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PHYSICAL REVIEW B 96,024445 (2017)

J1-J2square lattice antiferromagnetism in the orbitally quenched insulator MoOPO4

L. Yang,1,2M. Jeong,1,*P. Babk e v i ch, 1V. M . K a tu ku r i , 3B. Náfrádi,2N. E. Shaik,1A. Magrez,4H. Berger,4J. Schefer,5

E. Ressouche,6M. Kriener,7I. Živkovi ´

c,1O. V. Yazyev,3L. Forró,2and H. M. Rønnow1,7,†

1Laboratory for Quantum Magnetism, Institute of Physics, Ecole Polytechnique Féderale de Lausanne, CH-1015 Lausanne, Switzerland

2Laboratory of Physics of Complex Matter, Institute of Physics, Ecole Polytechnique Féderale de Lausanne, CH-1015 Lausanne, Switzerland

3Chair of Computational Condensed Matter Physics, Institute of Physics, Ecole Polytechnique Féderale de Lausanne,

CH-1015 Lausanne, Switzerland

4Crystal Growth Facility, Institute of Physics, Ecole Polytechnique Féderale de Lausanne, CH-1015 Lausanne, Switzerland

5Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institute, CH-5232 Villigen, Switzerland

6Université Grenoble Alpes, CEA, INAC, MEM, F-38000 Grenoble, France

7RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan

(Received 17 May 2017; published 28 July 2017)

We report magnetic and thermodynamic properties of a 4d1(Mo5+) magnetic insulator MoOPO4single crystal,

which realizes a J1-J2Heisenberg spin-1/2 model on a stacked square lattice. The speciﬁc-heat measurements

show a magnetic transition at 16 K which is also conﬁrmed by magnetic susceptibility, ESR, and neutron

diffraction measurements. Magnetic entropy deduced from the speciﬁc heat corresponds to a two-level degree of

freedom per Mo5+ion, and the effective moment from the susceptibility corresponds to the spin-only value. Using

ab initio quantum chemistry calculations, we demonstrate that the Mo5+ion hosts a purely spin-1/2 magnetic

moment, indicating negligible effects of spin-orbit interaction. The quenched orbital moments originate from the

large displacement of Mo ions inside the MoO6octahedra along the apical direction. The ground state is shown

by neutron diffraction to support a collinear Néel-type magnetic order, and a spin-ﬂop transition is observed

around an applied magnetic ﬁeld of 3 .5 T. The magnetic phase diagram is reproduced by a mean-ﬁeld calculation

assuming a small easy-axis anisotropy in the exchange interactions. Our results suggest 4 dmolybdates as an

alternative playground to search for model quantum magnets.

DOI: 10.1103 /PhysRevB.96.024 4 4 5

I. INTRODUCTION

The 4 dtransition-metal oxides naturally bridge the two

different regimes of the strongly correlated 3dcompounds

and the 5 dcompounds with strong spin-orbit coupling (SOC)

[1]. To what extent the 4 dcompounds represent either regime

or display original properties is largely an open question of

current interest [2]. Most notably, for instance, it is intriguing

that seemingly similar Ca2RuO4and Sr2RuO4display totally

different behaviors: the former is a Mott insulator [3–6],

while the latter is a metal and becomes superconducting at

low temperature [5–8]. Despite great interest, however, purely

4dquantum (spin-1/2) magnets are rather rare [9–12]asthe

electronic structure is often complicated by the presence of

other types of 3 dor 4 fmagnetic orbitals [13 ].

Among the few known 4d1magnets [9,11,12]themolyb-

denum phosphate MoOPO4is reported [14 ]. The MoO6

octahedra with Mo5+ions are corner shared to form a chain

along the crystallographic caxis of the tetragonal structure

[Fig. 1(a)], and these chains are further coupled to each

other via corner sharing PO4tetrahedra [Fig. 1(b)][14 ,15 ].

Previous susceptibility data on a powder sample of MoOPO4

shows a Curie-Weiss behavior with antiferromagnetic !CW =

−14 .5Kandamagnetictransitionat18K[16]. The 31P

NMR on a powder evidences a substantial exchange through

the PO4tetrahedra, and a sharp powder ESR line infers a

rather isotropic gfactor [16]. However, so far there have not

*minki.jeong@gmail.com

†henrik.ronnow@epﬂ.ch

been any studies on the magnetic structure in the ordered

state or magnetic properties of a single crystal. Moreover,

any discussion on the possible interplay between the crystal

electric ﬁeld and SOC is absent.

Here we report the magnetic and thermodynamic properties

of a MoOPO4single crystal using speciﬁc heat, susceptibility,

magnetization, ESR, and neutron diffraction experiments. We

also elucidate the electronic states and magnetic aspects in

light of SOC and crystal-ﬁeld effects, with the help of ab initio

quantum-chemistry calculations.

II. EXPERIMENTAL DETAILS

High-quality single crystals of MoOPO4were grown

following the procedure described in Ref. [14 ]. H2MoO4

was mixed with concentrated phosphoric acid and heated

up to 1000 ◦C for reaction in an open platinum crucible.

After being cooled to room temperature, the resulting dark-

blue solid was dissolved in a large amount of hot water.

The yellow transparent crystals were obtained in a platelike

shape [Fig. 1(c)]. Large crystals have a typical dimension

of 3 ×2×0.4mm

3with the caxis normal to the plate.

The crystal belongs to the space group P4/n,withlattice

parameters of a=b=6.204 4 ˚

Aandc=4.3003 ˚

A, obtained

by single-crystal x-ray diffraction, in agreement with Ref. [14 ].

Speciﬁc heat was measured using a physical properties

measurement system (PPMS, Quantum Design, Inc.), and

magnetization was measured using a magnetic properties

measurement system (MPMS, Quantum Design, Inc.). ESR

measurements were performed using a Bruker X-band spec-

trometer with a TE102 resonant cavity around 9.4 GHz.

24 69-995 0/2017/96(2)/024 4 4 5 (8) 024 4 4 5 -1 ©2017 American Physical Society

L. YANG et al. PHYSICAL REVIEW B 96,024445 (2017)

FIG. 1. Crystal structure of MoOPO4projected onto (a) the ac

planes, showing a chainlike arrangement of MoO6octahedra (yellow),

and (b) the ab planes, showing the coupling between the chains via

PO4tetrahedra (blue). Dashed lines represent the unit cells. Possible

in-plane (J1and J2) and out-of-plane (Jc)exchangecouplingsare

also shown. (c) Photograph of a representative single crystal.

Neutron diffraction experiments were performed on TRICS

and D23 beamlines at the Paul Scherrer Institute and Institut

Laue-Langevin, respectively. An incident neutron wavelength

of 2.3 109 ˚

A was employed.

III. RESULTS

A. Speciﬁc heat

Figure 2(a) shows the speciﬁc heat Cpmeasured from 2

to 15 0 K in zero ﬁeld and in a magnetic ﬁeld of 14 T. Cp

above 25 K for both ﬁelds is essentially the same, increasing

monotonically with increasing temperature. In zero ﬁeld a

pronounced peak is found at 16.1 K, while the peak is

shifted to a slightly lower temperature of 15.4 K at 14 T.

These peaks correspond to a transition into a magnetically

long range ordered phase, as evidenced by other experimental

measurements discussed in later sections.

In order to extract the magnetic part of the speciﬁc heat Cmag

and to deduce the corresponding entropy Smag,wesimulatethe

lattice contribution from the high-temperature data by taking

into account the Debye and Einstein contributions. We ﬁt the

Cpdata above 3 0 K by a lattice-only model, Cp=CD+

!iCE,i, where CDand CE,i represent the Debye and Einstein

terms, respectively. The Debye term is expressed as

CD=9nDR"T

!D#

3$!D/T

0

x4ex

(ex−1)2dx, (1)

and the Einstein term is expressed as

CE=3nERy2ey

(ey−1)2,y≡!E/T, (2)

where Rdenotes the gas constant, !Dand !Eare the Debye

and Einstein temperatures, and nDand nEare the numbers of

FIG. 2. (a) Speciﬁc heat Cpas a function of temperature in zero

ﬁeld (circles) and at 14 T (squares). The solid line represents the best

ﬁt of the simulated lattice contribution using the Debye (dash-dotted

line) and Einstein (dotted line) terms. The inset provides an enlarged

view of the low-temperature region. (b) Left axis: magnetic part of

the speciﬁc heat Cmag divided by temperature (circles). Right axis:

the solid line is the entropy calculated from Cmag.

the corresponding modes, respectively; the sum nD+nEis the

total number of atoms per formula unit. For our purpose, we

consider that a phenomenological ﬁt using nD,nE,!D,and

!Eas free parameters is sufﬁcient. The best ﬁt for the zero ﬁeld

was obtained when using one Debye and two Einstein terms,

which yields the characteristic temperatures !D=1177 K,

!E,1=372K, and !E,2=15 4 K and the numbers nD=4,

nE,1=2, and nE,2=1. The solid line in Fig. 2(a) is the

best-ﬁt result for the total lattice contribution, while the

dash-dotted and dotted lines are the corresponding Debye and

Einstein contributions, respectively. While the parameters in

the phenomenological phonon ﬁt may not be directly physical,

they provide a parametrization of the lattice contribution to the

speciﬁc heat, which can be substracted to estimate the magnetic

speciﬁc heat.

Figure 2(b) shows the resulting Cmag/T in zero ﬁeld (cir-

cles, left axis) obtained by subtracting the lattice contribution

from the measured Cp. The solid line in Fig. 2(b) plots

Smag(T)obtainedbyintegratingCmag/T over temperature

(right axis). Smag(T)isfoundtoreachandstayatRln 2 at

high temperatures, indicating two-level degrees of freedom.

The thin colored band in Fig. 2(b) represents the entropy range

obtained when ﬁtting the Cpdata by varying the lower bound

of temperature between 25 and 3 5 K to conﬁrm the negligible

dependence of the result on the chosen ﬁt range. A similar

analysis for the 14 T data (not shown) indicates negligible

ﬁeld effects.

024 4 4 5 -2

J1-J2SQUARE LATTICE . . . PHYSICAL REVIEW B 96,024445 (2017)

FIG. 3. (a) The dc magnetic susceptibility χ(T) in a ﬁeld of H=0.1 T applied parallel (circles) and perpendicular (squares) to the c

axis. The dashed line represents the Curie-Weiss ﬁt for H∥c, and the solid line shows the high-temperature series expansion using the Padé

approximant (see the text). The inset shows an enlarged view of the low-temperature region. (b) Isothermal magnetization M(H)forH∥c

(solid symbols) and H⊥c(open symbols) at several different temperatures. The inset plots the ﬁeld derivative dM/dH versus Hfor H∥c.

(c) Magnetic phase diagram from the susceptibility (squares), speciﬁc heat (upward triangles), magnetization (circles), and neutron diffraction

(downward triangles) data. Lines are guides to the eye. The colored background represents the result from the mean-ﬁeld calculations (see

text).

B. Susceptibility and magnetization

Figure 3(a) shows the dc magnetic susceptibility χ=

M/H, where Mis magnetization, in a ﬁeld of H=0.1T

applied parallel and perpendicular to the caxis. For both cases,

χ(T)showsalmostidenticalbehaviorfrom300downto20K.

However, for H∥c,χ(T) exhibits a sharp drop toward zero as

temperature is decreased across 17 K, while the one for H⊥c

remains only weakly temperature dependent. This is indicative

of an antiferromagnetic transition where the ordered moments

at low temperatures are collinear to each other and parallel to

the caxis.

The nearly isotropic, high-temperature part of χ(T)could

be well ﬁt by the Curie-Weiss formula, χ(T)=C/(T−

!CW )+χ0,where!CW is the Curie-Weiss temperature and

χ0is a temperature-independent diamagnetic and background

term that may arise from the plastic sample holder or the small

amount of grease used. The best and stable ﬁt is obtained

in the 5 0–3 00 K range, which yields the effective moment

µeff =1.67(1)µBper Mo5+ion, !CW =−6(1) K, and χ0=

2.2(1) ×10−4emu/mol for H∥cand µeff =1.69(1)µB,

!CW =−4(1)K,andχ0=4.6(1) ×10−4emu/mol for H⊥

c.ThebestﬁtforH∥cis shown as a dashed line in

Fig. 3(a). The negative !CW indicates that antiferromagnetic

interactions are dominant. The effective moments indicate a

spin-only value consistent with the speciﬁc-heat results.

The isothermal magnetization M(H)forH∥cand H⊥c

at several temperatures is shown in Fig. 3(b).At5 K,M(H)

increases slowly with the ﬁeld H∥cup to 3 T but then sharply

increases in a narrow ﬁeld range of 3 –4 T until it eventually

converges to the high-temperature M(H)dataobtainedat16or

20 K. This stepwise increase of M(H)becomessmearedoutas

temperature is increased. On the other hand, no such stepwise

behavior was observed at any temperatures for H⊥c.These

are typical signatures of a spin-ﬂop transition which occurs

when the ﬁeld is applied along an easy axis, along which

the ordered moments align: the spins on the two sublattices

rotate to attain components perpendicular to the applied ﬁeld

direction as a result of competition between antiferromagnetic

coupling, magnetic anisotropy, and the Zeeman energy.

The magnetic phase diagram is thus mapped out by combin-

ing the above bulk magnetic and speciﬁc-heat results, as shown

in Fig. 3(c).Theantiferromagnetictransitiontemperaturesin

different ﬁelds are obtained from the peaks in χ(T)andCp(T),

and the spin-ﬂop transition ﬁelds at different temperatures are

obtained from the peak positions in the dM/dH versus Hplot

[inset of Fig. 3(b)].

C. Electron spin resonance

In order to gain microscopic insight into the magnetic

properties, we have performed ESR measurements as a

function of ﬁeld orientation and temperature. Figure 4(a)

plots the obtained room-temperature gfactor as the ﬁeld

direction is rotated by φin the ab and ac planes. The g

factor in the ac plane shows a φvariation as large as 2%

with characteristic cos2φangular dependence. On the other

hand, the gfactor in the ab plane remains essentially constant,

as expected from the tetragonal symmetry, within the error of

0.08%, which might have arisen from a slight misorientation

of the crystal. We obtain the gfactor along the principal

axes as ga=1.926(2) and gc=1.889(2). The average value

g=(2ga+gc)/3=1.913 (2) agrees with the one previously

obtained by powder ESR [17]. These gvalues correspond to

the effective moments of 1.64 µBand 1.66µBfor spin-1/2for

H∥cand H⊥c,respectively,whichareveryclosetothe

effective moment values obtained from the Curie-Weiss ﬁt in

the previous section.

For a system with tetragonal symmetry with short distances

between the transition-metal and ligand ions, one would expect

ga<g

c[18]. However, we ﬁnd an opposite structure for the g

024 4 4 5 -3

L. YANG et al. PHYSICAL REVIEW B 96,024445 (2017)

FIG. 4. (a) Angular dependence of the gfactor at room temperature from the ESR measurements, where solid symbols are for the ﬁeld

orientation varied on the ac plane and open symbols are for the ﬁeld orientation varied on the ab plane. (b) Resonance ﬁeld B0(solid circles,

right axis) and linewidth $B0(open circles, left axis) of the ESR spectrum as a function of temperature. The inset plots normalized spin

susceptibility χs(T)/χs(3 00 K) as a function of temperature. (c) Temperature evolution of the spectrum for B∥cacross the transition (open

circles). The solid line is a sum of two contributions from intrinsic (dotted line) and defect (dashed line) susceptibilities. At 5 K, the data for

B⊥c(solid circles) are overlaid.

factor in MoOPO4, even though the orbital energy diagram for

the Mo5+ion is expected to be similar to that of tetragonally

compressed octahedron with a stabilized dxy orbital (see Fig. 7

below). As explained in Sec. III E,themultiorbitalcharacterof

the ground state in MoOPO4results in the observed gvalues.

Figure 4(b) shows the temperature dependence of the

resonance ﬁeld B0and the linewidth $B0of the ESR spectrum.

B0slowly decreases as temperature is lowered from 300 down

to 24 K, which may be attributed to a lattice contraction.

As temperature is further lowered below 24 K, B0starts

increasing sharply, which indicates that a magnetic transition is

approached. Similarly, $B0slowly decreases as temperature is

lowered down to 25 K but then starts broadening signiﬁcantly

as temperature is further lowered down to 15 K due to critical

spin ﬂuctuations. The inset of Fig. 4(b)plots the temperature

dependence of the local spin susceptibility, which is obtained

from the spectral area at each temperature normalized by

the one at 3 00 K, χs(T)/χs(3 00 K). The data could be ﬁt

to the Curie-Weiss formula with !CW =−8.9 K, which is in

reasonable agreement with the bulk susceptibility result shown

in Fig. 3(a).

Across the transition, the ESR line changes in shape and

intensity as shown in Fig. 4(c).Thelinesustainsaperfect

Lorentzian shape down to 16 K. On the other hand, the

line below 16 K close to the transition ﬁts better to a

sum of two Lorentzians: one corresponds to the intrinsic

sample susceptibility, while the other may correspond to

some defects. Indeed, the ESR signal at the paramagnetic

resonance ﬁeld position below 15 K corresponds to about

0.1% concentration of paramagnetic impurities. The response

below 15 K represents the summation of the possible defect

contribution and the intrinsic susceptibility. The tiny intrinsic

response below the transition temperature may represent

clusters of spins that continue to ﬂuctuate within the ESR time

window, which essentially disappears at lower temperatures

below 14 K. At 5 K, a broad hump of weak signal is observed

around 0.27 T for B∥cwhich is absent for B⊥c.Thissignal

may correspond to an antiferromagnetic resonance.

D. Neutron diffraction

To determine the microscopic magnetic structure, we

have performed neutron diffraction measurements. Magnetic

intensity appears at the position of the k=(100) wave vector at

5K,asshownintherotationscaninFig.5(a).Noappreciable

change in scattering is found close to (001) between 5 and 25 K,

as shown in Fig. 5(b). A small shoulder of the (001) reﬂection

is likely to originate from a closely oriented secondary grain.

Anonzero(100)reﬂectionwouldbeconsistentwithMospins

FIG. 5 . Neutron diffraction measurements of rotation scans

through (a) (100) and (b) (001) reﬂections recorded at 5 and 25 K.

(c) Temperature evolution of the Bragg peak integrated intensity. The

solid line is a power-law ﬁt I(T)∝(TN−T)2βwith the parameters

β=0.23 and TN=16.17 ±0.06 K. (d) The (010) peak counts as a

function of magnetic ﬁeld parallel to the caxis.

024 4 4 5 -4

J1-J2SQUARE LATTICE . . . PHYSICAL REVIEW B 96,024445 (2017)

FIG. 6. Schematic phase diagram of the spin-1/2J1-J2Heisen-

berg model on a square lattice with corresponding compounds

[12]. Different ground states are expected depending on the J2/J1

ratio as represented in the diagram, where CAF, NAF, and FM

refer to columnar antiferromagnetic, Néel antiferromagnetic, and

ferromagnetic ground states, respectively. The present compound,

MoOPO4,extendsthematerialsinvestigationfarintotheNAFregime.

related by a spatial inversion being antiparallel. Due to the

dipolar nature of the magnetic interaction, only magnetization

perpendicular to the scattering wave vector gives a nonzero

structure factor. As no change is observed for the (001)

reﬂection upon cooling below TN,wecanconcludethatthe

moments are parallel to the caxis. To verify that this is

consistent with the symmetry of the lattice and rule out any

other magnetic structures, we utilize BASIREPS [19]andoutline

the results here.

The magnetic representation is decomposed into six one-

dimensional irreducible representations &νwhose resulting

basis functions are shown in Table I. Examining the results

of the irreducible magnetic representations, we ﬁnd that only

&2is consistent with our observations. These results are in

contrast to the closely related AMoO(PO4)Cl (A=KandRb)

materials. Unlike the tilted arrangement of MoO6octahedra

d

xy

dyz,x

z

d

x2

−y2

d

3z

2

−r

2

t

2g

eg

FIG. 7. Single-particle energy-level diagram of dstates in an

octahedral arrangement of the oxygen ligands (in red) and Mo ion (in

yellow) in MoOPO4.

TABLE I. Basis functions of irreducible representation &νfor

k=(100) separated into real (Re) and imaginary (Im) components

and resolved along the crystallographic axes. The two equivalent Mo1

and Mo2ions are related by an inversion through the origin.

νMo1Mo2

1 Re (0,0,1) (0,0,1)

2 Re (0,0,1) (0,0,¯

1)

3 Re (1,0,0) (1,0,0)

3 Im (0,¯

1,0) (0,¯

1,0)

4 Re (1,0,0) (¯

1,0,0)

4 Im (0,¯

1,0) (0,1,0)

5 Re (1,0,0) (1,0,0)

5 Im (0,1,0) (0,1,0)

6 Re (1,0,0) (¯

1,0,0)

6 Im (0,1,0) (0,¯

1,0)

and PO4tetrahedra in MoOPO4,AMoO(PO4)Cl possesses a

higher symmetry where the octahedra and tetrahedra are ar-

ranged untilted in the abplane [20]. Powder neutron diffraction

measurements on AMoO(PO4)Cl reveal an antiferromagnetic

structure where Mo moments are instead conﬁned to the ab

plane [12].

Figure 5(c) shows the temperature dependence of the

(100) Bragg peak integrated intensity. By ﬁtting a power-law

dependence to the intensity, we ﬁnd TN=16.17 ±0.06 K,

which is consistent with the magnetization and speciﬁc-

heat measurements. The order parameter exponent is found

to be β=0.23 , corresponding to the two-dimensional XY

universality class. However, dedicated measurements with

better resolution and separating critical scattering would be

needed before any conclusions could be drawn from this. In

Fig. 5(d) we show the magnetic Bragg peak intensity as a

function of applied ﬁeld along the caxis recorded at 2 K. Above

3T,weﬁndasharpdecreaseinintensitywhichthenappears

to saturate above 5 T. The change in the Bragg peak intensity

is consistent with a spin-ﬂop transition that is observed in

the magnetization measurements shown in Fig. 3(b). This

corresponds to a tilt of the moments by approximately 3 5 ◦

away from the caxis for the ﬁelds above 5 T.

E. Model calculations

In order to gain insight into the magnetic interactions,

we ﬁt the experimental susceptibility shown in Fig. 3(a)

using a high-temperature series expansion [21] assuming a

J1-J2spin-1/2 Heisenberg model on a square lattice. The

best ﬁt [solid line in Fig. 3(a)] returns J1=11.4(0.4)K

and J2=−5.2(1.0) K, corresponding to J2/J1=−0.46.This

ratio supports a collinear Néel order for the ground state (see

Fig. 6) in agreement with the neutron diffraction result. Using

the mean-ﬁeld expression for the Curie-Weiss temperature,

!CW =−S(S+1)

3kB%

i=1,2

ziJi,(3 )

where ziis the number of neighbors for the corresponding

couplings (4 for both J1and J2in the present case), the high-

temperature expansion ﬁt yields !CW =−6.2K,whichagrees

with the value obtained from the simple Curie-Weiss ﬁt. Next,

024 4 4 5 -5

L. YANG et al. PHYSICAL REVIEW B 96,024445 (2017)

we simulate the phase diagram using a mean-ﬁeld calculation.

The results are presented by the colored background in

Fig. 3(c).Aslightexchangeanisotropy,$=0.02, has been

introduced in the Hamiltonian,

H=J1%

⟨i,j⟩&Sx

iSx

j+Sy

iSy

j+(1 +$)Sz

iSz

j'

+J2%

⟨i,k⟩&Sx

iSx

k+Sy

iSy

k+Sz

iSz

k',(4 )

where ⟨i,j⟩and ⟨i,k⟩refer to the nearest and the next-nearest

neighbors in the abplane, to account for the spin-ﬂop transition

in a spin-1/2systemwheresingle-ionanisotropyisnot

expected to be present. We note that the mean-ﬁeld calculation

reproduces the temperature dependence of the spin-ﬂop ﬁeld.

From the mean-ﬁeld expression for the Néel temperature,

TN=−S(S+1)

3kB%

i=1,2

zi(−1)iJi,(5 )

we obtain TN=16.6(1.4)K, whichis in excellent agreement

with the actual value from the experiments. In the above

analyses, we do not include Jcexplicitly: although an ar-

bitrarily small Jcis necessary in the actual system to give

rise to the (three-dimensional) long-range magnetic ordering,

including this parameter in the mean-ﬁeld calculation produces

an insigniﬁcant change in the phase diagram. In addition,

Jcconnects only two neighbors instead of the four of the

other couplings in the ab plane, and thus its effect should be

correspondingly weaker.

Our methods of analysis do not necessarily select the best

model, but rather test the validity and consistency of a proposed

one. For instance, ferromagnetic Jccomparable in strength

to antiferromagnetic J1, with negligible J2, may similarly

reproduce our experimental data. However, the ground-state

wave function from our ab initio quantum-chemistry calcula-

tions (see the next section) indicates zero contribution from

the out-of-plane orbitals and thus no direct virtual hopping

channels for Jcto be appreciable, in contrast to the other

coupling on the ab plane.

With strong ferromagnetic second-nearest-neighbor inter-

actions, MoOPO4populates a region of the J1-J2phase

diagram which has so far seen rather few investigations (see

Fig. 6). In the context of (π,0) zone-boundary anomalies on

the square lattice, linear spin-wave theory would for MoOPO4

predict a dispersion with signiﬁcantly higher energy at (π,0)

than at (π/2,π/2), opposite the case of weak antiferromagnetic

J2in Cu(pz)2(ClO4)2[22]. Compared to the 3 9% reduction in

ordered moment due to quantum ﬂuctuations for the nearest-

neighbor Heisenberg model, the estimate for J2/J1=−0.46

is only a 24 % reduction of the ordered moment. Adding

the weak anisotropy for MoOPO4yields a 21% reduction in

ordered moment. Hence quantum ﬂuctuations are likely much

weaker in MoOPO4than in, e.g., Cu(DCOO)2·4D

2O, called

CFTD for short [23 –26]orSr

2CuTeO6[27], and it would be

interesting in future investigations to examine whether this

leads to a similar suppression of the quantum dispersion and

continuum around (π,0).

TABLE II. Relative energies of d-level states of the Mo5+ion

obtained from CASSCF/NEVPT2 calculations. The corresponding

wave functions without (coefﬁcients) and with (weights) SOC at

the CASSCF level are also provided, where the up and down arrows

signify the Szvalues of +1

2and −1

2,respectively[41]. At the NEVPT2

level, the wave function would also contain contributions from the

inactive and virtual orbitals. For simplicity only the weights of the

SOC wave function are provided as the coefﬁcients are complex.

t1

2gStates Relative Wave function (CASSCF)

without SOC energies (eV) coefﬁcients

|φ0⟩00.95 |xy⟩−0.32|x2−y2⟩

|φ1⟩1.79 0.98 |yz⟩+0.21 |zx⟩

|φ2⟩1.79 0.21 |yz⟩−0.98 |zx⟩

|φ3⟩3.68 0.32|xy ⟩+0.95 |x2−y2⟩

|φ4⟩4.42 1.00 |z2⟩

t1

2gStates Relative Wave function (CASSCF)

with SOC energy (eV) normalized weights (%)

|ψ0⟩086.0|φ0,↑⟩ +14 .0|φ0,↓⟩

|ψ1⟩1.75 5 0.0|φ1,↑⟩ +50.0|φ2,↓⟩

|ψ2⟩1.82 4 6.0|φ1,↑⟩ +46.0|φ2,↑⟩

+4.0|φ1,↓⟩ +4.0|φ2,↓⟩

|ψ3⟩3.70 88.0|φ3,↑⟩ +12.0|φ3,↓⟩

|ψ4⟩4.44 100.0|φ4⟩

F. Ab initio calculations

An interesting feature in MoOPO4is that the axial position

of the Mo4+ion inside the MoO6octahedron is heavily shifted

such that the short and long distances to the apical oxygens

are 1.65 2 and 2.64 1 ˚

A, respectively. As a consequence, the

octahedral symmetry around the Mo ion is reduced, resulting

in the removal of octahedral orbital degeneracies and an

orbitally mixed electronic ground state. To elucidate the

electronic levels of the Mo5+ion in low-symmetry crystal

ﬁelds in MoOPO4, we performed ab initio quantum-chemistry

calculations using the cluster-in-embedding formalism [28].

AclusterofasingleactiveMoO

6octahedron along with

surrounding nearest-neighbor (NN) PO4tetrahedra within the

plane and the out-of-plane MoO6octahedra embedded in an

array of point charges that reproduces the effect of the solid

environment [29] was considered for many-body calculations.

The NN polyhedra were included within the cluster region

to better describe the charge density within the active MoO6

region. Such calculations have provided excellent insights into

the interplay of crystal ﬁeld and SOC effects for several 4d

and 5 dtransition-metal compounds [30–33].

A perfect octahedral arrangement of the oxygen ligands

around the transition-metal ion splits the dlevels into high-

energy egand low-energy t2gmanifolds (see Fig. 7). In

MoOPO4,thelow-symmetrycrystalﬁeldsfurthersplitthe

t2gand eglevels of the Mo5+ion, resulting in an orbital singlet

ground state. In Table II the ground-state wave function and the

d-dexcitations of the Mo5+ion are summarized. These were

obtained from many-body multiconﬁgurational self-consistent

ﬁeld (MCSCF) [34]andN-electron valence-state perturbation

theory (NEVPT2) [35]calculationsfortheatomsintheactive

cluster region. All-electron Douglas-Kroll-Hess (DKH) basis

sets of triple-zeta quality [36]wereusedtorepresentthe

024 4 4 5 -6

J1-J2SQUARE LATTICE . . . PHYSICAL REVIEW B 96,024445 (2017)

TABLE III. Computed gfactors of MoOPO4at the NEVPT2

level of theory. The ground-state multiconﬁguration wave function as

shown in Table II produces the correct structure for the gfactors.

CASSCF gagc

active orbital space

t2g1.91 1.99

t2g+eg1.92 1.84

Mo and oxygen ions in the central MoO6octahedron, and

for the Mo and P ions in the NN polyhedra we employed

effective core potentials [37,38]withvalencetriple-zeta[37]

and a single basis function, respectively. The oxygen ions

corresponding to the NN MoO6and PO4polyhedra were

expanded in two sand one patomic natural orbital type [39]

functions. All the calculations were performed using the ORCA

quantum-chemistry package [40].

In the complete active space formalism of the MCSCF

(CASSCF) calculation, a self-consistent wave function was

constructed with an active space of one electron in ﬁve Mo

dorbitals. On top of the CASSCF wave function, NEVPT2

was applied to capture the dynamic electronic correlation.

Table II shows that the ground state is predominantly of dxy

character but has signiﬁcant contributions from the dx2−y2

orbital. The ﬁrst orbital excitations are nearly degenerate at

1.79 eV and are composed of dyz-anddzx -like orbitals. This

scenario is in contrast to the situation in other t2g-active class

of compounds with regular transition-metal oxygen octahedra

where the t2gmanifold remains degenerate with an effective

orbital angular momentum ˜

l=1. In the latter scenario the

spin-orbit interaction admixes all the t2gstates to give rise to

atotalangularmomentumJeff ground state [42,43]. Due to

the large noncubic crystal-ﬁeld splittings in the t2gmanifold in

MoOPO4,thespin-orbitinteractionhasanegligibleeffecton

the Mo5+ground state ψ0(see the with SOC results in Table II).

However, the orbital angular momentum is unquenched in

dzx and dyz, and hence the SOC results in the splitting of

the high-energy states ψ1and ψ2. Our calculations result in

excitation energies of 3.68 and 4 .42 eV into the egstates.

To understand the unusual structure of gfactors deduced

from the ESR experiments, we computed them from the

ab initio wave function as implemented in ORCA [44]. In

Table III,thegfactors obtained from CASSCF calculations

with two different active orbital spaces, only t2gand t2g+eg,

are presented. With only t2gorbitals in the active space, we

ﬁnd ga<g

cas expected for tetragonal symmetry with the

dxy -like orbital occupied in the ground state. By enlarging the

active space, the wave function now contains conﬁgurations

involving the egorbitals as well, and this is crucial to produce

the experimentally observed gfactors with ga>g

c.

IV. CONCLUSION

We have show n with a va r iety of exp e rimental a n d

computational techniques that MoOPO4realizes a spin-1/2

magnetic system of 4 d1electrons, with the quenched orbital

moment due to the large displacement of the Mo ions inside

the MoO6octahedra. The magnetic ground state supports a

Néel-type collinear staggered order on the square lattice with

the moments pointing normal to the plane, while the moments

align ferromagnetically along the stacking axis. The compound

likely realizes a spin-1/2HeisenbergmodelonaJ1-J2square

lattice, with an unfrustrated conﬁguration of antiferromagnetic

J1and ferromagnetic J2, while a small interlayer coupling

Jcwould lead to the observed magnetic ordering transition.

The spin-ﬂop transition suggests a small easy-axis anisotropy

in the dominant antiferromagnetic exchange, and the mean-

ﬁeld calculation reproduces the experimental magnetic phase

diagram. The small anisotropy in the gfactor observed in ESR,

which is reproduced by the quantum-chemistry calculations,

indicates that the ground state involves the higher-energy eg

orbitals in addition to the t2gorbitals. Our results suggest that

4dmolybdates provide an alternative playground to search for

model quantum magnets other than 3 dcompounds.

ACKNOWLEDGMENTS

We tha n k R. Scopell i ti and O. Zaha r ko for thei r h elp with

x-ray and neutron diffraction, respectively. We also thank

V. Fav re a n d P. H u a n g f o r th e i r h e l p wi t h t h e s p ec i ﬁ c - h ea t

analysis. V.M.K. is grateful to H. Stoll for discussions on

effective core potentials. This work was supported by the

Swiss National Science Foundation, the MPBH network,

and European Research Council grants CONQUEST and

TopoMat (No. 3 065 04 ). M.J. is grateful for support from the

European Commission through the Marie Skłodowska-Curie

Action COFUND (EPFL Fellows). M.K. is supported by a

Grant-in-Aid for Scientiﬁc Research (C) (JSPS, KAKENHI

No. 15 K05 14 0). The ab initio calculations were performed

at the Swiss National Supercomputing Centre (CSCS) under

project s675 .

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