Strong Frustration Due to Competing Ferromagnetic and Antiferromagnetic Interactions: Magnetic Properties of M(VO)2(PO4)2 (M=Ca and Sr)

PHYSICAL REVIEW B 78, 024418 ͑2008͒

Strong frustration due to competing ferromagnetic and antiferromagnetic interactions: Magnetic properties of M(VO)2(PO4)2 (M=Ca and Sr)

R. Nath,1,* A. A. Tsirlin,1,2,† E. E. Kaul,1,‡ M. Baenitz,1 N. Büttgen,3 C. Geibel,1 and H. Rosner1 1Max-Planck-Institut für Chemische Physik fester Stoffe, Nöthnitzer Strasse 40, 01187 Dresden, Germany 2Department of Chemistry, Moscow State University, 119992 Moscow, Russia 3Experimentalphysik V, Elektronische Korrelationen und Magnetismus, University of Augsburg, D-86135 Augsburg, Germany ͑Received 29 April 2008; published 18 July 2008͒ We present a detailed investigation of the magnetic properties of complex vanadium phosphates ͑ ͒ ͑ ͒ ͑ ͒ 31 M VO 2 PO4 2 M =Ca and Sr by means of magnetization, specific heat, P NMR measurements, and band-structure calculations. Experimental data evidence the presence of ferromagnetic and antiferromagnetic ͑ ͒ ͑ ͒ ␪ Յ interactions in M VO 2 PO4 2, resulting in a nearly vanishing Curie-Weiss temperature CW 1 K that con- trasts with the maximum of magnetic susceptibility at 3 K. Specific heat and NMR measurements also reveal weak exchange couplings with the thermodynamic energy scale Jc =10–15 K. Additionally, the reduced maximum of the magnetic specific heat indicates strong frustration of the spin system. Band-structure calculations ͑ ͒ ͑ ͒ show that the spin systems of the M VO 2 PO4 2 compounds are essentially three-dimensional with the frustration caused by competing ferromagnetic and antiferromagnetic interactions. Both calcium and strontium compounds undergo antiferromagnetic long-range ordering at TN =1.5 and 1.9 K, respectively. The spin model reveals an unusual example of controllable frustration in three-dimensional magnetic systems.

DOI: 10.1103/PhysRevB.78.024418 PACS number͑s͒: 75.50.Ee, 75.40.Cx, 71.20.Ps, 75.30.Et

/ Ͻ I. INTRODUCTION netic ordering is formed for J2 J1 −0.25, and the pitch / angle of the spiral depends on the J2 J1 ratio. Recently, a / Ͻ Low-dimensional and/or frustrated spin systems are a number of frustrated chain systems with J2 J1 0 have been subject of thorough studies due to their variety of unusual studied,7 and even the vicinity of the quantum critical point 1,2 / 8 ground states and quantum phenomena. One of the most at J2 J1 =−0.25 was accessed experimentally. In these sys- striking effects in these systems is the formation of a spin tems, the possibility to tune the pitch angle by varying the liquid—a dynamically disordered ground state with short- exchange couplings may be particularly attractive due to an- range magnetic correlations. Spin liquids are caused by other intriguing property—the magnetoelectric effect ob- 2,9–12 quantum fluctuations that impede or even suppress long- served in LiCu2O2 and LiCuVO4. range magnetic ordering In two dimensions, we will consider the square lattice. Quantum fluctuations are enhanced in systems with low Frustration can be introduced by diagonal NNN interactions ͑ ͒ spin value, low dimensionality, and/or magnetic frustration. J2 that compete with NN interactions J1 running along the At present, numerous spin models ͑and the respective struc- side of the square. The resulting model is known as a frus- ͒ 3,4 ͑ ͒ tural types for the frustrated spin systems are known. trated square lattice FSL or J1 −J2 model. The phase dia- Most of these systems are geometrically frustrated magnets gram of the FSL has been extensively studied theoretically since the frustration is caused by the topology of the lattice. ͑see, e.g., Refs. 13–16͒. It shows three ordered phases ͑fer- However, several models reveal frustration due to a specific romagnet, Néel antiferromagnet, and columnar antiferromag- topology of magnetic interactions, while the lattice itself net͒ as well as two critical regions near the quantum critical / Ϯ does not prevent the system from long-range spin ordering. points at J2 J1 = 0.5. Basically, these regions are supposed An important advantage of the latter systems is the possibil- to reveal spin-liquid ground states, although there is also a / 17 ity to vary the degree of frustration by tuning individual proposition of a nematic order at J2 J1 =−0.5. Yet both the exchange couplings. Hence, the formation of different conjectures are not verified experimentally since most of the ground states within one model and the access to quantum FSL systems studied so far show columnar magnetic order- critical points are possible. Below, we focus on the respec- ing and do not fall to the critical regions. Nevertheless, the tive models and briefly review both one-dimensional ͑1D͒ detailed investigation of the respective compounds was and two-dimensional ͑2D͒ cases. highly helpful and disclosed important methodological as- The simplest 1D lattice is a uniform chain. To frustrate pects of studying the frustrated spin systems and understand- such a system, one should introduce next-nearest-neighbor ing their physics.18–21 ͑ ͒ NNN interaction J2. Typically, J2 is antiferromagnetic and Based on the above representative examples, one may ex- competes with either ferromagnetic ͑FM͒ or antiferromag- pect that similar changes of the ground state due to the alter- ͑ ͒ ͑ ͒ netic AFM nearest-neighbor NN interactions J1. The re- ation of the exchange couplings could be realized in other sulting physics strongly depends on the sign of J1. Thus, for frustrated models—either in one, two, or even three dimen- Ͼ ͑ ͒ J1 0 i.e., for AFM NN coupling the system has a dimer- sions. This consideration and the recent investigation of an / Ͼ ized ground state for J2 J1 0.241 that may be further sta- interesting FSL system, the complex vanadium phosphate ͑ ͒ 22 bilized by lattice distortion as experimentally observed in the Pb2VO PO4 2, motivated us to start a systematic research 5,6 Ͻ spin-Peierls compound CuGeO3. If J1 0, helical mag- of the related compounds. Below, we present the study of

1098-0121/2008/78͑2͒/024418͑13͒ 024418-1 ©2008 The American Physical Society ing transition-metal compounds mainly depend on two factors: the orbital state of the magnetic cation and the possible paths for superexchange interactions. In the following, we will discuss both of these factors. The orbital state of the magnetic cation is determined by ͑ ͒ ͑ ͒ the local environment. In the M VO 2 PO4 2 compounds, vanadium has distorted octahedral coordination with one short bond ͑see bottom part of Fig. 1͒. This type of environment is typical for V+4 ͑electronic configuration 3d1͒ and gives rise to a nondegenerate ground state with half-filled dxy orbital ͑ ͒ see Refs. 25 and 26 and Sec. V . The dxy orbital is located in the plane perpendicular to the short V-O bond. Short bonds are aligned along the structural chains; therefore the overlap of the half-filled orbital with p orbitals of the two bridging oxygen atoms is very weak, and V-O-V is not an efficient path for superexchange interactions. On the other hand, the dxy orbital may overlap with the p orbitals of four other oxygen atoms providing more complicated V-O-P-O-V paths for the superexchange. The chains of the VO6 octahedra in the structures of the ͑ ͒ ͑ ͒ M VO 2 PO4 2 compounds are not parallel, and the V-P-O framework looks quite complicated. In order to simplify the situation, one should take into account two considerations: ͑ ͒ i Every vanadium atom is connected to four different PO4 ͑ ͒ tetrahedra. ii Every PO4 tetrahedron is connected to four different vanadium atoms. Thus, every vanadium atom may have 3ϫ4=12 different V-O-P-O-V connections at the most. ͑ ͒ ͑ ͒ ͑ ͒ M VO 2 PO4 2 M =Ca, Sr —two representatives of the same compound family that reveal three-dimensional ͑3D͒ spin systems with the frustration caused by the specific topology and magnitudes of the exchange interactions. In the following, we show clear evidence of the frustration in the results of several experimental techniques: magnetization, specific heat, and nuclear-magnetic-resonance ͑NMR͒ measurements. In order to understand the microscopic origin of the frustration, we use band-structure calculations. The outline of the paper is as follows. In Sec. II, we ͑ ͒ ͑ ͒ analyze the crystal structure of the M VO 2 PO4 2 compounds focusing on possible paths of the exchange interactions. Methodological aspects are reviewed in Sec. III. Sec- tion IV presents experimental results including magnetization, specific heat, and 31P NMR. In Sec. V, we report the results of band-structure calculations and construct a microscopic model of the exchange interactions in ͑ ͒ ͑ ͒ Ca VO 2 PO4 2. Experimental and theoretical results are compared and discussed in Sec. VI followed by our conclu- sions.

II. CRYSTAL STRUCTURE ͑ ͒ ͑ ͒ ͑ ͒ Complex vanadium phosphates Ca VO 2 PO4 2 Ref. 23 ͑ ͒ ͑ ͒ ͑ ͒ and Sr VO 2 PO4 2 Ref. 24 are isostructural and crystallize in a face-centered orthorhombic unit cell ͑space group Fdd2, Z=8͒. The structure presents one vanadium site that shows +4 distorted octahedral coordination. The V O6 octahedra share corners to form chains running along the ͓101͔ and ͓101¯͔ directions. The chains are joined into a 3D framework ͑ ͒ by PO4 tetrahedra Fig. 1 . Magnetic interactions in insulat- STRONG FRUSTRATION DUE TO COMPETING… PHYSICAL REVIEW B 78, 024418 ͑2008͒

III. METHODS ͑space group Fdd2͒. The results of this calculation enabled ͑ ͒ ͑ ͒ the choice of the orbital states relevant for the magnetic in- Polycrystalline samples of M VO 2 PO4 2 were prepared ͑ ͒ teractions. The respective bands were analyzed using a tight- by solid-state reaction technique using CaCO3 99.9% , ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ binding ͑TB͒ model, and the resulting hoppings were applied SrCO3 99.999% ,VO2 99.99% , and NH4 2HPO4 99.9% as starting materials. The synthesis involved two steps. In the to estimate the antiferromagnetic contributions to the exchange integrals. Such an analysis accounts for all reason- first step, the intermediate compounds MP2O6 were prepared able paths of superexchange and provides a reliable basis for by firing the stoichiometric mixtures of MCO3 and ͑ ͒ the microscopic model of the magnetic interactions. How- NH4 2HPO4 at 850 °C for 1 day in air with one intermediate grinding. In the second step, the intermediate products ever, LDA does not give any information about ferromagnetic couplings in the system under investigation. were mixed with VO2 in the appropriate molar ratio and heated for 4 days in dynamic vacuum ͑10−5 mbar͒ with sev- To account for ferromagnetic interactions in the micro- eral intermediate grindings and pelletizations. The annealing scopic model, we performed LSDA+U calculations and ͑ ͒ ͑ ͒ compared total energies for four different patterns of spin temperatures were 900 and 850 °C for Ca VO 2 PO4 2 and ͑ ͒ ͑ ͒ ordering. Exchange couplings in the M͑VO͒ ͑PO ͒ com- Sr VO 2 PO4 2, respectively. The resulting samples were 2 4 2 single phase as confirmed by x-ray diffraction ͑STOE pow- pounds are very weak; therefore one has to use the smallest ͑ der diffractometer, Cu K␣ radiation͒. Lattice parameters unit cell i.e., the primitive cell—one-fourth of the face- ͒ were calculated using a least-squares-fit procedure. The ob- centered unit cell in order to achieve the highest possible ͑ ͒ ͑ ͒ ͓ accuracy. The primitive cell is triclinic ͑space group P1͒ and tained lattice parameters for Ca VO 2 PO4 2 a =11.774͑2͒ Å, b=15.777͑3͒ Å, and c=7.163͑1͒ Å͔ and includes four inequivalent vanadium atoms; hence, a number ͑ ͒ ͑ ͒ ͓ ͑ ͒ ͑ ͒ of different spin configurations can be formed. Sr VO 2 PO4 2 a=12.008 2 Å, b=15.924 3 Å, and c =7.207͑1͒ Å͔ are in agreement with the previously reported Following the results in Ref. 25, we employed several ͑ values.23,24 values of Ud Coulomb repulsion parameter of the LSDA ͒ Magnetization data were measured as a function of tem- +U method in our calculations. According to the previous 25 perature and field using a superconducting quantum interfer- study, Ud =6 eV provides a reasonable description of the +4 ence device magnetometer ͑Quantum Design magnetic prop- octahedrally coordinated V within the FPLO calculations. erty measurement system͒. Heat capacity data were collected Yet one should keep in mind that the optimal value of Ud ͑ 29 with Quantum Design physical properties measurement sys- depends on numerous factors basis set, local environment ͒ tem on pressed pellets using the relaxation technique. All the of the transition-metal cation, and objective parameters , and measurements were carried out over a large temperature the unique choice of Ud remains a subtle issue. Below, we ͑ Յ Յ ͒ ␮ use several representative values of U ͑4, 5, and 6 eV͒ and range 0.4 K T 400 K in magnetic fields 0H up to 14 d T. The low-temperature measurements were done partly us- carefully check their effect on the results. The exchange pa- ing an additional 3He setup. rameter of LSDA+U was fixed at J=1 eV since usually it 1 The NMR measurements were carried out using pulsed has minor importance compared to Ud, especially for the 3d NMR techniques on 31P ͑nuclear spin I=1/2 and gyromag- configuration. netic ratio ␥/2␲=17.237 MHz/T͒ nuclei in a large temperature range ͑1KՅTՅ300 K͒. We have done the measure- IV. EXPERIMENTAL RESULTS ments at a radio frequency of 70 MHz which corresponds to an applied field of about 4.06 T. Low-temperature NMR A. Bulk susceptibility and magnetization measurements were partly done using a 3He/ 4He dilution Bulk magnetic susceptibilities ͑␹͒ of the M͑VO͒ ͑PO ͒ refrigerator ͑Oxford Instruments͒ with the resonant circuit 2 4 2 compounds are presented in Fig. 2. With decreasing tempera- inside the mixing chamber. Spectra were obtained either by ture, the susceptibility increases in a Curie-Weiss manner Fourier transform of the NMR echo signals or by sweeping ␹ and passes through a broad maximum at T Ӎ3 K. The the field at fixed frequency of 70 MHz. The NMR shift max maximum is characteristic of low-dimensional spin systems K͑T͒=͓␯͑T͒−␯ ͔/␯ was determined by measuring the ref ref and indicates a crossover to a state with antiferromagnetic resonance frequency of the sample ͑␯͑T͒͒, with respect to a correlations. The change in slope observed at T =1.5 and 1.9 standard H PO solution ͑resonance frequency ␯ ͒. The 31P N 3 4 ref K ͓for Ca͑VO͒ ͑PO ͒ and Sr͑VO͒ ͑PO ͒ , respectively͔ can spin-lattice relaxation rate ͑1/T ͒ was measured by using 2 4 2 2 4 2 1 be attributed to the onset of long-range magnetic ordering. either a 180° pulse ͑inversion recovery͒ or a comb of satu- To fit the bulk susceptibility data at high temperatures, we ration pulses. use the expression Scalar-relativistic band-structure calculations were performed using the full-potential local-orbital scheme C ͑ ͒͑ ͒ ␹ = ␹ + , ͑1͒ FPLO7.00–27 Ref. 27 and the parametrization of Perdew 0 T + ␪ and Wang for the exchange and correlation potential.28 A k CW ␹ mesh of 1728 points within the first Brillouin zone ͑510 in where 0 is the temperature-independent contribution that the irreducible part for the space group Fdd2͒ was used. accounts for core diamagnetism and Van Vleck paramagnet- Convergence with respect to the k mesh was carefully ism, while the second term is the Curie-Weiss law with the ␮2 / checked. Curie constant C=NA eff 3kB. The data above 20 K were First, a local-density approximation ͑LDA͒ calculation fitted with the parameters listed in Table I. The resulting ␮␹ was done using the full symmetry of the crystal structure effective moments eff are in reasonable agreement with the

024418-3 NATH et al. PHYSICAL REVIEW B 78, 024418 ͑2008͒

1200 1200 4

+4 4 1000 1000 T T max 2 max

41 2 T TN + 800 N

+4 1 800

(10 emu/mole V )

0 0 600 1 10 100 600 1 10 100 T (K) T (K) Ca(VO)242 (PO ) Sr(VO)242 (PO ) 400 400

H = 0.5 T H = 0.5 T 0 0

1/ (emu/mole V ) 200 1/ (emu/mole200 V )

0 0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 T (K) T (K)

͑ ͒ ͑ /␹͒ ͑ ͒ ␮ FIG. 2. Color online Reciprocal magnetic susceptibility 1 vs temperature T measured at 0H=0.5 T down to 0.4 K for ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ␹ Ca VO 2 PO4 2 left panel and Sr VO 2 PO4 2 right panel . The solid lines are best fits of the data to Eq. 1 . In the insets, vs T is plotted ␹ in the logarithmic scale to focus on the Tmax and TN. ␮ spin-only value of 1.73 B, while the Curie-Weiss tempera- of the system. Thus, the values of Hs can be used to estimate ␪␹ tures CW for both the compounds are rather small as com- exchange couplings if the ground state of the system and the ␹ ͑ ␹ pared to Tmax we use the superscript to denote the values leading exchange interactions are known. We will further related to the analysis of the bulk susceptibility data͒. discuss this point in Sec. VI and employ the Hs values to get ␹ Basically, Tmax characterizes the energy scale of the mag- quantitative information about exchange couplings in the ␪ netic interactions, and CW is a linear combination of all the systems under investigation. 30 ␪ exchange integrals. Therefore, the reduction in CW as compared to T␹ implies the presence of both FM and AFM max B. Specific heat interactions in the system under investigation. Moreover, TN ␹ ͑ ͒ is considerably lower than Tmax, and this effect may be The specific-heat Cp results at zero field are shown in caused by either low dimensionality and/or frustration of the Fig. 4. At high temperatures, Cp is completely dominated by spin system. As we will show below ͑Sec. V͒, the spin sys- the contribution of phonon excitations; therefore, both the ͑ ͒ ͑ ͒ ͑ ͒ tems of the M VO 2 PO4 2 compounds are three dimen- compounds reveal similar Cp T curves. With decreasing ␹ sional. Thus, the reduction in TN as compared to Tmax indi- temperature, Cp starts to increase below 12–15 K indicating ͑ ͒ ͑ ͒ ͑ ͒ cates magnetic frustration in M VO 2 PO4 2. that the magnetic part of the specific heat Cmag becomes ͑ ͒ Field-dependent magnetization data for the prominent. With a further decrease in temperature, Cp T ͑ ͒ ͑ ͒ M VO 2 PO4 2 compounds are shown in Fig. 3. At low shows a broad maximum at around 3 and 4 K due to the fields, the curves reveal linear behavior, while a positive cur- correlated spin excitations and a sharp peak at TN =1.5 and vature is observed at higher fields. Further increase in the 1.9 K associated with the long-range magnetic ordering ͓the ␮ Ӎ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ field results in the saturation at 0Hs 8 and 11.5 T for values are given for Ca VO 2 PO4 2 and Sr VO 2 PO4 2, ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͔ 31 Ca VO 2 PO4 2 and Sr VO 2 PO4 2, respectively. The satu- respectively . ͑ ͒ ␮ / ration magnetization Ms is about 0.9 B mole, i.e., In order to get a quantitative estimate of Cmag, the phonon ␮ slightly below the expected value of 1 B. The underesti- part Cphon was subtracted from the total measured specific mate of Ms may be caused by nonmagnetic impurities in the heat Cp. The general procedure is similar to that reported in samples under investigation. This explanation is further sup- Refs. 32 and 33. The phonon part was estimated by fitting ␮ ͑ ͒ ͑ Յ Յ ͒ ported by the slight reduction in eff and the magnetic en- Cp T at high temperatures 15 K T 200 K with a sum tropy ͑see Sec. IV B͒. of Debye contributions. The additional term A/T2 accounted The saturation field shows the energy difference between for the magnetic contribution, and the final fit was performed the ground state and the fully polarized ͑ferromagnetic͒ state using the equation TABLE I. The parameters of the Curie-Weiss fit ͓Eq. ͑1͔͒ of the bulk susceptibility and the NMR shift ͑ ␹ ͒ ␹ ␮ data marked by the superscripts and K, respectively . 0 is the temperature-independent contribution, eff ␪ is the effective magnetic moment, and CW is the Curie-Weiss temperature. The listed standard deviations originate from the least-squares fitting.

␹ ␮␹ ␪␹ ␮K ␪K Sample 0 eff CW eff CW −6 ͑10 emu/mole͒ ͑␮B͒ ͑K͒ ͑␮B͒ ͑K͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Ca VO 2 PO4 2 2.1 5 1.630 1 −0.4 3 1.71 6 −0.3 1 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Sr VO 2 PO4 2 −1.5 5 1.653 1 1.0 1 1.70 5 −0.4 1

024418-4 STRONG FRUSTRATION DUE TO COMPETING… PHYSICAL REVIEW B 78, 024418 ͑2008͒

1.0 T = 1.8 K chain 0.8 0.4 sq. lattice

+4 triangular 0.6 0.3 Ca(VO)242 (PO )

B Sr(VO) (PO ) 0.4 242

mag

( /mole V )

CR 0.2

M Ca(VO)242 (PO ) 0.2 Sr(VO)242 (PO ) 0.1 0.0 0 2 4 6 8 10 12 14

m0 H (T) 0.0 0 1 2 3 ͑ ͒ ͑ ͒ TJ FIG. 3. Isothermal magnetization of the M VO 2 PO4 2 com- / c pounds measured at 1.8 K. FIG. 5. ͑Color online͒ Magnetic contribution to the specific heat n=4 3 ␪͑n͒/ 4 x ͑ ͒ ͑ ͒ A T D T x e of M VO 2 PO4 2 and theoretical curves for the uniform chain ͑ ͒ ͩ ͪ ͵ ͑ ͒ Cp T = +9R͚ cn ͑ ͒ dx, 2 ͑Refs. 30 and 34͒, nonfrustrated square lattice ͑Refs. 34 and 35͒, T2 ␪ n ͑ex −1͒2 n=1 D 0 and triangular lattice ͑Ref. 34͒ models. The reduced temperature / ␪͑n͒ scale T/J with J =ͱ͚ J2 is used. where R=8.314 J mole K is the gas constant, D are the c c i i characteristic Debye temperatures, and cn are integer coeffi- ͑ ͒ ͑ ͒ ͑ ͒ cients indicating the contributions of different atoms or teractions of type Ji . The structures of M VO 2 PO4 2 yield ͒ ͑ ͒ ͑ / ͒ 2 groups of atoms to the specific heat. The phonon contribu- zi =2 for any i see Sec. II , and we find A= 3R 16 Jc with tion was extrapolated down to 0.4 K and subtracted from the the thermodynamic energy scale of the exchange couplings ͑ ͒ measured Cp T . The reliability of the whole procedure was defined as J =ͱ͚ J2. Using the experimental values A=142 / c i i justified by integrating Cmag T. The resulting magnetic en- and 147 J K/mol, we estimate JC =9.6 and 9.8 K for the / ͓ ͑ ͒ ͑ ͒ c tropies are 5.30 and 5.58 J mole K for Ca VO 2 PO4 2 and ͑ ͑ ͒ ͑ ͒ ͔ calcium and strontium compounds, respectively the super- Sr VO 2 PO4 2, respectively consistent with the expected script C denotes the values obtained from the analysis of the value of R ln 2. specific-heat data͒. High-temperature magnetic contribution A/T2 is the To interpret the specific-heat data, we compare C with lowest-order term in the high-temperature series expansion mag the simulated curves for the representative spin-1/2 models for the specific heat. According to Ref. 30, in both one dimension ͑uniform chain͒ and two dimensions 3R ͑nonfrustrated square lattice, triangular lattice͒͑see Fig. 5͒. 2 ͑ ͒ A = ͚ ziJi , 3 The maximum value of the magnetic specific heat ͑Cmax͒ and 32 i mag the shape of the maximum are characteristic of the magni- where integers zi indicate the number of interactions Ji for a tude of quantum fluctuations. The enhancement of quantum single magnetic atom ͑i.e., coordination number for the in- fluctuations suppresses correlated spin excitations; therefore 5 10 90 90 Ca(VO)242 (PO ) Sr(VO)242 (PO ) C C 4 p 60 8 p 60 C C

1+41 phon phon 1+41

30 41 30 + C C

(J K (mole V ) ) 3 mag 6 mag (J K (mole V ) )

C 0 C 0 0 70 140 0 70 140 T (K) T (K)

1+41

2 4

(J K (mole V ) )

C 1 2

0 0 0 10 20 30 0 10 20 30 T (K) T (K)

͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ FIG. 4. Color online Temperature dependence of the specific heat measured at zero field for Ca VO 2 PO4 2 left panel and ͑ ͒ ͑ ͒ ͑ ͒ Sr VO 2 PO4 2 right panel . The open circles are the raw data, the solid lines show the phonon contribution Cphon as found from the fit to ͑ ͒ ͑ ͒ Eq. 2 , and the dashed lines denote the magnetic contribution Cmag. The insets show the fits with Eq. 2 in the wide temperature range.

024418-5 NATH et al. PHYSICAL REVIEW B 78, 024418 ͑2008͒

max Cmag is reduced, and the maximum gets broader. One may 3 0T Ca(VO)242 (PO ) follow this effect in Fig. 5. The nonfrustrated square lattice 1T maxӍ 34,35 max shows a high Cmag 0.46R. In the 1D case, Cmag is de- 2T 8 Paramagnetic 30,34 creased to 0.35R due to stronger quantum ﬂuctuations. 2 4T ͑ ͒ The triangular lattice a frustrated 2D system shows an even 5T ͑ max ͒ lower Cmag=0.22R and broader maximum as compared to 6T 4 AFM 34 the uniform chain. (T)

H According to the results of band-structure calculations 1 0

41 ͑Sec. V͒, the spin systems of M͑VO͒ ͑PO ͒ are three di- + 0 2 4 2 6.5 T mensional. The high dimensionality should result in a narrow 0.0 0.5 1.0 1.5 7T T (K) maximum of the magnetic speciﬁc heat with a high absolute 0 ͑ ͒ ͑ ͒ value at the maximum. However, the M VO 2 PO4 2 com- 1 9 Sr(VO) (PO ) 12 pounds reveal broad maxima with the absolute values of 242 Paramagnetic about 0.25R and 0.23R for M =Ca and Sr, respectively. Such (J K (mole V ) ) 0T

p 8

(T) absolute values are well below that for the uniform chain C 4T

H ͑ ͒ 6 7T 0 AFM 1D nonfrustrated system and nearly match the value for the 4 triangular lattice ͑2D frustrated system͒. This result points to 9T 11 T the presence of strong quantum fluctuations in the 0 M͑VO͒ ͑PO ͒ compounds. The spin systems are 3D; there- 3 0.0 0.5 1.0 1.5 2.0 2 4 2 T fore the fluctuations should be entirely caused by the frustra- (K) tion. The different breadth of the maxima for the calcium and strontium compounds ͑see Fig. 5͒ may also be an indication 0 1 2 3 4 of the stronger frustration in Sr͑VO͒ ͑PO ͒ . 2 4 2 T The strong frustration evidenced by the specific-heat data (K) is consistent with the considerable reduction in the ordering ͑ ͒ ͑ ͒ ͑ ͒ ͑ ␹ ͑ FIG. 6. Color online Specific heat of Ca VO 2 PO4 2 upper temperature TN as compared to Tmax see the previous sub- ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͒ panel and Sr VO 2 PO4 4 bottom panel measured at different ap- section . However, in contrast to the geometrically frustrated plied fields ͑H͒. The insets show the respective H−T phase systems, TN does not vanish completely, and at sufficiently diagrams. low temperatures, long-range magnetic ordering is estab- lished. To understand the nature of the ordered state, we ͑ ͒ studied field dependence of the specific heat Fig. 6 .At The hyperfine Hamiltonian for 31P can be written in the large, the transition temperature gradually decreases with the ˆ ␥ប increase in the field and finally gets suppressed below 0.4 K form H=− IAhfS, with I and S being dimensionless ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ nuclear and electron spins, respectively. The NMR shift K is at 8 and 11.5 T for Ca VO 2 PO4 2 and Sr VO 2 PO4 2, ␹ respectively.36 The H−T phase diagrams shown in the insets a direct measure of the uniform spin susceptibility spin. of Fig. 6 are typical for antiferromagnets and point to anti- Quite generally, their relation is written as follows: ͑ ͒ ͑ ͒ ferromagnetic ordering in the M VO 2 PO4 2 compounds.

205 K 100 K 53 K C. 31P NMR 1.0 For both the compounds, the 31P NMR spectra consist of a single and narrow spectral line as is expected for I=1/2 0.8 Ca(VO)242 (PO ) nuclei.37,38 The single spectral line implies that both ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 31 31P NMR Ca VO 2 PO4 2 and Sr VO 2 PO4 2 have a unique P site 0.6 consistent with the structural data. Additionally, a narrow H =4T line is also a signature of good sample quality. Figure 7 max 0

/ ͑ ͒ ͑ ͒ II 0.4 shows the representative spectra for Ca VO 2 PO4 2 above the transition temperature. The line width and the line shift were found to be temperature dependent. 0.2 The temperature dependence of the NMR shift K͑T͒͑Fig. 8͒ behaves similar to the bulk susceptibility ␹ ͑see Fig. 2͒. 0.0 NMR has an important advantage over ␹ for the determina- 70.2 70.4 70.6 70.8 tion of magnetic parameters. At low temperatures, extrinsic (MHz) Curie-type paramagnetic contribution often affects the bulk susceptibility, while in case of NMR this contribution broad- ͑ ͒ 31 ͑ ͒ ͑ ͒ FIG. 7. Color online P NMR spectra for Ca VO 2 PO4 2 ens the spectral line but does not contribute to the line shift. measured at 4 T by the Fourier transform of the spin-echo signal Therefore, it is sometimes more reliable to extract the mag- above TN. The representative spectra show the line shift with tem- netic parameters from the temperature dependence of the perature. All the spectra are normalized to unity by dividing the ͑ ͒ ͑ ͒ NMR shift rather than from the bulk susceptibility. intensity I by the maximum intensity Imax .

024418-6 STRONG FRUSTRATION DUE TO COMPETING… PHYSICAL REVIEW B 78, 024418 ͑2008͒

12 10 =70MHz 4

(%) 6 8

K Sr(VO) (PO ) 0 242 2 1 10 100 6 T (K) Ca(VO)242 (PO ) )

% ( Ca(VO)242 (PO ) K 4

1 0 6 =70MHz

K 2

1/ (%) 4

4 (%)

K 2 0 0 0 4 1 10 100 2 6 810 2 T (K) (102 emu/mole) Sr(VO)242 (PO ) FIG. 9. ͑Color online͒ 31P NMR shift ͑K͒ vs bulk susceptibility 0 ͑␹͒ with temperature as an implicit parameter. We employ the ␹ 0 100 200 300 data collected at the applied field of 4 T since this field was used for T (K) the NMR measurements. The solid and dashed lines show the linear fits with Eq. ͑4͒. ͑ ͒ 31 ͑ / ͒ FIG. 8. Color online Inverse of P NMR shift 1 K vs T for ͑ϱ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 1 ͑ M −M t ͒ ͑ / ͒ Ca VO 2 PO4 2 upper panel and Sr VO 2 PO4 2 bottom panel . a single exponential, 2 M͑ϱ͒ =A1 exp −t T1 +C, where Circles and squares indicate the experimental data, while the solid M͑t͒ is the nuclear magnetization at a time t after an invert- lines show the Curie-Weiss fits. The insets present the K vs T ing pulse, while A1 and C are time-independent constants. / curves. Temperature dependences of 1 T1 are presented in Fig. 10. The spin-lattice relaxation rates are temperature independent A above 20 K and rapidly increase with the reduction in tem- K = ␦ + ͩ hf ͪ␹ , ͑4͒ perature below 20 K. The temperature-independent behavior ␮ spin NA B / of 1 T1 is typical for the paramagnetic regime with fast and random fluctuations of electronic spins.42 The critical diver- where ␦ is the temperature-independent chemical shift, gence in the vicinity of T corresponds to the slowing down which is almost negligible, N is the Avogadro constant, and N A of fluctuating moments and indicates the approach to the A is the transferred hyperfine coupling between the elec- hf state with long-range magnetic ordering. trons and the probing nuclei. Therefore to calculate A , one hf Furthermore, the 1/T data enable to estimate exchange should use K vs ␹ plot with T as an implicit parameter. 1 spin couplings in the system under investigation. According to Since above 3 K the extrinsic paramagnetic contribution is Moriya,42 the high-temperature limit of 1/T for a system of negligible for both the compounds, we use the bulk suscep- 1 ␹ ␹ tibility instead of the spin susceptibility spin. The resulting 50 ͑ Ϯ ͒ 2.3 K plots are shown in Fig. 9. The fits yield Ahf= 4920 200 Sr(VO)242 (PO ) ͑ Ϯ ͒ /␮ ͑ ͒ ͑ ͒ 1200 2K and 5125 200 Oe B for Ca VO 2 PO4 2 and =70MHz ͑ ͒ ͑ ͒ ␹ 40 Sr VO 2 PO4 2, respectively. The linearity of the K vs spin 1.85 K 1.7 K plots confirms that by measuring K͑T͒ we can trace ␹ ͑T͒ 800 spin 1.25 K properly. The obtained hyperfine couplings for 31P are of the 1.45 K

1 30 same order as the values previously reported for vanadium 400 phosphates39–41 and indicate a sizeable hybridization of P Intensity (a.u.)

and V orbitals mediated by 2p orbitals of oxygen. T 20 ͑ ͒ 1/ (ms ) 0 To extract the magnetic parameters, we ﬁtted K T curves 3.7 3.8 3.9 ␹ /͑ ␪ ͒ above 15 K with the Curie-Weiss law =C T+ CW . The 0 H (T) ͑␮K ͒ 10 resulting values of the effective moment eff and the Curie- ͑␪K ͒ Weiss temperature CW are listed in Table I. These values Ca(VO)242 (PO ) Sr(VO)242 (PO ) are in good agreement with that obtained from the analysis 0 of the bulk susceptibility. 0 50 100 150 200 / For the 1 T1 experiment, the frequencies of the central T (K) positions of the corresponding spectra at 70 MHz have been ͑ ͒ ͑ / ͒ excited. For a spin-1/2 nucleus, the longitudinal magnetiza- FIG. 10. Color online Spin-lattice relaxation rate 1 T1 vs ͑ ͒ ͑ ͒ ͑ ͒ tion recovery is expected to follow a single exponential be- temperature T for the M VO 2 PO4 2 compounds. The inset ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ havior. In M VO 2 PO4 2, the recovery of the nuclear mag- shows the ﬁeld sweep spectra of Sr VO 2 PO4 2 measured at the netization after an inverting pulse can indeed be described by transmitter frequency of 70 MHz and temperatures close to TN.

024418-7 NATH et al. PHYSICAL REVIEW B 78, 024418 ͑2008͒

localized moments can be expressed as follows: 100 V dxy 75 Total 2 ͱ 1 ␥ S͑S +1͒ 2␲ V ͯͩ ͪͯ = ͚ ͉Ak ͉2, ͑5͒ 50 ␻ ij T1 T→ϱ 2 3 E k,i,j 25 P k ͑ ͒ O where A i, j=x,y,z are the components of the hyperﬁne 1

ij tensor due to the kth magnetic atom. The Heisenberg ex- 0.2 0.1 0.0 0.1 ␻ ␻ change frequency E is defined as E 50 ͑ /ប͒ͱ ͑ ͒/ ͱ͚ 2 =Jc kB 2zS S+1 3, Jc = iJi is the thermodynamic en- ͑ DOS (eV ) ergy scale of the exchange couplings, and z=zi =2 as introduced in Sec. IV B͒.19 Using the relevant parameters and the experimental high-temperature relaxation rates 8 and −1 ͑ ͒ T1 Ӎ 8.5 ms see Fig. 10 ,wefindJc 13.4 and 14.1 K for ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Ca VO 2 PO4 2 and Sr VO 2 PO4 2, respectively. These val- 0 C 4 0 4 ues are in reasonable agreement with the estimates Jc Ӎ10 K from the specific heat ͑see Sec. IV B͒. E (eV) In the spectral measurements at low temperatures ͑slightly ͒ above TN , a broad background signal appears along with the FIG. 11. ͑Color online͒ LDA density of states for ͑ ͒ ͑ ͒ central peak for both the compounds. At TN, the central peak Ca VO 2 PO4 2. The Fermi level is at zero energy. The calcium vanishes, and the broad background signal becomes promi- contribution is negligible in the whole energy range; therefore it is nent extending over a large field range of about 5 T. This not presented in the figure. The inset zooms the image near the ͑ ͒ ͑ ͒ Fermi level, only V 3d and oxygen contributions are shown. effect is illustrated in the inset of Fig. 10 for Sr VO 2 PO4 2. xy The central line disappears at 1.7 K consistent with the ordering temperature of 1.8 K as determined by the specific- ertheless, we succeed to establish a reasonable spin model heat measurements ͑see the inset of Fig. 6 and note that we and provide a plausible explanation of the frustration ͑Sec. use the TN value at 4 T since the NMR measurements are VI͒. carried out at this applied field͒. The broadening of our spec- Experimental data show the similarity of the magnetic ͑ ͒ ͑ ͒ tra upon approaching TN from above is the usual behavior properties of the M VO 2 PO4 2 compounds with M =Ca and expected for the magnetic ordering. Below TN, the spectral Sr. We calculated band structures for both the compounds, intensity melts into a broad background signal which gives analyzed exchange couplings, and did not find any consider- strong evidence of a large distribution of internal static fields able differences between the two systems. Therefore, in the in the ordered state. Similar broadening of the spectral line following we will discuss the calcium compound only. The ͑ ͒ ͑ ͒ has been observed for many other materials in the magneti- whole discussion is applicable to Sr VO 2 PO4 2 as well. cally ordered state ͑see, e.g., Refs. 43 and 44͒.

A. LDA and tight-binding model V. BAND STRUCTURE ͑ ͒ ͑ ͒ The LDA density-of-states plot for Ca VO 2 PO4 2 is ͑ ͒ ͑ ͒ The M VO 2 PO4 2 compounds reveal quite complicated shown in Fig. 11. Valence bands below −3 eV are mainly spin systems with numerous superexchange paths; therefore formed by oxygen orbitals, while the states near the Fermi the presented experimental data do not allow to determine level have predominantly vanadium character with an admix- the individual exchange couplings unambiguously. To over- ture of oxygen. The phosphorous and calcium contributions come this difficulty, we turn to band-structure calculations, to these states are tiny and hardly visible in the figure. Note estimate individual exchange couplings, and construct a mi- that the energy spectrum is gapless in evident contradiction ͑ ͒ ͑ ͒ croscopic model of the exchange interactions. As we will with the green color of Ca VO 2 PO4 2. This is a typical show below, the computational analysis of the failure of LDA due to an underestimate of strong electron- ͑ ͒ ͑ ͒ M VO 2 PO4 2 compounds is also rather difficult, and one electron correlations in the V 3d shell. Local-spin-density can hardly expect the reliable quantitative estimates of all the approximation ͑LSDA͒+U calculations readily reproduce exchange couplings in the systems under investigation. Nev- the insulating spectrum with an energy gap of 2.5–2.8 eV.

TABLE II. The hopping parameters ti of the tight-binding model and the resulting antiferromagnetic AFM ͑ ͒ ͑ ͒ couplings Ji for Ca VO 2 PO4 2. Only the interactions involving PO4 tetrahedra are listed. The effective on-site Coulomb repulsion potential Ueff=4.5 eV.

t1 t2 t3 t4 t5 t ͑meV͒ −12 −33 36 −3 −3

J1 J2 J3 J4 J5 JAFM ͑K͒ 1.5 11.3 13.4 0.1 0.1

024418-8 STRONG FRUSTRATION DUE TO COMPETING… PHYSICAL REVIEW B 78, 024418 ͑2008͒

TABLE III. LSDA+U estimates for the exchange integrals in ͑ ͒ ͑ ͒ ␪calc 0.1 Ca VO 2 PO4 2 and the resulting Curie-Weiss temperatures CW.

␪calc Ud J1 +J2 J3 J4 +J5 CW 0.0 ͑eV͒ ͑K͒ ͑K͒ ͑K͒ ͑K͒

)

eV ( 4 2.9 4.4 6.0 6.7 E 0.1 5 −14.0 4.1 9.2 −0.4 6 −40.5 3.7 14.0 −11.4 0.2

volving PO tetrahedra and the respective exchange X Z1 Y Z 4 integrals.47 The strongest AFM coupling runs between non- ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ FIG. 12. Color online LDA band structure of Ca VO 2 PO4 2 parallel structural chains J3 , the other strong coupling oc- ͑ ͒ near the Fermi level ͑thin light lines͒ and the fit with the tight- curs between parallel chains J2 , and all the other AFM binding model ͑thick darker lines͒. The Fermi level is at zero en- couplings are weaker at least by a factor of 8. The magnitude ergy. The notation of k points is as follows: ⌫͑0,0,0͒, X͑0.5,0,0͒, of J’s is consistent with the experimental data, although the ͑ ͒ ͑ ͒ ͑ ͒͑ Z1 0.5,0.5,0 , Y 0,0.5,0 , and Z 0,0,0.5 the coordinates are precise values are somewhat overestimated. For example, given along kx, ky, and kz in units of the respective reciprocal-lattice considering J2 and J3 only, we find Jc =17.7 K that slightly parameters 4␲/a,4␲/b, and 4␲/c͒. The points Z and Z are T1 Ϸ 1 exceeds the Jc 13 K estimate from NMR and is well equivalent due to the face-centered symmetry of the unit cell. C Ϸ above the estimate Jc 10 K from the specific heat. The latter values are in reasonable agreement with the sample color. B. LSDA+U: results To study exchange interactions, we construct an effective The spin system formed by J2 and J3 is 3D and nonfrus- model that includes the relevant half-filled bands only. Thus, trated. To get a frustrated scenario, one has to consider addi- we focus on the bands that are close to the Fermi level and tionally FM interactions using LSDA+U calculations. The have predominantly vanadium character. According to the regular approach deals with the construction of a supercell discussion in Sec. II, distorted octahedral coordination of enabling different patterns of spin ordering, the mapping of vanadium gives rise to a nondegenerate dxy ground state. the resulting energies onto a Heisenberg model, and the es- ͑ ͒ Indeed, we find four bands formed by V dxy orbitals in the timate of individual exchange integrals see, e.g., Ref. 25 . energy range between −0.2 and 0.1 eV ͑see Fig. 12 and the However, this approach seems to be inappropriate for ͒ ͑ ͒ ͑ ͒ inset of Fig. 11 . These bands correspond to four vanadium Ca VO 2 PO4 2, as the magnetic interactions are very weak, ͑ ͒ ͑ ͒ atoms in the primitive cell of Ca VO 2 PO4 2 and are used and the change in the total energy due to the variation of spin for a tight-binding fit of the transfer integrals ͑hoppings͒ rel- ordering is of the order of 10 K ͑i.e., 10−5 Hartree or ϳ10−9 evant for the AFM exchange interactions. The resulting of the total energy͒. Thus, we have to use the smallest unit transfer integrals ͑t͒ are introduced to the extended Hubbard cell in order to achieve the highest possible accuracy, and model, and the correlation effects are taken into account ex- only sums of the exchange couplings can be estimated. Nev- 45 AFM plicitly via an effective on-site repulsion potential Ueff. In ertheless, one may expect that the TB results on Ji will Ӷ our case t Ueff, hence the Hubbard model at half-filling can enable to resolve FM interactions ͑at least, qualitatively͒. be reduced to a Heisenberg model for the low-lying ͑i.e., The results of the LSDA+U calculations are listed in spin͒ excitations. Thus, we are able to estimate AFM contri- Table III. The most stable spin configuration is formed by AFM 2 / butions to the exchange couplings as Ji =4ti Ueff. Similar ferromagnetic structural chains that are coupled both ferro- ͑ ͒ to Ref. 25,weuseUeff=4.5 eV—a representative value for magnetically for parallel chains and antiferromagnetically vanadium oxides. ͑for nonparallel chains͒. We find the same ground state for An advantage of the TB approach is the possibility to Ud =4–6 eV, although the exchange couplings strongly de- estimate all the exchange couplings in the system under in- pend on the Ud value. This problem will be addressed below vestigation. Yet the TB fit is sometimes nonunique, and this ͑Sec. V C͒ after a comparison of the LSDA+U results with ͑ ͒ ͑ ͒ is the case for Ca VO 2 PO4 2 due to the presence of numer- the experimental data and the TB estimates. ous NN and NNN hoppings. To get an unambiguous solu- As we have mentioned in Sec. III, Ud is an adjustable tion, we assume that the leading interactions run via PO4 parameter that depends on the basis set, crystal structure of ͑ ͒ tetrahedra i.e., correspond to t1 −t5 introduced in Sec. II . the compound under investigation, and the objective param- This assumption looks reasonable since magnetic interac- eters of the calculation. Usually, one has to fit Ud to some tions in transition-metal phosphates are usually mediated by observable quantity ͑energy gap, magnetic moments, etc.͒.It ͑ ͒ PO4 groups see, e.g., Refs. 25, 32, 38, and 46 . Moreover, is preferable to use the quantity that is related to the objec- phosphorous gives larger contribution to the states near the tive parameters since different objective parameters may re- Fermi level as compared to calcium. Thus, the V-O-P-O-V quire the application of different Ud values. Below, we will ␪ superexchange paths should be favorable. employ the Curie-Weiss temperature CW for this purpose. The resulting TB fit is in perfect agreement with the LDA The Curie-Weiss temperature is the second-order term in the band structure ͑see Fig. 12͒. Table II lists the hoppings in- high-temperature series expansion of the magnetic suscepti-

024418-9 NATH et al. PHYSICAL REVIEW B 78, 024418 ͑2008͒

␪ bility. According to Ref. 30, for spin-1/2 system CW / ͚ / ͚ ␪ =1 4 iziJi =1 2 iJi. Thus, CW is directly related to the exchange couplings and provides reasonable justification for the choice of the Ud value. Clearly, the results for Ud =5 eV are in good agreement ␪ Ӎ with the experimental estimate CW −0.3 K, while for Ud =4 and 6 eV, the compliance of the experimental and com- J putational results is quite poor ͑see Table III͒. Thus, the op- 3 timal value of U for Ca͑VO͒ ͑PO ͒ is5eV.48 Yet in con- d 2 4 2 J J1 <0< 0 8,25,46 2 trast to other spin-1/2 systems, the slight variation of Ud results in a huge change in the exchange couplings ͑see Table J III͒. This effect will be further discussed in Sec. V C. In the 3 rest of this section, we will focus on the exchange couplings c calculated with Ud =5 eV. Ӷ AFM AFM According to Tables II and III, J1 +J2 J1 +J2 indi- b cating considerable FM contribution to either J1 and/or J2. J3 AFM is smaller than J3 and positive; hence there is also a FM a contribution to J3, but the overall interaction remains AFM. Finally, J +J ӷJAFM+JAFM, i.e., the LSDA+U and TB es- FIG. 13. ͑Color online͒ Enlargement of the crystal structure of 4 5 4 5 ͑ ͒ ͑ ͒ ͑ ͒ timates for these couplings are inconsistent. One may think Ca VO 2 PO4 2 see Fig. 1 that shows the proposed scenario of the that additional interactions contribute to the LSDA+U value frustration. Antiferromagnetic interchain couplings J2, J3 favor an- J J ͑ ͒ tiparallel orientation of spins for neighboring atoms in the chain of 4 + 5, but such contributions yield 2–3 K only Ref. 49 ͑ ͒ ͑ ͒ and do not explain the controversy. Our experience shows marked by large circles , while the in-chain coupling J1 is ferromagnetic and would favor parallel alignment of the respective that the TB approach is highly reliable for estimating ex- spins. change couplings in spin-1/2 systems.25,46,50 In contrast to LSDA+U, the TB approach includes a simple explicit rela- convinced that the agreement between the band-structure re- tion between Ji and Ueff, while the underlying LDA calculation is ab initio and does not include any adjustable param- sults and the experimental data is reasonable, especially tak- eters. Therefore, we have to admit the failure of LSDA+U to ing into account the complexity of the spin system and the provide valid quantitative estimates of the weak exchange weakness of the exchange couplings. ͑ ͒ ͑ ͒ ͑ couplings in Ca VO 2 PO4 2 see Sec. V C for further dis- cussion͒. Nevertheless, the LSDA+U results are helpful to C. LSDA+U: The influence of U get a qualitative understanding of the system under investi- d gation. To understand the J vs Ud trends, one should consider To get an idea about the FM interactions in exchange integrals as composed of the AFM and FM contri- ͑ ͒ ͑ ͒ AFM FM Ca VO 2 PO4 2, we consider vanadium-vanadium separa- butions: J=J +J . Within one-band Hubbard model, ͑ ͒ AFMϳ 2 / FM tions for J1 −J3. The shortest separation 3.46 Å corre- J t U, while J is independent of U. Therefore, the sponds to J1, while J2 and J3 reveal larger distances of 6.15 overall J should be reduced with the increase in U. The re- and 4.35 Å, respectively. FM interactions are short range; pulsion potential Ud is a parameter of the computational therefore it is natural to suggest the strongest FM contribu- method rather than the true on-site Coulomb repulsion, and AFM AFM / tion to be that of J1. Since J1 is close to zero, the overall J is not necessarily proportional to 1 Ud. Yet the increase interaction J1 should be FM. Unlike J1, J2 and J3 are AFM, in Ud tends to improve the localization of 3d electrons hence although the FM contributions may also be non-negligible suppressing AFM interactions. Thus, one would expect that and reduce the absolute values of the exchange couplings. antiferromagnetic exchange couplings should be reduced The presence of the ferromagnetic in-chain interaction J1 is with the increase in Ud. In fact, the strongest coupling usu- ϳ / consistent with our studies of other vanadium phosphates ally shows 1 Ud behavior, while other exchange integrals 51 ͑ ͒ having similar chains of corner-sharing octahedra. are nearly independent of Ud see, e.g., Refs. 25 and 52 .If ͑ ͒ ͑ ͒ AFMӷ FM Thus, the basic microscopic model for Ca VO 2 PO4 2 J J , the overall J is antiferromagnetic for any reason- AFMϷ FM includes three exchange interactions: ferromagnetic J1 and able value of Ud. However, for small J J the situa- antiferromagnetic J2 and J3 interactions. The resulting spin tion will be different, and the resulting J may be positive for system is 3D and frustrated ͑see Fig. 13 and Sec. VI͒. Un- small Ud and negative for large Ud. fortunately, the computational results do not enable quantita- The above considerations naturally explain why FM inter- ͑ ␪ ͒ tive estimates of individual exchange couplings ͑see the actions manifest themselves hence reducing CW at Ud above discussion and Sec. V C͒. Therefore, we do not make Ն5 eV only. However, the FM interactions are further en- further numerical comparisons with the experimental data. hanced with the increase in Ud, and this effect has a different Clearly, the presence of both the FM and AFM couplings is origin. LSDA+U treats electronic correlations within mean- ␪ qualitatively consistent with the strongly reduced CW. More- field approximation; therefore the filled states of vanadium over, band-structure calculations suggest rather weak ͑about are merely shifted to lower energies as the repulsive potential ͒ ͑ ͒ ͑ ͒ 10 K exchange interactions in Ca VO 2 PO4 2 consistent Ud is applied. At lower energies, oxygen states dominate; with the experimental energy scale Jc =10–15 K. We are hence vanadium-oxygen hybridization is enhanced with the

024418-10 STRONG FRUSTRATION DUE TO COMPETING… PHYSICAL REVIEW B 78, 024418 ͑2008͒

increase in Ud. Indeed, our calculations reveal the decrease structure calculations and estimated the individual exchange in the magnetic moment of vanadium and the partial spin couplings. Despite the problems discussed in Sec. V C, we polarization of oxygen atoms as Ud is increased. In case of succeeded to construct a reasonable microscopic model that ͑ ͒ ͑ ͒ Ca VO 2 PO4 2, this effect is even more pronounced as com- provides a plausible scenario of the frustration. The coupling pared to other vanadium compounds.25,52 Moreover, the ex- within the structural chains ͑the chains of corner-sharing ͒ ͑ ͒ change couplings are very weak, and the overestimated VO6 octahedra, see Fig. 1 is FM J1 , while interchain cou- ͑ ͒ vanadium-oxygen hybridization results in unrealistic results plings are AFM J2 and J3 . The interactions form a quad- ͑ ͒ rangle with one FM and three AFM sides ͑Fig. 13͒. The FM for Ji note the J1 +J2 value of −40 K at Ud =6 eV . In summary, we should point to two complications that in-chain interaction favors the parallel alignment of the spins arise during LSDA+U calculations for vanadium oxides within the chain, while the interchain AFM interactions favor with very weak magnetic interactions. First, one has to apply the antiparallel alignment. Thus, the spin system is frustrated. a relatively large Ud in order to suppress AFM couplings and ͑␪ Below, we will use several experimental quantities CW, to reveal FM contributions. Second, large Ud tends to over- ͒ Jc, and Hs in order to estimate the individual exchange cou- estimate vanadium-oxygen hybridization and to produce un- plings of the proposed model. Assuming the ground state realistic enhancement of the exchange couplings. There is an revealed by the LSDA+U calculations ͑see Sec. V B͒,we ͓ ͑ ͒ ͑ ͒ ͔ optimal Ud value 5 eV in case of Ca VO PO that gives ␮ / ͑ 2 4 2 find Hs =2J3g B kB the Heisenberg model is treated in a rise to a reasonable solution. However, even at this value ͒ ͑ C classical way . Using the averaged estimates of Jc = J ␹ c both problems are retained. In general, we think that T1͒/ ␪ ͑␪ ␪K ͒/ +J 2 and CW= + 2 and assuming negative J1, LSDA+U estimates of weak exchange couplings in highly c CW CW one arrives at J1 =−9.6 K, J2 =3.5 K, and J3 =5.4 K for complex structures are not accurate enough to provide quan- ͑ ͒ ͑ ͒ Ca VO 2 PO4 2 and J1 =−8.9 K, J2 =1.8 K, and J3 =7.7 K titatively correct results. Therefore, we mainly rely on the TB ͑ ͒ ͑ ͒ for Sr VO 2 PO4 2. These values should be considered as model in our analysis. Nevertheless, LSDA+U estimates fa- rough estimates since our model includes three interactions cilitate a qualitative understanding of the coupling scenario ␪ only, and the results of different methods for Jc and CW are resulting in a reasonable microscopic description of the mag- simply averaged. The estimates emphasize the similarity of netic behavior. the calcium and strontium compounds, although J2 and J3 values are somewhat different for M =Ca and Sr. The latter VI. DISCUSSION result may be relevant for the different breadth of the specific-heat maxima in the two compounds ͑see Fig. 5͒. ͑ Our experimental results show that the complex vanadium We should emphasize that at least two couplings J1 and ͑ ͒ ͑ ͒ ͒ ͑ ͒ ͑ ͒ phosphates M VO 2 PO4 2 are strongly frustrated spin sys- J3 in the M VO 2 PO4 2 compounds have similar magni- tems with competing FM and AFM exchange couplings. tude, and the spin system is essentially 3D rather than 1D ␹ Magnetic susceptibility data reveal a maximum at Tmax with frustrated interchain couplings. The frustration is con- Ӎ ␪ 3 K and nearly vanishing Curie-Weiss temperatures CW trolled by the magnitudes of the competing exchange inter- Յ 1 K. For simple spin systems with a single leading ex- actions. For example, the decrease of the absolute value of J1 ␹ ␪ change coupling, Tmax and CW are usually close to each will reduce the frustration, while for AFM J1 the system is ␪ Ӷ ␹ other, while the CW Tmax regime implies the presence of nonfrustrated at all. To the best of our knowledge, no theo- several different magnetic interactions in the system under retical results for this model are available. Basically, one ␪ investigation. As we have stated above, CW is equal to half may expect the presence of strongly frustrated regions ͓one ␪ ͑ ͒ ͑ ͒ ͔ of the sum of all the exchange integrals. Thus, the low CW is revealed by the M VO 2 PO4 2 compounds and quantum value points to the presence of both FM and AFM interac- critical points in the respective phase diagram. Further stud- tions that compensate each other. Further on, the low Néel ies of the model and the appropriate materials are desirable. ͑ Ϸ ␹ / ͒ temperatures TN Tmax 2 indicate that long-range ordering Finally, we will focus on the structural aspects of the in the systems is impeded by either low dimensionality present study. The main structural feature of the ͑ ͒ ͑ ͒ and/or magnetic frustration. M VO 2 PO4 2 compounds deals with the chains of corner- Low-temperature specific heat provides the information sharing VO6 octahedra. Such chains are typical for a wide on the spin excitations and presents one of the substantial variety of vanadium oxides.53 Within a very straightforward characteristics of quantum magnets. In systems with strong and naive approach, one can identify the chains of vanadium quantum fluctuations, the specific-heat maximum is reduced polyhedra as spin chains and arrive at 1D spin system with due to the suppression of correlated spin excitations. The weak interchain couplings. According to Sec. II, such an ap- maximum of the magnetic specific heat of the proach is inconsistent with the orbital state of vanadium, and ͑ ͒ ͑ ͒ ͑ max͒ M VO 2 PO4 2 compounds Cmag equals to 0.25R only. This the actual spin system may have any dimensionality depend- value is comparable to that for the spin-1/2 triangular lattice, ing on the exchange couplings via PO4 tetrahedra or other ͑ ͒ ͑ ͒ a system with strong geometrical frustration, and lies well side groups. In the case of M VO 2 PO4 2, the actual spin max ͑ ͒ below Cmag typical for nonfrustrated low-dimensional spin system is three dimensional. Yet in Sr2VO VO4 2 the spin systems. Thus, strong quantum fluctuations are present in system is 1D ͑consistent with the naive expectations͒, but the ͑ ͒ ͑ ͒ M VO 2 PO4 2, and these fluctuations are likely caused by spin chains are perpendicular to structural ones ͑in contrast the magnetic frustration. to the naive expectations͒.26 In general, the structures formed ͑ To reveal the origin of the frustration in the by VO6 octahedra and nonmagnetic tetrahedral groups PO4, ͑ ͒ ͑ ͒ +5 ͒ M VO 2 PO4 2 compounds, we have performed band- V O4, SiO4, etc. provide a promising way for studying

024418-11 NATH et al. PHYSICAL REVIEW B 78, 024418 ͑2008͒

different spin systems, and further investigations could be heat. The thermodynamic energy scale of the exchange cou- interesting. However, we should point to the importance of plings is Jc =10–15 K as shown by the speciﬁc-heat and careful thermodynamic measurements and electronic- NMR data. Band-structure calculations suggest a plausible structure-based microscopic modeling for proper understand- scenario of the frustration with the FM interaction along the ing of the respective systems. Thus, the interesting physics of structural chains and the AFM interactions between the ͑ ͒ ͑ ͒ the M VO 2 PO4 2 compounds is well hidden behind the 1D chains. In the resulting three-dimensional spin system, the features of the crystal structure and weak magnetic interac- frustration is controlled by the magnitudes of the competing tions resulting in the paramagnetic behavior above 4–5 K. exchange couplings. The proposed spin model deserves fur- Only the combination of experimental and computational ap- ther experimental and theoretical studies. proaches enables us to unravel the relevant microscopic mechanisms and to achieve an insight of the underlying ACKNOWLEDGMENTS physics. ͑ ͒ ͑ ͒ In conclusion, we have shown that the M VO 2 PO4 2 The authors are grateful to P. Carretta for his critical sug- compounds reveal strongly frustrated 3D spin systems with gestions on the NMR results, and to Nic Shannon for the competing FM and AFM interactions. The presence of both fruitful discussions. Financial support of GIF ͑Grant No. FM and AFM interactions is indicated by the vanishing I-811-257.14/03͒, RFBR ͑Grant No. 07-03-00890͒, and the ␪ Curie-Weiss temperature CW, while the strong frustration is Emmy-Noether program of the DFG as well as the compu- evidenced by the reduced maximum of the magnetic speciﬁc tational facilities of ZIH Dresden are acknowledged.

*[email protected] 17 N. Shannon, T. Momoi, and P. Sindzingre, Phys. Rev. Lett. 96, †[email protected] 027213 ͑2006͒. ‡Present address: Low Temperature Group, Bariloche Atomic 18 R. Melzi, P. Carretta, A. Lascialfari, M. Mambrini, M. Troyer, P. Centre—National Commission of Atomic Energy, Av. Bustillo Millet, and F. Mila, Phys. Rev. Lett. 85, 1318 ͑2000͒. 9500 ͑C. P. 8400͒ S. C. de Bariloche, Argentina. 19 R. Melzi, S. Aldrovandi, F. Tedoldi, P. Carretta, P. Millet, and F. 1 P. W. Anderson, Science 235, 1196 ͑1987͒. Mila, Phys. Rev. B 64, 024409 ͑2001͒. 2 S. Park, Y. J. Choi, C. L. Zhang, and S.-W. Cheong, Phys. Rev. 20 H. Rosner, R. R. P. Singh, W. H. Zheng, J. Oitmaa, S.-L. Drech- Lett. 98, 057601 ͑2007͒. sler, and W. E. Pickett, Phys. Rev. Lett. 88, 186405 ͑2002͒. 3 J. E. Greedan, J. Mater. Chem. 11,37͑2001͒. 21 A. Bombardi, J. Rodriguez-Carvajal, S. Di Matteo, F. de Ber- 4 A. Harrison, J. Phys.: Condens. Matter 16, S553 ͑2004͒. gevin, L. Paolasini, P. Carretta, P. Millet, and R. Caciuffo, Phys. 5 M. Hase, I. Terasaki, and K. Uchinokura, Phys. Rev. Lett. 70, Rev. Lett. 93, 027202 ͑2004͒. 3651 ͑1993͒. 22 E. E. Kaul, H. Rosner, N. Shannon, R. V. Shpanchenko, and C. 6 G. Castilla, S. Chakravarty, and V. J. Emery, Phys. Rev. Lett. 75, Geibel, J. Magn. Magn. Mater. 272–276, 922 ͑2004͒. 1823 ͑1995͒. 23 K. H. Lii, B. R. Chueh, H. Y. Kang, and S. L. Wang, J. Solid 7 S.-L. Drechsler, N. Tristan, R. Klingeler, B. Büchner, J. Richter, State Chem. 99,72͑1992͒. J. Málek, O. Volkova, A. Vasiliev, M. Schmitt, A. Ormeci, C. 24 F. Berrah, A. Leclaire, M.-M. Borel, A. Guesdon, and B. Loison, W. Schnelle, and H. Rosner, J. Phys.: Condens. Matter Raveau, Acta Crystallogr., Sect. C: Cryst. Struct. Commun. 55, 19, 145230 ͑2007͒, and references therein. 288 ͑1999͒. 25 8 S.-L. Drechsler, O. Volkova, A. N. Vasiliev, N. Tristan, J. Rich- A. A. Tsirlin, R. Nath, C. Geibel, and H. Rosner, Phys. Rev. B ter, M. Schmitt, H. Rosner, J. Málek, R. Klingeler, A. A. Zvya- 77, 104436 ͑2008͒. 26 gin, and B. Büchner, Phys. Rev. Lett. 98, 077202 ͑2007͒. E. E. Kaul, H. Rosner, V. Yushankhai, J. Sichelschmidt, R. V. ͑ ͒ 9 Y. Naito, K. Sato, Y. Yasui, Y. Kobayashi, Y. Kobayashi, and M. Shpanchenko, and C. Geibel, Phys. Rev. B 67, 174417 2003 . 27 ͑ ͒ Sato, J. Phys. Soc. Jpn. 76, 023708 ͑2007͒. K. Koepernik and H. Eschrig, Phys. Rev. B 59, 1743 1999 . 28 45 ͑ ͒ 10 F. Schrettle, S. Krohns, P. Lunkenheimer, J. Hemberger, N. Büt- J. P. Perdew and Y. Wang, Phys. Rev. B , 13244 1992 . 29 Note that we use FPLO7 in the present work. The basis sets em- tgen, H.-A. Krug von Nidda, A. V. Prokofiev, and A. Loidl, ployed in this code are slightly different from that of FPLO5 Phys. Rev. B 77, 144101 ͑2008͒. ͑applied in Ref. 25͒. Thus, the use of a different U value may 11 Y. Yasui, Y. Naito, K. Sato, T. Moyoshi, M. Sato, and K. Kaku- d be required. rai, J. Phys. Soc. Jpn. 77, 023712 ͑2008͒. 30 12 D. C. Johnston, R. K. Kremer, M. Troyer, X. Wang, A. Klümper, S. Seki, Y. Yamasaki, M. Soda, M. Matsuura, K. Hirota, and Y. S. L. Budko, A. F. Panchula, and P. C. Canfield, Phys. Rev. B ͑ ͒ Tokura, Phys. Rev. Lett. 100, 127201 2008 . 61, 9558 ͑2000͒. 13 ͑ ͒ P. Chandra and B. Doucot, Phys. Rev. B 38, 9335 1988 . 31 In the strontium compound, the transition anomaly is more pro- 14 O. P. Sushkov, J. Oitmaa, and Z. Weihong, Phys. Rev. B 63, nounced, and the origin of this effect is not clear. Basically, the 104420 ͑2001͒. specific-heat anomalies are affected by both intrinsic ͑dimen- 15 L. Siurakshina, D. Ihle, and R. Hayn, Phys. Rev. B 64, 104406 sionality of the spin system͒ and extrinsic ͑sample quality͒ fac- ͑2001͒. tors. At present, we can not distinguish which of these factors 16 N. Shannon, B. Schmidt, K. Penc, and P. Thalmeier, Eur. Phys. J. are relevant for the difference of the transition anomalies for the ͑ ͒ ͑ ͒ ͑ ͒ B 38, 599 2004 . M VO 2 PO4 2 compounds.

024418-12 STRONG FRUSTRATION DUE TO COMPETING… PHYSICAL REVIEW B 78, 024418 ͑2008͒

32 N. S. Kini, E. E. Kaul, and C. Geibel, J. Phys.: Condens. Matter 46 M. D. Johannes, J. Richter, S.-L. Drechsler, and H. Rosner, 18, 1303 ͑2006͒. Phys. Rev. B 74, 174435 ͑2006͒. 33 R. Nath, A. A. Tsirlin, H. Rosner, and C. Geibel, 47 The TB model included all the NN and NNN hoppings as well as ͑ ͒ arXiv:0803.3535 unpublished . a number of long-range hoppings. The largest transfer integral 34 ͑ ͒ B. Bernu and G. Misguich, Phys. Rev. B 63, 134409 2001 . not listed in Table II is 10 meV. The respective JAFM is about 1 35 M. Hofmann, T. Lorenz, K. Berggold, M. Grüninger, A. Fre- K, i.e., well below the strongest interactions J and J . There- imuth, G. S. Uhrig, and E. Brück, Phys. Rev. B 67, 184502 2 3 fore, in the present discussion we neglect all the exchange cou- ͑2003͒. 36 plings that do not run via PO4 tetrahedra. The careful inspection of the TN vs H dependence reveals ͑ ͒ ͑ ͒ 48 The optimal value of U for estimating exchange couplings in slightly different behavior of Ca VO 2 PO4 2 and d ͑ ͒ ͑ ͒ M͑VO͒ ͑PO ͒ ͑5eV͒ is different from one for Ag VOP O ͑6 Sr VO 2 PO4 2 at low fields. The ordering temperature of the 2 4 2 2 2 7 calcium compound is nearly field independent below 4 T, while eV, see Ref. 25͒. The difference is likely caused by the change of TN of the strontium compound is somewhat below the zero-field the basis sets in the computational program, see Ref. 29. 49 value even at low fields. In general, one may expect that low The value J4 +J5 in Table III sums all the interactions between ͑ fields suppress quantum fluctuations hence increasing TN such the atoms in positions ͑x,y,z͒ and ͑¯x,¯y,z͒ or ͑x+1/4,¯y behavior was observed in another vanadium phosphate, +1/4,z+1/4͒ and ͑¯x+1/4,y+1/4,z+1/4͒ of the Fdd2 space ͑ ͒ ͒ BaCdVO PO4 2, see Ref. 33 for details . However, the manifes- group. The TB model indicates two leading hoppings between tation of this effect depends on the magnetic anisotropy, the these atoms: t =−9 meV and t =−7 meV. The hoppings pro- magnitude of quantum fluctuations, and possibly, some extrinsic 6 7 vide an overall antiferromagnetic interaction of about 2 K that factors. Thus, the slightly different behavior of the two com- AFM AFM pounds is explainable, although at present we are unable to sug- exceeds J4 +J5 . However, such an interaction is still con- gest a unique reason of the effect. siderably lower than the LSDA+U result. 50 37 R. Nath, A. V. Mahajan, N. Büttgen, C. Kegler, A. Loidl, and J. O. Janson, R. O. Kuzian, S.-L. Drechsler, and H. Rosner, Phys. Bobroff, Phys. Rev. B 71, 174436 ͑2005͒. Rev. B 76, 115119 ͑2007͒. 38 R. Nath, D. Kasinathan, H. Rosner, M. Baenitz, and C. Geibel, 51 A. A. Tsirlin, R. Nath, C. Geibel, and H. Rosner, unpublished ͑ ͒ Phys. Rev. B 77, 134451 2008 . results on the magnetic properties of NaVOPO4 and 39 Y. Furukawa, A. Iwai, K. Kumagai, and A. Yakubovsky, J. Phys. Sr VO͑PO ͒ . The crystal structures of these compounds include ͑ ͒ 2 4 2 Soc. Jpn. 65, 2393 1996 . the chains of corner-sharing VO octahedra similar to that in 40 6 J. Kikuchi, K. Motoya, T. Yamauchi, and Y. Ueda, Phys. Rev. B M͑VO͒ ͑PO ͒ . According to Sec. II, the V-O-V superexchange 60, 6731 ͑1999͒. 2 4 2 in the chain is impossible, since the half-filled d orbital of 41 J. Kikuchi, N. Kurata, K. Motoya, T. Yamauchi, and Y. Ueda, J. xy vanadium is located in the plane perpendicular to the respective Phys. Soc. Jpn. 70, 2765 ͑2001͒. V-O bonds. Thus, the hopping should run via the PO tetrahe- 42 T. Moriya, Prog. Theor. Phys. 16,23͑1956͒; 16, 641 ͑1956͒. 4 43 J. L. Gavilano, S. Mushkolaj, H. R. Ott, P. Millet, and F. Mila, dron connecting two neighboring octahedra. The resulting trans- ͑ϳ ͒ Phys. Rev. Lett. 85, 409 ͑2000͒. fer integral is very weak 10 meV, similar to t1 in Table II ; 44 P. Vonlanthen, K. B. Tanaka, A. Goto, W. G. Clark, P. Millet, J. therefore the ferromagnetic in-chain coupling dominates. In- ͑ ͒ Y. Henry, J. L. Gavilano, H. R. Ott, F. Mila, C. Berthier, M. deed, in NaVOPO4 and Sr2VO PO4 2 we find ferromagnetic in- Horvatic, Y. Tokunaga, P. Kuhns, A. P. Reyes, and W. G. Moul- chain couplings of −3 K and −8 K, respectively. 52 ton, Phys. Rev. B 65, 214413 ͑2002͒. A. A. Tsirlin, A. A. Belik, R. V. Shpanchenko, E. V. Antipov, E. 45 Note that Ueff Ud and these two parameters have fairly differ- Takayama-Muromachi, and H. Rosner, Phys. Rev. B 77, 092402 ͑ ͒ ent meaning. Ueff is the effective Coulomb repulsion in the 2008 . 53 mixed V 3d-O 2p band, while Ud is the repulsive potential ap- S. Boudin, A. Guesdon, A. Leclaire, and M.-M. Borel, Int. J. plied to V 3d orbitals only. Inorg. Mater. 2, 561 ͑2000͒.

024418-13