A pseudotetramer in the geometrically frustrated spinel system CdV₂O₄

Noriaki Nishiguchi and Masashige Onoda^Note1

Institute of Physics, University of Tsukuba, Tennodai, Tsukuba 305-8571, Japan

Email: onoda@sakura.cc.tsukuba.ac.jp

Received 8 May 2002
Published 5 July 2002

Abstract. The structural and magnetic properties of the geometrically frustrated spinel system CdV2O4 with V3 +  (S = 1), whose lattice constant and oxygen u-parameter differ significantly from those of isomorphous MgV2O4 and ZnV2O4, have been explored by means of x-ray diffraction and through measurements of the magnetization and nuclear magnetic resonance. CdV2O4 undergoes a cubic-tetragonal structural transition with ct/at = 0.990 as well as a magnetic transition with a jump of the susceptibility at Tc1 = 97 K. Another magnetic anomaly appears at Tc2 = 35 K which may be a transition to the antiferromagnetic state. The analysis in terms of the high-temperature series expansion of up to eighth order precisely indicates the nearest-neighbour exchange-coupling constant for the cubic phase to be J = 44 K. In the vanadium spinel with S = 1, the spin-singlet V4-tetramer model with the exchange coupling Jtet inside V4, where Jtet = 2J, is applicable at temperatures above : for CdV2O4 with Jtet<Tc1, all of the susceptibility data for the cubic phase are explained using that model, while for MgV2O4 and ZnV2O4 with Jtet>Tc1, it is difficult to account for the data for between Jtet and Tc1. In this sense, the present bound state may be regarded as pseudotetramers.

1. Introduction

Geometrically frustrated spin systems have been investigated for a long time. A state where all pairs of nearest-neighbour spins are aligned in opposite directions is not allowed in such systems and spin correlation is suppressed, so the temperature of the transition to a possible ordered phase is reduced significantly. The system sometimes undergoes a structural phase transition to lower symmetry, since there is a gain of the orbital and magnetic energies on lifting the degeneracy. The triangular S = 1 system LiV O₂ with V^3 +  is one such material, where spin-singlet trimers are formed [1].

The normal spinel-type polaronic insulator MV₂O₄ with V^3 +  exhibits a frustration effect [2]. Here, M is a nonmagnetic divalent ion surrounded tetrahedrally by ions such as Mg and Zn (the so-called A site), and V has octahedral coordination (the B site). The network of the V ions is achieved by the linkage of a regular tetrahedron block that has a V ion at each apex, as in the case of pyrochlore-type compounds [3]. On the other hand, LiV₂O₄ with an apparent valence of V^3.5 +  is a metal and it has recently been considered as being a heavy-fermion system at low temperatures [4]. The mechanism of composition-induced metal-insulator transition in the Li_xMg_1 - xV₂O₄ and Li_xZn_1 - xV₂O₄ systems has been investigated from multiple viewpoints [5].

There have been many works on structural and magnetic properties of MgV₂O₄ and ZnV₂O₄ [2, 5, 6, 7, 8, 9, 10, 11]. These compounds have similar crystal data and oxygen u-parameters to each other, as shown in table 1. For MgV₂O₄ [6, 2], two magnetic anomalies appear, at T_c1 = 65 and T_c2 = 42 K. At both temperatures, specific-heat anomalies are also present, and at T_c1, the cubic-tetragonal transition essentially due to the Jahn-Teller effect occurs. The peak of the specific heat at T_c2 seems to be too small for a simple magnetic transition. In addition, the susceptibility of the magnetically diluted system Mg(V_0.85Al_0.15)₂O₄ with Al^3 +  (S = 0) suggests the existence of spin glass. Since this compound does not undergo structural transitions, the anomaly at T_c1 should be attributed to not only the Jahn-Teller effect, but also a certain kind of spin order. On the other hand, the paramagnetic behaviour at high temperatures was explained in terms of the high-temperature series expansion (HTSE) of up to sixth order with an exchange-coupling constant of about 100 K. It is noted that the nuclear magnetic resonance (NMR) Knight shift for ⁵¹V has a temperature dependence different from that of the susceptibility and the signal seems to disappear at a temperature close to T_c1, possibly due to the spin order [2].

ZnV₂O₄ has properties similar to those of MgV₂O₄ [7, 2, 8]. There is another interpretation in which the magnetic anomalies at T_c1 and T_c2 are attributed to the structural transition and the antiferromagnetic one, respectively [7], although the mechanism of change in the paramagnetic state via structural distortion had been unclear until the recent proposal of the theoretical work [11] as introduced below. For the magnetic structures of MgV₂O₄ [9] and ZnV₂O₄ [10], the magnetic moment of the V ion is half the expected value, , being the Bohr magneton, and its direction is parallel to the tetragonal c-axis.

Recently, a tetrahedral mean-field (TMF) theory of the magnetic susceptibility for the spinel sublattice B or the pyrochlore lattice was proposed [12]. Here, the use of the exact solution for the susceptibility for a set of four interacting spins with coupling J_tet in the corners of an isolated tetrahedron, in other words, the spin-singlet tetramer, is essential, and thus the appearance of a maximum in the susceptibility against temperature has been predicted. The ground-state properties for S = 1 were also investigated by breaking up each spin into a pair of -spins [11]. It has been postulated that the twofold degeneracy of the spin singlets of the tetrahedron is lifted by the Jahn-Teller effect, which leads to a cubic-tetragonal transition with magnetoelastic interaction.

The present work is a first report on the structural and magnetic properties of CdV₂O₄, which is isomorphous to MgV₂O₄ and ZnV₂O₄, based on measurements of x-ray diffraction, magnetization, and NMR. Here, both the cubic lattice constant a and the oxygen u-parameter of CdV₂O₄ are significantly different from those for MgV₂O₄ and ZnV₂O₄. Therefore, the exchange coupling of CdV₂O₄ may be modified substantially as compared with those of isomorphous compounds. Thus it will become possible to consider physical properties of MV₂O₄ as a function of the exchange coupling. In this work, our attention is mainly focused on the magnetic properties for the cubic phase, and the mechanism of magnetic anomalies is not discussed. In order to estimate the exchange coupling precisely, HTSE up to eighth order is performed, and the application of the spin-singlet V₄ tetramer is examined in detail. These analyses are also done for MgV₂O₄ and ZnV₂O₄.

2. Experiments

The polycrystalline specimens of MV₂O₄ (M = Cd, Mg, and Zn) were prepared by the solid-state reaction method using V₂O₅ (99.99% purity), CdO (99.99% purity), MgO (99.99 % purity), and ZnO (99% purity). First, V₂O₃ was made according to the procedure described in [13]. The mixtures of V₂O₃ and MO were pressed into pellets and sealed in evacuated silica tubes, and then heated at 993 K (M = Cd), 1173 K (Mg), and 1073 K (Zn). For MgV₂O₄, preheating at 993 K was necessary in order to obtain single-phase specimens. The x-ray powder diffraction patterns were taken with radiation and the - scan method in the temperature region between 80 and 300 K. The magnetizations were measured by the Faraday method with a field of up to 1 T at temperatures between 4.2 and 750 (900) K for M = Cd (Mg and Zn). The magnetic susceptibility was deduced from the linear part of the magnetization-field curve with a decreasing field. The ⁵¹V NMR measurements for CdV₂O₄ were also performed with a continuous-wave spectrometer at the Larmor frequency of 14.5 MHz at temperatures from 77 to 400 K.

3. Results and discussion

The cubic lattice constants a of MV₂O₄, where M = Cd, Mg, and Zn, determined in this work are listed in table 1 with the oxygen u-parameters revealed previously [2, 5, 14]. For M = Zn, the results based on x-ray four-circle analysis are added [15]. Selected interatomic distances and angles are also presented. The V-O bond length does not depend on M, since the V valence originates from the critical overlap integral of the V and O atoms. On the other hand, the V-V distance and the V-O-V angle both increase in the sequence Zn, Mg, Cd. These composition dependences may be partly attributed to the difference in ionic radii at the A site: 0.78, 0.57, and 0.60 Å for the four coordinations of Cd^2 + , Mg^2 + , and Zn^2 + , respectively [16].

Parts of the x-ray diffraction patterns at 82 and 296 K are shown in figures 1(b) and 1(c). The (440) peak at 296 K splits into two peaks at 82 K with an intensity ratio of about 2:1, whereas the (111) peak at 296 K does not change at 82 K. This result indicates that the crystal system at 82 K may be tetragonal with the axial ratio c_t/a_t = 0.990. The temperature dependence of the lattice constant a is shown in figure 1(a). In the heating process, the first-order cubic-tetragonal transition appears at T_c1 = 97 K. This structural anomaly with c_t/a_t< 1 is similar to those of MgV₂O₄ [6] and ZnV₂O₄ [7].

The temperature dependences of the magnetic susceptibilities for MV₂O₄ with M = Cd, Mg, and Zn are shown in figure 2(a) and the low-temperature parts are indicated in figure 2(b). For CdV₂O₄, a jump in the susceptibility appears at 97 K, which corresponds to the cubic-tetragonal transition temperature T_c1. At temperatures below T_c1, the susceptibility has a broad maximum at 73 K, which is significantly different from the cases of MgV₂O₄ and ZnV₂O₄. Another anomaly with a change of the temperature derivative occurs at T_c2 = 35 K, which may correspond to the antiferromagnetic order. For M = Mg and Zn, two anomalies appear, at T_c1 and T_c2, that agree with the previous reports [6, 7]. The present results are summarized in table 2. For all of the compounds, no difference was observed between the susceptibilities for the zero-field-cooled and the field-cooled processes at 1 T.

On the basis of the Heisenberg model,

where J is the exchange-coupling constant, S is the spin-1 operator, and denotes a nearest-neighbour pair, the magnetic susceptibility of the spinel sublattice B in terms of HTSE up to eighth order is given as follows:

Here, C is the Curie constant and corresponds to the temperature-independent susceptibility of the Van Vleck orbital and diamagnetic components. The full, dotted, and dot-dashed curves in figure 2(a) denote the results calculated for HTSE up to eighth, seventh, and sixth order, respectively, using parameters listed in table 2. For all of the compounds, C agrees with the value calculated from the spin with ; that is,  emu K/(mol V), which may be consistent with a temperature-independent behaviour of the orbital component at the temperatures of interest. The -values are comparable to those estimated previously [2]. J for CdV₂O₄ is half of those for MgV₂O₄ and ZnV₂O₄.

Figure 2. (a) The temperature dependences of the magnetic susceptibilities of MV2O4, where M = Cd, Mg, and Zn; (b) the behaviours at low temperatures. The full, dotted, and dot-dashed curves in (a) denote results calculated in terms of HTSE up to eighth, seventh, and sixth order, respectively; and the dashed curves indicate fits to the tetramer model, where the parameters used are listed in table 2.

Table 2. The transition temperatures, Tc1 and Tc2 (K), and the HTSE and TMF parameters for the magnetic susceptibilities of MV2O4, where M = Cd, Mg, and Zn; the Curie constant C (emu K/(mol V)), the constant susceptibility (10 - 4 emu/(mol V)), and the exchange coupling J (K) from equation (2); the exchange coupling inside the tetrahedron Jtet (K) based on equation (3).

M	Tc1	Tc2	C		J	Jtet
Cd	97	35	0.98(2)	1.8(2)	44(1)	82.9(1)
Mg	64.5	45	0.93(6)	1.2(3)	92(7)	175.8(1)
Zn	52	44	0.96(6)	1.1(3)	92(7)	174.6(1)

For the nearest-neighbour interaction between V ions, the direct exchange coupling and the superexchange coupling should be considered. From the V-O distances and the V-O-V angles listed in table 1, if the Anderson-type superexchange coupling is dominant, J for CdV₂O₄ would become more antiferromagnetic. This is not the case. On the other hand, for the case where the direct exchange coupling is the most effective, the composition dependence of J is explained qualitatively, since the V-V distance of CdV₂O₄ is larger than those of MgV₂O₄ and ZnV₂O₄.

Let us see how the V₄-tetramer model [12] explains the susceptibility data at temperatures above T_c1, although they do not exhibit the maximum phenomenon in the region where the crystal system is cubic. The susceptibility of the tetramer for the present system is written as

with x = exp ( - J_tet/T), J_tet being as defined before. Here, the interaction outside the tetrahedron is finally assumed to be zero as will be explained later. The dashed curves in figure 2(a) represent fits to equation (3) with J_tet listed in table 2, where C and are fixed at the HTSE values. For M = Cd, the agreement between the experimental and calculated results in the cubic phase is satisfactory. In order to clarify the origin of the susceptibility maximum that appears to be reproduced roughly with this model, the low-temperature structure should be determined precisely. On the other hand, for M = Mg and Zn, the data above are explained by this model, but it is difficult to fit those below J_tet. For all of the compounds, J_tet is twice J within a standard deviation, which is equivalent to the condition for the high-temperature limit in the TMF model. It should be noted that the finite interaction outside the tetrahedron that has been introduced in [12] does not improve the agreement significantly.

A single resonance line of the ⁵¹V nuclei in CdV₂O₄ was observed, although the intensity was very weak. A plot of the Knight shift K against the magnetic susceptibility is shown with an implicit parameter T in figure 3. The Knight shift consists of the d-spin component K_d and the Van Vleck orbital one, K_orb, which are related to their susceptibilities (i = d and orb) through the hyperfine fields A_i as follows:

where N is the number of V ions. The full line in figure 3 provides A_d = - 7.1 T , A_orb = 45 T , and K_orb = 1.78%, using listed in table 2 and diamagnetic susceptibilities given in [17]. The magnitude of A_d is comparable to the value for V^3 +  of LiV O₂ [1] and A_orb is nearly equal to the free-ion value of the radial average  [18]. This result indicates that the susceptibility presented here is intrinsic and the orbital contribution is temperature independent above T_c1, which is consistent with the susceptibility result described before.

No ⁵¹V NMR has been detected at temperatures below T_c1. Unfortunately, it is not clear whether the signal disappears or just becomes weaker.

Figure 3. The 51V Knight shift K against the magnetic susceptibility of CdV2O4 at temperatures above 100 K, where the full line denotes the least-squares fit.

4. Conclusions

The structural and magnetic properties of the spinel-type CdV₂O₄ have been revealed. The cubic-tetragonal structural phase transition with c_t/a_t = 0.990 and the magnetic transition with a jump of susceptibility take place at T_c1 = 97 K, and another magnetic anomaly occurs at T_c2 = 35 K, probably corresponding to the antiferromagnetic transition. The HTSE analysis up to eighth order precisely indicates the nearest-neighbour exchange-coupling constant to be J = 44 K. The susceptibility behaviour in the cubic phase agrees well with that of the spin-singlet tetramer with J_tet = 2J. On the other hand, for the isomorphous compounds MgV₂O₄ and ZnV₂O₄ with J_tet>T_c1, it is difficult to fit all of the susceptibility data in the cubic phase in the framework of the present tetramer model. Therefore, for the vanadium spinel with S = 1, the application of this model is limited to the temperature region above , and the present bound state may be regarded as pseudotetramers. In other words, this restriction may justify the use of an apparently isolated tetramer model for the regular spinel B sublattice. In order to understand the magnetic properties of the cubic phase at temperatures below J_tet, further theoretical investigations are necessary.

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Note1