3 Magnetization Plateau Induced by Magnetic Field in Monoclinic

Article pubs.acs.org/JPCA

1/3 Magnetization Plateau Induced by Magnetic Field in Monoclinic CoV2O6 Xiaoyan Yao* Department of Physics, Southeast University, Nanjing 211189, China ABSTRACT: It was recently observed by He et al. [J. Am. Chem. Soc. 2009, 131, 7554] that the monoclinic CoV2O6 shows an unusual magnetic behavior with a magnetization plateau at the height of 1/3 saturated magnetization, and the critical magnetic field for the second jump in magnetization is about 2 times as large as that for the first one. By using the Wang−Landau simulation, we show that this stepwise magnetic behavior can be well reproduced in a distorted antiferromagnetic triangular model with anisotropic exchange interactions. The 1/3 magnetization plateau here originates from the same ferrimagnetic state as observed in the regular triangular system, but the critical fields show different features due to the frustration relaxed by anisotropy. The relative value of the weakest interaction plays a key role in this stepwise magnetic behavior, and hereby the critical fields of the magnetization plateau can be accurately modulated by tuning exchange interactions, which provides a wide-use principle for the phenomena of 1/3 magnetization plateau observed in experiments.

T

interesting phenomenon of CoV2O6 cannot be explained by the regular triangular model, but we note that the monoclinic structure with quasi-one-dimensional feature can be regarded as a spin-chain system on a distorted triangular lattice, in which the frustration is relaxed by anisotropy. By using Wang−Landau simulation on such a triangular model with anisotropic interactions, we can well reproduce the stepwise M of CoV2O6 and thus this puzzle is solved well on the basis of the density of states (DOS). It is indicated that this M0/3 plateau is still induced by the competition between the interchain antiferromagnetic exchange interactions on the distorted triangular lattice, but the frustration here is relaxed by anisotropy, and hence the magnetization process shows the different feature from the case with strong frustration in regular triangular lattice. Furthermore, it is revealed that hc1 and hc2 can be modulated accurately by varying exchange interactions, and the weakest interaction plays a key role in this stepwise magnetic behavior, whereas the height of M plateau, namely M0/3, always remains unchanged. The results obtained are believed to cast more light on the h-induced steplike magnetic behavior observed in experiments, especially for the systems without regular triangular lattice. CoV2O6 crystallizes in the monoclinic structure with the lattice constants of a = 9.220(7) Å, b = 3.494(3) Å, c = 6.601(5) Å, and β = 111.71(3)°.7 In this compound, the edgeshared CoO6 octahedra with magnetic Co2+ ions form linear chains, whereas nonmagnetic V5+ ions in VO5 square pyramids

he steplike magnetic behavior induced by the magnetic field (h) has been widely observed in various materials with magnetic frustration and has long attracted considerable attention. In particular, it is well-known that an h-induced M0/3 (where M0 is the saturated magnetization) plateau can occur in the frustrated systems with a regular triangular structure, such as Ca3Co2O6,1−3 Sr3Co2O6,4 Sr3HoCrO6,5 and SrCo6O11.6 It is surprising that the recent experiment reported that a similar M0/3 plateau also occurs in other structures without regular triangles; that is, the monoclinic CoV2O6 single crystal clearly shows a staircase magnetization (M) curve against h with an M0/3 plateau when h is applied along the magnetic easy axis.7 This experimental discovery at low temperature (T) challenged the applicability of previous explanation on M0/3 plateau, and it was suggested to be caused by quantum reason or unusual competition between interchain antiferromagnetic and ferromagnetic interactions.7 However, these suggestions have not been confirmed up to now, and thus this fascinating phenomenon remains puzzling. This stepwise magnetic phenomenon looks similar to the typical M0/3 plateau in the regular triangular lattice with strong frustration but exotically occurs in the monoclinic CoV2O6. It should be mentioned that although the M0/3 plateau observed in CoV2O6 shows the same height to the M plateau in the regular triangular system, it also exhibits a very different feature. He et al. reported that this M0/3 plateau emerges at the first critical magnetic field hc1 ≈ 1.6 T, and then jumps to M0 at the second critical field hc2 ≈ 3.3 T, so hc2 ≈ 2hc1.7 In particular, M remains about zero up to hc1 ≈ 1.6 T for CoV2O6. On the contrary, the M0/3 plateau jumps at an infinitesimal h, namely hc1 ≈ 0, in the case of regular triangular lattice. Thus, this © 2012 American Chemical Society

Received: October 12, 2011 Revised: January 5, 2012 Published: February 27, 2012 2278

dx.doi.org/10.1021/jp209830b | J. Phys. Chem. A 2012, 116, 2278−2282

The Journal of Physical Chemistry A

Article

algorithm,12−16 a joint DOS g(E,M) is evaluated in E and M space. Starting from the initial value f 0 = 2.718281828459, the modification factor f is reduced according to the recipe f i+1 = f i1/2. For every f i, the histogram HM(E, M) for all possible E and M is required not less than 80% of the average histogram. At last, the random walks stop with the final modification factor f final = 1.000000119. For convenience, kB is chosen to be unity, and h is applied along the orientation of upward spin. On the basis of the DOS obtained, the magnetization as a function of T and h is given by

construct zigzag chains. All the chains are along the b-axis, resulting in a quasi-one-dimensional structural arrangement. The intrachain Co−Co distance is much shorter than the distance between the two nearest magnetic chains, which implies a much stronger intrachain coupling than the interchain one. Moreover, a very strong Ising-like magnetic anisotropy was also confirmed by the experiment of CoV2O6.7 In the ac-plane normal to the chains, CoV2O6 shows a lattice composed of parallelogram cells, as shown in Figure 1. It is worth noting that

M(T ,h) =

∑E , M Mg (E ,M )e−(E − hM )/ T ∑E , M g (E ,M )e−(E − hM )/ T

(3)

The magnetic susceptibility (χ) can be estimated in the following way χ(T ) =

each parallelogram cell can be split into four irregular triangles. Hereby, the structure in the ac-plane can be regarded as a distorted triangular lattice consisting of irregular triangles with three different edges. The anisotropy in structure means the anisotropic interchain exchange interactions. Considering the structural features mentioned above, a two-dimensional (2D) Ising model in a triangular lattice with anisotropic interactions can be applied to qualitatively explore this magnetic behavior at low T, where each single magnetic chain is regarded as a rigid chain-spin approximately. In the similar way, the stepwise magnetic behavior observed in Ca3Co2O6 had been well understood.8−11 The Hamiltonian can be written as

∑ Jij SiSj − h·M [i , j]

(1)

where Si = ±1 represents the Ising chain-spin on the ith site of lattice. The first term on the right of eq 1 is the exchange energy (E) where [i,j] denotes the summation over all the nearest-neighboring chain-spin pairs, and the antiferromagnetic coupling (Jij > 0) between Si and Sj can adopt three values in three different directions, namely J1, J2, and J3, as illustrated in Figure 1. Thus, for each irregular triangle, three edges are a/2 ≈ 4.610 Å, d ≈ 6.505 Å, and c ≈ 6.601 Å, corresponding to J1, J2, and J3 respectively. Here J1 is taken to be 1 for convenience, and the other two coupling constants are evaluated in units of J1. The second term is the magnetic field energy where M is evaluated as M=

∑ Si i

(4)

Because d approximates c and both are longer than a/2, we assume J2 = J3 < J1. As illustrated in Figure 2a, when J1 = J2 = J3 = 1 without anisotropy, a typical M0/3 plateau emerges with hc1 ≈ 0 and hc2 ≈ 6, which is just the well-known phenomenon in the regular triangular lattice. As J2 (J3 = J2) is weakened, namely the anisotropy is considered, the M(h) curve presents an interesting variation. It is seen that although the height of the M plateau, M0/3, remains unchanged, its width is shortened greatly. When J2 weakens from 1 toward 0, hc1 increases from 0 toward 2, while hc2 decreases from 6 also toward 2. Finally, when J2 reaches 0, the M0/3 plateau disappears completely, and M jumps from 0 to 1 directly at hc1 = hc2 ≈ 2. Thus, the width and position of M0/3 plateau can be modulated by tuning anisotropic exchange interactions, and consequently the ratio of hc1 to hc2 can be changed arbitrarily from 0 to 1. It was particularly mentioned in ref 7 that hc2 ≈ 2hc1 for CoV2O6. In fact, this puzzling ratio of critical fields can be well reproduced in the present model. Figure 2b plots the ratio of hc1 to hc2 as a function of J2. It is seen clearly that the curve reaches 1/2 at about J2 = 0.25. As also confirmed in Figure 2c, the simulation with J2 = J3 = 0.25 demonstrates the h-induced M0/3 plateau with hc2 ≈ 2hc1, and the T dependence of magnetic susceptibility (χ) along the direction of Ising spins upon h = 0.1 also shows a peak at low T, very consistent with the experimental phenomenon of CoV2O6.7 The striking advantage of the Wang−Landau algorithm is its straightforward connection with DOS. On the basis of DOS, the variation of the M plateau can be well understood. As plotted in Figure 3a, when J1 = J2 = J3, the DOS presents a horizontal bottom edge with the lowest E and M ranging from −M0/3 to M0/3, namely a nontrivial high degeneracy of ground states originated from the strong frustration in regular triangular lattice. In this case, an infinitesimal positive h can lift the degeneracy to reach the ground state with the highest M, namely point B with M0/3, and thus produce the M0/3 plateau at hc1 ≈ 0. And hc2 ≈ 6 just corresponds to the slope (ks) of the straight edge BC where C is the ferromagnetic state with the highest E and M = M0. When J2 (J3 = J2) is weakened, as plotted in Figure 3b at J2 = 0.6, a cusp with M = 0 appears at the bottom of DOS, which means that the ground states shrink to one point A. The nontrivial degeneracy of the ground states is released, reflecting that the frustration is relaxed. However, point B remains as a turning point. It divides the side edge AC

Figure 1. Sketch of one parallelogram cell in the ac-plane of the crystal structure of CoV2O6. Each empty circle represents a magnetic chain. This cell can be split into four irregular triangles with three different edges (denoted by solid, dashed, and dotted lines) in lengths of a/2, d, and c, corresponding to J1, J2, and J3.

H=

⟨M 2⟩ − ⟨M ⟩2 T

(2)

Our simulation is performed on an L × L triangular lattice (L = 12 and the total number of spins is N = 144) with periodboundary conditions applied. Following the Wang−Landau 2279



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they are actually different to some extent. In the regular triangular system (J1 = J2 = J3) with strong frustration, the M0/3 plateau originates from the lifting of degenerate ground states, namely, no energy is required to reach the first transition, and thus its characteristic is hc1 ≈ 0. The other case in the anisotropic system with relaxed frustration results from h-induced transition, which shows the characteristic hc1 > 0; namely, some energy must be spent to achieve this transition. Releasing the restriction of J3 = J2, the simulations with different sets of antiferromagnetic exchange interactions indicate that the M0/3 plateau exists widely in this anisotropic triangular system. In fact, all M(h) curves in such systems show the similar stepwise behavior at low T, namely, M = 0 from h = 0 to hc1, M = M0/3 from h = hc1 to hc2, and M = M0 above h = hc2. These three values of M, corresponding to points A, B, and C in the DOS, just result from the antiferromagnetic, ferrimagnetic, and ferromagnetic states, as shown in Figure 3d−f. These three states emerge sequentially with increasing h, separated by hc1 and hc2. Thus, the values of hc1 and hc2 can be calculated according to the spin configurations. Here the single-site energies of antiferromagnetic state (EA), ferrimagnetic state (EB), and ferromagnetic state (EC) can be written as EA =

1 ( −2J1 − 2J2 + 2J3) 2

(5)

EB =

1 1 1 · ( −2J1 − 2J2 − 2J3) − ·h 3 2 3

(6)

EC =

1 (2J + 2J2 + 2J3) − h 2 1

(7)

hc1 is just the h point where EA = EB, and hc2 is that where EB = EC, from which the values of hc1 and hc2 can be obtained as follows, hc1 = 2J1 + 2J2 − 4J3

(8)

hc2 = 2J1 + 2J2 + 2J3

(9)

According to eqs 8 and 9, the critical magnetic fields can be evaluated on the basis of the interchain exchange interactions, and thus the magnetic phase diagram at T = 0 can be presented in Figure 4a where the half-folded brighter plane below shows the location of hc1 whereas the darker plane above is that of hc2. Two planes divide this phase space into three parts, namely FO (ferromagnetic), FI (ferrimagnetic), and AFM (antiferromagnetic) states as illustrated in Figure 3d−f. Here the M0/3 plateau emerges in the FI area. Especially, the diagonal section of Figure 4a along the line of J2 = J3, namely, the phase diagram with J2 = J3 shown in Figure 4b, is completely consistent with the simulation results mentioned above. Furthermore, on the basis of eqs 8 and 9, the width of the M0/3 plateau (W) is given by

Figure 2. (a) M/M0 as functions of h at T = 0.1 with J2 (J3 = J2) ranging from 1 to 0. (b) Ratio of hc1 to hc2 as a function of J2, where hc2 ≈ 2hc1 at about J2 = 0.25 denoted by dotted lines. (c) M/M0 as a function of h with J2 = 0.25 at T = 0.1. The inset presents the T dependence of χ upon h = 0.1 with J2 = 0.25.

into two parts, namely, straight lines AB and BC with different slopes, i.e., ks ≈ 0.8 for AB and ks ≈ 4.4 for BC, just corresponding to hc1 ≈ 0.8 and hc2 ≈ 4.4 in the case of J2 = 0.6. During the variation of J2, point B always keeps its position at M0/3, which determines the unchanged height of the M plateau between hc1 and hc2. At the same time, the slopes of AB and BC are changed by reducing J2; namely, ks of AB increases while ks of BC decreases, well reflecting the variation in hc1 and hc2. When J2 reaches 0, as illustrated in Figure 3c, the turning point B disappears. The only one slope of edge AC, namely, ks ≈ 2, indicates hc2 = hc1 ≈ 2. It should be mentioned that although the M0/3 plateaus in the distorted and regular triangular systems all result from the same ferrimagnetic state on point B,

W = hc2 − hc1 = 6J3

(10)

and the ratio of hc1 to hc2 could be expressed as 3J3 J + J2 − 2J3 =1− hc1/hc2 = 1 J1 + J2 + J3 J1 + J2 + J3 3 =1− J1/J3 + J2 /J3 + 1

(11)

It is seen that W depends on J3, and the ratio of hc1 to hc2 is decided by the comparison between these three interactions. Here the weakest interaction plays the key role for this steplike 2280



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Figure 3. Logarithm of joint DOS, i.e., ln g(E,M) for (a) J2 = 1, (b) J2 = 0.6, and (c) J2 = 0 with J3 = J2. The sketches of (d) antiferromagnetic state with M = 0, (e) ferrimagnetic state with M = M0/3, and (f) ferromagnetic state with M = M0, corresponding to A, B, and C points in DOS. Here J3 is assumed to be the weakest coupling. The white and black solid circles represent spin-up and -down, respectively.

of critical fields. Thus, a stronger anisotropy with a weaker J3 means a shorter plateau with the ratio closer to 1. In particular, if hc1/hc2 = 1/2 is required, the three exchange interactions should satisfy the restriction below, J1 + J2 − 5J3 = 0

(12)

Moreover, if J2 = J3, then J2 = J3 = 0.25J1 is obtained, which has been confirmed above to show the M0/3 plateau with hc2 = 2hc1. In such an anisotropy, we can get hc1 = 1.5J1 on the basis of eq 8. Considering the Bohr magneton μB, Lande factor g, and the effective value of chain-spin Se, the real value of J1 can be evaluated as J1 =

g μBhc1 1.5Se

(13)

In summary, a distorted triangular antiferromagnetic model with the anisotropy of J2 = J3 = 0.25J1 can well reproduce the puzzling steplike magnetic behavior observed in the monoclinic CoV2O6, namely, the M0/3 plateau with hc2 ≈ 2hc1. It is revealed that such an h-induced M0/3 plateau still results from the same ferrimagnetic state as observed in the regular triangular system but simultaneously shows a different feature due to the frustration relaxed by anisotropy. The relative value of the weakest interaction plays the key role in the stepwise magnetic behavior of this system, and thus the exchange interactions can be tuned to accurately modulate the width and position of M plateau, namely the values of hc1 and hc2, with the height of M plateau kept unchanged, which provides a wide-use principle for the phenomena of M0/3 plateau observed in experiments.

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Figure 4. (a) Three-dimensional phase diagram with J2, J3, and h as three axes. The planes of hc1 and hc2 divide the whole phase space into three parts, namely, the FO (ferromagnetic) state, FI (ferrimagnetic) state, and AFM (antiferromagnetic) state. (b) Diagonal section of (a) along the line J2 = J3.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

ACKNOWLEDGMENTS This work is supported by the research grants from the National Natural Science Foundation of China (Grant Nos.

behavior; namely, its magnitude determines the width of plateau and its relative value to the other two interactions decides the ratio 2281



Article

10904014 and 11091240280) and by the computational center from the Department of Physics, Southeast University. X.Y. also thanks The Abdus Salam International Center for Theoretical Physics (ICTP) for the support.

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3 Magnetization Plateau Induced by Magnetic Field in Monoclinic

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