MÖSSBAUER STUDY ON THE CRITICAL DYNAMICS OF THE TWO-DIMENSIONAL ANTIFERROMAGNET

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MÖSSBAUER STUDY ON THE CRITICAL DYNAMICS OF THE TWO-DIMENSIONAL

ANTIFERROMAGNET

A. Ito, M. Horiike

To cite this version:

A. Ito, M. Horiike. MÖSSBAUER STUDY ON THE CRITICAL DYNAMICS OF THE TWO-

DIMENSIONAL ANTIFERROMAGNET. Journal de Physique Colloques, 1979, 40 (C2), pp.C2-290-

C2-296. �10.1051/jphyscol:19792102�. �jpa-00218471�

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JOURNAL DE PHYSIQUE Cblloque C2, suppI6ment au n ^O3, Tome 40, mars 1979, page C2-290

MOSSBAUER STUDY ON THE

CRITICAL DYNAMICS

OF THE

TWO-DIMENSIONAL ANTIFERROMAGNET

A. Ito and M. Horiike

Department of Physics, Faculty of Science, Ochanomizu University, Otsuka-machi, Bunkyo-ku, Tokyo, Japan.

Rhsum6.- On utilise ici la spectrom6trie Mgssbauer pour htudi~r le comportement magndtique de compo- sds du type Ising 2D antiferromagnstique RbzCoFc: 5 7 ~ e au voisinage de la temp6rature de Ndel. Le spectre de relaxation observd est analysh sur la base du modsle stochastique de M. Blume et J.A.Tjon.

Les phdnomznes de relaxation sont ainsi expliquds par le comportement des spins li6s 1 la dynamique de "clusters" antiferromagnhtiques r6sultant de correlation bidimensionnelles des spins.

Abstract.- Mgssbauer spectroscopy is applied to investigate the magnetic behavior near the NEel temperature of a typical two-dimensional Ising antiferromagnet Rb2CoF~:57~e. The relaxation spectra observed in the temperature region near T were analyzed using the formulation given by M. Blume and J.A. Tjon on the basis of a stochastic moiel. The results show that the relaxation phenomena are explained as a result of the behavior of spins tied to the dynamics of antiferromagnetic clusters created by the two-dimensional spin correlations.

1. Introduction.- A great deal of interest has re- cently been taken in magnetic behavior of the low- dimensional magnets. Much work has been done using various methods. Of those, neutron diffraction stu- dies on the low-dimensional magnets have shown that one-dimensional or two-dimensional spin correlations exist in a wide temperature range above their TN(TC), although most of the low-dimensional magnets have been known to show three-dimensional long-range order at low temperatures. The experimental results obtained so far have been mainly concerned wibh the spatial correlations of spins, and only qualitative observations have been reported on the dynamics of the critical fluctuations

.

The Mossbauer spectroscopy of 5 7 ~ e has been applied to investigate the magnetic properties of one- and two-dimensional magnets, and many results have been reported. However, relaxation spectra near TN have been observed only in the limited compounds, and details of the temperature dependence of the spin correlation time near the cirtical point have not been obtained, so far, con- trary to expectation : there is a possibility to get informations on the spin correlation time of order I O - ~ Q J ~ O - ' ~ ~ near the critical point. Wertheim et al.

studied a two-dimensional antiferromagnet Rb2FeFs (TN%55 K) and observed the line broadening within a range of a few degrees around TN /l/. They gave two alternative interpretations. According to one, the line broadening was caused by a distribution of TN over 5 K. According to the other, qualitative interpretation, the spin correlation time of the short- range order is increased with decreasing temperature

toward TN. B.D. Rumbold et al. observed the spectra showing the magnetic splitting up to%90 K in a one- dimensional antiferromagnet FeOHSOt, (TN=56 K) 121.

They interpreted the phenomena qualitatively as the relaxation effect associated with the one-dimensional spin correlation time. H. Keller et al. observed the line broadening near TN in a two-dimensional antiferromagnet (CH3NH3)2FeCli, (TN=94 .46 K) 1 3 1 , ^and J.L. Schurter et al. in the two-dimensional antiferro- magnets (MA) 2FeClr. (TN=95 .5 K) and (BA) 2FeClr

(TN=75.6 K) 141. They have not discussed the observed phenomena in connection with the low-dimensionality.

V. Petrouleas et al. and E. Gurewitz et al. indepen- dently observed the line broadening near TN in a one-dimensional antiferromagnet KFeC13 (TN= 16%18 K)

.

V. Petrouleas et al. explained the phenomena on the assumption that the relaxation among the three lowest spin-orbit levels of Fe2+ is slowed down by the appea- rance of the exchange field /5/.They considered that the line broadening observed above TN indicated the presence of the exchange field originating in the short-range order present in the ~arama~netic region.

On the other hand, E. Gurewitz et al. interpreted the phenomena as a result of the fluctuation of spins near TN 161. The idea is that each chain consists of domains of parallel spins arising from the finite correlation length, the direction of the spins in adjacent domains is antiparallel and these domains move along the chain, inducing a random spin flip. Both Petrouleas' and Gurewitz' groups used the equations as formulated by M. Blume and J.A. Tjon 171 and fitted the calculated line shapes fairly

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19792102

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well to their observed spectra, each group assuming a different relaxation mechanism, as described above.

The interpretation that the relaxation phenomena observed near T are to be attributed to the dynami-

N

cal properties of spins caused by the movement of the magnetic clusters of the short-range order along the chains is attractive. However, the mechanism proposed by V. Petrouleas et al. has been known to be a cause of the relaxation phenomena observed in certain ferrous compounds 181. Both the mechanisms taken into account by the two groups should be discussed carefully with respect to the compound KFeC13.

In this article we shall report our observa- tion of the relaxation spectra in the vicinity of TN=101.96 K in a typical two-dimensional Ising antiferromagnet Rb2CoF4: 5 7 ~ e . The electronic relaxation of ~ e ions in RbzCoFs is considered to be fast and ~ + the relaxation phenomena are possibly attributed to the spin dynamics of the short-range order.

2. Experimenta1s.- Single crystals of RbzCoFs:

0.2at%"Fe and Rb2coF1,: latzs7~e were grown by the flux method. The crystals obtained have excellent (001) cleavage planes. In our ~Gssba~ler measurements cleaved single crystals were used. y-rays were de- tected in the direction parallel to the tetragonal c-axis which is perpendicular to the cleavage plane.

The MGssbauer spectra were taken with a con- ventional constant-acceleration spectrometer. The velocity was calibrated with an Fe absorber. The line width of 0.24 mm/s was obtained for the two inner

lines of a 25pm Fe-foil. The Massbauer measurements were carried out at temperatures between 4.2 K and room temperature. Especially in the vicinity of the N6el temperature, the temperature variation of the spectra was observed in detail at small intervals of 0.1U.2 K. Temperature was kept stable throughout

the experiment to less than 0.05 K. The temperature gradient across the absorber was estimated to be less than 0.05 K.

3. Electronic properties of pe2+ in Rb2CoFs

.-

^The

compound RbnCoFb has the K2NiFs structure. Each successive Cop2 layer is separated by two layers of RbF and, in addition, the magnetic interactions between adjacent layers of CoF2 vanish from considerations of sywmetry. In such a situation, two- dimensionality obtains. Furthermore the strongly anisotropic character of the magnetic exchange between CO'+ ions makes the compound Rb2CoFk a good example of the two-dimensional Ising model. Doped

~ e " ions substitute for CO'+ ions. An Fe2+ ion is surrounded by a slightly distorted octahedron of

six F- ions. The syumetry around Fe2+ ions is tetragonal, the distance of the two F- ions along the tetragonal c-axis being shorter than those along the other directions. In this tetragonal field, the ground state of Fe2+ ions is an orbital singlet with a positive eZqq. This is clearly seen in Figure 1 where the Mgssbauer spectra observed at room temperature and 4.2 K with incident y-rays parallel to the c-axis are shown.

- 2 - 1 0 1 2 3 4 5 6 7 8

VELOCITY (mm,%)

Fig. I : Miissbauer spectra of single crystal Rb2CoF4:

~at%'~Fe. The direction of y-rays is parallel to the c-axis, These are the characteristic spectra : (a) in a paramagnetic region,(b) in an antiferromagnetic region.

These spectra are well defined and are characteristic of a and an antiferromagnetic region, respectively, except in the critical region near TN. The intensity ratio of the quadrupole lines is about 1:3, and the higher velocity line corresponds to the transitions of (312, '3/2b(1/2, +1/2)

(Fig.l(a)), and hence the sign of e2qq is known to be positive. This fact reveals that the orbital singlet is a ground state, in accord with the expectation from symmetry considerations. The spectrum shown in figure I (b) consists of four lines, indica- ting the absence of the lines corresponding to the transitions with Am=O. This fact shows that Fe2+

spins are in the direction of the c-axis which is the easy axis of co2+ spins, and that the direction of the magnetic hyperfine field Hhf is parallel to the major axis of the EEG tensor. In tnis situation a sub-level mixing of the excited state of "Fe in R ~ ~ C O F ; is not caused by the magnetic hyperfine interaction; this makes it easy to assign spectra and to treat critical phenomena theoretically. The temperature variation of e2qq/2 (Fig. 2(a)) indicates that the upper orbital doublet with negative e 2 q ~ is mixed into the ground orbital singlet to a conside-

rable ektent by the spin-orbit interaction ha.

Using the Hamiltonian

1 1

X = vcubic+ +(L:

-

7 L(L + +))l h 3 (1) we solved the secular equation for a 25x25 matrix in

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C2-292 JOURNAL DE PHYSIQUE

order to estimate the tetragonal splitting A which least in the vicinity of TN as will be mentioned could explain the temperature variation of e2qQ/2. later, the small value of B=0.145 is considered to Assuming that 10Dq=6000 cm-' and X=-90 cm-', the reflect the two-dimensionality of the compound.

value of

A

was estimated to be -400 cm-'

.

The temperature variation of the M6ssbauer spec-

L 1

OO 100 2 0 0 3 0 0

TEMPERATURE ( K )

Fig. 2 : Temperature variations of (a) (I /2)e2qQ and (b) Hhf which is obtained from the apparent splitting of the two absorption peaks corresponding to the transitions (312. f3/2)+(1/2, +1/2).

It is known from the wave function of each level that the level mixing by the spin-orbit interaction is considerable and the electronic relaxation between the spin-orbit levels is expected to be fast. In the region where an exchange field exists, we added a term for the exchange interaction in the molecular field approximation, to the equation (1).

The electronic relaxation is expected to be fast also in this situation. Therefore, the values of e 2 q ~ and Hhf at a given temperature are calculated as the thermal average over the electronic states.

The temperature variations of e2qQ/2 and

sf

^shown

in figure 2 are explained on the basis of the fast relaxations.

4. Mgssbauer study in the vicinity of TN.- The N6el temperature was determined by fitting the data to the power law Hhf=(I-TIT ) B with f3=0.145 and

N

TN=101.96 K in the range 0.002<(1-T/TN)<O.l. The values of Hhf were obtained from the apparent splitting of the two absorption peaks corresponding to the transitions (312, f3/2)-+(1/2, f 112). If one assumes that Hhf is proportional to a sublattice magnetization, the value of f3=0.145 should be compa- red with B=0.119 I91 and k0.123 1101 obtained by the neutron diffraction study and with 8=0.125 theoretically predicted for the 2D Ising system with spin 112. Although the assumption is violated at

trum observed for RbzCoFk: lat%"~e near TN is shown in figure 3.

- 1 0 1 2 3 4 5 6 - I 0 1 2 3 4 5 6 7

V E L O C I T Y ( m m / s )

Fig. 3 : Temperature variation of Mhsbauer spectra in the vicinity of TN=101.96 K.

The asymmetric line broadening becomes visible at 106.5 K and becomes remarkable as the temperature decreases. The reason why the line broadening of the higher velocity line is more pronounced than that of the lower velocity line is easily understood by the fact that the higher velocity line corresponds to the transitions (312, +3/2)+(1/2, f 112) and the lower one to the transitions (312, +1/2)+(1/2, ;1/2).

It should be noted that the higher velocity line is already split into two broad lines at 102.00 K just above TN. These facts reveal the presence of hyperfine interactions due to the short-range order in a paramagnetic region. As temperature is decreased below TN, the line width is decreased rapidly and the magnetic splitting becomes well resolved. The full width at half maximum absorption of the spectrum was about 0.3 mm/s in the temperature range between 91 K and 77 K and was 0.27 mm/s at 4.2 K. No diffe- rence in the temperature variation of the spectra was observed between specimens Rb2CoF~: lat%"Fe and Rb2CoF~:0.2at%57~e. Therefore, the Fe is sufficiently 'dilute for substituted E'e2+ spins to reflect the magnetic behavior of the pure system RbzCoFk.

It is clear from careful observations of the temperature variation of bhe spectra (Fig. 3) that the line broadening observed near TN is not due to a distribution of T in the sample. For reference,

N

we show in figure 4 the temperature variation of the spectral shape as obtained by calculations on the assumption that the NCel temperature has a spread

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with a Gaussian distribution exp(-(TN-<T >)2/2a2) N with a=O.l K and a=1.0 K.

Fig. 4 : Temperature variation of Mgssbauer spectral line shapes near <T > calculated on the assumption that the NEel temperature has a spread with a Gaus- N sian distribution ~ X ~ ( - ( T ~ - < T ~ > ) ~ / ~ U ~ ) . (a) U=0.1 K. (b) U=1.0 K. on15 four transitions, (312, +3/2)+(1/2, '112) and (312, +1/2)+(1/2, +1/2), are taken into account for the comparison with the spectra observed for the single crystal in our experiment

.

The relaxation spectra observed in the vicinity of TN cannot be understood on the basis of the properties of the electronic state of I?e2+ ions such as in the case of KFeC13, since the electronic relaxation is expected to be fast in the present case as we have mentioned already. We interpreted the observed phenomena as follows : antiferromagnetic clusters are possibly created in a CoF2 layer by the two-dimensional spin correlations in a paramagnetic region near TN. The situation is shown schematically in figure 5(a). Up-spins and down-spins are indicated by + and

-,

respectively. A boundary of clusters is found between spins of the same sign. The different clusters adjoin at the expense of antiferromagnetic exchange energies and the boundary of clusters is expected to move about in the layer. There-

fore, a spin at a given lattice point changes its direction up and down as a random function of time.

(a)

Fig. 5 : Schematic antiferromagnetic clusters presu- med to exist in (a) a paramagnetic region near TN, (b) an antiferromagnetic region just below T Up- spins and down-spins are indicated by + and respectively.

The magnetic hyperfine field Hif at a nucleus fluctuates according to the direction of the associated spins. At temperatures not very near TN, an average cluster size and a value of exchange field are small.

In consequence, a small hyperfine field at a nucleus fluctuates rapidly. As the temperature is lowered and approaches TN, the average cluster size and the value of exchange field grow, and the hyperfine field is increased and its fluctuations are slowed down. At the Ndel temperature the long-range order sets in. The situation below TN is shown shhemati- cally in figure 5(b). Below TN one kind of cluster, say, the cluster I, grows rapidly and the cluster 11 becomes a minor one. With decreasing temperature the proportion of the minor clusters will be decreased.

A nucleus at a given lattice point with the up-spin (down-spin) is immersed in the hyperfine field

+

H;lf (-Hif) and it feels

-Xf

(+HAf) only in a small time- interval, during which a minor cluster passes through the lattice point. The model mentioned above is considered to be a good approximation in an Ising system such as Rb2CoF*.

Keeping the model mentioned above in mind, we used the stochastic treatment of the relaxation phenomena formulated by H. Blume and J.A. Tjon 171 in order to estimate the spin correlation time. The present case is the one that a magnetic field fluctuates along the maximum principal axis of the EFG tensor. The cluster size will be in reality statis- tically distributed at any temperature, but we neglect this distribution. We assume for simplicity that a magnetic field jumps at random between the two values +Hif and -Hif along the maximum principal axis of the EFG tensor. Hif depends on temperature through the thermal average over the electronic

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c2-294 JOURNAL DE PHYSIQUE states depending on the value of the exchange field.

Since each of the static hyperfine fields +Hif and -HLf gives the same Mgssbauer spectrum, hereafter we shall not distinguish in a static limit between the lattice point with up-spins and that with down-spins.

Under these assumptions, the Hamiltonian for the nucleus is

=

a,

+ ~ ( 3 1 ~ - I ( I + l ) ) + ~ ~ ; l ~ ( ~ ) I ~ f ( t ) , (2) where f(t) is a random function of time, the value of which is either +l or -I. The transition rate of f(t) from the value of +l(-1) to that of -l(+\) is specified by the quantity W(w). The line shape of absorbers is calculated using the following equation,

x.?pi(j

1

(p-Wia3-I

1

i), 1J

where

p=-i(w-W -~(3m:-15/4))+ 7 1 ra, a=(gomo-gl m1)IJNHhf(T),

T=(;

&+=(-W W

W -W 1 and

We shall not give here any further details of the notations used (see Ref. / 7 / ) . According to the geo- metry in our experiment, only four lines corresponding to the transitions (312, f3/2)-+(1/2, "12) and (3/2, fl/2)+(l/2,

-

*1/2) were taken into account. In order to fit the calculated line shape to the ~ s s s - bauer spectra, the following integration was carried out,

I(V)=/~W(W)W~(W,V)~W. (4)

W (w,v) is a single-line source spectrum at the Doppler velocity of v;

rs

I

Ws(~*v)=

m

(w-Wo-(v/c)a)2 + (rs/2)2

.

We fitted the calculated line shape to that of the observed spectra by adjusting the parameters HAf(T), W(T) and w(T). We took W=w in the paramagnetic re- gion. Below TN the result of W<w was automatically obtained. We used the values rs=0.12 m / s and

ra

= 0.15 mm/s. The best fit to the observed spectra was achieved with the parameters listed in table I. The calculated line shapes are shown in figure 6 toge- ther with the observed spectra. The agreement between them is excellent. The range of acceptable values for the best fit was very narrow for the spectra at the temperatures except for 102.5 K and 99.0 K. The spectrum at 102.5 K could be practically explained with a set of parameters between (Hhf=85 kOe, W=2.5 mm/s) and (H;lf=65 kOe, W=1.4 m/s). For the

spectra observed at higher temperatures, the range of acceptable values of (HAf, W) was extended further.

I ' ' ' ' ' i ' ' ~ l

d

I I I I I I

- 1 0 1 2 3 4 5 6 7

V E L O C I T Y ( m m h )

Fig. 6 : Fits between the observed spectra (dots) and the lineshapes calculated with the best-fit values listed in table I (solid lines).

Table I

As a general tendency, Hif became small and W became large with increasing temperature. For the spectra observed at temperatures below 99 K, the value of Hif was coincident with that of Hhf which was obtained by the apparent splitting of the two absorption peaks corresponding to the transitions (3/2, '3/2)+

(1/2, f 1/21, b& the values of W and W could not be determined definitely because the line broadening was rather small.

The characteristic times T" and 'rw are listed

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in table 11, where T =I/v and Tw=1/vW (W=hv

W W W' (table 111) exists at the nucleus and it fluctuates w=hvW).

Table I1

up and down in an average time of order 4x10-'s.

Table 111

T (K) Hhf(kOel Hif W e ) 102. So unresolved 8 0 102. Z5 unresolved 85

102. O0 6 8 93

101.75 9 9 104

101.Oo 123 125.

100.Oo 137 137.5

99.00 144 144.

The temperature variations of TW and Tw are plotted in figure 7.

TEMPERATURE (K)

Fig. 7 : Temperature variation of the characteristic times .rW and T associated with the two-dimensional spin correlations. W

TW or T~ is a measure of the period during which the value +Hif or -Hif is kept. That is, T and T give

W W

the average times that a given spin is found in a cluster I and a cluster 11, respectively. In the region T>TN, T ~ = T ~ . At 102.5 K, only 0.5 K above

0

TN, a rather large hyperfine field of 80 kOe

As the temperature is lowered toward TN, the value of Hif is increased and the fluctuation of H'

hf is slowed down. The line broadening is striking at 102.00 K where the characteristic time T~ (=T,) is of the same order of the Larmor precession time T

L Of

nuclear moments of S 7 ~ e : ~ =T = 1.2x10-~s and T =

w w L

1.4x10-'s which corresponds to Hhf=93 kOe. These features suggest that the cluster size grows with decreasing temperature, because the dynamics of large clusters is expected to be slow. As the temperature is lowered below TN, TW is increases rapidly as seen in figure 7, where T > T ~ . This suggests

W

that a region of major clusters is expanded rapidly in accord with the growth of the long-range order.

On the other hand, Tw is decreased rather slowly and is nearly constant between 101.0 K and 99.0 K. This may suggest that the size of minor clusters does not change considerably down to 99 K which is the lowest temperature where rW and rw can be estimated, though the major clusters gain on the minor ones and accor- dingly the number of minor clusters is reduced rapidly with decreasing temperature.

In conclusion, the relaxation phenomena observed in RblCoFs: 5 7 ~ e near and above TN have been success- fully explained as the result of the behavior of spins tied to the dynamics of antiferromagnetic clusters. The temperature dependence of 'cW and -rw has suggested that the cluster size grows as the temperature approaches TN from above, and that below T

N one kind of clusters grows rapidly and gains on the other (minor cluster), but the size of minor cluster does not change considerably at least down to 99 K.

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C2-296 JOURNAL DE PHYSIQUE

In a low-dimensional magnet the critical phenomena References are known to be observed over a wider temperature

range than in a three-dimensional system. Eowever, the relaxation phenomena near TN have been observed by the Msssbauer spectroscopy only in a limited number of low-dimensional compounds. The compound RbsCoFb is an almost ideal two-dimensional Ising antiferromagnet; this is shown by the value 0.119 of the critical index B obtained by neutron diffraction / g / . The relaxation spectra are observed in a wide temperature range around T probably because of

N

this nearly ideal property of RbzCoFs. It therefore seems that interesting, although qualitative results concerning the cluster dynamics in the critical region can be obtained from these experiments.

Acknowledgments.- We would like to express our thanks to Professor Natsuki Hashitsume for valuable suggestions and his warm encouragements throughout this work. We are grateful to Professor Fumiaki Shibata for stimulating and invaluable discussions.

We are indebted to Professor Hironobu Ikeda for many helpful discussions.

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