critical behavior of tb: a physical realization of a lifshitz point?

ELSEVIER Physica B 215 (1995) 286 292

Critical behavior of Tb: A physical realization of a Lifshitz point?

Marcia C. Barbosa

lnstituto de Fisica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil

Received 7 June 1994; revised 23 May 1995

Abstract

We consider a X Y spin system in a hexagonal crystal lattice with lattice constants a and c and with competing interactions J1 and - J2 along the z direction. The model is in the same universality class as the ANNNXY one that exhibits an helicoidal structure characterized by a period qc. Using renormalization-group analysis carried out to first order in e = d - 4.5, we show that qc oc (c/a - (c*/a*)) ~ with an exponent flq characteristic of the Lifshitz point vicinity. Applications to Tb are shown to be in good agreement with the experiments.

1. Introduction

There has recently been a growing interest in helimagnets. It is known that many of these magnetic materials, besides the paramagnetic to ferromagnetic phase transition, also undergo a phase transition from a paramagnetic phase to a phase with a modulated superstructure [1, 2]. This struc- tures can be of many types. The metal erbium, Er, for example, exhibits a uniaxial modulation [3]. In that case, theory indicates that, even though the paramagnetic to ferromagnetic phase transition is Ising-like, the paramagnetic to modulated phase transition belongs to the same universality class as the XY model [4].

Experiments with other materials, such as Tb [5-7], Dy [8-10], and Ho [11-13] show that, differently from the Er, they form hexagonal closed- packed lattices, where hexagonal planes with lattice constant a are separated by a distance c. Those materials exhibit two successive phase transitions as follows. At low temperatures, T < T*, the spins are ferromagnetically aligned in the hexagonal

planes. At T = T*, there is a transition from this ferromagnetic phase to a modulated structure. Within this phase, the spins also lie in the hexagonal planes like in the ferromagnetic phase, but, differently from there, they turn through an angle 0 from plane to plane. This phase is still an ordered phase, but exhibits a modulation along one specific direction. This modulation is represented by a critical wave vector qc. At the transition point the lattice constants exhibit small discontinuities which actually characterize this transition to be first-order [7]. Within the modulated phase, as the temperature is increased, both the critical wave vector and the angle 0 increase.

At T = Tc > T* the second transition arises. It is a canonical transition from the modulated phase to a paramagnetic phase. The nature of this last transition is not well known; some experiments indicate weak first-order phase transitions [6, 9], while others indicate a continuous transition. In this last case, the critical exponent for the specific heat is ~t = 0.20 + 0.03, 0.24 + 0.02, 0.27 ___ 0.02 for Tb, Dy and Ho, respectively [7, 10, 13].

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M.C Barbosa/Physica B 215 (1995) 286-292 287

Indeed, even when the transition is considered to be first-order, such exponents are observed as defining a "pseudo-critical" behavior [9].

In this article, we study the region between those two transitions where experimental reports indicate that

(1) The lattice constants a and c vary linearly with the temperature (a increases, while c decreases when one increases the temperature) [7, 10, 13, 14].

(2) The angle 0 between two consecutive planes decreases smoothly with the temperature. The lattice constant ratio c/a, otherwise, increases with the decreasing temperatures with a scaling form 0 oc (c /a - c*/a*)~. Here the ratios a*/c* specify the values of the lattice constants of the modulated phase at the first-order phase boundary [5].

In order to explain this last point, it was sugges- ted recently that the correct behavior for the turn up angle 0 should be associated to an electron topological transition 1-15].

In our analysis, we follow a different approach. We basically propose that both, the continuous phase transition from the paramagnetic to the modulated phase and the first-order transition from the ferromagnetic to modulated phase, for those rare earth elements can be explained in the framework of the ANNNXY model (XY model with anisotropic next-nearest-neighbors competing interactions).

First, we assume that the lattice constants and, consequently, the coupling constants of the spin model vary with temperature in such a way that places the system close to the Lifshitz point (LP) of the ANNNXY model. From that, it follows that the continuous transition from the disordered to the helical phase must be in the universality class of the LP point which is shown to be in good agreement with experimental results.

Next, we extend this analysis to the region close to the first-order transition between the ferromagnetic phase and the modulated phase. All our analysis is based on the assumption that the behavior of the thermodynamic functions in that region is actually governed by the critical fluctuations of the neighborhood of the Lifshitz point. Accepting the plausibility of this last assertion, the renormalization-group theory to second-order in e = 4.5 - d expansion is used and the behavior of the angle

0 and of the critical wavevector qo are explicitly obtained. We then show that the uncertainty of our basic surmise find support in the experimental data.

Now, in order to make clear our point, in Section 2, we review some well known properties of the ANNNXY model. This model, originally proposed by both Garel and Pfeuty and Droz and Coutinho- Filho [16, 17], has already been studied extensively [4, 1621]. Its phase diagram, obtained using vari- ous methods and sketched in Fig. 1, exhibits three phases, namely, paramagnetic, ferromagnetic and modulated phases. Continuous lines separate the paramagnetic phase from the ferromagnetic phase, and the paramagnetic phase from the modulated phase, while a first-order phase boundary separates the ferromagnetic from the modulated phase. Those three lines join at a Lifshitz point. Even though, these continuous lines were originally be- lieved to be XY-like and 4-vectorial-like, respectively [4, 18], we propose that, in applications to rare earth elements, they should be in the universality class of the LP.

Then, we check this assumption with experiments [5, 7, 10, 13] and with other theoretical results [15] by noting, for the first time to our knowledge, that the rare earth metals exhibit critical exponents/~ (for Tb only) and ~ (for Tb, Dy and Ho) LP-like [22].

Finally, at the end of Section 2, we state the crucial assumption of this paper, that is, the quasi- critical behavior near the phase boundary between the ferromagnetic and modulated phases. We suggest that the thermodynamics of this region is in fact governed by a new scaling form for the free energy dominated by the Lifshitz point fluctuations. It then follows that at T ,~ T*, the helicoid rotation angle 0 vanishes as O~:(c/a- c*/a*)/~q. Applying renormalization-group to first-order in e = d - 4.5 expansion, we found that flq = 0.54, which is in agreement with other theoretical results [15]. Comparison with experimental data for Tb gives support to our arguments.

Since the above result is universal, we can anti- cipate that it is also valid for applications to many other rare earth materials such as Dy, Ho or compounds with Y that we do believe belong to the same universality class of Tb. Our results are sum- marized in Section 3.

288 M.C Barbosa/Physica B 215 (1995) 286-292

2. The model

Let us consider the ANNNXY model where each lattice site, i, is occupied by a two component spin si. These spins are situated in a three-dimensional lattice formed of two-dimensional hexagonal layers. Each hexagon has lattice constant a and nearest neighbor exchange interaction Jo. Those planes are separated by a lattice distance c. Along this direction, the spins interact with their axial nearest neighbors through a ferromagnetic interaction J1 and with their next-nearest neighbors with a antiferromagnetic interaction - J2 , On going to a continuous spin representation by adding a weighing term for each spin, we obtain the effec- tive Hamiltonian

2 H = -- ½~_, ~ s")(q)uz(q)sO)(-- q)

q i = l

- . , 2 2 is,','+ 2 (1) {q,}i ,s {q~} i

where, as usual, we have

u2 = k a T - J(q) (2)

and, for this model

, , . ,

+ 2Jx cos(qzC) - 2J2 cos(2qzc). (3)

This model, already studied in the literature using different methods [4, 16, 17, 21], exhibits the following phase diagram sketched in Fig. 1. Three phases are present: ferromagnetic, modulated and paramagnetic phases. For low values of JE/J1 < ¼ and at high temperatures, the usual disordered phase is present. As the temperature is lowered, one finds a paramagnetic to ferromagnetic continuous phase transition with critical exponents in the X Y model universality class. The special feature of this model arises for J2/J l > ¼. In that region, the Fourier transform J(q) exhibits a maxima at qz = qc. The ordered phase, in that case, exhibits a modulation along one fixed direction and, consequently, at kaT~/Jo = 6 + 2J1/Jo - 2 J z / J o and a continuous transition separates the paramagnetic from this modulated phase. A first-order line 4J 2 = Jl is the boundary between the two ordered

phases. At J 2 / J 1 = 1 and T = Tc = T L the two continuous lines and the first-order phase boundary meet at a Lifshitz point.

For the study of the modulated phase, fluctuations around qc play an essential role. In order to take them into account properly, one has to employ a rescaling to the original Brillouin zone into a new zone given by [qz [ <~ q¢/2. As a consequence of this, the relevant degrees of freedom are also duplicated and the originally X Y model becomes a four-component-like [4, 16, 17].

Unfortunately, in the vicinity of the Lifshitz point, this strategy cannot be applied since the wave vector q¢ goes to zero. In that case, J(q) can be expanded in powers ofq and one can rewrite Eq. (1) in the continuum limit as

H =

-- ½ r + q2 + pq2 a + q~ s(i)(q)s(i)(_ q) i=1

(4) i # j

where

r = kBT - (6J0 + 2 J 1 - 2 , ]2 ) , ( 5 )

x / 6 ( a J 2 -- J l ) (6)

P = - - 4 1 4 J z - J 1

On doing that, we loose the hexagonal symmetry. Consequently, the transition between the ferromagnetic and modulated phases becomes continuous 1-21]. Now, in order to regain this symmetry, sixth- order terms like

;ffff' H6 = Wl E (s(i)2 "~ s(J)2)3 i~.j

ffff; + w2 Z ( s"13 s~3) 2 (7) i c j

should be added to the Hamiltonian given by Eq. (4) and no cubic anisotropy term must be present, so v4 = 0. Mean-field analysis in that case indicates that the very inclusion of the sixth-order terms makes the transition between the two ordered phases first-order, in agreement with Fig. 1. Of course, small changes in the positions of the phase boundaries in Fig. 1 are also introduced. This last

M.C. Barbosa/Physica B 215 (1995) 286 292 289

KaTE Jo

K BT * Jo

Ke T Jo

PARA

FERRO

L .T= T c

o,, e

MODULATE D

0.25

J2 Ji

Fig. 1. Schematic phase-diagram of the three-dimensional ANNNXY model. The solid line represents a first-order transition and the two dashed lines are continuous transitions. The three lines meet at the Lifshitz point L. The dotted line represents a possible behavior of a rare-earth element as the temperature is lowered.

effect can be made irrelevant by taking the coup- lings w's as small perturbations. Regarding the paramagnetic to modulated phase transition, the sixth-order term is actually irrelevant and, consequently, it does not contribute to the scaling near the LP. In that region, one might just consider the Hamiltonian of Eq. (4) which we do next.

On analyzing Eq. (4), one might note that the parameter p plays an important role. Mean-field analysis indicates that, for p > 0, the paramagnetic and ferromagnetic phases are separated by a continuous line at r = 0. For p < 0, the ferromagnetic phase is unstable against the modulated phase and a continuous transition at r = rc separates the paramagnetic phase from the modulated phase. At p = 0, a "quasi-critical" line separates the ferromagnetic phase from the modulated phase (this line becomes first-order when the hexagonal symmetry is restored). These three phases meet at the Lifshitz point located at p = 0 and r~ = 0.

At the vicinity of this Lifshitz point, near the modulated to paramagnetic phase transition, Mukamel introduced a scaling form for the singular part of the free energy given by Ref. [22]:

Fs ~ Ipl(z-~)/(°W+ ]p-(1/4~' ] p ~ , ' (8)

where t = (T - T L ) / T L. Here t = p = 0 locates the LP.

From Eq. (4), it was shown that for p ~ 0, fluctuations are dominated by the term qn 4 and, consequently, the upper critical dimension dc = 4.5. In that case, standard renormalization-group to second-order in e = 4 . 5 - d expansion was used to study the p = r = 0 vicinity with cubic anisotropy [4, 22]. In those works, the RG analysis indicates that the symmetric u* :~ 0, v* = 0 fixed point is stable and a continuous phase transition with exponents

= 2 - v4(2d - 1), (9)

fl = ½(2 - ~ - 7), (10)

~ (11) v4 = ~ = ¼ + 20'

V4 7~2 (12) flq = 7 = ½ + 40~

was found [22]. The application of a scaling form similar to Eq.

(8) for the ferromagnetic to modulated phase transition is not a priori correct since it is a first- order transition. But given that this last transition is weakly first-order and that it occurs near the Lifshitz point, we will assume that this first-order phase boundary is actually dominated by the LP fluctuations. Accepting that, in order to obtain the correct scaling form for the free energy, we need to establish how the coupling constant varies with temperature. In order to do it, we will assume that the lattice constants a and c and, consequently, the coupling constants Ji are analytical functions of T given by

J, = j , + j / ( T - T*). (13)

The parameters i = O, 1, 2, Ji, J[ are nonuniversal fixed parameters and T* is some reference temperature chosen in order to have

1 J2 --J2(j2--Jt~I](T * - T). (14) P ~ 4 J1 ja \ j 2 j t /

From Eq. (13) one can see that, under our approximation, the ratio J2/J1 varies linearly with temperature for fixed (Ji, J[) values. For

290 M.C. BarhosajPh~vsica B 215 (1995) 2&Y--292

T * < T < T,, the system exhibits modulation with q = qC. Inside that region, the decrease of temper-

ature, (T + T*) implies the parameter p vanishes. For T < T*, only the ferromagnetic phase is present. Given that, one readily sees that our T *, the

“reference temperature”, locates the first-order

phase boundary. Using the Hamiltonian Eqs. (4) and (7) within

the mean-field approximation, the ferromagnetic to modulated phase transition was explained in two

different ways: (i) In the presence of a sixth-order contribution

given by Eq. (7) Michelson found that for negative values of p given by 0 > p > p*, even though the

modulated phase still exists, the ferromagnetic

phase exhibits lower free energy. At p = p* < 0, there is, then, a first-order transition between those

phases. At this first-order boundary, both the critical wave vector, q, = q* # 0, and the magnetization are discontinuous [21].

(ii) In the absence of the sixth-order term, the transition occurs at p = 0. In that case, at the phase boundary, the critical wave vector vanishes, qC + 0, the turn up angle 8 -+ 0 and no discontinuity in the

magnetization is observed. One might infer at a first sight that this would be a continuous transition. But, as pointed out by Michelson, since the magnetization is not uniform throughout the system as in the usual continuous phase transitions, one might infer that it is actually a weak first-order

transition [21]. Indeed, since we know that for the discrete model this transition is already first-order [19], we will accept that as true for the continuous model.

In both points of view, one can pose the following question: how does qC scale with T - T*? In order to answer to that point, let us first suppose that T*=T where T = TId is specified by

t = p = 0. In t?hat case, the scaling relation Eq. (8)

states that qC x (T - TJBq [22]. This assumption, even though possible, is not realized [20, 211. One must, then consider the case T* # TL. Since. at T = T *, only a first-order transition is present, the thermodynamic functions are finite and no singu- larity is usually expected. Therefore, the scaling relation (8) cannot be applied.

Note, however, from Eqs. (4)-(14) that, even though in the ferromagnetic to modulated phase

transition no singularities in the thermodynamic functions should be expected, the correlation func-

tion exhibits a peak as p -+ 0. This peak becomes sharper as f + 0 (or equivalently as T* + TL) where the real critical phenomena arises.

In order to study that region of the phase-dia-

gram, where the first-order phase boundary is asymptotically close to the Lifshitz point, we postulate a new scaling form for the free energy namely

where r = (T,_ - T * )/T * and where ,$ = ,~I,&/cP~.

The sixth-order parameters show up in Eq. (15) in the scaling forms wi/ 1 p 1 1i4w with c$++ < 0. Those

terms are irrelevant variables in RG sense and they did not affect the scaling of qC that will be due to the singular part of the free energy.

Of course, should the arguments of Michelson be correct, q, would not be zero at the first-order transition and, in that case, wi would well be “dan- gerous irrelevant variables”. Even in that case, the

behavior associated with the modulated phase would still be given by Eq. (15), assuming that the parameters p and r stay small. Scaling with a small nonvanishing parameter is a usual procedure for studying weak first-order transition and actually gives good results in studying this sort of systems

[23,24]. We must, otherwise, say that this is not the case for all the systems that exhibit helical phases. For physical realizations for the ANNNI model, the ferromagnetic to modulated phase is not weak first-order and, consequently, the scaling form Eq. (15) could not be applied [25].

Most of the rare earth compounds are quite similar. Let us, then, consider Tb as an example. It is known through experiments that like other heavy rare-earth metals, Tb exhibits a magnetic structure within a temperature range from 216 to 226 K. The transition between paramagnetic to modulated phase at T = 226 K exhibits a magnetic long range- order with fi = 0.25 f 0.01 and CI = 0.20 f 0.03. As one lowers the temperature from T = 226 to 216 K, the lattice constant c increases linearly. Besides, the wave vector decreases as qC cc 8 cc (c/a - c*/a*)pq where 8, M 0.5 (this value was simple obtained

M.C. Barbosa/Physica B 215 (1995) 286 292 291

by fitting the Dietrich and Neilson data). At T = 216 K, there is a first-order transition from this modulated phase to a ferromagnetic phase. The ferromagnetic phase exhibits long range magnetic order with /~ = 1/3, while the modulated phase exhibits/3 = 0.25.

Our explanation of the above results, goes as follows. For T < T* = 216K, using Eqs. (4)-(14), one sees that the spins are ferromagnetically aligned with magnetization M oc t a. In that case, even in the vicinity of the Lifshitz point, the critical dimension is de = 4 and consequently the exponent /~ ~ 0.36 is in the X Y model universality class [4]. This result is not far from the experimental ½ value.

At T = T¢ = 226K, there is a continuous transition from a paramagnetic phase to modulated phase. Within this helical phase, the magnetization M ~ ( T - T ¢ ) ~ exhibits an exponent given by /3 ~ 0.275, while the specific heat C ~ (T - T~) -~ exhibits an exponent ~ = 0.15. Both exponents were obtained by means of renormalization-group to first-order in d = 4.5 - e expansion as given by Eq. (9) [22]. Series analysis give /3 = 0.2 _+ 0.02 [20]. Again we can say that both theoretical results are in good agreement with the experiments that give ~ = 0.20 _+ 003 and/3 = 0.25 for Tb when com- pared with other theoretical results that indicate /3 = 0.39 and ~ = - 0.17 [16, 18].

Similarly to what happens for Tb, experimental results for the specific heat for Dy and Ho give

= 0.24 + 0.02 and ~ = 0.27 _+ 0.02, respectively. It seems, consequently, that at least in the critical region, the critical behavior of these elements is well explained, in the framework of the A N N N X Y model close to the Lifshitz point.

As a next step, one might be interested in the behavior of the critical wave vector and the turn up angle near the first-order phase boundary between the ferromagnetic and modulated phases. Within that region, experiments were done, using the lattice constant as the temperature parameter and they suggest that the critical wave vector q~ scales as qc ~ ( a / c - a * / c * ) pq, were a* and c* are the lattice parameters at T = T*. Now, let us try to put our variables in terms of this new "temperature scaling field". This can be easily done by assuming the lattice constants, for T* < T < T~, are analytic

functions of temperature given by

a = a* + a ' ( T * - T ) (16)

and by

c = c* + c ' ( T * - T ) . (17)

Now, using the above results, as well as Eq. (14), one sees that theoretical results indicate that the critical wave vector and, similarly, the turn up angle 0 approach to zero as the lattice constant ratio a / c approaches the value a * / c * with a scaling

qc = Qc - (18)

with the exponent/3q ,~ 0.54 1-22] or [3 = 0.5 _ 0.15 given by high temperature series expansion [20]. Note that this result is again in good agreement with the experimental result for Tb.

3. Conclusions

The main result of this paper is that we found support for the use of the A N N N X Y model near the

LP for the study of rare earth compounds. First, we assumed, following Garel and Pfeuty,

that such systems should be described by a model with competing interactions [-16,18]. Differently from their original work, we propose that the physical parameters related to the rare earth elements vary with temperature in a way that the paramagnetic to modulated phase transition occurs near the Lifshitz point [4, 16]. Experimental results for magnetization and specific heat of Tb, Dy and Ho close to the paramagnetic to modulated phase transition give support to our assumption [5, 7, 10].

Next, we analyzed what happens within the helical phase close to the phase boundary between the modulated phase and the ferromagnetic phase. We propose that even near the first-order transition the thermodynamics is governed by the critical fluctuations of the Lifshitz point. Accepting the plausibility of that assertion, we postulate a scaling form for the free energy and we find that, even close to the first-order phase boundary, the critical wave vector, qc, scales with p ~ c /a - c* /a* oc T * - T with the Lifshitz point exponent /3q ~ 0.54 (RG) or /3 = 0.5 4- 0.15 (series) [20]. These values are well borne

292 M.C. Barbosa/Physica B 215 (1995) 286 292

out by the exper imenta l value flq ~ 0.5 from Ref. ( [5] ) and by o ther theoret ical results [15].

One could argue that our s ta tement regarding the value of flq is suppor t ed only by da t a for Tb. O u r answer to that cri t icism is that, at least at the cri t ical region, exper iments indicate that Dy and H o exhibit the same behav ior as Tb. Besides, within the modu la t ed phase, the lat t ice cons tan ts of those elements are quite s imilar and varies with temper- a ture in the same way. Unfor tunate ly , the avai lable da t a for the angle 0 as a function of the lat t ice cons tan t for both Dy and Ho, in our poin t of view, is not close enough to the phase b o u n d a r y in o rder to give a precise es t imate for flq [14].

In s u m m a r y the A N N N X Y model cons idered at the Lifshitz po in t vicinity is not only able to give the correct qual i ta t ive phase d i ag ram for ra re-ear th metals, but also to predict bo th crit ical behaviors , as well as to give the correct scaling form for qc.

Acknowledgements

We are par t i cu la r ly grateful to Dr. Fe l ipe Barbedo Rizzato and to Prof. Car los Cast i l la Be- cerra for useful and interest ing suggestions. We also thank Claud io M a c h a d o Rizzato for helping with the figure. This work was suppor t ed in par t by CNPq Conse lho Nac iona l de Desenvolv imento Cientifico e Tecnol6gico and F I N E P - F inan- c i adora de Es tudos e Projetos , Brazil.

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critical behavior of tb: a physical realization of a lifshitz point?

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