Crossover from logarithmically relaxing to piezomagnetically frozen ...

PHYSICAL REVIEW B

VOLUME 49, NUMBER 9

Crossover from logarithmically

1

MARCH 1994-I

relaxing to piezomagnetically frozen magnetic remanence Feo 47Zno 53Fz

in low-field-cooled

J. Kushauer Angetoandte

and W. Kleemann

Physik, Universitat Duisburg, D-47048 Duisburg, Germany

J. Mattsson

and

P. Nordblad

Institute of Technology, Uppsala University, S 7512-1 Uppsala, Sweden (Received 29 September 1993) Weak-field-induced magnetization M and temporal relaxation of its remanence p are investigated in quantum interference device (SQUID) Fe047Zn053F2 by using Faraday rotation and supercouducting Stretched logarithmic decay as predicted by Nattermann and Vilfan is confirmed at magnetometry. T=2. 8 K for freezing fields 3~B ~4. 9 T. At very low fields, 0.001~B ~1.5 T, virtually constant remanence, p-44 A/m, is observed. It is due to immobile domains, which form via piezomagnetic coupling to built-in random stress fields.

Owing to the extreme critical slowing down of the random-field Ising inodel (RFIM), the order-parameter fluctuations become frozen in the vicinity of the phasetransition temperature T, . The quenched random fields (RF's) are, thus, causing a metastable nonequilibrium domain state on cooling to below T, . Experimentally, the domain-state conjecture'* was verified on diluted anisotropic antiferromagnets in an axial uniform magnetic field (DAFF}. As is well known, its critical behavior can be mapped onto that of the RFIM. ' On the one hand, finite correlation lengths of the antiferromagnetic (AF) order parameter were observed on the field-cooled (FC) prototypic DAFF system Fe& Zn„Fz. On the other hand, excess magnetization, hM, of the domain walls was demonstrated on Feo7Mgo 3C12 (Ref. 6), Feo 4~Zno»F2 (Ref. 7},and in Monte Carlo simulations. After removing the external field, a large part of bM decays on a time scale well below 1 s. An appreciable magnetifraction survives as long-lived thermoremanent zation (TRM) p. In the absence of RF pinning at B =0, the domains evolve in time t via random-exchange Ising model (REIM) dynamics. Pinning is due to the distribution of magnetic vacancies and possibly influenced by dipolar interactions between different domains. The temporal law, p vs t, for the case of weak freezing fields, B J, and low temperatures, T Tz, is particudenotes the leading magnetic Here, larly interesting. exchange interaction and Tz is the zero-field ordering temperature. Nattermann and Vilfan (NV)' assume that the fractal domain walls rearrange in time only on small length scales. The original domain sizes do not vary within finite times. NV' predict

'

«

«

J

p(t)=a(ln(t j~)] ~+b,

t

&r,

where a refers to the initial excess magnetization and b denotes a small volume contribution. ~-10 ' s corre4 is expected in the sponds to a local spin-flip time. weak-B, low-T limit from scaling arguments. ' In fact, data obtained on Feo 7Mgo 3C12 (Ref. 11} after FC to T =4. 5 K with B =0.2 T seem to obey relation (1) 55. However, at large values of B and T the with

$-0.

$-0.

0163-1829/94/49(9}/6346(4}/$06. 00

49

exponent lit increases considerably. Similarly, too large values, g& 1, were recently' obtained on Feo 47 Zno 53F2 when using quite large fields, 5 B 7 T, and relatively high temperatures, 5 ~ T 20 K. Moreover, a search for the best fitting analytic decay function' turned out to be clearly in favor of power laws, either in the conventional form, p,

(t)=ct

(2)

or in a generalized formulation,

p(t) =d exp[ —xln(t lr)

],

t &

r, re

1

.

(3)

Intuitively, the relevance of fractal temporal laws (2) or (3} seems justified in the strong-8, high-T limit. As rethe DAFF sysvealed by Monte Carlo simulations, ' tem decays into a network of interpenetrating fractal AF

'

domains dispersed over an extremely wide size distribution. Within this spin-glass-like structure, relaxation clearly leads to domain growth and substantial wall displacements. ' Hence, the basic conditions leading to the Eq. (1) are invalidated. The present paper is devoted to the relaxational behavior of the prototypic DAFF system Feo 47Zno 53F2 in weak-B (B & 5 T) and low T(T & 5 K) lim-it. Analysis of our very precise Faraday rotation (FR) data, obtained at T=2. 8 K after FC with 3&B &4. 9 T, supports Eq. (1), with an exponent close to the expected one. quantum inSurprisingly, however, superconducting terference device (SQUID) magnetometry in the very weak FC limit, 0. 001 B 1.5 T, does not improve the agreement with Eq. (1). Instead, virtually constant p, independent of both t and B, is observed. This peculiar behavior seems to be related to the quasispontaneous creation of large AF domains. They are stabilized at T & T, (B) by internal stress via the piezomagnetic efFect. ' Application of an arbitrarily small field then easily induces a net magnetization. This does not change with increasing field unless domains smaller than the piezomagnetic ones are created via the RF mechanism. Very probably this mechanism also explains the weak-B remanence previously observed on FeF2 (Ref. 16) and

'

'

6346

1994

The American Physical Society

BRIEF REPORTS Mn&

„Zn„F&.'

6347

'

The experiments were carried out on the same very homogeneously diluted sample which was previously inwith the FR technique. Again, in order vestigated to measure p vs t the FR angle 8 was measured with an so as to obtain a resolution of improved apparatus 58=5X10 deg. The linearly polarized light beam with wavelength A, =442 nm travels, precisely aligned parallel to the tetragonal c axis, through the sample and is detected after passing a rotating combination of an elasto-optic modulator with an analyzer as described preData were taken after cooling the sample from viously. about 70 K (-2T+) to T=2. 8 K, stabilized to within hT =1 mK. The switchoff time to of the superconducting solenoid from the initial FC value 8 to zero (e.g. , rp = 194 s for 8 = 4. 86 T) was measured prior to each relaxation experiment and taken into account' when fitting p vs t to model functions. The temperature dependence of the magnetization M (Quantum was measured with SQUID magnetometers Design MPMS2 and MPMS5S) in fields ranging between B =0. 1 rnT and 5 T. The usual measurement protocol comprises zero-field cooling (ZFC) from T-70 to 10 K, subsequent field heating (FH) up to T =48 K and FC back to T = 10 K. Data points were taken after changing T by steps of AT =2 K and subsequent stabilization of T. Then, upon zero-field heating, the TRM was measured between T =10 and 48 K. Figure 1 shows the time dependence of the TRM as measured by the FR angle after cooling in B =4. 86, 4.0, 3.5, and 3 T (curves 1 —4, respectively) to T=2. 8 K. Contrary to previous attempts, ' very small but finite decreases are clearly observed. Within 10 and 10 s, the FR only drops by less than 0. 1 deg (i.e., & 5%). The scatter of the data is of the order of 1 mdeg, hence, not affecting the shape of the resulting relaxation curves. In order to find out the best fitting relaxation law, we first consider Eqs. (1) and (2), since both of them essentially involve two parameters and thus are, comparable. First of all, it seems reasonable"' to use Eq. (1) with ~=10 ' s, which corresponds to a typical spin-Hip atwith previous time. ' Moreover, in agreement the b term is neglected. Finally, the relaxatempts, tion occurring during the decrease of the magnetic field is taken into account' by starting the time scale (i.e. , t =0) after passing 80% of the switchoff period to (see Table I). Table I lists the best fitting parameters to Eqs. (1) and (2) for the data in Fig. l. It is seen that the reliability numbers of the fits, g~&~ and g~2~, respectively, are sensitively better in the case of the logarithmic decay law, Eq. (1). The corresponding curves 8 vs t (Fig. 1, solid lines) are perfectly fitting, in particular for t 10 s. Hence, contrary to the higher field region, B ~ 5 T, ' the power law (2) does not seem to be the better choice. If, instead, the generalized power law (3) is chosen, the fit is improved. However, this is a consequence of enhancing the number of fitting parameters by including the exponent I'Wl. By setting v= 1 s (Ref. 12) we obtain I =1.76, 1.18, 1.06, and 1.02 and slightly improved reliability numbers, 10 y(3) 1.09, 0.40, 0.48, and 0.57 for the data The improvements are not sets (1)—(4), respectively.

6.4

'"'

"'

)

6.2

0-

4.

4-

2.

4—

1,

1.2 10

10

10

10

t [s]

FIG. 1. Semilogarithmic plot of the Faraday rotation angle 8 T =2.8 K after FC with

vs time t measured on Fe047Zno 53F2 at

B =4. 86 (1), 4(2), 3.5(3), and 3 T(4), respectively. The

solid lines

are best fits to Eq. (1).

essential, however, at least in the cases of curves (1), (2), and (4). We thus conclude that the NV prediction, Eq. (1), T. seems to be correct in the low-T, low-B region, B This corroborates previous" assumptions and seems to confirm the underlying model of large compact domains thus with fractal surfaces. ' In particular, the exponent obtained agrees with the scaling theoretical prediction, ' 4. The only exception, /=0. 49 for the lowest field B =3 T, is, on the one hand, due to the poor signal-tonoise ratio. On the other hand, as will be outlined below, one approaches the very low B regime where the TRM is no longer relaxing at all because of secondary pinning forces. In this extreme case, Eq. (1) is no longer expected to hold. Nevertheless, at B = 3 T most of the TRM seems to be due to relaxing RF-induced domains as judged from the essential 8 dependence (Fig. 2 inset, solid line) of the

(5

f

$-0.

TABLE I. Freezing fields and fitting parameters referring to 1-4 of Fig. 1 and to Eqs. (1) and (2), respectively (see

curves

text).

B(~)

&o(&)

4.86 4.00 3.50 3.00

194 160 140 120

10 x

24.9 14.8 10.5 7.4

0.39 0.38 0.42 0.49

1.14

0.40 1.40

0.65

6.74 4. 17 2.55 1.44

1.2 1.2 1.2 1.7

10 y

1.59 1.14 1.42 1.47

BRIEF REPORTS

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49

-2

place upon FC to below T limit if er, godicity upon field cycles (see below). Coalescence of all p vs T curves is found at T 27 K-0. 7 T~ (Fig. 3), similarly as observed on Feo7Mgo ~Clz. This is due to the rapid relaxation of the RF-induced domain state in the vicinity of T~. ' However, the novel bias TRM, induced by arbitrarily weak fields, persists above 0.7 T~ and vanishes only at Tz-37 K. Obviously, a small, but finite ferromagnetic moment remains after FC in the very weak-B limit. Within waiting times up to 1 h, virtually no decrease of this moment is observed at T = 10 K. Very probably, LMo is due to the dominating inhuence of AF domains, which are not induced by the RF mechanism. En the present case of Fe, „Zn, F2 and for Mn, „Zn„Fz, (Refs. 17, 18) we assume "spontaneous" formation of stable and reproducible domains of millimetric size upon cooling to below Tz, similarly as observed on another rutile-type AF, MnF2. ' Peculiar to this crystal structure, we propose the piezomagnetic effect to be at the origin of the constant low-field-induced remanence. As pointed out by Borovik-Romanov, ' a reversal of the longitudinal moment piezomagnetic m, ~B, +Aztr„ I, i~t, by inverting B from [001] to [001] under a fixed shear stress o, should be accompanied by a rotation of the AF vector I through 180' ( A, 2 = piezomagnetic coupling constant). Here, by symmetry, we argue that the same rotation of I must occur when fixing the direction of B, but changing the sign of cr„. Hence, randomly distributed built-in stress fields with alternating cr components will give rise to AF domains with altertial relaxation

1000—

T, (B)

T—

taking

(B). ' T, (B) T, (B) signifies the

)

10

bD

'

aV

cs

l00B [T] p

p p p

p p

p p

p

p

p

p

0. 1

0.01

0.001

p

10

B [T]

FIG. 2. Log-log plots of the TRM, p, and of the NV parameter a (inset; see Table I) vs freezing field B at T =10 K. The interpolated solid lines represent B' dependences. a (Table I). ' The T dependence of the TRM obtained at various field values, 0.001 B & 5 T, is shown in Fig. 3. Surprisingly, nearly identical curves, p vs T, emerge for weak fields, 0. 001 B ~1.5 T. The B dependence of p at T = 10 K, starting with the low Bvalue )tto-=44. 3 A/m, is shown in Fig. 2. Only at B & 2 T, the TRM is steeply risWe find ing and follows the usual RF behavior. ' — p =aB p, (Fig. 2, solid line), where a is a constant and p' probably accounts for the parthe negative intercept — amplitude

~

500

400

400

300

300 200

2 200

0—

100

100

10

20

30

40

20

30

40

50

T [K]

FIG. 3. p (

), 0.03

(

interpolated

vs

T on ZFH after FC with B =0.001 (0), 0.01 (6), (V), 3.5 (C'), and 5 T (H) and

3 ), 1.5 (o ), 2 by eye guiding lines.

FIG. 4. M vs T measured within 10 ~ T ~ 50 K after ZFC on FH and subsequent FC (arrows) in fields B =0.001 (0), 0.01 (o ), and 0.03 T ( ), respectively.

BRIEF REPORTS

49

nating orientations when cooling through T, in arbitrarily weak fields B~~[001]. All domains contribute to m, with the sign of B.' Their orientations are interchanged when inverting 8 as observed previously. ' ' ' This effect is demonstrated in Fig. 4. It shows M vs T curves obtained after initial ZFC to T=10 K on FH and subsequent FC with 8 =1, 10, and 30 mT, respectively. As usual, ' splitting of the curves, MFH &M„c, is observed at T & T, (B). Unexpectedly, however, the FH curves for 8 =1 and 10 mT start with negatiue M values at T=10 K. Inspection shows that this is due to a weak negative external field, 8& &0 with ~Bo~ & 1 mT, created by the remanence of the superconducting solenoid. It is impressive to see nearly perfect inversion of M when changing from FH to FC at the lowest applied field, 8 =1 mT. Here, M is nearly exclusively due to piezomoments magnetic m, ~ A, 2t7„„1,/~ l, ~, controlled by 0 upon FC, respective80 &0 upon "ZFC" and ly. At larger fields, 8 =10 and 30 mT, the usual fieldis superimposed. In all cases, induced moment, m, however, d!M=MFC MFH =2po at T =10 K, since only the sign, but not the amplitude of the ferromagnetic bulk moment changes with different signs of the controlling field upon "ZFC" and FC, respectively. Once formed, the AF domains retain these moments, even in 8 =0. The remanence found for Feo 47Zno 53Fz is comparable to that observed in CoF2 after relieving both B and the uniform external stress o ' By means of a simple experiment we have shown that po is most probably due to random shear stress. By chemical etching and thus, partially removing surface strains from a mechanically polished of sample Feo 6Zno 4F2 (resembling our above sample) we succeeded in reducing po by 20%. Note that real ZFC under random-stress fields also creates piezomagnetic moments, however, with zero average, (m, ) =0.' A slight excess, ( m, ) %0, due to incomplete domain statistics is occasionally' observed. The crossover between RF-induced and piezomagneti-

8+Bo) ~8„

„.

'J. Villain, J. Phys.

(Paris) 46, 1843 (1985).

D. S. Fisher, Phys. Rev. Lett. 56, 416 (1986). S. Fishman and A. Aharony, J. Phys. C 12, L729 (1978). 4J. L. Cardy, Phys. Rev. B 29, 505 (1984). 5D. P. Belanger et al. , Solid State Commun. 54, 79 (1985). 6U. A. LeitXo and %'. IGeemann, Phys. Rev. B 35, 8696 (1987). P. Pollak et al. , Phys. Rev. B 38, 4773 (1988). U. Nowak and K. D. Usadel, Phys. Rev. B 39, 2516 (1989). T. Nattermann, J. Phys. A 21, L635 (1988). t T. Nattermann and I. Vilfan, Phys. Rev. Lett. 61, 223 (1988). "U. A. LeitXo et al. , J. Phys. (Paris) Colloq. 49, C8-1217 (1988). S. J. Han et al. , Phys. Rev. B 45, 9728 (1992). U. Nowak. and K. D. Usadel, Phys. Rev. B 44, 7426 (1991). S. J. Han and D. P. Belanger, Phys. Rev. B 46, 2926 (1992). A. S. Borovik-Romanov, Zh. Eksp. Teor. Fiz. 38, 1088 (1960)

6349

cally driven domain formation occurs in the vicinity of Although both mechanisms are active at any field value, for energetic reasons that one dominates which produces the smallest AF domains. Below 8& random-stress-induced domains, probably of millimetric size, ' prevail with a ferromagnetic volume moment po. Above B &, however, the large piezomagnetic domains decay into small RF-induced domains. Analysis of our data (Fig. 2) assuming wall-dominated magnetization reveals domain sizes as small as 2. 2 R 0. 3 pm for 3 ~8 ~4. 9 T in good agreement with neutron data. Since these values are still large enough to fulfill the NV topological requirements, ' we are able to confirm the expected stretched logarithmic decay law (1) with the predicted exponent, 4. This, in conclusion, is the primary issue of this paper. Second, we have clarified the limiting conditions of random-bond and random stress AF domain pinning, whose crossover properties have still to be explored. added in proof Rec.ently, we became aware of a paper by M. Lederman, J. Hammann, and R. Orbach [Physica B 1654166, 179 (1990)] dealing with "Net spontaneous magnetization in the dilute Ising antiferromagnet The authors presume a deformation perFe46Zn54F2. pendicular to the c axis to appear below T, due to magnetoelastic coupling, which has the same T dependence as the staggered magnetization. In contrast with the extrinsic stress-induced mechanism described above, however, they claim that the intrinsic magnetoelastic deformation might break the equivalence of the AF sublattices and thus give rise to a net magnetization.

8 t = 1.5 T.

$-0.

¹te

"

Thanks are due to V. Jaccarino for the high-quality crystals, to C. Binek for some of the SQUID measurements, and to U. Nowak for discussions and technical This work was Deutsche help. supported by SonderforschungsForschungsgemeinschaft through bereich 166.

[Sov. Phys. JETP 11, 786 (1960)]. ~6M. Chirwa et a/. , J. Magn. Magn. Mater.

15-18, 457 (1980). H. Ikeda, J. Phys. C 17, 1221 (1984). T. Fries et al. , J. Phys. Condens. Matter 5, L107 (1993). W. Kleemann et al. , Phys. Rev. B 34, 479 (1986). 2oF. C. Montenegro et al. , Phys. Rev. 8 44, 2255 (1991). M. Schlenker and J. Baruchel, J. Appl. Phys. 49, 1996 (1978); Acta Crystallogr. Sect. A 34, S255 (1978); J. Baruchel et al. , J. Magn. Magn. Matter. 15-18, 1510 (1980). ~~I. E. Dzialoshinskii, Zh. Eksp. Teor. Fiz. 32, 1547 (1957) [Sov. Phys. JETP 5, 1259 (1957)]; T. Moriya, J. Phys. Chem. Solids ii, 73 (1959). 15 min with 1 HNO3+14 H20 at room temperature (see Ref. 21). R. A. Cowley et al. , Z. Phys. B 58, 15 (1984).