Non-linear excitations in Ising-type magnetic chain systems II

Physica B 154 (1989) 254-266 North-Holland, Amsterdam

NON-LINEAR EXCITATIONS IN ISING-TYPE MAGNETIC CHAIN SYSTEMS II FREQUENCY-DEPENDENT

SUSCEPTIBILITY

RELAXATION

STUDIES

ON PURE

AND IMPURE

FERROMAGNETS*

M. ELMASSALAMI** Kamerlingh Received

and L.J. DE JONGH

Onnes Laboratorium

3 October

der Rijksuniversiteit Leiden,

Postbus 9506, 2300 RA Leiden,

The Netherlands

1988

The dynamics of non-linear excitations (kinks) in pure and impure Ising-type ferromagnetic chain systems is studied by means of frequency-dependent (AC) differential susceptibility studies in the frequency range 10Hz to 50Hz. The experiments are explained in terms of a coupling of the AC field with the domain walls (kinks) in the ferromagnetic chain systems. Both single kinks and kink-antikink pairs can be distinguished. The associated kink-lattice and kink-kink relaxation phenomena are shown to be analogous to the well-known spin-lattice and spin-spin relaxation phenomena in paramagnetic systems.

1. Introduction In recent years non-linear physics has become increasingly important [l-9], not only because it is a highly interesting research subject in itself, but also due to its applicability to a variety of seemingly different phenomena. Using the same mathematical tools, it offers a unification in terminology and understanding for seemingly quite different subdisciplines. Above all, it offers new and transparent interpretations for phenomena that were in the past intractable by the familiar methods of linear physics, even when perturbation theory to high orders was applied. In that respect low-dimensional magnetic systems occupy a special position [lo] and as an example of the unification theme, we recall that the nonlinear excitations in Ising-type chains which are called kinks, spin clusters, magnon-bound states, etc. bear the same features as their solitonic counterpart in the less anisotropic systems, the so-called rr walls, Bloch walls, wall pairs, breathers, etc. * This

paper is the second of a series of three, all published in this same issue. ** Present address: Physics Department, University of Khartoum. P.O. Box 321, Khartoum, Sudan.

In this paper we will study the time-dependent effects (relaxation processes) associated with the exchange of energy between the constituents of the total thermodynamical system (the lattice and the magnetic non-linear system), namely between the kinks and the phonon system, as well as within the kink system itself. These effects have generally received little attention, which is rather surprising since in principle all the experimental measuring techniques depend on these time-dependent effects. We shall show theoretically, and provide abundant experimental evidence from frequency-dependent susceptibility studies, that with the kink-kink interactions and the kink-lattice interactions there are associated, respectively, the kink-kink relaxation and kink-lattice relaxation processes. Moreover, the characterization of the temperature dependence of both relaxation processes shall be considered. We shall point out that there are quite illuminating analogies between the kinkkink relaxation and kink-lattice relaxation processes on the one hand, and the well-known conventional spin-spin and spin-lattice processes in paramagnetic substances on the other. We shall also show that the possibility to excite or annihilate the kinks either as single-kinks or in pairs leads to the presence of single-kink-lattice

0921-4526/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

M. ElMassalami and L.J.

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I Non-linear excitations in Ising-type systems II

relaxation and pair-kink-lattice relaxation, respectively. The format of this paper is as follows: In section 2 we shall review the excitation spectra of Ising-type chains and define a temperature of the kink system, T,, in terms of the statistical properties of the kink system. We shall elaborate, in particular, on the cooling mechanism between the elementary constituents of the kink-phonon system [ll] and shall consider how this coupling may give rise to time-dependent effects,. and show further how the latter are being reflected in (or influencing) the magnetic responses of the kink system. The description of the experimental set-up and the materials used will be considered in section 3, while the experimental results are given in section 4 and discussed in section 5. Finally, some conclusions can be found in section 6.

255

model possesses no dynamics and the energy spectrum is given simply by (see fig. 1) [12-171 E, = 25 + 2ng,,p.,H,

(4)

where J is the z-component of the intra-chain coupling, and n is the number of oppositely flipped spins. The elementary excitations are the well-known spin-cluster excitations (SCE) [12, 15-171 and spin-cluster resonances (SCR) (see fig. 1) [14-161. A spin-cluster of size m is the set of m adjacent spins that are flipped opposite to the majority of the spins on the chain [14]. Let n,,,,) denote the total number of these m-sized clusters, and II, the total number of all clusters, i.e. n, = C nlm), and finally let n, denote the total number of oppositely flipped spins. Further, if the chain is open-ended the energy spectrum includes another type of excitation, which corresponds to breaking the magnetic

2. Theory

IT) 2.1. Excitation spectra of the quasi 1 D king-type

14> 13, 12) II)

spin system 2J

It is well known that most of the features of Ising-type cyclic chains can be accounted for by employing a Hamiltonian of the type (see e.g. refs. [12, 131) X=X~+X,+X,+X~,

(1)

where %?=is the Zeeman term, X1 is the Ising part, and the last two perturbations are (i) the transverse exchange Hamiltonian

tttttt

(b)

Xh = c Jr&

* s,,

Jh = (J,, + JJ

volume spincluster multlplet

+ s: * S,7+,);

12 7

and (ii) the transverse

(2)

(co

%a = c J.&q * si+,, + s; * S,,); J, = (J,, - JY,)/2 ,

(3)

where all the terms have their usual meanings. In the Ising limit where Xg = X;, = 0, the resulting

ttttt

ttttujjttttt

(C) Heff I

anisotropy

Heff

_._.__

10)

‘3)

____..

I ttttirJl~itttr J

‘4)

Fig. 1. Field dependence of the energy levels for an n-fold spin cluster in a ferromagnetic Ising chain. H,,, is the total field while H,,, is the exchange field (denoted by the heavy line) due to the interchain forces. Notice the SCE (dashed line) and SCR (arrow) excitations which couple (0) + 13) and (3) + (4), respectively.

256

M. EIMassalami and L.J.

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I Non-linear excitations in king-type

bonds only once. The energy spectrum of these single kinks is then [ll] (see fig. 1) E, = .I + 2ng,, /+H

.

(5)

The excitations described by eqs. (4) and (5) have been called volume (pair-kink) excitations and surface (single-kink) excitations, respectively

[111. It is obvious that in the presence of the X,, perturbation the dynamics will be introduced through the terms St+ . S,:+1, which turn the spinclusters of size m > 2 into magnon-bound states that have a dispersion character. On the other hand, the X, term makes it possible to excite the magnetic transition through spectroscopic means and, moreover, leads to an admixture of evenfolded clusters into the ground state even at T=O [12]. Finally, for quasi-1D Ising-type systems there are finite interchain interactions which. similar to the above, can be decomposed into Ising and non-Ising parts [12, 181. The Ising part can be taken into account through an effective field, while the non-Ising parts would cause the spin deviations (and thus fluctuations) in one chain to be coupled to the deviations on the neighbouring chains. 2.2. Thermodynamical Ising-type chains

properties of

Generally, for obtaining the equilibrium properties of Ising-type chains (e.g. the equilibrium values of quantities like nzc, n, or nlmj, etc.), the method of constant temperatures is used 1191. This implicitly assumes the availability of some thermal reservoir and the existence of some coupling mechanism between the degrees of freedom of the reservoir and the spin system. Thus, energy fluctuations in the spin system are not isolated from the phonon system and vice versa, and so an equilibrium state can be defined for the combined spin-lattice system. As usual, this will be denoted as the external equilibrium, in contrast to the internal equilibrium which involves only the spin system (on the assumption that there is a mechanism of establishing the internal

systems II

equilibrium). Obviously, internal fluctuations within the spin system would involve intra-band excitations (SCR) while external equilibrium is established by inter-band excitations, SCE (pairkink or single-kink excitations). However, at thermal equilibrium, the principle of detailed balance requires a balance between all these transitions [19-211, as can be seen in fig. 2, such that, for example, II, and n,,) retain their required values. In fact, for a given T and H, equilibrium values of n,, it, can be obtained from thermodynamical considerations, while n,,,,) , which is a measure of the thermal population of 2J spin-cluster multiplet, can be obtained by using combinatorial techniques [ 141, and is found to be “Irn) = (n,‘ln,)(l

- n,ln,)“‘-’

(6)

It is well known that in chemically pure Isingtype chains, the density of kinks at T = 0 is governed only by the admixture of even-fold clusters in the ground state, while for T > 0 the kink density will be mainly governed by the thermal excitation of kinks, as can be argued from entropy considerations. Obviously, the presence of impurities will affect the kink density [22]. In particular, the ratio of surface to volume activation (single- and pair-kinks) processes depends on the degree of non-magnetic dopants that create free ends. Below, we will assume that for T > 0 there are a finite (and appreciable) number of kinks in the cyclic chain at thermal equilibrium and that these are distributed according to eq. (6). This is the same as assuming that it is possible to use the temperature functional dependence of nlmj to define a temperature for the kink system. This is equivalent to the use of the Boltzmann distribution in paramagnetic relaxation processes to define a temperature for the spin system [19-211. This definition of the temperature in terms of the Boltzmann distribution is commonly adopted, even when the spin system is not in equilibrium with the lattice, and we shall follow the same path here. At this stage it might be worthwhile to consider the stability condition for the equilibrium state. We know that for all finite temperatures

M. ElMassalami and L.J.

Elm)

m "im)

+3)

90

!lm)

T nlm)

: :

de Jongh

I Non-linear excitations in king-type systems II

257

“loo) : :

: ;

i

El2)

2

n‘I 2)

Eli)

’

“I

.\ \

I> T

ber of sptn-clusters -

EIO)

10)

ground

a

state

b

Fig. 2. (a) shows schematically the thermal population of the 2J spin-cluster multiplet for an king-type chain. Notice that the energy separations are non-equidistant (the perturbations in eq. (1) are non-zero). The heavy line in the insert gives the energy separation at zero external field. (b) shows the transitions involved to or from the energy level E,,, (see text).

and fields, the chain response functions are nonnegative (C, > 0 and x > 0); thus the equilibrium state will be stable, since the chain entropy is non-positive [19]. Furthermore, from the principle of Le Chatelier, we can state that “once the chain system is in stable equilibrium then any spontaneous (inter-band or intra-band) fluctuations which tend to bring the system out of equilibrium must bring about processes which tend to restore the system to equilibrium” [19]. 2.3. The coupling

mechanisms for external and internal equilibrium

establishing

In order to appreciate the need for the existence of a coupling mechanism between the kink system and the lattice, let us consider the following simple and well-known argument. In Isingtype systems two spectroscopic resonances have been observed, SCR in the microwave region [14, 161 and SCE in the infrared region [12, 171. In both cases, energy is absorbed from the exciting electromagnetic field by the spin system, and its temperature will rise accordingly. The presence of a coupling mechanism now follows di-

rectly from the fact that these spectroscopic measurements are possible without saturation. Let us recall that, in an open-ended chain, there are two channels for energy exchange between the lattice and the kink system, namely the pair-kink to lattice and single-kink-lattice channels. Let us denote both the associated relaxation processes by the generic name kinklattice relaxation (KLR) process. Furthermore, let us call the relaxation involved in the surface activation (volume activation) by single-kinklattice relaxation, SKLR (pair-kink-lattice relaxation, PKLR) and denote the relaxation time constant as 7sroR (7rk&. Generally speaking, there is no direct coupling between the spin moments and the lattice system. This is particularly valid for S-state ions. In such a case, the energetic contact between spin and lattice may arise from the fact that thermal vibrations of the atoms around their equilibrium position will modulate the inter-ionic magnetic interactions (Waller mechanism), whether of dipolar or exchange origin. On the other hand, for the non-S state, the coupling between the spins and the lattice may in addition be achieved via

M. ElMassalami and L.J.

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/ Non-linear excitations in Ising-type systems II

the (modulation of the) spin-orbit coupling (van Vleck mechanism). In principle, both the possible energy exchange mechanisms should be considered; however, the van Vleck mechanism will be dominant in the compounds under investigation because of the strong spin-orbit interactions in the employed 3D transition metals. This is commonly accepted, and is confirmed by the observation that at the same temperatures the magnetic ion situated in the same crystal field environment without inter-ionic magnetic bonding shows a fast relaxation rate. It is obvious that the temperature behaviour of the relaxation time constant (the reciprocal of the relaxation rate) should reflect the magnetic energy gap (25 or J), independent of whether we choose the Waller or van Vleck model. Here the magnetic energy gap corresponds to the excitation of a single kink (J), or a kink-pair (2J), as is explained in fig. 1. We shall take the relaxation time constant to be (7)

TPKLR

a exp(2JikB

T,

,

(8)

which is justified since we consider only the direct relaxation process, at temperatures low compared to the magnetic energap gap (T + J/k,). At such low temperatures, the Orbachand Raman-type process can indeed safely be neglected. 2.4. Probing the relaxation processes by AC susceptibility measurements Now we can ask how these relaxation processes will influence the magnetic response of the kink system if the magnetic field is varied. In general, any field variation will perturb the energy level distribution and thus their thermal populations. Therefore, when the field is varied, say from H, to HZ, a new kink density as well as a new kink distribution compatible with the new field value must be attained, and this cannot be realized instantaneously. Rather, intra-band (SCR) transitions as well as inter-band (SCE) transitions (with the simultaneous exchange of

the required energy) have to occur in order to re-establish equilibrium. The relaxation time constant will then be a measure of the time needed to reach this new equilibrium state, and the magnetic response of the kink system will depend on the inequality relations between the following time constants; at, rKLR, and 7KKR, where 6t is the time window of the instrument, and rKLR and rKKR are the kink-lattice and the kink-kink relaxation time constants, respectively. It is worthwhile recalling that a change in the kink density will involve the creation or the annihilation of a kink pair (or a single kink in the open-ended chain), which will cost an energy of 2J (resp. J). However, to change the kink distribution of nk, i.e. n,,), at a constant kink density, a much smaller energy is needed, of the order of the interchain coupling. It is important to notice that, in these differential AC susceptibility experiments, the energy needed to effect the spin flips required to change the nzk,or the magnetization of the system from its initial to its final value, is provided through the interaction with the lattice, and not through the absorption of an external electromagnetic wave. This is to be contrasted with the infrared spectroscopic experiments, where the transitions (SCR and SCE) are actually induced by the externally applied infrared signal. Although in both cases the net result is a change in the population of the levels, the source of energy is of course quite different. These are crucial points, since they explain how the kink-lattice relaxation can be observed with the differential susceptibility technique. Generally there are two practical methods of measuring the spin-lattice relaxation by varying the magnetic field [23]: the step-field method and the induction method. We will discuss here only the induction method, where a small modulating, sinusoidally oscillating magnetic field of the type H(t) = h exp(iwt) (the symbols have their usual meanings) will modulate the distribution of the nlrn), and with that the T,(T,). Assuming the validity of the linear response approximation, the induced magnetization will have the same sinusoidal time dependence as the driving field, except for a phase shift that depends on the

M. ElMassalami and L.J.

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I Non-linear excitations in king-type systems II

259

process involved, i.e. M(t) = M + where m(o) is a complex quantity. The complex susceptibility is then given by relaxation

m(w) exp(iwt), m(o)lh

= x’(o)

-ix”(w).

If one needs to obtain an analytical expression for x’ and x” in terms of o, T and H, one can resort to the model of Casimir and du PrC [24] which was originally introduced to describe the spin-lattice and spin-spin relaxation influences on the susceptibilities of “paramagnetic systems”. The formalism is based on general thermodynamical arguments, the only extra assumptions involved being [24,25] that the energy transfer is proportional to the temperature difference between spin system and lattice, and that the spin system responds linearly to the applied field. Then, the frequency dependence of x’ and x” is described by the Debye relations X’IX, = 1 - F + iV[1+ W’r’] )

XI/X, = 1 - c { Fi - FJ[l + (o . TV)*]}, i

{<&Vi/[1 + (w

c

log

RSKLR

RKKR

(frequency)

(10)

where F = (1 - x,/x,) and ,yt and x, are the isothermal and the adiabatic susceptibility, respectively. These equations describe a single relaxation process with a single relaxation time constant. In the case of a multirelaxation process, described by several mutually independent relaxation channels, the susceptibility can be expressed as a superimposition of separate Debye functions [25], as given by

=

R PKLR

(9)

g//y, = Fwr/[l + W”7’] )

/f/x*

R LROR

* Ti)‘]}

)

(11) (12)

i

where the .Fi. and 7i refer to the ith relaxation process. Trivially, the same holds for the Argand diagrams, which will also reflect the superposition of these relaxation processes. Fig. 3 is a simulation of eqs. (11) and (12)) where a superposition of four relaxation processes is shown, as will be encountered below. In fact, we have used eqs. (11) and (12) extensively to fit the experimental Argand curves, and from them, the values of the oh’s and Fi’s are obtained and associated with the various relaxation processes.

Fig. 3. (a) A simulation of eqs. (11) and (12) in the frequency scan method; superposition of individual Debye curves. The insert is the so-called Argand diagram. Notice that when the relaxation rates are similar, highly asymmetric Debye curves (or a highly flattened Argand diagram) will result. (b) Simulation of eqs. (11) and (12) in the temperature scan method; the SKLR (low-T peak) and the PKLR process (high-T peak).

3. Apparatus and materials 3.1. The AC susceptometer The zero-field xAC measurements have been carried out in the temperature range of 1.2 K to 20 K, with a mutual inductance system [23], in which the absolute value and the phase of the xAC are determined by measuring the suscep-

260

M. EIMassaIami and L.J.

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I Non-linear excitations in king-type systems II

tibility of a known calibrant (Mn-tuton salt). The frequency range available was 10 Hz to 50 kHz and the range of the values of the peak-to-peak amplitude of the oscillating field is approximately 0.2 to 20 Oe. Within experimental error, we have observed no change in the magnitude of x’ and x” for all the samples upon varying the amplitude of the applied oscillating field. This verifies the linear response condition. We have used the two well-known measuring procedures, namely the frequency-scan and the temperature-scan method [23]. In the frequencyscan (temperature-scan) method, the temperature of the sample (frequency of the driving field) is kept constant while the frequency (temperature) is varied, and consequently the results are given in terms of the x’ vs. o and x” vs. w (x’ vs. T and x” vs. T). In most cases the x”(T) peaks (the semicircle) in the temperature- (frequency-) scan method are quite sharp (circular), and the relaxation time constants are then determined by locating the temperature and the angular frequency at which x”(T) attains its maximum value: 7 = w -‘. It was found, experimentally that both methods give the same results, which should be expected, since it may be seen from eqs. (11) and (12) that o and T appear as products, and to the same power. So the equivalency of the relaxation times as obtained from the two scans follows if the temperature dependence of the Fi’s are weak, which is the case in our experiments. Finally, the ,yAc data presented below have not been corrected for demagnetization effects or diamagnetic contributions, since both effects are negligible or at most quite small, and do not change any of the conclusions drawn from this study. 3.2. Materials The magnetically one-dimensional (1D) isostructural series MCl,Py, (M = Mn, Fe, Co, Ni) is one of the best known examples of how the 1D magnetic properties are correlated to the structural characteristic of a compound. The crystal structure [26] consists of bundles of stacked chains, where each chain consists of dichlorine-

bridged M 2+ ions in a orthorhombically distorted octahedral coordination. The bulky pyridine ligands, which are situated above and below the plane formed by the M 2+ ions and the bridging chlorine ions, keep the chains far apart, leading to the good 1D magnetic character of these systems. The nature of the low-temperature magnetic behaviour of these compounds [27] is determined by two important characteristics, namely the above mentioned 1D character and the strong magnetic anisotropy arising from the crystal field splittings (due to the distorted trans-MCl, configuration). For T ~20 K, the magnetic properties of the Fe2’ ions can be approximated by an effective spin S = 1 Hamiltonian with Isingtype exchange interactions. Furthermore, it has been shown that for all 3d metal ions (M”) considered, the anisotropy axis has the same orientation in the Cl and Py octahedron, namely along the N-M-N axis [27]. This special property means that if any one of the magnetic compounds is doped with any other of the three magnetic ions, the resulting mixed system has no competing anisotropy. Furthermore, this same axis is actually the main axis of the z-component of the EFG tensor. These findings have been checked and are supported by the Mossbauer spectroscopic results on the above-mentioned series [28].

4. Results 4.1. Introduction The zero-field AC susceptibility studies have been carried out on powder samples of the above-mentioned series and their dopants. The most characteristic features, common to all compounds, are the following. In all of the doped samples, and in particular those with non-magnetic impurities, we have observed a lowering of T,, as compared to the pure samples. This is in fact reassuring, since it indicates that the distribution of the impurities along the chain is indeed random. Below we shall denote the critical relaxation process as the long

M. ElMassalami and L.J.

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I Non-linear excitations in king-type systems II

range order relaxation (LROR) process (in some cases the associated LROR relaxation process is evident as a very small x”(T) peak). We have noticed that in most of the doped samples, the x” which is associated with critical phenomena extends over a considerable temperature range, approximately 2 K, whereas in the pure sample the peak in x” is much sharper. This relatively large temperature region can be associated with the distribution in length of the unperturbed chain segments in the doped materials. In many cases we have been able to discern very clearly three types of relaxation processes in the frequency and temperature range covered, and the following simple critieria have been used for their identification: a relaxation is identified as a long range order relaxation (LROR) process when the position of the x”(T) peak is not sensitive to the variation of the frequency and, moreover, the peak occurs at the same temperature of which dx’/dT attains its maximum. The other two relaxation processes were found to have strong temperature and frequency dependence, from which we can identify a relaxation process as a pair-kink-lattice relaxation (or a single-kink-lattice relaxation) process when the associated relaxation rate corresponds to a thermally activated process with an activation energy 25 (J). The theoretical arguments that support these simple criteria have already been given. It was found that the presence or absence of the different types of relaxation processes are strongly connected to the presence, type, and amount of the impurities: the purer the sample, the lower the intensity of the relaxation processes. In particular, it was observed that magnetic impurities enhance the intensity of the

PKLR process, while the non-magnetic impurities appear to enhance the intensities of both the SKLR and PKLR processes. When comparing the x’(T) curve of a doped sample with the corresponding curve of the pure sample, both measured under the same experimental conditions, one notices that for T < T, the x’(T) of the doped sample is always larger than that of the pure one. The limited frequency window (10 Hz-50 kHz) for the AC susceptibility has prohibited studies on the kink-kink relaxation (KKR) process with this technique. The KKR process has relaxation times of the order of lo-* s, as has been shown by means of 57Fe Mossbauer spectroscopy relaxation studies [27, 28, 301, for which the frequency window is of the order of 106-lo9 Hz. 4.2. Selected examples For all the samples studied, with the exception of the X:MnCl,Py, (X = dopant) system, typical and characteristic relaxation effects have been observed in the probed o and T region (see table I). Below, because of space limitations, we shall consider only the results obtained for the system X:CoCl,Py,. Three samples were studied: CoCP, the nominally pure sample; (Cd)CoCP, the nonmagnetically doped sample, and (57Fe)CoCP, the magnetically doped sample. Starting with the pure CoCl,Py,, we show in fig. 4 the curves found for the xAC for some representative frequencies. Remarkably enough, when the x”(T) plot is seen on an enlarged scale (see insert of fig. 4), a characteristic structure is noticed which corresponds to the typical peaks associated with the above-discussed three relaxation processes. Although the intensities of these

Table I Experimentally determined T, and intrachain parameters, Miissbauer relaxation studies Sample

COCP (Cd)CoCP (Fe)CoCP

Doping content

T, (K)

0.06 0.03

3.15 2 0.05 2.8 2 0.1 2.8 2 0.2

261

as determined

Activation energy PKLR (K)

SKLR (K)

21.5 rc 1.0 21.5 f 1.0 21.5 2 0.5

10.6 k 1.0 10.6 5 1.0 10.6 + 1.0

from xAc and Reported J/k, (K) 11 [16-B]

M. ElMassalami and L.J.

262 ‘5

.

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I Non-linear excitaticns in king-type

,

systems II

106.

,

co Cl2 Py* G

4-

82.86

0

_.&

? 2 -%

3. 2.

Co CL2 Py, series

2.x

\\

w/2rr

1

!“*‘k;

Hz

\

f

’. 2.651 kHz

:

IO5

i *:

cl X1-T

i’ ! t

I f

4” .

K

t,=3.17

. ..dJ”

I

c b X”-T

0

T(K)

01

0

L...L

1.0

1

p,

3.0 2.0 temp. (K)

4.0

5.0

Fig. 4. Susceptibility of CoCl,Py, as a function of temperature, shown for selected frequencies. (a) x’(w) vs. T, and (b) x”(o) vs. T. The insert is an enlargement of (b).

J-

%4/

0.4

0.6 inverse

relaxation processes in the pure COCP sample are quite low, and for some frequencies fall within experimental error, we have been able to determine the relaxation rates from the ~“(7’) peaks for a few cases. The resulting values are plotted in fig. 5 together with those from the doped samples. Upon doping CoCl,Py, with non-magnetic impurities, we obtained most examples of the relaxation effects discussed above. The two different kink-lattice relaxation processes are clearly evident, because their relaxation rates are far enough apart. In fig. 6, some representative results for x’(T) and x”(T) are plotted. The two x”(T) peaks at lower temperatures have a very strong frequency and temperature dependence, and the two obtained relaxation rates are plotted versus the inverse temperature in fig. 5, where one observes very clearly the exponential temperature behaviour of the relaxation processes over a wide range. According to the abovementioned criteria, we have identified them as PKLR and SKLR processes. From fig. 6, in particular the insert, it is seen

temp.

0.8 (K-l)

1.0

Fig. 5. The experimentally determined relaxation rates vs. the inverse temperature for the series M,:Co,_,Cl,Py, (M, = Co,, Cd,,,, Fe,,,).

that the intensities of the relaxation processes (4,,, and 4KLR ) vary monotonically with T and w. It will be shown below that the increase of the intensities of the relaxation processes can be accounted for in terms of the corresponding increase in the isothermal susceptibilities. Finally, for the sake of comparing the xAC results obtained on CoCP, (Cd)CoCP, and (Fe)CoCP, we have given in table II the absolute value of the x’ and ,$I at T= 1.2 K and for ~127~ = 82.86 Hz, which shows clearly the dramatic influence of the non-magnetic doping on the susceptibilities. For the magnetic impurities we have chosen the “Fe isotope, in order to enable the relaxation effects to be studied in addition by means of Mossbauer effect measurements. The xAc relaxation studies on this sample reflect the presence of the three typical relaxation processes, however less dramatically than for the (Cd)CoCP, as can

M. ElMassalami and L.J.

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I Non-linear excitations in king-type systems II

Fe, CO(,-~& 4.

+

b,

. 2.651 kHz

82.86 HZ

. 2.651 kHz

0 E

i s

PYZ

wl2Tc

4,

c 82.86

263

. 3.75

kHz

4

2.

t

0

-x

Tc,(2.8*0.2)

K

XI-T

_ 0.2Or L

I

temp.

(K)

Fig. 7. Some representative curves of (a) x’ vs. T and (b) x” vs. T for the magnetically doped sample Fe,,O,:Co,,,,Cl,Py,. temp.

(K)

Fig. 6. Some representative plots of (a) ,$ vs. T and (b) ,$’ T, for several frequencies, for the compound Cd o.oa:Co,,,,Cl*PY,’

Fe,

VS.

be seen from fig. 7. The LROR process is seen around T, = (2.8 2 0.2) K. For T < T,, a strong dominant peak is evident which, from the temperature and the frequency dependence of the x” peak, has been associated with the PKLR process. It is tempting to assume that the asymmetry on the low-temperature side in this peak arises from an additional, but less intense, SKLR process. In fact, a clearer separation of these two kink-lattice relaxation processes is provided by the Argand diagram, as measured at T = 1.22 K and shown in fig. 8. Here the solid curves repreTable II Comparison of the susceptibility values, as seen for the series M,Co,_,Cl,Py, (M = Co, Fe, Cd), at w/2n=82.&Hz and T=1.2K Susceptibility X value

CoCP + 0.06

(Cd)CoCP -c 0.06

(Fe)CoCP f 0.06

x’ (emu/mole) x” (emu/mole)

0.13 0.006

0.95 0.146

0.33 0.09

CO(,-x)CL2 PY2

T= (1.22?0.05)

h

0’

0

’

’

0.2

D X’(W)/x;-log

w

0 X”(w)Ix~-log

w

1”0.4 X’(w)

Fig. 8. Shows the Argand diagrams Fe,.,,: Co,,,,Cl,Pyz quencies in (b) are

K

’

’

0.6

’

’

0.8

’

1

1.0

I XL.

fitted Debye curves (a), while the fitted are given in (b), for the compound at T = 1.22 K. The indicated angular frein kHz.

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I Non-linear excitations in lsing-type systems II

sent the calculated behaviour, on the basis of the two mutually independent Debye-type relaxation processes of eqs. (11) and (12). The average relaxation rates that are obtained for the PKLR and the SKLR processes are depicted versus T ’ in fig. 5. Anticipating the discussion of the Mossbauer data, we note here that a quite satisfactory agreement between the present ,yAc relaxation results and the Mossbauer relaxation results is found. Furthermore, it is worth mentioning that the intensity of the x” has a strong temperature and frequency dependence. Finally, we have given in table II the values of the susceptibility components at T = 1.2 K and 0.~12~= 82.86 Hz, from which we can conclude that the magnetic doping also increases the magnitude of the susceptibility, although the effect is less pronounced than with the non-magnetic dopant.

5. Discussion Let us first discuss the observation that in “chemically-pure” Ising systems, the x”(T) is found to be (almost) zero. It is clear that in pure 1D antiferromagnetic systems (with pure Ising interactions) a kink-structure is not expected to occur in zero field for T < T,, because the kinks are driven out of the system. Obviously, the absence of kinks means that each chain would be completely ordered. This can also be visualized as a complete depopulation of the spin-cluster energy levels. Therefore, applying a small perturbing field (of the order of 1 Oe) will not be able to modulate the magnetization of such a system; the system first has to be excited from the ground state in order to have some population of the energy levels. Then, through modulating nlm) and due to the presence of the relaxation processes which will cause a phase shift, the magnetization would not follow the perturbing field and hence x” woud not be zero. On the other hand, the presence of a kink structure in the impure system would mean that either the energy levels of the spin-cluster multiplets are thermally populated, or that the kinks have been pinned at (or near) the impurity sites. For pure cyclic chains at low T, the populated

levels comprise the 23 multiplet, while for the segmented chain, in addition the J multiplet is also populated. Therefore by applying a perturbing field, one would be able to modulate the spacing as well as the thermal population of these levels: hence a relaxation process can be observed. Furthermore, for T -=cT,, the presence of the kink structure means that the sublattice magnetization is not fully saturated (the individual chains are not fully ordered) and therefore, a contribution to the susceptibility from the xi, should be expected, in addition to the x1, and consequently the isothermal powder susceptibility for an impure sample should be larger than that for the pure system, in particular when gll + g,. Let us assume, for the sake of simplicity, that there are only two relaxation processes involved, namely the PKLR and the KKR. We shall take the height of the x”(T) peak as a measure of the intensity of the associated relaxation process. Furthermore let us assume that the frequency and the temperature dependence of these susceptibility components is given by the Debye relations as in eqs. (9) and (10). Then at w -’ = these equations will reduce to ‘PKLR) x’(w, T) = f(x, +x,)

9 x”(w,

T) = %xt -x,>

3

(13) which shows clearly that the temperature behaviour of the height of the peak of x”(w, T) would follow the temperature behaviour of the isothermal susceptibility (at such low temperatures, the adiabatic susceptibility has no temperature dependence). It is an experimental fact that the isothermal susceptibility of these materials decreases monotonically with T for T < T,. Therefore one would expect the peak in x” to decrease monotonically with T. This is seen experimentally in fig. 6. Furthermore, eq. (13) shows that if the doping influences the isothermal or the adiabatic susceptibility, then it will also influence the height of the x”(T) peak. Since we have just argued that for T < T,, a doped sample should have a larger isothermal susceptibility than the pure one (see table II), we

M. ElMassalami and L.J.

de Jongh

I Non-linear excitations in king-type systems II

expect that doping will increase the height of the x” peak (compare fig. 4 for CoCP with fig. 6 for CdCoCP) .

In addition, we can argue that the enhancement of the intensity of the relaxation process is actually coupled to an increase of the kink density in the sample. The increase of the kink density with T( > T,) can be easily understood from entropy considerations. On the other hand, as mentioned before, the introduction of an impurity into the chain is also a means of increasing the kink density. In the non-magnetic impurity case, in addition to the introduction of a kink structure, the impurity also enhances the surface activation process [ 11, 221. Contrastingly, since the magnetic impurity is still coupled magnetically to the neighbouring host spins, it should not be expected to introduce surface activation, but to serve mainly as a pinning centre. It is worthwhile to mention that the presence of these relaxations is not confined to the ordered phase of the quasi 1D Ising-type system. Rather, field dependent xAc relaxation studies on similar systems have already revealed their presence for T > T, [29, 311 and, moreover, also at or near metamagnetic phase transitions [31, 321. Finally we would like to add that similar relaxation behaviour has been noticed in higher dimensional magnetic systems [32].

6. Conclusions We have shown, starting from the well-known Ising character of the quasi 1D Ising-type systems, how the relaxation effects can be explained. We have discussed the origin of the exponential activation character which is reflected in the temperature dependence of the relaxation rates. The presence of this Arrhenius-type activation behaviour should not be surprising, since the magnetic energy gaps in the energy spectrum of the spin system have an energy quanta of J or 2J, depending on the boundary conditions. Moreover, theoretical studies on the excitation spectra and the low-T thermodynamics of Ising-Heisenberg chains [13] have shown that the zero-field behaviour of the iso-

265

thermal susceptibility is governed by the existence of an effective gap. Furthermore, we have shown how and under what conditions the AC susceptibility techniques can probe these relaxation processes. In particular, the effect of the boundary conditions on the observability, the presence, type, and the intensity of the relaxation processes have been also considered. Combining the results of the xAc experiments with the Mossbauer effect data published in the following paper [28], we have been able to study the temperature dependence of the kink relaxation rates over the frequency range of 10 to lo9 Hz.

Acknowledgements We are indebted to J. Reedijk of the University of Leiden for providing all the samples for this study. We are also indebted to A.J. van Duyneveldt and J. Verstelle for allowing us the use of the AC susceptometer. The stimulating and sometimes challenging discussions with R.C. Thiel, H.J.M. de Groot, H.H.A. Smit, T.A.M. Haemers, and M.W. Dirken of the Mossbauer Group of the Kamerlingh Onnes Laboratory are greatly acknowledged. M.W. Dirken has also participated enthusiastically in the measuring program. This work is part of the research program of the Foundation for the Fundamental Research on Matter (FOM) and was made possible by financial support from the Netherlands Organization for the Advancement of Pure Research (ZWO). M.E. would like to thank the Sudanese Government for financing the Ph.D. scholarship.

References [l] A.R. Bishop, J.A. Krumhansel and S.E. Trullinger, Physica D 1 (1980) 1. [2] L.J. de Jongh, J. Appl. Phys. 53 (1982) 8018 and reference therein. [3] H. Yoshizawa, K. Hirakawa, S. Satija and G. Shirane, Phys. Rev. B 23 (1981) 2298. [4] H.J. Mikeska, J. Appl. Phys. 52 (1981) 1950. [S] J. Villain, Physica B 79 (1975) 1; in: Ordering in

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I Non-linear excitations in king-type

Strongly Fluctuating Condensed Matter Systems, Riste, ed. (Plenum, New York, 1980) p. 81. [6] J.F. Currie, J.A. Krumhansel, A.R. Bishop and S.E. Trullinger, Phys. Rev. B 22 (1980) 477 and references cited therein. [7] J.A. Krumhansel and J.R. Schrieffer, Phys. Rev. B 11 (1975) 3535. [8] K. Maki, Phys. Rev. B 24 (1981) 335. [9] N. Ishimura and H. Shiba, Prog. Theor. Phys. 63 (1980) 743. [lo] (a) L.J. de Jongh and A.R. Miedema, Adv. Phys. 23 (1974) 1; (b) M. Steiner, J. Villain and C.G. Windosr, Adv. Phys. 25 (1976) 87. [ll] M. Tinkham, Phys. Rev. 188 (1969) 967. [12] (a) J.B. Torrance Jr. and M. Tinkham, Phys. Rev. 187 (1969) 587; Phys. Rev. 187 (1969) 595; (b) D.F. Nicoh and M. Tinkham, Phys. Rev. B 9 (1974) 3120. [13] J.D. Johnson and J.C. Bonner, Phys. Rev. B 22 (1980) 251. [14] M. Date and M. Motokawa, Phys. Rev. Lett. 16 (1966) 1111; J. Phys. Sot. Jpn. 24 (1968) 41. [15] G.O. Berim, M.M. Zapriov and A.R. Kessel, Zh. Eksp. Teor. Fiz. 66 (1974) 734; Sov. Phys. JEPT 39 (1974) 355. [16] W.J.M. de Jonge, C.H.W. Swuste and K. Kopinga, in: Physics in l-d, J. Bernasconi and T. Schneider, eds. (Springer-Verlag, Berlin, 1981) (Springer Series in Solid State Sciences) p. 169 and references therein. [17] L.A. Bosch, G.J.P.M. Lauwers, K. Kopinga and W.J. de Jonge, J. Magn. Magn. Mat. 54-57 (1986) 1161. (181 A. Pines and D. Hones, J. Phys. Sot. Jpn. 44 (1978) 43; (b) J.W. Stout and R.C. Chisholm, J. Chem. Phys. 36 (1962) 979. [19] SK. Ma, Statistical Mechanics (World Scientific, 1985) pp. 297-307; (b) L.E. Reich], A Modem Cause in Statistical Physics (Edward Arnold Ltd., 1980) pp. 4554; (c) B.M. McCoy and T.T. Wu, The Two-Dimensional Ising Model (Harvard Univ. Press, Cambridge, Mass., 1973) pp. 31-49. [20] J.C. Verstelle and D.A. Curtis, Handbuch der Physik (1968); (b) A. Morrish, The Physical Principle of Magnetism (John Wiley & Sons, Inc., 1965) p. 144; (c) C.P.

systems II

Slichter, Principle of Magnetic Resonances (Harper & Row, 1963) p. 115. [21] (a) R.L. Carlin and A.J. van Duyneveldt, Magnetic Properties of Transition Metal Compounds; (b) C.J. Gorter, in: Paramagnetic Relaxation (Elsevier, Amsterdam, 1974). [22] T. de Neef, J. Phys. Sot. Jpn. 37 (1974) 71. [23] (a) A.J. de Vries and J.W.M. Livius, Appl. Sci. Res. 17 (1967) 31; (b) J. Soeteman, PhD. Thesis, Leiden (1974); (c) H.A. Groenendijk, PhD. Thesis, Leiden (1981); (d) H.A. Groenendijk and A.J. van Duyneveldt, Physica B 115 (1982) 41; (e) Physica B 115 (1982) 63. [24] H.B.G. Casimir and F.K. du Pre, Physica 5 (1938) 507. [25] (a) A.J. van Duyneveldt, J. Soeteman, A. van der Biit and L.J. de Jongh, Colloques Intern. C.N.R.S. N 242 Phys. Sous Champs Magn. Int., Grenoble (1974); (b) A.J. van Duyneveldt, J. Soeteman and L.J. de Jongh; J. Phys. Chem. Solids 6 (1975) 481; (c) J. van den Broek, PhD. Thesis, Leiden (1960) (in Dutch). 1261 K. Takeda, S. Matuskawa and T. Haseda, J. Phys. Sot. Jpn. 44 (1978) 43. [27] M. ElMassalami, PhD. Thesis, Leiden (1987). [28] M. ElMassalami, H.H.A. Smit, R.C. Thiel and L.J. de Jongh, Paper III, Physica B 154 (1989) 267 (this issue). [29] L.J. de Jongh, C.A Mulder, R.A. Cornelisse and A.J. van Duyneveldt and P. Renard, Phys. Rev. Lett. 23 (1981) 1672. [30] (a) H.J.M. de Groot, L.J. de Jongh, M. ElMassalami, H.H.A. Smit and R.C. Thiel, Hyp. Int. 27 (1986) 93;. (b) R.C. Thiel, H. de Graaf and L.J. de Jongh, Phys. Rev. B 47 (1981) 1414; (c) H.J.M. de Groot, L.J. de Jongh, R.C. Thiel and J. Reedijk, Phys. Rev. B 30 (1984) 4041; (d) H.H.A. Smit, H.J.M. de Groot, L.J. de Jongh, R.C. Thiel, C.E. Johnson and M.F. Thomas. Solid State Commun. 53 (1985) 573. [31] J.C. Scott, C.B. Zimm and A. Sarhangi, J. Appl. Phys. 53 (1982) 2768; (b) T.W. Hijmans, A.J. van Duyneveldt and L.J. de Jongh, Physica B 125 (1984) 21. [32] (a) W.L.C. Rutten, PhD. Thesis, Leiden (1985); (b) A. van der Bilt and A.J. van Duyneveldt, Physica B 95 (1978) 305.