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Physica A
journal homepage:www.elsevier.com/locate/physa
Magnetic behaviors of frustrated Ising spin-chain system:
Wang–Landau simulation for three-dimensional lattice
T.H. Yang
a, W.S. Lin
a, X.T. Jia
b, M.H. Qin
a,∗, J.-M. Liu
c,∗aInstitute for Advanced Materials and Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China
bSchool of Physics and Chemistry, Henan Polytechnic University, Jiaozuo 454000, China
cLaboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China
h i g h l i g h t s
• The magnetic behaviors of Ca3Co2O6are investigated using the Wang–Landau method.
• The study is based on a well accepted three-dimensional Ising model.
• The equilibrium state produces the two-step magnetization curve at low temperatures.
• The four-step magnetization curve must arise from the non-equilibrium magnetization.
a r t i c l e i n f o
Article history:
Received 11 December 2013
Received in revised form 10 March 2014 Available online 20 May 2014
Keywords:
Step-like magnetization Wang–Landau algorithm Frustrated system
a b s t r a c t
The magnetic behaviors of triangular spin-chain system Ca3Co2O6have been investigated using the Wang–Landau method based on the Ising model for three-dimensional lattice.
Our simulation shows that the equilibrium state of the model produces the two-step mag- netization curve at low temperature even when the intra-chain interaction is considered.
As a consequence, this work indicates that the four-step magnetization curve observed in experiments must arise from the non-equilibrium magnetization, which is consistent with earlier report.
©2014 Elsevier B.V. All rights reserved.
1. Introduction
The so-called ‘‘frustrated’’ spin system is a system in which one cannot find a configuration of spins to fully satisfy the interaction between every pair of spins [1]. During the past decades, frustrated spin systems have attracted widespread interest because very rich physics can appear in these systems. For instance, interesting step-like magnetization behaviors have been reported in a number of frustrated spin systems, such as triangular Ising spin-chain system Ca3Co2O6[2–5] and Sr5Rh4O12[6,7]. A lot of theoretical and experimental explorations have been devoted to these interesting phenomena so far [8–22].
It is well known that Ca3Co2O6has a rhombohedral structure composed of Co2O6chains running along thecaxis [2]. Each chain is built by alternating CoO6trigonal prisms and CoO6octahedral, and surrounded by six equally spaced chains. One of the most intriguing results reported in Ca3Co2O6is the step-like magnetization(M)vs. external magnetic field(B)applied along the chains. Specifically, two steps are observed when temperature(T)is decreased to about 25 K.Mquickly reaches
∗Corresponding authors. Tel.: +86 13632457166.
E-mail addresses:[email protected](M.H. Qin),[email protected](J.-M. Liu).
http://dx.doi.org/10.1016/j.physa.2014.05.018 0378-4371/©2014 Elsevier B.V. All rights reserved.
the first plateau atM=M0/3 (M0is the saturated magnetization) whenBis increased from zero. Then the step is replaced by that ofM=M0with the perfect ferromagnetic (FM) order asBfurther increases to about 3.6 Tesla (T). More interestingly, theM0/3 step splits into three equidistant steps at a lowBsweep rate forT < 10 K, constituting a four-step magnetiza- tion pattern [9]. The origin of such magnetization behavior has been extensively studied based on a two-dimensional (2D) rigid-chain model on a triangular lattice [11,12]. This model is proposed based on the fact that the intra-chain FM interac- tion is much stronger than the inter-chain antiferromagnetic (AFM) coupling, and the chain can be assumed to be in two ordered states at lowT. The static magnetization behavior of this model was investigated, and the four-step magnetization curve was predicted when a random-exchange term was considered [12]. However, in our earlier work, the Wang–Landau (WL) simulation of the 2D rigid-chain model has demonstrated that the equilibrium state of the rigid spins produces the two-step magnetization curve at lowT [19]. On the other hand, the simulation of the non-equilibrium evolution has been performed based on the Glauber-type dynamics and well reproduces the experimental observations [16]. It is suggested that the formation and the motion of domain walls may play an important role in the emergence of the magnetization plateaus.
As a consequence, it is strongly indicated that the four-step magnetization behavior may be caused by the non-equilibrium magnetization. In some extent, these two approaches above supplement each other and the joint investigation of frustrated magnetic systems is rather convincing.
Most recently, the response of Ising spin-chain system to the external magnetic field has been investigated in the three- dimensional (3D) Glauber dynamics [21,22]. It is shown that the experimentally observed relaxation process at intermediate magnetic fields in Ca3Co2O6is related to the growth of a domain size. In addition, the step-like magnetization behaviors have been reproduced by the dynamic simulation. However, the static magnetization curve studied by Metropolis algorithm for perfect 3D lattice also exhibits the four-step magnetization behavior [13]. As pointed out in earlier works, the conventional Metropolis algorithm may fail to relax a trail state into the equilibrium one in frustrated spin models. Thus, it is still an un- solved issue demonstrating the equilibrium state of the 3D model in order to completely understand the magnetic behavior of Ca3Co2O6. After all, the four-step magnetization curve has been observed even at a very lowBsweep rate. Furthermore, such study does also contribute to the development of statistical mechanics. However, the number of the metastable states will be rapidly increased when the intra-chain interaction is considered, preventing the system from relaxing to its equilib- rium state. In other words, the CPU time for WL simulation of such a 3D model is greatly increased, in some extent, leading to the fact that few works on this subject have been reported, as far as we know.
In order to make clear this question, we shall study the magnetic properties of the Ising model for 3D lattice at the equilibrium state using the standard WL method. The calculated results demonstrate that the equilibrium state of the model produces the two-step magnetization curve at low temperature, strengthening the conclusion that the two additional steps in Ca3Co2O6magnetization curve arise from non-equilibrium states.
The remainder of this paper is organized as follows: In Section2the model and the simulation method will be described.
Section3is attributed to the simulation results and discussion. At last, the conclusion is presented in Section4.
2. Model and method
Considering the 3D anisotropic spatial arrangement and ignoring the phase difference between neighboring chains in Ca3Co2O6, we assume that the Co3+ions are coplanar and the lattice structure is composed of 2D triangular lattices stacked along thecaxis. The model can be expressed as:
H= −J1
[i,j]
SiSj−J2
[i,k]
SiSk−BµBg
i
Si, (1)
where [i,j] denotes the summation over all the nearest-neighboring (NN) pairs in the chain, while [i,k] is that over all the NN pairs in theabplane.J1 = 6.9035×10−23J (Joule) is the FM-intrachain coupling,J2 = −5.5228×10−24J is the AFM-interchain interaction,Si=2 is the effective spin of cobalt ion at sitei,g =2 is the Lande factor,kBis the Boltzmann constant, andµBis the Bohr magneton. It is noted that almost all these parameters are fixed to the same values as those used previously, making our analysis simple [13].
In order to investigate the magnetic properties of a system with the WL algorithm, one has to calculate the density of state (DOS)g(E,M)in energy and magnetization space whereEis the energy of a given spin configuration of the Hamiltonian in the absence ofB. The simulation procedure is chosen to be the same as that stated in the pathbreaking work of Wang and Landau [23,24].
For every possible (E,M) states, we set all entries to the DOSg(E,M) =1 and a histogramRH(E,M)= 0 at the very beginning. Then we flip spins randomly to begin random walk in the energy and magnetization space. The transition prob- ability from state (E1,M1) to state (E2,M2) is
p(E1,M1→E2,M2)=min
g(E1,M1) g(E2,M2),1
. (2)
Each time a state (Ei,Mi) is visited or retained, the histogramRH(Ei,Mi) (the number of visits) is accumulated and the existing DOS is modified by a modification factorf0; i.e.,g(Ei,Mi)=g(Ei,Mi)f0. In this report, the initial modification factor is set to f0=exp(1), and the factor is reduced to a finer one according to the recipefi+1=fi1/2when the histogram becomes ‘‘flat’’.
-20 -10
0 10
20 1.0 0.5
0.0 -0.5
100
80
60
40
20 0
In g(E, M)
E/Nk
B
M/M0
Fig. 1. (Color online) Calculated density of state in the (E,M) plane.
After finishing the initial run we perform 27 cycles, resulting in a final modification factor of 1.00000000745. For simplicity, the ‘‘flat’’ histogram means that histogramRH(E,M) for all possible (E,M) is not less than 80% of the average histogram.
Afterg(E, M)has been obtained, we may calculate the thermodynamic and magnetic quantities at anyT andB. For example, the internal energy can be calculated by
U(T,B)=
E,M
Hg(E,M)exp(−H/kBT)
E,M
g(E,M)exp(−H/kBT) ≡ ⟨H⟩T,B, (3) and the magnetizationM(T,B)can be calculated from
M(T,B)=
E,M
Mg(E,M)exp(−H/kBT)
E,M
g(E,M)exp(−H/kBT) . (4)
Our simulation is performed on a 6×6×4 supercell with period-boundary conditions in theabplane and open chain ends along thecaxis were used, similar to earlier report [22]. Here, open chains are used in order to have a clear comparison with earlier work conveniently [21]. In addition, the finite size effect has also been checked by simulating a system of 6×6×6 cubic lattice with period-boundary conditions applied in all directions. Similar results are observed, strengthening our conclusion significantly, although the corresponding results are not given in this report.
3. Simulation results and discussion
Fig. 1shows the simulatedg(E,M)of the 3D model for Ca3Co2O6. It is observed thatg(E, M)shows a parabolic shape in the low-energy range and reaches its single maximum value atM=0 for a given energy. In addition, the calculated DOS at the lowest energyEminis rather large, indicating that the ground state is highly degenerate. Similar behaviors have been reported for some other systems such as the 2D triangular magnets and Mo72Fe30[19,25]. In the positiveErange, both the g(E,M)and the possibleMrange are decreased with the increase ofE. At the highest energy, only the so-called A-type AFM order (M=0) in which all the in-plane spins are parallel with each other while the out-of-plane NN spins are anti-parallel with each other is possible. In addition, one may note that such a phenomenon of the 3D model is completely different from that of the 2D triangular AFM model in which only the FM orders are possible at the highest energy [19].
Normally,Mand some other parameters can be accurately calculated because the obtained DOS covers all possible (E,M) space. For example, the simulated magnetization curves are shown inFig. 2which clearly exhibits two steps at lowT. Similar to the earlier report, theM0/3 plateau results from the homogeneousab-plane ferrimagnetic order due to the AFM interaction between the chains [26,27]. In such a state, all spins in each chain preserve the same orientation and flip together, which confirms earlier assumption in the 2D model. Specially, one may refer to the earlier report in which the phase diagram and critical behavior of the 2D triangular Ising model with the nearest and the next-nearest neighbor interactions have been investigated in detail with Monte Carlo simulations [28]. WhenBincreases to about 3.6 T,Mswitches toM0. The perfect FM state in which all the spins are arranged along the+caxis is stabilized byB. On the other hand, the steps can be progressively washed out due to the thermal activation with the increase ofT, as clearly shown inFig. 3(a). WhenTis increased to about
1.0 0.8 0.6 0.4 0.2 0.0
M/M0
0 1
2 3
4 5
T(K) B(T) 0
10 20
30 40
50
Fig. 2. (Color online)M/M0as a function ofTandB.
a b
Fig. 3. (Color online)M/M0as a function ofB(a) at variousTand (b) for variousJ1.
30 K, theM0/3 step disappears completely and the system exhibits the paramagnetic (PM) property that theM−Brelation is linear. As a consequence, our simulation results show that the equilibrium state of the Ising model for the 3D lattice also produces a two-step magnetization curve at lowT.
In addition, the effect of the intra-chain FM interaction has also been investigated in our simulation, and the correspond- ing results are shown inFig. 3(b). At lowT (T = 5 K), both the magnetization curves for different FMJ1clearly show two steps, demonstrating that the intra-chain FM interaction cannot assist in generating additional steps. However, near the crit- ical fields (B=0 and 3.6 T), the perfect ferrimagnetic order may be easily destabilized for smallJ1, leading to the smoothness of the magnetization curve with the decrease ofJ1, as revealed in our simulation.
In earlier work, the four-step magnetization curve was predicted by means of Monte Carlo simulation of the same model based on the Metropolis algorithm [13]. However, the equilibrium state obtained by the WL algorithm gives a different magnetization curve, suggesting that the Metropolis algorithm cannot relax the system into the equilibrium state. This argument has been verified in our simulation by comparing the magnetization and internal energy respectively obtained by the WL and Metropolis algorithms. It is worthy to note that for Metropolis algorithms, the initial 1×105Monte Carlo steps are discarded and another 1×105steps are retained for statistic averaging.
The two simulations are well coincident with each other atT = 15 K, as clearly shown inFig. 4(a) and (b), indicating that the Metropolis algorithm also allows the equilibrium state to be reached. However, a big discrepancy between the two simulated magnetization curves is observed belowB = 3.6 T atT = 5 K, as shown inFig. 4(c). On one hand, theM0/3 step splits into three equidistant steps in the Metropolis simulation, the same as the earlier report. On the other hand, the M0/3 step keeps almost invariant in the WL simulation. It is observed that the relevant internal energies obtained from the Metropolis simulation are obviously higher than those corresponding WL simulation results, as clearly shown inFig. 4(d).
At the same time, it is noted that the temperature concerned in this work is so low that it contributes little to the values of the free energies of the system. As a consequence, the above results allow us to argue that the intra-chain FM interaction
a
b c
d
Fig. 4. (Color online) Comparison of (a), (c)M/M0and (b), (d)Uas a function ofBrespectively calculated from the WL method and the Metropolis algorithm atT=15 K andT=5 K.
does not change the two-step shape of the magnetization curve, and the two additional steps in Ca3Co2O6magnetization curve must arise from non-equilibrium magnetization.
After all, it is well known that geometrically frustrated spin systems such as Ca3Co2O6can be easily trapped into meta- stable state at lowT and are hard to relax to the equilibrium state. In other words, the time available experimentally may not be sufficient for the spin rearrangement, even though the energy difference between plateaus may be very small. Thus, a reliable experimental approach in these frustrated systems becomes extremely challenging in terms of understanding their equilibrium states. The present WL simulation on 3D lattice clearly shows the equilibrium magnetization of Ca3Co2O6, and strongly strengthens the conclusion obtained from the Glauber dynamic simulation.
4. Conclusion
In conclusion, we have calculated the magnetization of triangular spin-chain system with the WL method based on the Ising model on a three-dimensional lattice. It is demonstrated that the equilibrium state of the model produces the two-step magnetization curve at low temperature even when the intra-chain interaction is taken into account. Thus, this work indi- cates that the four-step magnetization curve observed in experiments must be due to the non-equilibrium magnetization.
This conclusion is in agreement to the results of earlier Glauber dynamics.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (11204091, 11274094, 51332007), and the National Key Projects for Basic Research of China (2011CB922101).
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