ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 322 (2010) 1917–1922

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials

0304-88

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/jmmm

Magnetic properties of a ferrimagnetic mixed ð1;3=2Þ spin chain withinhomogeneous crystal-field anisotropy

E. Solano-Carrillo, R. Franco, J. Silva-Valencia �

Departamento de Fısica, Universidad Nacional de Colombia, A.A. 5997 Bogota, Colombia

a r t i c l e i n f o

Article history:

Received 18 December 2009

Received in revised form

4 January 2010Available online 11 January 2010

Keywords:

Mixed-spin Ising chain

Magnetic plateau

Molecular-field theory

DMRG

53/$ - see front matter & 2010 Elsevier B.V. A

016/j.jmmm.2010.01.007

esponding author.

ail address: jsilvav@unal.edu.co (J. Silva-Valen

a b s t r a c t

Using molecular-field theory and density-matrix renormalization group calculations we investigated

the magnetic properties of a ferrimagnetic mixed ð1;3=2Þ Ising spin chain with inhomogeneous crystal-

field anisotropy. Our analysis introduces a clear physical mechanism for the appearance of the magnetic

plateaus in the system and for the quantum phase transitions which are present. We consider two cases

of interest: when the crystal field anisotropy D1 is present only on the spin-1 ions, and when D3=2 is

present only on the spin�3=2 ions. This latter case turns out to be the more interesting one since a

plateau at 15 of the saturation magnetization is formed by means of two physically distinct mechanisms.

The magnetic change between these two phases is gradual, varying over the region 1=2oD3=2 o1. We

also found that the case where the crystal field anisotropy is present only on the spin-1 ions is favorable

since the overall free energy of the system is lower.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Low-dimensional quantum spin systems have remained at theforefront of research in condensed matter physics for severalyears [1,2]. In these striking systems, the quantization of themagnetization is one of the most interesting phenomena, and hasattracted considerable attention recently [3–5]. It is detected as aplateau in the magnetization curve at low temperature or at theground state. Using the Lieb–Schultz–Mattis theorem [6], ageneral necessary condition for the presence of the plateau wasderived [7] which, for ferrimagnetic systems consisting of twocomplementary sublattices a and a0 reads [8]

ðSaþSa0 Þð1�M=MsatÞ ¼ integer; ð1Þ

where M is the magnetization at which the plateau appears, Msat

is the saturated magnetization and Sa is the spin of the ions in thesublattice a.

One of the mechanisms (along with dimerization, frustration,periodic field and so on) for the appearance of the magneticplateaus in one-dimensional spin chains is produced when theaxial crystal-field (or single-ion) anisotropy D is relevant in thesystem. Chen et al. [9] employed the classical Monte Carlotechnique to investigate the magnetization plateaus of one-dimensional classical spin-1 and spin-3=2 antiferromagnetic Isingchains with single-ion anisotropy under an external magneticfield at low temperatures. They showed that the homogeneoussystems with spin S and positive single-ion anisotropy have 2Sþ1

ll rights reserved.

cia).

magnetic plateaus. This was precisely verified recently using thetransfer-matrix technique [10], and for general spins, using theformulation of the problem on the Bethe lattice [11]. The interestin magnetic systems with crystal-field anisotropies thus in-creases; for instance, for the spin-1 Heisenberg antiferromagneticchain with exchange and single-ion anisotropies, evidence hasbeen presented recently for various types of supersolid andsuperfluid structures for chains of finite length [12,13]. Further-more, since this magnetic anisotropy is closely related to thespin–orbit effect, an exhaustive investigation is needed due to thenew discovery of the possibility of using the spin–orbit couplingto manipulate the direction of spin polarization induced by acurrent, which is at the heart of efforts to merge spintronics withsemiconductor technology [14].

On the other hand, among quantum spin systems, ferrimag-netic mixed-spin chains have lately attracted a lot of interest.Recent synthesizing techniques have produced a lot of thesematerials, such as the bimetallic chain [15] and the organic one[16]. In these systems two different spins Sa and Sa0 are arrangedalternately in a line and coupled by the nearest-neighborantiferromagnetic exchange interaction. Aydiner [17] has studiedthe ferrimagnetic ðSa; Sa0 Þ ¼ ð1;3=2Þ Ising chain with homogeneoussingle-ion anisotropy using classical Monte Carlo technique buthe obtained three plateaus which do not obey the 2Sþ1 rule. Toour knowledge, few theoretical studies have been done on one-dimensional systems in which an inhomogeneous crystal-fieldanisotropy is considered, even though this often happens inrealistic scenarios. Sakai and Okamoto [18] have investigated themagnetization curve of the ðSa; Sa0 Þ ¼ ð1;1=2Þ Heisenberg systemwith the numerically exact diagonalization of finite clusters andsize-scaling analyses, considering the single-ion anisotropy only

ARTICLE IN PRESS

E. Solano-Carrillo et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1917–19221918

on the spin-1 sites. This was motivated by the recentlysynthesized ð1;1=2Þ chain NiCuðpbaÞðD2OÞ32D2O which possiblyhas the anisotropy D on the Ni ion ðS¼ 1Þ. Motivated by Miyasakaet al.’s recent success [19] in synthesizing one-dimensionaltransition metal complexes composed of both Ni2þ and Mn3þ ,Tonegawa et al. [20] investigated the magnetization curve of theðSa; Sa0 Þ ¼ ð1;2Þ Heisenberg system using density-matrix renorma-lization group (DMRG) [21] techniques and considering differentcrystal field anisotropies on both ions.

Ferrimagnetic Ising systems are studied not only out of purelytheoretical interest but also because they have been proposed aspossible models to describe a certain type of molecular-basedmagnetic materials which are of current experimental interest[22–24]. For the case ðSa; Sa0 Þ ¼ ð1=2; SÞ with S41=2 and crystal-field anisotropies D1=2 ¼ 0; DSa0 an extensive study has beendone in the past for lattices with coordination number zZ3, whichincludes exact solutions on a honeycomb lattice ðz¼ 3Þ [25,26], aswell as on a Cayley tree [27]. However, since most magnetic ionstake a spin greater than 1=2, recent interest in the study of spinmixtures both greater than 1=2 has arisen. The most investigated,within effective-field techniques, is the ðSa; Sa0 Þ ¼ ð1;3=2Þ spinmixture in square, honeycomb and simple-cubic lattices, whichshows multicritical points in the temperature vs. anisotropy plane[28–34]. To our knowledge no study has been done on thisferrimagnetic Ising spin mixture with inhomogeneous crystal-fieldanisotropies for one-dimensional systems ðz¼ 2Þ.

In this paper we study the ground state magnetic properties ofthe ferrimagnetic Ising chain with ðSa; Sa0 Þ ¼ ð1;3=2Þ when thesingle-ion anisotropies Da and Da0 are different for each sublatticea and a0 of the system. Motivated by the recently synthesizedquasi-one-dimensional heterotrinuclear complex ½NiCr2ðbipyÞ2ðC2O4Þ4ðH2OÞ2�H2O, which shows a rare case [35] of antiferro-magnetism between Ni(II) S¼ 1 and Cr(III) S¼ 3=2, we investigatean alternating system using a molecular-field theory and DMRGnumerical calculations, obtaining very intuitive explanations forthe results.

Fig. 1. Representation of the effect of the spin–orbit and Zeeman interactions on

an orbital singlet (a) with spin 1 (b) with spin 3=2. The spin–orbit interaction is a

byproduct of the crystal field felt by each ion.

2. The model

The model used here is particularly useful for magneticcompounds containing transition–metal ions. This is because in theseions the unpaired electrons lie in the outermost shell, and thereforethey are easily influenced by any external (crystal) field produced byneighboring ligands. The effect of this crystal field, if it is ofsufficiently low symmetry, is the removal of any orbital degeneracy,which manifests itself as a splitting of the spectroscopic energy levels,the lowest of which is determined by Hund’s rules. We proposedividing our system into two sublattices a¼ 1;3=2. Thus, in theabove-mentioned non-degenerate situation, if we assume axialsymmetry and an external field H applied along the z-axis, the localspin Hamiltonian (spin–orbit + Zeeman) for the ion ia is given by [36]

ðHlocÞia ¼Da½ðSziaÞ2�1

3SaðSaþ1Þ��gmBHSzia; ð2Þ

where Da is proportional to the squared spin–orbit couplingparameter and g is the zz element of the g-tensor. Notice that thesequantities reflect the symmetry of the crystal.

The spin interaction with the environment is considered as ananisotropic exchange interaction of the Ising type

ðHintÞia ¼ JXd

Szia

Sziaþ d

; ð3Þ

where d refers to nearest neighbors. Then we can write the totalHamiltonian of our system as

H¼Xi;a½ðHlocÞiaþðHintÞia �: ð4Þ

In the following we consider a linear array of a- ions distributed inan alternating way and interacting antiferromagnetically ðJ40Þ.

3. Molecular-field theory

In order to obtain an initial physical idea of the magneticresponse of our system we propose using an argument similar tothe one given by Blume [37] for the study of the first-ordermagnetic phase transition in UO2. For this, we decided to take arepresentative ion ia and consider only a single term in (4). Notethat the first term in (2) gives just a zero-field splitting of theenergies for different spin microstates. For a given Sa and Da thisis fixed and therefore we only have to consider the furthersplitting of these energy levels by a magnetic field (see Fig. 1). Thismagnetic field will turn out to be an effective field which is acombination of H and an average molecular field due to theexchange interactions between the various ions. The magneticenergy associated with this effective field is described from (2)and (3) by the Hamiltonian

ðHeff Þia ¼ JXd

Szia

Sziaþ d�gmBHSz

iaCgmB

Xd

J

gmB

/Sziaþ d

S�H

!Sz

ia

¼ gmBðlMa0�HÞSzia; ð5Þ

where the effective magnetic field is identified as

Ba ¼ lMa0�H: ð6Þ

Here l¼ 2J=NðgmBÞ2 is the Weiss constant, N is the number of

ions in each sublattice and Ma the corresponding magnetization.Then in the local basis jSa;maS in which the relationSz

iajSa;maS¼majSa;maS holds, the free energy per a- ion is given

by

Uma ¼Da½m2a�

13SaðSaþ1Þ�þgmBBama: ð7Þ

For our mixed-spin system (7) becomes

Um1¼D1ðm

21�

23ÞþgmBðlM3=2�HÞm1; ð8Þ

Um3=2¼D3=2ðm

23=2�

54ÞþgmBðlM1�HÞm3=2; ð9Þ

where m1 ¼ 0;71 and m3=2 ¼ 71=2;73=2. This pair of equationscan be solved for Ma in a self-consistent way. For this, we proposeto keep track of the spin microstates (see Fig. 1) which minimizeenergy when we vary the applied magnetic field H. In the nexttwo subsections we do this for each sublattice a separately,keeping Da0 ¼ 0.

3.1. Magnetization process of spin-1 ions

If l is sufficiently large and we assume that M3=2 is positive,then from (6) and (8) we have that for H¼ 0 the energy of spin-1ions is minimized in the m1 ¼�1 microstate. This situation isdepicted in Fig. 2(a). In this phase, the magnetization of spin-1ions is M1 ¼�gmBN. Substituting this in (9), with H¼ 0, we seethat the energy of spin-3=2 ions is minimized in the m3=2 ¼ 3=2

ARTICLE IN PRESS

Fig. 2. Magnetization process of a spin-1 ion. The thin solid line represents the

energy of the m1 ¼ 0 microstate of the singlet, while the thick solid lines are those

of the m1 ¼ 0;71 microstates of the triplet (see the text).

Fig. 3. Magnetization process of spin-3=2 ions. The thin solid lines represents the

energies of the m3=2 ¼ 71=2 microstates, and the thick solid lines are those of the

m3=2 ¼ 73=2 microstates (see the text).

E. Solano-Carrillo et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1917–1922 1919

microstate. Therefore M3=2 ¼ 3=2gmBN which is consistent withour assumption of M3=2 being positive. Thus, the totalmagnetization in this phase in units of gmB (called the totalmagnetization hereafter) is

Mð�1;3=2Þ ¼1

gmB

ðM1þM3=2Þ ¼1

2N: ð10Þ

When we increase the applied magnetic field, the splitting of thetriplet starts to decrease since from (6) the effective magneticfield B1 decreases. When B1 ¼D1=ðgmBÞ the splitting of the tripletequals the zero-field splitting due to the crystal field (seeFig. 1(a)). From this point, there is a collapse of spin-1 ions tothe singlet state. Solving for H in (6) for this particular value of B1,this quantum phase transition is determined to occur at thecritical field

Hc ¼3J�D1

gmB

: ð11Þ

This is valid as long as Hc 40; since our model is considered forH40. If H4Hc we have a situation such as the one shown inFig. 2(b). In this phase the energy is minimized in the m1 ¼ 0microstate which gives M1 ¼ 0. Substituting in (9) we see that thespin-3=2 ions remain in the same state M3=2 ¼ 3=2gmBN; thereforein this phase the total magnetization is

Mð0;3=2Þ ¼32N: ð12Þ

If we continue increasing the applied field there is a pointH0 ¼ 3J=ðgmBÞ for which the splitting of the triplet disappears, as isshown in Fig. 1(c). At this point, the m1 ¼ 71 microstates changeroles. Then for H4Hs ¼H0þD1=ðgmBÞ the spin-1 ions leave thesinglet state and the m1 ¼ 1 microstate becomes the mostenergetically favorable. The critical field of this saturated phase is

Hs ¼3JþD1

gmB

: ð13Þ

Here the magnetization of spin-1 ions is M1 ¼ gmBN and that ofspin-3=2 ions remains the same. The total magnetization in thiscase is

Mð1;3=2Þ ¼52N: ð14Þ

3.2. Magnetization process of spin-3=2 ions

The situation for spin-3=2 ions is more elaborate. It dependsstrongly (see Fig. 1(b)) on the ratio 3:1 in the separation betweenthe m3=2 ¼ 73=2 energy levels and that between them3=2 ¼ 71=2 ones, together with the crystal-field splitting2D3=2. For a given separation between the m3=2 ¼ 71=2 levels(taken as unity) we see from Fig. 3(a) that for 0rD3=2o1=2 theonly topologically allowed ground state is that with m3=2 ¼ 3=2,where we have assumed again that M3=240. This is notnecessarily the case when D3=2 ¼ 1=2, as is shown in Fig. 3(b).Then for small H we have B140, which makes the m1 ¼�1

microstate the more energetically favorable for spin-1 ions. Thus,in this phase the total magnetization is given by (10).

Increasing the applied field we see from (8) that whenH4Hs ¼ lM3=2, then B1 changes sign, which makes the m1 ¼ 1microstate for spin-1 ions the more energetically favorable. Thissaturated phase is characterized by the transition field

Hs ¼3J

gmB

; ð15Þ

which is independent of D3=2, and a total magnetization given by(14). total magnetization is given by (10).

When D3=241=2, a ground state with m3=2 ¼ 71=2 is possible.This is shown in Fig. 3(c) for D3=2 ¼ 1. For D3=241, and assumingM140 we have from (9) that B3=240 which makes them3=2 ¼�1=2 microstate for spin-3/2 ions the more energeticallyfavorable. This situation is depicted in Fig. 3(d). This implies from(8) that the m1 ¼ 1 microstate minimizes the energy of spin-1ions. Then in this phase the total magnetization is

Mð1;�1=2Þ ¼12N: ð16Þ

Increasing the applied field, we see from (9) that B3=2 decreasesand when H4Hc ¼ lM1, then B3=2 changes sign (see Fig. 3(e))which makes the m1 ¼ 1=2 microstate for spin-3=2 ions the moreenergetically favorable. The critical field for this phase to occur is

Hc ¼2J

gmB

; ð17Þ

which is independent of D3=2, and the total magnetization of thesystem in it is

Mð1;1=2Þ ¼32N: ð18Þ

From Fig. 3(f) we note that the saturated phase with totalmagnetization given by (14) is reached when the applied fieldovercomes the transition field

Hs ¼2ðJþD3=2Þ

gmB

: ð19Þ

To summarize, we expect that in the two cases considered hereðDaa0;Da0 ¼ 0Þ, the total magnetization (denoted simply as M

hereafter) per site shows magnetic plateaus at the quantizedvalues

M

2N-

1

4;3

4;5

4: ð20Þ

ARTICLE IN PRESS

E. Solano-Carrillo et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1917–19221920

4. DMRG results

We carried out DMRG calculations to study our system. As iswell-known, this algorithm yields very accurate results at zerotemperature. Here we consider a 200 site chain with openboundary conditions (OBC), taking J¼ 1. Periodic boundaryconditions (PBC) have also been used to analyze end effects. Wemaintained up to 80 states during up to 10 sweeps, with a totaltruncation error on the order of 10�15. This extremely low error isdue to the fact that we are dealing with an Ising system, and thusonly up to two eigenstates of the system’s block density matrixare the most probable.

We have calculated physical quantities of interest like theprofile of the z component of the magnetization /Sz

iaS, where the

brackets denote quantum mechanical expectation values, and thefree energy of the system in terms of the applied magnetic field. InFigs. 4 and 5 we show the total magnetization per site as afunction of the applied magnetic field, for the cases D1 ¼ 0:5,D3=2 ¼ 0 and D1 ¼ 0, D3=2 ¼ 0:5, respectively, and using OBC. Asexpected from the discussion given in the previous section,specifically from (20), the magnetization per site shows a step-like structure with plateaus at the quantized values 1=4, 3=4 and

0 1 2 3 4 5 6

0.25

0.5

0.75

1

1.25

M/2

N

H

D1 = 0.5, D3/2 = 0

Fig. 4. Total magnetization per site as a function of the applied magnetic field (in

units of gmB) for the crystal-field anisotropies D1 ¼ 0:5, D3=2 ¼ 0.

0 1 2 3 4 5 6

0.25

0.5

0.75

1

1.25

M/2

N

H

D1 = 0, D3/2 = 0.5

Fig. 5. Total magnetization per site as a function of the applied magnetic field (in

units of gmB) for the crystal-field anisotropies D1 ¼ 0, D3=2 ¼ 0:5.

5=4, which were found also by Aydiner [17] for the homogeneoussingle-ion anisotropy case. We emphasize that the two differentinhomogeneous anisotropic cases considered in these figures giverise to a different number of magnetic plateaus in each case,which reveals the influence of inhomogeneity on macroscopicquantities. In addition to these plateaus in the magnetizationcurve, we have found some perturbations caused by end effects.Measuring the /Sz

iaS profile we realized (see Table 1) that these

end states are localized at the open end of the sublattice whereDaa0. They manifest themselves as if they were a byproduct ofthe creation of a two-spinon localized excitation on thetranslational invariant phases shown in Table 2, which increasesthe total magnetization by one, and generate magnetic plateauswhich do not satisfy (1). We varied the chain length from 60 to200 sites and the end states always appear for the even-sizelength considered with OBC.

It is worthy to note that end effects have been obtained withDMRG, as effective spin-1=2 objects at each end of open spin-1Heisenberg chains [38]. These have been observed in real systemscontaining spin-1 chains, such as NiðC2H8N2Þ2NO2ClO4 (NENP),using magnetic resonance techniques [39], and explained quali-tatively [40] with the valence-bond picture introduced by Afflecket al. [41] and with Schwinger-boson mean-field theory [42].

We also studied our ferrimagnetic mixed-spin system usingPBC with an even 2N number of sites, and we observed that whenN is odd the end states disappear and when N is even they persist(the frontiers between translational invariant phases in thequantum phase diagrams remaining unaltered in both cases).The strong dependence of magnetization profiles on the totalnumber of sites being odd or even has been reported recently forspin-1 Heisenberg antiferromagnetic chains with exchange andsingle-ion anisotropies, using DMRG with OBC [13]. This can beunderstood since a residual effective interaction � JE�L=x (L is thelength of the chain, x is the correlation length) between the twoend states exists, which is believed to be antiferromagnetic forchains with an even number of sites and ferromagnetic for chainswith an odd number of sites, and thus the corresponding groundstates are fundamentally different [42]. Here the results with PBCshow this kind of behavior with N, which we believe is particular

Table 1Representation of the magnetization profiles of the system in the various phases

with broken translational symmetry.

M=2N D1 Z0, D3=2 ¼ 0 D1 ¼ 0, D3=2 Z0

49=200 kmkmkm � � �kmkmkm

51=200 -mkmkm � � �kmkmkm mkmkmk � � �mkmkmm

52=200 mmkmkm � � �kmkmkm mmkmkm � � �kmkmkm

151=200 mm-m-m � � � -m-m-m mmmmmm � � �mmmmmm

The scaled arrows represent m1 ¼ 71 and m3=2 ¼ 71=2;3=2 spin microstates, and

the small horizontal line represents the m1 ¼ 0 microstate.

Table 2Representation of the magnetization profiles of the system in the various

translational invariant phases.

M=2N D1 Z0, D3=2 ¼ 0 D1 ¼ 0, D3=2 Z0

1=4 kmkmkm � � �kmkmkm kmkmkm � � �kmkmkm

mkmkmk � � �mkmkmk

3=4 -m-m-m � � � -m-m-m mmmmmm � � �mmmmmm

5=4 mmmmmm � � �mmmmmm mmmmmm � � �mmmmmm

The scaled arrows represent m1 ¼ 71 and m3=2 ¼ 71=2;3=2 spin microstates, and

the small horizontal line represents the m1 ¼ 0 microstate. In the case D1 ¼ 0,

D3=2 Z0 two different mechanisms generate the plateau at 1=4. The transition

region is 1=2oD3=2 o1.

ARTICLE IN PRESS

E. Solano-Carrillo et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1917–1922 1921

to the mixed-spin ferrimagnetic system with an even number ofsites.

In Fig. 6 we show the quantum phase diagram when D1Z0,D3=2 ¼ 0 using PBC with a 202-sites chain. The translationalinvariant phases, which satisfy (1), are in agreement with themagnetization process described from (10) to (14). This is also thecase when D1 ¼ 0, D3=2Z0, which is depicted in the quantumphase diagram in Fig. 7 and described by the magnetizationprocess from (15) to (19). One may gain a better insight into theseprocesses by referring to Table 2. There, we represent themagnetization profiles of the system in the various translationalinvariant phases when we vary Da and H, keeping Da0 ¼ 0. We seethat in the magnetization process of spin-1 ions ðD1Z0;D3=2 ¼ 0Þthere exists Neel order along H for small D1. This shows featuresof the well-known Haldane phase [43], which has exponentiallydecaying correlation functions and a gap (plateau at 1=4). Thisphase disappears for all H when D143, as expected frommolecular-field theory. Before reaching saturation (plateau at5=4), the spin-1 ions make a discontinuous turn towards the xy

plane when D1o3. In this phase, the system shows amagnetization plateau at 3=4. For D143 the system is in the

0 1 2 3 40

1

2

3

4

5

6

7

H

5/4

3/4

1/4

D1

Fig. 6. Ground state quantum phase diagram for the case D1 Z0; D3=2 ¼ 0 with

PBC. The numbers in each region correspond to the total magnetization per site.

0 1 2 3 40

2

4

6

8

10

H

5/4

3/4

1/4

D3/2

Fig. 7. Ground state quantum phase diagram for the case D1 ¼ 0;D3=2 Z0 with

PBC. The numbers in each region correspond to the total magnetization per site.

large-D phase [18], and this latter spin configuration is measuredeven in the ground state until saturation occurs.

The situation is even more interesting when we considerD1 ¼ 0;D3=2Z0. In this case we encounter two different mechan-isms for the appearance of the same magnetic plateau at 1=4. Asimilar phenomenon was observed recently by Sakai andOkamoto [18] for the plateau at 1=3 of the saturation magnetiza-tion in the ferrimagnetic system ðSa; Sa0 Þ ¼ ð1;1=2Þ with D1Z0,D1=2 ¼ 0. The two mechanisms in this system reveal a quantumphase transition between the Haldane phase (Neel ordered) andthe large-D phase (not Neel ordered), separated byðD1Þc ¼ 1:11470:001. Tonegawa et al. [20] have also found threemechanisms for the appearance of the plateau at 1=3 of thesaturation magnetization in the system ðSa; Sa0 Þ ¼ ð1;2Þ withD1a0, D2a0.

As may be seen from Table 2, we have for our system amagnetic change between two Neel ordered phases. The region1=2oD3=2o1 is a transition region where the system ‘‘decays’’gradually from one phase to the other. Note that, in contrast to theD1Z0, D3=2 ¼ 0 case, the plateau at 1=4 appears for all values ofD3=2. For D3=241 all the spin-3=2 ions are first fixed to m3=2 ¼ 1=2(plateau at 3=4) with increasing field before the saturation phase(plateau at 5=4).

Finally we studied the free energy per ion as a function of theapplied magnetic field. In Fig. 8 we show this for the highestcrystal-field anisotropies considered. From this thermodynamicalproperty we can obtain the same information of the quantumphase diagrams because of the formula relating free energy andmagnetization M¼ ð@E=@HÞT , where we have T ¼ 0 in our system.That is, the critical and saturation fields are obtained at thosepoints where the slopes of the curves change. We have verifiedthis by calculating the numerical derivative in the above formula.An interesting feature that can be observed from this figure is thefact that the system ‘‘prefers’’ the situation where the crystal-fieldanisotropy is present only at the spin-1 ions rather than only atthe spin-3/2 ions. This is because the latter circumstancemanifests an overall higher energy. When the crystal-fieldanisotropy is homogeneous throughout the lattice D1 ¼D3=2 ¼D,the overall energy is even higher (a case not shown in this paper).Thus the consideration of the inhomogeneous crystal-fieldanisotropy is of critical importance from both a theoretical andexperimental point of view.

0 2 4 6 8 10

−8

−6

−4

−2

0

E/2

N

H

D1 = 4, D3/2 = 0

D1 = 0, D3/2 = 4

Fig. 8. Free energy per ion as a function of the applied magnetic field (in units of

gmB) for the cases D1 ¼ 4;D3=2 ¼ 0 and D1 ¼ 0;D3=2 ¼ 4. The system ‘‘prefers’’ the

crystal-field anisotropy only on the spin-1 ions.

ARTICLE IN PRESS

E. Solano-Carrillo et al. / Journal of Magnetism and Magnetic Materials 322 (2010) 1917–19221922

5. Conclusions

We investigated magnetic properties such as the magnetiza-tion curve and the free energy as a function of an appliedmagnetic field in a ferrimagnetic mixed ð1;3=2Þ Ising spin chainwith inhomogeneous crystal-field anisotropy, using molecular-field theory and density-matrix renormalization group calcula-tions. Molecular field theory allows us to understand in a simpleand intuitive way the physical mechanisms for the appearance ofplateaus in the magnetization curve as well as the quantum phasediagrams. The DMRG algorithm gives the magnetic propertieswith extreme accuracy, efficiently diagonalizing the many-bodyHamiltonian without considering the entire Hilbert space. Ouranalysis was divided into two cases of interest: when the crystalfield anisotropy D1 is present only on the spin-1 ions, and whenD3=2 is present only on the spin-3=2 ions. This latter case turns outto be the more interesting since a plateau at 1=5 of the saturationmagnetization is formed by means of two physically distinctmechanisms. In terms of lowering the energy, we found that thecase where the crystal field anisotropy is present only on the spin-1 ions is favorable, which makes the consideration of inhomoge-neous crystal-field anisotropy of vital importance.

Acknowledgments

This paper was supported by the Direccion de InvestigacionesSede Bogota of the Universidad Nacional de Colombia under theproject DIB 8003316 and from resources obtained with the‘‘Estudiante Sobresaliente de Posgrado’’ scholarship awarded bythe University.

References

[1] L.J. de Jongh, A.R. Miedema, Adv. Phys. 23 (1974) 1.[2] I. Bose, Frontiers in condensed matter physics, vol. 5, Allied Publishers Pvt.

Ltd., New Delhi, 2005.

[3] D.C. Cabra, M.D. Grynberg, A. Honecker, P. Pujol, Condensed Matter Theories,vol. 16, Nova Science Publishers, New York, 2001.

[4] J. Silva-Valencia, J.C. Xavier, E. Miranda, Phys. Rev. B 71 (2005) 024405.[5] J. Silva-Valencia, E. Miranda, Phys. Rev. B 65 (2002) 24443.[6] E. Lieb, T. Schultz, D. Mattis, Ann. Phys. (N.Y.) 16 (1961) 407.[7] M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett. 78 (1997) 1984.[8] A. Langari, M.A. Martın-Delgado, Phys. Rev. B 62 (2000) 11725.[9] X.Y. Chen, Q. Jian, W.Z. Shen, C.G. Zhong, J. Magn. Magn. Mater. 262 (2003) 258.

[10] F. Litaiff, J.R. de Sousa, N.S. Branco, Solid State Commun. 147 (2008) 494.[11] C. Ekiz, H. Yaraneri, J. Magn. Magn. Mater. 318 (2007) 49.[12] P. Sengupta, C.D. Batista, Phys. Rev. Lett. 99 (2007) 217205.[13] D. Peters, I.P. McCulloch, W. Selke, Phys. Rev. B 79 (2009) 132406.[14] D. Awschalom, N. Samarth, Physics 2 (2009) 50.[15] O. Kahn, Magnetism of the Heteropolymetallic Systems, Springer, Berlin,

1987.[16] A. Markosyan, T. Hayamizu, H. Iwamura, K. Inoue, J. Phys. Condens. Matter 10

(1998) 2323.[17] E. Aydiner, Chin. Phys. Lett. 21 (11) (2004) 2289.[18] T. Sakai, K. Okamoto, Phys. Rev. B 65 (2002) 214403.[19] H. Miyasaka, R. Clerac, K. Mizushima, K. Sugiura, M. Yamashita, W.

Wernsdorfer, C. Coulon, Inorg. Chem. 42 (2003) 8203.[20] T. Tonegawa, T. Sakai, K. Okamoto, M. Kaburagi, J. Phys. Conf. Ser. 51 (2006)

151.[21] U. Schollwock, Rev. Mod. Phys. 77 (2005) 259.[22] O. Kahn, Molecular Magnetism, VCH, New York, 1993.[23] T. Kaneyoshi, Y. Nakamura, J. Phys. Condens. Matter 10 (1998) 3003.[24] T. Kaneyoshi, Y. Nakamura, S. Shin, J. Phys. Condens. Matter 10 (1998) 7025.[25] C. Domb, Adv. Phys. 9 (1960) 149.[26] L.L. Goncalves, Phys. Scr. 32 (1985) 248.[27] N.R. de Silva, S.R. Salinas, Phys. Rev. B 44 (1991) 852.[28] A. Bobak, Physica A 258 (1998) 140.[29] Z.H. Xin, G.Z. Wei, T.S. Liu, J. Magn. Magn. Mater. 188 (1998) 65.[30] A. Bobak, Physica A 286 (2000) 531.[31] O.F. Abubrig, D. Horvath, A. Bobak, M. Jascur, Physica A 296 (2001) 437.[32] J.W. Tucker, J. Magn. Magn. Mater. 237 (2001) 215.[33] A. Bobak, O.F. Abubrig, D. Horvath, J. Magn. Magn. Mater. 246 (2002) 177.[34] C. Ekiz, J. Magn. Magn. Mater. 307 (2006) 139.[35] N. Stanica, C.V. Stager, M. Cimpoesu, M. Andruh, Polyhedron 17 (10) (1998)

1787.[36] R.M. White, Quantum Theory of Magnetism, Springer, Berlin, 1983.[37] M. Blume, Phys. Rev. 141 (2) (1966) 517.[38] S.R. White, Phys. Rev. Lett. 69 (19) (1992) 2863.[39] S.H. Glarum, S. Geschwind, K.M. Lee, M.L. Kaplan, J. Michel, Phys. Rev. Lett. 67

(1991) 1614.[40] S.R. White, D.A. Huse, Phys. Rev. B 48 (6) (1993) 3844.[41] I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Phys. Rev. Lett. 59 (7) (1987) 799.[42] T.K. Ng, Phys. Rev. B 45 (1992) 8181.[43] F.D.M. Haldane, Phys. Rev. Lett. 50 (1983) 1153.