragonal pyramids are attached to the A sheet on both sides, along with the Ca(OH)2(H2O)4Cl2 eight-cornered polyhedra, which provide the linkage of the two adjacent layers via long Ca−Cl bonds. The Cu(4) and Ca polyhedra form the B sheet. The difference between the 2M and 2O polytypes arises as a result of different stacking of layers along the c axis. The cation array of the layer corresponds to the capped kagomé lattice that is also observed in several other natural Cu hydroxychlorides: atacamite, clinoatacamite, bobkingite and avdoninite. The mineral is named after Dioskouri, the famous inseparable twin brothers of ancient Greek mythology, Castor and Polydeuces, the same in face but different in exercises and achievements; the name is given in allusion to the existence of two polytypes that are indistinguishable in appearance but different in symmetry, unit cell configuration and XRD pattern.

• Fiber fracture, matrix crack, delamination of face sheets, rib inclining/fracture of lattice core, and debonding are the active failure modes of sandwich panel. • The sandwich panel struck by the flat impactor nose has the highest energy absorption among three impactor noses. • The large impactor nose size brings about the high peak load, the long response time and the high energy absorption. • The energy absorption of andwich panel struck at interlace point of lattice core is highest among four impact positions. Dynamic response and failure of carbon fiber reinforced plastic (CFRP) Kagome lattice core sandwich panels subjected to low-velocity impact were investigated experimentally and numerically. The geometrical shapes of the impactor noses are flat, hemispherical, and conical. The active failure modes of fiber fracture, matrix crack, delamination of face sheets, rib inclining/fracture of lattice core, and debonding were observed in experiment. The impactor nose shape has significant influence on the failure modes of sandwich panels. The energy absorption of sandwich panel increases with increasing impact velocity. The sandwich panel struck by the flat impactor has the highest energy absorption among three impactor noses. Finite element (FE) simulations implementing a user-defined material subroutine (VUMAT) of Hashin criterion and progressive degradation were performed and agree well with the experimental results. It is shown that both dynamic response and energy absorption of the sandwich panels are sensitive to the impactor size and impact position. [ABSTRACT FROM AUTHOR]

[Display omitted] • A new family of 2D semiconductors A 2 B 3 C 4 (A = K, Rb, Cs; B = Ni, Pd, Pt; C = S, Se) has been predicted based on first-principles calculations. • A perfect Kagome lattice formed by transition metal atoms have been revealed in these fifteen monolayers. • The A 2 B 3 C 4 monolayers have fairly large power factor due to the loop formed by the conduction band minima. • Ultralow thermal conductivities at room temperature have been confirmed in these monolayers. Two-dimensional (2D) materials hold promising application potential in the future, so it is worthy to explore new 2D materials with fascinating structures and functionals. Here we predict a series of 15 stable 2D semiconductors, A 2 B 3 C 4 (A = K, Rb, Cs; B = Ni, Pd, Pt; C = S, Se), using first-principles calculations. The feasibility of mechanical exfoliation from their bulk phases has been confirmed by low cleavage energies. Interestingly, a perfect Kagome lattice formed by transition metal Ni/Pd/Pt atoms have been revealed in these fifteen monolayers. All A 2 B 3 C 4 monolayers exhibit indirect band features (1.82 eV to 2.76 eV). More encouragingly, and it is theoretically proven, based on constant relaxation time (CRT) approximation, that the A 2 B 3 C 4 monolayers possess large n-type power factors (1.77 ∼ 2.10 × 103 μWm-1K−2), which is mainly contributed from the loop formed by the conduction band minima in different directions in energy band structure. Furthermore, ultralow lattice thermal conductivity (0.20 Wm-1K−1 to 3.04 Wm-1K−1) has been confirmed in 2D A 2 B 3 C 4 at room temperature. Finally, the thermoelectric figure of merit (ZT) of A 2 B 3 C 4 monolayers has been evaluated based on CRT approximation. These interesting findings make A 2 B 3 C 4 monolayers promising candidates for low-dimensional electronic devices. [ABSTRACT FROM AUTHOR]

The rare-earth based magnetic kagome materials RMn 6 Sn 6 have attracted tremendous interest for their potential to host exotic topological quantum states. Here, we report the observation of the anomalous Hall effect in the kagome ferromagnet YbMn 6 Sn 6 single crystal. YbMn 6 Sn 6 has an easy-plane ferromagnetic structure till 2 K without magnetic ordering of the R ions. The low-temperature magnetoresistance is positive and does not exhibit a tendency to saturate up to 5 T. The magnetoresistance becomes negative with the increasing temperature. An intrinsic anomalous Hall conductivity with the value of 40.2 Ω−1·cm−1 is obtained in YbMn 6 Sn 6 , which is associated with the Berry curvature of electronic band structure in momentum space. Our results indicate that YbMn 6 Sn 6 may be an excellent platform to study the relationship between the magnetic and electronic structure and to explore novel quantum phenomenon. • The magnetic and transport properties of YbMn 6 Sn 6 single crystal have been investigated systematically. • Both positive and negative MR is observed in YbMn 6 Sn 6. • The intrinsic AHC deriving from Mn-based kagome lattice is comparable with that for DyMn 6 Sn 6. [ABSTRACT FROM AUTHOR]

• A theoretical model that is able to self-assemble into Archimedean lattices is proposed. • Each lattice site is represented by a single interacting particle with a specifically designed shape. • Principles of surface complimentarity and minimal frustration are employed. • Possible experimental realizations of the proposed model are discussed. Today, low-dimensional structures are finding an increasingly widespread use in various technological applications such as sensors, catalytic converters etc. Kagome lattice, along with other two-dimensional lattices generated from Archimedean tilings, are of particular interest because of their unique electronic, optical and magnetic properties. Here, based on two simple design principles: shape complementarity and minimization of frustration, we introduce a theoretical model that is able to self-assemble into all Archimedean lattices except one. Within this model each lattice site is represented by a single interacting particle with a specifically designed shape. The required shapes are triangles, squares, rhombi and other relatively simple geometrical figures. Each particle carries a single attractive patch designed to minimize the potential energy of the target structure. As the first step of the lattice assembly, the attractive interactions induce the formation of clusters that have space-filling shapes. At the second step, the clusters coalesce into a superlattice with the desired structure known as the cluster crystal. Monte Carlo simulations are carried out to prove that each target lattice is a) easy to assemble by the model and b) constitutes the minimum free energy conformation. Possible experimental realizations of the proposed model are discussed. [ABSTRACT FROM AUTHOR]

High pressure electric transport and synchrotron x-ray diffraction (XRD) measurements together with the first-principles calculations are performed on a shandite compound Pd _3 Pb _2 Se _2 which contains the Kagome lattice of the transition metal Pd. A pressure-induced superconducting transition is observed above 25 GPa, for the first time in the shandite compounds, although the crystal structure of the compound seems to be very robust and persists up to the highest pressure in the XRD study (76.3 GPa). The superconducting transition temperature is about 2.2 K and almost does not change with pressure. The carrier density suddenly increases around 20 GPa possibly due to the emergence of two electron pockets at the Γ point. Our work indicates that the superconductivity in Pd _3 Pb _2 Se _2 is strongly correlated to its electronic structure.

An effective-field theory based on the single-spin cluster has been used to study of a diluted spin-1/2 Ising antiferromagnet on the kagome lattice with nearest-neighbor interactions. We observe five plateaus in the magnetization curve of the diluted antiferromagnet when a magnetic field is applied which is in agreement with Monte Carlo calculation. The effect of the site-dilution on the magnetic susceptibility is also investigated and discussed. In particular, we have found that the frustrated kagome lattice inverse susceptibility falls to zero at 0 K. [ABSTRACT FROM AUTHOR]

Owing to a high degree of geometrical frustration an Ising antiferromagnet on a kagome lattice is known to exhibit no long-range ordering at any temperature, including the ground state. Nevertheless, at low temperatures it shows a strongly correlated, highly fluctuating regime known as a cooperative paramagnet or classical spin liquid. In the ground state it is characterized by a macroscopic degeneracy which translates to a relatively large value of the residual entropy. It has been shown that the presence of a macroscopic degeneracy associated with geometrical frustration below the saturation field can facilitate an enhanced magnetocaloric effect, which can exceed that of an ideal paramagnet with equivalent spin by more than an order of magnitude. In the present study we investigate magnetic and magnetocaloric properties of the Ising antiferromagnet on a kagome lattice by Monte Carlo simulation. In particular, we calculate the entropy of the system using the thermodynamic integration method and evaluate quantities which characterize magnetocaloric effect, such as the isothermal entropy and adiabatic temperature changes in a varying magnetic field. It is found that the Ising antiferromagnet on a kagome lattice shows the most interesting magnetocaloric properties at low temperatures and moderate magnetic fields, suggesting that its potential can be used in technological applications for low-temperature magnetic refrigeration. [ABSTRACT FROM AUTHOR]

Kagome lattice materials are layered two-dimensional (2D) materials in which atoms are arranged in a trihexagonal tiling lattice pattern. It has been suggested that the Kagome lattice can possess topologically non-trivial band structures. By performing atomistic quantum transport simulations, we show that the topological edge modes of a Fe3Sn Kagome nanoribbon have excellent carrier transport properties, with a mean free path several orders of magnitude larger than that of the bulk modes. Vertical stacking of intercalated Kagome layers can further boost the conductance per unit width. As a result, the 2D Kagome lattice materials offer low resistivity and promising potential for interconnect applications in the sub-10nm regime. [ABSTRACT FROM AUTHOR]

Two new members of the A2B′Cu3F12 family of kagome-related materials have been prepared, in order to further understand the crystal-chemical relationships, phase transitions and magnetic behaviour within this family of potentially frustrated S = ½ two-dimensional quantum magnets. Cs2TiCu3F12 adopts a crystal structure with the ideal kagome lattice topology (space group R m) at ambient temperature. Diffraction studies reveal different symmetry-lowering structural phase transitions in single crystal and polycrystalline forms at sub-ambient temperatures, with the single crystal form retaining rhombohedral symmetry and the powder form being monoclinic. In both cases, long-range antiferromagnetic order occurs in the region 16–20 K. Rb2TiCu3F12 adopts a distorted triclinic structure even at ambient temperatures.

Kagome lattice provides a distinctive platform to investigate various correlated electron orders. Recently, an unconventional charge density wave (CDW) with novel chirality is observed in the kagome metal A V 3 Sb 5 ( A = K, Rb, Cs), and the origin of which is still unclear. Here, using a tight-binding model and the mean-field method, we calculate the electron order in the quasi-two-dimensional kagome lattice with 1/3 electron filling, and show that the chiral CDW emerges under a set of parameters withC6rotational symmetry but without mirror symmetry. Physically, the reason why we choose this set of parameters is based on the possible tangential distortion of the kagome lattice. Our results provide a fresh insight to understand the microscopic origin of the unconventional CDW in A V 3 Sb 5 ., (© 2023 IOP Publishing Ltd.)

We report on the reentrant canonical semi spin-glass characteristics and controllable field-induced transitions in distorted Kagomé symmetry of (GeMn)Co 2 O 4 . This B -site spinel exhibits complicated, yet interesting magnetic behaviour in which the longitudinal ferrimagnetic (FiM) order sets in below the Néel temperature T FN ∼ 77 K due to uneven moments of divalent Co (↑ 5.33 μ B ) and tetravalent Mn (↓ 3.87 μ B ) which coexists with transverse spin-glass state below 72.85 K. Such complicated magnetic behaviour is suggested to result from the competing anisotropic superexchange interactions ( J AB / k B ∼ 4.3 K, J AA / k B ∼ -6.2 K and J BB / k B ∼ -3.3 K) between the cations, which is extracted following the Néel's expression for the two-sublattice model of FiM. Dynamical susceptibility ( χ ac ( f, T )) and relaxation of thermoremanent magnetization, M TRM ( t ) data have been analysed by means of the empirical scaling-laws such as Vogel-Fulcher law and Power law of critical slowing down. Both of which reveal the reentrant spin-glass like character which evolves through a number of intermediate metastable states. The magnitude of Mydosh parameter (Ω ∼ 0.002), critical exponent zυ = (6.7 ± 0.07), spin relaxation time τ 0 = (2.33 ± 0.1) × 10 -18 s, activation energy E a / k B = (69.8 ± 0.95) K and interparticle interaction strength ( T 0 = 71.6 K) provide the experimental evidences for canonical spin-glass state below the spin freezing temperature T F = 72.85 K. The field dependence of T F obtained from χ ac ( T ) follows the irreversibility in terms of de Almeida-Thouless mean-field instability in which the magnitude of crossover scaling exponent Φ turns out to be ∼2.9 for the (Ge 0.8 Mn 0.2 )Co 2 O 4 . Isothermal magnetization plots reveal two field-induced transitions across 9.52 kOe ( H SF1 ) and 45.6 kOe ( H SF2 ) associated with the FiM domains and spin-flip transition, respectively. Analysis of the inverse paramagnetic susceptibilityχp-1χp=χ-χ0after subtracting the temperature independent diamagnetic termχ0(=-3 × 10 -3 emu mol -1 Oe -1 ) results in the effective magnetic momentμeff= 7.654 μ B / f.u . This agrees well with the theoretically obtainedμeff= 7.58 μ B / f.u . resulting the cation distributionMn0.24+↓A[Co22+↑]BO4in support of the Hund's ground state spin configurationS=3/2andS= 1/2of Mn 4+ and Co 2+ , respectively. The H-T phase diagram has been established by analysing all the parameters ( T F ( H ), T FN ( H ), H SF1 ( T ) and H SF2 ( T )) extracted from various magnetization measurements. This diagram enables clear differentiation among the different phases of the (GeMn)Co 2 O 4 and also illustrates the demarcation between short-range and long-range ordered regions., (© 2023 IOP Publishing Ltd.)