Exotic magnetic field-induced spin-superstructures ina mixed honeycomb triangular lattice system

V. Ovidiu Garlea,1, ∗ Liurukara D. Sanjeewa,2, 3 Michael A. McGuire,3 Cristian D. Batista,4, 5 Anjana

M. Samarakoon,4 David Graf,6 Barry Winn,1 Feng Ye,1 Christina Hoffmann,1 and Joseph W. Kolis2

1Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA2Department of Chemistry and Center for Optical Materials Science and Engineering Technologies (COMSET),

Clemson University, Clemson, South Carolina 29634-0973, USA3Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

4Shull-Wollan Center, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA5Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA

6National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA(Dated: September 7, 2018)

The temperature-magnetic-field phase diagram of the mixed honeycomb triangular lattice systemK2Mn3(VO4)2CO3 is investigated by means of magnetization, heat capacity and neutron scatter-ing measurements. The results indicate that triangular and honeycomb magnetic layers undergosequential magnetic orderings and act as nearly independent magnetic sublattices. The honeycombsublattice orders at about 85 K in a Neel-type antiferromagnetic structure, while the triangularsublattice displays two consecutive ordered states at much lower temperatures, 3 K and 2.2 K.The ground state of the triangular sublattice consists of a planar “Y” magnetic structure thatemerges from an intermediate collinear “up-up-down” state. Applied magnetic fields parallel orperpendicular to the c-axis induce exotic ordered phases characterized by various spin-stacking se-quences of triangular layers that yield bilayer, three-layer or four-layer magnetic superstructures.The observed superstructures cannot be explained in the framework of quasi-classical theory basedonly on nearest-neighbor interlayer coupling and point towards the presence of effective second-nearest-neighbor interactions mediated by fluctuations of the magnetic moments in the honeycombsublattice.

PACS numbers: 75.25.-j, 75.50.Ee, 75.30.Ds, 75.10.Jm, 71.27.+a, 61.05.F-

I. INTRODUCTION

Field-induced magnetic states that occur in layeredtriangular lattice antiferromagnets (TLA) have been ex-tensively studied and discussed in the context of bro-ken discrete symmetries of the lattice and spin-rotationin the plane perpendicular to the applied field. It iswidely recognized that thermal and quantum fluctuationslift the degeneracy of the classical spin configurations ofthe triangular antiferromagnetic Heisenberg model in amagnetic field. This well-known realization of order-by-disorder [1] leads to specific planar states: a “Y”-stateor 120◦ spin configuration with two spins canting “up”and one pinned in a “down” direction, a collinear “up-up-down” (uud) state, and a canted “2:1” phase that isan oblique version of the uud state.[2–9] These planarspin configurations are sketched in Fig. 1. An easy-axisanisotropy can also remove the degeneracy and stabi-lize the same coplanar arrangements. The collinear uudstate, which gives a one-third of the saturation magneti-zation plateau at intermediate fields (M=Ms/3), breaksthe discrete Z3 symmetry of the lattice. The canted pla-nar “Y” and “2:1” states that involve Sx − Sy spin com-ponents also break the continuous U(1) symmetry of spinrotations about the field axis. The latter spin states with

∗ garleao@ornl.gov

H < Hsat/3 H = Hsat/3 Hsat/3 < H < Hsat

HY - state up-up-down canted 2:1

FIG. 1. Planar spin configurations of single-layer triangularlattice antiferromagnet at 0 K and finite magnetic fields.[4]

broken mixed symmetries Z3 ⊗ U(1), can be viewed asmagnetic supersolid phases that combine superfluid prop-erties with long-range periodicity of solids, as proposedby Liu and Fisher. [9, 10]

Most of existing theoretical studies on TLA assumenegligible interplanar interactions. However, the pres-ence of such coupling, as expected in real materials, cancause additional exotic field-induced phase transitionswhere the relative spin orientations change between adja-cent planes to produce interesting magnetic superstruc-tures. The spin-stacking pattern is expected to dramat-ically depend on the manner of stacking of neighboringlayers which can be either eclipsed (adjacent layers re-lated by a simple translation perpendicular to the layer

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plane) or staggered (shifted by certain in-plane latticetranslation) near-neighbor planes, with the latter also en-abling frustrated out-of-plane interactions. Possible mag-netic structures of weakly-coupled eclipsed TLA were dis-cussed by Gekht,[5] Chubukov[4] and Yamamoto.[8] Yet,most theoretical predictions remained unconfirmed dueto the limited number of good experimental realizationsof equilateral TLA.

Some of the most explored series of TLA compoundsare V X2 and ABX3, where A = Cs, Rb, B = Ni, Mn, Cu,and X = Cl, Br, I; as well as ACrO2, with A = Li, Cu,Ag, or Pd.[11] Unfortunately, these systems either haveinterplanar interactions that are larger than the intra-planar ones, or too strong intraplanar nearest-neighborcoupling that makes the field-induced transitions inac-cessible to the currently available magnetic-fields at theneutron scattering facilities. Improvements in samplesynthesis techniques allowed in the recent years the dis-covery of new triangular systems with transitions at ac-cessible magnetic fields. Detailed temperature-magnetic-field (T −H) phase diagrams have been reported for sev-eral S = 5/2 TLA systems with planar (XY) anisotropy,RbFe(MoO4)2,[12–15] and RbAg2Fe[VO4]2, [16] or withweak easy-axis (Ising) anisotropy, Rb4Mn(MoO4)3.[17,18] Another exciting class of TLA that has recentlyemerged is that of 6H-perovskites Ba3 M

′M ′′2 O9 withM ′ = Ni, Co, Mn, and M ′′= Nb, Sb or Ta.[19–32] Thesecompounds can possess either easy-plane or easy-axisanisotropies and adopt at low temperatures the expectedplanar 120◦ magnetic structure. For all aforementionedsystems experimental evidences for the three predictedfield-induced states, the “Y”, the one-third magnetiza-tion plateau uud, and the “2:1” canted configuration,have been found. However, the lack of sufficiently largesingle crystal samples required for detailed neutron scat-tering studies has, in most of the cases, hindered theunderstanding of the impact of interlayer coupling in sta-bilization of field-induced ordered phases.

The vanadate - carbonate system K2Mn3(VO4)2CO3

has been recently identified as a very promising proto-type for studying magnetic frustration. [33, 34] Its struc-ture, shown in Fig. 2, consists of alternately stackedtriangular and honeycomb magnetic layers. Previousmacroscopic measurements indicated complex physicalproperties with a presumed Jahn - Teller transforma-tion at about 80 K and two successive magnetic phasetransitions, at about 3 K and 2 K, into a weakly fer-romagnetic ground state. It was inferred that divalentMn is present in a high-spin state (S = 5/2) in theoctahedral environment of the honeycomb layer and alow-spin state (S = 1/2) in the trigonal bipyramidalcoordination of the Mn2+ ions occupying the triangu-lar layer. [33] The low-temperature magnetic transitionswere attributed to the ordering of the S = 5/2 ions of thehoneycomb lattice , while the S = 1/2 triangular lay-ers were thought to remain paramagnetic. Subsequentfirst-principles density functional theory-based analysisshowed that, contrary to the previous suggestion, both

FIG. 2. (Color online) Polyhedral view of K2Mn3(VO4)2CO3

crystal structure. The structure consists of alternate stackingof two types of layered subsystems: a honeycomb layer madeof edge-sharing MnO6 octahedra, and a triangular layer madeof MnO5 trigonal bipyramids linked by CO3 triangles. Pro-jections along c-axis of the two distinct layers are shown inthe right panel.

inequivalent Mn ions occupying the two layers are inhigh-spin S = 5/2 state. [34] The calculations predictthat both layers exhibit antiferromagnetic orders withvastly different strengths of magnetic interactions. In-trigued by the richness of the magnetic phase diagramfeatured by this material we have undertaken a compre-hensive magnetization and neutron scattering study us-ing high-quality single crystal samples. Our study clar-ifies the magnetic ground-states of two magnetic layers,and establishes that K2Mn3(VO4)2CO3 represents an ex-cellent candidate TLA system for studying the effectof interplanar interactions in field-induced states. Ap-plied magnetic fields induce new magnetic superstruc-tures characterized by various spin-stacking sequences oftriangular layers. Because the observed magnetic super-structures cannot be explained by any existing theoriesbased only on nearest-neighbor interlayer coupling, theycompel a closer look at effective second-nearest-neighborinteractions mediated by fluctuations of the magneticmoments in the honeycomb sublattice.

II. EXPERIMENTAL DETAILS

Single crystals of K2Mn3(VO4)2CO3 were grown usinga high-temperature hydrothermal technique. The chemi-cals used in this synthesis were used as received, withoutfurther purification: Mn2O3 (Alfa Aesar, 98%), V2O5

(Alfa Aesar, 99.6%) and K2CO3 (Alfa Aesar, 99.997%).In a typical reaction, Mn2O3 and V2O5 were mixed ina 3:2 molar ratio with 0.8 mL of 5 M K2CO3 mineral-izer, giving approximately 0.4 g of reactants. Reactionswere performed in silver ampules with a diameter of 9.5

3

mm, with approximately a 70% fill of free volume. Afterloading the reactants and the mineralizer, the ampouleswere welded and loaded in a Tuttle cold-seal style auto-clave and filled with distilled water at 80% of free volumeto provide suitable counter pressure. The autoclave washeated to 580 ◦C for two weeks at a typical pressure of1.5 kbar. Brown hexagonal crystals were isolated usingsuction filtration. The single crystal specimens used forphysical properties characterization and neutron scatter-ing experiments were physically examined and selectedunder an optical microscope equipped with a polariz-ing light attachment. The powder sample used for thepowder neutron diffraction measurements was generatedby grinding crystals produces in the reaction describedabove.

Temperature and field-dependent magnetic measure-ments were carried out using a Quantum Design Mag-netic Property Measurement System (MPMS). The mea-surements were carried out on a single crystal specimenwith the crystallographic c-axis aligned either parallel orperpendicular to the applied magnetic field. The tem-perature dependence of static susceptibility (M/H(T ))was measured over a temperature range of 2 to 700 Kfor applied fields µ0H = 0.01 T and 1 T. The isothermalmagnetization measurements were performed for fieldsup to 8 T. Additional isothermal magnetization curveswere recorded using a vibrating sample magnetometer(VSM) in magnetic fields up to 16 T, applied along c-axis. Heat-capacity measurements were performed witha Physical Property Measurement System (PPMS, Quan-tum Design) in zero and magnetic fields up to 6 T appliedeither parallel and perpendicular to the crystal c-axis.

Neutron powder diffraction measurements were carriedout using the HB2A high-resolution diffractometer at theHigh Flux Isotope Reactor (HFIR),[35] using the 2.41 Awavelength neutron beam. A powder sample with a totalmass of approximately 5 grams was compacted in pellets,loaded into a cylindrical aluminium can, and placed in-side a cryostat with 3He insert. Data were collected attemperatures from 150 K to 0.3 K. Single-crystal neu-tron diffraction measurements were performed at T =150 K using the TOPAZ time-of-flight diffractometer atthe Spallation Neutron Source (SNS). The integratedBragg intensities were obtained using the 3-D ellipsoidalQ-space integration method and were corrected for back-ground using the MANTID software.[36] Data reductionincluding, Lorentz and absorption corrections as well asspectrum, detector efficiency, data scaling and normaliza-tion was carried out with the ANVRED3[37] program.

Elastic neutron scattering measurements at tempera-tures down to 1.6 K were performed at the fixed-incident-energy (14.6 meV) HB1A triple-axis spectrometer at theHFIR, and at CORELLI[38] and HYSPEC[39] spectrom-eters at the SNS. Two separate crystals with the ap-proximated dimensions 1 x 1 x 4 mm3 and 4 x 3 x 1mm3 were used for collecting data under magnetic fieldsapplied parallel and perpendicular to the c-axis, respec-tively. At CORELLI, white-beam Laue diffraction mea-

surements were taken at 1.6 K and selected magneticfields up to 5 T, applied along c axis or [1, 1, 0]. For eachfield configuration the sample was rotated in steps of 3◦

over ranges of 60◦. MANTID software was utilized tocarry out the Lorentz and spectrum corrections, as wellas to merge the full volume of the scattering data.[36]At HYSPEC, the elastic measurements were performedin applied magnetic fields up to 8 Tesla, using a fixedincident energy Ei = 15 meV and a Fermi chopper fre-quency of 120 Hz. Measurements were concentrated overnarrow range reciprocal lattice volumes, centered aroundthe relevant magnetic reflections.

Inelastic neutron-scattering (INS) measurements wereperformed at the HYSPEC spectrometer, operated withthe incident energies Ei = 25 meV and 3.8 meV, and theFermi chopper frequency of 360 Hz. For these measure-ments, multiple single crystals where coaligned along thec-axis, while the in-plane directions were distributed ina quasi-random manner to provide a total mass of about0.3 g.

Structural and magnetic data refinements were carriedout with the FullProf Suite program. [41] Possible mag-netic structures models have been explored by represen-tation analysis using the program SARAh,[42] and by themagnetic symmetry approach using the tools available atthe Bilbao Crystallographic Server.[43] The INS data re-duction and visualization was done with the MANTIDand DAVE [40] software packages. Spin-wave calculationwere performed using the linear spin wave theory withthe program SpinW.[44]

To reproduce experimental data, we have performedclassical Monte Carlo simulations using a standardmetropolis sampling algorithm on the triangular latticesubsystem. The simulations were performed on finitelattice of 12 x 12 x 4 unit-cells (containing 1152 spins)with periodic boundary conditions. Starting from a com-pletely random and disordered configuration, a spin sys-tem was annealed down to a finite temperature in finitenumber of intermediate temperature steps. Then, themagnetic field was increased up to 22 T with 300 in-termediate field points. At each temperature/field step,the magnetization (M) and heat capacity (C) were calcu-lated by taking the ensemble average over the 2000 sam-ples followed by thermalization sampling with adaptivestep size, while the static spin structure factor S(Q) wascalculated by Fourier Transforming captured spin config-urations with the frequency of 500 samples. For betterstatistics, all the measurements were averaged over 120independent simulations.

III. EXPERIMENTAL RESULTS ANDDISCUSSION

A. Crystal structure

The structural model of K2Mn3(VO4)2CO3 proposedby Yakubovich et al, [33] has been confirmed by the re-

4

TABLE I. Refined structural parameters and selected bonddistances of K2Mn3(VO4)2CO3 from single-crystal neutrondiffraction data collected at T = 150 K.

Atom Wyck. x y z Ueq

K 4f 1/3 2/3 0.6578(1) 0.0144(4)

Mn1 4f 1/3 2/3 0.0046(1) 0.0081(3)

Mn2 2a 0 0 1/4 0.0092(5)

V 4e 0 0 0.0800(4) 0.0057

C 2c 1/3 2/3 1/4 0.0070(3)

O1 12i 0.3210(1) 0.3098(1) 0.0550(1) 0.0080(1)

O2 4e 0 0 0.1552(1) 0.0138(3)

O3 6h 0.0682(1) 0.6255(1) 1/4 0.0133(3)

Mn1–O1 (x 3): 2.1444(6)A; Mn2–O2 (x 2): 2.1234(8)A

Mn1–O1 (x 3): 2.1912(7)A; Mn2–O3 (x 3): 2.1452(7)A

Mn1–Mn1 (x 3): 3.006(1)A; Mn2–Mn2 (x 6): 5.195(1)A

Space group: P63/m, a=b= 5.1959(3)A, c = 22.405(2)A

Rf = 0.051, χ2 = 2.04

finements of single crystal neutron diffraction data. Therefined structural parameters, such as atomic coordinatesand displacement parameters along with selected bondlengths involving the magnetic Mn atoms, are given inTable I. As previously described, the structure consistsof two types of Mn-O layers alternating along the c-axisof the hexagonal unit cell. One layer consists of a honey-comb web made of edge sharing MnO6 octahedra, whilethe second consists of MnO5 trigonal bipyramids that arelinked together by CO3 coplanar triangle groups to forman equilateral triangular lattice. Each layer is composedby a single Mn crystallographic site: the honeycomb isbuilt of Mn1 ions occupying 4f Wyckoff position of theP63/m space group, and the triangular layer is definedby Mn2 ions located at the 2a Wyckoff position. TheMn1 atoms are coupled via double oxygen (O1) bridgesand are spaced at about 3.006 A apart. Inside the trian-gular layer the interatomic distance between neighboringMn2 atoms is approximately 5.195 A. There are twice asmany Mn1 ions in the honeycomb layer as compared toMn2 located in the triangular layer. The Mn2 ions arelocated exactly on top or underneath the hollow centerof the Mn1 honeycomb. Considering that there are twolayers of each type per unit cell, the distance between twoconsecutive triangular layers is about c/2 = 11 A, whilethat between the honeycomb and triangular layer is c/4= 5.5 A. The interlayer space is occupied by K+ cationsand V5+O4 tetrahedra that share oxygen vertices withmanganese polyhedra. Due to the non-magnetic natureof V5+ cation, the magnetic interactions between adja-cent layers are expected to be subdominant comparedto the intralayer interactions. A perspective view of thecrystal structure and the two types of Mn-O layers are

depicted in Fig. 2.It is instructive to compare here the structural proper-

ties of our compound with other recently studied trian-gular lattice systems, where Mn2+ magnetic ions adopteither six-fold or five-fold oxygen coordinations. Forinstance, the Rb4Mn(MoO4)3 features Mn2+O5 poly-hedra forming equilateral triangular lattices separatedby MoO4 tetrahedra, with intralayer and interlayer dis-tances between Mn2+ ions of 6.099 A and c/2= 11.856 A,respectively.[17] In Ba3MnNb2O9 the Mn2+ ions have aoctahedral coordination and form triangular lattices withMn-Mn intralayer distance of 5.773 A and interlayer dis-tances c = 7.0852 A.[21]

The evolution with temperature of K2Mn3(VO4)2CO3

lattice parameters has been investigated by powder neu-tron diffraction and single-crystal X-ray measurements.A smooth temperature dependence has been observed,suggesting that no noticeable structural change takesplace down to 1.7 K. This would appear to disprove theearlier speculations in Ref. [33] of a Jahn - Teller distor-tion taking place at around 80 – 100 K.

B. Macroscopic magnetic behaviour

The temperature dependence of magnetic susceptibil-ity (χ = M/H) measured with a magnetic field appliedalong and perpendicular the c-axis is shown in Fig. 3.In this figure we choose to plot the temperature axis ina logarithmic scale to increase the visibility of the mag-netic phase transitions that occur over a large tempera-ture interval. A first magnetic transition near TN1 ≈ 85K is revealed by a subtle drop in the susceptibility curvemeasured with field applied along c direction. A secondmagnetic order transition is discernible as a rise of the−→H ‖ c susceptibility at TN2 ≈ 3 K. This is immediatelyfollowed by a third transition, which appears as a kinkin the susceptibility curve at TN3 ≈ 2.2 K. The large dif-ference between the TN1 and TN2 ordering temperaturesdenotes that the honeycomb and triangular magnetic lay-ers possess magnetic interactions of completely differentenergy scales and act as nearly independent magneticsublattices.

The low-temperature region of the zero-field heat ca-pacity data, together with magnetic susceptibility datacorresponding to two different magnetic field orienta-

tions,−→H ‖ c and

−→H ⊥ c, are displayed in Fig. 4. The

two successive magnetic transitions at approximately 3K and 2.2 K can be clearly seen in heat capacity data(cP ) as two distinguishable lambda-shaped peaks. Onecan also notice in Fig. 4 that the magnetic susceptibilityexhibits significant anisotropic behavior. When the fieldis applied parallel to the c-axis the susceptibility under-goes changes at both TN2 and TN3 transitions. However,when the field is applied perpendicular to the c-axis, onlythe lower transition near 2.2 K (TN3) is visible. This be-haviour is indicative of a spin-canting taking place at thelowest temperature.

5

1 0 1 0 020 . 0 00 . 0 20 . 0 40 . 0 60 . 0 80 . 1 00 . 1 20 . 1 4

0 1 5 0 3 0 0 4 5 0 6 0 00

5 01 0 01 5 02 0 0

6 0 8 0 1 0 0 1 2 0

0 . 0 1 5

0 . 0 1 8

0 . 0 2 1

cH ⊥� T N 1T N 2

( b )

cH ⊥�

M / H

(cm3 m

ol-Mn

-1 )

T ( K )

� 0 H = 1 T

( a )

T N 3

cH / /�

C u r i e - W e i s s f i t

H / M

(mol-

Mn cm

-3 )T ( K )

T N 1 cH / /�

M / H

(cm3 m

ol-Mn

-1 )

T ( K )

FIG. 3. (a) Evolution of the magnetic susceptibility (M/H)as a function of temperature, measured in a field of 1 T ap-plied parallel (open symbol) and perpendicular (solid symbol)to the c-axis. The temperature axis is given in a logarithmicscale to increase the visibility of the magnetic phase transi-tions. The inset shows the inverse susceptibility that featurestwo linear regimes at 200 K < T < 700 K and 10 K < T < 50K. The Curie-Weiss fits are shown as red lines. (b) Expandedview of magnetization curves around the TN1 transition tem-perature. The magnetic order is revealed by a subtle dropin the magnetization curve measured with field applied alongthe c direction.

The inverse magnetic susceptibility is shown in the in-set of Fig. 3 along with fits using the Curie-Weiss model(χ = χ0 + C/(T - ΘCW )). The 1/χ data shows two lin-ear regimes: one for the temperature range 200 to 700 Kwhere all spins are paramagnetic, and a second at lowertemperatures ranging from 10 to 50 K. For the high tem-perature range, the Curie-Weiss fit yields a Curie con-stant C = 4.44 cm3/mol-Mn/K, a Weiss temperatureof -215 K, and a temperature-independent term χ0 =-0.00027 cm3/mol-Mn. The negative experimental valueof χ0 can be attributed to the diamagnetic backgroundfrom the sample holder, as the core diamagnetism cor-rection is estimated to be about an order of magnitudesmaller that the obtained value. The negative Weiss tem-peratures indicates dominant antiferromagnetic interac-tions. The obtained effective moment 5.95 µB/Mn is very

close to that expected for S = 5/2 Mn2+, g√S(S + 1)

2 . 0 2 . 5 3 . 0 3 . 5 4 . 00 . 0 1 2

0 . 0 1 6

0 . 0 2 0

0 . 0 2 4c P ( H = 0 T )

T ( K )

c P (J/ g

K)

0 . 0 6

0 . 0 8

0 . 1 0

0 . 1 2T N 2

T N 3

( H = 0 . 0 1 T )M / H cH / /� cH ⊥

�M / H

M / H

(cm3 m

ol-Mn

-1 )

FIG. 4. Low-temperature region of the zero-field specificheat, and low-field static magnetic susceptibility (M/H) mea-sured with µ0H = 0.01 T for two different field orientations,−→H ‖ c and

−→H ⊥ c, revealing the two successive magnetic

transitions at about 3 K and 2.2 K.

= 5.91 µB/Mn, suggesting that Mn ions have the samespin state in both honeycomb and the triangular planes.A Curie-Weiss fit performed using a constrained χ0 =0, yields a slightly lower effective moment 5.72 µB/Mnand ΘCW = -188 K. Note that previous study[33] re-ported much smaller values for both the effective mo-ment (2.75 µB/Mn) and the Curie-Weiss temperature(114 K), which were likely caused by an overestimationof the temperature-independent susceptibility term. Al-though the temperature interval of the second Curie-Weiss regime, 10 K < T < 50 K, is too narrow for ex-tracting definite information, the Curie-Weiss fit yieldsa Weiss temperature of about -19 K and Curie constant1.62 cm3/mol-Mn/K. One could remark that this Curieconstant represents about 36% of the value obtained forthe high temperatures and it correlates reasonably wellwith the 1/3 fraction of spins located in the triangularlayer. This suggests that the honeycomb and triangularlayers act as nearly independent magnetic sublattices.

The isothermal magnetization curves measured at se-lected temperatures ranging from 2 K to 300 K are shownin Fig. 5. Magnetization data was taken for magneticfields up to 16 T applied along c-axis, and up to 8 T forfields applied perpendicular to the c-axis. In agreementwith the previous report,[33] the magnetization curve

measured with−→H ‖ c at T = 2 K exhibits a plateau

at about 4.5 Tesla, followed by a sharp upturn at ap-proximately 7 T. Another step-like transition is observednear 11 T, and then a tendency towards saturation asthe field approaches 16 T. The magnetization value ofthe plateau state (≈ 0.55 µB/Mn) appears to correspondto about 1/9 of total saturation value of Mn2+ moments(MS = gS = 5 µB), or 1/3 of saturation value of Mn2+

6

0 2 4 6 8 1 0 1 2 1 4 1 60 . 0

0 . 3

0 . 6

0 . 9

1 . 2

1 . 5

1 . 8

cH / /�

cH ⊥� 1 0 K

3 0 0 K

2 K

8 5 K5 0 K3 0 K

1 5 K

4 KM S T = 1 . 6 6 � B

M (� B/M

n)

� 0 H ( T )

1 / 3 M S T

( T = 2 K )

FIG. 5. (Color online) Isothermal magnetization (M vs H)curves measured at selected temperatures ranging from 2 Kto 300 K for magnetic fields applied along c-axis (black andblue curves). Magnetization curve measured at 2 K for mag-netic fields applied perpendicular to the c-axis is shown inred color and indicated by arrow. The magnetization valuescorresponding to the plateau state (1/3 MST ) and saturationMST of the triangular magnetic sublattice are indicated bya dashed line. As described in the text, the triangular layercontains only one third of the total magnetic ions of the sys-tem.

moments located in the triangular layers. The magneti-zation plateau value, labeled as 1/3 MST , is marked by adashed line in Fig. 5. We note that the saturation of mag-netization of the triangular layers, MST = 1.66 µB/Mn,is found to be reached at a magnetic field of about 14.3

Tesla. At T = 4 K, the plateau-like state in the µ0−→H

‖ c magnetization curve is no longer present, suggest-ing that this is related to the low-temperature orderingstates observed below 3 K (TN2). On the other hand,the 11 T step-like transition persists up to 50 K, appear-ing to be related to the higher-temperature ordered statethat emerges at TN1 ≈ 85 K. There is no apparent ratio-nal number correlation between the magnetization valuecorresponding to this transition and the total saturationvalue MS . In contrast to the magnetization curve mea-

sured with−→H ‖ c, the curve measured with

−→H ⊥ c shows

a smooth increase with increasing the magnetic field upto the highest measured value of 8 T.

Heat capacity measurements in applied magnetic fieldshave been carried out to construct the T −H phase di-agram around the two low-temperature magnetic tran-sitions. The results are summarized in Fig. 6. For thefield oriented along the c-axis direction, the TN3 mag-netic transition shifts quickly towards lower temperatureswith increasing the magnetic field, while the intermedi-ate transition TN2 extends first to higher temperaturesand then diminishes to define a dome-like shape with thetip at approximately µ0H = 4 T and 3.75 K. The tran-

0123456

2 . 0 2 . 5 3 . 0 3 . 5 4 . 0

2 . 0 2 . 5 3 . 0 3 . 5 4 . 00 . 0 1

0 . 0 2

0 . 0 3

0 . 0 4

2 . 0 2 . 5 3 . 0 3 . 5 4 . 0

0 . 0 2

0 . 0 5

0 . 0 8

H / / c

T N 3

cH ⊥�

cH ⊥�

T *N 3

T N 2

T N 3

T N 3

T ( K )� �H

(Tesla

)

T N 2H / / c

T N 2

T *N 3

T N 3

cH ⊥�cH / /�

0 T1 T2 T3 T4 T5 T6 T

T N 2

( c )

( b )

0 T1 T2 T3 T4 T5 T6 Tc P (J

/ g K

)

( a )

FIG. 6. (Color online) Low-temperature region of the heat-capacity data measured in applied magnetic fields orientedparallel (a) and perpendicular (b) to the c-axis direction. (c)The partial T − µ0H magnetic phase diagram deduced from

heat-capacity data. For−→H ‖ c, denoted by solid symbols, the

lowest transition (TN3) shifts to lower temperatures with in-creasing the magnetic field, while the TN2 transition displays

a dome-like shape. For−→H ⊥ c, represented by open symbols,

the TN3 transition evolves into two distinct transitions (TN3

and T ∗N3) while the TN2 transition increases slightly.

sition points for−→H ‖ c are represented by filled sym-

bols in Fig. 6(c). It is also interesting that the heatcapacity peak corresponding to the intermediate tran-sition displays a dramatic increase in intensity as thefield increases to 4 T, after which it remains relatively

flat. This−→H ‖ c phase diagram is reminiscent of that of

a triangular lattice antiferromagnet with weak easy-axisanisotropy, and it is strikingly similar to that observedfor Rb4Mn(MoO4)3[17] and Ba3MnNb2O9 [21].

Upon applying the magnetic field perpendicular to c-axis the heat capacity peak corresponding to the TN2

transition shifts only slightly to higher temperatures,whereas the peak denoting the TN3 transition splits intotwo components that display dome-shaped profiles as afunction of magnetic field. The higher-temperature com-ponent that emerges from the zero-field heat capacitypeak, labeled as T ∗N3 in Fig. 6, is much broader and re-duced in amplitude. The phase diagram for this fielddirection is unexpectedly more complicated than that ob-served in other triangular lattice antiferromagnets.

7

FIG. 7. (Color online) Contour plot of the evolution of theneutron powder diffraction patterns recorded on cooling from110 K to 50 K. The data reveals the appearance of magneticBragg reflections of the type(1, 0, L=2n) corresponding tothe long-range magnetic ordering of the Mn1 ions occupyingthe honeycomb layers.

C. Zero-field magnetic order

1. Magnetic order of the honeycomb sublattice

Powder and single crystal neutron diffraction data col-lected below TN1 reveal additional scattering at low an-gles reflections, of the type(1, 0, L=2n). The evolution ofthe powder diffraction pattern across this first magnetictransition is shown in Fig. 7. Magnetic structures modelscompatible with the space group P63/m and the propa-gation vector k=(0, 0, 0) have been explored using boththe magnetic symmetry approach, using MAXMAGNprogram, [43] and the representation analysis with theprogram SARAH. [42]. Among the four possible max-imal magnetic space groups, the P6′3/m (#176.145) isthe only one that fits well all observed magnetic inten-sities. The magnetic structure at intermediate tempera-tures, 3 K 6 T 6 85 K, consists on a Neel-type antiferro-magnetic order characterized by antiparallel alignment ofnearest-neighbor Mn1 moments in the honeycomb layer.The magnetic moments at the Mn2 sites of the trian-gular layer remain disordered. The Mn1 moments areoriented parallel to the c-axis and the antiferromagnetichoneycomb layers are stacked ferromagnetically along thec-axis direction. The refined amplitude of the static mo-ment is 2.9(1)µB at 50 K , and it converges to 5.0(1)µB

at 1.7 K. The magnetic structure is depicted in Fig. 8.The magnetic moment orientation for each atomic po-sition is explicitly given in Table II. It is worth notingthat the Neel-type AFM ground state of the honeycomblattice is susceptible to undergo a spin-flop transition fora magnetic field applied parallel to c-axis.[45] Thus, thestep-like anomaly observed at about 11 T in the isother-mal magnetization measurements can be interpreted asa spin-flop transition.

TABLE II. Magnetic structure configuration ofK2Mn3(VO4)2CO3 for the temperature range 3 K 6 T6 85 K, described by k=(0, 0, 0) and the magnetic spacegroup P6′3/m.

Atom ( x, y, z ) ( ma, mb, mc )

Mn1 ( 1/3, 2/3, 0.00460(5) ) ( 0, 0, mc )

( 1/3, 2/3, 0.4954(5) ) ( 0, 0, mc )

( 2/3, 1/3, -0.00460(5) ) ( 0, 0, −mc )

( 2/3, 1/3, 0.50460(5) ) ( 0, 0, −mc )

Mn2 ( 0, 0, 1/4 ) ( 0, 0, 0 )

( 0, 0, 3/4 ) ( 0, 0, 0 )

FIG. 8. (Color online) Magnetic structure for the tempera-ture range 3 K 6 T 6 85 K, defined by a Neel-type antifer-romagnetic arrangement of Mn1 moments in the honeycomblayer, while the Mn2 in triangular layers remain paramag-netic. The ordered moments are aligned along the c direction.The successive antiferromagnetic planes are stacked ferromag-netically. The right panel display a projection of the structurealong c-axis, emphasizing the location of paramagnetic Mn2site on top or underneath the zero molecular field created bythe six surrounding Mn1 ordered moments.

2. Magnetic order of the triangular sublattice

A new set of magnetic reflections with propagation vec-tor k = ( 1

3 , 13 , 0) appear upon cooling below approxi-

mately 3.2 K. The temperature evolution of the ( 13 , 1

3 , 1)peak intensity is shown in Fig. 9. The observed transitionpoint agrees well with TN2 obtained from macroscopicmeasurements. Furthermore, a kink in the order param-eter curve is observed near 2.2 K, that matches the posi-tion of the second peak at TN3 in the heat capacity data.No change in intensity of the k = (0, 0, 0) -type magneticpeaks associated with the ordering of Mn1 honeycombsublattice was observed to occur at these low temper-atures. The spin reorientation suggested by the staticsusceptibility data to take place at TN3 is best capturedby following the temperature dependence of the ratio be-tween ( 1

3 , 13 , 1) and ( 1

3 , 13 , 3) peak intensities, shown in

the insert of Fig. 9. It is noticeable that the ( 13 , 1

3 , 3) ex-

8

1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 00 . 0

0 . 1

0 . 2

0 . 3

1 . 8 2 . 1 2 . 4 2 . 7 3 . 01 . 3

1 . 7

2 . 1

2 . 5T N 3

( 1 0 1 )

Inten

sity (a

rb. un

its)

T ( K )

T N 2

)3()1(

31313131

IIR =

)1( 3131 Inten

s. Ra

tio

T ( K )

T N 3

FIG. 9. (Color online) Temperature dependence of ( 13, 1

3,

1) magnetic peak intensity revealing the long-range magneticorder of Mn2 ions on the triangular layer at TN2 ≈ 3.2 K, followed by a spin reorientation at TN3 ≈ 2.2 K. The ab-sence of magnetic scattering contribution to the (1,0,1) re-flection across the two low-temperature transitions suggeststhat the intermediate magnetic order consists of a collinearup−up−down structure with modulated-amplitude (see textfor details). The insert displays the temperature dependenceof the ratio of the intensities ( 1

3, 1

3, 1) and ( 1

3, 1

3, 3). The

larger gain in intensity of ( 13, 1

3, 3) below TN3 demonstrates

that magnetic moments are rotating away from the c-axis toproduce a “Y”-type spin structure.

hibits a larger gain in intensity below 2.2 K. Consideringthat only the moment component perpendicular to themomentum transfer Q contributes to the magnetic in-tensities, the abrupt change in the relative intensity gainindicates that some magnetic moments are rotating awayfrom the c-axis. A rough estimate of the canting angleΘ = 54(5)◦ is obtained from the intensities ratio at T =1.7 K.

The new satellite peaks associated with the wavevectork = ( 1

3 , 13 , 0) that developed below 3 K have been at-

tributed to the long-range magnetic ordering of the man-ganese atoms (Mn2) in the triangular layer. The neutronscattering data is well described by a magnetic structurewith the static magnetic moments pointing along the c-axis, and amplitudes which follow the k-wavevector mod-ulation such as: mi = m0<[cos(2πk·ri+φ)].[46] Here, m0

represents the amplitude of the ordered moment in thezeroth cell, and φ is a phase factor. The moment distri-bution for a phase φ = 0 along the a axis is m0, −m0/2,−m0/2, while for a choice of φ = π/2 the sequence be-

comes 0, −√

3/2m0,√

3/2m0. The moments are thusfully compensated inside the plane. Furthermore, the se-lection rule L=2n+1 indicates that successive triangularlayers are stacked antiferromagnetically along the c di-rection. It is important to realize that we are discussingthe ordering of the static moment. Other possible disor-dered or strongly-fluctuating magnetic components thatmay exist do not contribute to the Bragg intensities. The

(a)

(b)

FIG. 10. (Color online) (a) Three-sublattice static magneticorder of Mn2 atoms (inside the triangular layer) in the inter-mediate temperature 2.2 K 6 T 6 3 K. The magnetic mo-ments are aligned parallel to the c-axis in an up− up− downconfiguration. The moment amplitude follows the k=( 1

3, 1

3,

0) modulation, mi = m0<[cos(2πk · ri], resulting in a fullycompensated magnetization inside the plane. Successive tri-angular layers are stacked in an antiparallel manner. The hon-eycomb layers maintain the AFM structure shown in Fig. 8but are omitted for clarity. A view along the c-axis of a three-sublattice collinear magnetic order is shown in the right panel.(b) Three-sublattice magnetic structure below 2.2 K. Two ofMn2 atoms develop in-plane (ab ordered spin components toform a planar canted Y-type structure, where moments arerotated by about 120◦ between neighboring sites. The struc-ture remains bilayer with the moments of adjacent layers be-ing antiparallel to each other. The right panel of the figuredisplays the projection of the spin structure on the ab basalplane.

only possibility to obtain a uniform amplitude across allMn2 magnetic sites is if an additional k = (0, 0, 0) compo-nent would be present. This is important to understand,since most publications refer to this intermediate state,which is specific to the triangular lattice systems withweak easy-axis anisotropy, as the uud phase, without anyfurther description of its single-k or double-k character.In our case, the lack of k = (0, 0, 0) contribution (andthus of net magnetization) is obvious from the magneti-zation measurements. Furthermore, a careful measure-ment of the temperature evolution of the (1,0,1) reflec-tion across the two low-temperature transitions, shown inFig. 9, indicates no additional magnetic scattering as one

9

TABLE III. Magnetic spin configuration of Mn2 site for thetemperature range 2.2 K 6 T 6 3 K, and below 2.2 K, de-scribed by k=( 1

3, 1

3, 0) and magnetic space groups P6′3/m

and P2′1, respectively. The atomic coordinates (x′, y′, z′)are defined for the 3 x 3 x 1 magnetic super-cell. In the ex-panded lattice there are three non-equivalent magnetic sitesand moments projections in the ab plane and c-direction fol-low the modulation imposed by k wave-vector (as described inthe text). We constrained the moment directions to form anideal 120◦ structure with the in-plane projection mab ‖[1,1,0]direction. The refined magnitude of the static moment m0 at1.7 K is 3.7(2)µB .

2.2 K 6 T 6 3 K T < 2.2 K

P6′3/m P2′1

Atom (x′ , y′ , z′ ) (ma,mb,mc) (ma,mb,mc)

Mn21 (0,0,1/4) (0,0,m0/2) (m0/2,−m0/2,m0/2)

(0,2/3,3/4) (0,0,−m0/2) (m0/2,−m0/2,−m0/2)

(1/3,1/3,3/4) (0,0, −m0/2) (m0/2,−m0/2,−m0/2)

(1/3,2/3,1/4) (0,0,m0/2) (m0/2,−m0/2,m0/2)

(2/3,0,3/4) (0,0,−m0/2) (m0/2,−m0/2,−m0/2)

(2/3,1/3,1/4) (0,0,m0/2) (m0/2,−m0/2,m0/2)

Mn22 (0,0,3/4) (0,0,−m0/2) (−m0/2,m0/2,−m0/2)

(0,2/3,1/4) (0,0,m0/2) (−m0/2,m0/2,m0/2)

(1/3,1/3,1/4) (0,0,m0/2) (−m0/2,m0/2,m0/2)

(1/3,2/3,3/4) (0,0,−m0/2) (−m0/2,m0/2,−m0/2)

(2/3,0,1/4) (0,0,m0/2) (−m0/2,m0/2,m0/2)

(2/3,1/3,3/4) (0,0,−m0/2) (−m0/2,m0/2,−m0/2)

Mn23 (0,1/3,1/4) (0,0,−m0) (0,0,−m0)

(0,1/3,3/4) (0,0,m0) (0,0,m0)

(1/3,0,1/4) (0,0,−m0) (0,0,−m0)

(1/3,0,3/4) (0,0,m0) (0,0,m0)

(2/3,2/3,1/4) (0,0,−m0) (0,0,−m0)

(2/3,2/3,3/4) (0,0,m0) (0,0,m0)

would expect from an uncompensated magnetic compo-nent. Our amplitude modulated uud (single-k) magneticstructure can be described by the magnetic space groupP6′3/m in a 3 x 3 x 1 magnetic supercell. Within this ex-panded unit cell, there are three non-equivalent magneticsites (Mn2i, i=1,2,3) that follow the modulation imposedby the k wavevector. Note that the same magnetic spacegroup can be used to describe a uniform uud double-kstructure. The proposed magnetic structure model forMn2 site in the temperature range 2.2 K 6 T 6 3 Kis shown in Fig. 10(a), and the site-specific orientationof magnetic moments is given in Table III. In Fig. 10we omitted the Mn1 honeycomb layers that preserve thesame spin arrangement as shown in Fig. 8.

As evidenced by the order parameter curve in Fig. 9, anin-plane spin component develops below 2.2 K resultingin a canting of the ordered moment away from the c direc-tion. The in-plane spin component exhibits a modulationin amplitude in accord to k = ( 1

3 , 13 , 0). The natural ten-

dency toward an uniform static moment magnitude forall sites is fulfilled by the development of in-plane com-ponents only on the two sites with reduced mc staticmoments. This can be realized by considering an offsetbetween phase factors φ of the two moment components,mc and mab of π/2. Ideally, this produces a 120◦ spinconfiguration in the plane containing the c-axis, and isknown in literature as the planar “Y” canted structure.Similar to the intermediate phase, the successive Mn2triangular layers remain aligned antiparallel with respectto each other. This low-temperature ordered state inher-its the broken lattice symmetry while also breaking thespin-rotational symmetry within the ab plane and thuscan be viewed as an magnetic analogue to a supersolidstate.[9, 10] The magnetic structure can be described us-ing the monoclinic magnetic subgroup P2′1 (#4.9) op-erating on a 3 x 3 x 1 magnetic supercell. We notethat the in-plane spin component can align along anydirection and, experimentally, this cannot be uniquelydetermined. For fitting the single-crystal neutron datawe selected a model were the spins form an ideal 120◦

structure in the (1,1,0) plane. The refined magnitude ofthe static moment is 3.7(2)µB , significantly smaller thanthat expected for spin S = 5/2. Similar reduced orderedmoment of about 3.9(5)µB was reported for the S=5/2TLA system RbFe(MoO4)2.[12] One possibility is thatthe static moment does not reach full saturation at themeasured temperature of 1.7 K, but it could also be thatthe geometrical frustration are causing strong quantumfluctuations. The magnetic moments arrangement in thiscanted structural model for the temperatures below 2.2 Kis summarized in Table III and illustrated in Fig. 10(b).

D. Spin-wave excitations

1. Magnetic excitation spectrum of the honeycombsublattice

Inelastic neutron-scattering measurements performedwith the incident energy Ei = 25 meV revealed a welldefined spin-wave branch emerging from the (1,0,0) mag-netic peak and extending to an energy transfer of approx-imately 22 meV (see Fig. 11(a)). There is no discernabledispersion along the c-axis indicating very weak couplingbetween magnetic layers. As visible in Fig. 11(b), thetwo-dimensional correlations is evidenced by rods of scat-tering along [0,0,L] direction in the contour plot of the re-ciprocal lattice plane corresponding to the energy trans-fer range 6 - 10 meV. The scattering intensity along Qfollows the decay expected for the magnetic form factorof Mn2+ magnetic ion. As described in Section II thesample used for these measurements was relatively small(0.3 g) and consisted of coaligned crystals only along c-axis direction with random in-plane orientation. This hasnegatively impacted the data quality and the amount ofinformation that can be extracted from it. Thus, cer-tain assumptions which simplified the analysis had to be

10

FIG. 11. (color online) (a) Energy-momentum slice of the inelastic neutron scattering data measured with Ei = 25 meV atthe HYSPEC spectrometer. A spin-wave branch emerges from the (1,0,0) magnetic peak and extends to an energy transfer ofapproximately 22 meV. (b) Contour plot of the (H,0,L) reciprocal plane for energy transfer integrated between 6 and 10 meV.The rod of scattering along [0,0,L] direction indicates very weak coupling between magnetic layers. (c) Calculated spin-wavespectrum using a Heisenberg model that includes first- and second-neighbor in-plane exchange interactions (J1 = 1.55 meV,J2 = -0.45(3) meV) in the S=5/2 honeycomb lattice. (d) Low-energy magnetic excitations that develop around the magneticpeaks at (1/3, 1/3, 1) and (2/3, 2/3,1). (e) 2D slice of momentum space corresponding to energy transfer range 0.2 - 0.3 meVrevealing the quasi-two-dimensional character of the excitations. (f) The calculated spin-wave spectrum of S=5/2 triangularlattice Heisenberg antiferromagnet characterized by nearest-neighbor exchange interactions (J= 0.08 meV) and a easy-axisanisotropy (Dt = 0.03meV). The spectrum is averaged over all in-plane Q directions to reproduce the experimental conditions.

made. Excitations from a honeycomb lattice are typicallydescribed by using a Heisenberg model that includes first-, second-, and third-neighbor in-plane exchange interac-tions (J1, J2, J3), an exchange interaction between planesJc, and an anisotropy term Dh. [48–50]. Based on our ex-perimental resolution we can estimate that the interplanecoupling Jc and the anisotropy are two order of mag-nitude smaller than the in-plane exchange interactions.The J3 is not expected to have a significant contribution,and is beyond our ability to determine with the availabledata. The strength of nearest-neighbor exchange interac-tion can be estimated from the Curie-Weiss temperature(ΘCW = -215 K), J1 = 3kB ΘCW /ζ S(S+1). [47] Consid-ering the number of nearest-neighbor ζ = 3 and the spinvalue the S =5/2 , one obtains J1 = 2.1(3) meV. Fix-ing the J1 value, the measured inelastic spectrum can bewell reproduced by using an additional second-neighborexchange J2 = -0.3(1) meV. The ratio between the J1and J2 is consistent with that expected for a Neel-typemagnetic ground state in a honeycomb lattice.[48, 49]

In the absence of measurable spin-wave excitation gap,the magnitude of the axial anisotropy term Dh, responsi-ble for the spins alignment along c-axis, can be obtained

from the value of the field-induced transition observednear 11 T. For a honeycomb lattice ordered in a Neel-typeAFM magnetic structure a spin-flip transition is expectedto occur at a field Hsf = 2S

√Dh(3J1 −Dh)/gµB . This

yields an anisotropy Dh of approximately 0.010(2) meV.The calculated spin-wave excitation spectrum using theaforementioned exchange parameters over an averagedin-plane momentum transfer is shown in Fig. 11(c).Our estimates show that the dipole-dipole interactioncan present an important contribution to the uniax-ial anisotropy of Mn2+ (S = 5/2) ions in the honey-comb layer. The intralayer and interlayer distances,3.01 A and 5.5 A, yield an anisotropy comparable inmagnitude (10−2 meV) to those observed and calcu-lated for the classical antiferromagnets MnO [51, 52] andMnF2 [53, 54], where the nearest-neighbors distances areabout 3.1 A and 3.3 A, respectively.

2. Magnetic excitation spectrum of the triangular sublattice

The T = 1.7 K inelastic data contains additional low-energy magnetic excitations that are due to cooperative

11

fluctuations of magnetically ordered spins in the triangu-lar layer. As shown in Fig. 11(d) spin-wave excitationsdevelop around the magnetic Bragg peak positions ( 1

3 ,13 , 1) and ( 2

3 , 23 , 1) and have a bandwidth of approx-

imately 0.8 meV. The quasi-two-dimensional characterof the excitations is revealed by the rod-like scatteringalong [0,0,L] direction in the 2D slice plot of momentumspace corresponding to energy transfer range 0.2 - 0.3meV, displayed in Fig. 11(e). Keeping in mind that onlyone-third of magnetic ions occupy the triangular layer, itis not surprising that the INS data quality for these low-energy excitation is even more affected by the small sam-ple mass and its random in-plane orientation. In addi-tion, a diffusive quasielastic component seems to overlapwith the spin-waves, accounting for some of the missingordered moment. Such diffuse scattering was previouslyobserved in other triangular systems and was attributedto uncorrelated trimers that could survive well below theordering temperature. [55] Nonetheless, from the band-width of the spin-wave spectrum one can estimate thevalue of a nearest-neighbor exchange interaction (J) asapproximately 0.08(1) meV. As pointed out in the previ-ous sections, the magnetic behaviour observed from thetriangular layer suggests the existence of a weak easy-axis anisotropy (Dt). The energy gap expected from suchanisotropy could not be resolved in our INS data. How-ever, one can determine the value of Dt from its relation-ship to the two-step transition temperaturesDt ≈ J (TN2

- TN3)/TN2, or from the the saturation magnetic field ofthe triangular magnetic sublattice HS = (9J-2Dt)S/gµB

= 14.3 T. [3] The estimated value of the axial anisotropyis Dt ≈0.028(2) meV, which is nearly twice as large thanthat in the honeycomb layer Dh ≈0.010(2) meV. Consid-ering that the interatomic distance for Mn ions locatedin the triangular layer are much larger, ∼ 5.2 A, a muchsmaller dipolar contribution of the order of 10−3 meV isestimated for this site. It therefore appears that the Dt

magnetic anisotropy is mainly due to higher-order termsin the interplay between crystalline-field and spin- or-bit couplings and covalency effects, as demonstrated forother 3d5 systems [56–58] For instance, spin-orbit cou-pling effects were shown to produce an anisotropy D ofabout 0.06 meV in Mn2+WO4 . [59] One shall also stressthat the trigonal bipyramidal coordination of Mn2+ inthe triangular layer is anticipated to induce a strongerspin-orbit-induced anisotropy than the more-regular oc-tahedral environment of the magnetic ions in the honey-comb layer. [60] The calculated in-plane averaged spin-wave spectrum using the estimated J and Dt parametersis shown in Fig. 11(f). The inelastic magnetic spectrumof the intermediate ordered state uud has also been mea-sured at 2.8 K and it shows no discernible change withrespect to that observed at the base temperature.

E. Field-induced magnetic phases in triangularsublattice

The effect of applied magnetic fields on the magneticorder of K2Mn3(VO4)2CO3 has been studied using theCORELLI and HYSPEC instruments for fields up to 8T. For such magnetic fields, only the triangular mag-netic layer shows a notable magnetic response. Fig-ure 12 shows the evolution of the elastic scattering in the(H,H,L) reciprocal plane for the magnetic fields of 0, 0.5T and 5 T, applied along the c-axis direction. The datareveal changes in intensity of satellite magnetic peaksassociated with the propagation vector k=( 1

3 , 13 , 0), as

well as the development of strong diffuse scattering in-tensity along the L-direction. The diffuse scattering be-comes progressively stronger with increasing field, andhas a well-structured profile. A cut along the L directionof the diffuse scattering with prominent peaks at L=0,±0.5, ±1 and ±1.5 is displayed in Fig. 12(d). A morethorough survey of the field effect on the intensities of( 13 , 1

3 , 0) and ( 13 , 1

3 , 1) magnetic Bragg peaks has beencarried out using HYSPEC instrument and is shown inFig. 13. One can observe an immediate increase in in-tensity of ( 1

3 , 13 , 0) peak as the field is raised above zero.

Furthermore, the ( 13 , 1

3 , 1) peaks intensity undergoes aslight decrease with a minimum near 1.5 T, followed by arecover with a local maximum at about 5.5 T, and thena strong decrease to a nearly complete disappearance atthe highest measured field µ0H = 8 T. The profile of thefield dependence reveals the existence of at least four dif-ferent spin configurations before reaching the saturation.

When the magnetic field is applied along the [1,1,0]direction, the evolution of magnetic peaks is strikinglydifferent. Representative slices of the (H,H,L) recipro-cal plane obtained at fields of 0, 1.5 T, 3.5 T and 5 T arepresented in Figure 14. The 1.5 T data reveals the ap-pearance of well-defined ( 1

3 , 13 , 0) and ( 2

3 , 23 , 0) magnetic

peaks and no diffuse scattering along the L-direction. At3.5 T the magnetic peaks appear to relocate along L-direction to a slightly incommensurate lattice vector ξ =0.654(1). As the magnetic field is ramped up to 5 T, themagnetic phase recovers its commensurate state with k= ( 1

3 , 13 , 0). The order-parameter profiles of the commen-

surate and incommensurate magnetic states as a functionof the field at 1.7 K, are shown in Fig. 15. One can seethat the incommensurate phase forms at about 2 T anddisappears near 4.5 T. There is no detectable change inthe incommensurability for this field range. Above 4.5T, the commensurate (1

3 , 13 , 1) peak reappears but does

not recover all its intensity, and exhibits a slow decreasewith increasing the field. The phase-diagram revealedby neutron scattering measurement sheds light on theorigin of the split into two components of the heat ca-pacity low-temperature peak, presented in Fig. 6. It isplausible that the heat capacity peak at T ∗N3 is relatedto the commensurate-incommensurate phase transition.The intermediate-field incommensurate state appears tobe stable only over a finite temperature range, which sug-

12

- 2 - 1 0 1 2

0 . 1

0 . 2

0 . 3

0 . 3 0 . 5 0 . 7- 2- 1012 � H I I c , 0 T

[0,0,L

] (r.l.u

)( a ) ( b )

0 . 3 0 . 5 0 . 7

5 T( c )

0 . 3 0 . 5 0 . 7[ H , H , 0 ] ( r . l . u . )

0 . 5 T

Inten

sity (a

rb. un

its)

[ 0 0 L ] ( r . l . u . )

5 T 0 . 5 T 0 T

( d )

FIG. 12. (Color online) (a-c) Evolution of scattering intensity in the (H,H,L) reciprocal plane for different magnetic fieldsapplied along the c-direction. Data demonstrates the development of strong structured diffuse scattering intensity along theL-direction. (d) Cut along along the diffuse scattering at ( 2

3, 2

3, L) position revealing prominent peaks centered at L = 0, ±0.5,

±1 and ±1.5.

0 1 2 3 4 5 6 7 80 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0 cH / /�

( i v ) ( i i i ) ( i i ) ( i )

( 1 / 3 , 1 / 3 , 1 ) ( 1 / 3 , 1 / 3 , 0 )

� � H ( T )

Relat

ive In

tensity

, T = 1 . 7 K

FIG. 13. (Color online) Field dependence of ( 13, 1

3, 0) and

( 13, 1

3, 1) peak intensities at T = 1.7 K and

−→H ‖ c exposing

the presence of at least four intermediate magnetic states be-tween the zero field ground state and the full magnetizationsaturation.

gests that thermal fluctuations play a key role in its for-mation.

Based on the recorded integrated intensities, modelsof the magnetic structures corresponding to each stage

of the−→H ‖ c or

−→H ⊥ c phase-diagrams have been con-

structed. The available models were further scrutinizedby considering the results of Monte-Carlo simulationsperformed on a spin Hamiltonian of a TLA described us-ing the determined nearest-neighbor exchange coupling(J = 0.08 meV) and easy-axis anisotropy (Dt = 0.028meV), as well as an approximate weak inter-layer cou-pling Jz1 = 0.005 meV. The calculated magnetization

curves for T = 0 K and 1.7 K, along with the predictedmagnetic configurations at 0 K are presented in Fig. 17.The zero-field heat-capacity curve, shown in the inset ofFig. 17, reproduces the two successive magnetic transi-tions at low-temperatures, while the calculated isother-mal magnetization curve captures the experimental datavery well.

The appearance of L = even reflections in neutrondata suggests the change in the spin-stacking sequencebetween triangular layers, with part of the spins tendingto align parallel to each other. Therefore, in the low-field

regime of µ0−→H ‖ c, the “Y” magnetic configurations of

the adjacent triangular layers are no longer compensat-ing each other as in the zero-field structure but followthe field direction to produce a net magnetization alongthe c-axis. The determined low-field magnetic structureis shown in Figs. 17(a) and 18(i). It consists of two tri-angular layers, as the chemical crystal structure, withthe stacked spins rotated by approximately 2π/3 rela-tive one to another. Upon further increasing the mag-netic field, the spins in each layer are continuously ro-tating to arrange parallel to the c-axis into a 1/3 MST

plateau state uud. This collinear configuration is dis-played in Fig. 18(ii). Note that such ordered state has adouble-k nature, as k = (0, 0, 0) contribution adds to thezero-field k =( 1

3 , 13 , 0). Due to the geometrical degener-

acy, there are three possible magnetic configurations foreach triangular layer: uud, udu or duu, that we will re-fer to in the following as A , B and C magnetic layers.These magnetic configurations are presented in Fig. 16.The existence of stacking disorder of neighboring layersis apparent in the presence of the structured diffuse scat-tering along the L-direction. The relative intensities at( 13 , 1

3 , L) peaks positions for L = 0, 1 and 0.5 can beexplained by an admixture of two magnetic polytypes:the bilayer AB − AB and a four-layer superstructure

13

0 . 3 0 . 6[ H , H , 0 ] ( r . l . u . )0

3

6

Intensity (arb. units)( d ) 5 T

0 . 3 0 . 6

0

1

2

3cH ⊥

�( a ) 0 T

[0,0,L

] (r.l.u

.)

0 . 3 0 . 6

( c ) 3 . 5 T

0 . 3 0 . 6

( b ) 1 . 5 T

FIG. 14. (Color online) (a-d) Representative plots of the (H,H,L) reciprocal plane obtained at T = 1.7 K for magnetic fieldsapplied along the [1,1,0] direction. The µ0H = 3.5 T data shows a shift along the L-direction of magnetic peaks to a slightlyincommensurate position δL=0.654. At 5 T, the magnetic phase recovers its original commensurate state with the wavevectork = ( 1

3, 1

3, 0).

0 1 2 3 4 5 6 7 80 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0cH ⊥

� ( 1 / 3 , 1 / 3 , 1 ) ( 2 / 3 , 2 / 3 , 2 . 6 5 )

� 0 H ( T )

Relat

ive In

tensity

(1/3,

1/3,1)

T = 1 . 7 K

( i v ' ) ( i i i ' ) ( i i ' ) ( i ' )

0 . 0

0 . 1

0 . 2

0 . 3Int

ensity

(2/3,

2/3,2.

265)

FIG. 15. (Color online) Field dependence of ( 13, 1

3, 1) and

( 23, 2

3, 2.654) magnetic peak intensities at T = 1.7 K mapping

out the phase diagram for−→H ‖[1,1,0]. One observes that

the intermediate incommensurate phase separating two otherfield-induced commensurate states begins at about 2 T anddisappears near 4.5 The dashed line is a guide to the eye.

AABB − AABB. The phase ratio of the two observedpolytypes at T = 1.7 K is found to be nearly 4:1. Figure18 displays only the dominant AB − AB bilayer mag-netic polytype. While the spin-stacking disorder couldbe explained by the presence of small structural imper-fections in the crystal, the four-layer polytype is indica-tive of the presence of second-nearest-neighbor inter-layerinteractions. This is very surprising since the separationbetween second-nearest-neighbor triangular layers is verylarge (≈ 22.4 A), while each Mn atom is located in a zero-molecular-field of the adjacent honeycomb sublattice. Aspresented in Fig. 16, a bilayer structure is expected forAFM nearest-neighbor interlayer coupling (Jz1) and FM

A B C

Jz1(interlayer NN)

Jz2(interlayer NNN) Stacking sequence

AFM FM AB - AB (bilayer)AFM AFM ABC - ABC (three-layer)FM AFM

(|Jz2|>|Jz1|/2)AABB - AABB (four-layer)

FIG. 16. The three possible plateau state configurationsdue to geometric degeneracy of triangular layer: A (uud), B(udu) or C (duu). The spin-stacking sequence of successivetriangular layer is determined by the nature (antiferromag-netic (AFM) or ferromagnetic (FM)) and relative strengthsof nearest-neighbor (Jz1) and second-nearest-neighbor (Jz2)interlayer interactions.

second-nearest-neighbor interlayer interaction (Jz2). Incontrast, a four-layer superstructure AABB −AABB isonly possible for FM nearest-neighbor interlayer interac-tion (Jz1) weaker than twice the second-nearest-neighborinterlayer interaction (2 | Jz2 |>| Jz1 |). The experimen-tal restraints prevented from carrying out measurementbelow 1.7 K to determine if the disorder and polytypesmixing ratio exhibits any temperature dependence. Thismight be expected if thermal and quantum fluctuationsare involved in the effective interlayer coupling via orderfrom disorder mechanism (effective interaction mediatedby fluctuations of the magnetic moments in the honey-

14

FIG. 17. (Color online) Calculated magnetization curves for−→H ‖ c (a) and

−→H ⊥ c (b) from a triangular magnetic lattice

characterized by the NN exchange interaction and easy-axisanisotropy determined experimentally. Temperature effect onthe magnetization plateau is shown by comparing the T =0 K (black curve) with T = 1.7 K (red curve) calculations.The inset shows the computed zero-field heat-capacity curvethat reproduces well the observed successive transitions 2.2K and 3 K. Predicted field-induced bilayer spin states fromour classical Monte Carlo simulations are sketched along themagnetization curves, with the planar ac structures of theadjacent layers plotted on top of each other.

comb sublattice). The potential of quantum fluctuationsto generate interlayer coupling via the mechanism of or-der from disorder has so far been considered for the caseof the body-centered tetragonal lattice [61, 62] but ourexperimental observations raise the possibility that sim-ilar mechanism can occur in triangular layers mediatedby a honeycomb lattice. We recall that in our systemthe Mn ions at the triangular layer are located exactlyon top or underneath the hollow center of the Mn hon-eycomb lattice. It is worth noting that evidence of twonon-equivalent modulations with L = 0 and 0.5 and un-derlaying disorder has been also reported in the staggeredthree-layer TLA compound LiCrO2. [63]

At magnetic fields above 5.5 T the collinear state startsto evolve towards an oblique “2:1” structure, as shownin Fig. 18(iii). As the field increases the magnetic mo-ment that is pointing opposite to the field direction iscontinuously rotating to align with the field. In the ab-sence of neutron data for fields larger than 8 T, we used

classical Monte Carlo simulations to define the magneticstate. The calculations indicate that the moments arepass through a canted “V” spin configuration before thefull saturation (MS), as shown in Fig. 18(iv). The field-induced magnetic configurations within the layer are con-sistent with the theoretical predictions and with thoseobserved in similar triangular systems. The dominant bi-layer magnetic polytypes (i),(ii) and (iii) found at low andintermediate fields are consistent with the configurationspredicted by Gekht and Bondarenko for weakly-coupledtriangular antiferromagnets. [5] However, the presence ofmultiple magnetic polytypes including the four-layer su-perstructure is indicative of more complicated interlayercoupling and calls for more thorough theoretical investi-gations.

In the case of the field applied perpendicular to c-axis,along the [1,1,0] direction, the low-field magnetic struc-ture can be looked as a distortion of the zero-field “Y”structure where the two canted spins are rotating to-wards the field direction. This reorientation does notalter the lattice periodicity along the c-direction andthe magnetic structure remains bilayer up to about 2T. The low-field magnetic structure that describes wellthe low-field (0 < µ0H ≤ 2 T) neutron data is shownin Fig. 18(i′). The rotation of the spins continues untilthey reach a nearly-orthogonal arrangement, at about 2T, with one spin pointing along the in-plane field direc-tion and the other two aligned nearly parallel to c-axis(i.e. up-right-down or urd magnetic structure). Similarlyto the plateau case described by A, B, C configurations(Fig. 16), any permutation of the three spin orientationsis possible for such an orthogonal model. Our diffrac-tion data clearly indicates that a three-layer superstruc-ture is stabilized at intermediate fields between approx-imately 2 T and 4.5 T. This superstructure is shown inFig. 18(ii′). The stacked spins are alternating their ori-entations across the three layers as, udr, rud and dru,equivalent to a ABC−ABC configuration that becomesstable when both nearest-neighbor and second-nearest-neighbor are antiferromagnetic. There is, however, asmall incommensurability of the modulation along c-axis(k=( 1

3 , 13 , 0.654)), and therefore the spin orientations un-

dergo slight undulations around the ideal directions. Thiscould arise from a competition between nearest-neighborand second-nearest-neighbor inter-layer interactions. Itis interesting to point out that the c-axis modulation ofthe magnetic order appears to be locked-in to a constantvalue for the entire range (2 < µ0H < 4.5 T), despitethe fact that the magnetization curve shows a uniformincrease suggesting a continuum rotation of the spins to-wards the field direction. It is also important to note thatno stacking disorder is observed, which indicates thatthis intermediate-field incommensurate magnetic phaseis well stabilized. Our classical Monte Carlo simulation,which only accounts for a weak nearest-neighbor inter-layer exchange, predicts for this intermediate-field rangeonly a bilayer spin arrangement in the form of udr – rud(see Fig. 17(b)). Considering the stability of the three-

15

FIG. 18. (Color online) Models of the static magnetic orders induced by magnetic fields applied parallel to c (i-iv) and along[1,1,0] (i’-iv’) as determined from neutron diffraction data. The figure only presents the dominant bilayer structure but the (i-iii)states are susceptible to spin-stacking disorder and four-layer superstructure formation as described in the text. A three-layerstacking sequence (ii’) was found for the in-plane field orientation where the moments form a nearly orthogonal configurationand are permuting orientations across the three layers.

layer phase over a finite temperature interval, one mayinfer that the second-nearest-neighbor interlayer couplingare strengthened by quantum-mechanical coupling viaorder-by-disorder mechanism. [1, 61, 62]. Upon increas-ing the magnetic field above 4.5 T the magnetic structurerecovers its original bilayer form, with a tilted variant ofudr – rud configuration, labeled as “W”, illustrated inFig. 18(iii′). Our Monte Carlo calculation predicts thatbefore saturation the spins may experience an additionalchange in the stacking pattern as drawn in Figs. 17(b)and18(iv′).

IV. SUMMARY

The structural and magnetic properties of the vana-date carbonate K2Mn3(VO4)2CO3 have been studied bymeans of magnetization, specific-heat and neutron scat-tering measurements. The structure consists of an alter-nate stacking of honeycomb and triangular layers madeof edge sharing MnO6 octahedra and MnO5 trigonal-bipyramids, respectively. Contrary to what was previ-ously reported in Ref. 33, and in agreement with thefirst-principles calculations of Ref. 34, we found that bothlayers consists of Mn2+ in in high-spin state. The two

magnetic layers act as nearly independent magnetic sub-lattices with magnetic interactions of completely differ-ent energy scales. The honeycomb magnetic sublatticeorders at about 85 K in a Neel-type AFM magnetic struc-ture, with the Mn moments oriented parallel to the c-axis.This AFM state produces a zero-molecular-field on Mnatoms located in the triangular sublattice, which ordermagnetically in two steps at much lower temperatures.Analysis of neutron diffraction data show that the firsttransition, at ≈ 3 K, is to an amplitude-modulated AFMcollinear uud magnetic structure described by a propa-gation vector k = ( 1

3 , 13 , 0). The second magnetic transi-

tion, at 2.2 K, corresponds to the development of in-planespin components that leads to the formation of a canted“Y” magnetic structure in a plane containing the c-axis.Successive triangular layers are aligned antiferromagneti-cally along the c-direction. Magnetization measurementsperformed under applied magnetic fields revealed a 1/3magnetization plateau and a saturation near 14.3 T ofthe triangular sublattice, and a spin-flop reorientationof the honeycomb sublattice at about 11 T. On the ba-sis of these field-induced transitions, the axial magneticanisotropy of the octahedrally coordinated Mn ions inthe honeycomb lattice was estimated to be 0.010(2) meV,while the anisotropy in the pentagonal-bipyramidal en-

16

vironment of the triangular lattice was found to be ap-preciably larger, about 0.028(2) meV. Inelastic scatter-ing measurements revealed spin-wave excitations with astrong two-dimensional character. The excitation branchassociated to the honeycomb ordered state was describedby using a Heisenberg model that includes first-neighborJ1 = 2.1(3) meV, and second-neighbor J2 = -0.3(1) meVin-plane exchange interactions. On the other hand, thespin-wave spectrum of the triangular sublattice yieldedan estimate of the nearest-neighbor exchange interactionvalue of 0.08(1) meV. The determined exchange interac-tions were used for Monte Carlo simulations to obtain thetemperature – magnetic field phase diagram. A system-atic neutron diffraction study for applied magnetic-fieldalong the c-axis revealed that the triangular magnetic lat-tice undergoes at least four intermediate magnetic phasesbefore reaching the saturation. These phases are gener-ally consistent with the previous theoretical predictionsfor TLA and with our Monte Carlo simulations and in-clude the canted “Y”, collinear uud, canted “2:1” and“V” in-layer spin configurations. Nevertheless, some dis-crepancies have been identified with regard to the spin-stacking sequence of neighboring layers. A stacking dis-order and a mixture of bilayer and four-layer magneticpolytypes was revealed by the presence of highly struc-tured magnetic diffuse scattering along the L-direction.An applied magnetic field perpendicular to c-axis is alsofound to produce at intermediate fields a novel magneticstate that exhibits a three-layer periodicity along the c-direction. In the three-layer structure the spins are al-ternating orientations in a quasi-orthogonal arrangementwith two perpendicular and one parallel to the field di-rection. The appearance of such magnetic superstruc-tures indicate that at finite temperature and interme-diate fields the second nearest-neighbor interlayer inter-actions cannot be ignored, and may become instrumen-tal in generating new magnetic orderings. This raisesthe possibility that subtle thermal and quantum fluctua-tions may generate effective interlayer tunneling betweenthe triangular layers separated by honeycomb lattices.Furthermore, the formation of multiple spin-stacking se-quences in a single material is very remarkable since eachsuperstructure requires different types (AFM or FM)

and different relative strengths of the interlayer nearest-neighbor and second-nearest-neighbor interactions. Wehope that our findings will stimulate further theoreticalstudies and launch new considerations of field-inducedmagnetic phase diagrams in weakly-coupled triangularlattices.

ACKNOWLEDGMENTS

Work at the Oak Ridge National Laboratory, was spon-sored by US Department of Energy, Office of Science,Basic Energy Sciences, Scientific User Facilities Division(neutron scattering) and Materials Sciences and Engi-neering Division (magnetization and heat capacity analy-sis). The authors also acknowledge the financial supportfrom the National Science Foundation under grant no.DMR-1410727. A portion of this work was performed atthe National High Magnetic Field Laboratory (NHMFL),which is supported by NSF Cooperative Agreement No.DMR-1157490 and the State of Florida. We acknowledgeStan Tozer for use of his 16 T PPMS which is supportedby the NHMFL and the Center for Actinide Science andTechnology (CAST), an EFRC funded by the DOE - BESunder Award Number de-sc0016568. VOG thanks Xiaop-ing Wang for the help provided with the TOPAZ singlecrystal data refinement, and Igor Zaliznyak for useful dis-cussions.

Notice: This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725with the U.S. Department of Energy. The UnitedStates Government retains and the publisher, by ac-cepting the article for publication, acknowledges thatthe United States Government retains a non-exclusive,paid-up, irrevocable, world-wide license to publish or re-produce the published form of this manuscript, or al-low others to do so, for United States Government pur-poses. The Department of Energy will provide pub-lic access to these results of federally sponsored re-search in accordance with the DOE Public Access Plan(http://energy.gov/downloads/doe-public-access-plan).

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