Static magnetization and critical behaviour of EuxSr1−xS single crystals

Journal of Magnetism and Magnetic Materials 45 (1984) 157-174 North-Holland, Amsterdam

157

STATIC MAGNETIZATION AND CRITICAL BEHAVIOUR OF EuxSr l_ xS SINGLE CRYSTALS U. KOBLER, K. FISCHER, W. ZINN Institut fur Festk~rperforschun~ KFA Jhfich, D-5170 Jfdich, Fed. Rep. Germany

and H. PINK Forschungslaboratorium Siemens A G, D-8000 Mhnchen, Fed. Rep. Germany

The critical behaviour of the diluted Heisenberg ferromagnetic system EuxSrl_xS has been investigated with static magnetization measurements. The well-known effects associated with magnetic dilution are also confirmed for this system: 1) increase of the effective critical exponents fl, "t and 8, 2) increase of the critical range, 3) invariance of the Widom relation 3' = fl(8 - 1 ) . Most surprising is a linear decrease of the Schofield critical temperature b* -1 = ((/$ _ 1)(1 - 2 f l ) / ( 8 - 3)) 1/2 with dilution. This quantity obeys, approximately, the simple empirical relation b * - 1 = T~(x)/O(x).

1. Introduction

In the last few years the diluted ferromagnetic system EuxSra_xS has been the subject of numerous experimental [1] and theoretical [2] investigations. EuxSr~_xS is considered to be a model system for the diluted Heisenberg ferromagnet because of its many favourable properties such as composition independent pure spin moments provided by the Eu2 +-ions and well known exchange interactions for the compact material EuS [3] which decrease only slightly on dilution [4]. A strong competition between ferromagnetic nearest and antiferromagnetic next-nearest neighbour interaction shifts the critical concentration for a long range ferromagnetic order to a value as high as x c --0.57 and separates this ordered phase from the paramagnetic one by an intermediate spin glass range at low temperatures. The nature of the spin glass state is still a matter of discussion [5] as well as the proper experimental methods by which its boundaries may safely be established [6]. In particular, it is questionable whether the spin glass constitutes a real thermodynamic phase for finite temperatures [7]. Various experimental techniques have been applied for the investigation of the composition-

driven transition from spin glass to ferromagnet. In particular, elastic and inelastic neutron scattering [6,8,9], ac-susceptibility measurements [8,10], specific heat [11,12] and dc-magnetization measurements [13]. Very recently the magnetic Bragg line profiles have been carefully analyzed in order to localize the transition into the ordered state by a diverging spin correlation length [6,14]. These experiments indicate that a long range ferromagnetic order does not rise at a sharply defined composition at low temperatures but is rather a gradual process. In such a situation the numerical result for the critical concentration depends strongly on the experimental care and the adopted method in the data analysis. Here we investigate the static magnetic properties of EuxSrl_xS single crystals in more detail and present a rough analysis of the critical behaviour along To(x). Critical exponents have already been measured on sintered EuxSra_xS material by Siratori et al. [15]. Since somewhat different results are obtained here using selected single crystalline samples, we report on this part of our investigation at length. In mixed magnetic systems an analysis of the critical behaviour is much impeded by the presence of inevitable compositional imperfections in

0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

158

u. KObler et al. / Critical behaviour of EuxSrI xS single crystals

the samples. Among these stoichiometric defects, precipitations and deviations from a statistical distribution of diamagnetic and paramagnetic ions are the most serious ones. We have therefore started by checking our samples with high-temperature susceptibility measurements against the presence of such deficiencies and selected single crystal pieces for a detailed investigation for which these are a minimum. A stringent requirement, which must be fulfilled by all samples, is that the paramagnetic ordering temperature O(x) is a smooth and a very nearly linear function of composition. Stronger deviations of O ( x ) from linearity or even a discontinuous behaviour of O ( x ) are indicative for chemical short-range order [16]. On the other hand, this method of characterization is limited to a relative accuracy in the paramagnetic ordering temperature of + 0.2 K and this is not sensitive enough to exclude the influence of residual sample imperfections on the numerical values of the critical parameters. A further quality criterion seems to be that x ( T - T~) must obey a real power law in the critical temperature range. Although this cannot be postulated a priori, the occasionally observed phenomenon that a log-log plot of x ( T T~) vs. T - T~ did not result in a straight line for any choice of T~ does point strongly to some deficiency structure for these samples. We have therefore selected such single crystals for which X ( t ) - t -~ is fulfilled throughout the whole critical temperature range. It should be noted that if X ( t ) deviates from a power law it is always such to give increasing effective "/values for decreasing reduced temperatures. This imposes a misleading trend towards too large "r exponents. Using the above mentioned criteria in selecting appropriate samples it is observed that critical scaling holds well for all magnetic compositions, but the obtained critical parameters still strongly depend on the individual sample. This shows that the observation of the expected scaling behaviour does not provide a sufficient criterion for the quality of the sample. It is therefore difficult to identify a clear quantitative composition dependence of the critical parameters. The critical behaviour of many diluted ferromagnetic systems has already been investigated and we refer to an article by Westerholt and

Sobotta as a thorough compilation of existing experimental data [17]. All these systems show increasing effective critical exponents with diamagnetic dilution. In the EuxSrl_xS system investigated here this effect is also observed but it starts to be pronounced only for sufficiently strong dilutions with x _< 0.7. As we shall see below, the critical amplitude of X does, in addition, increase with dilution and both features together result in an exceedingly strong divergence of the susceptibility on approaching the critical concentration x c between ferromagnet and spin glass. This makes an accurate evaluation of Tc(x ), and hence of all critical parameters, very difficult in practice but also the spontaneous magnetization m s cannot be evaluated with high precision for x---x c. It is therefore necessary to evaluate T~(x) by a fitting procedure along the usual lines of assuming critical power laws for susceptibility X, spontaneous magnetization m s, and critical isotherm Bc ( T = Tc ). The critical parameters are then evaluated by the requirement that the scaled magnetization is a unique function of the scaled magnetic field for both T > ~ a n d T < T c. Due to the limited temperature and field range (T>~ 1.8 K, B 0 >/10 -3 T) of our Faraday magnetometer the spin glass range could only be investigated in part. This range is characterized by practically infinite susceptibility values near the ferromagnetic to spin glass boundary xc. Only for x < 0.5 the susceptibility is small enough to assume a measurable order of magnitude. The experimental limits for the largest measurable susceptibility are of the order -- 10 3 (SI-units!) if only relative changes with field or temperature are considered. The evaluation of absolute susceptibility values is limited mainly by the asphericities of the samples if "t is very large. Assuming that the effective demagnetization factor is uncertain by a few percent, the largest measurable absolute susceptibility is of the order ---10 2. This source of error makes the low-temperature susceptibility measurements particularly difficult in the composition range 0.5 ~
159

U. Kabler et aL / Critical behaviour o f E u x S r 1 _ x S single crystals

x = 0.3. Near this composition we expect the interesting change back to a diverging susceptibility for T---, 0. This means that below temperatures of about 0.5 K, larger susceptibilities may occur for smaller magnetic concentrations. Another problem associated with large, but finite, susceptibilities in the spin-glass range is the fact that a state with a finite internal field is frozen-in w~th the field-cooled magnetization curves as long as X < oo. Such a state may differ from that one prepared with zero-field cooling. In fact, from apparent discrepancies between our resuits and true zero-field spin-correlation measurements [6] we conclude that fields of B 0 = 10 - 3 T may be too large to sample the intrinsic zero-field spin glass state, in particular in the composition range 0.5 < x < 0.6.

2. Chemical short range order

EuxSrt_xS single crystals are grown from the melt by directional solidification in closed tungsten crucibles using temperatures up to 2700 °C. The observed melting temperatures of EuS and SrS are 2500 and 2600 o C, respectively. The starting material for crystal growth is EuxSrl_xS powder prepared as outlined in ref. [18]. All EuxSrl_xS single crystal pieces used for magnetization measurements have first been characterized by high-temperature susceptibility measurements, x(T) is analyzed by the method outlined in ref. [4] which results in 0 = 21.8 K for EuS. The paramagnetic ordering temperature 0 is a most revealing quantity for mixed magnetic systems since it gives the sum of all magnetic interactions. This sum may be too small as well as too large due to short range order effects [16]. This depends on whether the Eu ions have within their first and second coordination shells more Sr or Eu neighbours than that which corresponds to the ideal random distribution. Unfortunately the growth of single crystals (s.c.) from the melt occasionally resulted in bulk material with O~.c.values considerably larger as compared to the powder samples. Since the starting EuxSrl_xS powder has been prepared by chemical methods [18] a random distribution of Eu and Sr ions can better be as-

sumed for the powder. This is also suggested by susceptibility measurements on the powder samples reported in ref. [4] which revealed a relatively smooth curve for 6powder (x). In the presence of chemical short-range order, O(x) should be strongly non-linear and may even assume a non-monotonous behaviour [16]. It is interesting to note that only 6s.c. >/6powder is observed and that the slope of the Curie-Weiss line always reveals the correct average number of Eu2+-ions irrespective of the actual 0s.c.-Value. Together both features are indicative of chemical short-range order with relatively more Eu-ions within the first and second coordination shells (unmixing) then corresponds to the statistical average. Since the scatter of the 0s.c. values due to short-range order effects is much larger than for the powder samples, we use the a v e r a g e 0powder values as standard and select single crystal pieces with 0s.c. = Opowd= for further magnetization measurements. Fig. 1 shows the agreement between 0s.c. and

25

I

I

I

20 - 8

r

I

I

r

I

~ ° °°

o

"

15

/1 I

E~xSr,.~s


,powder sornptes osingle crystols

5

I

SrS

I

0.2

I

I

I

0.4

I

0.6

I

I

0.8

I

EuS

X ~

Fig. 1. Paramagnetic ordering temperature 0 divided by composition x of EuxSrl_xS powder samples (full circles) and single crystals (open circles). The 0-values of the chemically prepared powder samples s e r v e a s r e f e r e n c e in selecting single crystal pieces which are free from chemical short range order. The small quadratic x dependence of 0 is due to decreasing magnetic interactions with the expansion of the EuS lattice by incorporation of Sr.

160

U. K~bler et al. / Critical behaviour o f E u x S r l _ ~ S single crystals

0powder for those bulk samples which have been selected for the magnetization measurements. In the presentation O / x vs. x of fig. 1, a constant value of 21.8 K is expected if the magnetic interaction parameters would be independent of x. This does not hold strictly. The lattice expansion between EuS with a 0 = 0.597 nm and SrS with a 0 = 0.602 nm gives rise to a relatively strong decrease of the exchange parameters with dilution. Therefore 8 decreases faster than linear according to 8 = 21.8x - 7.3x(1 - x ) ,

(1)

t

as was discussed in ref. [4] previously. Below x = 0.2 an even stronger decrease of 8 / x on dilution is revealed by the data of fig. 1. However, one should realize that these deviations from eq. (1) are still small and compare with the experimental error of + 0.2 K.

3. Static susceptibility

0

I

2

3

4

5

6 T~

All susceptibility data x ( T , x ) have been evaluated from magnetization measurements on spherical single crystals. For a sphere the relevant internal field is given by B i = B o - D J with D = ½ as the demagnetization factor. The numerical values of the susceptibility are given here in SI-units and hence are larger by a factor of 4~r as compared to cgs-units. Furthermore, we prefer the dimensionless volume susceptibility X = J / B i , which we divide by the magnetic concentration x. Using this normalization the slope of the Curie-Weiss line is independent of x and changes in x / x with x are only due to changing magnetic interactions. Fig. 2 shows a set of magnetization curves measured in an external field of B 0 = 0.01 T as a function of decreasing temperature. The ordinate gives the dimensionless ratio of the magnetic polarization J and the external field B 0. J is defined as the density of the magnetic moments given by J = r n / V with m the measured magnetic m o m e n t of the sample and V its volume. The volume of the sample has been evaluated from its weight using the known X-ray density. For the spherical samples used here, a value of J / B o = 3 marks an infinite susceptibility viz. a state with a vanishing internal field. For samples with x > 0.54

7

8

9

I0

11

12

[K]

Fig. 2. Magnetic polarization divided by external field B 0 of EuxSr 1_x S single crystal spheres measured for decreasing temperatures at B0 = 0.01 T. For x > 0.54 a constant value of J / B o = 3 is reached at low temperatures, indicative for a diverging susceptibility. J / B o rests gradually below a value of 3 for x < 0.54 which is typical for the spin-glass range. The paramagnetic Curie susceptibility (x = 0) must again reach J / B = 3 for T---, 0.

the magnetization assumes a value of J = 3B 0 at low temperatures indicative for a state with a long range magnetic order with a resulting spontaneous magnetic moment. However, for x < 0.54 the magnetization still reaches a plateau at low temperatures but the plateau values fall very gradually below J = 3B 0. This quantitative change in behaviour sets in extremely slowly and it is therefore impossible to accurately define the composition where this feature begins. For x = 0.4 we could not observe whether the magnetization will ultimately reach a real plateau, since much lower temperatures would be required to observe this, but a trend to a saturation behaviour at the lowest temperature can still be seen. A change from a concave temperature dependence of the magnetization

161

U. KObler et aL / Critical behaviour o f E u x S r 1 _ x S single crystals

at very small x-values to a convex one at intermediate concentrations is the most prominent difference in the curves of fig. 2. Again it is difficult to derive an accurate composition value from the present experimental data where this qualitative change occurs but we suggest that it is near x --- 0.3. For x > 0.3 we have evaluated the approximate inflection points of the corresponding x ( T ) curves and use these as the characteristic temperature for a beginning magnetic saturation at low temperatures. The dash-dotted line in fig. 2 gives a connection of these points and thus, marks, an approximate boundary for the spin glass range SG. It must, however, be emphasized that we have magnetization values of J < 3B0 in the spin glass range and therefore a situation with a finite internal field. Besides the problem of whether or not the inflection points of the x ( T ) curves give an unequivocal definition for the spin glass boundary, our results for the spin glass range in fig. 2 may generally differ from those obtained for a vanishing magnetic field. In fact, a plateau-like magnetization value for 0.50 ~_-0.005 T. The conclusion to be drawn from these measurements is that the characteristic temperature at which magnetic saturation sets in does not depend much on the applied field in the range 0.005 ~
j

S/Bo

X Bo - D J - 1 - D J / B o"

Fig. 3 shows the results given as x/x vs. temperature. For high temperatures all these curves reach the Curie-Weiss straight line asymptotically, which has a slope independent of x, since we have normalized the volume susceptibility by x. These Curie-Weiss asymptotes intersect the temperature axis at the higher temperatures the higher x, viz. the magnetic interaction, is. For x > 0.54 the x/xcurves touch the abscissa indicating a diverging susceptibility and hence a magnetically ordered state. However, for x < 0.54 very large susceptibilities occur at low temperatures but no real divergence is observed for T>~ 1.8 K. This behaviour must eventually change since for the very lowest magnetic concentration a Curie-type susceptibility is to be expected with a diverging susceptibility for

0.7

0.6 :0.48 0.5 :=0.50 O.t,, : :0.52 X x

0.3

::0.54

0.2

: =0.57

I =0.60 0.1

0

1

2

3

k T -~,'--

5

6

7

8 [K]

Fig. 3. Reciprocal susceptibility divided by magnetic composition x vs. temperature for EuxSr 1 _x S samples. The high-temperature asymptotes of these curves (Curie-Weiss lines) are parallel to the calculated Curie-line for x = 0. A diverging susceptibility is seen as x / x ~ 0 for samples with x > 0.54. For samples with x < 0.54 very large b u t finite susceptibilities are observed down to the lowest temperature of 1.8 K. A tentative boundary of the spin glass range is indicated by dash-dotted line.

162

u. Kabler et aL / Critical behaviour of EuxSr1_ xS single crystals

T - ~ 0. This is i n d i c a t e d b y the line x = 0 which r e p r e s e n t s the c a l c u l a t e d Curie law for n o n i n t e r acting Eu 2÷ ions. A tentative e x t r a p o l a t i o n of the x / x curves to T - ~ 0 reveals t h a t the susceptibility decreases f r o m a n infinite value at the b o u n d a r y b e t w e e n spin glass a n d f e r r o m a g n e t to a finite value of a p p r o x i m a t e l y X / X -- 25 n e a r to x -- 0.3 a n d returns then b a c k to an infinite value for smaller m a g n e t i c c o n c e n t r a t i o n . It is evident f r o m fig. 3 that t e m p e r a t u r e s d o w n to 0.5 K w o u l d b e necessary to verify this interesting x ( T , x ) beh a v i o u r a r o u n d x = 0.3. Fig. 4 shows the reciprocal susceptibility as a f u n c t i o n of c o m p o s i t i o n . A typical feature of these curves is that the susceptibility diverges stronger the lower the m a g n e t i c c o n c e n t r a t i o n , viz. Tc(x), is. U n f o r t u n a t e l y , the available x-values are n o t close e n o u g h to resolve the divergence of X / X

1.6

14

accurately. The c o m p o s i t i o n s at which x / x = 0 have therefore been e v a l u a t e d f r o m m o r e accurate m e a s u r e m e n t s of x ( T ) for c o n s t a n t x. A s we shall see later the stronger divergence of X on going d o w n the Curie-line Tc(x ) is due to b o t h increased critical e x p o n e n t s a n d increased critical a m p l i tudes. The results of fig. 4 restrict the critical c o n c e n t r a t i o n for a long range f e r r o m a g n e t i c o r d e r to the range 0.5 < x < 0.6, in a c c o r d a n c e with earlier results [8]. T h e different divergence b e h a v i o u r of X for c o m p a c t EuS a n d for the diluted samples can be seen qualitatively in fig. 5. W h i l e a critical e x p o n e n t of y = 1.33 for EuS p r o d u c e s a sharp k i n k at T = Tc in the p l o t x / x vs. T, the s a m e curves for the diluted s a m p l e s are r o u n d e d i n d i c a t i n g a s t r o n g e r divergence of X / X . However, the influence of c o n s i d e r a b l e p r e p a r a t i v e effects can also be seen in fig. 5 since the x / x curves are n o t shifted parallel in strict a c c o r d a n c e to their n o m i nal c o m p o s i t i o n s n o r is a c o n t i n u o u s l y increasing r o u n d i n g of the curves very n e a r to the abscissa noticeable.

°°51//#/

1.2

x=0.57 0.04

t

1.0

x ×

J

.59 0 63 065 " 0.68

70

0.03

08

0.6

1.0

0.02

04 0.01

EuxSr1_×S

02

4 SrS

02

0.4

0.6

0.8

EuS

S

6

7

8

i

i~

I

I

I

9 T~

10

16

17

18 [K]

X ~

Fig. 4. Reciprocal susceptibility divided by magnetic composition x vs. x. X diverges successively stronger for lower critical temperatures viz. compositions due to increasing critical exponents and critical amplitudes.

Fig. 5. Reciprocal susceptibility divided by magnetic composition for EuxSr 1_xS samples from the ferromagnetic range. The different divergence behaviour of X is noted by the different rounding of the curves in the range X/X > 100.

163

U. K~bler et aL / Critical behaviour o f E u x S r 1 _ x S single crystals

4. Critical behaviour

I

i

p

~

r

w

EuS

As can be seen from fig. 5 an accurate evaluation of the Tc(x) values for the diluted samples is complicated by the stronger divergence of the susceptibility with magnetic dilution. The experimental critical exponent "y = 1.33 (theoretical value: 1.38 + 0.01) for EuS is low enough so that the Curie temperature can directly be noticed from a plot of the reciprocal susceptibility vs. temperature (fig. 5). For the diluted samples, however, the susceptibility assumes very large values appreciably before divergence and T~(x) must be evaluated by a fitting procedure. We proceed therefore in the usual way by applying critical power laws to X, B~ and m s and evaluate Tc through the condition that the scaled magnetic field is a unique function of the scaled magnetization for the two regions T < T~ and T > T~. Fig. 6 compares the experimental magnetization data near Tc for one EuS and one Eu0.65Sr0.35S single crystalline sphere. All magnetization curves are normalized to a maximum value of one in the magnetically ordered state by dividing the magnetic polarization J through 3B 0. In this representation any non-proportionality of the curves can directly be seen. Note that the curves drawn through the points have no theoretical significance and that the reduced temperatures on the abscissa are not to scale for both samples. There are two apparent differences between both sets of magnetization curves. While the critical range of EuS is restricted to a span of approximately 3% in t = ( T ~ - T ) / T ~ , it seems to spread over nearly 10% in the case of Eu0.65Sr0.35S. Secondly, the critical exponent "¢ is considerably higher for the diluted sample compared with compact EuS. These observations are in keeping with results on the very similar system EuxSr I _xSo.sSeo.5 and seem to be a common feature of diluted magnetic systems [17]. Aharony and Stauffer [19] studied the width of the critical range in diluted magnetic systems and concluded that the width of the critical range should increase with decreasing T~(x). This prediction is in qualitative agreement with the result of fig. 6 and the estimation given in ref. [19] for the increase of the critical range for Tc(x ) ~ 0 is in

TE:

16.37K

10

t

09

0.8

I

-008

I

-0.06

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-0.02

0

002

OOL,

0.06

008

t i

O l 09

i

i

i

i

- ~o~o

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T~=66~K

37

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o 07

-03

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-012

I

-011

0 t

011

012

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Fig. 6. Normalized magnetization curves in the critical temperature range for one EuS and one Eu0.65Sr0.35S single crystalline sphere. Division of J by 3B0 shows the non-proportionality of these curves clearly. For the diluted sample the width of the critical range is larger as compared to EuS and the susceptibility (B 0 ---, 0) is considerably higher at the same value of the reduced temperature t.

reasonable quantitative accord with our observations. The second point concerning increased effective critical exponents in random magnetic systems is now a quite common experimental observation [17]. Theoretical approaches to this problem gave, however, conflicting results, but recent investigations arrive at the conclusion that the critical behaviour may change in random magnetic systems [20]. Strong changes occur, however, only for x = x c and this is a theoretically very difficult composition range. The most recent theory is due to Sobotta [21] who predicts limiting values for the exponents y ~ 2, fl ~ 0 . 5 and ~ ~ 5 for x - - , x c.

164

u. Kfbler et al. / Critical behaviour of E u x S r 1 x S single crystals _

These values are, however, subject to m a t h e m a t i c a l a p p r o x i m a t i o n s m a d e in the r e n o r m a h z a t i o n g r o u p t r e a t m e n t of this p r o b l e m b u t our results c o n f i r m fl ---, 0.5 nicely while we c a n n o t rule out that 8 a n d y d o increase even s t r o n g e r for x --* x c. T h e result of a scaling analysis of m a g n e t i z a t i o n curves, as in fig. 6, is best visualized b y a scaling plot, as is shown in fig. 7, for a s a m p l e with x = 0.65. Such an analysis has a l r e a d y b e e n pres e n t e d for EuS [22] a n d is therefore n o t shown here. If scaling b e h a v i o u r holds, all m a g n e t i z a t i o n d a t a for T > Tc fall on one single curve ( l e f t - h a n d e d b r a n c h ) a s s o c i a t e d with relatively smaller m a g n e t i -

I

I

z a t i o n values, whereas the m a g n e t i z a t i o n d a t a for T < Tc are relatively larger a n d fall on the righth a n d e d branch. F o r T---* Tc b o t h b r a n c h e s reach a s y m p t o t i c a l l y the critical i s o t h e r m e B~ = d ( J / J o ) 8 which has the right slope 8 in the scaling plot. The s p o n t a n e o u s m a g n e t i z a t i o n d a t a b e l o n g to a state with B i = 0 a n d are on the vertical a s y m p t o t e of the T < T~ b r a n c h while the susceptibility d a t a fall on the l e f t - h a n d side a s y m p t o t e of the T > Tc b r a n c h which has a slope of u n i t y in the chosen scaling r e p r e s e n t a t i o n . T h e scaling plot therefore gives o n l y a selection of the e x p e r i m e n t a l m a g n e t i z a t i o n

I

I ÷

Euo.65Sro.35S

I

T =6.64K

÷

:050

i

"6 =2.01 5 =6.37

0

• • •

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•

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a

* 0.06 K

• 0.06K

• 0.11 K

• 0.09 K

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a ÷ a

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• 0.26K

• 0.26 K

• 036 K

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_

• 0.36K d'

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• 056 K

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I

0 [T/K °5]

Fig. 7. Scaled representation of magnetization data from fig. 6 showing one branch for T < Tc and one for T > T~. The critical power laws for susceptibility, critical isotherm and spontaneous magnetization appear here as the asymptotes of the two branches having the slopes 1, 8 and ~ , respectively•

u. K6bler et al. / Critical behaviour of EuxSr1_ x s single crystals

data in the critical range since it contains the critical isotherm, the spontaneous magnetization, and the susceptibility data only asymptotically. As will be shown below, the values of the critical exponents fl and 8 for the x = 0.65 sample are considerably outside of the expected average value at this composition. The scaling relation 3" = fl(8 - 1) does however hold rather well. Inserting 3'=2.01, f l = 0 . 5 0 and 8 = 4 . 3 7 results in 3 " / f l ( 3 - 1 ) = 1.19, which is rather satisfactory in view of an error of a few percent for each exponent. The scaling relation does hold even better for most of the other investigated samples even though the numerical values of all exponents are strongly influenced by a fluctuating quality of the samples. This shows that the scaling relation must necessarily be fulfilled but cannot be taken as a sufficient criterion for the quality of the samples. A further point of interest is the composition dependence of the critical amplitudes. As will be shown below, the critical amplitude for the spontaneous magnetization goes to zero at the ferromagnetic to spin-glass boundary and provides, therefore, a criterion for the critical concentration x¢ of a long-range ferromagnetic order. The critical exponents can be related to the critical amplitudes under the conditon that the linear magnetization model proposed by Schofield holds [23]. We have therefore analyzed our magnetization data following the lines of Schofield, Litster and H o [24]. The linear model of Schofield involves a coordinate transformation which is given in parametric representation by n i =

ar¢~O(1 - 02),

t = r(1 - b'202),

J/Jo = ram ( 0 )

(1)

with re(O) as the new magnetization and 0 as the new state parameter. In this representation the susceptibility X / x = gt -v corresponds to 0 = 0, the spontaneous magnetization rns/mo=bt/~, characterized by B i = 0, corresponds to 0 = 1 and the critical isotherm B~ = d( J/Jo)8 corresponds to 0 = 1 / b * with b* defined by b .2 = (3 - 3 ) / ( 3 - 1)(1 - 2fl).

(2)

If the linear model holds we can write re(O) = kO and the following relation between critical ampli-

165

tudes and critical exponents results db(8-1)g

b*(8-3)Jo(x = 1)

(b .2

-

(3)

1) v-1

This relation can easily be verified by an elimination of the constants k and a from equations (1) using the critical power laws for X / X , Bc and ms/mo. A result for a data transformation according to eq. (1) is given in fig. 8 for a sample with x = 0.59. Although the corresponding plot for EuS has a better quality, the hnear relation r e ( O ) = k 8 also seems to be realized rather well for the diluted samples so that we can make use of eq. (3) in comparing critical exponents and amplitudes. Interestingly, the constant k virtually does not change with dilution in contrast to the constant a which decreases strongly with dilution as well as b * - 1. A comprehensive survey of the concentration dependence of both critical amplitudes and effective critical exponents is given in fig. 9. As can be seen, the scatter in the data for the critical exponents is nearly as large as the possible change of the exponents with dilution and therefore only a rough qualitative trend can be inferred from these results. Even this can only be done subject to the restriction that all sources of systematic errors with dilution can be excluded. However the observation that some combinations of the exponents exhibit a smooth X-dependence leads us to believe that the critical exponents are really changed with dilution. The dashed lines drawn through the points of all six quantities indicate rather arbitrarily a possible trend, but they are drawn in accordance to the scaling relation y = fl(8 - 1) and in agreement with the condition of the linear model given by eq. (3). The smallest change with dilution seems to occur for the critical exponent ft. This exponent describes a spontaneous property of the system and this does not change much with dilution as compared to the exponents 3' and 8, which describe the behavior of the system under an external perturbation. Our results admit the possibility that 8 increases by a factor of two while y may even increase by a factor of three towards the ferromagnetic to spin-

166

U. K6bler et aL / Critical behaviour o f E u x Sr 1 _ x S single crystals

I

I

1

Euo.59Sro.~,1S 1.2

-

Tc = 5.33 K

o

1 =o.45

o

"6 = 1.7t~

1.0

8 :/,.5

T-rc 08

Tc-T • 0.03 K

" O. 0 2 K u

• 0.07 K

I

o 0.17

o

K

• 0.08 K • 0.13 K • 0.23 K

• 0.27 K

CD E

/.// • 0.37 K

0.6-

• 0.57 K

• 0.67K

0.6-

0.2

• 0.33 g

trc-533K

• O.&7 K

• O.t~3 K

• 0.53 K

• 0,63K • 0.73K

i

.0.83K

I

I

I

I

0.2

0.6

0.6

0.8

O----~ Fig. 8. Schofield plot of magnetization data from fig. 6. In this representation the state of magnetization depends only on one parameter O which has the values 0, 0.483 (at x = 0.59) and 1 for susceptibility, critical isotherm and spontaneous magnetization respectively. A linear dependence m(O) = k0 implies a relation between the critical exponents and amplitudes.

glass b o u n d a r y . Such large values for the expon e n t s 8 a n d ), are n o t e x p e c t e d a c c o r d i n g to the t h e o r y of S o b o t t a [21] whereas a limiting value of fl ---, 0.5 for x ~ x~ is consistent with o u r results. It is, however, i m p o r t a n t to b e a r in m i n d that an alternative i n t e r p r e t a t i o n of o u r critical m a g n e tization d a t a w o u l d b e to a s s u m e that the true

critical e x p o n e n t s stay c o n s t a n t along the whole p h a s e line To(x) excluding the e n d p o i n t Tc(xc) = 0 where the e x p o n e n t s could j u m p d i s c o n t i n u o u s l y to new values. T h e p o i n t Tc(xc) w o u l d then be a m u l t i c r i t i c a l p o i n t and, as a consequence, the exp o n e n t s e v a l u a t e d in this analysis w o u l d b e effective e x p o n e n t s in terms of a crossover p h e n o m e -

167

U. KObler et aL / Critical behaviour of Eu~Sr 1 _ ~S single crystals

3

I

'

i

Eu,Sq_,S

i

T

B

L°

05

~.

5

i"

i 6

I

I

I

I

J

03

,

,

~

,

15

i • q~

t, ,

,

i

I

i

i'--'1 ....

t~

i

i

q

12

i

°

b

•

10

i

.~,

1

\

I 06

°

°~•

~ 05

~ -

I

05 ,-~ ......

i

2

i

05 06 017

017 018 019 10

018

019

10

05 0i6 017 0i8 019 10

Fig. 9. Possible composition dependence for effective critical exponents and amplitudes. The dashed lines are drawn arbitrarily but such that they are in accordance with the scaling relation ~/=fl(8-1) and with Schofields's relation between exponents and amplitudes. The critical exponent fl for the spontaneous magnetization increases the less with dilution and the critical amplitude b of ms is the only one to decrease with dilution.

n o n familiar from other multicritical points. I n this case the t e m p e r a t u r e range where the true ( x - i n d e p e n d e n t ) e x p o n e n t s are observed would shrink to zero as x ~ x c a n d this would make the o b s e r v a t i o n of the true critical b e h a v i o u r impossible in practice. However the c o n t r a r y o b s e r v a t i o n of strongly increased critical t e m p e r a t u r e ranges in diluted systems provides one a r g u m e n t i n favour of the analysis here presented.

It is f u r t h e r m o r e interesting to note that the critical a m p l i t u d e s for susceptibility a n d critical isotherm increase with d i l u t i o n while the amplitude for the s p o n t a n e o u s m a g n e t i z a t i o n does decrease strongly with dilution. I n other words, the system becomes more susceptible to a n external field with d i l u t i o n b u t the order p a r a m e t e r weakens c o n s i d e r a b l y j u s t for high magnetic concentrations. T a b l e 1 summarizes all n u m e r i c a l results for the critical parameters. It is i m p o r t a n t to distinguish b e t w e e n those parameters or c o m b i n a t i o n s thereof which are i n v a r i a n t against magnetic d i l u t i o n a n d those which are not. A m o n g the i n v a r i a n t ones the most p r o m i n e n t is the c o m b i n a t i o n V / f l ( 8 - 1) = 1 k n o w n as W i d o m relation. Westerholt a n d Sobotta [17] have already c o n f i r m e d the validity of this relation in m a g n e t i c diluted systems. This m e a n s that critical scaling also holds in r a n d o m - s i t e systems. A second quantity, which is virtually inv a r i a n t against dilution, is Schofield's p r o p o r t i o n ality c o n s t a n t k i n the linear f u n c t i o n m ( 0 ) . Both p a r a m e t e r s k a n d a i n the parametric transformation of eq. 1 involve all measured m a g n e t i z a t i o n d a t a a n d should therefore be evaluated with a high precision. This c a n be n o t e d for k ( x ) a n d also for a ( x ) , b u t the latter q u a n t i t y does decrease strongly with d i l u t i o n reaching a ~ 0 for x ~ 0.57. This strong decrease of a ( x ) with d i l u t i o n reflects the strongly i n c r e a s i n g susceptibility values with decreasing T~(x). This can easily be shown from eq. (1) which gives X / x = (kJo(x = 1 ) / a ) t -~ for the susceptibility. A further very interesting q u a n t i t y is the trans-

Table 1 Experimental critical exponents and amplitudes for EuxSr1_xS single crystals m s / m o = bt#; X / x = gt-V; Bc = d( J/Jo)a; re(O) = kO; b.2 = (8 - 3 ) / ( 8 -1)(1-2fl); b = k / ( b .2 - 1 ) # ; g = kJ0(x = 1)/a; d = a(b .2 - 1 ) / k a b . 3 - 8 x

Tc (K)

fl

b

~

g

8

d (T)

~ #(8 - 1 )

k

a

~ = b *-1

1 0.70 0.68 0.65 0.63 0.59 0.57

16.37 7.88 7.60 6.62 6.05 5.34 3.90

0.41 0.38 0.41 0.50 0.41 0.45 -

1.22 0.83 0.69 0.82 0.64 0.73 -

1.33 2.67 2.14 2.01 2.97 1.74 4.6

0.215 0.175 0.403 0.60 0.38 1.92 1.6

4.1 12.3 5.9 4.37 8.1 4.5 .

3.08 5.08 7.5 12.4 7.6

1.05 1.07 1.19 1.02 1.10

1.23 1.08 1.09 1.23 1.08 1.28 .

7.9 5.5 3.3 2.65 3.43 0.9

0.71 0.54 0.55 0.50 0.48

.

.

.

168

U. KObler et al. / Critical behaviour o f E u x S r 1 _ x S single crystals

formed critical temperature Oc _ b . _ 1 = ( ( 8 - 1 ) (~1- - -: ~

2fl))1/2

1.0

.

(4)

This new critical temperature is universal in the sense that it is a function of only the critical exponents but it does not depend on the experimental critical temperature Tc(x ). Our data indicate however a rather smooth and linear decrease of b * - a with dilution from b * - l = 0.71 for EuS down to b * - l = 0.48 for x = 0.59. This shows an unequivocal composition dependence for the combination of the critical exponents according to eq. (4). On the other hand if the prediction /3--* 0.5 for x--* xc is correct then eq. (4) gives b * - l - - * 0 for x--* x~. This means that b * - I must decrease very strongly in a narrow composition interval between x = 0.59 and x~ = 0.57. This strong change of b * - I m a y well be induced by a discontinuous j u m p of/3 --* 0.5 very near to x c. Such a behaviour would be consistent with a conjecture by Sobotta and Wagner [25] that the critical exponents may develop discontinuously very near to x c. The interesting behaviour of b * - a can best be visualized if the experimental b *-1 values are plotted vs. T J 0 . The quantity Tc/0 can be taken as a measure for the lability of a ferromagnet with Tc/O--* 1 for a strong ferromagnet and TJO ~ 0 for the most labile one. It can be seen from fig. 10 that b *-1 follows the simple linear relation b *-1 - T J 0 with the experimental precision. Since critical scaring holds for the diluted samples as well as for the compact ones we may include the data of other ferromagnets such as EuO in fig. 10. The problem in doing this are the often too small experimental 0-values. The Curie-Weiss law applies only asymptotically for high temperatures and most experimental determinations of 0 have not been performed at sufficiently high temperatures. These problems have been discussed for EuS in ref. [4] and for EuO in ref. [30] in terms of consistent values for the ratios between nearest and next-nearest neighbour exchange constants on the one hand and TJ0-values on the other hand. At a first glance the linear relation b * - 1 = Tc/0 is very surprising. The ending points b * - 1 ___>0 and b *-1---, 1, however give reasonable results, b *-1 --, 0 for T J 0 -~ 0 requires/3 ~ 0.5 which is well

I

I

I

0.8

t

I

t

[

I

I

I

[

I

-

0.6

~ Euo68sro32s Euo~3Sro37S~

- Euo~Sro~S

0.2 I

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0~

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08

10

Tc/O Fig. 10. Schofield critical temperature b* - ] as a function of the ratio To/0 which characterizes the stability of a ferromagnet. For the most stable ferromagnet with b * - l = T ~ / O = l all critical exponents have a minimum. With increasing lability go b *-1 and To/0 to zero and the critical exponents reach their maximum. Most compact ferromagnets fall into the narrow range 0.76 < b * - 1 < 0.82.

understood in the light of Sobotta's theory for a ferromagnet at the critical composition for which T J O --* 0. For T J O --* 1 a necessary consequence is that 7 = 1 and this results from definition of b *-1 if the scaling relation y = f l ( 8 - 1 ) is included. Fig. 10 suggests, therefore, a more general validity of Schofields critical temperature b *-1. Most compact ferromagnets have b * - 1-values which fall within the very narrow interval 0.76 < b *-1 < 0.82 [24]. The advantage of diluting a ferromagnet with strongly competing positive and negative exchange interactions such as EuS is that the ratio Tc/8 can be varied in this way in a wider range than is found in the family of the compact ferromagnets. This competition in the interactions leads to a much stronger decrease in T~ with dilution as compared to 8. The diluted samples therefore reach parameter regions which are not found in the compact materials. On the other hand dramatic changes in the critical parameters are only observed for high degrees of randomness. It must be questioned whether b*-l-values of 0 or 1 are realized in practice. Fig. 10 suggests that if b *-1 -* 0 will be reached this is likely to occur by a

169

U. K~bler et aL / Critical behaviour o f Eu~ S r z - x S single crystals

discontinuous jump of b *-1 from b * - 1 --0.45 down to b * - 1 = 0. The further consequences which can be drawn from eq. (1) if fl ~ 0.5 and therefore b .2 ~ o¢ for x---, x o are that the amplitudes for the spontaneous magnetization b = k / ( b .2 - 1)/~ goes to zero but the amplitude for the critical isotherm diverges according to d = a ( b . 2 - 1 ) / k S b . 3 - 8 as ab . 8 - 1 , even though a goes weakly to zero for x --> xc. In the Schofield relation between the critical amplitudes and critical exponents given by eq. (3) both sides seem to be invariants against dilution. This can be inferred from our results within relatively large experimental uncertainty limits. Finiteness of the right side of eq. (3) requires 8 = 27 + 1 which agrees with Sobotta's pedictions 8 --> 5, 7 --* 2 but this condition is, of course, only a specialisation of Widom's scaling relation 7 = f l ( 8 - 1) for fl = 0.5. A further consequence of b . 2 --~ oO for x ~ x~ is that there exists only O - - 0 as a reasonable value for the Schofield coordinate 0. This means that only a strong diverging susceptibility and a critical isotherm do occur but no spontaneous magnetization which would require B i --> 0 and hence O ~ 1 according to the first part of eq. (1). We may summarize the discussion of this section by the following suggestive conclusions. All physical ferromagnets, diluted or compact ones, should have b * - 1-values in the range 0 ~< b*-1 ~ 1. For b*-1 ~ 0 all critical exponents have a maximum with f l = 0 . 5 and 8 = 2 7 + 1 while b = 0 , d = oo and g = oo. For b *-1 ~ 1 all critical exponents have a minimum with 7 = 1 and fl = 1 / ( 8 1) while b = o¢, d = 0, but g is finite and identical with the Curie-constant.

magnetization. Near to the critical temperature it is appropriate to evaluate rn s from measurements of the magnetization as a function of temperature. Such measurements have been made in the critical temperature range and are shown in fig. 6. It is evident from these curves that it becomes more difficult to locate the transition into the ordered state characterized by J = 3B0 the lower the temperature is. This problem gets even worse in the second type of measurements, which is commonly employed to evaluate m s in the temperature range T << To. Here the magnetization is measured as a function of field for different temperatures. This type of measurement is shown in fig. 13; however for a sample with x = 0.48 which does not order magnetically. Nevertheless, the experimental problem of evaluating finite m / m o - values for B i --* 0 from the very steep low-temperature magnetization curves can clearly be noticed. In order to get a rough impression on the behaviour of m s ( T , x ) we have extended the ms(T ) curves from the critical temperature range down to T = 1.8 K in as quantitative way as possible considering the enormous experimental uncertainties. Fig. 11 gives the results. Roughly speaking the x-dependence of the m J m o curves is for all temperatures very similar in the sense that the behaviour of m s ( x ) / m o at T = 1.8 K is very similar to that of the critical amplitude b ( x ) . This means

- ~ x i

1.0

i

i

i

i

i

i

i

,

I1=

I

i

i

I

I

i

i

I

I

S

r

08

5. Spontaneous magnetization ~

As we have seen in the preceding section the susceptibility increases strongly in the vicinity of the paramagnetic to ferromagnetic phase boundary with diamagnetic dilution. This means that the transition into the ordered state is gradually less associated with a sharp structure in the magnetization curves. A smooth magnetization behaviour applies for both types of measurements normally employed for the evaluation of the spontaneous

x x x\ \xx\ x \ x\ \ \ \x x\

0.2

I

I 2

=

I /*

II

! 6

II

8 T - -

10

12

I

I

1/~

II 16

[K]

Fig. 11. Spontaneous magnetization curves normalized to the theoretical saturation value. For the diluted samples these curves cannot be evaluated very accurately below the critical temperature range due to the enormous large susceptibility values for very small internal fields.

170

U. Kbbler et aL / Critical behaviour o f E U x S 0 _ x S single crystals

that as for b(x), shown in fig. 9, the extrapolated spontaneous magnetization for T---, 0 is likely to decrease initially linearly with dilution. This is a stronger x-dependence than those resulting from computer simulation experiments [27].

6. M e a s u r e m e n t s

f o r B i = 0.1 T

We have also investigated the magnetization for a fixed internal field of B i = 0.1 T. At this high field value one is outside the problematic region where the susceptibility has enormously large values. This means that the magnetization is a slow function of the field around B i = 0.1 T compared with the demagnetization line J = 3B0. These magnetization data can therefore be evaluated convehiently with high precision. Fig. 12 shows the results, giving the magnetization in units of the theoretical saturation value as a 1.0 0.9 0.8

i

~

I I

I:°

•

EuxSr1-xSl

~

.

o.

',.'_'_/__Z //

/ /'7,,/

oO, 05

--.

o. /yO,o T=2.

0.3

6K 81<

!

I

SrS

0.2

I

O.t+

I

I

0.6

08

EuS

x

Fig. 12. Relative magnetization as a function of composition for a fixed internal field of 0.1 T. The Curie-line To(x), evaluated by the divergence of the susceptibility, has been inserted as a dashed line. Tc ( x ) intersects the m ( x ) / m 0 curves at their approximate inflection points for high temperatures but at low temperatures the significance of this point has changed and characterizes the approximate boundary of the spin glass range.

function of composition for different temperatures. For all temperatures the relative magnetization increases with increasing magnetic density viz. increasing magnetic interactions without showing any sharp structure. This is not to be expected if a ferromagnet is subjected to a finite magnetic field. The increase of the m ( x ) / m o curves should, however, be strongest near to those concentrations at which magnetic order sets in. The inflection points of the curves should mark this event. This is, however, correct only at high temperatures. The inflection points of the low-temperature curves have not this significance but mark rather the transition into the spin glass state apart from a possible shift between the B i = 0 position of this boundary and the measured one here at B i = 0.1 T. For a better discussion of the results of fig. 12 we have inserted the T~(x) phase line as it was evaluated from the divergence of the susceptibility in the preceding sections. This dashed line gives the value of the relative magnetization at an internal field of 0.1 T along Tc(x ). It can be seen that just starting from EuS the relative magnetization at T~(x) increases with diamagnetic dilution as is characteristic for increasing susceptibility values along T~(x). For T = 10, 8 and 6 K, T~(x) intersects the m ( x ) / m o curves approximately at their inflection points, but this is evidently not the case for T = 4 K and below. For temperatures lower than 5 K, T~(x) increases dramatically in fig. 12 and this reflects the very strong susceptibilities on approaching the critical concentration for ferromagnetism. We have also inserted the approximate boundary of the spin glass range into fig. 12. This boundary was defined in fig. 2 by the inflection points of the field-cooled susceptibility x ( T ) / x . At T = 2.25 K this P - S G boundary also coincides well with the inflection point of the r n ( x ) / m o curve in fig. 12, but at 4 K the situation is not as clear. The branching point between T~(x) and P - S G boundary should, however, be near to x 0.57 and T~ = 4.5 K as was found with the susceptibility measurements. The results of fig. 12 raise the interesting question about the behaviour of the m ( x ) / m o curve for T = 0 K. For the whole ferromagnetic phase

U. Kbbler et al. / Critical behaviour of E u x S r 1 _ x S single crystals

one can extrapolate a value of m ( x ) / m o - - 1 for T ~ 0 K and we expect, therefore, that the Tc(x )line will also reach m ( x ) / m o = 1 for T = 0. A diverging magnetization at T~ for a finite field of 0.1 T is untypical for the ferromagnetic state and hence this event marks the end of the ferromagnetic phase. At the P - S G phase boundary surprisingly small m ( x ) / m o values are observed and finite m ( x ) / m o values can be extrapoled for T ~ 0. This behaviour must, however, change for very low magnetic concentrations. Since the paramagnetic susceptibility (x---0) does diverge for T ~ 0 we expect m ( x = 0, B i = 0.1 Z ) / m 0 = 1 for T ~ 0. The P - S G boundary must therefore also end at m ( x ) / m o = 1 for T ~ 0 and we expect this event at an approximate composition of x --~0.3.

7. Spin glass range The accesible temperature and field range in this work does not allow a very extended investigation of the spin glass state. We have therefore measured only one sample with x = 0.48 in more detail. According to the susceptibility measurements the spin glass transition temperature for this sample is approximately 2.6 K, which is quite high, so that hysteresis effects are small. On the other hand the susceptibility of this sample is still small enough at T = 2.6 K to be well measurable. Fig. 13 gives in the left-hand part the results of magnetization measurements as a function of the internal field B i. As can be seen, the linear susceptibility does not diverge in the temperature range down to 1.8 K. Assuming an experimental uncertainty in the field calibration of 0.001 T, the largest measurable susceptibility amounts to X/X --500. This must be compared with the experimental value of X / x ~ 80 for T = 2.6 K! It has been postulated that the higher coefficients in the series expansion of the magnetization should diverge at the spin glass transition temperature [28]. We have therefore fitted the low-field part of the magnetization curves in fig. 13 by the equation J = x B i - X3 B3. The fit results are shown by the solid lines in fig. 13. It is evident from these curves that the higher terms in the series expansion of J ( B i ) become increasingly important at lower tem-

oA

'

'

t

r

.,.."'2./+ ' ..30

T=18K .." ..",-'"

0.3

171

i

i

i

i

..,36

...~2.~-" .................... -':." ..'" - ..-

...'"

..... /,i~

.......

....~i~

0.2 SLOPE:-20"

01

001

002

0.03 Bi ~

004

005

006 [TI

0/,5 050

055 060 (ogT~

065

0.70 [K]

Fig. 13. Relative magnetization vs. internal field for a sample with x = 0.48 which does not exhibit long-range ferromagnetic order. The solid curves fit the low-field part of the measurements by a s u m of a linear and a cubic field-term and show the increasing importance of the higher order terms with decreasing temperature. Right-hand part: log-log plot of cubic susceptibility X3 vs. absolute temperature. X3 diverges so strongly that the critical temperature cannot be evaluated accurately. Assuming that X3 diverges for T--, 0 an exponent of - 2 0 is obtained asymptotically.

peratures. Yet the convergence of a power series for J(Bi) may be questioned for sufficiently low temperature. The behaviour of x 3 ( T ) has been investigated more closely and the data show directly the often observed phenomenon that x 3 ( T ) diverges very strongly [29]. In such a mathematically and experimentally difficult situation the assumed critical temperature can be widely varied without any associated loss in the quality of the fit. The only effect is that t h e critical exponent shifts slightly. A divergence of X3 at T - - 2 . 6 K can, however, be ruled out and it is evident that the divergence is at a much lower temperature, probably at T-- 0. The right-hand part of fig. 13 shows a log-log plot of X3/X vs. the absolute temperature. If X3 diverges for T = 0 our experimental temperatures are much too high to fall into the power law region of the T-dependence of X a, but for lower temperatures X 3 becomes excessively large and cannot reasonably be measured. An asymptotical evaluation of the slope in fig. 13 gives an approximate exponent of - 2 0 . This is so much for an exponent that neither its accurate value nor the location of the divergence can be measured directly but must be believed from a fit.

172

U. KObler et al. / Critical behaviour of EuxSr 1 _ xS single crystals

The experimental problems are so tremendous here that they will hardly be overcome even if more subtle experimental techniques are applied. However the definite observation that X3 diverges at a much lower temperature than was evaluated in the preceding sections for the spin glass transition constitutes a serious strike against these methods of evaluation. The transition into the spin glass state is a weak event in the sense that if it is at all associated with a critical thermodynamic behaviour, the observed exponents are so large that they smear the phenomenon over a considerable temperature interval, which even includes T = 0. Furthermore the question as to whether or not the spin glass constitutes a phase in a thermodynamical sense at finite temperatures remains open. In a computer simulation analysis of a three dimensional Ising spin glass an approximate exponent of - 1 5 was suggested for the divergence of X3 assuming that divergence occurs at T = 0 [31]. This estimate is at least in a rough agreement with our results.

8. Conclusions With detailed static magnetization measurements on EuxSr l_xS single crystals we could show that the divergence of the susceptibility at T~(x) becomes successively stronger with decreasing Tc(x ). This effect is relatively unimportant for the dilution range x > 0.7 but below x = 0.7 both the exponent 3' and the critical amplitude g increase acceleratedly. For x = x~ = 0.57 the divergence of X is so tremendous for the present system with a high spin quantum number that no accurate measurements are possible. This is due to a diverging critical amplitude g and a critical exponent 3' which is considerably increased over the value of 3, = 1.33 for EuS. Our data are consistent with a 3,-value of 3 for x ~ x c, but conflict, however, with a value of 3, = 2 predicted by Sobotta [21]. The experimental and theoretical situation seems to be clear as concerns fl which reaches fl ~ 0.5 for x ~ x c. In conjunction with the decreasing critical amplitude b this means that the spontaneous magnetization rises gradually slower in the diluted samples. The critical isotherm behaves similar to

the susceptibility in that it also exhibits an increasing exponent and an increasing critical amplitude with diamagnetic dilution. It therefore seems appropriate to distinguish between the behaviour for the field induced quantities susceptibility and critical isotherm and the one for spontaneous magnetization. The critical amplitude of Bc also diverges for x ~ x c , but our data indicate that 8 may approximately have doubled like y as x --->x~. This is also in variance with the theoretical value of 8 = 5 given by Sobotta [21]. Taking the validity of Schofield's linear model also for granted in the diluted systems we find a composition independent proportionality constant k. The scaling constant a for the internal field decreases, however, with dilution and reaches a ---,0 for x --->x c. This is identical with a diverging critical amplitude g for the susceptibility as x ~ x c. The scaled product of the three critical amplitudes db(8-l)g also belongs to the composition-invariant quantities. According to Schofield's relation (eq. (3)) finiteness of the right side requires 8 = 23, + 1 for x---, x c or b * Z ~ oo. This is in accord with Sobotta's prediction but constitutes nothing more than Widom's scaling relation y = 1 3 ( 8 - 1) for /3 ~ 0.5. This relation seems to be valid in m a n y diluted systems even though the actual values of the individual exponents may vary strongly [17]. The evaluation of the spontaneous magnetization m s becomes a more and more difficult problem with high degrees of dilution owing to the very large susceptibility values in the critical temperature range. These problems turn even worse for the evaluation of ms(T ) at low temperatures. Our results indicate that ms(x, T = 0) decreases nearly in the same way with dilution as does the critical amplitude b(x). In particular a weak linear decrease of both quantities just for high magnetic densities cannot be ruled out by our data. This is a stronger decrease than resulted from the computer simulation experiments by Binder, Kinzel and Stauffer [27]. A convenient and practicable definition for the spin glass boundary consists in choosing the inflection point of the field-cooled susceptibility. This definition agrees with the inflection point of the low-field magnetization as a function of composition for constant temperatures. Although this

U. KObler et aL / Critical behaviour of Eu x Sr 1

method is straightforward from an experimental point of view, it must be noted that there is no theoretical justification for this practice and an inflection point is overmore a magnetically weak event. For the upper composition limit of the spin glass range this inflection point coincides with X--' oo for x c ---, 0.57 and T - 4.5 K, but on the lower composition limit a finite susceptibility value is found for x - - 0 . 3 and T---, 0. On both sides extrapolations are therefore necessary in evaluating the spin glass boundary. Choosing the divergence of the third order susceptibility as a criterion for the spin-glass boundary does not result in a more accurate way of evaluation although this definition is theoretically better substantiated [28]. X3 diverges with an exponent as large as 20, which gives rise to considerable ambiguities in fitting the critical temperature. The discrepancies between both methods of defining the spin glass range are 18

I

~

S single crystals

173

enormously large, as X3 is likely to diverge only for T---) 0 [31] in contrast to conclusions of ref. [29] where a divergence of X3 at a finite temperature Tf was found. The magnetic phase diagram contains only the linearly decreasing Tc(x ) phase boundary as an undisputed detail (fig. 14). This line has a slope of dTc/dx = 27.5 K and ends at the approximate coordinates x = 0.57, T = 4.5 K. At lower compositions this T~(x) line is followed nearly steadily by the spin glass boundary defined by the inflecting susceptibility. This spin-glass boundary seems to end at approximately x--0.3 and T = 0 and passes across one data point of Ferr6 et al. [26]. At the temperature available here (T >/1.8 K), a reentrance of the spin glass phase into the ferromagnetic one at very low temperatures [8] could not be investigated.

Acknowledgements I

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f

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l

I

I

We want to thank K. Binder for many illuminating discussions and for a critical reading of the manuscript. We also gratefully acknowledge several valuable discussions with D. Wagner, G. Sobotta and K. Westerholt of the Ruhr University Bochum during the final state of this work. The expert technical assistance of F. Deloie is acknowledged with gratitude. Thanks are also extended to Miss I. Apfelstedt for valuable assistance in the data analysis.

16

EuxSr1S -x T¢(x~) /

14

_

12

~a 6

T,(

4

References

Fer~et at./ /SG o

2

//

P

SrS

I

I

0.2

I

I

I

0.4

I

0.6

I

I

0.8

I

EuS

X ~

Fig. 14. High-temperature part of magnetic phase diagram for EuxSrl_xS as evaluated here with static magnetization measurements. The linear T~(x) line has a slope of d T ~ / d x = 27.4 K. This line transforms nearly imperceptibly into a line Tf(x) where the x ( T ) curves do no longer diverge but have an approximate inflection point. The data point of Ferr6 et al. from ref. [26] is obtained by extrapolating the ac-susceptibility cusp to a frequency of zero and falls nicely onto the extrapolated Tf(x)-line. F: ferromagnetic, P: paramagnetic, SG: spin glass and SP: superparamagnetic range.

[1] H. Maletta, in: Excitations in Disordered Systems, ed. M.F. Thorpe (Plenum, New York, 1982). [2] W. Kinzel and K. Binder, Phys. Rev. B24 (1981) 2701. [3] H.G. Bohn, W. Zinn, B. Dorner and A. Kollmar, Phys. Rev. B22 (1980) 5447. [4] U. KObler and K. Binder, J. Magn. Magn. Mat. 15-18 (1980) 313. [5] K. Binder and W. Kinzel, in: Lecture Notes in Physics, vol. 149 (Springer, Heidelberg, 1981) p. 124. [6] H. Maletta, G. Aeppli and S.M. Shapiro, J. Magn. Magn. Mat. 31-34 (1983) 1367. [7] I. Morgenstern and K. Binder, Phys. Rev. B22 (1980) 288. [8] H. Maletta and W. Felseh, Phys. Rev. B20 (1979) 1245. [9] H. Maletta, W. Zirm, H. Scheuer and S.M. Shapiro, J. Appl. Phys. 52 (1981) 1735.

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[10] H. Maletta and W. Felsch, Z. Phys. B37 (1980) 55. [11] D. Meschede, F. Steglich, W. Felsch, H. Maletta and W. Zinn, Phys. Rev. Lett. 44 (1980) 102. [12] A. Scherzberg, H. Maletta and W. Zinn, J. Magn. Magn. Mat. 24 (1981) 186. [13] H.G. Bolm, K.J. Fischer, U. K6bler, H. Liatgemeier and Ch. Saner, J. Magn. Magn. Mat. 15-18 (1980) 667. [14] H. Maletta, G. Aeppli and S.M. Shapiro, Phys. Rev. Lett. 48 (1982) 1490. [15] K. Siratori, K. Kohn, H. Suwa, E. Kita, S. Tamura, F. Sakai and A. Tasaki, J. Phys. Soc. Japan 51 (1982) 2746. [16] K. Binder, Solid State Commun. 42 (1982) 377. [17] K. Westerholt and G. Sobotta, J. Phys. F13 (1983) 2371. [18] H. Pink, Z. Anorg. Allgem. Chemie 364 (1969) 377. [19] A. Aharony and D. Stauffer, Z. Phys. B47 (1982) 175. [20] D. Mukamel and G. Grinstein, Phys. Rev. B25 (1982) 381. [21] G. Sobotta, J. Magn. Magn. Mat. 28 (1982) 1.

[22] D.D. Berkner, Phys. Lett. 54A (1975) 396. [23] P. Schofield, Phys. Rev. Lett. 22 (1969) 606. [24] P. Schofield, J.D. Litster and J.T. Ho, Phys. Rev. Lett. 23 (1969) 1098. [25] G. Sobotta and D. Wagner, J. Phys. C 11 (1978) 1467. G. Sobotta and D. Wagner, J. Magn. Magn. Mat. 15-18 (1980) 257. [26] J. Ferr6, J. Rajchenbach and H. Maletta, J. Appl. Phys. 52 (3) (1981) 1697. [27] K. Binder, W. Kinzel and D. Stauffer, Z. Phys. B36 (1979) 161. [28] M. Susuki, Progr. Theor. Phys. 58 (1977) 1151. [29] R. Omari, J.J. Prejean and J. Souletie, J. de Phys. 44 (1983) 1069. [30] N. Menyuk, K. Dwight and T.B. Reed, Phys. Rev. B3 (1971) 1689. [31] K. Binder and A.P. Young, Phys. Rev. B (1984) in press.