U K Voma

S Bhattacharya

Edwin Kermarrec

J Alam

Y M Jana

B Sana

P Khuntia

S K Panda

B Koteswararao
email: koteswararao@iittp.ac.in

Electronic structure and magnetic properties of the effective spin S = 1/2 2D triangular lattice K 3 Yb(VO 4 ) 2

Keywords:

come

I. INTRODUCTION

Geometrically Frustrated Magnetism in twodimensional (2D) magnetic materials draw significant attention to the scientific community due to their storming and unique magnetic properties. The strong quantum fluctuation enhanced by competing interaction between spins leads to unusual ground states.

In S = 1/2 Heisenberg 2D triangular spin systems, the quantum fluctuations are more robust, which prevents the magnetic long-range order (LRO) even at T = 0 K in the presence of anisotropic exchange interactions arises from spin-orbit coupling (SOC), and can exhibit the novel disordered ground state called Quantum Spin Liquid (QSL) as contemplated by P. W. Anderson [START_REF] Anderson | Resonating valence bonds: a new kind of insulator?[END_REF]. The spins in the QSL state can be highly entangled with each other, even at long distances. The QSL state has been proposed experimentally for various 2D geometrically frustrated magnetic systems (GFMS), such as triangular and Kagome lattices, with gapped and gapless fractionalized spin excitations without a conventional symmetry breaking phase transition [START_REF] Balents | Spin liquids in frustrated magnets[END_REF][START_REF] Li | Spin liquids in geometrically perfect triangular antiferromagnets[END_REF][START_REF] Broholm | Quantum spin liquids[END_REF]. A few well-known S = 1/2 QSL materials in 2D GFMS are κ-(BEDT-TTF) 2 Cu 2 (CN) 3 [START_REF] Shimizu | Spin liquid state in an organic Mott insulator with a triangular lattice[END_REF][START_REF] Isono | Quantum criticality in an organic spin-liquid insulator κ-(BEDT-TTF)2Cu2(CN)[END_REF], EtMe 3 Sb[Pd(mit) 2 ] 2 [START_REF] Yamashita | Thermodynamic properties of a spin-1/2 spin-liquid state in a κ-type organic salt[END_REF][START_REF] Yamashita | Gapless spin liquid of an organic triangular compound evidenced by thermodynamic measurements[END_REF], and ZnCu 3 (OH) 6 Cl 2 [START_REF] Han | Fractionalized excitations in the spin-liquid state of a kagomelattice antiferromagne[END_REF][START_REF] Helton | Spin Dynamics of the Spin-1/2 Kagome Lattice Antiferromagnet ZnCu3(OH)6Cl2[END_REF][START_REF] Khuntia | Gapless ground state in the archetypal quantum kagome antiferromagnet ZnCu3( OH )6Cl2[END_REF].

Recently, the interest in understanding the physics of S = 1/2 quantum magnetism is expanded from the 3d to 4d, 5d, and 4f rare-earth-based magnetic systems. The effective S = 1/2 can be realized in some of the Ru 3+ , Ir 4+ , Ce 3+ , Yb 3+ based materials due to the strong spinorbit coupling and crystal electric field (CEF) effects, which lead to exotic quantum phenomena. A few exemplar materials of this class are 4d honeycomb lattice α-RuCl 3 [START_REF] Banerjee | Neutron scattering in the proximate quantum spin liquid α-RuCl3[END_REF], 5d honeycomb lattices A 2 IrO 3 , (A=Li, Na, Cu) [START_REF] Singh | Antiferromagnetic Mott Insulating State in Single Crystals of the Honeycomb Lattice Material Na2IrO3[END_REF][START_REF] Choi | Exotic Low-Energy Excitations Emergent in the Random Kitaev Magnet Cu2IrO3[END_REF][START_REF] Mehlawat | Heat capacity evidence for proximity to the Kitaev quantum spin liquid in A2IrO3 ( A = Na , Li)[END_REF] and H 3 LiIr 2 O 6 [START_REF] Kitagawa | A spin-orbital-entangled quantum liquid on a honeycomb lattice[END_REF], 4f -Yb 3+ based triangular lattices YbMgGaO 4 [START_REF] Li | Gapless quantum spin liquid ground state in the two-dimensional spin-1/2 triangular antiferromagnet[END_REF][START_REF] Li | Rare-Earth Triangular Lattice Spin Liquid: A Single-Crystal Study of YbMgGaO4[END_REF][START_REF] Li | Muon Spin Relaxation Evidence for the U(1) Quantum SpinLiquid Ground State in the Triangular Antiferromagnet YbMgGaO4[END_REF][START_REF] Paddison | Continuous excitations of the triangular-lattice quantum spin liquid YbMgGaO4[END_REF][START_REF] Li | Crystalline Electric-Field Randomness in the Triangular Lattice Spin-Liquid YbMgGaO4[END_REF][START_REF] Shen | Fractionalized excitations in the partially magnetized spin liquid candidate YbMgGaO4[END_REF][START_REF] Li | Rearrangement of uncorrelated valence bonds evidenced by low-energy spin excitations in YbMgGaO4[END_REF], YbZnGaO 4 [START_REF] Ma | Spin-Glass Ground State in a Triangular-Lattice Compound YbZnGaO4[END_REF], and NaYbX 2 ( X=O, S, Se), etc. [START_REF] Ding | Gapless spin-liquid state in the structurally disorder-free triangular antiferromagnet NaYbO2[END_REF][START_REF] Ranjith | Field-induced instability of the quantum spin liquid ground state in the J ef f = 1/2 triangular-lattice compound NaYbO2[END_REF][START_REF] Bordelon | Field-tunable quantum disordered ground state in the triangular-lattice antiferromagnet NaYbO2[END_REF][START_REF] Sarkar | Quantum spin liquid ground state in the disorder free triangular lattice NaYbS2[END_REF][START_REF] Dai | Spinon fermi surface spin liquid in a triangular lattice antiferromagnet NaYbSe[END_REF][START_REF] Zhang | Pressure induced metallization and possible unconventional superconductivity in spin liquid[END_REF].

In the compound YbMgGaO 4 , the magnetic, specific heat (C p ), Muon spin relaxation (µSR), and inelastic neutron scattering (INS) measurements revealed the absence of static magnetic LRO down to several mK temperatures, despite an antiferromagnetic coupling strength with the Curie-Weiss temperature (θ CW ) of -4 K. The magnetic specific heat C m data in the low-temperature region follow the power-law temperature dependence as C m ~T α . Furthermore, the value of α close to 2/3 suggests the possible realization of a gapless QSL state for the triangular Heisenberg antiferromagnetic lattice [START_REF] Li | Gapless quantum spin liquid ground state in the two-dimensional spin-1/2 triangular antiferromagnet[END_REF]. Electron spin resonance (ESR) study revealed the anisotropic nature of the spin interactions [START_REF] Li | Rare-Earth Triangular Lattice Spin Liquid: A Single-Crystal Study of YbMgGaO4[END_REF]. The µSR relaxation rate exhibits a temperature-independent plateau behavior at low temperatures suggesting persistent spin dynamics [START_REF] Li | Muon Spin Relaxation Evidence for the U(1) Quantum SpinLiquid Ground State in the Triangular Antiferromagnet YbMgGaO4[END_REF]. The INS measurements revealed continuous diffusive magnetic excitations suggesting that YbMgGaO 4 might host a gapless QSL with a spinon Fermi surface [START_REF] Paddison | Continuous excitations of the triangular-lattice quantum spin liquid YbMgGaO4[END_REF][START_REF] Shen | Fractionalized excitations in the partially magnetized spin liquid candidate YbMgGaO4[END_REF]. While the isotropic perfect S = 1/2 2D triangular lattice model with nearestneighbor (NN) interactions yield a 120 • Néel state theo-1 retically at T = 0 K [START_REF] Capriotti | Long-Range Néel Order in the Triangular Heisenberg Model[END_REF][START_REF] White | Neél Order in Square and Triangular Lattice Heisenberg Models[END_REF], the origin of QSL state in the compound YbMgGaO 4 could be due to strong spinorbit coupling (SOC) effects. From the recent arguments, in comparison with YbZnGaO 4 , it was also pointed that YbMgGaO 4 showed a spin glass-like ground state from the observations of frequency-dependent magnetic susceptibility [START_REF] Ma | Spin-Glass Ground State in a Triangular-Lattice Compound YbZnGaO4[END_REF], which was alternatively explained as a tiny spin freezing [START_REF] Li | Rearrangement of uncorrelated valence bonds evidenced by low-energy spin excitations in YbMgGaO4[END_REF]. As the ground state of this system is still under debate due to the site mixing of Ga 3+ and Mg 2+ ions, the exploration of suitable candidate materials is required to understand the physics behind the 2D S = 1/2 triangular lattice whether the strong SOC or disorder-induced anisotropy drives the QSL state.

The study of rare-earth-based 2D triangular magnetic materials without site disorder has attracted attention in searching the novel QSL states. The series of rare-earth chalcogenides NaYbX 2 were identified as promising QSL candidates. In the case of NaYbO 2 , no magnetic LRO down to 50 mK is revealed. Furthermore, a field-induced quantum phase transition was observed above an applied field of 2 T [START_REF] Ranjith | Field-induced instability of the quantum spin liquid ground state in the J ef f = 1/2 triangular-lattice compound NaYbO2[END_REF]. On the other hand, the pressure-induced metallic states were observed in NaYbSe 2 [START_REF] Zhang | Pressure induced metallization and possible unconventional superconductivity in spin liquid[END_REF], indicating that these rare-earth chalcogenides offer an ideal venue to discuss QSL nature and superconductivity as predicted by P. W. Anderson. To understand further interesting properties of 2D triangular GFMS, there is a necessity to discover promising materials of this category.

Herein, we introduce another structurally perfect triangular lattice system K 3 Yb(VO 4 ) 2 [START_REF] Kimani | Hydrothermal Synthesis and Spectroscopic Properties of a New Glaserite Material, K3RE[END_REF], with the 2D triangular lattices of Yb 3+ ions. Yb 3+ hosts effective S = 1/2 moments as found below. Based on the measured magnetic and specific heat data, we find that the system does not exhibit any LRO down to 0.5 K. The obtained θ CW of -1 K indicates weak antiferromagnetic coupling between the Yb 3+ ions. This is further supported by the computed exchange interactions using the LSDA+U method. We analyzed the detailed electronic structure based on LSDA+U and LDA+DMFT methods and found that the insulating state in K 3 Yb(VO 4 ) 2 arises due to the combined effects of SOC and electronic correlations. Our results indicate that this system possesses a large magnetic crystalline anisotropy. We also found that the dynamical correlation effects are crucial for accurately describing the f -multiplet spectra.

II. EXPERIMENTAL AND COMPUTATIONAL DETAILS:

We have synthesized the polycrystalline material K 3 Yb(VO 4 ) 2 using the conventional solid-state synthesis method. Initially, the rare-earth oxide Yb 2 O 3 was fired at 1000 • C for 6 hours to remove the moisture and absorbed CO 2 . It was then mixed with K 2 CO 3 , V 2 O 5 in stoichiometric proportion and ground thoroughly. The pressed pellets are placed in an alumina crucible and fired at 750 • C for 48 hours with several intermediate grindings. To confirm the single-phase, the X-ray diffraction theory (DFT) [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF][START_REF] Jones | The density functional formalism, its applications and prospects[END_REF] with local density approximation (LDA) and local spin-density approximation+Hubbard U (LSDA+U) [START_REF] Liechtenstein | Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators[END_REF] approaches as implemented in the allelectron, full potential code WIEN2K [START_REF] Schwarz | Solid state calculations using WIEN2k[END_REF]. The Brillouin-Zone integration is performed with a 12 × 12 × 8 kmesh. To achieve energy convergence of the eigenvalues, the wave functions in the interstitial region were expanded in plane waves with a cutoff R M T k max = 8, where R M T denotes the smallest atomic sphere radius and k max represents the magnitude of the largest k vector in the plane wave expansion. The valence wave functions inside the spheres are expanded up to l max = 10, while the charge density is Fourier expanded up to a large value of G max = 12. Based on the converged non-spin polarized LDA solutions, we constructed the maximally localized Wannier function for the f like bands, using the Wien2wannier [START_REF] Kunes | Wien2wannier: From linearized augmented plane waves to maximally localized Wannier functions[END_REF] and WANNIER90 codes [START_REF] Mostofi | An updated version of wannier90: A Tool for Obtaining Maximally-Localised Wannier Functions[END_REF] to compute the hopping parameters between the effective Yb-f orbitals.

We also carried out LSDA+U calculations using the full potential linearized augmented plane wave method (FP-LAPW) as implemented in the RSPt code [START_REF] Andersen | Linear methods in band theory[END_REF][START_REF] Wills | Synthesis of band and model Hamiltonian theory for hybridizing cerium systems[END_REF]. The consistency of the results from RSPt and Wien2k provides further credence to our calculations. Using the LSDA+U solutions of RSPt, we employed the magnetic force theorem [START_REF] Liechtenstein | Local spin density functional approach to the theory of exchange interactions in ferromagnetic metals and alloys[END_REF][START_REF] Katsnelson | First-principles calculations of magnetic interactions in correlated systems[END_REF] to extract the effective inter-site exchange parameters (J ij ). A detailed discussion of the implementation of the magnetic force theorem in RSPt is provided in Ref. [START_REF] Kvashnin | Exchange parameters of strongly correlated materials: Extraction from spin-polarized density functional theory plus dynamical mean-field theory[END_REF]. The J ij are extracted in a linearresponse manner via Green's function technique.

Considering that the many-body effects are extremely significant for the highly correlated f -states, we also performed LDA+DMFT simulations using the same RSPt code [START_REF] Wills | Synthesis of band and model Hamiltonian theory for hybridizing cerium systems[END_REF]. In this implementation [START_REF] Kvashnin | Exchange parameters of strongly correlated materials: Extraction from spin-polarized density functional theory plus dynamical mean-field theory[END_REF][START_REF] Grechnev | Theory of bulk and surface quasiparticle spectra for Fe, Co, and Ni[END_REF][START_REF] Marco | Correlation effects in the total energy, the bulk modulus, and the lattice constant of a transition metal: Combined local-density approximation and dynamical mean-field theory applied to Ni and Mn[END_REF][START_REF] Thunstr Om | Multiplet effects in the electronic structure of intermediate-valence compounds[END_REF][START_REF] Grån As | Charge selfconsistent dynamical mean-field theory based on the fullpotential linear muffin-tin orbital method: Methodology and applications[END_REF] LDA+DMFT, the many-body corrections appear in a form that depends on a self-consistently calculated density matrix and on the correlated orbitals, which are 4f states on the Yb atoms in the present case. The effective impurity problem in the LDA+DMFT calculations has been solved through the Hubbard I (HIA) solver.

The reported results are based on a single-shot simulation where the convergence of the self-energy has been achieved with 5000 Matsubara frequencies, and the temperature is taken as 150 K.

To describe the electronic correlation effect within LSDA+U and LDA+DMFT approaches, we have taken U = 4.0 eV, J = 0.5 eV for Yb-f states which are obtained from a constrained random phase approximation (cRPA) calculations [START_REF] Aryasetiawan | Frequency-dependent local interactions and low-energy effective models from electronic structure calculations[END_REF] using the implementation of Ref. [START_REF] Vaugier | Hubbard U and Hund exchange J in transition metal oxides: Screening versus localization trends from constrained random phase approximation[END_REF]. This method has been successfully used for several transition metal oxides. The first-principle estimation of U using cRPA ensures that the reported electronic and magnetic properties are completely parameter-free and thus, our approach is truly ab-initio.

III. STRUCTURAL DETAILS:

The structure of K 3 Yb(VO 4 ) 2 belongs to the family of Glaserite, having the general formula A 3 RE(VO 4 ) 2 (A -alkali metal, RE -rare-earth ) with trigonal crys- The bond-length of Yb-O2 in YbO 6 octahedra is 2.19 Å. The environment of Yb 3+ holds D 3d point group symmetry, and less distorted in comparison to NaYbO 2 , YbMgGaO 4 . The presence of the inversion center between the bonds that precludes antisymmetric Dzyaloshinskii-Moriya (DM) exchange The 2D triangular layers are separated by K atomic layers. The possible magnetic exchange path, in the titled compound is through Yb-O2-V-O2-Yb while it is through only O atoms in NaYbO 2 and YbMgGaO 4 [START_REF] Li | Gapless quantum spin liquid ground state in the two-dimensional spin-1/2 triangular antiferromagnet[END_REF][START_REF] Ranjith | Field-induced instability of the quantum spin liquid ground state in the J ef f = 1/2 triangular-lattice compound NaYbO2[END_REF]. In this path the bond angles are Yb-O2-V ≈ 164.4 • , O2-V-O2 ≈ 108.7 •

IV. RESULTS:

A. Magnetic properties:

Temperature-dependent magnetization measurements were performed in the presence of a magnetic field 10 kOe. The magnetic susceptibility χ and inverse-magnetic susceptibility χ -1 as a function of temperature (T ) down to 5 K is shown in Fig. 3. The χ -1 data are fitted with the expression (T -θ HT ) /C HT on a high-temperature region from 150 K to 340 K (see Fig. 3(b)). The value of Curie constant at high temperature (C HT ) is obtained to be 3.2 cm 3 K/mol, which gives J = 7/2 and g J = 1.28. The obtained value of g J is close to the expected value g J = 8/7

1.14. The high-temperature Curie-Wiess temperature is found to be θ HT ≈ -110 K. The large value of θ HT is not due to the magnetic exchange couplings, but it results from the CEF effect on the Yb atom. As the total angular momentum J = 7/2 (holds eightfold degeneracy) is in the CEF environment of D 3d symmetry, the energy levels split into four Kramers doublets by preserving the time-reversal symmetry. The energy difference between the ground state doublet and the first excited state of Kramers doublet, is defined as the CEF gap (∆ CEF ). We have tabulated II the ∆ CEF values for a few of Yb 3+ based triangular materials along with the high-temperature values of θ HT (see table II). As the value of θ HT of K 3 Yb(VO 4 ) 2 is close to the value of NaYbO 2 compound, the ∆ CEF of K 3 Yb(VO 4 ) 2 could also be similar to that of NaYbO 2 (404 K) [START_REF] Ding | Gapless spin-liquid state in the structurally disorder-free triangular antiferromagnet NaYbO2[END_REF]. While the ∆ CEF is expected to be large (as we found θ HT is about -110 K), the spin tends to lie in the ground state and hosts the effective spin-1/2 (S eff = 1/2) at the low-temperature region.

The zero field cooled (ZFC) and field cooled (FC) susceptibility with temperature variation down to 0.5 K is shown in Fig. 4(a). No bifurcation is noticed, which represents the absence of spin freezing or spin glass transition. As shown in Fig. 4(b), the fit to the formula 1/ [χ 0 + C LT / (T -θ LT )] in the temperature range 1.8 -20 K gives θ LT ≈ -1 K. The temperatureindependent susceptibility is found to be χ 0 ≈ 4.6 × 10 -3 cm 3 /mol. After subtracting the core diamagnetic susceptibility χ dia ≈ -1.61 × 10 -4 cm 3 /mol calculated from the individual ions values [START_REF] Selwood | Magnetochemistry[END_REF], the obtained Van-Vleck susceptibility is χ VV ≈ 4.76×10 -3 cm 3 /mol. The Curie con- stant C LT provides the magnetic moment µ eff ≈ 2.41 µ B . As compared to the value µ eff ≈ 4.53 µ B for free ions, the reduced value of the magnetic moment in the lowtemperature region is due to the presence of S = 1/2 doublet as the ground state of Yb 3+ ion with the effective spin S eff = 1/2 with g ≈ 2.75. The nonlinear variation of the field-dependent magnetization plot at 1.8 K is shown in Fig. 5. The saturation magnetic moment is about 1.3 µ B /Yb 3+ obtained from the intercept of the linear fit as shown in Fig. 5 The specific heat after subtracting the zero field data with the fit to Schottkey anomaly (see the text). The two energy level Schottky fit after subtracting the specific heat at 0 T. The inset shows the energy gap variation with applied magnetic fields.

B. Specific heat measurements:

The specific heat data as a function of temperature (C p (T )) were measured down to 1.8 K in different magnetic fields upto 90 kOe. As shown in Fig. 6(a), we did not observe any sharp transition in zero-field data. Field-dependent behavior is observed. A broad peak is seen in the presence of magnetic fields. The broad peak moves towards high temperatures with increasing magnetic fields due to the Zeeman splitting of the ground state Kramers doublet (S eff = ±1/2). The two-level Table III: Hopping integrals t (in meV) between the NN Yb-f states.

Schottky fit C = N k B (∆/T ) 2 e -∆/T (1+e -∆/T ) 2 to the C p (H kOe) - Γ M K Γ A H L A -4 -2 0 2 4 6 Energy(eV) 0 4 8 12 PDOS (1/eV) (a) (b) Yb-f V-d O-p Γ MK Γ A H L A -0.2 -0.1 0 Energy (eV)
Yb-f xz 2 Yb-f yz 2 Yb-f xz 2 -1.5 -0.38 Yb-f yz 2 -0.38 -1.5
C p (0 kOe) data is shown in Fig. 6(b). The extracted Zeeman energy gap (∆) between the energy levels of the ground state doublet S eff = ±1/2 increases linearly with the applied magnetic fields. The g value yielded from the slope of the linear fit to the expression ∆ = gµ B H is about 2.5, is good agreement with the value estimated from the magnetic measurements.

C. Basic Electronic Structure:

To begin with, we have analyzed the band structure and partial density of states (PDOS), obtained using the LDA method. The results of our calculations are presented in Fig. 7. Close to the Fermi level, we find seven almost dispersion less bands which give rise to the sharp peak observed in the PDOS of Yb-f (Fig. 7(b)), implying the strongly localized nature of the f states. These states are also mostly occupied and thus consistent with the nominal 3+ (f 13 ) charge state of Yb-ions. The occupied O-p states appear in the energy range between -4 eV to -0.5 eV, while the V-d states are fully unoccupied, located above 3 eV from the Yb-f states. The hybridization between Yb-f and O-p states looks very weak.

Since the Yb-f states are well separated from the rest of the bands, and they only contribute to the Fermi surface, we constructed the effective Wannier functions for these bands using the WANNIER90 formalism [START_REF] Mostofi | An updated version of wannier90: A Tool for Obtaining Maximally-Localised Wannier Functions[END_REF] where only the Yb-f states were kept in the basis, and the rest of the orbitals were downfolded. In these calculations, we primarily find out the relevant orbitals for the magnetism and estimate the inter-site hopping strengths between them. The Wannier interpolated bands along with the LDA bands are shown in the inset of Fig. 7(b), and the agreement is quite remarkable. The diagonalization of the obtained onsite blocks of the Yb-f Hamiltonian, gives the eigen-energies -147, -128, -80, -80, -80, -56, -56 meV, corresponding to the eigen-states hav-

ing predominant f y(3x 2 -y 2 ), f z 3 , f xyz, f z(x 2 -y 2 ), f x(x 2 -3y 2 ), f xz 2 , f yz 2 characters, respectively. Since
Yb is in f 13 state, the above onsite energies indicate that all the lower-lying states (f y(3x

2 -y 2 ), f z 3 , f xyz, f z(x 2 -y 2 ), f x(x 2 -3y 2
)) are completely filled up and thus magnetically inactive. However, the topmost degenerate states which have predominant f xz 2 , f yz 2 character, are partially filled up and would contribute to the local moment formation. The computed inter-site hopping integrals between these two relevant orbitals is reported in table III. Interestingly, the magnitudes of the hopping parameters come out to be very weak. This also implies that the magnetic exchange interactions (J ij ) between the Yb-ions would be small. The estimation of the (J ij ) is discussed in the later part of the manuscript.

D. LSDA+U analysis of magnetic properties :

Next, in order to provide a microscopic understanding of the experimentally observed magnetization, we estimated the different terms necessary for constructing the spin Hamiltonian for this system. Apart from the external magnetic field-dependent Zeeman term, spin dynamics of a system are governed by the isotropic Heisenberg exchange, anisotropic Dzyaloshinskii-Moriya (DM) interaction, and the single-ion anisotropy or the magnetocrystalline anisotropy of the system. Using the formalism of Ref. [START_REF] Kvashnin | Exchange parameters of strongly correlated materials: Extraction from spin-polarized density functional theory plus dynamical mean-field theory[END_REF] , the estimated Yb-Yb magnetic-exchange interactions (J ij ) come out to be 0.1 meV and antiferromagnetic in nature. This is consistent with the lowtemperature value of Curie-Weiss temperature (θ LT ) obtained from the fitting of experimental susceptibility. We note that such small antiferromagnetic exchange in the triangular Yb network might promote magnetic frustration at a very low temperature which will oppose long range ordering. The experimental magnetic data also shows that the system does not have LRO down to 0.5 K. Although the Yb-f electrons posses large SOC, but the presence of inversion symmetry in this system makes DM interactions to be zero.

To determine the magneto-crystalline anisotropy, we have computed total energies by fixing the spin axis along various possible directions within the LSDA+U approach incorporating SOC. The computed energies, spin, and orbital moments are displayed in Table IV. Our calculations suggest that the Yb spins favor ab-plane as the preferred plane of magnetization. We also find that the spin moment for the prefer spin-axis of the Yb 3+ ion is 0.80 µ B , and the orbital moment is almost 2.4 times that of the spin moment, making the total moment (2.73 µ B ).

Another key finding of our LSDA+U+SOC calculations is that the system posses a large easy plane magnetocrystalline anisotropy energy (MAE). As displayed in Table IV, the ab-plane is energetically favorable with the energy -98.4 meV.

The partial density of states of the Yb-f , V-d , and O-p states for both the spin channels obtained using LSDA+U with, and without SOC has been shown in Fig. 8. We clearly see that SOC is crucial to open up the insulating gap in this system. Our results show that the majority Yb-f states are completely occupied, and located between -4.5 eV to -1.8 eV, and the gap opens up in the minority channel. This is expected since Yb is in a nominal 3+ charge state (f 13 ), the first 7 electrons occupy the majority channel, while the rest 6 electrons fill up the minority f states, keeping only one level empty. The value of the insulating gap comes out to be 1.9 eV which is a p-f gap as the p states appear just below the Fermi level. Therefore according to the LSDA+U theory, this system is a charge-transfer insulator which is an unlikely electronic state for f -electron systems. However, it is well known that LSDA+U has limited success in describing the electronic structure of f -electron systems. Moreover, LSDA+U method works with symmetry broken solutions, and thus appropriate description of the paramagnetic electronic structure is outside its capabilities, necessitating the need of investigation using methods that are beyond the static mean-field approach.

E. Electronic structure from LDA+DMFT:

In view of the fact that an accurate treatment of the many-body electronic correlation is crucial for describing the correlated paramagnetic state of K 3 Yb(VO4) 2 , we have employed LDA+DMFT with HIA incorporating SOC. In Fig. 9, we report the spectral function obtained in LDA+DMFT, which for simplicity, we will also label as PDOS. The obtained band-gap 2.2 eV, which is slightly larger than that of LSDA+U. Most importantly, contrary to the LSDA+U results, the gap is observed between the f -states, and the origin of this gap is a combined effect of SOC, and electron-electron correlation. Thus we can conclude that K 3 Yb(VO 4 ) 2 belongs to the very interesting class of spin-orbit driven Mott-Hubbard insulators. We note here that our LDA+DMFT calculations not only find out the correct nature of the insulating ground state but also provide the correct position of the f -multiplets.

V. DISCUSSION AND CONCLUSION

The ideal 2D Heisenberg triangular lattice hosts 120 o Néel ordered state at absolute temperature T = 0 K [START_REF] Capriotti | Long-Range Néel Order in the Triangular Heisenberg Model[END_REF][START_REF] White | Neél Order in Square and Triangular Lattice Heisenberg Models[END_REF]. In the presence of 3D couplings, the system will order at finite temperature. For example, the 2D triangular lattice materials Ba 3 CoSb 2 O 9 [START_REF] Zhou | Successive Phase Transitions and Extended Spin-Excitation Continuum in the S =1/2 Triangular-Lattice Antiferromagnet Ba3CoSb2O9[END_REF], and Ba 3 CoNb 2 O 9 [START_REF] Yokota | Magnetic phase diagram of the S = 1/2 triangular-lattice Heisenberg antiferromagnet Ba3CoNb2O9[END_REF] shows magnetic LRO below the temperatures 3.7 K, and 1.4 K, respectively. Recently there have been extensive theoretical studies on the effect of strong SOC coupling on a triangular lattice. As per the spin-orbit coupled triangle lattice model, the possibility of various QSL phases was discussed. QSL state would be achieved due to strong SOC, which mainly generates highly anisotropic interactions. In the reference [START_REF] Iaconis | Spin Liquid versus Spin Orbit Coupling on the Triangular Lattice[END_REF], the detailed study was carried on the ground state phase diagram, the effects on spinon Fermi surface of the QSL in the presence of SOC, and due to the presence of 2 nd NN & 3 rd NN couplings. The 2 nd NN interactions create a Dirac QSL as the lowest energy state. The inclusion of even a very small 3 rd NN interaction can greatly stabilize the QSL phase. In this line of theoretical studies, it is essential to synthesize, and investigate 2D triangular quantum spin materials with possibly the large SOC. Our title material is another type of material in addition to those of previously published materials YbMgGaO 4 , and AYbX 2 . The theoretical band structure calculations also confirmed that this material has large SOC, and in-plane anisotropy.

In conclusion, we have synthesized, and studied the magnetic properties, and performed the electronic structure calculations on a compound K 3 Yb(VO 4 ) 2 having 2D triangular-lattice of Yb 3+ ions. Our results confirm that Yb 3+ holds S eff = 1/2 moments with an antiferromagnetic coupling with θ CW about -1 K. No magnetic LRO was found down to 0.5 K. Electronic structure calculations confirm that the system is a Mott insulator with large SOC, and sizeable magneto-crystalline anisotropy energy. Further low-temperature experiments, such as µSR, and INS, would be required to fully explore the nature of its ground state.

Figure 1 :

 1 Figure 1: (color online). (a) Crystal structure built with the YbO6 octohedra, VO4 tetrahedra, and K atoms. (b) The 2D triangular layers of Yb atoms are connected through V and O atoms [34].

Figure 2 :

 2 Figure 2: (color online). Refinement of the X-ray diffraction data measured at room temperature for K3Yb(VO4)2 polycrystalline sample.

  of

Figure 3 :

 3 Figure 3: (color online). (a)The magnetic susceptibility χ(T ) of K3Yb(VO4)2 at the applied magnetic field 10 kOe, (b) The inverse-magnetic susceptibility data with the fit to the expression 1/[χ0 + C HT / (T -θ HT )].

Figure 4 :

 4 Figure 4: (color online). (a) Temperature-dependent ZFC and FC χ(T ) plot in the range from 0.5 K to 20 K. (b) Inverse-χ versus T plot with the Curie-Weiss fit (represented by solid line).

Figure 5 :

 5 Figure 5: (color online). (a) Magnetic field-dependent magnetization at T = 1.8 K. The solid straight line represents the extrapolation to linear fit in the range 60 -70 kOe. (b) The data with simulation to Brillouin function for S = 1/2 and g ≈ 2.6.

  (a). The Fig. 5(b) displays the M (H) data matches with the simulation to the Brillouin function for S eff = 1/2, g ≈ 2.6 at T = 1.8 K( > θ LT ).

Figure 6 :

 6 Figure 6: (color online). (a) Temperature-dependent specific heat Cp at different applied magnetic fields from 0 kOe to 90 kOe. (b)The specific heat after subtracting the zero field data with the fit to Schottkey anomaly (see the text). The two energy level Schottky fit after subtracting the specific heat at 0 T. The inset shows the energy gap variation with applied magnetic fields.

Figure 7 :

 7 Figure 7: (color online). Non-spin-polarized LDA. (a) Band dispersion along various high symmetry directions, and (b) orbital projected partial density of states of Yb-f , V-d, and O-p states. For clarity, the f states are plotted after scaling the PDOS by a factor of 0.2. Inset of (b) shows the Wannierinterpolated bands superimposed on the LDA bands, plotted in the narrow energy region around the Fermi level.

Figure 8 :

 8 Figure 8: (color online). The partial density of states of Ybf , V-d, and O-p in the spin polarized ground state obtained from LSDA+U and LSDA+U+SOC calculations.

Figure 9 :

 9 Figure 9: (color online). The partial density of states of Ybf , V-d, and O-p in the paramagnetic state, obtained from LDA+DMFT+SOC simulations.

Table I :

 I The atomic coordinates for K3Yb(VO4)2

	Atom type Wyckoff position Site	x	y	z	Occupancy
	Yb(1)	1b	-3m	1	0	0.5	1
	K(1)	2d	3m 0. 6667 0.3333 1.2961	1
	K(2)	1a	-3m	0	0	1	1
	V(1)	2d	3m 0. 6667 0.3333 0.7474	1
	O(1)	2d	3m 0. 6667 0.3333 0.9595	1
	O(2)	6i	m 0.3499 0.1750 0.6693	1
	(XRD) measurements were carried out at room temper-
	ature. The temperature (T ) and magnetic field (H) de-
	pendent magnetization (M ) were done using Vibrating
	Sample Magnetometer (VSM) down to 5 K and super-
	conducting quantum interference device (SQUID) mag-
	netometer used to do the measurements down to 0.5
	K, attached to Physical Property Measurement System
	(PPMS), Quantum Design, Inc. Specific heat as a func-
	tion of temperature C p (T ) measurements were performed
	in several applied magnetic fields using PPMS down to
	2 K.
	We have carried out both non-spin and spin-polarized
	electronic structure calculations using density functional
	2

well consistent with Ref. [34]. The position coordinates and occu- pancies of different atoms in the crystal structure are mentioned in the table

• , γ = 120 • are I. The crystal structure is constituted with VO 4 tetrahedra, YbO 6 octahedra, and K atoms. Fig.

1(a)

shows the crystal structure of the K 3 Yb(VO 4 ) 2 in which Yb atoms are placed in a wellseparated 2D triangular lattice with NN distance 5.85 Å. The inter-layer separation 7.58 Å is larger than the intralayer distance of Yb-Yb.The ratio between the interlayer and intra-layer distance (d inter /d intra )

is only 1.29. This ratio for the K 3 Yb(VO 4 ) 2 compound is smaller than the values of 1.7 for NaYbO 2 and 2.4 for YbMgGaO 4 . However, it is some- what similar to those of other 2D triangular lay- ered materials with the ratio of

  d inter /d intra 1.1 for NaBaYb(BO 3 ) 2 and d inter /d intra ~1.

43 for Rb 3 Yb(PO 4 ) 2 , which show the magnetic frustra- tion and QSL features [35, 36].

Table II :

 II The crystal electric filed gaps(∆CEF) of various Yb-based compounds

	2D triangular lattice YbO6 Environment θHT( K ) ∆CEF( K )
	NaYbS2	D 3d	-65	197 [28]
	NaYbSe2	D 3d	-60	183 [31]
	NaYbO2	D 3d	-100	404 [25]
	K3Yb(VO4)2	D 3d	-110	to be measured

Table IV :

 IV Energies of different magnetization axis with respect to the energy of c-axis magnetization (∆E). The corresponding spin moments µs, orbital moments µo, and total moments µtot (µs+µo) are tabulated.

		010 axis 110 axis 001 axis
	∆E (meV/cell)	-98.4	-98.4	0.00
	µs (µ B /Yb)	0.80	0.80	0.95
	µo (µ B /Yb)	1.93	1.93	1.89
	µtot (µ B /Yb)	2.73	2.73	2.84

VI. ACKNOWLEDGMENT

B. K. thanks DST INSPIRE faculty award scheme, and NFIG grant from IIT Tirupati. E. K. acknowledges financial support from the labex PALM for the QuantumPy-roMan project (ANR10-LABX-0039-PALM)


C N Abadie

Byron Byrne

Byron Byrne Cyclic

Cyclic loading response of monopile foundations in cohesionless soils

come

INTRODUCTION

The design of offshore piles has been developed over many years by the oil and gas industry. In the absence of guidelines for offshore wind turbine structures, designers use the practices for conventional fixed offshore platform, such as those published by the American Petroleum Institute API (2011). For lateral loading of piles these rely on the p-y approach, introduced by [START_REF] Reese | Analysis of laterally loaded piles in sand[END_REF] and [START_REF] O'neill | An evaluation of p-y relationships in sands[END_REF]. In this method, the soil-pile interaction is modelled by a series of independent springs, characterised by a non-linear empirical behaviour law between the lateral pile deflection and the soil resistance. This method was originally calibrated against full-size pile tests in sand, under lateral loading. The piles were long, slender and flexible and intended to replicate those used for oil and gas platform foundations. This empirical approach appears to be reasonably efficient since few platform pile foundation failures have been recorded.

However, offshore wind turbine piles have both a different geometry and loading conditions compared to oil and gas structure foundations. Thus, it is becoming apparent that the current p-y methodology may not be appropriate for the design of offshore wind turbine monopiles, without additional calibration [START_REF] Kallehave | Modification of the API p-y formulation of initial stiffness of sand[END_REF], and particularly for the effects of long-term cyclic lateral loading [START_REF] Long | Effects of cyclic lateral loads on piles in sand[END_REF].

This paper outlines current design guidance used for offshore wind turbine monopiles, and presents a new methodology being proposed to address the issue of cyclic lateral loading. Additional experimental results are presented to further validate the methodology.

CURRENT DESIGN METHODOLOGIES

Guidelines from the American Petroleum Institute

The current guidelines for the design of offshore wind turbine piles (for example DNV 2011) are based around the standards published for the design of oil and gas structures, such as those by the American Petroleum Institute (API 2011). The recommended relationship between the lateral pile deflection y and the soil resistance p is:

         u Ap y K 0 u tanh Ap p
where pu is the static ultimate lateral pile resistance, A is an empirical parameter that takes into account the loading type (static or cyclic) and K0 is the initial reaction modulus of the soil which depends on the angle of friction , the relative density Rd and the depth z.

For the case of cyclic loading, A is taken to be 0.9. However, the value of A stays the same irrespective Whilst there is some knowledge transfer from oil and gas design there are also a number of key differences, which means new design guidelines are needed. This paper outlines some of the key issues confronting the offshore wind turbine foundation designer and concentrates on the effect of cyclic loading. It presents experimental results from a series of 1g model tests, following on from the work of Leblanc et al. (2010a). The tests aim at further exploring a framework for calculating the long term accumulated rotation. The results confirm the phenomenological laws proposed by Leblanc et al. (2010a) for the accumulated rotation and the cyclic secant stiffness. The results also highlight that in addition to the relative density and load characteristics, the accumulated rotation and the secant stiffness appear to be dependent on the sand properties.

of the number of cycles, N. Many authors underline this gap in the method [START_REF] Little | Cyclic horizontal load tests on six piles in sands at Houston Ship Channel[END_REF][START_REF] Long | Effects of cyclic lateral loads on piles in sand[END_REF][START_REF] Verdure | Lateral cyclic loading on single piles in sand[END_REF] and propose to modify Equation 1 in order to integrate the number of cycles N into the model. One such methodology is the degradation stiffness model where a reduction factor is applied to the soil reaction modulus (e.g. [START_REF] Reese | Analysis of laterally loaded piles in sand[END_REF][START_REF] O'neill | An evaluation of p-y relationships in sands[END_REF][START_REF] Long | Effects of cyclic lateral loads on piles in sand[END_REF][START_REF] Verdure | Lateral cyclic loading on single piles in sand[END_REF]. A key problem in the development of these models is that the data are usually limited to small numbers of cycles (<500), and may therefore not be easily extrapolated. This method is not discussed further in this paper but the results presented could be recast within in this framework.

Framework of response for long-term cyclically loaded piles, Leblanc et al. (2010a)

More recently, a framework of response for the be- The test programme involved a laboratory scale monopile in dry Yellow Leighton Buzzard sand at two different relative densities (Rd = 4 % and Rd = 38 %), subjected to a wide range of load conditions (b and c). The results led to the identification of laws characterizing the evolution of the pile tilt as well as the soil stiffness with cycle number. Leblanc et al. (2010b) completed this framework by an analysis of the effect of random cyclic loading on the pile response, though this is not discussed here. Leblanc et al. (2010a) evaluated the displacement of the pile via the magnitude of rotation  = N max -0 max caused by cyclic loading. They related  to the rotation S that would occur in a static test when the applied load is equal to the maximum cyclic load. Their test results show that the accumulated ro-tation follows a power law relationship with the number of cycles, according to Equation 4.

Accumulated displacement over the cycles

     N T R T c c d b b S     ) ( ) , (
Here Tb and Tc are dimensionless empirical functions depending on the relative density Rd and the load characteristics, whilst  is an empirical evolution parameter equal to 0.31.

Evolution of the stiffness

The test results also highlight a logarithmic evolution of the cyclic secant stiffness with the number of cycles given by Equation 5.

) ln( ) ( ) ( ~N C k k k c c b b N       In this equation, ' ~2 / 5  a N N p D L k k  is the non- dimensional stiffness, b k ~and c k ~ are empirical func-
tions that depend on the load characteristics. C is a dimensionless empirical constant equal to 8.02.

Definition of the stiffness

The analysis of cyclic phenomena requires a careful definition of the term stiffness. This clarification is essential and is illustrated in Figure 1.

Absolute stiffness K

The absolute stiffness KN is the parameter that relates the soil state at cycle N to the initial state. It describes the progressive evolution of the soil-pile interaction and is defined as the ratio of applied load versus the absolute deflection of the pile.

Cyclic secant stiffness k

The cyclic secant stiffness kN characterizes the state of the soil-pile interaction after N cycles, and is de-Figure 1. Definition of the absolute (K i ) and cyclic secant (ki) stiffness scribed by Equation 5. Its evolution is critical since it drives the change in natural frequency of the structure with cycle number (Byrne 2011, Equation 6).

           N nN k h EI h m f 2 3 3 1 2 1 
Here, m is the mass and h the height of the centre of gravity of the superstructure. E and I are the young's modulus and second moment of area of the superstructure respectively. In this paper, the evolution of the secant stiffness is addressed, though the results could be interpreted to provide information about the absolute stiffness.

EXPERIMENTAL MODEL

Experimental equipment

The laboratory tests were carried out with a rig developed by [START_REF] Rovere | Cyclic loading test machine for caisson suction foundations[END_REF]. It is based on a suspended mass system that enforces a cyclic lateral load at an eccentricity e from the mud line (Figure 2).

The mass m3 balances the weight of the main beam and the motor (m1 = m2 = 0). The values of m1 and m2 directly set the value of the sinusoidal force H(t) = H0+Ha.sin(t) applied at the top of the pile. When the motor is running, it carries m1 along a circular path which makes the gravity centre of the main beam change sinusoidally. The wire connecting the main beam and the pile transmits loads to the top of the pile. The pulsation is enforced by the motor frequency (f = 0.106 Hz). This rig offers the possibility of applying thousands of repeatable cycles in a realistic time.

The values of the loading and pile dimensions have been chosen according to the non-dimensional framework introduced by Leblanc et al. (2010a). As an illustration, two non-dimensional parameters that result from this model and are used in this paper are given by Equations 7 and 8.

' ~   L p a   ' ~3  D L M M 
Where pa is the atmospheric pressure, ' the effective soil unit weight and where non-dimensional parameters are denoted by a tilde.

The size of the pile is scaled to 1/50 of a typical 2 MW offshore wind turbine monopile (Table 1), although it is geometrically similar to larger wind turbine piles. The dimensions mean that it is stiff, compared with the soil stiffness and it is made of copper. The dimensions of the sand container are 600 mm x 600 mm x 527 mm which allows for a clearance of at least 2.75 D between the pile and the container wall. This minimises the disturbing effect of the walls on the soil-pile interaction.

Test programme

The results presented in this paper are aimed at exploring the effect of the sand particle size on the framework introduced by Leblanc et al. (2010a).

For this purpose, tests were carried out with two different sands, Yellow Leighton Buzzard 14/25 (YLB) and RedHill 110 (RH110). Both are very uniform silica sand with an angular grain shape. The major difference between them is their particle size (Table 2). The sands were tested dry so that only the drained behaviour was investigated.

For both sands, the samples were prepared by carefully pouring the sand from a very low drop height with a scoop. This method enables to prepare The characteristics of the applied cyclic load were defined in terms of b and c. The properties of each test have been summarized in Table 3.

RESULTS

Static moment capacity

For each sand sample, a static test was carried out to determine the non-dimensional ultimate capacity of the pile R M ~. This parameter is used as a reference for the choice of the load magnitude for the cyclic tests (Equation 2). The results of the static tests are given in Figure 3.

The failure limit of a laterally loaded pile is not trivial since no clear failure behaviour appears on the curve (Figure 3). The guidelines (DNV, 2011) advise to determine the value of R M ~ via, first, a theoretical calculation of the lateral pile resistance and, second, by an evaluation of the lateral displacement at the pile head. Since the pile is very stiff, the theoretical lateral pile resistance is too high to be chosen as a limit. It would indeed lead to a pile-head displacement of over 50 % of the diameter of the pile. The failure limit of the pile-head deflection has therefore been fixed to 5 % of the diameter of the pile, which is equivalent to a non-dimensional shows that Tb depends on the relative density and the particle size. Though, more tests would be required to fit a mathematical expression for Tb. Figure 4.d shows that Tc is rather independent of the sample properties and solely varies with c.

Evolution of the cyclic secant stiffness

Limitations

The framework presented in this paper relies on the assumption that the results obtained for the initial few thousands of cycles can be extrapolated to millions of cycles. Moreover, the completed work only accounts for unidirectional loading in a single layer of dry sand. Further work would involve the investigation of the response of the pile in more realistic conditions. Data from full-scale monitored monopiles would also be useful for the validation of the model.

CONCLUSION

A series of 1g tests have been conducted in order to supplement Leblanc et al.'s (2010a) research as well as the current methodologies used for the design of piles subjected to lateral cyclic loading. The tests involved 1,000 to 24,000 cycles in dry sand. The nondimensional framework developed by Leblanc et al. (2010a) has been used in order to properly scale the laboratory model. The test results show that the accumulated rotation of the pile is well fitted by the exponential law proposed by Leblanc et al. The results also confirm the logarithmical law proposed for the evolution of the cyclic secant stiffness. They demonstrate that both the accumulated rotation and the secant stiffness depend on the sand particle size.

This framework constitutes a good approach for improving the current guidelines concerning the design of offshore wind turbine piled foundations. It provides new data concerning the response of the pile to large number of cycles in a range of site conditions.

  haviour of monopile foundations subjected to large numbers of cycles (over 8000) has been introduced by Leblanc et al. This work was based on a comprehensive laboratory testing program.The loading is characterized in terms of two parameters: the magnitude b and amplitude c (Equations 2 and 3).

where

  Mmin and Mmax is the minimum and maximum moment over a load cycle and MR is the static moment capacity of the pile. b ranges between 0 and 1 when c∈[-1,1]. c = 1 for a static test, 0 for a oneway loading test and -1 for a two way loading test.

Figure 2 .

 2 Figure 2. Schematic of the experimental rig

  Figure 4 illustrates the influence of the sand particle size on the framework introduced by Leblanc et al.. On graphs 4.a and 4.b, the dotted lines have been obtained using Equation 4 with  = 0.31. They show a good fit to the results and prove that the value of  = 0.31 is well-adapted for the results obtained with RH110. Therefore, the value of  appears independent of the particle size, the load characteristics and the relative density. Figures 4.c and 4.d display the evolution of Tb and Tc with the load characteristics, relative density and particle size. Given that Tc = 1 when c = 0, Tb and Tc can be analysed separately by carrying out one-way loading tests, and then two-way loading tests for the same value of b.Figure 4.c clearly

  Figure 4.c clearly

Figure 5

 5 Figure 5 shows the influence of the sand particle size on the cyclic secant stiffness. The dotted lines on Figures 5.a, b have been fitted using Equation 5 with C = 8.02. The trend lines show a good fit to the test results and highlight that the phenomenological law proposed by Leblanc et al. (2010a) appears to describe the evolution of the secant stiffness. In addition, the value C = 8.02 does not show to depend on the sand particle size. Figures 5.c and 5.d display the results obtained for the functions b k ~ and c k ~. These graphs show that the function b k ~ depends on the sand particle size when the function c k ~ is independent of this parameter. Therefore, the empirical law proposed by Leblanc et al. (2010a) for c k ~ is appropriate but com-

Figure 4 .

 4 Figure 4. Measured accumulated rotation as a function of N, particle size and load characteristics -Results of the one-way (a) and two -way (b) loading tests wit h RH110 (dotted lines obtained from Equation4) -Functions relating Tb (c) and Tc (d) to the particle size

  Cyclic loading response of monopile foundations in cohesionless soils C.N. Abadie & B.W. Byrne University of Oxford, United-Kingdom ABSTRACT: Most offshore wind farms around Europe are being constructed with monopile foundations.

Table 1 .

 1 Properties of the copper pile

	Property	value
	Pile diameter (mm)	80
	Thickness (mm)	2.0
	Penetration depth (mm)	360
	Load eccentricity (mm)	430
	Table 2. Properties of YLB (Schnaid 1990) and RH110 (Vil-
	lalobos et al. 2005, Schupp 2009)
	Property	YLB RH110
	Mean particle size D50 (mm)	0.80	0.137
	Specific Gravity GS	2.65	2.63
	Minimum dry unit weight min (kN/m 3 )	14.65	12.64
	Maximum dry unit weight max (kN/m 3 )	17.58	15.72
	Critical angle of friction cr (degrees)	34.3	36
	Coefficient of uniformity Cu	1.35	1.82
	Curvature coefficient Cd	0.91	1.05

Table 3 .

 3 Test Programme

	No.	Type of test	Sample	d50 (mm)	Rd (%)	b	c	N
	1	Static	YLB	0.80	4	-	-	-
	2	Static	RH110	0.137	18	-	-	-
	3	Cyclic, 1-way	YLB	0.80	4	0.29	0	13400
	4	Cyclic, 1-way	YLB	0.80	4	0.37	0	15000
	5	Cyclic, 1-way	YLB	0.80	4	0.42	0	7400
	6	Cyclic, 1-way	RH110	0.137	18	0.37	0	1000
	7	Cyclic, 1-way	RH110	0.137	18	0.38	0	24100
	8	Cyclic, 1-way	RH110	0.137	18	0.42	0	1000
	9	Cyclic, 2-way	RH110	0.137	18	0.37	-0.45	1000
	10	Cyclic, 2-way	RH110	0.137	18	0.37	-0.95	12000

b k ~ in details.


C N Abadie

Byron Byrne

Sarah Levy-Paing

Sarah Levy

Paing Model

Model pile response to multi-amplitude cyclic lateral loading in cohesionless soils

INTRODUCTION

Offshore wind turbines are being installed in ever greater numbers around the UK and Europe. The most common foundation design is the large diameter monopile; a single pile upon which the turbine is located. These piles are substantially larger than piles used for oil and gas applications, with diameters up to 10 m being considered for future designs. Consequently there are concerns about whether current design approaches imported from oil and gas design, particularly for lateral loading, are robust. In addition there is limited guidance for assessing the effects of cyclic loading on pile response, both during storm loading and also over the lifetime of the structure.

To respond to these concerns a number of recent research programs have focused on cyclic loading of piles, principally through laboratory model testing. The aim of these testing programs has been to explore pile response under constant amplitude cyclic loading, and in particular the evolution of the foundation response (e.g. rotation, stiffness) with the number of cycles (e.g. Leblanc et al., 2010a;[START_REF] Peralta | Investigations on the Behavior of Large Diameter Piles under Long-Term Lateral Cyclic Loading in Cohesionless Soil[END_REF][START_REF] Klinkvort | Centrifuge modelling of drained lateral pile-soil response[END_REF]Cuéllar, 2011). It is recognised that the loading on the foundation, caused by the wind and wave environment, is likely to comprise a range of amplitudes rather than a single, or constant, amplitude. Therefore further experimental work is needed to determine the equivalence be-tween multi-amplitude and constant amplitude cyclic loading.

This paper presents experimental results, from model scaled pile tests in sand that explores multiamplitude loading. The experimental equipment is described along with the framework for interpreting both constant amplitude and multi-amplitude test results. The results from a number of constant amplitude cyclic loading tests along with two multiamplitude tests are presented. They are interpreted using currently available methods, and provide a basis against which future work can be developed.

LATERAL LOADING MODEL PILE TESTS

Background and motivation

Recent research on pile response to cyclic lateral loading (e.g. Leblanc et al. 2010a;[START_REF] Peralta | Investigations on the Behavior of Large Diameter Piles under Long-Term Lateral Cyclic Loading in Cohesionless Soil[END_REF][START_REF] Klinkvort | Centrifuge modelling of drained lateral pile-soil response[END_REF]Cuéllar, 2011;[START_REF] Abadie | Cyclic loading response of monopile foundations in cohesionless soils[END_REF] demonstrates that constant amplitude cyclic loading can cause significant increases in pile deflection and rotation over time. This effect is typically described as accumulated displacement with cycle number, and experimental data from such experiments have been fitted using both power and logarithmic laws by a range of Authors.

For example Leblanc et al. (2010a) show that the evolution of the accumulated rotation  for constant amplitude cyclic loading can be described by

S. Levy-Paing

Département THEMIS, EDF R&D ABSTRACT: Monopile foundations for offshore wind turbines are subjected to many cycles of loading during their lifetime. This loading consists of a range of amplitudes, applied in various sequences, of many different cycle numbers. Calculation of the accumulated rotation experienced by the monopile as a result of this cyclic loading, and whether this exceeds allowable limits, is an important part of the design process. This paper provides an overview of recent research exploring laterally loaded pile response relevant to the design of offshore wind turbine monopiles. Experimental equipment for carrying out cyclic lateral loading tests is introduced, along with considerations of scaling for model testing. Results from a series of small scale model tests covering realistic multi-amplitude testing are then presented, providing new insight into the behaviour of rigid piles subjected to cyclic loading. The results are interpreted using a linear superposition method, as typically used for structural fatigue calculations, and this shows a good fit to the experimental results. cycle number for both tests. An interesting outcome of test MA2 is that the increase in rotation due to cyclic load amplitude C0 reduces to approximately zero following the fourth peak load event ( b =0.46, corresponding to a load magnitude of the SLS). Similar tests with larger cycle number would be helpful to assess the above statement for larger cycle numbers (10 7 ), though it would require a higher cyclic frequency for tests to run in a reasonable time frame.

× ×C1 -100×C -100×C2 -1 = 0.27) -1 × C0 -1 × ( b = C0 -1 × ( b = 1 × ( b =0.
(N x  b  - × ( b = 0.29) × ( b = 0.47) × ( b = 0.67) C2 -1×C3 - 1×C3 -100× × ( b = 0.31) - = 0.42) -500 = 0.51) -500 ) -500×C0 -
The two tests indicate that the pile displacement response, following a change in load, depends on whether there is an increase in load or decrease. Figure 6 demonstrates this by plotting the differences in pile head rotation against differences in load, when a change in load amplitude occurs. Decreases in load magnitude are displayed on the left, and increases on the right. It also shows that the order in which loads occur matters, by highlighting a sharper increase in pile rotation during the first storm events (top right ellipse) compared with the subsequent events (bottom right ellipse). The graph shows that there is significant non-linearity involved, corresponding to substantial plastic deformations occurring.

METHOD FOR PREDICTION OF PILE ACCUMULATED ROTATION

Linear superposition model description

The investigation of the relationship between spectrum loading and fatigue lifetime is important to accurately predict the evolution of pile deflection through random cyclic loading. First, the random loading history is decomposed into an equivalent set of uniform load reversals. This is a common procedure in fatigue life assessment and can be performed with extended rain-flow counting [START_REF] Rychlik | A new definition of the rainflow cycle counting method[END_REF].

The decomposition of the load series enables a dam-age rule to be applied for predicting the fatigue lifetime. A common method is the linear cumulative damage rule, initially proposed by [START_REF] Palmgren | Die lebensdauer von kugellagern (life length of roller bearings, in german)[END_REF], and then popularised by [START_REF] Miner | Cumulative damage in fatigue[END_REF]. This concept was applied to laterally loaded piles by [START_REF] Lin | Permanent strains of piles in sand due to cyclic lateral loads[END_REF], following the work of [START_REF] Stewart | Permanent strains from cyclic variable amplitude loadings[END_REF]. They demonstrated a good fit to their experimental results; however, their experiments only involved 50 cycles maximum. Extending this, [START_REF] Peralta | Investigations on the Behavior of Large Diameter Piles under Long-Term Lateral Cyclic Loading in Cohesionless Soil[END_REF] performed tests of up to 45,000 cycles, finding that the method underestimated the final pile deflection. In contrast Leblanc et al. (2010b) found that such a method provided a good prediction to their test results. The superposition method adopted here is described in detail by Leblanc et al. (2010b). If there are two load sequences subscripted a and b, in the order N a then N b , then firstly the accumulated rotation  a caused by N a cycles of lateral load a is de-

. 0 ) ( ) ( a a s c b a N T T      (4)
This can be made equivalent to eq ab N cycles of load type b using the calculation:

31 . 0 ) (           b s c b a eq ab T T N   (5)
If N b cycles of load b are then applied to the pile, the overall accumulated rotation is given by:

31 . 0 ) ( ) ( eq ab b b s c b tot N N T T       (6)
The total pile rotation is then calculated as: In this Equation,  is the slope of the fitting law for descending load sequences. The results of such a calculation are given in Figure 5(a), showing a better prediction of the final rotation (error of 0.16%). Additional work is needed to develop a better understanding of the term  th . New methods must capture the non-linearity of pile response more accurately, recognising that the cyclic accumulated pile loaddisplacement response is unlikely to be calculated precisely using linear superposition methods.

  b a tot b , 0 , 0 , max        (

CONCLUSION

This paper presents a series of laboratory floor model tests exploring pile response under multi amplitude cyclic loading, representing storm loading on the pile. An important observation from this work is that the pile rotation appears to reach a limiting value following a series of maximum storm type loads. The paper shows that the pile response to multi amplitude cyclic loading involves significant nonlinearity, particularly when large plastic defor-mations occur. A linear superposition method, such as described by Leblanc et al. (2010b), is shown to provide a reasonable but conservative approximation to the final pile rotation. This is modified to predict the pile response more accurately.

To further develop these methods research must focus on understanding the non-linearity highlighted in section 3.2, for both loading and unloading. Targeted laboratory floor testing and field testing, along with theoretical development, will be needed. The resulting model would improve design guidance for cyclic loading of offshore wind turbine monopiles.

Fig

  Fig(a)

Figure 6 .

 6 Figure 6. Analysis of the change in rotation during the loading sequence changing phases Decreasing -Test MA1 first jump Decreasing -Test MA1 second jump Decreasing -Test MA2 Fitting curve Increasing -Test MA1 first jump Increasing -Test MA1 second jump Increasing -Test MA2 Fitting Curve

  Modification based on experimental resultsTests C1 to C3 make it possible to deduce the corresponding values of T b , T c and  s for each load type applied in the multi-amplitude tests. The above linear superposition method is applied to test MA1 with the results shown in Figure5(a). The model broadly captures the results, slightly over-predicting the final pile rotation but with an acceptable error of 3.7%. We propose modifying the calculation, taking account of observations from Figure6for decreases in load, by changing the term max 0,a ,  0,b } in Equation (7) to:

ACKNOWLEDGEMENTS

The first author thanks the generous support from EDF R&D and EDF Energies Nouvelles.


K Boya

K Nam

K Kargeti

A Jain

R Kumar

S K Panda

S M Yusuf

P L Paulose

U K Voma

Edwin Kermarrec

Kee Hoon Kim
email: khkim@phya.snu.ac.kr

B Koteswararao
email: koteswararao@iittp.ac.in

Signatures of spin-liquid state in a

I. INTRODUCTION

Geometrically frustrated antiferromagnetic (GFAFM) systems have attracted considerable attention in condensed matter physics due to their exotic magnetic properties, such as the spinliquid state. In those GFAFM systems, spins are put into a bizarre situation that doesn't allow a single conventional antiferromagnetic (AFM) ground state because they fail to satisfy all the AFM interactions simultaneously. Moreover, it has been known that the materials with a small spin value of S = ½ bear strong zero-point quantum fluctuation even at T = 0 K [1, 2]. Therefore, the spin lattices with geometrically frustrated interaction, with low dimensions and low spin The Ir 4+ (5d 5 ) bears the magnetic moment of Jeff = ½ due to the strong spin-orbit coupling.

Initially, magnetic data analysis shows no long-range order (LRO), and the heat capacity follows T 2 behavior, indicating a formation of a spin-liquid phase with gapless excitations [8, 9]. The recent nuclear magnetic resonance (NMR) and neutron scattering studies reveal slowing down of spin fluctuations and a spin frozen state in the system [17, 18]. On the other hand, the system PbCuTe2O6 has an S = ½ (Cu 2+ ), 3D corner-shared triangular lattice [11]. Bulk magnetic susceptibility suggests that it is a paramagnetic system with a Curie-Weiss temperature (θCW) of -22 K. Neither magnetic ordering nor spin freezing is seen down to 20 mK. Electronic structure calculations suggest that it has a 3D hyperkagome network with additional frustrated couplings.

Low-temperature NMR, muon-spin rotation, and inelastic neutron scattering studies have revealed that it is a spin-liquid system with gapless, fractionalized spinon excitations [12, 13].

Recently, a ferroelectric ordering with quantum critical behavior has been observed at 1 K due to the strong lattice distortion [14].

In search of 3D QSLs, recently, interest has been raised in another type of 3D frustrated system called the trillium lattice, which constitutes a 3D network of corner-shared triangles with z = 6. The trillium lattice is usually found in the B20 structures. A few example materials for B20 structures are MnSi,FeGe,. Due to the lack of inversion symmetry, these trillium lattice systems exhibit skyrmion-type ordering [24]. A few more frustrated trillium lattice network systems include CeIrSi [25], demonstrating a spin-ice ground state with Ising interactions, and EuPtX (Ge, Si) [26], showing strong magnetic frustration with a finite LRO at 4.1 K. Recently, the field-induced quantum spin liquid behavior has been found in an S = 1 3D frustrated system K2Ni2(SO4)3. Magnetic susceptibility data follow the Curie-Weiss behavior with θCW = -18 K, suggesting the AFM interactions. Heat capacity data (Cp) show the 2 nd order transition at 1.14 K and the 1 st order transition at 0.74 K. The magnetic field H suppresses these transitions, and Cp data, following T 2 behavior, suggest the field-induced QSL ground state with gapless spinon excitations. It was claimed that K2Ni2(SO4)3 is an inter-connected trillium lattice of two Ni 2+ ions; however, the non-trillium coupling is indeed more dominant than trillium interactions, according to the reported electronic structure calculations [16]. The presence of dominant non-trillium couplings could render the system exhibit LRO at finite temperatures.

To explore exotic properties in the class of 3D complicated network systems, we have investigated a compound KSrFe2(PO4)3 that has a similar structure to K2Ni2(SO4)3 but has a spin S = 5/2. Quite interestingly, no long-range order (LRO) is found down to 0.19 K, although it has relatively strong AFM correlation with θCW = -70 K. Magnetic heat capacity follows the power law T α (α ~ 2.3) behavior, suggesting that our titled compound is a possible candidate of forming a spin-liquid state with gapless excitations in the family of 3D frustrated spin network with S=5/2.

II. METHODS

The polycrystalline samples of KSrFe2(PO4)3 were synthesized using the conventional solid-state heating method at 1000 °C [27]. The non-magnetic compound KSrSc2(PO4)3 with the same structure as KSrFe2(PO4)3 was prepared at 1100°C [28]. The polycrystalline powder samples were used to measure X-ray diffraction (XRD), magnetization, and heat capacity measurements. The magnetic and heat capacity measurements on the polycrystalline samples were carried out using the Physical Properties Measurement System (PPMS, Quantum Design).

The low-temperature heat capacity was measured on a tiny pellet using a CF-150 dilution fridge (Leiden cryogenics). The Rietveld refinement of the XRD profile has been done using FULLPROF SUITE software (see Fig. 1(a)). The extracted lattice parameters are a = 9.780 Å and χ 2 = 2.30, Rp = 20.2 %, Rwp = 19.2 %, and Rexp = 12.63 %. As reported previously [27], there is nearly 20 % of site-exchange between K and Sr atoms.

III. STRUCTURAL DESCRIPTION

The compound KSrFe2(PO4)3 holds a cubic crystal system with space group P213 (space group number 198). The 3D framework is built by Fe1O6, Fe2O6 octahedral units, and PO4 tetrahedral units, K, and Sr atoms, as shown in Fig. 1(b). The Fe1O6 octahedral unit is more distorted than the Fe2O6 octahedral unit. The magnetic interactions are exchanged among Fe1 and Fe2 atoms through PO4 units. The 1 st , 2 nd , and 3 rd NN couplings (J1, J2, and J3) are between Fe1 and Fe2 atoms. The combination of 1 st NN and 2 nd NN couplings forms a J1-J2 anisotropic diamond lattice, known to be a non-frustrated system. After adding the 3 rd NN coupling (J3), the system still remains non-frustrated, as shown in Fig. 2(a). On the other hand, the 4 th and 5 th NN couplings J4 and J5 are through Fe1-Fe1 and Fe2-Fe2 bonds, respectively (see Table I). The couplings J4 and J5 alone built up two independent trillium lattices, respectively (see Fig. 2(b)), which are highly frustrated 3D triangular networks. Overall, the compound KSrFe2(PO4)3 has the combination of all exchange couplings J1, J2, J3, J4, and J5 and stands for a complicated 3D frustrated network system, as shown in Fig. 2(c). The super-super-exchange interaction between Fe 3+ ions is through the O-P-O path. The details of various magnetic couplings are listed in Table I. The magnetic exchange energies were extracted from the DFT calculations, which will be discussed in the later sections.

A. Magnetic measurements

Temperature-dependent magnetic susceptibility χ(T) measurements were carried out on the polycrystalline samples of KSrFe2(PO4)3 down to 2 K H =10 kOe, as shown in Fig. 3 as observed in most of the geometrically frustrated magnets [36,37]. The Cm data are compared with the T 2 and T 3 power-law fits. The Cm(T) data fall between T 2 and T 3 behavior, meaning that the power is between 2 and 3. To understand further, we have fitted the Cm data with the power-law, T α , over the temperature range from 0.19 K to 1.7 K, resulting in power-law behavior with a coefficient of α ≈ 2.3 (see Fig. 4(c)). In general, the Cm(T) data follow the T 3 behavior for conventional AFM LRO systems due to 3D magnon excitations [38], which thus rule out the presence of AFM LRO in our system. Conversely, the sublinear (T

C. Neutron diffraction measurements

To confirm the absence of magnetic LRO, neutron diffraction (ND) experiments were performed on the polycrystalline samples of KSrFe2(PO4)3 down to 2 K. The measurements were carried out using the neutron powder diffractometer PD-I (λ = 1.094 Å) at various temperatures from 6 K to 300 K. The ND data 2, and 20 K were obtained at another neutron powder diffractometer PD-II (λ = 1.244300 Å) with three linear position-sensitive detectors at the Dhruva reactor, Bhabha Atomic Research Center, India. Rietveld refinement on ND data was performed using the FULLPROF software package using the inputs from the structural parameters obtained from the powder XRD data. Neither the additional magnetic Bragg peaks nor enhancement in the intensity of the fundamental nuclear Bragg peaks from 300 K to 2 K were found (see Fig. 5).

The measured ND pattern could be fitted by considering only the nuclear phase. In addition, no significant difference was found upon high-temperature ND data at 300 K being subtracted from that at 6 K. This suggests that the broad maximum in the heat capacity may stem from shortrange correlations. Similarly, no significant difference is seen in the subtracted ND data at 20 K from that at 2 K, ruling out the presence of magnetic LRO down to 2 K.

D. Electronic structure calculations

The electronic structure calculations were performed using a full-potential linearized muffintin orbital method [44, 45] as implemented in the RSPt code [46]. For the exchange-correlation functional, we have employed a generalized gradient approximation + Hubbard U (GGA+U)

approach. The Brillouin zone integration is done using the thermal smearing method with an 8 x 8 x 8 k-mesh. For charge density and potential angular decomposition inside the muffin tin sphere, maximum angular momentum is considered to be lmax = 8. To account for electronic correlation within rotationally invariant GGA+U approach [47], we have considered Coulomb

Interactions U = 4.0 eV [48] and Hund coupling JH = 0.8 eV for Fe-3d states.

To begin with, we have checked the relative stability of a few possible magnetic states within the GGA+U approach and found that an AFM state where the NN Fe spins are oppositely aligned comes out to be the lowest in energy. The computed total and partial density of states for Fe-d and O-p states are displayed in Fig. 6. The total density of states (Fig. 6(a)) clearly shows that the ground state is insulating with a gap of 2.2 eV. From the projected density of states (Fig. 6(b) and (c)), we find the majority Fe-t2g and eg, states are completely occupied while corresponding minority states are completely empty. According to Hund's rule, such distribution of Fe-states is expected for Fe 3+ (d 5 ). The gap primarily arises between the majority and minority d-states which are exchange split due to the formation of local moments and the correlation effect of the Fe-3d

subshell. The magnetic moment on the Fe-site is estimated to be 4.3 μB, which is smaller than the moment expected from the nominal d 5 configuration. This also implies strong Fe-d and O-p hybridization in this system, as evident from the spectral distribution of the O-p states (Fig. 6(c)), which appears in a broader energy range between -4 eV and Fermi energy. We will see below that such strong d-p hybridization will facilitate a super-super-exchange mechanism for the magnetic interactions between the Fe-spins, which are even quite far from each other.

Further, we have attempted to understand the nature of magnetism and employed the magnetic force theorem to extract the magnetic exchange energies [49-51]. The computed intersite exchange interactions are displayed in Table-I. Surprisingly, the 5 th NN coupling (J5) is dominant among all five exchange couplings, although the bond-distance of J5 is larger than other couplings, which forms the trillium lattice by the Fe2 atoms (see Fig. 2b). As per the structure, the 4 th NN coupling J4 also forms another trillium lattice by Fe1 atoms. The magnitude of J4 is slightly smaller than that of J5, which could be due to the significant distortion in Fe1O6 than that of Fe2O6 environments. In addition to these J4 and J5 couplings, there are strong and compatible AFM couplings J1, J2, and J3. All these spin interactions make a complicated 3D spin system.

IV. DISCUSSION

To understand the origin of the highly frustrated magnetism and spin liquid features in the complicated 3D spin network, one needs to look into the details of structural features and magnetic coupling strengths. It is clear from Fig. 2(a) that the system is non-frustrated upon the magnetic couplings of only J1, J2, and J3 being considered. Hence, we need to look at other couplings, J4 and J5, to trace the origin of magnetic frustration. Fig. 2(b) clearly shows that J4

and J5 interactions built two uniform equilateral triangular networks (3D perfect trillium lattices).

From the DFT electronic structure calculations, it is noticed that J5 is a dominant coupling, which is the starting point to explain the geometrical frustration in this compound as the J1-J2 anisotropic diamond lattice is not frustrated even with the inclusion of the J3 coupling. However, the isolated trillium lattice model is insufficient to describe the geometrical frustration of this compound. There exist two trillium lattices as formed by J4 and J5 couplings, respectively. The connections between two trillium lattices are made by the three interactions of J1, J2, and J3.

These are AFM, of whose strengths are almost comparable to those of J4 and J5. Importantly, these AFM inter-trillium connections again produce the triangular-type of interactions between J4 and J5 triangles (see Fig. 2(c)). These connections make the J4-J2 & J5-J3 anisotropic tetrahedral environments, which are another type of frustrating motifs (see Fig. 2(c)). Overall, the compound forms a more complicated 3D frustrated spin network than the trillium lattice.

The formation of a complex 3D frustrated network with competing and compatible magnetic interactions (J1, J2, J3, J4, and J5) is likely the primary source of high frustration and the related gapless spin liquid feature in the compound.

Since the crystal structure and space group of KSrFe2(PO4)2 are the same as those of K2Ni2(SO4)3, it will be worthwhile to compare the structural and magnetic features of the two compounds. There are a few differences noticed, leading to different magnetic ground states.

While the K2Ni2(SO4)3 shows magnetic LRO at zero magnetic fields, the titled compound KSrFe2(PO4)3 shows the spin-liquid feature. In the case of K2Ni2(SO4)3, one of the interactions (J2) is indeed ferromagnetic; the combination of AFM and FM couplings likely reduces the levels of magnetic frustration. In addition, one of the inter-trillium couplings (J3) in the compound K2Ni2(SO4)3 is nearly three times more dominant than the intra-trillium couplings [16]. Unlike K2Ni2(SO4)3, all the magnetic couplings in the KSrFe2(PO4)2 are AFM and comparable with each other, which may enhance the levels of magnetic frustration. The elevated levels of magnetic frustration in the compatible magnetic interactions could lead to the robust spin-disorder in KSrFe2(PO4)3. Moreover, the system has a site-disorder between K and Sr atoms and thus induces some randomness in the magnetic exchange couplings. The randomness could destabilize the LRO and help the system stay in a spin liquid ground state [56,57]. The experimental and theoretical results thus reveal that the titled compound is a suitable candidate for forming the spin-liquid state with gapless excitations.

V. CONCLUSION

We have investigated the thermodynamic properties and spin interactions by the electronic structure calculations in the compound KSrFe2(PO4)3, which forms a 3D frustrated lattice with S = 5/2 magnetic moments. The absence of a conventional magnetic LRO is confirmed down to 0.19 K, despite the relatively sizable Curie-Weiss temperature θCW = -70 K. The power-law (T 2.3 ) behavior of the magnetic heat capacity Cm(T) at lower temperatures, followed by a broad maximum at 9 K, supports the formation of a gapless ground state. The appearance of geometrical spin frustration is understood as due to the complicated frustrated AFM exchange couplings, rendering the system remain in an unusual spin-liquid state, at least down to 0.19 K.

Our results suggest that the compound KSrFe2(PO4)3 is an appealing candidate for forming a gapless spin-liquid state among the family of S = 5/2, 3D spin networks.

  values, have been actively investigated in the search for exotic spin liquid states. A few ideal examples for the two-dimensional (2D) edge-shared triangular lattice and 2D corner-shared triangular (kagome) lattice are κ-(BEDT-TTF)2Cu2(CN)3, and ZnCu3(OH)6Cl2, respectively [3-7]. These materials are promising examples of the quantum spin-liquid (QSL) state with S = 1/2 moments and exhibit strong quantum fluctuations. As the quantum fluctuations are known to be dominant in the low-dimensional materials and diminish in higher dimensions, it has been challenging to discover a QSL candidate with no static spin ordering in three-dimensional (3D) systems, especially with large spins like S = 5/2. Recently, emergent magnetic behavior has been observed in 3D geometrically frustrated magnets due to the interplay between competing spin interactions and quantum fluctuations. A few examples of 3D GFAFM systems can be found in the corned-shared triangle (hyperkagome and trillium) lattices and corned-shared tetrahedral (pyrochlore) lattices; the 3D corner-shared triangular networks include Na4Ir3O8 [8-10], PbCuTe2O6 [11-14], Yb3Ga5O12 [15], and K2Ni2(SiO4)3 [16]. Na4Ir3O8 has a 3D hyperkagome structure with Ir 4+ ions forming a 3D cornershared triangular network with z = 4, where z is the number of nearest neighbor (NN) atoms [8].

  (a). No conventional magnetic LRO is observed in the χ(T) data down to 2 K. A minor splitting between zero-field-cooled (ZFC), and field-cooled (FC) susceptibility data is noticed at 3.5 K in the low fields, as shown in the inset ofFig. 3(a). However, this splitting disappears at a small H =5 kOe.It could be due to the site exchange between K and Sr atoms, as also noticed in many geometrically frustrated magnets like Na4Ir3O8 [8] and LiGaCr4S8 [32] due to the disorder. The inverse χ(T) data follow the Curie-Weiss behavior at high temperatures, as shown in Fig.3(b).The data fit to the expression 1/χ =1/[ χdia + χvv + C/(T-θCW)]. The extracted θCW is about -70 K, suggesting the dominant AFM couplings. The calculated diamagnetic susceptibility (χdia) from the individual ions of K + , Sr 2+ , Fe 3+ , P 5+ , O 2-is -9.75 ⨯ 10 -5 cm 3 /mol [33]. The yielded Van-Vleck susceptibility χvv from the fit is 1 ⨯ 10 -5 cm 3 /mol-Fe. The calculated effective magnetic moment from the Curie Constant (C) = 4.43 cm 3 K/mol-Fe is about μeff = 5.95 μB, which is consistent with the expected value for S = 5/2 of the Fe 3+ . The χ(T) data shows field-independent behavior down to 2 K in the magnetic fields up to 50 kOe (see Fig.3(c)). Interestingly, the loglog scale of χ(T) data shows nearly temperature-independent behavior at low temperatures, as seen in the gapless spin-liquid materials such as PbCuTe2O6, EtMe3Sb[Pd(dmit)2]2, κ-H3(Cat-EDT-TTF)2 [12, 34, 35]. The H-dependence of magnetization M(H) was also measured at 1.9 K by varying H up to 160 kOe, as shown in Fig.3(d). The magnetization did not saturate and reached only 1.42 μB even at 160 kOe, indicating the presence of AFM correlations in this system.The absence of hysteresis indicates the antiferromagnetic nature of the samples.B. Heat capacity measurementsTemperature-dependent heat capacity Cp(T) measurements were carried out down to 0.19 K at H= 0 and 30 kOe on KSrFe2(PO4)3 samples. The Cp(T) for the nonmagnetic analog KSrSc2(PO4)3 is also measured down to 2 K at zero fields, as shown in Fig.4(a). The Cp(T) data of KSrSc2(PO4)3 is well-matched with its Cp(T) data of KSrSc2(PO4)3 at high temperatures. It deviates at low temperatures due to the presence of magnetic contribution of KSrFe2(PO4)3.Following the analysis on KSrSc2(PO4)3 [16], the Cp(T) data fit the polynomial AT 3 +BT 5 in the range from 1.8 K to 7 K. It is extrapolated down to 0.19 K. The magnetic heat capacity Cm(T) is obtained after subtracting the Cp(T) of KSrSc2(PO4)3 from that of KSrFe2(PO4)3. The Cm(T) data does not show any sharp peak down to 0.19 K, suggesting the absence of magnetic ordering. In particular, Cm(T) does not have an anomaly at 3.5 K, where ZFC-FC magnetic susceptibility shows the bifurcation, which rules out the possibility of magnetic LRO. However, the possibility of spin-glass behavior and/or its percentage of the spin frozen state is to be further investigated through the detailed Muon spin rotation measurements. The frustration parameter (f), the ratio of Curie-Weiss temperature (θCW) to Neel temperature (TN), is greater than 370. Cm(T) data show a broad maximum at 9 K, indicating the presence of short-range spin correlations [seeFig. 4(b)]

ACKNOWLEDGMENTSB

  Figures and captions:

Figure 1 :

 1 Figure 1: (Colour online) (a) The refinement of X-ray diffraction of the titled compound KSrFe2(PO4)3. (b) Crystal structure of the KSrFe2(PO4)3 in a unit cell with Fe1O6, Fe2O6 octahedral and PO4 tetrahedral units.

Figure 2 :

 2 Figure 2: (Colour online) (a) The non-frustrated 3D network formed by J1, J2, and J3 couplings between Fe1 (magenta) and Fe2 (blue) atoms. (b) The formation of two trillium networks by J4 and J5, respectively. (c) The complicated 3D network is formed by J1, J2, J3, J4, and J5 couplings.

Figure 3 :

 3 Figure 3: (Colour online) (a) Temperature dependence of magnetic susceptibility χ(T) at 10 kOe of KSrFe2(PO4)3. The inset (a) shows the ZFC and FC χ(T) data at various magnetic fields H=500 Oe, 5 kOe, and 10 kOe. (b) Inverse magnetic susceptibility χ -1 (T) with Curie-Weiss fit. (c) χ(T) data measured in different fields 10, 20, 30, and 50 kOe. (d) Isotherm magnetization as a function of H up to 160 kOe at T = 1.9 K.

Figure 4 :

 4 Figure 4: (Colour online) (a) Variation of heat capacity Cp of KSrFe2(PO4)3 and KSrSc2(PO4) measured at various magnetic fields. (b) Temperature dependence of magnetic heat capacity Cm and the normalized magnetic entropy Sm . The magnetic entropy reaches nearly a maximum Run(6) (R: universal gas constant) expected value for the S = 5/2 system. (c) The log-log scale Cm versus T data with different power-law fits.

Figure 6 :

 6 Figure 6: (Colour online) Total and partial density of states in the magnetic ground state using GGA+U calculations (U = 4 eV).

Table I :

 I Details of magnetic exchange couplings in KSrFe2(PO4)3.

	Magnetic	Magneti	Bond	Exchange Path	Exchange	Relative
	couplings	c path	distance		energy (K)	energy
			(Å)			(Ji/J5)
	J1	Fe1-Fe2	4.55	Fe1-O4-P-O1-Fe2	-4.87	0.91
	J2	Fe1-Fe2	4.92	Fe1-O2-P-O3-Fe2	-2.67	0.5
	J3	Fe1-Fe2	5.97	Fe1-O2-P-O4-Fe2	-4.99	0.93
	J4	Fe1-Fe1	6.03	Fe1-O2-P-O1-Fe1	-3.94	0.73
	J5	Fe2-Fe2	6.10	Fe2-O4-P-O3-Fe2	-5.34	1
	III. RESULTS

Table II :

 II Details of 3D GFAFM spin-liquid materials

							2/3 ) dependent behavior is
	expected for the 2D QSL with spinon Fermi surface [39]. As seen in Table II, Cm ∝ T 2 behavior
	is observed in many experimental 3D spin-liquid systems [16, 40-42]. The peculiar power-law
	behavior of Cm(T) could be due to the spin liquid nature of the system. Further, detailed
	theoretical and experimental studies are needed to understand the precise nature of excitations.
	Despite the application of H= 110 kOe at temperatures above 2 K, no field dependence was
	observed in this compound, suggesting that the low-energy excitations are robust in this spin-
	liquid candidate [8, 43].
	Compound	Spin		Lattice	θCW	T max in	α ( in Cm	Ground	Ref
					(K)	Cm(K)	~ T α )	state
	Na4Ir3O8	Jeff =	3D		-650 30	2	Gapless	[8-10]
		½	Hyperkagome				QSL
	PbCuTe2O6	S = ½ 3D		-22	1.15	1.9	Gapless	[11-14]
			Hyperkagome				QSL
			network with
			additional
			interactions
	K2Ni2(SO4)3	S = 1	3D	coupled	-18	5	2	Field-	[16]
			trillium				induced
								QSL
	NaCaNi2F7	S = 1	3D pyrochlore -129 18	2.2	SL	[41, 58]
	LiGa0.2In0.8Cr4O8 S = 3/2 3D Breathing	-386 70	1.9	QSL	[59]
			Pyrochlore
	KSrFe2(PO4)3	S = 5/2 3D frustrated	-70	9	2.3	SL	This
			network					work