Dynamic magnetic properties of ferromagnetic and antiferromagnetic Heusler alloys

Journal of Magnetism and Magnetic Materials 50 (1985) 7-18 7 North-Holland, Amsterdam

D Y N A M I C M A G N E T I C P R O P E R T I E S O F F E R R O M A G N E T I C A N D A N T I F E R R O M A G N E T I C H E U S L E R A L L O Y S

P.J. W E B S T E R

Department of Civil Engineering, Unwersity of Salfora~ Salford M5 4WT, UK

and K.R.A. Z I E B E C K

lnstitut Max yon Laue-Paul Langevin, 156X, 38042 Grenoble Cedex, France

Received 5 November 1984

Inelastic neutron scattering measurements on a ferromagnetic Heusler alloy, Pd 2 Mnlnl-x Snx at the composition x = 0.75, have established the spin wave dispersion in the three principal symmetry directions. The results have been interpreted using a simple Heisenberg model in which the exchange constants are of long range, extending beyond 12 .~. The Curie temperature, spin wave stiffness constant and the thermal variation of the magnetisation, calculated using the derived exchange parameters are in close agreement with observation.

Anomalies in the spin wave dispersion, which are also present to a lesser degree in Pd2MnSn, have been interpreted as precursor effects associated with the onset of antiferromagnetic ordering, type AF3A, which is the magnetic structure observed in the range 0.2 ~< x ~< 0.6.

1. Introduction

Heusler alloys are ternary intermetallic compounds based on the composi t ion X2YZ that have the cubically ordered L2 t structure shown in fig. 1. In general X represents a transition metal element f rom group VIII , or noble metal; Y is normal ly

Pd Mn Pd In/Sn !

Fig. 1. The Heusler, L21, structure.

manganese and Z is an element f rom group III , IV or V. The majori ty of the compounds order ferromagnetically and, with the exception of those for which X represents cobalt or perhaps nickel, the momen t is substantially confined to the manganese a toms and is close to the 4/~ B expected for a doubly ionised a tom (Mn 2÷) [1].

Since the manganese atoms, ordered on an fcc superlattice, are separated by distances greater than 4 ,~ there is negligible overlap of their d wave functions and exchange coupling must therefore be indirect. I t has been suggested that the d electrons responsible for the magnetism do not participate in the Fermi surface. Support for this opinion is provided by ' s ' like character of the t ransport properties and the apparent localised behaviour of the magnetic properties. The importance of long-range interactions has been under- lined by inelastic neutron scattering measurements of the spin wave dispersion [2,3]. Several different exchange mechanisms have been proposed, for example indirect mechanisms such as R K K Y or double resonance involving s - d or d - d coupling,

0304-8853/85 /$03 .30 © Elsevier Science Publishers B.V. (Nor th -Hol land Physics Publishing Division)

8 P.J. Webster, K.R.A. Ziebeck /Dynamic magneticproperties of Heusler alloys

or superexchange [4-8]. The signs of the exchange constants, derived from the neutron measurements, were found to change in a manner expected for Friedel oscillations thus lending support to the indirect mechanism. The importance of the conduction electrons, and the electron to atom ratio, in stabilising the crystal structure has long been recognised following the work of Hume-Rothery. The effect on the magnetic properties of varying the conduction electron concentration has been more recently demonstrated [9,10] on the quater- nary Heusler alloys P d 2 M n ( I n / S n ) and Pd2Mn(In/Sb ). These measurements, together with evidence of systematic trends derived from considering all the compounds, have led to specific roles being attributed to the X and Z sites. It has been conjectured that the principal role of the X atoms is to determine the lattice parameter and it is the Z atoms that provide the valence electrons important in mediating the magnetic interactions [5,8]. However, the behaviour of other related series [11,12] and measurements in progress indicate that this hypothesis is too specific.

The series Pd2Mnln~_xSn x is of particular in- terest since it provides the possibility of studying the transition from antiferromagnetism to ferromagnetism as the electron concentration changes. Whereas P d2M nIn is an antiferromagnet, Pd2MnSn orders ferromagnetically and a second antiferromagnetic structure is observed at an intermediate composition. The dynamics of magnetic phase transitions in Pd2Mn(In/Sn) will be the subject of a series of papers. Reported here are the results of an inelastic neutron scattering investigation of the spin wave dispersion in the ferromagnet Pd2Mnln0.25Sn0.75.

2. Properties of Pd2Mnln 1_ xSn~ 'alloys'

2.1. x = O

Depending upon the heat-treatment the degree of atomic order can be altered from largely L21, the Heusler structure as shown in fig. 1, to predominantly B2 in which there is preferential disorder between Mn and In sites. When slow-cooled Pd2MnIn is 80% ordered in the L2~ structure with

some 20% B2 type disorder. On quenching the compound the proportions are essentially reversed with smaller regions of L21 order in a matrix 80% B2 ordered. The lattice parameter, 6.373 A, is insensitive to the degree of chemical order. Static susceptibility and neutron diffraction measurements indicate that both the slow-cooled (sc) and quenched (q) alloys order antiferromagnetically below 142 and 112 K, respectively. Above these temperatures the susceptibilities are Curie-Weiss but with slightly different Curie constants. Effec- tive moments of 4.9 and 4.8 /~B per manganese atom were determined for the (sc) and (q) samples, respectively.

Early neutron diffraction powder measurements [13] had shown that the alloys order antiferromagnetically in both L21 and B2 regions but with different magnetic structures. The L2~ structure regions were found to have the fcc type 2 antiferromagnetic order whereas the B2 structure gave rise to simple cubic type 1 order.

2.2. x = 1

Pd2MnSn forms a single phase Heusler compound with a very high degree of L21 order, and a lattice parameter 6.380 A. Below 189 K the compound orders ferromagnetically with a moment equivalent to 4.23#B at the manganese sites. Early polarised neutron diffraction measurements [14] suggest an upper limit of 0.2~t B may be associated with the palladium sites. Improved measurements on isostructural PdEMnSb [15] indicate that an upper limit of only 0.05/~ B could be associated with palladium atoms. It was found, from inelastic neutron scattering experiments [2,16], that at 5 K spin waves disperse out to the zone boundary in the three principal symmetry directions. A slight kink was observed in the dispersion along the (00~) direction at a wavevector of ~ = 0.5. Ad- ditionally there was a tendency for the dispersion to turn over, i.e. to adopt a negative gradient, at the (110) zone boundary. A least squares fit of a Heisenberg model to the observed data could only be made satisfactorily by incorporating exchange interactions out to the eighth nearest neighbour. Qualitative agreement of the derived exchange parameters, which were found to oscillate in sign,

P.J. Webster, K.R.A. Ziebeck / Dynamic magnetic properties of Heusler alloys 9

was obtained with the double resonance model. Improved agreement was obtained by extending the model calculation beyond the asymptotic approximation. Subsequent band structure calculations have yielded good agreement with experiment. The spin waves were found to renormalise with temperature and to disappear close to T c. The thermal variation of the magnetisation could be explained entirely by the thermal population of the spin waves. Above T c the static uniform susceptibility exhibited Curie-Weiss behaviour, yield- ing an effective moment of 4.96/~ a per manganese atom and an atomic moment of/~p = 4.0/~ B, where pfff = #p(/~p + 2). The Rhodes-Wohlfarth ratio is close to unity as expected for localised behaviour.

Further justification for the employment of the Heisenberg model was provided by neutron diffraction measurements of the paramagnetic response [16]. Complementary inelastic neutron scattering measurements have confirmed that the second moment of the response is in close agreement with the mean field prediction of the Heisen- berg model. Polarised neutron and polarisation analysis measurements at room temperature and up to 4T c revealed that the wave vector depen- dence of the response follows that expected for randomly aligned localised moments having a 3d form factor. At each wavevector the integrated response corresponded to S ( S + 1).

4.0

= 80

u 60 u - -

g, 4o

"~ 20

g o

In

Smp EIecfron concentration n

4.4 4.8 52 5.6 6.0

a / I 20 - / AF3 / FERIROMAGNETIC 40

- x , ~ , , i 80 : i ~ i I 0 0

(12 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 Sn Sb

(In~. x Sn x) lSnl- x Sb x 1

ATOMIC [OMPOSITION X

4.0 4.4 4.8 5.2 5.6 6.0 p u i n [ I i I u

200

150 , ~ o Bo

1 O0 ~ , B N 50 A 8 F

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 In (In1_ x Sn x ) Sn (Sni. x Sb x } Sb

ATOMI[ [OMPOSITION X

Fig. 2. (a) Magnetic phase diagram; (b) paramagnetic and ferromagnetic Curie temperatures Op and O F and Nrel temperatures 9 N versus atomic composition x and s - p electron 9~oncentration n for the alloys Pd2Mnln l_xSn x and Pd2 MnSn ] _xSbx.

2.3. O < x < l

The series Pd2Mnlnl_xSn x forms a solid solu- tion of chemically single phase alloys which are predominantly ordered in the L21 structure [9]. Close to the composition Pd2Mnln the degree of chemical order becomes progressively sensitive to heat-treatment but when slow-cooled all the alloys have a substantial degree of L21 order. The magnetic phase diagram and the variation of the bulk magnetic properties with composition x are shown in fig. 2. The related isostructural series Pd2Mn- In l_ySby [10] exhibits similar behaviour. The increase in electron concentration, obtained by re- placing In with Sn or Sb, is accompanied by an overall increase in the strength of the ferromagnetic exchange interaction. The evolution towards the ferromagnetic state was found to be twice as

rapid when Sb was added rather than Sn. In effect, similar magnetic structures are observed in the two series when y = 2x. Powder neutron diffraction measurements revealed that the size of the magnetic moment remained essentially constant as the electron concentration was varied. However, the magnetic order changed from fcc type 2 antiferromagnetism (AF2) to fcc type 3 (AF3A) and finally to ferromagnetism (F), as shown in fig. 2a, as the electron concentration was increased. Mixed magnetic phase regions, (AF2 + AF3A) and (AF3A + F), were observed at intermediate compositions between the single phase regions. Recent high precision neutron, magnetisation and N M R measurements have confirmed that the intermediate (AF3A + F) phase is indeed mixed and not a canted magnetic structure. The unmixed antiferromagnetic structures observed ~re shown in fig. 3.

10 P.J, Webster, K.R.A. Ziebeck / Dynamic magnetic properties of Heusler alloys

z [111]

a)

z[001]

×

b)

Fig. 3. Face-centred cubic antiferromagnetic structures (a) type 2; (b) type 3A.

The present paper is concerned with inelastic neutron scattering measurements on the single phase ferromagnet Pd z MnIn025 Sn o.75.

3. Experiment

3.1. Sample preparation

A single crystal of Pd 2 Mnln 0.25Sn 0.75 was grown by Perrier de la Bathie, of the CNRS Grenoble, using the Bridgman technique and dements of purity 99.99%. The specimen was approximately cylindrical, 30 mm long by 10 mm diameter, with a (110) axis parallel to the axis of the cylinder. Subsequent neutron diffraction measurements confirmed that the crystal was of excellent quality, highly ordered in the L21 structure, with a mosaic spread of 0.3 ° .

3.2. Crystal structure

The Heusler L21 structure may be conveniently considered as four interpenetrating fcc sublattices A, B, C, D with origins at (0, 0, 0), (1/4, 1/4,

1/4), (1/2, 1/2, 1 /2) and (3/4, 3/4, 3/4). In Pd2Mn(In/Sn) , Pd atoms may be considered to occupy the B and D sites and Mn and I n / S n the A and C sites, respectively. The symmetry of the space group is Fm3m and gives rise to Bragg reflections with non-zero structure amplitudes when the indices are unmixed. The structure amplitudes are:

h, k, l all odd

F ( l l l ) = 41[(SA - / c ) 2 +(f,,-fo)2]'/~l, (h + k + / ) / 2 odd

F(200) = 4lfA --fB + f c --fol ,

(h + k + l ) / 2 even

F(220) = 4If g +fa +fc +fo l ,

where fABCD are the average scattering amplitudes for the respective sublattices.

F(111) and F(200) correspond to superlattice reflections and as such are order dependent whereas F(220) represents the fundamental bcc reflections that are unaffected by the state of order. For an alloy X2YZ ordered in the Heusler structure the neutron nuclear structure amplitudes become:

F ( l l l ) = 41 b v - b z i,

F(200) = 41 b v + b z - 2b x l,

F(220) = 4 Ib v + b z + 2b xl,

where bxv z are the neutron nuclear scattering lengths for the X, Y and Z atoms.

In the B2 structure Y and Z atoms occupy the A and C sites equally and randomly, reducing F ( l l l ) to zero but leaving the even reflections unaffected. Neutron diffraction measurements confirmed both that the alloy investigated was highly ordered in the L2 i structure and that the magnetic moment was essentially confined to the manganese atoms. Thus, as only the manganese atoms have non-zero magnetic scattering amplitudes, the magnetic structure amplitudes corresponding to the three types of reflection, odd and even superlattice and principal are identical with:

V(Mag) = 4[ PM, [,

where the magnetic scattering amplitude PMn =


0.269#Mnfun, and /~ and f are the magnetic mo- sketched out by performing measurements out at ment and form factor, respectively, large wavevectors.

3.3. Measurements

The spin dynamics of Pd2Mnln0.25Sn0.75 were investigated between 6 and 200 K, using the triple-axis spectrometer IN8 at the ILL, Grenoble. The crystal was mounted, with a (110) axis verti- cal, in a variable temperature cryostat. A silicon diode in thermal contact with the sample was used to measure the sample temperature which was maintained stable to ___0.2 K. The spectrometer was operated with fixed final wavevectors of between 2.662 and 4.0 ,g-1 depending upon the magnitude of the energy transfer. When the final wavevector used was 2.662 ,~-t a pyrolytic graphite filter was aligned in the scattered beam to minimise higher order contamination. The incident neutron beam was monochromated by the (111) planes of a copper crystal and the scattered beam was analysed by the (002) planes of pyrolytic graphite. Soller collimation of successively 50', 40', 40' and 40' was employed between the reactor and the detector. The experimental arrangement en- abled the instrumental resolution to be easily changed dependent upon the observed features of the dispersion surface. The dispersion curves were obtained by carrying out both constant q and constant ~0 scans.

The dynamics of the ferromagnetic fcc lattice were investigated primarily in the Brillouin zones centred on the reciprocal lattice points (111) and (002). Since the moment is essentially confined to the manganese atoms, the magnetic structure factors characterising the dynamic behaviour are expected to be similar in both zones. Confining the measurements to relatively small wavevectors relatively enhanced the magnetic cross-section since the form factor fMn was near maximum. Ad- ditionally the phonon cross section, which varies as Q2, was minimised. Furthermore the structure factors for acoustic phonons in the superlattice Brillouin zones (111) and (200) are small. Any ambiguity, between magnons and phonons, was checked by repeating the scans in different zones. Conversely the acoustic phonon dispersion was

4. Results and discussion

The detailed investigation of the spin wave dispersion at 6 K was confined to the three principal symmetry directions (001), (110) and (111). Less extensive measurements were also made at several temperatures between 6 K and the Curie temperature 173 K. As in PdEMnSn the spin waves were found to propagate out to the zone boundary, as would be expected for a system in which the density of single particle modes in the thermal region is low. Thus it may be anticipated that Stoner excitations are of negligible importance in characterising the thermal properties of Pd 2 Mn(In/Sn).

The spin wave dispersion curves for the three principal symmetry directions are shown for PdEMnIn025Sn075 in fig. 4. For wavevectors up to = 0.4 A-a ' the spin wave dispersion was found to be essentially isotropic. Distinct anomalies are observed in the dispersion at (0, 0, 0.67) in the (00~) direction, and at the zone boundary, the K symmetry point, in the (~0) direction. Furthermore there is a definite tendency for the spin wave dispersion to 'flatten' to give a significant region

6

4

g

u.

2

I

0.5

P

I I O.5

00~ x K ~0

r /

O.5

Fig. 4. Spin wave dispersion in Pd2MnIno.25Sno.75 showing marked anomalies in the (00~) and ( ~ 0 ) directions. The continuous line represents a least squares fit using eight exchange interactions.

12 P.J. Webster, K.R.A. Ziebeck / Dynamic magnetic properties of Heusler alloys

with near zero slope as the zone boundary is approached in the ( ~ ) direction.

Although a distinct anomaly was also observed at the K point in Pd2MnSn only a slight inflection was observed for spin waves dispersion in the (00~) direction [2].

Least squares fits to the observed dispersion curves were carried out using a Heisenberg model, hw= 2S[Jo-Jq], with varying numbers of exchange constants. It is evident that, with the structure observed in the dispersion curve, a large number of Fourier components is required in order to obtain a satisfactory fit. Noda et al. [2] also used a Heisenberg model to analyse the spin waves in Pd 2MnSn and concluded that exchange constants extending up to the eighth neighbour were required. The analysis for Pd2MnIn0.25Sn0.75 led to the requirement of nine exchange parameters even though the anomalies were much more pro- nounced. The quality of agreement, R, between the observed and calculated spin wave frequencies was indicated by a X 2 test. The R factor was found to improye from 0.11 with six exchange parameters to 0.04 with nine parameters. Extend- ing the number of interactions beyond nine nearest neighbours did not significantly improve the quality of the fit. The exchange parameters derived from the best fit are listed, and compared with those for PdzMnSn, in table 1. The calculated dispersion is compared with the measured points in fig. 4. It may be seen from the figure that the agreement between calculation and observation is good, and any discrepancies are small, i.e. within

Table 1 Calculated exchange interactions for the Heusler alloys

Pd2Mnlno.25Sno.75 and Pd2MnSn [2]

Neighbour z i Ri(A ) Pd2Mnlno.25Sno.75 Pd2MnSn

J(THz) J(THz)

1 12 4.5153 0.037 0.055 2 6 6.3830 0.025 0.027 3 24 7.8175 0.005 0.013 4 12 9.0269 - 0.017 -0.032 5 24 10.0924 0.016 0.011 6 8 11.0557 -0.019 -0.010 7 48 11.9415 0.006 0.003 8 6 12.7660 -0.022 --0.003 9 12 13.5404 -0.007 -

the statistical error. However, it should be noted that the fit is systematically less good in the (00~) direction. This feature will be discussed later. The exchange constants, listed in table 1, may be seen to oscillate in sign both in Pd2Mnlno.zsSno.75 and Pd2MnSn. The exchange constants are initially positive and decrease in magnitude with distance as expected ( J = t2/w, where t is the hopping integral and w is the bandwidth). At the fourth neighbour distance there is a change in sign and beyond both positive and negative exchange constants occur. These' features underline the relative importance of the higher order Fourier components required to fit the observed data from the Pd2Mn(In /Sn ) series.

In the long wavelength limit for cubic crystals

ho% = O ( T ) q 2 (1)

and the spin wave stiffness constant D, at low temperatures, is given by

D ( T ) = D(0) [1 - D , ( T / T c )5/2], (2)

where the constant D l is

D I 3" D(0) . 4~D(0) (3)

and

D(O) = ½SY'ziJ~ri 2, (4) i

where V 0 is the atomic volume per Mn atom and ~(5/2) is the Riemann zeta function of values 1.341.

Using eq. (4) the calculated spin wave stiffness constant, D = 21 THz A2, was found to be in close agreement with the mean value, D = 19.4 THz ~2, obtained from the observed spin wave dispersion in the three principal symmetry directions at low q. Although the thermal variation of the spin wave stiffness constant can be fitted to a function of the form given by eq. (2) the constant D 1 obtained, 0.6 T H z / T 5/2, is significantly different from that calculated using eq. (5), which gives D 1 = 0.077 T H z / T 5/2. A similar discrepancy occurs also between the observed thermal variation of the static magnetisation and the low temperature approxi-


mation

M ( T ) = M(O)(1 - AT3~2), (5)

where

Vo " 3" [ kB 13/2 A = ~-~'[=l [ 4,r~-(O ) ~ (6)

Reasonable agreement was obtained with the observed thermal variation of the spin wave dispersion and magnetisation using a Heisenberg modal and considering only the dynamics of two magnon processes as represented by the equations:

l htdq ---- htoq(O) -'-'~ E <n> [ J(0) - J ( q)

K

+ J ( t c - q ) - J ( r ) ] (7)

and

n~ = [ e x p ( h o ~ ( T ) / r T ) - 1]-1 (8)

The spin wave anomalies, particularly in the (00~) and ($~0) directions, give rise to a high density of magnon states above 2.3 THz. The density of magnon states was determined by computing 511988 lattice sums using a cubic mesh in the positive quadrant of the truncated octahedra. At the resolution employed, distinct peaks were re- solved at 2.66 and 3.54 THz in the magnon density of states. The short wavelength fluctuations thus significantly contribute to the thermal variations of the magnetic properties which to first order are accounted for by simple Bloch description. Such behaviour could have been anticipated since the maximum spin wave frequencies are of the same magnitude as that corresponding to the Curie temperature. A simple mean field approximation using the derived exchange parameters was used to estimate the asymptotic Curie temperature

k~T c = ( 2 / 3 ) ( S ( S + 1))E4. (9) i

The calculated value of 169 K is in surprisingly good agreement with the experimental value of 178 K even though the higher order exchange constants are anomalously large.

At low temperatures the magnon groups were well defined although the instrumental resolution

200

I0C

200 i 100

~200 t_t ,

o ~ 100

z 2 0 0

100

I I I I

o o o o 0 Oooo~ooOooo°Oo°O °oo°Oo°Oo°oOO°oO

2WK i o

o 0 OooO°OOO~OOOo°O°° °°O°°OOooOooooOo(

2 5 0 K

° ° o OOooOoo( oO°oo 2~K

OooO° o o o oOoO°oooo¢o °o

oo o )°°°°° °°°°°°°°4 °°° o o

°°°OOOOoOOOoOo o(

1 8 5 K o°o o

200 oo o OoOoooOooooO~oO o 100Lo °°°°°°o oooO°o(

O~ 150K o°o o 20 oo o 0~)000000000000~)00 0000000000000(

10 l O O K

0 / I i I I 0.6 0.65 0.7 025

(oo~)

Fig. 5. Thermal variation of the elastic peak at (0, 0, 0.67) for Pd 2 MnIn o.25 Sn 0.75.

precluded any conclusion regarding their intrinsic life times. As the temperature was increased the groups remained well defined up to = 0.7T c be- fore collapsing close to T o The anomalies in the dispersion curves also persisted up to at least 0.7T c. Above T c the response was diffusive in nature consistent with randomly aligned local moments of fixed amplitude. Since in the system Pd 2 MnInl-xSnx there is a move towards antiferromagnetism as x decreases it is tempting to inter- pret the observed spin wave anomalies as possible soft modes. However, the observed positions of the anomalies at approximately (0, 0, 0.67) and (0.7, 0.7, 0) do not correspond to rational fractions of the chemical unit cell, and neutron diffraction measurements did not reveal the existence of any additional magnetic structure.

A wavevector scan along the (00~) direction, with the spectrometer set at the elastic condition, revealed a small peak at (0, 0, 0.67). The intensity of this peak was low, several orders of magnitude


0 1 0 0.5,1,0 110

1 , 0 . 5 , 0

000 I00

Fig. 6. Encrgy contours (in THz) for Pd2Mnln0.25Sno.75 in the (001) plane showing depressions extending towards the position of the (1, 1/2, 0) and (I/2, 1, 0) antifcrromagnctic AF3A reciprocal lattice points.

smaller than the intensity of the nuclear Bragg peaks associated with the L21 structure. Without prior indication and special attention such small peaks would not easily be detected. The thermal variation of the weak additional peak is shown in fig. 5. It is seen that the peak persists beyond the Curie temperature up to room temperature. How- ever, the peak is better defined at the lower temperatures.

0]1 0,0.5,1 010

Fig. 7. 3 dimensional representation of the dispersion surface for Pd2MnIno.25Sno.75 in the (001) plane.

The antiferromagnetic structures observed, AF3A and AF2, give rise to magnetic Bragg reflections of the type (h, k / 2 , 2/) and (h /2 , k / 2 , / / 2 ) , respectively, where h, k and ! represent odd in- tegers. Since the AF2 structure only occurs at the end of the series close to x = 0 it may be expected that precursor effects due to it will be small at the measured composition x = 0.75. This is indeed the case. Only a small effect, flattening of the dispersion close to the L point in the (~(~) direction, is observed at the (1/2, 1/2, 1 /2) position of the first reflection expected for the AF2 structure. At the measured composition precursor effects due to the AF3A structure should dominate. However they may not be immediately evident as the AF3A magnetic lattice points are not accessible in the (110) scattering plane that was used. It is possible though to construct the dispersion surface in the neighbourhood of the first AF3A magnetic lattice point (1, 1/2, 0), the symmetry point W, by using the derived exchange parameters and a different projection. Fig. 6 shows a contour map of equal energy surfaces for the (001) zone axis and fig. 7 a three dimensional representation. It may be seen from this projection how the anomalies observed in the (00~) and (~$0) directions combine to form a distinct valley which extends to the (1, 1/2, 0) position. Furthermore the calculated dispersion

001

000

\ \,

I 110

Fig. 8. Energy contours (in THz) for Pd2Mnlno.25Sno.75 in the (110) plane.


000

Z _'Y I l l 001

Fig. 9. 3 dimensional representation of the dispersion surface for Pd2Mnlno.2sSn0.75 in the (110) planes.

between (1, 0, 0) and (1, 1, 0) shows a minimum at (1, 1/2, 0). Equivalent energy contours in the measured (110) zone are shown in figs. 8 and 9. It may be anticipated, from the systematic deviation of the simple model calculation from the observed dispersion in the (00~) direction, that the softening is underestimated by this model.

Since the Mn atoms are separated by more than 4 A, (a /¢~) , the predominant interactions giving rise to the magnetic order cannot be direct exchange couplingsbetween the manganese atoms. There exist several different exchange mechanisms which may account for the observed magnetic order and whose relative importance will depend upon the nature of the X and Z atoms. For example, each manganese atom is at the centre of a simple cubic environment of Pd atoms (at a distance of av~ /2 ) and therefore electron correla- tions between the Pd (4d) and Mn (3d) electrons may be expected to be important. The interaction would be mediated by orbitals of t zg symmetry [17] which, since the 4d band of the Pd atoms is more than half full, would lead to ferromagnetism. However, the absence of spin density on the Pd atoms (<0.1~tB) has been confirmed using polarised neutrons, suggesting that the Pd atoms are magnetically 'inert', most likely being in a 4d a° configuration. It may be expected that the 4s and 4d bands are well separated with the latter lying

well below the Fermi level. Measurements of the transport properties of Palladium-based alloys indicate that the carriers are valence electrons of s or p character [20,21]. Thus neither the Pd 4d electrons nor the magnetic 3d electrons participate in the Fermi surface. It may be concluded that the coupling between the manganese atoms is mediated by the s -p electrons. The next-nearest neighbours of each Mn atom, at distances of a/2, are Z atoms which form an octahedral environment. Since the 3d eg orbitals, which overlap with the p orbitals of the Z atoms, are less than half full this covalent interaction would lead to ferromagnetic alignment. However, the polarised neutron measurements indicate the absence of significant spin density of the Z atoms, even for the compound containing Sb which is the most electro-negative Z element studied in the Pd series. Although superexchange plays a relatively minor role in the ferromagnetic compounds containing palladium this may not be the case in other Heusler alloys. Kiabler et al. [18,19], on the basis of band structure calculations, have suggested that the hy- brid p - d states are of particular importance in controlling the type of magnetic order and the magnitude of the manganese moment. Although this may not be of predominant importance in the Pd2Mnln0.25Sn0.T5 compound, polarised neutron measurements on other Heusler alloys suggest the presence of significant covalent p - d bonding. The possibility that in Pd 2Mnln025Sn0.75 the exchange coupling is mediated by a d-(sp) RKKY type of interaction is discussed later in this section.

Although the observed anomalies may be attributed to precursor effects associated with the incipient antiferromagnetic structures, particularly the AF3A structure, the underlying physical mechanism has not been established. However, since the magnetic properties are intimately related to the electron concentration, it may be expected that the anomalies are related to features of the band structure and the Fermi surface.

The spin wave dispersion in Pd2Mnln02sSn0.vs between (1, 0, 0) and (1, 1, 0), i.e. along a (010) direction reveals, as seen in figs. 6 and 7, a minimum in the neighbourhood of (1, 1/2, 0). A softening of the spin wave dispersion at this wave vector may be anticipated since it is known from


magnetisation measurements that the AF3A superlattice begins to form below x = 0.7. Incipient AF3A behaviour is detectable in the spin wave dispersion along (~00) at the stoichiometric composition Pd2MnSn. Comparison of the exchange constants for the two compositions Pd2MnSn and Pd2Mnln0.25Sn0.75, listed in table 1, reveals that J2 is only slightly reduced in magnitude whereas J1 and J3 are significantly reduced at x = 0.75. Anal- ysis based on mean field theory, and considering up to three neighbours, indicates that for ferromagnetic order.

kOp = l Z J 1 + 6J 2 + 24,/3, (10)

whereas for the AF3A structure

KON3 = 4J1 + 2 J 2 + 8./3, (11)

where k = 3kB/2S(S + 1) and 0p and 0N3 are the paramagnetic and Nrel temperatures, respectively. Since the exchange interactions of the AF3A structure couple neighbouring spins of mixed sign, for -/1 there are 8 negative and 4 positive alignments, a reduction in the magnitude, and possibly a change in sign, of the exchange constants may be anticipated as the indium content is increased and the AF3A structure is approached. An indirect mechanism, such as the RKKY model, can qualitatively account in a straightforward manner for the observed changes in sign and magnitude of the exchange constants. Although the separation of the manganese atoms hardly changes with indium content the electron concentration does and causes a corresponding change in the phase of the Friedel oscillations. However, the first two interactions J1 and J2 operate at a radius at which the asymptotic approximation, inherent in many theoretical mod- els, breaks down and only broad qualitative con- clusions can be drawn in the absence of reliable information on the band structure. For example, detailed information is required in order to under- stand why adding 25% indium has such a small effect upon the important exchange interaction J2-

A least squares fit of the RKKY exchange integral, based on the free electron approximation and in the asymptotic limit J ( R ) - ( N 2 F / Ef )cos(KR) / (KR) 3 to the observed exchange constants yielded K F ---1.52 .~-a. Reasonable

agreement was obtained except for the fourth nearest neighbour exchange parameter which was observed to be significantly more negative than that calculated. A slight improvement was obtained by introducing a phase shift as employed in the Caroli-Blandin double resonance exchange mechanism J(R) - (EF/S 2) sinZ~b - cos 2(KR + 2q~-)/(KR) 3. The phase shift q~- is related to the number of down spin electrons Z - or the magnetic moment, M of manganese atoms by ~ - = ( ~ r / 5 ) Z - = (nr/5)(5 - M). For Pd2MnInSn the phase shift was determined to be 29 ° .

Price [7] obtained good agreement with the observed exchange constants in PdzMnSn , using a model without asymptotic approximations, assuming a spherical Fermi surface and a wave vector of 1.9 ,~-1. This value is somewhat greater than the value of 1.3 .~-1 normally used assuming a total valence of 5 and Price emphasised the need for more information on the detail of the band structure.

Any microscopic theory must account not only for the moment formation in Heusler alloys but also the observed magnetic structures. Since there is negligible overlap of the d wave functions on the manganese atoms, and the manganese atoms carry the magnetic moment, the exchange coupling must involve an indirect process. The bulk properties, and in particular the paramagnetic Curie temperatures, depend sensitively upon the electron concentration. Thus the magnetic order may depend sensitively upon the geometry of the Fermi surface in determining the maximum in the wavevector dependent susceptibility x(q).

The indirect exchange coupling is related to the wavevector dependent susceptibility as a linear response of the conduction electrons to the effective field of the local manganese moments

N fk --fk+q N J(q) = -~-IV(q)12~ I t ( q ) 12~(q) Ok+ q -- C k 2

(12)

in which F is the contact interaction, often considered wavevector independent, and ~ and f are the energy and Fermi-Dirac distribution function, respectively.

For a spherical Fermi surface J(r i - ri) de-

P.J. Webster, K.R.A. Ziebeck / Dynamic

i -3 creases as ]r i - 9 and oscillates with a period corresponding to the calipering of the Fermi surface in the ( r i - ~) direction. An infinite slope in J(q) is obtained from x ( q ) at q = 2 K F. A Kohn construction I ~" + q l = 2KF, using K F = 1.3 ~ - 1 , yields a locus of anomalies close to the

~ wavevectors at which the discontinuities in the dispersion curve were observed. Although reasonable correlation is obtained it is unlikely that the Fermi surface is strictly spherical. For the special cases of plane or cylindrical regions of the Fermi surface the fall-off of J(r~- rj) is much slower, being proportional to R -~ and R -2 respectively. Thus the range of the interactions would be substantially increased. For plane portions of the Fermi surface J(q) has a logarithmic singularity producing a large maximum in x(q) , a feature important in rare earth elements and in chromium in which the nesting between electron and hole Fermi surfaces drives the magnetic order. Band structure calculations on PdEMnSn indicate that bands cross the Fermi surface at approximately the wavevectors at which the anomalies are observed [22,23]. For the majority band three hole- like surfaces have been proposed. The first surface is a ' rounded octahedron' and the other two touch the Brillouin zone boundary at the L point. The exact geometry of the surfaces depend crucially upon the Fermi level, particularly since the electron concentration of Pd2Mn( In /Sn ) is less than that of Pd2MnSn. The variation must be such as to produce a maximum in J(q) at q = 0 , q = (2,~/a)(1, 1 /2 , 0) or q = (2at/a) (1, 1, 1) for the compositions x = 1, 0.65 and 0 to yield the F, AF3A and AF2 structures, respectively. It must be remembered that a coupling exists between the magnetic and crystal lattices which produces a magnetostrictive distortion below T N in antiferro- magnets. This will slightly modify the stability criteria, particularly for the AF3A structure.

5. Summary

Spin waves have been observed to propagate out to the zone boundary in the ferromagnetic compound Pd2MnInl_xSn x at the composition x = 0.75. Anomalies observed in the dispersion

magnetic properties of Heusler alloys 17

can be correlated with precursor effects associated with the onset of the antiferromagnetic AF3A structure which is observed at 0.2 ~ x ~< 0.6. The spin dynamics are in excellent agreement with the prediction of a Heisenberg model in which the exchange interactions extend beyond 12 ,~. As the temperature is raised the spin waves renormalise and the response becomes entirely diffusive above T o consistent with disordered local moments of fixed amplitude. The variation of the magnetisation can be accounted for, assuming only the thermal population of magnon states and mag- n o n - m a g n o n interaction. A qualitative agreement was obtained between the observed exchange parameters and the predictions of an R K K Y model.

It may be concluded that Pd2MnIn l_xSnx represents a metallic system with localised magnetic properties. As such it provides a good model system for studying spin dynamics in materials having mixed interactions. Indeed the possibility of studying ferromagnetism and antiferromagnetism (both AF3A and AF2) in a single alloy series is perhaps so far unique to this system. Further measurements concerning the spin dynamics of the AF3A and AF2 structures are in progress, together with a neutron study of the mixed fe r ro -AF3A region ( x - - 0 . 6 ) which exhibits spin glass behaviour.

Acknowledgements

The neutron diffraction measurements were made at the Institut Laue-Langevin, Grenoble, experiment proposal numbers 4 -03-80 , 95. The authors are grateful to the staff at ILL who were responsible for the IN8 spectrometer and to Mr. R.M. Mankikar who assisted with some of the

i

measurements.

References

[1] P.J. Webster, Contemp. Phys. 10 (1969) 559. [2] Y. Noda and Y. Ishikawa, J. Phys. Soc. Japan 40 (1976)

690. [3] K. Tajima, Y. Ishikawa, P.J. Webster, M.W. Stringfellow,

D. Tocchetti and K.R.A. Ziebeck, J. Phys. Soc. Japan 43 (1977) 483.


[4] B. Caroli and A. Blandin, J. Phys. Chem. Solids 27 (1966) 503.

[5] T. Kasuya, Solid State Commun. 15 (1974) 1119. [6] M.I. Darby and P.J. Webster, AlP Conf. Proc. No. 29

(1976) 418. [7] D.C. Price, J. Phys. F 8 (1978) 933. [8] M.B. Stearns, J. Appl. Phys. 50 (1979) 2060. [9] P.J. Webster and M.R.I. Ramadan, J. Magn. Magn. Mat. 5

(1977) 51. [10] P.J. Webster and M.R.I. Ramadan, J. Magn. Magn. Mat.

13 (1979) 301. [11] P.J. Webster, J. Appl. Phys. 52 (1981) 2040. [12] P.J. Webster and K.R.A. Ziebeck, J. Magn. Magn. Mat.

31-34 (1983) 623. [13] P.J. Webster and R.S. Tebble, Phil. Mag. 16 (1967) 347. [14] Y. lshikawa, K. Tajima and P. Radhakrishna, J. Phys. Soc.

Japan 40 (1976) 1597. [15] S.M. Johnson, J.A.C. Bland, P.J. Brown, P.J. Webster and

K.R.A. Ziebeck, J. de Phys. Colloque C7 43 (1982) 119.

[16] K.R.A. Ziebeck, P.J. Webster, P.J. Brown and J.A.C. Bland, J. Magn. Magn. Mat. 24 (1981) 258.

[17] J.B. Goodenough, Magnetism and the Chemical Bond (J. Wiley, New York, 1966).

[18] J. Kubler, Proc. Workshop on 3d Metallic Magnetism, eds. D. Givord and K.R.A. Ziebeck, ILL, Grenoble [832118T] (1983) 99.

[19] A.R. Williams, J.L. Moruzzi, C.D. Gelatt and J. Kubler, J. Magn. Magn. Mat. 31-34 (1983) 88.

[20] A. Hamzic, R. Asomoza and I.A. Campbell, J. Phys. F 11 (1981) 1441.

[21] C.M. Hurd, I. Shiozaki and S.I. McAlister, Phys. Rev. B26 (1982) 701.

[22] S. Ishida, Y. Kubo, J. Ishida and S. Asano, J. Phys. Soc. Japan 48 (1980) 814.

[23] S. Ishida, H. Asato, E. Iwashima, Y. Kubo and J. Ishida, J. Phys. F 11 (1981) 1035.

Dynamic magnetic properties of ferromagnetic and antiferromagnetic Heusler alloys

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Transcript of Dynamic magnetic properties of ferromagnetic and antiferromagnetic Heusler alloys