This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 137.195.150.201

This content was downloaded on 28/10/2014 at 09:14

Please note that terms and conditions apply.

Magnetic Properties of One-Dimensional Ferrimagnetic Mixed (1,3/2) Spin Chain with Single-

Ion Anisotropy

View the table of contents for this issue, or go to the journal homepage for more

2004 Chinese Phys. Lett. 21 2289

(http://iopscience.iop.org/0256-307X/21/11/059)

Home Search Collections Journals About Contact us My IOPscience

CHIN.PHYS.LETT. Vol. 21, No. 11 (2004) 2289

Magnetic Properties of One-Dimensional Ferrimagnetic Mixed (1;3=2) Spin Chain

with Single-Ion Anisotropy

Ekrem Ayd�ner�

Department of Physics, Faculty of Sciences and Letters, Dokuz Eyl�ul University, TR-35160 Buca, Izmir, Turkey

(Received 9 June 2004)

We have carried out Monte Carlo simulations to study the magnetic properties of a mixed S = 1 and S = 3=2ferrimagnetic system interacting antiferromagnetically on a one-dimensional spin chain with single-ion anisotropy.

It has been shown that at suÆciently low temperatures, the system has magnetization plateaus near the ground

state under an external �eld. Other interesting physical quantities such as the speci�c heat and the N�eel order

at low temperatures are also discussed.

PACS: 76. 50.+g

In recent years, remarkable attention has been paidto quasi-one-dimensional gapped spin systems such asspin-Peierls systems, Haldane systems, and spin lad-ders. The most fascinating characteristics of thesesystems is that they show magnetic plateaus at lowtemperatures like topological quantization. Varioustheoretical and experimental e�orts have been madeto understand the magnetic behaviour of such low-dimensional magnetic spin systems.[1�9]

In this study, one-dimensional ferrimagnetic mixed(1; 3=2) spin chain with single-ion anisotropy is consid-ered. In this sense, (i) whether the magnetic plateausappear in this spin system at low temperature is re-searched, and (ii) other interesting physical quantitiessuch as the speci�c heat CV and the N�eel order ON�eel

at low temperatures are also discussed. The Hamilto-nian of the system constructed here in a quasi-classicalmanner allows use of the Ising type variable insteadof the quantum spin operators. Such a system in theexternal �eld with di�erent single-ion anisotropy is de-scribed by the Hamiltonian

H = JXhi;ji

SziASzjB +DA

Xi(odd)

(SziA)2

+DB

Xj(even)

(SzjB)2 + h

Xi;j

(SziA + SzjB);(1)

where J denotes the exchange coupling of antiferro-magnetic type (J > 0), DA and DB describe thestrength of the di�erent single-ion anisotropy corre-sponding to sublattice A and B respectively, and h isthe external �eld. Also, SziA refers to spin of magni-tude 1, which takes on 0, and�1 values, and SzjB refersto spin of magnitude 3=2, which takes on the �1=2 and�3=2 values. In the limited case, DA = DB = D, andEq. (1) can be reduced to

H = JXhi;ji

SziASzjB +D

� Xi(odd)

(SziA)2

+X

j(even)

(SzjB)2�+ hXi;j

(SziA + SzjB):(2)

The study in the limit of DA = DB = D may beassumed to be an extreme case. Of course, it is possi-ble to study the problem for di�erent values of single-ion anisotropy by taking DA 6= DB . This work maylook simple, however it will be useful to understandthe behaviour of the system depending on single-ionanisotropy in this limit.

In this Letter, the Monte Carlo simulation is em-ployed to investigate the magnetic properties of theabove de�ned system. The details of the Monte Carlosimulation is described, and the results are presented.

For convenience, the ith site was labelled as Si(i = 1; � � � ; N). With this labelling, it is set that oddi is associated with S = 1, and even i is associatedS = 3=2 corresponding to SziA and SzjB, respectively.Rede�ning Eq. (2), hence a simple Hamiltonian is setas follows:

H = J

NXi=1

Szi Szi+1 +D

NXi=1

(Szi )2 + h

NXi=1

Szi : (3)

The standard Monte Carlo method is used to simu-late the Hamiltonian described by Eq. (3) on a one-dimensional lattice with periodic conditions. For thesimulation are set J = 1 and N = 40. The con�gura-tion was generated by sequentially sweeping throughthe lattice and making single- ip attempts. The ipswere accepted or rejected according to the Metropo-lis algorithm. The data were generated with 2 � 106

Monte Carlo steps per site after discarding the �rst5� 105 steps per site. Although the system describedby Eq. (3) has ferrimagnetic order, we are interestedin the three expressions of (i) ferromagnetic order m,(ii) antiferromagnetic order ON�eel, and (iii) the speci�cheat CV , which respectively are

m =1

N

NXi=1

Szi ; (4)

� Email: ekrem.aydiner@deu.edu.trc 2004 Chinese Physical Society and IOP Publishing Ltd

2290 Ekrem Ayd�ner Vol. 21

ON�eel =1

N

NXi=1

(�1)iSzi ; (5)

CV =hH2i � hHi2

NT 2; (6)

where T denotes the temperature and h� � �i denotesthe ensemble averages, which come from the averageover 1000 independent con�gurations. It should benoted that the strong uctuations may appear in thisferrimagnetic system due to antiferromagnetic interac-tions between spins. Therefore, one may expect these uctuations to break down the ferromagnetic or anti-ferromagnetic order as noted in Ref. [8]. Fortunately,as shown in the following, the Monte Carlo is a verygood method to be used in a one-dimensional systemof the big spin owing to the strong uctuations inone-dimension depressed by large spin. Therefore, theMonte Carlo method is used to simulate this system.

Fig. 1. The N�eel order ON�eel and the speci�c heat CVplotted as a function of temperature T in the case of fer-rimagnetic mixed spin chain for D = 0 and h = 0.

The simulation data have been obtained fromMonte Carlo calculations. The N�eel order ON�eel andthe speci�c heat CV are plotted as a function oftemperature T in Fig. 1 for the single-ion anisotropyD = 0 and the external �eld h = 0. In this �gurethe N�eel order ON�eel, marked by the square-dot line,declines with the increasing temperature, and the spe-ci�c heat CV , denoted by the circular-dot line, shows aSchottky-like round hump. No phase transition occursin such a one-dimensional system. Figure 1 gives us afamiliar result. In addition, the speci�c heat CV andthe N�eel order ON�eel, which re ect the system's energyand spin uctuations respectively, are described wellat low temperature.

As is stated above, the uctuations could breakdown antiferromagnetic or ferromagnetic order sincethey are very strong in the low dimensional case. Thissituation might cause Monte Carlo to fail to reach theequilibrium state. However, the simulation results ob-tained in this research reveal that uctuations are re-

strained owing to the big value of spin even in thelow-dimensional system, and the N�eel order survivedat low temperatures.

Fig. 2. The magnetization m as a function of mag-netic �eld h for D = 0:1 in the ferrimagnetic mixed spinchain, where hs is the saturated magnetization �eld, andT = 0:01 (in units of J).

Fig. 3. The magnetization m as a function of magnetic�eld h for D = 0:3 in the ferrimagnetic mixed spin chain,where hc is the beginning point of the second step plateauand hs is the saturated magnetization �eld, and T = 0:01(in units of J).

In addition, under the external �eld at suÆcientlylow temperatures near the ground state, it was shownin a previous study that the one-dimensional ferri-magnetic mixed spin chain given by Eq. (3) takes onstep-like plateaus which are similar to S = 1 andS = 3=2.[8] However, two di�erent qualitative be-haviours were observed under the conditions givenabove. For example, while two plateaus appear forD � 0:11, three plateaus occur for D > 0:11. Toshow this interesting behaviour, D can be chosen asD = 0:1 and D = 0:3, and the magnetization m isplotted at T = 0:01 (in units of J) as a function ofexternal �eld h. Figures 2 and 3 show the results for

No. 11 Ekrem Ayd�ner 2291

D = 0:1 and D = 0:3. The step-like plateaus oc-cur at m = 0:25 and m = 1:25, while they appear atm = 0:25, m = 0:75, and m = 1:25 in Fig. 3. More-over, there are several interesting behaviours of thissystem di�erent from S = 1 or S = 3=2 antiferro-magnetic spin chain.[8] For example, the number ofthe plateaus of the system studied does not obey the2S + 1 rule[7;8] for all the D values. This point isnot very clear. The �rst magnetic plateaus are lo-cated at m = 0:25 here, whereas the �rst plateau inthe former work is found at m = 0 for both S = 1and S = 3=2.[8] This phenomenon probably originatesfrom ferrimagnetic nature of the system studied here.In a ferrimagnetic material, two inequivalent momentsinteracting antiferromagnetically can achieve a spon-taneous magnetization at low temperatures, since theinequivalent moments are antiparallel but do not fullycancel at low temperatures.

Fig. 4. The magnetization phase diagram of the groundstate of ferrimagnetic mixed Ising chain with single-ionanisotropy under �nite magnetic �eld.

On the other hand, in order to investigate thee�ect of single-ion anisotropy on the magnetizationplateaus, the magnetization m was calculated at �-nite h for a series of values of D (0:11 < D � 2:0),and the data were plotted as a magnetization phasediagram in Fig. 4. The magnetization phase diagramfor D � 0:11 has not been plotted since it is not in-teresting. For D > 0:11, three plateau lines placed atm = 0:25, m = 0:75, and m = 1:25 are divided by theinitial �eld and saturated �eld lines as shown in Fig. 4.The longitudinal coordinate of the square-dot line isthe beginning point of the �eld for the appearance ofthe plateau m = 0:75 which corresponds to critical�eld hc values, and its ending point is signed by thecircular-dot line which corresponds to the saturated�eld hs values. In addition, the distance between boththe lines for the same value of D is the width of the

plateau m = 0:5.Figure 4 shows detailed information of the mag-

netization plateaus in the ground state for D > 0:11,and is summarized as follows. In the one-dimensionalsystem stated above, it seems that the positive single-ion is an indispensable condition for the appearanceof plateaus. The nonzero magnetization in the groundstate of mixed ferrimagnetic spin chain begins from azero �eld and continues to the beginning of the secondmagnetization plateau for all the D values. However,the second plateau in the ground state of this systembegins from a �nite �eld, which is called the initial�eld hc. However, the second plateau does not appearfor D � 0:11. The value of hc decreases in the regimeof 0:11 < D < 1:5 with the increase of D, but in-creases for D > 1:5. Also, the width of the plateau form = 0:75 increases with D, while D increases in theregime of 0:11 < D < 1:5, but is independent of D forD > 1:5. In fact, the intermediate plateau m = 0:75(for D > 0:11) will decrease and disappear when thesingle-ion anisotropy D increases to in�nity for a given�xed h. It is obvious that the plateau comes probablyfrom interactions among spins. On the other hand, forD > 0:11, the value of the saturated �eld hs increasesmonotonically with the increase of D in the groundstate.

In conclusion, the ground state properties of one-dimensional classical ferrimagnetic mixed spin chainhas been investigated in the limit of DA = DB =D. The uctuations are suppressed by the big valueof spins even though they are strong. The long-range N�eel order remained at low temperatures nearthe ground state for weak single-ion anisotropy. Inthis system, two step-like plateaus occur when D �0:11. However, three step-like plateaus occur owingto the co-existence of antiferromagnetic interactionof (Szi S

zi+1) and positive single-ion anisotropy (Szi )

2

when the external �eld changes from zero to the sat-urated �eld.

Acknowledgements: The author would like tothank Professor G�ung�or G�und�uz and Filiz Tasdemirfor their useful comments and fruitful discussions.

References

[1] Haldane F D M 1983 Phys. Rev. Lett. 50 1153[2] Hida K 1994 J. Phys. Soc. Jpn. 63 2359[3] Totsuka K 1998 Phys. Rev. B 57 3454[4] Sakai T and Takahashi M 1998 Phys. Rev. B 57 R3201[5] Yamamoto T et al 2000 J. Phys. Soc. Jpn. 59 3965[6] Tonegawa T et al 1996 J. Phys. Soc. Jpn. 65 3317[7] Oshikawa M et al 1997 Phys. Rev. Lett. 78 1984[8] Chen X Y et al 2003 J. Magn. Magn. Mater. 262 258[9] Narumi Y et al 1998 Physica B 246 509