the magnetisation process of the ferrimagnetic spiral structure in dymn6−xcrxge6 compounds

iomai of

,44 N"'- magnetic ~1~ materials

ELSEVIER Journal of Magnetism and Magnetic Materials 171 (1997) 267-279

The magnetisation process of the ferrimagnetic spiral structure in DyMn6-xCrxGe6 compounds

J . H . V . J . B r a b e r s a'b, P . S c h o b i n g e r - P a p a m a n t e l l o s c, K . H . J . B u s c h o w a'*, F . R . d e B o e r a

a Van der Waals-Zeeman Laboratory, University of Amsterdam, 1018 XE, Amsterdam, The Netherlands b lnstitut ~ r Kristallographie, ETHZ, CH-8092 Ziirich, Switzerland

cPhilips Research Laboratories, P.O. Box 80 000 Eindhoven, The Netherlands

Received 20 December 1996

Abstract

The thermodynamical stability and magnetic-field response of the ferrimagnetic spiral (FS) structure in DyMn6Ge6 is investigated in terms of a simple model based on a competition between various magnetic interactions, including the R-sublattice anisotropy, antiferromagnetic Mn-Mn exchange interactions, and the R-Mn exchange interaction. It can be shown that the exchange interaction between the Mn moments belonging to different magnetic units acts as an effective Mn anisotropy, which plays a crucial role in the stabilization of the cone structure observed in DyMn6Ge6 at low temperature. The response of the FS structure to low applied magnetic fields consists of a rotation of the Mn and R moments towards the c-axis. At higher fields, transitions towards a (canted) structure of two collinear single sublattices (2SS or SS) are possible. Experimental high-field free-powder magnetisation data obtained on DyMn6-xCrxGe6, are interpreted in terms of the theoretical analysis. From the field regions corresponding to the 2SS structure, estimates for the na Mn mean-field intersublattice coupling parameter are obtained. The estimated nDy_mn values agree with nR 3d values obtained in other R-3d compounds. In addition, the temperature dependence of the free-powder magnetisation, established by experiment, is satisfactorily interpreted in terms of the theoretical analysis.

Keywords: Rare earth manganese germanides; Cr substitution; Crystal field induced anisotropy; Mn anisotropy

I. Introduction

Recently, various investigations have been re- por ted on the magnetic structure of DyMn6Ge6 [-1, 2]. This hexagonal c o m p o u n d crystallizes in the MgFe6Ge6 structure (space group 6/mmm), which

* Corresponding author. Tel.: + 31 20 525 5714; fax: + 31 20 525 5788; e-mail: [email protected].

consists of a layered ar rangement of R and Mn ions. The reported zero-field magnetic structure at 11 K is represented in Fig. 1. The R and Mn moments belonging to a single magnetic unit (shaded area in Fig. 1) are coupled antiparallel and make an angle of 55 ° with the c-axis. When going from one magnetic unit to another, the projection of the magnetic unit magnetisat ion on the basal plane rotates over an angle 58.7 ° at T = l l K .

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268 J.H V.J. Brabers et al. ,,Journal ()l'Magnetism and M.gnetic Materials 17/ (1997) _67 .79

I I I II

,m

I f~ .~" i l I i G i V l

,"7"4,'----',

I I I I I I

G !

I I I

- ,~D~'. Oy

~N' Mn

O I I

FS

= 58.7 o

0 = 5 5 °

magnitude smaller than the R anisotropy. In the present paper, we present a simple model for the magnetic interactions in DyMn6G%, explaining how exchange interactions between Mn moments belonging to different magnetic units can provide a large (effective) Mn anisotropy. Furthermore, a discussion of the magnetisation process will be given in terms of the same model. Computational details about the least squares fit of the cone angle as a function of temperature are also included in this paper.

2. Theoretical considerations

2.1. Magnetic interactions and their role in the magnetisation process

Within a single magnetic unit (labelled 1), the magnetic moments of both Mn layers can be considered as one single sublattice. The relevant free- energy expression for a single magnetic unit then becomes:

(a) (b)

Fig. 1. Ferrimagnetic spiral (FS) structure as observed in DyMn6G% at low temperatures. The Mn moments within a single magnetic unit (consisting of a layer of R-moments in between two layers of Mn-moments stacked along the c-axis) are coupled parallel to each other and antiparallel to the Dy moments (a). The total moment of a single magnetic unit makes a finite angle (0 ~ 55 ° at 11 K) with the c-axis. The projections of the magnetisations of the magnetic units turn over 58.7 ° when going from one magnetic unit to another (b). The net magnetisation vector at low temperatures is nonzero and points in the c-direction.

In an earlier investigation [2], we found that with increasing temperature the cone angle (the angle between the magnetic moments and the c- axis) changes, until at about 100 K all magnetic moments are confined to the basal plane. A satisfactory least squares fit of the cone angle as a function of temperature was obtained, taking into account the temperature dependence of the R-sublattice anisotropy and assuming a substantial negative Mn-anisotropy constant.

A large 3d anisotropy however, is exceptional, as usually the 3d anisotropy is at least one order of

F -~ ~ n~pn~mt + nRTmRmx -- B(mR + mr) o~

+ K1R s i n 2 0 s + K2R sin 4 0R, (1)

where the first term represents the 3d-3d exchange interaction with the Mn-moments of the other units, and the second term the R-3d exchange interaction. The third term stands for the Zeeman energy, whereas the last two terms represent the R-anisotropy energy. The index ~ runs over all units. The symbols mR, m~.l, and K, respectively, stand for the R-sublattice moment, the Mn moments, and the anisotropy constants, all per magnetic unit. Furthermore, n~ and nRx are mean-field coupling constants corresponding to the inter-unit 3d-3d interaction and the R-3d interaction, respectively.

The 3d-3d exchange term can be specified fur- ther. Therefore, we define the polar angles 0 and 4, as in Fig. 2. The vectors rnl and m~ can then be expressed as:

ms = m3d(sin 0 cos(~q~), sin 0 sin(~qS), cos 0), (2a)

m x = mad(sin 0, 0, cos 0), (2b)

J.H.KJ. Brabers et al. / Journal o f Magnetism and Magnetic Materials 171 (1997) 267-279 269

c-axis The angle ~b only occurs in the 3d-3d exchange interaction terms of F. In equilibrium, the angle ~b follows from the equation:

0 / M subl

Fig. 2. Definition of the polar angles 0 and q~.

where 0 is the angle between the Mn-moments and the c-axis, and q~ the rotation angle of the projection of the magnetic unit magnetisation on the basal plane, when going from one (magnetic) unit to another. By direct substitution of the expressions (2a) and (2b) in the first term on the right-hand side of Eq. (1), the 3d-3d exchange interaction of ml with moments belonging to the other units can be written as

~. n~m~ml = ~ n~m~d(sin 2 0 cos(oc~b) + cos 2 0) a 0~

= ~ n~m2d + sin 2 0 • n,mZd(COS(CtqS) -- 1). (3) Gt gt

From the right-hand side part of this equation it can be seen that the 3d-3d exchange interaction in fact acts as an effective 3d-anisotropy, of which the energy contribution consists of a 'constant' term Ko, and a term K3d sin 2 0. The anisotropy constants Ko and K3d are given by

Ko = ~ n~,m2d, (4a)

K3a = ~ n,m~d(cos(:¢~) -- 1). (4b)

~F Otk sin 2 0 ~ n,,m2aot sin(ct~b) = 0. (5)

~t

Solving this equation with respect to ~b, yields a solution which is independent of 0, i.e. ~b is unaf- fected by (field induced) changes of the orientation of the sublattice magnetisation vectors with respect to the c-axis. Therefore, when the field is applied along the c-axis at a certain temperature, q~ is a constant in the field region where the helix structure is stable. This holds for both the FS and the SS structure, as the analysis so far is applicable to both these magnetic structures. The magnetisation process in low and modest applied fields will presumably consist of a rotation of the magnetic moments towards the c-axis without a change of q~ [1]. As in general both the n~ mean-field parameters and the 3d-moments are temperature dependent, q~ may change however with temperature.

A quantitative analysis of the magnetisation process requires knowledge of the n~ mean-field parameters. Approximations for nl and n2 can be obtained by neglecting all n, parameters for ~ > 2. A justification for this is that the 3d-3d interaction is probably of a RKKY-type. The oscillating RKKY interaction strength (JRtKY) is proportional to the inverse of the third power of the distance (r) between the interacting moments (JRKKY "~

cos(kr)/ra), and as a consequence it drops very strongly with increasing r. Therefore, we assume that only the lowest order mean-field parameters are relevant. Eqs. (4b) and (5) then provide a set of two equations from which nx and n2 can be derived, provided that the effective 3d anisotropy constant K3a is known.

Anticipating the high field magnetisation measurements to be presented later in this paper, we already mention here the possibility of a field- induced magnetic phase transition from the ferrimagnetic spiral structure (FS) towards a two-sublattice structure in which all moments of a given sublattice are collinear (2SS). In the latter case, all Mn- and R-moments, respectively, couple as a single Mn- and a single R-sublattice moment.

270 .LH. K J, Brabers et al. 'Journal of Magnetism and Magnetic Materials 171 t1997) 267 279

From Fig. 2, it may be inferred that, for the same angles between the c-axis and the Mn- and R- moments, the 2SS configuration corresponds to a larger free-powder magnetisation than the FS configuration. Therefore, the Zeeman term pro- motes the 2SS configuration, whereas the inter-unit 3d-3d interaction suppresses it. The phase transition occurs when the free-energy minimum corresponding to the FS configuration equals the free- energy minimum corresponding to the 2SS configuration. In general, such a transition involves a dis- continuous change of the angles of the Mn and R moments with the c-axis. Therefore, the transition is of first order. An interesting observation is that in case of the collinear 2SS configuration (q5 = 0), the effective 3d anisotropy is zero, as K3d = 0 when ~b = 0, which may be seen from Eq. (4b).

2.2. Magnetic interactions in relation to the zero-field magnetic structure

The 'exchange contribution' to the 3d anisotropy can be quite high when the interlayer interactions are strong. In RMn6Ge6 compounds this is most likely the case, as may be inferred from high-field magnetisation data on YMn6Ge6 and LuMn6Ge6 [3]. In DyMn6Ge6, the R and 3d moments belonging to a single magnetic unit are rigidly antiparallel coupled in zero field [2]. For the system of 3d- and R-layer magnetisations, we can therefore define a joint anisotropy of the total magnetic unit moment, consisting of the R anisotropy and the (effective) 3d anisotropy.

The (effective) 3d anisotropy most likely favours a moment direction perpendicular to the c-axis (K3d < 0). The reason for this is the observation of a flat spiral (SS) structure in DyMn6Ge6 at room temperature [1]. Usually, only lowest-order anisotropy constants contribute at room temperature. Because M6ssbauer experiments were indicative of a first order R-anisotropy constant favouring a moment alignment parallel to the c-axis in case that c9 < 0 [4], the 3d sublattice must have an in-plane anisotropy to stabilize the SS configuration. (c~a stands for the Stevens factor [4].)

The negative electric field gradient observed from M6ssbauer experiments on GdMn6Ge6, imply a positive value of the second order crystal

field parameter A °. This means that for DyMn~,Ge~ the first-order anisotropy constant of the R-moments is positive (K1R > 0) [5]. The observation of the cone structure in DyMn6Ge6 suggests, however, that also higher order anisotropy constants are relevant in this particular case. Such higher- order anisotropy constants are necessary prerequi- sites for a cone structure. To stabilize the cone structure the joint anisotropy constants K~ = KIR q- K3d and K2 = K2R should then be such that [6]

0 < K1 < 2K2 (6)

and the relation between the angle 0 and K1 and K2 is given by

sin 2 0 - - K ~ (7) 2K2

Using crystal field theory K1R and K2R can in a lowest-order approximation be expressed as [7]

KIR 3 0 = - ~B2 (02 °) - 5B ° (O°), (8)

g 2 R = ~5 B 0 ( 0 o ) , (9)

where B20 and B ° are the crystal-field parameters (B°2 = a,A°i(r 2 ) and B ° = f lsA°(r 4) [4]), and (O ° ) and (O °) the thermal averages of the expectation values of the corresponding Stevens operators (thermal averaging is indicated through overlining of the physical quantities). The expectation values of the Stevens operators are defined by ( O , " ) = (J , mslO=,lJ, m j ) , where I J, m j ) represents a free-ion eigen-function of the total angular-momentum oper- ator. Expressions for the O~' operators and their expectation values are extensively tabulated for a number of different J-states in Ref. [4].

Generally, it is assumed that in the presence of a strong R 3d exchange interaction and strong applied magnetic fields, the 4f-electrons can still be well described through the corresponding free-ion wave functions. This allows straightforward analysis and calculation of the energy levels of the 4f shell. The level shifts due to crystal-field interactions can be calculated by diagonalizing the crystal-field hamiltonian matrix. As both the 020 and O ° matrices already have the diagonal structure, such calculations are comparatively easy in the particular case of the present investigation. As

J.H. KJ. Brabers et al. / Journal of Magnetism and Magnetic Materials 171 (1997) 267-279 271

a consequence of the diagonal structure of O ° and O °, the level scheme will consist of singlets [J, ms) in case of nonzero R-3d interaction and/or under an applied field. In case of Dy (J = ~) , a strong R-3d interaction therefore favours a 1 ~ , ~ ) ground state. As ( ~ , ~10°1~, ~) = 16 380 > 0 [4], B ° should be positive in order to satisfy Eq. (6) at low temperatures (K2 > 0). From Eq. (8) it can be inferred that a positive B ° value enhances the negative character of K1R (and thereby of K 0, necessary to stabilize the observed cone structure. An extra (presumably large) negative contribution to K1 comes from the inter-unit 3d-3d interaction. The presence of this negative contribution persists at higher temperatures and it is then responsible for the simple spiral structure observed above 100 K.

As a conclusion we can say that the cone structure in DyMn6Ge6 is most likely related to an interplay between second- and fourth-order contri- butions to the crystal-field hamiltonian, and the inter-unit 3d-3d interaction.

3. Experimental

The samples used in the present investigation were prepared by arc melting and subsequent vac- uum annealing at 800°C for at least 1 week. The quality of the samples was verified by X-ray diffraction. The crystalline specimens were approximately single phase and the crystal structure could be identified as the hexagonal MgFe6Ge6 structure (space group 6/mmm).

Magnetisation measurements in high magnetic fields up to 35 T were performed in the high-field installation of the University of Amsterdam.

Neutron-diffraction experiments were carried out at the facilities of the Reactor Saphir, Wiirenlingen.

Low-field measurements of the magnetisation as a function of temperature were performed on a SQUID magnetometer (Quantum Design).

4. Results and discussion

4.1. Neutron diffraction

Earlier investigations by neutron diffraction at T = 11 K revealed a ferrimagnetic spiral structure

in DyMn6Ge6. The R and Mn moments belonging to a single magnetic unit are rigidly antiparallel coupled. The angle between the magnetic unit magnetisation and the c-axis was estimated to be about 55 ° at T = l l K and by going from one magnetic unit to another, the projection of the unit magnetisations on the basal plane rotates by 58.7 ° .

As can be inferred from the previous section, such a magnetic structure can easily be understood in terms of a competition between the effective 3d anisotropy (originating from the inter-unit 3d-3d interaction), and the first- and second-order R- sublattice anisotropy constants. The antiparallel coupling of the R and Mn moments originates, of course, from a strong R-Mn interaction. In an earlier paper [2] we have already presented measurements by neutron diffraction of the angle between the magnetisation and the c-axis as a function of temperature (Fig. 3). The fit of the data points, which is represented by the solid line in Fig. 3, was obtained on basis of Eqs. (7)-(9), and on adjusting the crystal-field parameter B °, the 3d anisotropy constant K3d, and the R-Mn exchange field experienced by the R-moments. For B2 ° a value of B ° = - 0.6K/R-ion was used. This value was obtained from the electric field gradient Vzz determined by M6ssbauer experiments [51, and application of the relation B ° = c t j ( r 2 ) A ° with O~j(r 2) ---- - - 5 . 0 2 (10- 3ao 2) (values for ( r z ) taken from Ref. [8]), in combination with the empirical relation A ° ~ T V z z , with 7 = - 3 5 K a o 2 / V m -1 [9]. The temperature dependence of the R-Mn exchange field was not taken into account, and for all temperatures the R-Mn exchange field at low temperatures was assumed. Furthermore, no temperature dependence of the effective 3d anisotropy was considered. The role of temperature in the competition between the different anisotropy con- tributions was incorporated entirely in the temperature dependence of the R-anisotropy.

As may be seen from Fig. 3, the best fitting calculated curve matches the experimental data reasonably well. The obtained data for the fitting parameters are: B ° = 8 x 10 -4 (K/ion) n R T = 9.86 (T f.u./~tB) K3d = -- 112 (K/f.u.). Because of the neglection of the temperature dependence of both the R-Mn exchange field and the 3d anisotropy, these

272 J.H.V.J. Brabe~w et al. /Journal ()f Magnetism and Magnetic Materials 171 (1997) 267 279

100

90

80

Q

70

6O

50 . . . . 25 50 75 100 125

T (K)

Fig. 3. Variation of the angle 0 (indicating the angle between the magnetic-unit magnetisation and the c-axis) with temperature.

values should, however, be taken with some reser- vation.

By substituting g 3 d = - - 112 (K/f.u.), ~b = 58.7 ° and 0 = 55 ° into Eqs. (4b) and (5), estimates for nl and n2 were obtained. The estimated values are nl = - 4.1 (T f.u./gB) and n 2 = 1.9 (T f.u./I-tB). For m3d, the values obtained from neutron-diffraction data [-1] were used in the calculation (#M, = 11.3 gB/f.u.). The obtained nl and n2 values have opposite sign, which is consistent with the observed helix structure. The interaction between the Mn moments in neighbouring units is ferromagnetic (nl < 0), whereas the interaction between Mn moments in non-adjacent units is antiferromagnetic (nz > 0). The absolute values of the intersublattice coupling parameters nl and rt 2 are significantly smaller than the values of the intersublattice interaction parameters nRT found in an earlier investigation [3] for RMn6Ge6 compounds with R = Gd, Tb, Dy. The order of magnitude however is the same.

4.2. High field magnetisation

In earlier papers, we already presented measurements at 4.2 K of the free-powder magnetisation of DyMn6Ge6 as a function applied magnetic fields up to 35T [1,3]. This measurement is represented in Fig. 4a. In view of the theoretical

framework derived in the present paper a more solid interpretation of this measurement is possible n o w .

An important feature of Fig. 4a is the gradual increase of the magnetisation (M) as a function of the applied field, even for small B values. Similar phenomena have previously been observed in con- nection with magnetically diluted compounds, and in those cases they were explained as a result of non-homogeneity of the magnetic interactions on a microscopic scale, enabling some moments to rotate towards B at low fields already [10]. As DyMn6Ge6 is a non-diluted compound, this interpretation is not valid in this particular case. More likely, the low field increase of M with B corresponds to a rotation of the sublattice magnetisation vectors towards the c-axis, thereby gradually lifting the conical arrangement of the sublattice magnetisation vectors. The kink at B = 21 T possibly marks a transition towards a two single sublattices configuration (2SS). The possibility of such a transition was already mentioned in Section 2. Be- yond 21 T all Mn and R moments couple as two single Mn- and R-sublattices, respectively. With increasing applied field, the sublattice moments start to bend towards each other. The slope of the magnetisation curve for B > 21 T is equal to the inverse of the D y - M n coupling constant nDy Mn: m = B/tiDy Mn [1 1]. Therefore, the high-field part of the magnetisation curve can be used to obtain information on the D y - M n coupling strength. From the results presented in Fig. 4b, one obtains nDy Mn = 6.8 (T f.u./ktB) for D y M n 6 G e 6 .

In Section 2, an analytical expression was given for the effective 3d-coupling constant K3d by Eq. (4b). From this equation it follows that K 3 d is proportional to the square of the 3d sublattice moment. Therefore, dilution of the 3d sublattice moment with non-magnetic ions might have influence on the magnetisation process: because of the quadratic dependence of K3d o n rn3a , the anisotropy constant K3d is expected to decrease rapidly as function of the dilution rate.

Fig. 4b-Fig. 4d show high-field magnetisation measurements on various DyMn6_xCrxGe6 compounds with x = 1 (b), 1.5 (c), and 2 (d) at T = 4.2 K. Earlier investigations I-5,9] have shown that Cr is hardly magnetic in RMn6Ge6

J.H.V.J. Brabers et al. / Journal of Magnetism and Magnetic Materials 171 (1997) 267-279 273

5

% ~3

000 o o ° ° °

o o

DyMnoG%

0 O0 00 0

(a)

i e i 10 2o 30 40

B (T)

6

2 e e e e e e

e • i

lO

DyMnsCrGe 6

e •

I •

2'o ' 30 B (T)

(b)

40

6 DyMn4.sCrl.sGe 6 •

V V

V V

2 ,,,,,,,v (c) V T

0 I I I

0 10 20 30 40 B (T)

6

2

DvMn4Cr2Ge6

&&

A A

~ A A A A AAAA

10 2'0 3'0 B (T)

(d)

40

Fig. 4. Free-powder magnetisation at 4.2 K of various DyMn6-xCrxGe6 compounds with x = 0 (a), x = 1 (b), x = 1.5 (c), and x = 2 (d).

compounds, and therefore it is appropriate for the purpose of diluting the Mn sublattice.

Fig. 4b shows quite a steep increase of the magnetisation curve at low fields• This steep increase has two different origins. The first one is the loss of strength of the 3d anisotropy. However, as Cr- substitutions are expected to have only little effect on the crystal-field parameters in RMn6_xCrxGe6 compounds [5], the R anisotropy will most likely be almost the same as in the pure DyMn6Ge6 compound• Therefore, a rotation of the magnetic- unit moments towards the c-axis is easier in this case. A second mechanism leading to a steep increase of the magnetisation curve is the distur- bance, by Cr substitution, of the microscopic surroundings of the different moments mentioned earlier. Between 15 and 20 T, there is a magnetic phase transition, possibly again a transition towards a 2-sublattice structure. The phase transition

is of first order as can be inferred from the observation of a weak hysteresis. The slope of the high field part of the curve can provide information on the D y - M n coupling strength• Application of the expression M = B/nDy Mn tO the high-field part of the curve yields /IOy_Mn = 6.6 T f.u./gB.

Fig. 4c shows the measurement for the compound with x = 1.5. Little can be said about this curve, besides that the slope of the high-field part of the curve is almost the same as in the curve pictured in Fig. 4b. Therefore, it is likely that the high- field magnetisation process is dominated by the interplay between the Zeeman energy and the R - M n exchange energy• One obtains rtDy_Mn =

6.1 T f.u./gB. Fig. 4d represents the high-field data for

DyMn4Cr2Ge6. As in this compound the Mn-sublattice is subject to substantial dilution, a dramatic decrease of the 3d-anisotropy constant may be

274 J.H.V.J. Brabers et al. ,; Journal o f Magnetism and Magnetic' Materials 171 (1997) 267 279

expected. On the basis of the nominal composition one derives, in combination with Eq. (4b) a drop of K3d of 56% compared to the pure compound. It is very well possible that condition (5) for a cone structure to be stable, is no longer fulfilled. In that case, the magnetic moments would align at 4.2 K along the c-axis, or in the basal plane, if A ° and A ° are not affected by Cr substitution.

Information about the alignment of the magnetic moments at low temperatures can be obtained through measurements of the magnetisation of magnetically aligned powder samples• At room temperature, the material is still very weakly magnetic, and alignment of powder particles in a magnetic field appeared to be possible• From the X-ray diffraction pattern of a magnetically aligned powder sample it was found that, at room temperature, the net magnetisation component is parallel to the c-axis for x = 2. Conclusions about the easy magnetisation direction cannot be based on this observation only. The net magnetisation may result from a magnetic-moment configuration parallel to the c-axis. But it may also result from a field induced rotation of the magnetic moments present in a heli- cal arrangement, resulting in a net magnetisation component along the c-direction. To distinguish between these two possibilities, magnetisation data on aligned samples have to be compared to the free-powder magnetisation. The free-powder magnetisation is measured, at least at low fields, with the applied field along the easy direction. Fig. 5 shows the magnetisation data for an aligned powder sample at 5 K, together with the low-field part of the free-powder measurement shown in Fig. 4d. The measurement on the aligned powder was performed with the applied field along the c-direction as derived from X-ray diffraction. Both curves match reasonably well. The discrepancy between the data represented by both curves may be caused by inaccuracies in the mass of the aligned powder as well as by misalignment of the powder grains. The magnitude of the free powder magnetisation indicates a collinear antiparallel configuration of the Dy and Mn moments (when a Mn moment of 2~tB/Mn is assumed and the Dy moments is assumed to be equal to the free-ion moment). Com- parison of both curves presented in Fig. 5 then presents reasonable evidence for a very small or

2.50

2.00

=. 1.50 % 1.00

0.50

0.00 0

, , . •

B (T)

Fig. 5. Low-field magnetisation of DyMn4Cr2Ge6 at 5 K: (O) measurement on an aligned sample with BIIc-axis; (A) low field part the free powder measurement represented in Fig. 4d.

even zero cone angle in DyMn4Cr2Ge6, and an antiparallel configuration of the Dy and Mn moments.

The small cone angle implies that the FS structure becomes less stable and vanishes, and the 2SS structure becomes stable, even in very low applied fields. As a consequence, the 3d anisotropy vanishes [see Eq. (4b)], and the curve presented in Fig. 4d can be interpreted as arising from the normal be- nding process of two single sublattices in an applied field• Below about 15 T one basically has two collinear sublattices consisting of Dy and Mn moments coupled antiparallel. Above the critical field rtDy MnIMDy- MM,I-~ 15T, the Mn and R moments start to rotate gradually towards the applied field direction. The high-field linear part of the curve for B > 15 T is described through B = M~ nDy Mn" From the high field slope of the curve, one derives nR_Mn = 6.4 T f.u./~t~.

The observed nay Mn values of all compounds investigated are listed in Table 1. The noy-M, value for DyMn6Ge6 obtained from high-field data, is substantially lower than the value obtained from the fit of the temperature dependence of the angle 0, presented in Fig. 3. However, because of the rather crude model assumptions on which the fitting procedure was based, this discrepancy is not a real cause for concern.

Using a simple mean-field analysis [11], the nDy Mn values can be related to the microscopic JR M. coupling constants. The J a ~ , values are also listed in Table 1.

J.H.V.J. Brabers et al. / Journal of Magnetism and Magnetic Materials 171 (1997) 267-279

Table 1 nR Mn and JR Mn coupling parameters, ordering temperatures (TN), and K3d values

275

Compound nR M, (T f.u./l.tB) JR Mn/k (K) Tr~ (K) K3d (K/f.u.)

DyMn6Ge6 6.8 - 9.1 423 a - 90 DyMnsCrGe6 6.6 - 8.9 314 - 72 DyMn,.sCrl.sGe 6 6.1 - 8.2 235 - 65 DyMn4Cr2Ge6 6.4 - 8.6 174

Walue taken from Ref. 1-12].

4.3. Magnetisation vs. temperature

An additional tool to investigate the role of Cr substitutions on the stability of the FS structure is the measurement of the magnetisation (M) as a function of temperature (T). Fig. 6a~-Fig. 6dl show M vs. T for four different DyMn6_xCrxGe 6 powder samples (with x = 0, 1, 1.5, and 2, respectively), taken from the same ingots as those used in the high-field investigation described in the previous section.

For the pure DyMn6Ge 6 compound (Fig. 6a0, the magnetisation slowly increases with T between 5 K and 55 K. Above 55 K however, the magnetisation drops sharply with temperature and remains very low up to 350 K. Most likely, this temperature dependence of the magnetisation is related to the transition from the FS to the SS structure, an ex- planation which was mentioned already earlier in Ref. [1]. The magnetic ordering temperature (T ~ 423 K [12]), exceeds the temperature range covered by Fig. 6al.

For DyMnsCrGe6 (Fig. 6b~), the behaviour of M vs. T follows a similar scheme as in case of DyMn6Ge6. First, the magnetisation increases with T until a maximum is reached at 95 K. Then, at temperatures above 95 K, the magnetisation decreases rapidly with temperature, and remains very small at temperatures above 120 K. The peak at 315 K marks the breakdown of the magnetic ordering.

Most likely, the temperature dependence of the magnetisation of DyMnsCrGe6 is also related to the transition from an FS to an SS configuration. The low-temperature regime would then corre- spond to the FS structure, whereas at higher temperatures the SS structure is stable. As can be

inferred from Fig. 6bi, the net magnetisation at very low temperatures is not very much different from the magnetisation at higher temperatures where the SS structure is expected to be stable. Therefore, the compound DyMnsCrGe6 is probably close to compensation at low temperatures, in that sense that the antiparallel R and Mn moments belonging to the same magnetic unit cancel. The maximum in the magnetisation at 95 K can be explained as the result of an interplay between the decrease of the R-sublattice moments with temperature, causing an increase of the net magnetic moment per unit, and the transition from FS to SS, making the total net magnetisation gradually van- ish. The weak maximum at 55 K shown in Fig. 6al probably has the same origin, although, at this level, the possibility of domain-wall effects ob- structs sure statements on this matter.

Fig. 6cl shows the magnetisation as a function of temperature for DyMn4.sCrLsGe6. A compensation point at Tcomp ~ 85 K is clearly recognisable. A remarkable observation however, is that above Tcomp the net magnetisation does not show a gradual increase over a wide temperature range (like in many other compounds with a compensation point). Instead, there is only a very low, narrow peak around 95 K, and at temperatures above 100 K, the magnetisation remains rather low. This low magnetisation state persists up to the magnetic ordering temperature marked by the small cusp at about 230 K.

Most likely the temperature dependence of the magnetisation of DyMn4.5Crl.sGe 6 is also strongly dominated by a transition from the FS to the SS phase, just like in the previously mentioned cases of DyMn6Ge6 and DyMnsCrGe6. Such a transition may then result in a cut-off of the net magnetisation

276 J.H. V.J. Brabers et al. ,"Journal (?/'Magnetism and Magnetic Materials 17/ (1997) 267 279

0 . 4 0 [ ~ / " - D y M n e G e 6

a 1

0 . 0 0

0 . 2 0

1 . 2 0 [

0 . 8 0

0 . 4 0

0.00

K3d = -90 K/f.u.

a 2

0 . 8 0

0 . 4 0

E 0 . m C~ 0.00 (D E C~ 0 . 4 0

0 . 2 0

~ t

DyMn~CrGe 6

, ,

b 1

%. 4 " ' , " , , ~ . . ~ . . . , . - ~ " ~ ' ,%

DyMn4.sCr ~.~C~e e

0.80 [

0 . 4 0

0.00

1 . 2 0 ~ 0 . 8 0

O.4O

0 . 0 0 0 . 0 0

i/ K3d = -72 K/f.u.

b2

i r

K3d = -65 K/f.u.

C 2

0 1 0 0 2 0 0 3 0 0 4 0 0

2 . 0 0

1 . 0 0 *,%

".

v v

DyMn4Cr2Ge e

dl

0 . 0 0 " ' - ' ......... 0 1 0 0 2 0 0 3 0 0 4 0 0

Temperature [K] Fig. 6. Measured and calculated magnetisation ofDyMn6 ~CrxGe6, asafunctionoftemperature. Fig. 6al Fig. 6dl:experimentaldata for x = 0, 1, 1.5, and 2, respectively. Fig. 6a2-Fig. 6c2: calculated magnetisation curves for x = 0, 1, and 1.5. The data presented in Fig. 6al were taken from Ref. [1].

jus t above Tcomp , thereby l imit ing the M vs. T curve above 95 K to a small peak as observed in Fig. 6c 1.

In Fig. 6dz, the magne t i sa t ion of D y M n 4 C r z G e 6 is shown as a funct ion of tempera ture . In con t ras t

to the measurements presented in Fig. 6 a l - Fig. 6cl, the curve shows a s t rong m o n o t o n o u s decrease of the magnet i sa t ion , wi thout any peaks, up to 75 K. At t empera tu res h igher than 75 K, the

J.H.V.J. Brabers et aL / Journal of Magnetism and Magnetic Materials 171 (1997) 267-279 277

magnetisation remains very low and without any anomalies, except for a shallow maximum near 175 K, marking the magnetic ordering temperature.

The suggested physical picture behind the temperature dependence of the magnetisation, described in terms of a temperature dependence of the R-sublattice moments and a transition from the FS to the SS configuration, allows straightforward numerical calculation of M vs. T within the scope of the theoretical framework presented earlier. The R-sublattice moments can be calculated as thermal averages, using the combined energy level splittings obtained on the basis of the crystal-field interaction, and the R-Dy exchange interaction. The net magnetisation follows from the component (Mz) of the magnetic-unit magnetisation along the e-axis. The angle 0 between the magnetic-unit moments and the c-axis, needed for the calculation of Mz, can be obtained as a function of temperature from Eq. (7) by taking K~ equal to the joint anisotropy constant K1 = Kaa + K3d, and by using the expressions (8) and (9) for KIR=K1R(T) and K2a = KaR(T), respectively. In a first approximation, the influence of temperature on the 3d-sublattice moment can be neglected, as the temperature usually affects the R sublattice magnetisation much stronger than the 3d sublattice magnetisation. As in the present investigation K3a is also assumed to be independent of temperature, the temperature dependence of the magnetisation is then determined entirely by the temperature dependence of the aver- age R moment per unit, and by the temperature dependence of the R anisotropy.

The calculational procedure described above was used in an effort to reproduce the characteristic features of the curves represented in Fig. 6aa- Fig. 6c 2. For the calculation of KaR and K2R, the same values for A ° and A ° were taken as those obtained from the fit of 0 vs. T presented in Fig. 3. As the Cr substitutions are expected to have only little influence on the crystal field, these values were employed for all compounds. Appropriate choices for the nDy Mn coupling constants used in the calculation were obtained from the previous section. The Mn moment and the effective 3d anisotropies (K3d) were allowed to vary for each compound. The results of the calculations are represented in Fig. 6a2-Fig. 6C 2.

Fig. 6a2 and Fig. 6b2 represent the results for DyMn6Ge6 and DyMnsCrGe6, respectively. For DyMn6Ge6 the Mn moment obtained from neutron diffraction at 11 K was employed (11.3 ~tB/f.u.), whereas for DyMnsCrGe6 the Mn-moment was taken equal to 10 ~tB/f.u. The values for K3d were chosen in such a way that the maxima of M in the calculated curves agree with the maxima in the measured curves (Fig. 6al and Fig. 6bl). The curves in Fig. 6a2 and Fig. 6b2 match the curves in Fig. 6al and Fig. 6bl quite well. It is worth noticing that the results presented in Fig. 6a2 and Fig. 6b2 were obtained with a substantially higher K3d value for DyMnsCrGe6 ( -72 K/f.u.) than for DyMn6Ge6 ( -90 K/f.u.). This result is consistent with Eq. (4b), which predicts a decrease of the strength of the effective 3d anisotropy with decreas- ing 3d moment. It is worth noticing that the K3d- value found for DyMn6Ge6 on basis of the magnetisation vs. temperature data (K3d = - 90 K/f.u.) shows a slight deviation from the K3d-Value found from the fit of the temperature dependence of the angle 0 represented in Fig. 3 (K3d = --112 K/f.u.). The reason for this discrepancy may be found in the fact that the value of K3d---- --112 K/f.u. was obtained on the basis of a calculational procedure in which nDy Mn was used as an adjustable parameter, whereas the value of K3d = --90 K/f.u. was derived from a calculation taking a fixed value for nDy Mn' However, the differences between the Kad-Values obtained from both calculations are acceptable in view of the crude assumptions on which both calculations were based.

Fig. 6c2 represents the results for DyMna.sCrl.sGe6. For the calculation, the Mn moment was taken equal to 8.75g~, and the K3a constant equal to - 65 K/f.u. Both the experi- mentally observed compensation point and the small peak in the magnetisation just above Teomp appear in the calculation. However, the K3d value used in the calculation should be considered with great care as it is only slightly smaller than the value taken for DyMnsCrGe6, whereas Eq. (4b) suggests that it should be in fact much smaller due to the differences in the 3d moments of DyMnsCrGe6 and DyMn4.sCrl.sGe6.

The experimental results for DyMn4Cr2Ge6, presented in Fig. 6da, could not be reproduced in

278 J.H. k~J. Brabers et al. / Journal oj'Magnetism and Magnetic Materials 171 (199D 267 279

a satisfactory way. The reason for this is probably the neglection of a temperature dependence of the 3d-moments. Lowering the Mn concentration causes lowering of the magnetic ordering temperatures, and therefore the temperature dependence of the Mn moments becomes important at much lower temperatures for the lower Mn concentrations than for the higher Mn concentrations.

As can be seen in Fig. 6ai and Fig. 6bl the magnetic ordering temperatures Ty in DyMn6Ge6 and DyMnsCrGe6, are very well above and near room temperature, respectively. In the temperature regime of interest at present (T < 100 K), the effects of temperature on M can therefore be neglected in these cases. For DyMn4.sCrl.sGe6 and DyMn4Cr2Ge6, having TN far below room temperature, this is probably no longer the case, and in reality the Mn moments probably decrease strongly with temperature below 100 K. As a consequence the R-Mn exchange field strongly diminishes with temperature in these compounds, causing at low fields already a strong decrease of the thermal aver- age that determines the 4f moments. This implies a rapid breakdown of the R-anisotropy with temperature, which in turn facilitates the transition from the FS to the SS structure (see Section 2). An approach neglecting the temperature dependence of the Mn moments fails in this particular respect. A calculation procedure which does take into account the temperature dependence of the 3d moments, is expected to provide more satisfactory results. As may be inferred from Eq. (4b) K3d o~ mzd will also decrease with temperature in such an approach. The resulting low temperature values of K3d are therefore expected to be significantly less negative than the effective K3d values (K3d < 0), found in the present investigation for DyMn6Ge6 and DyMnsCrGe6, as the decrease of the R-Mn exchange field with temperature tends to increase (make less negative) the K3d value necessary to have a stable SS structure at a certain temperature.

In spite of the difficulties met in the calculations of DyMn4CrzGe6 and DyMn4.sCrl.sGe6, the satisfactory agreement between experiment and theory in case of the other compounds suggests that the basic physical principles that govern the temperature dependence of the magnetisation are quite well understood, and that it can be successfully

described by means of simple, phenomenological models.

5. Conclusions

In this paper, we have indicated how the 3d-3d exchange interaction between Mn moments belonging to different magnetic units contributes to the (effective) Mn anisotropy. This contribution can be fairly large, and probably plays a crucial role in the stabilization of the FS structure in DyMn6-xCrxGe6 compounds.

In case of free-powder magnetisation measurements at 4.2 K, field-induced transitions are possible from the FS structure to the 2SS structure. In the latter structure the Dy as well as the 3d sublat- rice have a collinear moment arrangement, and the two corresponding sublattice magnetisations are antiparallel but will bend towards each other ac- cording as the field strength increases. These transitions are of first order and arise from the interplay between the Zeeman energy and the 3d-3d exchange interaction. Possibly, the observed field induced phase transitions in DyMn6_xCrxGe6 with x = 0 and x = 1 are of this particular type. From the high-field parts of the free powder magnetisation we obtained estimates for the rtDy Mn mean field parameters of the respective compounds. The nDy Mn values obtained are roughly of the same magnitude as those found for other DyMn6_ xCr~Ge6 compounds.

The temperature dependence of the magnetisation of DyMn6_ xCrxGe6 compounds is dominated by a gradual transition from the FS to the SS structure, as can be derived from M vs. T measurements. Numerical calculations show that for low Cr concentrations the temperature dependence of the magnetisation can be reproduced in terms of the theoretical framework outlined in this paper.

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the magnetisation process of the ferrimagnetic spiral structure in dymn6−xcrxge6 compounds

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