Development of short and long-range magnetic order in the double perovskite based frustrated triangular lattice antiferromagnet Ba2MnTeO6

Oxide double perovskites wherein octahedra formed by both 3d elements and sp-based heavy elements give rise to unconventional magnetic ordering and correlated quantum phenomena crucial for futuristic applications. Here, by carrying out experimental and first principles investigations, we present the electronic structure and magnetic phases of Ba2MnTeO6, where Mn^2+ ions with S = 5/2 spins constitute a perfect triangular lattice. The magnetic susceptibility reveals a large Curie- Weiss temperature -152 K suggesting the presence of strong antiferromagnetic interactions between Mn^2+ moments in the spin lattice. A phase transition at 20 K is revealed by magnetic susceptibility and specific heat which is attributed to the presence of a sizeable inter-plane interactions. Below the transition temperature, the specific heat data show antiferromagnetic magnon excitations with a gap of 1.4 K. Furthermore, muon spin-relaxation reveals the presence of static internal fields in the ordered state and provides strong evidence of short-range spin correlations for T>TN. The DFT+U calculations and spin-dimer analysis infer that Heisenberg interactions govern the inter and intra-layer spin-frustrations in this perovskite. The inter and intra-layer exchange interactions are of comparable strengths (J1 = 4.6 K, J2 = 0.92 J1). However, a weak third nearest-neighbor ferromagnetic inter-layer interaction exists (J3=-0.04 J1) due to double-exchange interaction via the linear path Mn-O-Te-O-Mn. The combined effect of J2 and J3 interactions stabilizes a three dimensional long-range magnetic ordering in this frustrated magnet.

In this context, B site ordered double perovskites of general formula A 2 BB´O 6 , where A represents a divalent cations, B is a 3d transition metal ions and B´= Te 6+ , Mo 6+ or W 6+ offer an alternate route to realization of novel magnetism and spin dynamics as a result of intricate interplay between spin, lattice and charge degrees of freedom. It has been observed that many unconventional magnetic ground states are governed by planar structure of B-site ions. For example, Ba 2 CoTeO 6 is a unique case of B-site ordered double perovskite, where Co 2+ (S = 1/2) ions form two (triangular and honeycomb) subsystems. The spins on the triangular lattice behave as Heisenberg spins, while the spins on the honeycomb lattice show Ising like antiferromagnetic interactions [30,31]. Electron-spin resonance (ESR) and magnetization measurements show that applied magnetic field perpendicular to the easy-axis induces magnetization plateaus for both sub-lattices due to strong quantum effects of S = 1/2 spins [30][31][32]. Another interesting example is Sr 2 CuTeO 6 , a quasi-two dimensional Heisenberg antiferromagnet, where Cu 2+ (S = 1/2) ions form a planar square lattice and develop Néel order below 29 K [33]. Recently, magnetic susceptibility, specific heat and µSR studies on Sr 2 CuTe 1−x W x O 6 demonstrated a quantum disordered ground state for x = 0.5 [34]. This is a promising candidate to tune electron correlation by quenched disorder in the J 1 -J 2 Heisenberg model on a square lattice.
Recently, a new B site ordered double perovskite Ba 2 MnTeO 6 (henceforth BMTO), where Mn 2+ ions with spin S = 5/2 constitute a perfect spin lattice without anti-site disorder, has been reported [35][36][37]. While one of the study proposed a cubic space group (Fm3m), the other suggested a trigonal space group R3m for describing the structure of BMTO [35,36]. The same trigonal space group has been proposed for the BMTO in an earlier study [37]. The high-temperature magnetic susceptibility data follow the Curie-Weiss law with large Curie-Weiss temperature, which suggests the presence of strong antiferromagnetic interaction between Mn 2+ spins. An anomaly is observed in the magnetic susceptibility and specific heat data at T N = 20 K, which is an indication of a symmetry breaking phase transition in BMTO. However, a clear picture of the crystal structure and exchange interactions of BMTO is missing. Also, the presence of static internal fields, the development of the order parameter and spin dynamics above and below the Néel temperature of this novel antiferromagnet have not been yet explored. In this work, we report our results based on XRD, magnetization, specific heat, and muon spin relaxation (µSR) studies as well as density functional theory calculations in order to shed new insight into the crystal structure, magnetism, anisotropy and spin correlations in this novel frustrated triangular lattice antiferromagnet. We have found that the trigonal and cubic structures can both index the observed XRD peaks in BMTO, as also concluded previously [36]. All structural data thus indicate that the two structures are only marginally different, so that the exact structure likely has no significant effect on magnetism and spin dynamics. Indeed, it is suggested that the two structures are very close if one converts one space group to the other. As the trigonal space group offers an additional degree of freedom for the positions of Ba and O sites along the c-axis, this suggests that the trigonal space group may be advantageous over the cubic one as reported recently [32,36,38,39]. The Néel ordering at T N = 20 K is confirmed by local-probe µSR measurements revealing the appearance of static internal fields below T N . These measurements show that the whole sample enters a long-range magnetically ordered state below this temperature, while short-range correlations are observed all the way up to 35 K. Moreover, µSR reveals critical slowing-down of spin dynamics at T N and its persistence to the lowest temperatures, as well as tracks the development of the order parameter. Furthermore, we find that a broad maximum at 10 K in the magnetic specific heat data can be associated with the magnon gap due to magnetic anisotropy. Namely, the magnetic specific heat data below T N reveals magnon excitations with a gap of ∼ 1.4 K. The origin of magnetic ordering in BMTO is studied within the frame work of density functional theory (DFT) calculations for the trigonal space group. Our DFT results reveal that the Mn 2+ spins favor a dominant Heisenberg antiferromagnetic ordering consistent with the experimental results. Our calculations using the DFT +U formalism yield intra-layer exchange energy J 1 = 4.6 K and a comparable interlayer exchange coupling J 2 = 0.92 J 1 . In addition, a weaker ferromagnetic inter-layer interaction exists with third nearest neighbor (J 3 = − 0.04 J 1 ) due to doubleexchange interaction via the linear path Mn-O-Te-O-Mn.
Though the strength of this indirect interaction (J 3 ) is one order of magnitude smaller than the leading AFM interaction, the combined effect of J 2 and J 3 contributes towards stabilizing the long-range magnetic order in this frustrated magnet.

II. EXPERIMENTAL DETAILS
Polycrystalline samples of BMTO were prepared by a conventional solid state method. Prior to use, we preheated BaCO 3 (Alfa Aesar, 99.997 %), MnO 2 (Alfa Aesar, 99.996 %) and TeO 2 (Alfa Aesar, 99.9995 %) to remove any moisture. The appropriate stoichiometric mixtures were pelletized and sintered at 1200 • C for 30 hours with several intermittent grindings. The phase purity was confirmed by the Rietveld refinement of XRD taken employing a smartLAB Rigaku X-ray diffractometer with Cu Kα radiation (λ = 1.54Å). Magnetization measurements were carried out using a Quantum Design, SQUID VSM in the temperature range 5 K ≤ T ≤ 340 K under magnetic fields 0 T ≤ µ 0 H ≤ 7 T. Specific heat measurements were performed on a Quantum Design Physical Properties Measurement System (QD, PPMS) by thermal relaxation method, in the temperature range 2 K ≤ T ≤ 240 K. µSR measurements were performed using the GPS spectrometer at the Paul Scherrer Institute, Villigen, Switzerland, on a 1-gm powder sample in the temperature range 1.6 K ≤ T ≤ 50 K. The sample was put on a "fork" sample holder and the veto mode was employed, which ensured minimal background signal. The transverse muon-polarization was used in zero applied field (ZF) and in a weak transverse field (TF) of 5 mT.

A. XRD and structural details
To check the phase purity, we first measured the XRD pattern of polycrystalline BMTO samples. Fig. 1 (c) depicts the powder XRD pattern at room temperature. Rietveld refinement of XRD data using GSAS software [41] reveals that BMTO crystallizes in trigonal crystal structure with the space group R3m (No.166) and gives lattice parameters ( Table I) that are consistent with those previously reported [36,37]. Our analysis reveals the absence of any site disorder between constituent atoms in BMTO. In the BMTO crystal structure shown in Fig. 1 (a) Mn 2+ (3d 5 ; S = 5/2) and Te 6+ (4d 10 ; S = 0) ions form MnO 6 and TeO 6 octahedra with nearest-neighbor oxygen ions, respectively. The Mn-O bond length within the MnO 6 octahedra is 2.179Å and the Te-O bond length within the TeO 6 is 1.932Å. In the a-b plane, the nearestneighbor (5.81Å) Mn 2+ ions form equilateral triangular planes stacked along c axis (see Fig. 1 (b)). The consecutive Mn 2+ triangular planes are separated by nonmagnetic triangular planes of Te 6+ ( Fig. 1 (b)). The adjacent inter-layer Mn-Mn distance is 5.81Å. The nearestneighbor Mn 2+ ions of MnO 6 octahedra are connected via TeO 6 octahedra. The inter-planar Mn 2+ ions are connected through the linear path Mn (1) and Mn 2+ (2) refer to Mn 2+ ions in two adjacent planes. We found a similarity between the structure of BMTO with rare-earth based spin-liquid candidate YbMgGaO 4 , though the latter is composed of 4f magnetic ions [42]. YbMgGaO 4 crystallizes in the same space group R3m with lattice parameters a = b = 3.41Å and c = 25.14Å [42]. In YbMgGaO 4 , a single crystallographic site (3a) of Yb atoms and atomic coordinate (0, 0, 0) matches with coordinate of Mn 2+ in BMTO. Although both systems belong to the same crystal class, the spin orbit coupling plays an important role to host an exotic ground state in YbMgGaO 4 [43], whereas in BMTO the inherent physics of high-spin-state of the magnetic ions is expected to be different due to much smaller spin-orbit coupling and the presence of fi-nite inter-plane interactions. Fig. 2 (a) depicts the temperature dependence of the magnetic susceptibility (χ(T )) of BMTO in a magnetic field µ 0 H = 1 T. In order to estimate the effective magnetic moment µ ef f and the Curie-Weiss temperature (θ CW ), the inverse magnetic susceptibility, 1/χ(T ) was fitted (see right y-axis of Fig. 2 (a)) with the Curie-Weiss model

B. Magnetic susceptibility
where χ 0 is the temperature independent contribution due to core diamagnetism and van Vleck paramagnetism, C is the Curie-constant and θ CW is the Curie-Weiss temperature. The Curie-Weiss fitting in the hightemperature range 150 K ≤ T ≤ 340 K yields C = 4.45 cm 3 K/mol, χ 0 = 4.5 × 10 −5 cm 3 /mol and θ CW = − 152 K. The relatively large and negative value of θ CW reveals the presence of strong antiferromagnetic exchange interaction between Mn 2+ spins. The calculated effective moment µ ef f = √ 8C = 5.97µ B per Mn atom, which is very close to the expected moment µ ef f = g S(S + 1)µ B = 5.92µ B for the high-spin state (S = 5/2) of Mn 2+ assuming the g factor g = 2 [44]. The high spin state is further confirmed from the DFT calculations. The corresponding effective moment gives the Landé g factor g = 2.018, a similar g value was also determined in the triangular lattice Ba 3 MnSb 2 O 9 by ESR [45]. With decreasing temperature the χ(T ) data start deviating from the Curie-Weiss law and show an anomaly at 20 K which suggests an antiferromagnetic long-range order at this temperature. Similar behavior was also seen in several other frustrated triangular lattice systems [46]. Indeed, other compounds in this series of double perovskites also show a common feature of long-range antiferromagnetic ordering around 20 K including Sr 2 CuTeO 6 and Pb 2 MnTeO 6 [33,47]. The strength of frustration in the present antiferromagnet is quantified by the frustration parameter f = |θ CW | T N ≈ 7, which suggests the existence of moderate frustration in the host magnetic lattice. As shown in Fig. 2 (b), χ(T ) data for all fields up to 7 T are very similar in magnitude and we observed no shift in anomaly with the applied field up to 7 T. Absence of any hysteresis in magnetization curve at 5 K ( Fig. 2 (c)) excludes any ferromagnetic component, either being intrinsic or due to a tiny amount of impurity phase of Mn 3 O 4 [48]. The reduced magnetic moment compared to saturation moment 5.92 µ B (Mn 2+ , S = 5/2) at 7 T is consistent with the presence of strong antiferromagnetic exchange interactions between Mn 2+ spins.

C. Specific heat
In order to provide further evidence of long-range magnetic order, we have measured the temperature dependence of specific heat (C p (T )) of BMTO in zero field in the temperature range 2 K ≤ T ≤ 250 K. A lambdalike anomaly appears at T N = 20 K, which is the same temperature at which we observed an anomaly in χ(T ). This confirms the occurrence of an antiferromagnetic long-range order in BMTO at this temperature. The absence of any anomaly at 42 K, which is the transition temperature of Mn 3 O 4 , indicates BMTO is free from minor impurity phase of Mn 3 O 4 [49]. An esti-mate of the associated magnetic contribution to the specific heat data of BMTO is obtained after subtraction of lattice contribution from the total specific heat data, i.e., C mag (T ) = C p (T ) − C latt (T ), where C mag (T ) and C latt (T ) are the magnetic and lattice specific heat, respectively. In the absence of a suitable non-magnetic analog of BMTO, we model the lattice contribution as [50] which includes a Debye term and three Einstein terms. In Eq.
(2) θ D is the Debye temperature, θ i are the Einstein temperatures of the three modes, R and k B are the molar gas constant and Boltzmann constant, respectively. As depicted in Fig. 3 (a), the experimental data show good agreement with the model for temperatures above 40 K for θ D = 324 K, θ E1 = 128 K, θ E2 = 194 K, and θ E3 = 645 K. In the fit the coefficients were fixed in the ratio C D :C E1 :C E2 :C E3 = 1:1:3:5 as in BMTO the number of acoustic and optical modes of lattice vibration has the ratio of 1:9 [51]. The one Debye term corresponds to the acoustic mode and three Einstein terms approximate all optical modes. After subtracting the lattice contribution, the magnetic contribution to specific heat C mag (T ) is obtained and shown in Fig. 3 (b). There is a clear anomaly in C mag (T )/T at T N ∼ 20 K, which suggests that Mn-Mn exchange interaction connectivity in BMTO is essentially 3D. Next, we have calculated the entropy change (∆S(T )) by integrating C mag (T )/T over the temperature range from 2 K to 50 K as shown in Fig. 3 (b). It is noticed that the rise of entropy change with increasing temperature saturated to a value of 12.34 J/mol-K at 50 K, which is somewhat lower than the theoretical value of the total entropy 14.9 J/mol.K (R ln(2S + 1)) for the high-spin S = 5/2 state of Mn 2+ ions. Thus, we recovered 82 % of the expected total entropy and the missing 18 % is most likely due to over-estimation of the lattice contribution to total specific heat and thus underestimation of short-range spin correlations above T N . Below the transition temperature, the lattice contribution to the specific heat becomes practically negligible so the measured specific heat is of magnetic origin. At low temperatures up to T N , approximately 50 % of the entropy is recovered, suggesting that the other 50 % is due to short-range spin correlations that develop already above T N .
In order to investigate the nature of magnetic excitations in the ground state, the low temperature (∼T N /3) magnetic specific heat data are fitted with a phenomenological model [52][53][54] C mag (T ) = αT n exp(−∆/T ), where α and n are constants and ∆ is the gap between lower band and upper band of closely spaced energy levels. A similar empirical formula was employed to describe the gapped magnon excitations of α-RuCl 3 in the ground state [54,55]. The fit yields a gap ∆/k B ≈ 1.4 ± 0.1 K in the magnetic excitation spectrum. The presence of small gap is attributed to an easy axis anisotropy term in the spin Hamiltonian [53,56].

D. Muon spin relaxation (µSR)
TF µSR measurements are a very efficient probe of magnetic ordering and spin correlations. In the absence of static internal magnetic fields of electronic origin, the muon asymmetry precesses in a weak external transverse field B TF with the frequency γ µ B TF /(2π), where γ µ = 2π × 135.5 MHz/T is the muon gyromagnetic ratio. Muons experiencing additional static internal fields, which are in insulators usually in the range between a few tens and a few hundreds of mT [57], oscillate much faster and lead to a strongly damped signal that is observable only at very short times. Except from these short times, muon asymmetry follows the general form where the amplitude A 0 describes the volume fraction of the sample that experiences zero static internal fields and A 1 > 0 arises from the ordered part of the sample with the internal field being parallel to B TF .
The temperature dependence of the TF µSR asymmetry in BMTO with corresponding fits of the model (4) is shown in Fig. 4 (a). The relative amplitude of the signal oscillating with the frequency γ µ B TF /(2π), i.e., the volume fraction of the muons not experiencing sizeable static internal magnetic fields, starts decreasing from unity already below 35 K 2T N and quickly drops towards zero when the temperature approaches T N (Fig. 4 (b)). A 0 (T )/A 0 (50 K) < 1 indicates the presence of static internal fields, which we attribute to short-range ordering for T > T N and long-range ordering for T < T N . Also in BMTO diffuse neutron scattering originating from the same Q positions as spin waves below T N is found at temperatures far above T N [35], therefore our confirmation of the short-range order nicely complements these results. We note that the transverse muon spin relaxation rate λ T (inset in Fig. 4 (b)), which measures the width of the distribution of static fields for the component with no net internal field, increases when temperature approaches T N . Next, we determine the static internal magnetic fields B µ in BMTO below T N more precisely from ZF µSR measurements, where the frequency of oscillations in muon asymmetry is directly given by these fields, ν µ = γ µ B µ /(2π). Indeed, high-frequency oscillations develop in the muon asymmetry below T N due to magnetic ordering ( Fig. 5 (a)). The corresponding experimental curves can be fit at short times with a model including two dis- tinct muon stopping sites, Here, the constant "1/3-tail" for each site corresponds to the projection of the initial polarization in a powder sample on the internal magnetic field and, which does not precess, while the oscillating part is due to the perpendicular component [57]. The internal fields at the two muons stopping sites at 1.6 K amount to B µ1 = 0.63 T and B µ2 = 0.38 T, while large relaxation rates λ T 1 λ T 2 ∼ 60(10) µs −1 indicate relatively broad distributions of internal fields. The temperature dependence of the average internal fields (Fig. 5 (b)) corresponds to the evolution of the order parameter and the same ratio B µ1 /B µ2 = 1.7(1) is maintained up to T N . At T N the static internal fields vanish, contrary to the refined magnetic moment deduced from neutron diffrac- tion, which exhibits a smooth evolution across the transition temperature [35]. The fraction of the muon stopping site with the higher internal field value is f = 0.32 (5) and is temperature independent. On a longer time scale, the fast oscillations due to static internal magnetic fields below T N are averaged out, so that only the "1/3-tail" is seen in ZF muon asymmetry ( Fig. 6 (a)). This tail exhibits pronounced relaxation even at the lowest temperature of 1.6 K, i.e., well below T N , which is due to the dynamics of the local fields. In fact, the ZF muon asymmetry on the long time scale can be fit with the same model at all temperatures, where the two terms again correspond to the two muon stopping sites. The initial asymmetry falls from the high-temperature value of A 0 = 0.245 to about A 0 /3 (inset in Fig. 6 (a)), as expected. The decrease of this parameter is due to ordering effects. It is again observed already below 35 K and becomes very pronounced in close vicinity of T N , mimicking the change of the amplitude of the TF signal shown in Fig. 4 (b). The longitudinal muon relaxation rates λ L1 and λ L2 exhibit divergent behavior at T N ( Fig. 6 (b)), which is a typical fingerprint of critical slowing down of spin fluctuations. Above T N the ratio of the relaxation rates of the two components scales with the ratio of the squares of the internal fields in the long-range ordered phase, λ L1 /λ L2 = (B µ1 /B µ2 ) 2 = 2.8 ( Fig. 6 (b)). As the muon spin relaxation rate is proportional to the square of the fluctuating fields in the fast-fluctuation regime corresponding to the paramagnetic phase [57], this experimental scaling firmly validates our analysis with two muon stopping sites in BMTO. Below T N the longitudinal muon spin relaxation is due to collective excitations and the ratio λ L1 /λ L2 increases to about 10. Importantly, we find that the dynamics of local fields persists down to the lowest temperatures, as observed in various different frustrated spin systems [58][59][60].

A. Computational Methods
To further understand the magnetic interactions in BMTO, DFT calculations have been performed using the plane-wave-pseudopotential approach as implemented in Quantum ESPRESSO [61]. The experimentally obtained structure has been considered for the calculations. The ultra-soft pseudopotentials are used to describe the electron-ion interactions [62], in which the valence states of Mn include 15 electrons from 3s, 3p, 4s and 3d; Ba includes 10 electrons in 5s, 5p, and 6s; Te includes 10 electrons in 5s and 5p orbitals; and O includes 6 electrons from 2s and 2p shells. The exchange-correlation functional is approximated through PBE-GGA functional [63]. The convergence criterion for self-consistent energy is taken to be 10 −6 Ry. A k-mesh of 4 × 4 × 2 is used for the Brillouin zone integration of the supercell of size 2 × 2 × 2. The kinetic energy cut-off for the electron wave functions is set at 30 Ry and the augmented charge density cut-off is set to be 300 Ry. We have also performed test calculations with a higher energy cut-off of 40 Ry and charge density cutoff at 400 Ry as well as with a higher k-mesh of 8 × 8 × 4. As the results remain the same below the tolerance level, we have used the lower cut-off and lower k-mesh to reduce the computational time. The strong correlation effect is examined through Hubbard U formalism [64]. The magnetic coupling strengths are evaluated as a function of U in this strongly correlated system.

B. Electronic Structure
The crystal structure of BMTO can be described as alternate stacking of layers of TeO 6 and MnO 6 octahedra, and the neighboring layers are connected through corner sharing oxygen as shown in Fig. 1 (a). However, the electronic structure, presented in Fig. 7 through total and partial densities of states (DOS), shows that the Te-p state is almost completely occupied and lie around 7 eV below the Fermi energy (E F ). Therefore, from the electronic and magnetic structure point of view this compound can be treated as an open spaced structure in the sense that the minimum Mn-Mn separation, both interlayer and intra-layer, is ∼ 5.8Å which is roughly double than that of the closed packed transition metal perovskites. Here, the electronic structure of the system is supposed to be nearly a sum of the electronic structure of the individual MnO 6 octahedra [65]. To verify this we first examine the DOS within the independent electron approximation (U = 0) which are shown in Fig. 7 (a). Here, we observe that due to crystal-field splitting, the Mn-d states split into triply degenerate t 2g and doubly degenerate e g states. Due to stronger axial interactions, the e g states have reasonable overlapping with the O-p states, which dominate the energy window −5 to −2 eV with respect to E F . Due to 2+ charge state for Mn, the d-orbitals are half-occupied and stabilize in a high-spin states (S = 5/2) where the d-orbitals in the spin minority channel are completely unoccupied. The S = 5/2 state is also confirmed from the magnetization measurement discussed earlier. For such a spin-state, the spin splitting is strong enough to create a band gap even for U = 0 [65]. With inclusion of strong correlation effect (finite U ), the Mn-d states are pushed to lower energies in the valence band and to higher energies in the conduction band to widen the bandgap (see Fig. 7 (a-c)). The O-p states now dominate the valence band near E F which implies that BMTO is a charge transfer insulator [66] which favors antiferromagnetic (AFM) ordering. The total and Mn-d DOS for the stable AFM ordering (see Fig. 7 (e)) are shown in the lower panel of Fig. 7 (d-f). As both FM and AFM ordering makes the system insulating, we infer that this is primarily a weakly coupled classical spin system. The strength of the coupling is discussed next.

C. Magnetic Interactions in BMTO
The experimental results presented in this work imply dominant antiferromagnetic interactions through θ CW . There are three dominant exchange interaction paths (J 1 , J 2 , J 3 ) in this compound which demonstrates a hexagonal ABC stacking pattern as shown in Fig. 8 (a). The J 1 represents the intra-plane nearest-neighbor interaction for which the Mn-Mn distance is 5.81Å, whereas J 2 represents the inter-plane nearest-neighbor interaction (d M n−M n = 5.81Å). The 3rd nearest-neighbor interaction (8.22Å) is considered by the J 3 term. Here, we shall examine the strengths of these J i 's through a spin-dimer analysis using Noodlemann's broken-symmetry method [67,68]. According to this method, the energy difference between the high spin (HS) and low spin (LS) states for a spin dimer is given by where J is related to the spin-dimer Hamiltonian,Ĥ= Ĵ S 1 ·Ŝ 2 , with S max being the maximum spin of the dimer.
As we have a Mn-Mn spin dimer, both sites of the dimer have five unpaired electrons. Therefore, Eq. (7) reduces to where, E HS and E LS can be estimated from the DFT calculations as discussed below.
To evaluate exchange constants in the framework of DFT, one needs to design several possible magnetic configurations, and calculate the total energies. The relative energy differences among them are expressed in terms of J i 's leading to a set of linear equations. The magnetic configurations (FM, AFM1, AFM2, and AFM3), considered here are designed on a 2 × 2 × 2 supercell as shown in Fig. 8 (b-e). The total energies of each configuration is estimated with the sum of all exchange paths which yield the following set of equations: Hence, by solving the above equations, we have estimated the J i 's as a function of U and plotted them in Fig. 9. While the dominant interactions J 1 and J 2 are antiferromagnetic, J 3 corresponds to a weakly ferromagnetic coupling. This is due to the fact that J 3 is formed by a linear Mn-O-Te-O-Mn path, where the axial e g -Op-Te-p-O-p-e g covalent interaction is formed leading to a double-exchange ferromagnetic interaction. However, since Te-p states form a nearly closed shell configuration, the interaction is very weak. Although the exact values of the exchange constants J i depend on the Hubbard repulsion parameter U , we can evaluate their strength by comparing with the experimentally obtained Curie-Weiss temperature, as recently demonstrated for another frustrated lattice [69]. The Curie-Weiss temperature is given by where Z i represents the coordination numbers of Mn atom. By plotting θ CW as a function of U (see Fig. 9) we find a match with the experimental value of −152 K for U = 5.75 eV for which J 1 = 4.6 K, J 2 = 4.2 K and J 3 = − 0.2 K with J 1 :J 2 :J 3 = 1 : 0.92 : −0.04. These values are slightly larger than those (J 1 = J 2 = 3.1 K and J 3 = − 0.6 K) found from the fit of spin waves detected by inelastic neutron scattering [36]. The latter values however underestimate the Curie-Weiss temperature by yielding a bit smaller values of exchange couplings.

V. DISCUSSION
A conventional 3D antiferromagnet exhibits a symmetry breaking phase transition at a temperature close to the Curie-Weiss temperature, but the titled compound BMTO shows long-range order only at T N = 20 K despite much larger θ CW ∼ − 152 K. This suggests that BMTO is a moderately frustrated antiferromagnet with frustration index f = 7. In this system, two structural reasons could explain the observed long-range order: (i) Magnetic ions are in the high-spin state (S = 5/2) and quantum fluctuations are less pronounced even though the Mn 2+ ions are arranged in 2D triangular plane; (ii) The intra-plane and the inter-plane nearest-neighbor distances are the same which allows for sizeable inter-plane interactions that forces the system to undergo a long-range magnetic ordering. The origin of the magnetic ordering in this strongly correlated system is examined within the framework of DFT. Our calculations of exchange interaction reveal dominant antiferromagnetic intra-layer exchange coupling J 1 = 4.6 K and a comparable inter-layer J 2 = 0.92 J 1 . Furthermore, a very weak ferromagnetic interlayer interaction exists with third nearest neighbor (J 3 = − 0.04 J 1 ) due to double-exchange interaction via the linear path Mn-O-Te-O-Mn. The magnetic specific heat data below the AFM transition are well reproduced with Eq.(3) suggesting the presence of magnon excitations. A broad maximum at 10 K in magnetic specific heat data suggests the presence of gapped magnon excitation in the ground state. Similar types of broad maxima were also observed in BiMnVO 5 and MnWO 4 which indicates Mn 2+ ions are subjected to anisotropic magnetic interactions [53,70]. In BMTO, the estimated magnon excitation gap is 1.4 K, a similar value of magnon excitation gap is also observed in MnWO 4 [56]. The missing of entropy which is estimated as ∼ 18% of the expected entropy for S = 5/2 system is due to the presence of spin frustration and significant short-range spin correlations between Mn 2+ spins already above T N , which are however underestimated by our crude modeling of the lattice contribution to the specific heat. In fact, the evolution of µSR spectra in weak transverse field show that short-range ordering effects become apparent already at 35 K 2T N , and gradually increase as the temperature approaches to T N . The ratio of the µSR amplitudes A 0 (T )/A 0 (50 K) < 1 in the temperature range 20 K ≤ T ≤ 35 K suggests the presence of short-range spin correlation above the antiferromagnetic transition temperature. If there are no significant short-range spin correlations above the transition temperature, volume fraction of the sample does not change above T N as observed in weak transverse field µSR spectra of LiCrO 2 [71]. The short-range spin correlations reflect the presence of moderate spin frustration in the magnetic lattice of BMTO. The zero-field µSR spectra show that below 35 K the spin lattice relaxation rate gradually increases, which is commonly observed in the vicinity of magnetic phase transition temperature. At T > 35 K, the temperature independent initial asymmetry can be associated with the paramagnetic nature of Mn 2+ spin. The position of sharp maximum in the muon relaxation rate and the reduction of initial asymmetry (A 0 ) to A 0 /3 both occurs at T = 20 K, which confirms a phase transition at this temperatures. Finally, static internal fields are directly observed through the oscillations of muon asymmetry below T N and the evolution of the order parameter is reflected in the temperature dependence of these fields.

VI. CONCLUSION
The double perovskite BMTO crystallizes in the trigonal crystal symmetry R3m, wherein Mn 2+ ions form two dimensional triangular layers with sizeable inter-layer exchange coupling. Our comprehensive results, well supported by first principle calculations reveal the presence of antiferromagnetic long-range magnetic order below T N = 20 K. Below T N , magnetic specific heat data suggest the presence of magnon excitations with a gap of approximately 1.4 K, which indicates the presence of magnetic anisotropy as commonly observed in classical Heisenberg systems. Our zero-field and weak transverse field µSR results provide a concrete evidence of static internal fields in the long-range ordered state below T N and short-range spin correlations above T N . This frustrated triangular lattice antiferromagnet is also potentially interesting to uncover exotic ground state associated with quenched disorder in triangular lattice by substitution of less electronegative cations at the tellurium site. Further studies on single crystals are required to shed more insight into the low energy excitations of this double perovskite based frustrated magnet.