114
Journal of Magnetism and Magnetic Materials 61 (1986) 114-120 North-Holland, Amsterdam
G A L V A N O M A G N E T I C P R O P E R T I E S OF Ni 2 Mnln, Ni 2 MnSn A N D N i 2 M n S b H E U S L E R ALLOYS
C.M. H U R D and S.P. M c A L I S T E R Solid State Chernistry, Chemistry Division, National Research Council of Canada, Ottawa, Canada KIA OR9 Received 26 December 1985; in revised form 12 March 1986
The field and temperature dependences of the electrical resistivity, Hall effect and transverse magnetoresistance of Ni2Mnln and Ni2MnSb are described over the approximate ranges 4 300 K and 0 - 4 . 8 x 1 0 6 Am 1 together with measurements of the magnetization in the range 0-1.6 × 106 Am 1. Together with existing data for Ni2 MnSn, these results show the systematic effects of the shifting Fermi level in the In ---, Sn ---, Sb series. We give a qualitative interpretation of these electron transport effects in terms of a delocalized picture.
1. Introduction
The galvanomagnetic effects in Heusler alloys (XzMnY) reflect generally the scattering of conduction electrons by the Mn ions. These effects are dominated [1,2] by spin-orbit coupling and spin-flip processes as electrons interact with the split virtual bound state of Mn. Until recently, the usual model of such alloys invoked a local moment on each Mn ion due to the localised ' m a g netic' d electrons, and these moments were perceived to interact mutually through the nearly free electron gas produced by the other atoms. But this rigid distinction between 'magnetic' and 'conduction' electrons is a problem in the interpretation of the galvanomagnetic effects; for them the distinction has to be blurred by introducing the concept of s - d - s scattering [1,2]. A new interpretation [3] of energy-band calculations dispenses with the concept of strictly localised electrons at the Mn sites. It shows that the magnetisation is confined to the Mn ions, not because of localised d electrons but rather because of the exclusion of particular delocalised electrons from the Mn states. The potential of this new picture to describe the formation and coupling of the magnetic moments has been discussed in detail [4], but the question of how it could be recon-
ciled with the behaviour of the galvanomagnetic effects has not been considered. It would be worth looking at this aspect since the Hall and magnetoresistance effects are particularly sensitive to the interplay between the delocalised and localised nature of the electrons at the magnetic ions. We therefore measured the electrical resistivity, Hall effect and transverse magnetoresistance of the ferromagnetic alloys Ni2MnIn and Ni2MnSb. We combine these results with our previous ones [2] for Ni2MnSn to provide a systematic view of the homologous series (In ~ Sn ~ Sb). Thus, we study a series in which the Fermi level shifts with respect to the virtual bound-state while other parameters are held ostensibly constant. We are able to assess qualitatively the contribution of the different classes of electrons to the galvanomagnetic effects. The electrical measurements cover ~ 4-300 K with fields up to 4.8 × 10 6 Am - t (6 T). Magnetisation measurements were made up to 1.6 x 10 6 A m - t (2 T). Apart from the resistivity results of Schreiner et al. [5], the following appear to be the only electrical studies of either Ni2MnIn or Ni 2MnSb. Previous work on these alloys has been concerned either with the hyperfine fields [6,7] or with magnetic or structural properties [8]. Section 2 gives the experimental details, section
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C.M. Hurd, S.P. McA lister / Properties of Heusler alloys
115
Table 1 Summary of samples' characteristics Alloy Ni zMnln Ni2MnSn Ni 2MnSb
0(4.2 K) (£m)
/x a) 0%)
O b) (K)
I.p.
R0
(,~l)
(m3/As)
n (m 3)
5.98 × 10- 8 1.10 5( 10 -7 6.83 × 10- 7
4.57 +_0.04 3.88 + 0.04 3.70 + 0.04
314 + 1 346 + 2 360 c)
6.90 + 0.01 6.05 c) 6.01 + 0.01
- 3.67 x 10-12 2.40 × 10 n 7.84 × 10-11
(e)l.7 × 1030 (h)2.6 X 10 29 (h)8.0 × 1028
a) Saturation moment per Mn ion at 4.2 K. b) Ferromagnetic Curie temperature. c) Literature value; not measured in this work. e, h = electrons, holes. 3 shows the results a n d section 4 describes how these results fit q u a l i t a t i v e l y the m i c r o s c o p i c picture p r o p o s e d b y W i l l i a m s a n d c o w o r k e r s [3,4].
2. Experimental details Stoichiometric quantities of starting m a t e r i a l s ( > 99.999 at% purity) were m e l t e d u n d e r p u r e A r in an i n d u c t i o n furnace using h i g h - p u r i t y A1203 crucibles. T h e m o l t e n mixtures were chill cast into h i g h - p u r i t y C u m o l d s (two or three times to imp r o v e h o m o g e n e i t y ) to p r o d u c e b o t h cylindrical (3 m m d i a m ) ingots for m a g n e t i s a t i o n samples, a n d flat, p l a t e - s h a p e d ingots from which g a l v a n o m a g netic s a m p l e s with a p p r o p r i a t e side a r m s were cut b y s p a r k erosion using a template. Offcuts from each ingot were a n a l y s e d chemically to verify the c o m p o s i t i o n . Each s a m p l e was a n n e a l e d at 1073 K in purified A r to m a x i m i s e the h o m o g e n e i t y a n d order: N i 2 M n S b for 63 h a n d N i 3 M n I n for 24 h. X - r a y analysis c o n f i r m e d the fully o r d e r e d structure in each case with lattice p a r a m e t e r s (1.p.) given in table 1, which s u m m a r i s e s the s a m p l e s ' characteristics, i n c l u d i n g N i 2 M n S n from previous w o r k [1]. T h e m e a s u r e m e n t techniques a n d app a r a t u s were the s a m e as in ref. [2].
b u t a n o t a b l e difference is the t e n d e n c y at higher t e m p e r a t u r e s t o w a r d s a T" behaviour, where n is significantly less than 2. (The q u a d r a t i c value is expected when the resistivity is d o m i n a t e d b y m a g n o n scattering.) W e find n -- 1.8 for N i 2 M n I n a n d n = 1.6 for N i 2 M n S b . A n e x p o n e n t of a b o u t 1.7, which is seen [1,2] in other f e r r o m a g n e t i c H e u s l e r c o m p o u n d s (including N i 2 M n S n ) a n d in elemental ferromagnets, is a p p a r e n t l y characteris-
7 0
10
100
1000
I
I
I
INieMn?1 /
< I o
o.
g NizMnln]
o
o
5/
•
e*
3. Results ,2 3,1. Resistivity
Fig. 1 shows the t e m p e r a t u r e d e p e n d e n c e of the ideal resistivity o f N i 2 M n l n a n d N i 2 M n S b . T h e results are similar to those of Schreiner et al. [5],
t O
"='
I
,
I0 Temperature
I
I00
I000 (K)
Fig. 1. Showing the temperature dependent part of the resistivities. The open symbols refer to the upper abscissa.
:,..rj
116
C.M. Hurd, S.P. McA lister / Properties of Heusler alloys l~
i
]
4K /<> ~.~>
-5
\
I
"\
~/'¢~
- m~\k
g--. I
a
•
2
v
[
?
0 ~J
"~
: *< . . . . . . . .
<1
+_",~_o_+~_+_
-25 ° N 150K'K,
°
1
,
NiaMnln
\
200K
z\
~KN
g60K
\ -1
,
0
I
2
,
,
I
-45
~
4
6
,
0
I
,
l
2
4rrH (10? A / m )
L
I
4
47rH (10? A / m )
Fig. 2. The field dependences of the Hall resistivity and transverse magnetoresistance of Ni2 M n l n at different temperatures. H is the applied field. The abscissa unit corresponds to a flux density of 1 T.
I
I
{i(} ~o o o ~ ° ~ < ) - - °
t--,
g-.
,I 0 :p:,
.
.
v o o - o - ~ - o - ° - + - °
.
.
o
o
,-,,-I v
>~. . . . . . . . 22222:::=+=+ =+-+ o
°~°~
°
"\.
?
\\o, <1
%
"~
~,++
o
/
5-.
~
I
E
o ,=.,v--<--o--o--¢>
"~, \
-10
\,-
~0
\. \. I
I
2
4
4rrn (I0 ? A / m)
,
I
6
-20
,
I
I
2
4
4?rH (10 ? A / m)
Fig. 3. As in fig. 2, except here for Ni2MnSb. The symbols refer to the temperatures shown in fig. 2.
117
C.M. Hurd, S.P. McAlister/ Propertiesof Heusler alloys I
I
I
10
I
60 ~'~
MeMnSbJ,
zMn Sn ,.
° ~ o /
o
~_z.------'7:
"
"
Ni2Mnln
40
c:~
~.~" cx/
o
[
,
I
,
100 200 300 Temperature (K) Fig. 5. O E is the effectiveHall angle per microscopicscattering event, obtained as explained in the text. The plot is used to distinguish the microscopicorigins of the AHE.
20
L
0
,
2
J
I
4
i
6
4rrH (10 v A , / m ) Fig. 4. A comparison of the AHE in the three systemsNi2MnSb (B), Ni2MnSn fin) and NiEMnln (0).
reflecting the magnetisation [1,2]. These features are pronounced in figs. 2 and 3 since the A H E is dominant in the ranges studied. Fig. 4 shows how the A H E affects p21(B) at 260 K for the three systems. The anomalous Hall angle O E is derived from the data in figs. 2 and 3 using [1]: OE(T ) = [Ap2,(T)/po(T)](1/M;),
tic of ferromagnetic order but it presently has no theoretical explanation [1]. 3.2. Hall and transverse magnetoresistance effects Figs. 2 and 3 show the field dependences of the Hall resistivity P21 and transverse magnetoresistance A p / p at different temperatures for Ni2MnIn and NiEMnSb. (Corresponding results for Ni 2 MnSn appear in figs. 3 and 4 of ref. [2].) The transverse magnetoresistance for the three systems show the same qualitative behaviour, which is typical of a ferromagnet [1,2]: the negative component increases with increasing temperature, opposite to the usual behaviour of dilute magnetic alloys. The field dependence of P21 ( B ) has two parts: one that saturates below about 1 T, and another that varies linearly with field. The former is the anomalous Hall effect (AHE). It is prominent in ferromagnets, were it increases with temperature and shows a sharp " k n e e " in its field dependence,
where M ~ ( T ) = M s ( T ) / M s (4.2 K). The anomalous Hall angle, which is used [1,3] to distinguish the microscopic scattering mechanism at a Mn site, is shown in fig. 5. The ordinary Hall coefficient R 0 is estimated from the gradient of p2~(B) above saturation and at 4.2 K, where complications from phonon scattering are minimised. (Correction was made for the field dependence of the magnetization above saturation at 4.2 K; we estimate the uncertainty in R 0 as ___5%). The results are given in table 1, together with the corresponding effective carrier densities n from free-electron theory. 4. Discussion and conclusions 4.1. General remarks In the model of X 2MnY proposed by Williams and co-workers [3,4], the d-states from the Mn atoms hybridise with the p-states of the Y atoms
118
C.M. Hurd, S.P. McAlister / Properties of Heusler alloys
ENERGY
o : iii~iii,,~
~m o
~
4- m
N_ /-
mm
NUMBER OF ELECTRONS Fig. 6. Density of states for the d-level of Mn in X2MnY. The up-spin states form a common d-band with the X ions, but the spin-down electrons are excluded from the region of the 3d-shell of Mn. The spin-orbit coupling removes the degeneracy of the m substates in an external magnetic field.
to form the 'nearly free electrons' of the conventional picture. These same d-states also hybridise with the d-states of the X atoms to form a d-band. Both the up- and down-spin d-electrons are itinerant, but the down-spin ones are excluded from the region of the 3d shell of a Mn ion. Thus, the common d-band of itinerant electrons produces an apparently localised moment on a Mn ion. Here we show how this idea leads to a qualitative interpretation of the trends seen in the galvanomagnetic effects without introducing the contrived s - d - s scattering. The situation proposed for X RMnY by Williams and coworkers [3,4] is shown schematically in fig. 6, which is essentially fig. 7 or 8 of ref. [3] except that we include the separation in energy of the orbital states in a magnetic field. (This is required to understand the origin of asymmetric scattering, which leads to the AHE.) We work in terms of the rigid-band model so that the only
change we envisage in fig. 6 as we cross the series In~Sn~Sb is a shift upwards of the Fermi energy corresponding to the diminishing localised moment per Mn ion (table 1). Given that fig. 6 represents itinerant electrons in the vicinity of a Mn ion, we can deduce from it, and the experimental data given above, qualitative conclusions about the contributions from the different classes of electrons to the galvanomagnetic effects. These conclusions, which are considered separately below, are: 1. The A H E arises from the down-spin electrons in the hybridised d-orbitals of the X and Mn atoms. 2. The negative magnetoresistance arises from the inelastic transfer between up- and down-spin states in the hybridised d-orbitals of the X and Mn atoms. 3. The ordinary Hall effect reflects the competitive effects that the up- and down-spin Mn d-states have on states at the Fermi energy. 4. 2. A nomalous Hall effect This arises not from the Lorentz force but from spin-orbit coupling. It has two manifestations: so-called ' skew scattering' and ' side j u m p ' [10,11 ]. The first calculation of the A H E for a ferromagnetic band model, which is the situation appropriate to the Williams' model for X 2MnY, was made by Luttinger and colleagues [12,13]. Luttinger considered the effect of intrinsic spin-orbit coupling s . L that exists between an electron's spin s and angular m o m e n t u m L (fig. 6.5 of ref. [11]). The calculation produced two terms. The first gives skew scattering, which can be understood [9] from the separation of the substates in the applied field, as in fig. 6. With the Fermi level at the position illustrated, the spin-up states are essentially completely full and so contribute no asymmetric scattering. The minority states, on the other hand, are partially occupied and so more states are available to + m than to - m . There is therefore is a greater probability for the spin-down electrons to be scattered to one side of the Mn ion rather than to the other. So in the Williams' model the skew scattering arises entirely from the few
C.M. Hurd, S.P. McAlister / Properties of Heusler alloys
spin-down electrons that are not excluded from the d-shells of the Mn ions. Luttinger's second t e r m - s o - c a l l e d 'velocity ren o r m a l i s a t i o n ' - h a s a nonclassical origin and becomes important when h / T E F is large [13]. It is expected to have maximum influence when the mean free path of the electrons is short. Side j u m p is Berger's [14] physical picture of this nonclassical term. It envisages that whenever an itinerant electron encounters a Mn ion it experiences a discrete transverse displacement to one side of the ion rather than the other. Skew scattering and side j u m p can be distinguished [1,3] by the behaviour of OE(T), which is independent of temperature when the A H E is dominated by skew scattering but temperature dependent when side j u m p dominates. Fig. 5 shows that when Y = I n or Sn then 6)z(T ) is essentially independent of temperature compared with Y = Sb. From this we deduce that the A H E in N i 2 M n l n and Ni2MnSn it is dominated by skew scattering while in Ni2MnSb it is dominated by side jump. (The situation is reminiscent of the difference between N i z M n S n and Pd 2MnSn or Cu2MnAI seen in fig. 5 of ref. [1].) The relative magnitudes of the A H E components at 260 K can be judged from fig. 4; note how the side-jump contribution, which increases with reducing mean free path, is emphasised at this temperature. In terms of the simple rigid-band model used here, there seems to be no straightforward explanation for the dominant side j u m p when Y = Sb. A judgement based on the variation of h/'rE F is inconclusive because although ~-, as reflected in P4.2, decreases across the series In ~ Sn ~ Sb, E v must increase, and the net trend is not clear even qualitatively.
4.3. Transverse magnetoresistance The magnetoresistance (figs. 2 and 3) reflects a competition between a negative component, which generally depends strongly on applied field and temperature, and a normal positive one due to the Lorentz force. The negative component occurs [1,2] when kBT>__ glzBHi, where H i is the internal field. Involved behaviours can then arise because
119
increasing the temperature until kBT exceeds g#BHi opens spin-flip scattering channels while increasing the applied field strength, and hence Hi, closes them. Thus, we can understand qualitatively the behaviours in figs. 2 and 3 if we assert that at the lowest temperature (---4 K) gl~BHi is larger than kBT and the amount of spin-flip scattering is small, imperceptibly so for Ni2MnSb (fig. 3) and Ni2MnSn [2] since they show no negative magnetoresistance at 4 K. Whatever spin-flip scattering exists at 4 K in N i 2 M n l n is extinguished as the applied field is increased (fig. 2), leaving the normal positive magnetoresistance. As temperature is increased above 4 K (figs. 2 and 3), the positive Lorentz component is reduced by the decreasing mean free path. At the same time, the negative component grows as kBT tends to exceed gl~BHi throughout the range of fields studied. Hence the increasing negative magnetoresistance seen for all the alloys in the series In Sn ~ Sb, as in figs. 2 and 3.
4. 4. Ordinary Hall effect This arises from the Lorentz force acting on electrons near he Fermi energy. Given that the low-field condition prevails [11], the ordinary effect is determined by the dynamical properties of electrons at the Fermi energy, such as their effective mass, velocity and the variation, if any, of the relaxation time over the Fermi surface. If we assume, within the rigid-band model, that the shape of the Fermi surface stays fixed across the series In ~ Sn ---, Sb, then calculations for N i z M n S n [15] can be used to predict the tendency of the effective mass across the series. (According to Williams et al. [3], these calculations differ only quantitatively from their model.) Fig. 7 shows, for example, the salient features for Ni2MnSn near the L or F points. The majority-spin carriers near the Fermi level occupy electron-like states that are relatively unaffected by the lower lying d-bands of the Mn and X atoms (fig. 1 of ref. [15]). But the minority-spin carriers occupy states that are holelike mainly because of the influence at the Fermienergy of the upper d-levels of the Mn atoms. These lie above the Fermi energy and significantly
C.M. Hurd, S.P. McA fister / Properties of Heusler alloys
120
to a more hole-like contribution. The behaviour of R 0 across the series reflects the balance between these contributions.
ENERGY
Acknowledgements We thank L.D. Calvert and F. Lee of the X-ray group for work done on our behalf, and G.F. Turner for his careful sample preparation and other technical help.
References L,F
L, F
Fig. 7. Showing the shape of the majority- and minority-spin energy bands in the vicinity of the L and F points of Ni 2 MnSn. (After Ishida et al. [15].)
depress the E(k) curves for the minority carriers (compare figs. 1 and 3 of ref. [15]). Thus, the upand down-spin contributions to the ordinary Hall effect in Ni 2MnSn have opposite signs. Although we cannot predict the net sign of the Hall effect in X2MnY, we can deduce from fig. 7 the expected tendency across the series In ~ Sn --, Sb. For an upward movement of the Fermi level, the majority-spin contribution 'will remain electron-like, but the minority-spin contribution will become increasingly hole-like. Hence the tendency of R 0 seen across the series (table 1). Thus, we think the ordinary Hall effect reflects the degree of hybridisation in the Mn d-bands: the weaker hybridisation of the majority-spin electrons leads to an electron-like contribution, but the stronger hybridisation of the minority-spin electrons leads
[1] C.M. Hurd, I. Shiozaki and S.P. McAlister, Phys. Rev. B26 (1982) 701. [2] S.P. McAlister, I. Shiozaki, C.M. Hurd and C.V. Stager, J. Phys. F. 11 (1981) 2129. [3] A.R. Williams, V.L. Moruzzi, C.D. Gelatt and J. Kiibler, J. Magn. Magn. Mat. 31-34 (1983) 88. [4] J. Kiabler, A.R. Williams and C.B. Sommers, Phys. Rev. B28 (1983) 1745. [5] W.S. Schreiner, D.E. Brand,o, F. Ogiba and J.V. Kunzler, J. Phys. Chem. Solids 43 (1982) 777. [6] M. Elfazami, M. DeMarco, S. Jha, G.M. Julian and J.W. Blue, J. Appl. Phys. 52 (1981) 2043. [7] M.B. Stearns, J. Appl. Phys. 50 (1979) 2060. [8] P.J. Webster, Contemp. Phys. 10 (1969) 559. [9] G. Bergmann, Phys. Today 32 (1979) 25. [10] S.P. McAlister and C.M. Hurd, J. Appl. Phys. 50 (1979) 7526. [11] C.M. Hurd, The Hall Effect and its Applications, eds. C.L. Chien and C.R. Westgate (Plenum Press, New York, 1979) p. 1. [12] R. Karplus and J.M. Luttinger, Phys. Rev. 95 (1954) 1154. [13] J.M. Luttinger, Phys. Rev. 112 (1958) 739. [14] L. Berger, Phys. Rev. B 2 (1970) 4559. [15] S. Ishida, Y. Kubo, J. Ishida and S. Asano, J. Phys. Soc. Japan 48 (1980) 814.