Steplike magnetization of spin chains in a triangular lattice

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Steplike magnetization of spin chains in a triangular lattice: Monte Carlo simulations

X. Y. Yao, S. Dong, and J.-M. Liu*

Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China International Center for Materials Physics, Chinese Academy of Sciences, Shenyang, China 共Received 19 March 2006; revised manuscript received 10 May 2006; published 29 June 2006兲 A two-dimensional Ising-like model for a triangular spin-chain lattice, where each spin-chain is treated as a rigid giant spin, is proposed to investigate the magnetization of a triangular spin-chain lattice by Monte Carlo simulation. The simulations show the steplike evolution of the magnetization at low temperature against an external magnetic field, namely two steps above 10 K and four steps below 10 K, in quantitative agreement with experiments on a Ca₃Co₂O₆compound. It is argued that the interchain interaction and magnetic inhomogeneity of the lattice are two important ingredients to induce the intriguing steplike feature of the magnetization below 10 K.

DOI:10.1103/PhysRevB.73.212415 PACS number共s兲: 75.30.Kz, 75.60.⫺d, 75.40.Mg

It is known that dimensionality reduction and geometrical frustration can result in fascinating physical phenomena in strongly correlated electron and spin systems. Typical ex- amples are those compounds with well-aligned one- dimensional共1D兲 spin chains along the same direction, and the cross section forms a triangular lattice. They offer peculiar property and continue to attract interest. One example is CsCoX₃, where X is Cl or Br and both intrachain and interchain interactions are antiferromagnetic共AFM兲共Ref. 1兲. The partially disordered antiferromagnetic共PDA兲 state was first observed in these compounds and a ferrimagnetic state is attained below the transition temperature.

Recently, another family of 1D spin-chain compounds with the general formulaA₃

⬘

^ABO6 共whereA’ is Ca or Sr,A and B are transition metal elements兲 have attracted attention.^2,3 They have a rhombohedral structure consisting of parallel 1D ABO₆chains along the hexagonalcaxis, sepa- rated byA

⬘

²⁺^ions.⁴These chains are built by alternating and facesharing AO₆ trigonal prisms and BO₆ octahedra. Each chain is surrounded by six equally spaced chains, forming a triangular lattice in the ab plane. Generally, the interchain distance is about double of the intrachain A-B distance.

Among these, Ca₃Co₂O₆, as the only compound in which bothA and B sites are occupied by the same metallic ele- ment, has been intensively studied due to its complex magnetic behaviors.^4–12. In contrast to CsCoX₃ 共Ref. 1兲, the intrachain coupling in Ca₃Co₂O₆ is ferromagnetic共FM兲 along the c axis, while the interchain coupling 共much weaker兲 in the ab plane is AFM 共Ref. 5兲. Experimental^6,7,12 and theoretical¹³ investigations confirmed the strong Ising-like anisotropy of the chains. The fascinating feature observed in Ca₃Co₂O₆ is a steplike magnetization 共M兲 as a function of the external magnetic field共h兲applied along the chains.^6,8,11 However, beside the M₀/ 3 step共where M₀ is the saturated M兲 observable in other materials,³ the three substeps observed at the lower temperatureT^6,8,11were reported only for Ca₃Co₂O₆, while its origin is still a matter of debate.

Theoretically, effort was made to explain those peculiar effects using different models.^14–16 Recently Kudasov¹⁵ de- velop a two-dimensional共2D兲Ising model to investigate the steplike magnetization by an analytical method, regarding a spin-chain as a large rigid spin and assuming a quench at

T= 0. In that paper, the fourth approximation qualitatively reproduces the four equidistant steps of the M共h兲 curve as observed in experiments. In our work, as an approach to the 1D spin-chain magnets, we improve this 2D Ising-like model to study the multistep magnetic behaviors of a triangular lattice by Monte Carlo simulation. Our simulation results quantitatively give the variation of the M共h兲 curve over whole T range, well consistent with the experiments on Ca₃Co₂O₆. And the heights of steps are demonstrated to be correlative with the inhomogeneity of the system.

As revealed experimentally, in Ca₃Co₂O₆ the 1D spin chains align along thecaxis and form a triangle lattice in the ab plane. Because the intrachain FM interaction is much stronger than the interchain AFM coupling, when the temperature decreases, each ferromagnetic chain would behave like a magnetic moment, namely a rigid giant spin. There- fore, the resultant magnetic structure should have two- dimensional character, which has been evidenced by several previous investigations.^7,8,15As an approximation, the 3D is- sue is reduced into a 2D triangular lattice composed of giant chain spins. It should be mentioned that the chain spins only have two equivalent projections along the chain direction due to the strong Ising-like anisotropy. Between the two nearest-neighboring spin chains only the AFM coupling is considered. In addition, considering the inhomogeneity of the system, a random exchange term⌬m,n is taken into ac- count. The Hamiltonian can be written as follows

H=

兺

关m,n兴共J+⌬m,n兲Sm

eS_n^e−h␮Bg

兺

_m ^S^m^e^, ^共1兲

⌬m,n=span·J·RAM_m,n, 共2兲 whereJ⬎0 is the AFM interchain coupling,S_mê is the effec- tive spin moment of a spin-chain with the value ±Sê,gis the Lande factor, and ␮B is Bohr magneton; 关m,n兴 denotes the summation over all the nearest-neighbor pairs;RAM_m,nis the random number in 关−1 , 1兴, and span represents the magni- tude of random exchange term. As a rigid giant spin, Sê should be much more than the spin moment of a magnetic ion in the chain. On the other hand, the ion spin interaction along the chain is not long ranged, soSêshould be finite. For PHYSICAL REVIEW B73, 212415共2006兲

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different materials,S^eandJhave different values. However, the real values of the two parameters are not available from experiments and a reasonable choice of them is judged from a quantitative comparison between the simulated results and the experimental data. The values of these parameters for the simulation are shown in Table I.

The simulation starts from anL⫻L 共L= 100兲Ising triangular lattice with periodic boundary conditions. The proce- dure of the simulation is described as follows. At a givenT, the simulation starts fromh= 0 with a random spin configuration. The standard Metropolis algorithm is employed to reach the equilibrium, and then magnetization is evaluated.

Afterwardhis raised and the simulation is performed on the state obtained before to reach a new equilibrium. This pro- cess is repeated until high field. Therefore the simulation results obtained are for equilibrium state. The final results are obtained by averaging ten independent data sets with different seeds for a random number generation.

For Ca₃Co₂O₆, the intrachain FM ordering of Co ions forms at⬃40 K. We start our simulation from 40 to 2 K. For the convenience of discussion, the T range is divided into two subranges: intermediateTrange共40 K⬃10 K兲and lowTrange共10 K⬃2 K兲. The simulatedM共h兲curves in the intermediate T range are presented in Fig. 1共a兲, revealing two clear steps at 10 K. When h increases from zero, M rapidly reaches the first plateau, and then switches to M₀ above h_c⬇3.6 T. M⬃M₀/ 3 on the first plateau, resulting from the ferrimagnetic ordering of the spin chains due to the AFM interaction between the chains. The spin configuration is snapshot in Fig. 2共a兲at 10 K ash= 1.8 T on the first plateau, where the black and gray-white solid circles represent spin up and spin down, respectively, with the spin up along the direction ofh. This configuration shows a regular ferrimagnetic structure, namely two kinds of spin chains are observed: among the three spin chains in the hexagonal unit cell, one takes spin down and is surrounded by six chains of spin up, and the other two chains are spin up with half neigh-

bors spin up and half spin down. In other words, one of the three spin chains takes the spin down, while the other two take the spin up, leading to M⬃M₀/ 3. As T is raised, the steps are progressively washed out due to the thermal acti- vation. Above T= 35 K, the M₀/ 3 plateau disappears com- pletely and theM-h relation is linear.

Figure 1共b兲demonstrates the simulatedM共h兲curves in the low T range, showing that besides the jump to M₀ above hc⬇3.6 T, three magnetization substeps appear gradually with decreasing T. For the curve at 2 K, below h_c⬇3.6 T one observes three substeps maintaining in regular intervals of 1.2 T, namely h_S1⬇1.2 T and h_S2⬇2.4 T. The partial spin snapshots ath= 0.6, 1.8, and 3.0 T, respectively on the three substeps atT= 2 K, are shown in Figs. 2共b兲–2共d兲. Dif- ferent spin configurations correspond to the three substeps.

On the first substep, shown in Fig. 2共b兲, the spin configuration consists of ferrimagnetically ordered regions as shown in Fig. 2共a兲and chain-to-chain FM stripes either spin down or spin up. We denote this state as A. The spin configuration on the second substep is shown in Fig. 2共c兲, revealing the aggregation of the ferrimagnetically ordered regions into large regions. However, many irregular small FM regions appear too. This state is signified as B. Figure 2共d兲presents the spin configuration on the third substep. The small irregular regions mentioned above connect with each other, forming spin-up FM stripes, while the majority of whole area retains the ferrimagnetic ordering. We mark this state with C.

For Ca₃Co₂O₆, the measuredM共h兲curves indeed show a steplike shape.^6,8,11AtT⬃10 K, with increasingh, theM共h兲 curve first presents aM₀/ 3 plateau, then jumps toM₀above hc⬇3.6 T. The ferrimagnetic spin configuration on theM₀/ 3 plateau was revealed by neutron powder diffraction.⁵ AsT falls below 10 K, three substeps at h_S1⬇1.2 T and h_S2

⬇2.4 T below hc⬇3.6 T were observed. Our simulations TABLE I. System parameters chosen for the simulation.

Parameter Value Parameter Value

k_B共J/K兲 1.3807⫻10⁻²³ J共J兲 3.592⫻10⁻²⁵

␮B共J/T兲 9.274⫻10⁻²⁴ S^e 32

g 2 span 0.15

FIG. 1. 共Color online兲 M共h兲 curves in共a兲 the intermediate T range, and共b兲the lowTrange.

FIG. 2. 40⫻40 spin snapshots of the triangular lattice for共a兲 T= 10 K,h= 1.8 T;共b兲T= 2 K,h= 0.6 T;共c兲T= 2 K,h= 1.8 T, and 共d兲T= 2 K,h= 3.0 T.

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reproduce these steps quantitatively. And it is shown in the spin snapshots from our simulation that the long-range ferrimagnetic correlation still exists below 10 K. This phenom- enon was evidenced by neutron diffraction too.¹⁷ Therefore our model exhibits the intrinsic character of the structure.

But the height of the second substep in the experiments is lower than our simulation result to some extent, which is due to the complexity of real material. The equidistance of the substeps is ascribed to the competition between the exchange interaction and magnetic field, as shown in Eq.共1兲. Starting from the view of the mean field, ignoring the random exchange term⌬m,n, for a spinS_m^e the first itemJS_m^e兺Sn

erepre- sents the interaction energy and the second one –h␮BgS_m^e is the magnetic field energy. These two energies compete with each other, namely J兺Sn

e vs h␮Bg. When one of the six nearest-neighboring spins flips, the change ofJ兺S_nê is 2JSê. So 2JSê corresponds to the interval between two critical fields, which is estimated to be hint= 2JSê/共g␮B兲⬇1.2 T.

And the critical spin-flip fieldsh_S1= 1hint⬇1.2 T,h_S2= 2hint

⬇2.4 T,hc= 3hint⬇3.6 T, resulting in the equidistant distri- bution of these critical spin-flip fields.

According to the magnetic behavior shown above, we then construct the phase diagram as shown in Fig. 3. Several domains can be distinguished, including the paramagnetic 共P兲, ferrimagnetic 共Fi兲, ferromagnetic 共Fo兲 and disordered magnetic 共DIS兲 states, which coincides well with the experiments.^7,8 In addition, the three states presented above are added to the phase diagram, marked with A, B, and C.

For the M₀/ 3 plateau in the intermediate T range, the ferrimagnetic scenario was confirmed in many experiments.

But for the three substeps in the lowT range, the origin is still a matter of debate. Currently, three scenarios are proposed: 共i兲 h-induced transitions between different spin configurations;^8,15 共ii兲 quantum tunneling of the magnetization¹²; and 共iii兲 a combination of two relaxation pro- cesses: T-independent relaxation and thermally activated relaxation.¹¹In our simulation, a chain is regarded as a rigid giant spin, and the spin fluctuation inside the chain is negli-

gible and only the interchain interaction is considered. Even with this approximation, the complex magnetic behavior of Ca₃Co₂O₆ can be reproduced quantitatively. So the interchain interaction plays an important role in determining the three substeps of magnetization below 10 K, even if this behavior may be ascribed to a combination of several ingredients.

The substeps in the lowTrange reflect the coexistence of several spin configurations, and the random exchange term in addition toJenhances the inhomogeneity and consequently conduces to form different configurations. Figure 4共a兲illus- trates the spin snapshot withspan= 0.05, corresponding to a small random exchange term, atT= 2 andh= 0.6 T. Compar- ing Fig. 4共a兲with Fig. 2共b兲, besides the ferrimagnetic ordering, regions of the other spin configurations in Fig. 2共b兲oc- cupy a larger fraction than that shown in Fig. 4共a兲. On the other hand, asspanarises from 0 to 0.5 at 2 K, the difference ofM between neighboring substeps increases but the borders of the substeps become more and more faint, as shown in Fig. 4共b兲. Whenspan= 0.5, corresponding to a large random exchange term, the three substeps are almost smeared out.

The doping experiment on Ca₃Co₂O₆shows similar results.¹⁰ Therefore, it is argued that the inhomogeneity has an important effect on those substeps. Only when the inhomogeneity is in an appropriate range, theM共h兲curve in the lowTrange shows a distinct steplike pattern.

In summary, an Ising-like model has been employed to investigate the spin-chain structure in a triangular lattice by Monte Carlo simulation. The results show the different steplike behaviors in the different temperature ranges, consistent with experimental observations on Ca₃Co₂O₆. It is indicated that the interchain spin interaction and the inhomogeneity of the system are two important ingredients to influence the steplike feature in the lowTrange. Although the spin-chain interactions in real materials are far more complicated, the magnetic property and microscopic spin configurations can be well explained by such a simple Ising model.

The authors thank the Natural Science Foundation of China 共Grants Nos. 50332020, 10021001, 10474039兲 and National Key Projects for Basic Researches of China共Grants Nos. 2002CB613303, 2004CB619004兲.

FIG. 3. Simulated magnetic phase diagram. P, Fi, Fo, and DIS, are for paramagnetic, ferrimagnetic, ferromagnetic, and disordered magnetic states, respectively. A, B, and C represent the three states presented in the text.

FIG. 4.共Color online兲共a兲40⫻40 spin snapshot of the triangular lattice withspan= 0.05 forT= 2 K andh= 0.6 T.共b兲 M共h兲 curves for differentspanatT= 2 K.

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*Corresponding author; email address: [email protected]

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