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SOLIDS

THE QUADRUPOLE PHASE IN A MAGNET

WITH ANISOTROPIC BIQUADRATIC EXCHANGE

I.P. SHAPOVALOV

UDC 537.61

c2008

I.I. Mechnikov Odesa State University

(2, Dvoryanska Str., Odesa 270100, Ukraine; e-mail: [email protected])

ISSN 0503-1265. Ukr. J. Phys. 2008. V. 53, N 7 651

The quadrupole phase in a uniaxial magnet with single-ionanisotropy of the easy-plane type and the anisotropic biquadraticexchange interaction is investigated. The case where the spin at asite equals unity is considered. A new method for the calculationof the spectrum of spin excitation modes at finite temperaturesis proposed. It is shown that, in the limit case where T = 0,the results obtained are in full compliance with those reportedby other researchers. The expressions for two branches of thespin excitation spectrum at finite temperatures are obtained, andthe stability conditions for the spectrum modes are determined.The temperature, at which the stability of the spectrum modesbreaks, is determined analytically as a function of the Hamiltonianparameters. It is proved that, under certain conditions as thetemperature is lowered, the stability of the modes of the spinexcitation spectrum first breaks and then is restored. The existenceof a metastable state is predicted.

1. Introduction

Magnetic compounds, in which the constants of bothSIA and BEI are of the same order as the usual exchangeinteraction, were discovered in the 1970s [16]. However,the theoretical study of such systems met difficulties,because the inclusion of both SIA and BEI in theHamiltonian results in the fact that the algebra of theoperators describing a system state goes beyond thelimits of the SU(2) algebra. The attempts to overcomethese difficulties continue to date with a greater or lesssuccess [729]. Unfortunately in almost all cases wherethe researchers accounted for BEI, the calculations wererestricted to the consideration of isotropic BEI.

In works [79], the authors studied the magnetswith an isotropic BEI in the mean field approximationand came to a conclusion that, for a certain rangeof the Hamiltonian parameters, the ferromagneticand antiferromagnetic phases are unstable against the

appearance of the quadrupole phase (QP) with SZ = 0.As was shown in work [10], a new type of ordering occursin QP at T = 0, for whichSZ = 0 at each site to withinzero-point oscillations. The spin-wave approximationwas built for the S = 1 case, and it was shown that,for the case where the SIA and BEI constants, J0 andK0, respectively, satisfy the condition K0> J0> 0, QPbecomes stable.

In work [22], the finite-temperature studies of themagnets with SIA and isotropic BEI were carried outfor the case of S = 1. The possible kinds of orderedphases at T = 0 were determined, among which twophases were shown to display a quadrupole ordering withSZ = 0. At the same time, for the first phase, QP1,the equality(SZ)2 = 1 is valid, which means that theordering occurs in both directions along the z axis. Onthe other hand, it was shown that(SZ)2 = 0 for thesecond phase, QP2, which corresponds to the ordering inthe xy plane. At the zero temperature, the ordering inQP1 doesnt differ from the antiferromagnetic one, i.e.it is inhomogeneous. It is also proved that, apart fromthe usual ferromagnetic phase FMZ with the ordering

along the z axis, a phase QFMX can exist and ischaracterized by the ferromagnetic ordering along the xaxis and the quadrupole one within the plane that formsa certain angle with the xy plane, i.e. the phase with aspontaneously broken symmetry.

In work [25], the results of the low-temperaturestudies were reported for the magnets with ananisotropic BEI. The possible types of homogeneousordering were determined for the cases of the finite andzero external magnetic field h. It is demonstrated that,for the case of hZ = 0, a quadrupole-angular phase(QAP) can exist, for which all average S values are


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I.P. SHAPOVALOV

equal to zero, but the plane of quadrupole orderingforms a certain angle with the xy plane. This phase,just as the QFMX one, is a phase with spontaneouslybroken symmetry. However, contrary to the latter, QAPcannot be formed when BEI is isotropic. In the abovework, a spin excitation spectrum was found for eachof the ordered phases. It was shown that a quadraticdispersion law, (k) = +k2, was observed in thelong-wave limit for QP2. The phase diagrams, whichillustrated the field-induced phase transitions, werebuilt. However, since the investigations were limited tothe low-temperature region, it was impossible to observethe temperature evolution of the phase transitions.

The authors of works [28,29] considered the angular

phase which appears as a result of the competitionbetween the magnetic and strong crystalline fields ineasy-plane magnets. It was shown in [28] that, in suchsystems, the Landau phenomenological theory is suitablefor the description of phase transitions. The results ofwork [29] demonstrated that the phase transition fromthe singlet state to the antiferromagnetic one belongs tomagnetic phase transitions of the substitution kind.

This paper is aimed at the finite-temperature studyof QP2 for a uniaxial magnet, which is characterized bySIA and anisotropic BEI. Here, we propose the methodwhich makes it possible to find two branches of the

spin excitation spectrum, with one of these branchesbecoming soft at the center of the Brillouin zone forcertain values of the Hamiltonian parameters. At thesame time, in the long-wave limit, one observes thequadratic dispersion law, (k) k2, when hZ = 0,and the linear one, (k) k, when hZ = 0. It isdemonstrated that, with lowering the temperature, thestability of the spectrum modes becomes broken at firstunder certain conditions, and then it is restored. Thephase diagrams in the phase temperature coordinatesare constructed.

2. Operation Algebra and System Hamiltonian

In the case where the spin S at a site is equal to 1,the algebra of operators which describe all interactionsin the system is the ASU(3) algebra. The generators ofthis algebra are the tensor operators Oml , where l is thetensor rank, and m = 0,1, . . . ,l. A relation betweenthese operators and the spin ones is described by theformulae

O01 =SZ, O11 S+ =

12

SX iSY

,

O11 S = 1

2 SX + iSY

, O02 =

SZ

2 2

3,

O 12 =

SZS + SSZ

, O 22 =

S2

. (1)

The mean values of theOml operators determine the spinorder in the system, i.e. they form the multicomponentorder parameter.

The studies will be carried out basing on theHamiltonian which is invariant under the exp

iSZ

transformation (here, is an arbitrary angle):

H= hZi

SZi

i,j(i=j)

Jij

SZi S

Zj 2S+i Sj

+

+Di

O02i

i,j(i=j)

Kij

3O02iO

O2j

2O12iO12j + 4O22iO22j

, (2)

where hZ is the z-component of the external magneticfield,Jij are the exchange interaction constants,Kij arethe BEI constants, is the anisotropy constant of the

exchange interaction, and , are the BEI anisotropyconstants.In this work, the calculations are restricted to the

case where all the constants of the Hamiltonian arepositive. Under these conditions, only the homogeneous(one-sublattice) ordering occurs in the system. Inaddition, SIA is assumed to be of the easy-plane type.

In QP2, only the diagonal components of theSZandO02 order parameters are nonzero. Thus, in themolecular field approximation, Hamiltonian (2) takesthe form

H0=

hZ+2J0

SZ

iSZi +D

6K0

O02i

O02i,

(3)

whereJ0 and K0 are:

J0i

Jij , K0i

Kij .

The atomic energy levels depend on the value of thespin projection on the z axis (SZ = 1, 0, or1). Thus,we have

E1= hZ 2J0SZ+13

D 2K0O02,

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E0= 23

D+ 4K0O02,

E1= hZ+ 2J0SZ+13

D 2K0O02.. (4)

For QP2, it follows from the hZ = 0 condition thatSZ = 0 and, thus, E1 = E1. This means that thedoublet withSZ = 1lies above the singlet ground statewith SZ = 0. IfhZ= 0, the doublet splits (E1 < E1)by the value2(hZ+ 2J0SZ).

Thus, QP2 can be formed provided that E0 < E1,which is equivalent to the inequality

D

6K0

O02

> hZ+ 2J0

SZ

. (5)

It should be noted that inequality (5) is just thecondition of the QP2 stability against the transitions tothe FMZ phase. When hZ = 0, condition (5) is fulfilledautomatically, which means that the FMZphase is notrealized.

At the zero temperature (T = 0; hZ = 0), theequalitiesSZ = 0andO02 = 2/3are valid for QP2.On the other hand, when T = 0, theSZ andO02values are determined by the formulae

SZ

=

n

SZn exp(En/)

n

exp(En

/) ,

O20

=

n

O20exp (En/)

n

exp(En

/) ,

(6)

where is the temperature in the energy units (= kT).With regard for expressions (4), formulae (6) can be

rewritten as

SZ =2sh

hz+ 2J0SZ

exp6K0O02 D

1 + 2chhz+ 2J0SZ

exp

6K0O02 D

,

O02 = 1

3 1

1 + 2chhz+ 2J0SZ

exp

6K0O02 D

.

(7)

The values ofSZ and O02 grow with increase in.At the same time, the value ofO02remains negative.

For the case where hZ = 0, a relation between thetemperature and theO02value can be obtained fromthe second equation of system (7):

= D 6K0O02

ln [(2 6O02) / (2 + 3O02)]. (8)

3. Spin Excitations

For QP2at zero temperature, all spins are aligned in thexyplane, but their alignment within the plane is randomand, thus, SX = SY = 0. At finite temperatures, thespin excitations withSZ = 1andSZ = 1appear. Theyare generated by the Hubbard operators X10 andX10,respectively. Since, in the presence of a magnetic field(hZ = 0, T= 0), the excitation energy is smaller forSZ = 1 than that for SZ =1, a finite magnetization(SZ > 0) appears in the system and grows withincrease in the temperature, which completely agreeswith system (7). ForhZ = 0, the energies of excitationsof both kinds are equal, and, thus,

SZ

= 0.

To calculate the spin excitation spectrum, wepropose a method, whose essence is as follows. Wewill enumerate the atomic levels f, which correspondto the different values of spin projection on thez axis, by natural numbers n. Let the level withn = 1 be the lowest one. Then the Hubbardoperators X12f , X

13f , . . . , X

1nf , . . . are the annihilation

operators for the spin excitations at the site f, andX21f , X

31f , . . . , X

n1f , . . . are the creation operators for the

corresponding excitations. The set of the X1nf and Xn1f

(n = 2, 3, . . . ) operators forms an ensemble X. With

the use of the approximation, which will be describedbelow [see (13)], the commutators of operators fromthis ensemble with Hamiltonian (2) can be presentedas a linear combination of these operators. In the casewhere the commutators of annihilation operators withthe Hamiltonian give only the linear combinations ofannihilation operators, we can write

X12f , H

=

=a11X1(1+1)f + a12X

1(1+2)f + . . . + a1nX

1(1+n)f + . . . ,

X13f , H=

=a21X1(1+1)f + a22X

1(1+2)f + . . . + a2nX

1(1+n)f + . . . ,

............................................................X

1(n+1)f , H

=

=an1X1(1+1)f + an2X

1(1+2)f + . . . + annX

1(1+n)f + . . . ,

..............................................................

(9)

At the same time, aij are the expressions whichdepend on the Hamiltonian parameters. A set of all aijconstitutes a dynamical matrixAij . It is clear that, if the

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I.P. SHAPOVALOV

dynamical matrix is a diagonal one (aij = 0 for i=j),the a

ii values are the energies of spin excitations from

the ground state to the states with n = 2, 3,... at thesite f. On the other hand, they are the eigenvalues ofthe matrix Aij , since it is diagonal. In the k-space, theexpressions for these eigenvalues coincide with those forthe branches of the spin excitation spectrum. IfAij is anon-diagonal matrix, the expressions for its eigenvaluesin the k-space also coincide with those for the branchesof the spin excitation spectrum: n =n. To find theseeigenvalues, a secular equation can be used:

a11 k a12 ...a21 a22 k ...

......................................

= 0. (10)

If the dynamical matrix consists of the creationoperators for the spin excitations (Xk1f , X

l1f ,...), the

expressions for its eigenvalues in the k-space differ inthe sign from the expressions for the spectrum branches:n = n.

In the case where the construction of the dynamicalmatrix requires the use of both the annihilation andcreation operators, the expressions for the eigenvalues,which correspond to the former, coincide with those forthe corresponding spectrum branches. On the contrary,the expressions for the eigenvalues, which correspond to

the latter, differ in the sign from the expressions for thecorresponding spectrum branches.

To apply the proposed method to Hamiltonian (2),it is necessary to turn to the Hubbard operators. Therelation of the quadrupole operators Oml to the HubbardoperatorsXps can be expressed as

SZ =X11 X11, S+ = X10 X01,

S =X01 + X10, O02 =X11 + X11 2

3,

O12 =X

10

X01

, O12 = X

01

+ X10

,

O22 =X11, O22 =X

11. (11)

At the same time, the initial Hamiltonian (2) should berewritten as

H= hZi

X11i X11i

i,j

Jij

X11i X11i

X11j X11j

+

+2X10i + X

01i X

01j + X

10j

+

+Di

X11i + X

11i

2

3

i,j

Kij

3

X11i +X

11i

2

3

X11j +X

11j

2

3

2

X10i X01i X01j + X10j

+ 4X11i X11j

.

(12)

Calculating the commutators

X01p , H

andX10p , H

and using the approximation

XiXj =XiXj + XiXj

Xnm = 0 (n =m), (13)

we obtain the equation of motion for the operatorsX01p and X

10p , which acquires, after the Fourier

transformation, the formX01k , H

=

D hZ 2SZJ0 6O02K0+

+SZ+ 3O02 (Jk+ Kk)

X01k +

+SZ 3O02 (Jk+ Kk)

X10k ,

X10k , H

=

SZ 3O02 (Jk Kk) X01k +

+

D hZ 2SZJ0+ 6O02K0+

+SZ 3O02 (Jk+ Kk)

X10k , (14)

where Xnmk , Jk, and Kk are the Fourier transforms ofthe quantities Xnmp , Jij , and Kij , respectively.

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Consequently, the dynamical matrix has thedimension2

2and the solution of the secular equation

has two eigenvalues:

1, 2k =hZ+ SZ (2J0 Jk Kk)

SZ2 (Jk Kk)2 + D 6O02 (K0 Jk)

D 6O02 (K0 Kk)1/2

. (15)

The operatorX01k

, which is the annihilation operator,corresponds to the eigenvalue 1k. On the other hand,the operator X10k , which is the creation operator,corresponds to the eigenvalue 2k. For this reason,

the 1 , 2k spectrum branches are determined by theexpressions

1(k) = 1k; 2(k) = 2k, (16)

or

1,2(k) =

D 6O02 (K0 Jk)

D 6O02 (K0 Kk)+

+SZ2 (Jk Kk)21/2

hZ+ SZ (2J0 Jk Kk) . (17)In the low-temperature limit (T = 0), we obtain

SZ = 0 andO02 =2/3. In this case, expressions(17) take the form

1, 2(k) =

[D+ 4 (K0 Jk)]

[D 4 (K0 Kk)]1/2

hZ. (18)

Expressions (18) coincide with those obtained bythe method of transformation of spin operators to thesecondary quantization operators in work [25]. However,contrary to the results of work [25], the method proposedhere makes it also possible to easily study both the cases,T = 0 and T

= 0.

Since1(k) 2(k), the modes of spectrum (17) arestable provided that

2(k)> 0 for all values k.

In the system, the one-sublattice ordering occurs.For this reason for the corresponding values of theHamiltonian parameters, the2(k)branch becomes softat the center of the Brillouin zone, i.e. when k = 0.When k = 0, the inequality 2(k)> 2(0)is valid [30].Thus, the stability condition for the modes of spectrum(17) for all k values is 2(0)> 0 or

[D 6O02(K0 J0)][D 6O02K0(1 )]+

+SZ

2

(J0 K0)21/2 hZ

SZ (2J0 J0 K0)> 0. (19)The stability threshold can be calculated as

D 6O02 (K0 J0)

D 6O02K0(1 )

+

+SZ2 ( J0 K0)2

1/2=

=hZ+ SZ (2J0 J0 K0) . (20)For the case where condition (20) is valid and

hZ = 0, the spin alignments in the easy plane andalong the z axis are energetically equivalent, i.e. thecorresponding excitations have zero energy. The energyof the transitions to the state with the spin alignmentopposite to the z axis remains positive.

Since Jk and Kk are the even functions of the wavevector k, the soft mode displays the quadratic dispersionlaw, (k) k2, forhZ = 0.

If there is no external field (hZ = 0), two branches

coincide:

1,2

k

=

=

[D 6O02 (K0 Jk)][D 6O02 (K0 Kk)].(21)

In this case, the expressions for the stability condition(19) and the stability threshold (20) take the form

D 6O02 (K0 J0)

D 6O02K0(1 )

> 0,(22)

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I.P. SHAPOVALOV

Fig. 1. The stability diagram for an easy-plane magnet with

anisotropic BEI for min < 1 < max: 1 a phase with the

spontaneously broken symmetry; 2 QP2. The boundary between

the phases is calculated for max = 1.4. In the diagram, Dmax =

1.6 and 1 = 1.87

andD 6O02 (K0 J0)

D 6O02K0(1 )

= 0,

(23)

respectively. If condition (23) is fulfilled, all threepossible values ofSZ are energetically equivalent, and,thus, the energies of the transitions to the states withSZ = 1 equal zero.

For the casehZ= 0, the soft mode displays the lineardispersion law, (k) k.

4. Phase Boundaries

To analyze the phase boundaries for QP2, it is pertinentto rewrite condition (22) as

D 6O02K0(1

max)

D 6O02K0(1 min)> 0, (24)where the designations

max= max { ; J0/K0}, min= min { ; J0/K0}(25)

are introduced.For QP2, the inequalityO02 < 0 is valid. Thus,

under the conditions that

min 1 and max 1, (26)

the modes of spectrum (21) are stable for the arbitrarypositive values ofD.

In the case where only the former of inequalities (26)is fulfilled, the stability condition can be written as

O02 > D/6K0(1 max) .. (27)

Since O02 is a function increasing with thetemperature, the stability breaking occurs at

O02 = D/6K0(1 max) (28)

with decrease in the temperature.

With regard for (28), formula (8) gives therelation between the Hamiltonian parameters and thetemperature of the stability breaking for the modes ofspectrum (21):

max = max D

(max 1)ln4K0(max 1) + 2D4K0(max 1) D

. (29)

It is seen from (29) that, as D 4K0(max 1),max goes to zero, which means that there is a limitpoint in the case of the zero temperature:

Dmax = 4K0(max 1) . (30)

After the introduction of the dimensionlessparameters = /K0 and D = D/K0, we can buildthe phase diagram in the D coordinates, whichillustrates the case min< 1 < max (Fig. 1).

In the diagram, QP2 is formed in domain 2. Tospecify the kind of spin ordering in domain 1, we considerthe case of low temperatures in more details.

If

> J 0/K0, , (31)

formula (30) takes the form

Dmax = 4K0( 1) . (32)

On the contrary, if

< J 0/K0, (33)

we obtain

Dmax = 4 ( J0 K0) . (34)

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Expressions (31)(34) completely agree with theresults of work [25] which describes what occurs in thesystem with decrease in D in the case where T = 0 andhZ = 0. When condition (31) is fulfilled, QP2 passesinto QAP at the point which is given by expression(32). On the contrary, if condition (33) is valid, QP2transforms into the QFMX phase at the point which isgiven by expression (34). Thus, in domain 1, either theQAP or QFMX phase is realized, depending on which ofinequalities, (31) or (33), is valid.

It is noteworthy that, although Hamiltonian (2) isinvariant under a rotation around the z axis, the QAPand the QFMX phase do not possess such an invariance.This means that the QAP and the QFMX phase are the

phases with spontaneously broken symmetry [25].If

min> 1, and max> 1, (35)

inequality (24) is fulfilled, when both multipliers havethe same sign. In the case where both of them arepositive, the stability condition for the spectrum modesis condition (27), whereas the temperature of thestability breaking is given by formula (29). When boththe multipliers are negative, the stability condition takesthe form

O02 < D/6K0(1 min) . (36)Correspondingly, the stability breaking occurs withincrease in the temperature at the point

O02 = D/6K0(1 min) . (37)The temperature of the stability breaking is

determined by the expression

min = min D

(min 1)ln4K0(min 1) + 2D4K0(min 1)D

.(38)

As a result, if the external magnetic field is zero, themodes of the spin excitation spectrum are stable eitherfor < min or for > max, provided that condition(35) is fulfilled.

If= J0/K0, the equality min =max is valid. In

this case, the modes of spectrum (21) are stable for all values, except for the point = min =max.

Figure 2 shows the stability diagram in the Dcoordinates. Its lines illustrate the boundaries of thestability breaking and the recovery for the modes ofthe QP2 spectrum, i.e. the diagram reflects a reentrantbehavior.

Fig. 2. The stability diagram for an easy-plane magnet with

anisotropic BEI for min > 1 and max > 1: 1 a metastability

region for QP2;2 a phase with spontaneously broken symmetry;

3 QP2. The boundary between the phases is calculated for

max = 1.6 and min = 1.2. In the diagram, Dmax

= 2.4,

Dmin = 0.8, 1 = 2.13, and 2 = 1.6

A phase with spontaneously broken symmetry, eitherQAP provided that (31) is valid or the QFMX phase

if the condition (33) is fulfilled, is realized in domain2. The modes of the QP2 spectrum are stable indomains 1 and 3. However, in domain 1, QP2 can berealized as a metastable phase, since the ground stateenergy is greater for this phase than for the phase withspontaneously broken symmetry [25].

As is seen from the diagram, the situation thatthe stability firstly breaks at the point = max andsubsequently is restored at the point = min occursonly under condition that D < Dmin. IfDmin < D max.In the case where D > Dmax, the spectrum modes arestable at arbitrary temperatures.

Thus, the line max

D

coincides with the line of

the phase transition from QP2 to a phase with brokensymmetry (either QAP or the QFMX phase). Since thecomponents of the order parameter (the mean valuesof the spin and tensor operators) are the continuousfunctions of the field and temperature, these phasetransitions are of the second order. It is noteworthy that

the linemin

D

is only the line of the stability recovery

for QP2.

It is seen from (29) that ifD 4 (max 1), then

0. On the other hand, when D

0, then

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I.P. SHAPOVALOV

(4/3) max. This allows us to obtain the expressions forDmax and

1:

Dmax = 4 (max 1) , 1= (4/3) max. (39)Dmin and 2 can be found from Eq. (38):

Dmin = 4 (min 1) , 2= (4/3) min. (40)

5. Discussion of Results

As follows from the results of this paper, in the magnetswith anisotropic BEI, the value of the anisotropyconstant affects the stability condition for the modesof the spin excitation spectrum for QP2. At the same

time, if condition (35) is valid, with decrease in thetemperature, the stability first breaks at the point max

and then is restored atmin. However, ifD < Dmin, QP2can exist only as a metastable phase after the stabilityrecovery.

It should be noted that the situation described aboveis only possible if BEI is anisotropic in the system.Otherwise, if BEI is isotropic, i.e. = 1, the modesof spectrum (21) can be unstable only under conditionthat J0 > K0. When also = 1, the modes becomeunstable only for J0> K0. These conclusions are in fullcompliance with the results of work [22], where only theisotropic exchange interaction and BEI were considered.

Thus, the anisotropic BEI gives rise to theappearance of the reentrant effect in the system (see Fig.2) only for a certain range of the Hamiltonia parameters[see condition (35)]. If this condition is not fulfilled, thesituation shown in Fig. 1 is realized.

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26. Yu.A. Fridman, O.V. Kozhemyako, and B.L. Eingorn, Fiz.Nizk. Temp. 27, 495 (2001).

27. V.M. Kalita and V.M. Loktev, Fiz. Tverd. Tela 45, 1450(2003).

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30. R. Blinc and B. Zeks, Soft Modes in Ferroelectrics andAntiferroelectrics (North Holland, Amsterdam, 1974).

Received 27.08.07.Translated from Ukrainian by A.I. Tovstolytkin

I II I

I..

i i -i ii () i- i i i (). - , i i i. i - i , -i i i T=0. i i - i i-i . i i - ii i i i-i. , i i i ii i , i, , ii i. - i i .

658 ISSN 0503-1265. Ukr. J. Phys. 2008. V. 53, N 7

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