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Crystallographic and Mössbauer study of zinc blende type FeS
M. Wintenberger, B. Srour, C. Meyer, F. Hartmann-Boutron, Y. Gros
To cite this version:
M. Wintenberger, B. Srour, C. Meyer, F. Hartmann-Boutron, Y. Gros. Crystallographic and Mössbauer study of zinc blende type FeS. Journal de Physique, 1978, 39 (9), pp.965-979.
�10.1051/jphys:01978003909096500�. �jpa-00208839�
CRYSTALLOGRAPHIC AND MÖSSBAUER STUDY OF ZINC BLENDE TYPE FeS
M. WINTENBERGER
D.R.F./D.N. Centre d’Etudes Nucléaires de Grenoble, B.P. 85X, 38041 Grenoble Cedex, France (*)
B. SROUR, C. MEYER, F. HARTMANN-BOUTRON and Y. GROS, Laboratoire de Spectrométrie Physique (**), Université de Grenoble I,
B.P. 53X, 38041 Grenoble Cedex, France (Reçu le 4 avril 1978, accepté le 23 mai 1978)
Résumé.
2014Jusqu’à présent, on connaissait essentiellement deux phases de FeS : une phase hexa- gonale, antiferromagnétique et semi-métallique et une phase tétragonale non magnétique. Nous présentons ici l’étude cristallographique et par effet Mössbauer de la nouvelle phase cubique de type blende qui a été découverte il y a quelques années. Cette phase est instable à l’ambiante. Quand la température décroît, elle subit une transition cristallographique du premier ordre aux environs de 234 K, passant de la structure blende cubique à une structure orthorhombique. L’effet Mössbauer
a montré qu’à toute température la nouvelle phase est apparemment ionique et qu’elle contient des ions ferreux de haut spin. Au-dessus de 234 K, elle est paramagnétique et au-dessous elle subit une
transition magnétique du premier ordre vers une phase ordonnée colinéaire où tous les moments
magnétiques sont parallèles à l’un des côtés de la maille orthorhombique. Divers mécanismes ont été
envisagés en vue d’expliquer la transition magnétocristalline du premier ordre à 234 K : magnéto-
striction d’échange classique ; effet Jahn-Teller mettant en jeu le doublet orbital 03933, associé à de la
magnétostriction d’échange ou de l’échange généralisé, etc., mais nous n’avons pas pu arriver à une conclusion définie. Comme on pouvait le prévoir, le volume par motif FeS augmente lorsqu’on passe de la phase hexagonale semi-métallique à la phase tétragonale non magnétique et de celle-ci, à la phase cubique ionique de haut spin. Les propriétés de la phase tétragonale, que l’on a attribuées à de la covalence, pourraient peut-être aussi être décrites en termes d’ions ferreux bas spin (cas de champ
cristallin fort).
Abstract.
2014Until a few years ago two main phases of FeS were known : a hexagonal phase which
is antiferromagnetic and semi metallic, and a tetragonal phase which is non magnetic. We present here the results of a crystallographic and Mössbauer study of the new cubic zinc blende type phase
which was discovered recently. This phase is unstable at room temperature. When the temperature is decreased it undergoes a first order crystallographic transition (cubic ~ orthorhombic) at about
234 K. The Mössbauer effect has shown that at all temperatures the new phase is apparently ionic,
with high spin Fe++ ions. Above 234 K it is paramagnetic and below 234 K it exhibits a first order
magnetic transition to an ordered collinear phase in which the magnetic moments are all parallel to
one of the axes of the orthorhombic cell. Various mechanisms have been examined in order to inter- pret the first order magnetocrystalline transition at 234 K : standard exchange magnetostriction ;
Jahn-Teller effect involving the 03933 orbital doublet, in association with exchange magnétostriction or generalized exchange and so on, but we could not arrive at a definite conclusion. As expected the
volume per FeS formula increases in going from the semi metallic hexagonal phase to the non magne- tic tetragonal phase, and from the tetragonal phase to the ionic high spin cubic phase. The properties
of the tetragonal phase, which had been interpreted as covalent, could perhaps also be described in terms of low spin ferrous ions (strong crystalline field case).
Classification
Physics Abstracts
61.10
-75.30
-76.80
(*) Also, C.N.R.S., B.P. 166X, 38042 Grenoble Cedex, France.
(**) Associé au C.N.R.S.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003909096500
’
1. Introduction.
-The preparation of a new FeS . phase with zinc blende structure (B3 of Strukturbe- richt, space groupe F4 3m, no 216) (Fig. 1) with
a
=5.42 Á, was mentioned in 1970 by de Medicis [1]
and Takeno et al. [2]. The methods of preparation
of [1] ] and [2] are very similar, and yield only small
amounts of the cubic phase, which converts rapidly
to tetragonal FeS at room temperature, and to hexagonal FeS when slightly heated.
In the present work we have studied the influence of temperature on the crystal structure and on the magnetic properties of cubic FeS. Strong near neigh-
bour antiferromagnetic interactions can be expected,
as shown by magnetic susceptibility measurements on
(Fe, Zn)S solid solutions [3], and might lead to anti- ferromagnetic order as in cubic MnS [4] which is also of B3 type. Moreover Fe2 + ions have 4 3m (Td) symmetry, so that their ground state is the orbital doublet r 3, and they might exhibit some effects of Jahn Teller instability. Fe2+ in Td symmetry induces cooperative distortions in some oxides with spinel
structure [5], but not in the thiospinel FeCr2S4,
which remains cubic down to 4.2 K [6]. As for the zincblende lattice, some studies on Fe2+ impurities in
ZnS have shown that, if the concentration of iron atoms is low enough (so that the probability of
existence of iron pairs in nearest neighbour sites,
or of larger clusters, is negligible), the electric field
gradient at iron nuclei has cubic symmetry [7] and optical spectra are not affected by dynamical Jahn-
Teller effects [8]. On the contrary a dynamical Jahn-
Teller effect on the r 3 level has been observed in
iron-doped CdTe [9] by optical experiments.
2. Préparation of samples and experimental tech- niques.
-Cubic FeS has been prepared by the method
of référence [1]. Distilled water, deoxygenated by bubbling argon through it, is used to prepare a SH2
solution. Massive iron is immersed in this solution and kept at room temperature. After two or three days a thin coating of FeS is formed. X-ray diagrams,
recorded immediately after removing the sample
from the solution, show that this coating contains
three phases, namely cubic, tetragonal and hexagonal
FeS.
Three forms of iron have been used :
a) pure iron foils 12.5 pm thick ;
b) piano wire. In this case the sulfide coating can
be scraped off. An electron microprobe analysis of
this wire has shown the presence of a few 0/00 of Si, Ni,
Mn, Zn, in addition to a 1 % C content ;
c) Armco iron plates.
The X-ray study was made with CuKoe radiation on a diffractometer equipped with a monochromator in front of the detector, and with a low temperature device using gazeous nitrogen.
The coexistence of three different FeS phases in the
samples did not allow any magnetic investigation
with macroscopic techniques. On the contrary Môss- bauer spectroscopy can give much information thanks to the selectivity of its local observations. The 57 Fe Môssbauer experiments were performed on a classical
constant acceleration drive spectrometer, using a Cos57/Rh source. Because of the instability of the cubic
phase at room temperature, care has been taken to store the samples in liquid nitrogen between their
chemical préparation and the experiments. Several samples have been used : iron foils coated with FeS,
or powders scraped off from iron plates or piano wire.
Despite their 12.5 pm thickness, the FeS/Fe ratio was
too low for the iron foils, so most of the experiments
were performed on powders, which contained practi- cally no metallic iron. The powders were attached to
aluminium foils with vacuum grease. The spectra were recorded between 5 K and 250 K and the proportions
of the three phases (as derived from the analysis of
the spectra, see below) remained approximately
unaltered during these experiments. The analysis of
the spectra was performed by computer, using a least
square minimization program where the hyperfine
parameters are adjusted to obtain the best fit with observation.
3. Crystallographic properties.
-The sample which
has been studied in more detail was an iron foil coated with iron sulfides, and fixed on the diffrac-
tometer sample-holder with General Electric varnish.
The diagrams were recorded up to a Bragg angle
0
=370.
At 81 K some lines of the cubic phase are split.
The diagram can be indexed (Table I) with an ortho- rhombic, pseudo tetragonal unit cell
TABLE 1
Observed and calculated Bragg angles at 81 K
The intensities of the three lines of a hkl triplet are
not equal : the line with l highest is always the weakest,
or is even missing in the case of 002 and 004, which correspond to weak lines of the cubic phase. This
shows that there is some preferred orientation of the orthorhombic axes.
TABLE II
Lattice parameters and unit cell volume at different
temperatures
The orthorhombic distortion decreases with
increasing temperature (Table II and Fig. 2) and
vanishes suddenly at 234 K. Figure 3 shows the behaviour of the triplet 220/202/022. At 234 K the triplet and the cubic phase line are simultaneously present, with relative intensities which do not vary with ,time. This is probably due to a slight thermal gradient in the sample. The absence of line broaden-
ing shows that the variation of lattice parameters is discontinuous in this temperature range and the transition is first order. However no thermal hysteresis
has been observed.
FIG. 1.
-Coordination of iron in the zinc blende type structure.
We have checked that a sample prepared with powder scraped off from piano wire, like those used for Môssbauer experiments, has identical properties :
the crystallographic transition temperature and the unit cell parameters at 150 K are the same. The pre-
Fic. 2.
-Lattice parameters and unit cell volume versus tem- perature.
FIG. 3.
-Evolution of the 220/202/022 triplet with temperature.
ferred orientation effects mentioned above are reduced in this case.
The only orthorhombic space group compatible
with the observed unit cell and subgroup of F4 3m
is F222 (no 22) with
so that the crystallographic transition occurs via
an homogeneous strain, and the three strain compo- nents
(where ao is the lattice parameter of the cubic phase)
describe the change of shape of the FeS4 or SFe4 tetrahedra, as well as that of the unit cell. This strain involves the elastic coefficient (CII-C12)’
As the orthorhombic distortion belongs to the
irreducible representation T 3 of the Td group, and the symmetrized cube [T 33 ]
=r 1 + T 2 + T 3, the
transition is necessarily first order according to
Landau’s theory, but this theory says nothing about
the microscopic mechanisms which drive the transi- tion...
4. Fe 57 Mossbauer spectra and electronic properties
of the iron atom. - 4.1 ANALYSIS OF THE SPECTRA.
-Because of the instability of the new FeS phase above
250 K, the Môssbauer study was essentially performed
between 0 and 250 K ; this is indeed the interesting
zone since, as we will see later, the new phase exhibits
a first order magnetic transition around 237 K.
Two samples were studied, called 1 and II, corres- ponding to two independent preparations. Both of
them were a mixture of three phases : the new phase
of interest, the tetragonal phase studied in refe-
rence [ 1 O] and the standard hexagonal phase. We were
thus obliged to decompose the spectra into three sub- spectra with the help of the already known properties
of the tetragonal and hexagonal phases.
Tetragonal FeS [10] is diamagnetic at all tempe-
ratures with no apparent quadrupole splitting. Its
Mossbauer spectrum is a unique and fairly broad
line whose isomer shift (I.S.) at 5 K is 0.495 mm/s
with respect to metallic Fe.
Hexagonal FeS [11] has a Néel point at 595 K.
Môssbauer studies show that the hyperfine field at
5 K is 328 kOe, and that in the temperature range of interest to us, the angle between the hyperfine field
and the main axis of the axial (1) electric field gradient (E.F.G.) is 0
=480; the quadrupole parameter for the I
=3/2 state of the nucleus is
(1) Strictly speaking the E.F.G. in the hexagonal phase is not quite axial : see reference [11] for discussion.
Fie. 4. - a) Môssbauer spectrum of sample 1 at 5.1 K, prior
to analysis ; b) Môssbauer spectra of sample II at 78, 150, 236 and 240 K showing the decomposition into three subspectra corresponding to the three phases (the anomalous variation of the
total integrated area is instrumental).
The Môssbauer spectrum consists of six lines with intensities 3 : 2 : 1 : 1 : 2 : 3, and at 5 K the I.S. with respect to Fe metal is 0.91 mm/s.
With the help of these data relative to the hexagonal
and tetragonal phases, we have been able to computer analyse the spectra represented in figure 4 (spectrum
at 5 K obtained with sample I, spectra at 78, 150, 236 and 240 K obtained with sample II) and to
extract the spectra of the new phase.
According to the relative integrated areas of the subspectra, in sample 1 at 5 K we have 47 % of the
new phase, 35 % of the tetragonal phase and 18 %
of the hexagonal phase (the absolute precision being
about 2 %).
In sample II at 78 K the proportions are 56 % of
the new phase, 26 % of the tetragonal phase and 18 %
of the hexagonal one. These proportions did not seem
to vary during the whole period during which the
Môssbauer spectra were recorded. Since the amount of the new phase is highest in sample II, the Môss- bauer study was mainly performed with this sample.
The Môssbauer spectra of the new phase alone, as extracted from figure 4 are represented in figure 5.
Above 237 K, where the crystalline structure is cubic, they reduce to a single (and rather broad) line which overlaps with the tetragonal line (respective I.S. are
0.665 mm/s and 0.419 mm/s) ; this makes the analysis
difficult. Below 237 K this line disappears rapidly and
is replaced by a spectrum with a magnetic structure
characteristic of an ordered magnetic phase, which
is quite reminiscent of that obtained by Imbert
et al. [12] in briartite CU2FeGeS4 ; in that compound
the hyperfine field is perpendicular to the main axis Oz of the axial electric field gradient and V ZZ 0. In
our case the least square fit leads to similar conclu-
FIG. 5.
-Môssbauer spectrum of the new phase as extracted from
figure 4. The dashed curve corresponds to the cubic form, the full line curve to the orthorhombic form.
sions, except for the fact that the E.F.G. is not quite axial, with a small asymmetry parameter il. This is
clearly related to the transformation of the new phase
from cubic to orthorhombic when the temperature decreases : the single line observed above 237 K
TABLE III
Columns 2 to 5 and 8 : thermal variation of the hyperfine parameters of the new phase : the isomer shift is referred to metallic iron, as calculated from an I.S. of - 0.114 mm/s fôr Fe with respect to our rhodium source. Columns 6 and 7 give the percentage of the two crystallographic forms of the new phase
in sample II (except for the data at 5 K which are relative to sample I) ; the absolute precision is about 2 %.
corresponds to the cubic form and the magnetic spectrum below 237 K corresponds to the ortho- rhombic form.
The values obtained for the hyperfine parameters of the new phase are reported in table III.
As shown by the curve giving the hyperfine field Hn
versus T (Fig. 6), which exhibits a discontinuous jump
at 237 K, the magnetic transition is first order. This is to be related to the first order crystallographic
transition observed at 234 K. The différence between
FIG. 6.
-Thermal variation of the hyperfine field Hn in the new phase.
the two transition temperatures is not really signi-
ficant : indeed if we look at table 111, we see that at 230 K the proportion of the orthorhombic form has decreased to 45 % instead of 56 % at 78 K, and at 236 K it has decreased to 20 %, these decreases corres- ponding to transformation into the high temperature cubic form. This is suggestive of a distribution of transition temperatures similar to those observed for Fe2+ and Fe3+ in ZnCr204 and CdCr204, where
it was attributed to the combined effects of exchange magnetostriction and strains in the crystallites [13].
We may therefore say that, with reasonable cer-
tainty, we have a simultaneous crystallographic and magnetic first order transition around 234 K. The
possible origins of such a transition will be discussed in section 5.
4.2 RELATION BETWEEN THE HYPERFINE DATA AND THE ELECTRONIC PROPERTIES OF THE IRON ATOM IN THE NEW PHASE. - Both the values of the I.S. (which is
the same as for Fe+ + in ZnS) and the great similarity
of the magnetic spectra with those of briartite strongly suggest that in the new FeS phase, iron behaves like
a standard ferrous ion. We will see now that this
assumption does indeed provide a coherent interpre-
tation of the Môssbauer data of the low temperature phase.
4. 1. 1 Tetragonal approximation.
-As shown in
figure 2 in the orthorhombic phase the parameters b and a are not very different, so as a first approximation
we will assume b
=a. Then the immediate neigh-
bourhood of an iron atom is a flattened tetrahedron
as in FeCr204 and in briartite. We may therefore
use the results derived for this situation in refe-
rences [14] (FeCr204) and [12] (briartite). The orbital level scheme of Fe+ + (L
=2, S
=2) for such a case is represented in figure 7. In the absence of tetragonal
distortion we have a lowest doublet F 3 and a highest triplet T5 separated by the cubic field splitting A (d
=3 400 cm-1 see below). A tetragonal distortion along Oz (c axis of the crystal) splits T3 into two sin- glets separated by ô « J, with wave functions (2) 1 (Q > == ILz=o>and
the triplet r 5 is split into a doublet 1 Lz
=+ 1 ) and a singlet
whose relative positions A ’, A " with respect to the ground state are not known (Li’, A " - A ).
FIG. 7.
-Orbital level scheme of a high spin Fe++ ion (3d6 SD) in
a tetrahedral site with tetragonal and orthorhombic distortions.
When state 1 (Q > == I Lz
=0 ) is lowest and ô is large compared to the (second order) effects of the spin orbit coupling (ô > 200 cm-1), it has been found in reference [14] that the E.F.G. has axial symmetry around Oz with, at 0 K :
i.e. Vzz is negative and its magnitude is of the order
! [ V 3 mm/s (the exact value depending on the
(1) The wave functions are referred to axes Oxyz parallel to the
cell edges abc (see Fig. 1 ).
matrix because of shielding effects) (3). When the
temperature rises, Vzz varies according to a law :
Fitting our experimental data between 0 and 200 K
(which are not very precise) to such a law leads to
à - 400 cm-1 (tetragonal splitting of 13).
Also, with LZ
=0 > lowest, we have an easy
magnetization plane perpendicular to Oz with a spin
Hamiltonian : v
(Â
=spin-orbit coupling, p
=spin-spin interaction + second order effects of the spin-orbit coupling between terms).
Finally the magnetic hyperfine interaction has the form (e
=excited nuclear state, g
=ground nuclear state) :
with
(fi,,
=Bohr magneton ; A
=Fermi core contribution ;
A’
=distance of 1 (Q > to the doublet 1 ± 1 > ; in
what follows we will make the approximation
4 ’ = d cubic).
Experimentally the hyperfine field H. is perpendi-
cular to the tetragonal axis, in agreement with the prediction of an easy plane for the magnetization.
Therefore S is perpendicular to Oz and what we measure is
In particular at 0 K the experimental value is
We must compare this value to theoretical estimates.
For this, in order to evaluate U, V, D, we will use the data obtained in references [8, 15] for Fe+ + as an
impurity in ZnS (zincblende). This seems reasonable
since blende and the new phase of FeS in its cubic form have the same structure and very nearly the same
lattice parameter (5.40 A and 5.42 A respectively).
However, as we will see, there still remains some
(3) We neglect the lattice contribution to the gradient, which
is about 1/10 of the electronic contribution.
ambiguity relative to the value of the spin orbit coupling parameter À.
Indeed, according to reference [15], d
=3 400 cm-1.
On the othèr hand for the free atom Â
= -103 cm-1,
p
=+ 0.95 cm - 1. This would lead to :
while experimental infrared spectroscopy on ZnS : Fe+ + shows that :
which is appreciably smaller. As discussed in refe-
rence [15] this reduction may have two origins : covalency effects which should reduce 1 À to 73 cm-’,
,or the Jahn-Teller effect. Recent Môssbauer experi-
ments on ZnS : Fe++ [16] ,indicate the presence of small Jahn-Teller effects but it is not yet clear whether
they can produce such a reduction of D. Both effects
(covalency and J. T.) might be present.
For this reason we will evaluate the hyperfine tensor both with À -100 cm -1 and Â
= -80 cm -1.
Another source of uncertainty is the Fermi contact
field, which is around 450-400 k0e but is not precisely
known. It is likely that covalency effects, if any, will reduce both 1 À. 1 and the Fermi field ; therefore we
will try either Â
= -100 cm-1 and AS
=450 kOe,
or Â
= -80 cm-1 and AS
=400 kOe.
In this way we get
These values of U are slightly higher than the expe- rimental value 145 kOe but show that our interpre-
tation is basically correct, in view of the uncertainties
on the parameters.
A similar analysis was performed in reference [12]
for briartite where the cubic field splitting is not
known. The authors assumed that J
=4 000 cm-1.
Then, in order to obtain U
=170 kOe, they had to
choose either Â
= -100 cm-1, AS’
=435 kOe or
À
= -80 cm-1, AS
=390 kOe. These results are
consistent with ours.
We must now take account of the orthorhombic distortion.
4.1.2 Inclusion of orthorhombic distortion.
-It manifests itself primarily by the existence of a non zero
asymmetry parameter ’1 for the E.F.G. Let OXYZ be the principal axes of the E.F.G., defined in such a way
that 1 Vxx 1 I yYY 1 1 Vzz I ; inspection of the crystalline structure shows that OX YZ coincide with the orthorhombic axes of the cell ; due to the smallness
of the orthorhombicity, OZ is along c, but we cannot
tell whether OX, 0 Y are ab or ba. By definition of il :
The origin of the non zero 11 is then the following :
the main effect of the orthorhombic distortion is to
slightly mix the wave functions of the two lowest orbital states, i.e. the wave function of the ground
state becomes :
whence :
From the experimental value at 0 K, ’1
=0.23 we
deduce that :
We must now look at the hyperfine structure and at
the Spin Hamiltonian. This is a little more complicated
as both contain second order contributions which involve the excited orbital states originating from T5. We must therefore take account of the effect of the orthorhombic distortion on the energy lèvels. It appears that the doublet 1 ± 1 ) is split in first order into two states
separated by an interval 2 0 which is a measure of the orthorhombic distortion ; we have assumed the ortho- rhombic distortion is small compared to the tetragonal
one, therefore 0/ô « 1. The energy of the singlet
is unchanged. Finally it is found that
and are displaced by quantities
of order 92/d, while the mixing parameter f3/LX
~0/à (as it should).
In terms of these it is found that the hyperfine
structure has the form :
with, to order 0/ô (we neglect terms of order
On the other hand the Spin Hamiltonian takes the form :
with to order 0/à :
Numerical values for the Spin Hamiltonian : with
we get :
we get :
The main result is that E is small and positive :
the easy direction in the basal plane is 0 Y.
Numerical values for the hyperfine field :
the uncertainty in these values being probably of the
order of 20 k0e.
Experimentally what we measure is the hyperfine
field along the easy direction O Y i.e. :
the values obtained 140-133 kOe are in satisfactory
agreement with the experimental value 145 k0e, which in contrast would be incompatible with U’. In
addition the lowering of 20-25 kOe upon going from U
in tetragonal symmetry to U" in orthorhombic sym- metry, is also in qualitative agreement with the diffe-
rence between the hyperfine fields in briartite and in
FeS (170 and 145 kOe respectively).
Finally we have already mentioned that the ortho-
rhombic distortion affected in the same way the
neighbourhoods of all Fe+ + ions. This, together
with the fact that we have a unique Môssbauer spectrum, implies that the magnetic structure of the
new phase of FeS is collinear along the O Y axis, i.e.
along one of the two orthorhombic axes, a or b, of the basal plane (choosing between a and b would req uire knowledge of the crystalline potential, which cannot
be computed with certainty).
In agreement with these predictions, recent neutron diffraction experiments (to be published separately [17])
indeed show that the magnetic moments are parallel
to the a axis.
5. Possible origins of the first order magnetocrys- talline transition in the new phase of FeS. We must
now investigate the origin of the simultaneous first order magnetic and crystalline transitions. We notice that immediately below the transition temperature : a - b, that is, the crystal is pseudotetragonal ; the
small orthorhombicity only develops at lower tempe- ratures, which means that it is perhaps due to standard magnetostriction. The main phenomenon to be explained is then the cubic - tetragonal distortion
around 234 K ; as mentioned in § 3 it is an homoge-
neous distortion which can be described in terms of the
macroscopic strain components :
One must also notice that this transformation occurs
almost without volume change : Ac ~ 2 Aa. Another feature of the problem is that the r3 orbital state of the Fe+ + ion is subject to the Jahn-Teller effect ; however for isolated Fe++ ions in ZnS, Imbert et al. [16] only
observe a very small Jahn-Teller effect
which makes it unlikely that standard cooperative
Jahn-Teller effect could play an important role in the
problem. In contrast a fairly strong orbital splitting off3 (~ 100 cm-l) has been observed by the Môss-
bauer effect for Fe++-Fe++ pairs in ZnS [7] but its origin was not clear (see table 1 of [7] and discussion
p. 2099) ; it might be due to electric multipole interac-
tions and/or generalized exchange (see below).
Usually first order magnetic transitions are ascribed either to biquadratic exchange [18, 19] or to exchange magnetostriction [20, 21]. Biquadratic exchange does
not give rise to crystalline distortion and can be discarded. We will now discuss, first, exchange magnetostriction and, second, three other possible
mechanisms :
-
generalized exchange interactions in the pre-
sence of orbital degeneracy,
-
exchange magnetostriction in the presence of Jahn-Teller coupling,
-
spin orbit coupling of magnetization and Jahn-
Teller distortions.
5.1 EXCHANGE MAGNETOSTRICTION. - In this mechanism [18] a simple exchange Hamiltonian of the
Heisenberg form is used :
but it is assumed that the exchange integrals lij depend rather strongly on the distances between the atoms. In reference [18] this is replaced for simplicity by a volume dependence. Then, by minimization of the free energy (Eq. (2) of [18]) one gets a first order
magnetic transition (Eq. (3) of [18]) together with a
volume change. More generally the magnetic transi-
tion will be accompanied by a crystalline distortion.
Such first order magnetic transitions have been observed in MnAs [18], K MnF3, ZnCr204, CdCr204 [13] and exchange magnetostriction is certainly a possible mechanism in cubic FeS (in particular, as already mentioned, a distribution of strains in the
crystallites could explain the distribution of transition
temperatures).
5.2 OTHER MECHANISMS. INTRODUCTORY DISCUS- SION.
-Before discussing these mechanisms it is
interesting to look at the review by Gehring and Gehring on cooperative Jahn-Teller effects [22].
According to this paper, which is essentially devoted to
rare earth compounds, the occurence of simultaneous
magnetic and crystalline distortions in R.E. com-
pounds can be examined with the help of a Hamilto-
nian of the form (see § 4.7 of [22], and also equations (3.14), (2.19), (3.20)) :
where
S is the true spin or an effective spin,
R z 3 SZ - S(S + 1) describes the Jahn-Teller
splitting of the electronic levels. R is a spin quadrupole,
a. ak are phonon creation and destruction opera- tors,
e is the macroscopic strain associated with the
change of shape of the sample,
ç(k) is a coefficient given by equation (3.20) of [22].
In this Hamiltonian :
-
the first term is the standard exchange coupling
between spins,
-
the second and third terms represent the linear
coupling of the spin quadrupole respectively with the macroscopic strain and the phonons. In rare earths, this is the coupling which may give rise to Jahn-
Teller effect,
-
the fourth term is the elastic energy,
-
the fifth term is the phonon energy (a% being
the angular frequency of a phonon of wave vector k).
We now eliminate the linear terms in ak, at and E :
this will give rise to indirect couplings between the Rn.
The method is the following :
-
As it concems e, we minimize the Hamiltonian with respect to E ; then :
We put this value into the Hamiltonian and find that the two terms in e are replaced by an indirect coupling
(see [22], eqs (3.14), (3.15), (3.16)).
-
In the case of the phonon terms several methods
can be used : second order perturbation theory, dis- placed harmonic oscillator, canonical transformation
(see [22], § 3 . 3 . l, 3.3.2, 3.3.4). With the displaced
operator methods one introduces new phonon opera- tors (see [22], eq. (3.23)) :
where
and one finds that the two phonon terms in equa- tion (22) are replaced by :
where the second term is an indirect coupling which,
in a perturbation approach, could be interpreted as
due to virtual phonon exchange. It can also be put into the form (see eqs (3.26) and (3.29) of [22]) :
Finally when we gather all these results we find a new
Hamiltonian of the form :
with
=V2s /cl2 (compare the second and third terms with equation (4.2) of [22], in which S H R, J(n - m) H K(n - m)). It appears that the coupling
between Jahn-Teller distortions Rn; i.e. the second term of equation (29), is due, both to virtual phonon exchange (K(n - m)) and to the macroscopic strain (U).
It remains to look for solutions of (29) with S > 0 (magnetic ordering) { and} or R :0 0 (qua-
drupolar ordering) [23]. If R > :0 0 there will be a macroscopic distortion of the crystal associated with the macroscopic strain e :
and/or displacements of the atoms with respect to their initial equilibrium positions, which can be
obtained from equations (3.27) and (3.28) of [22] :
where - Q > is the average displacement of the optical phonon mode at the centre of the Brillouin
zone.
We tum now to iron compounds. This case is less simple because, whilst standard magnetic exchange
still involves the spins, the Jahn-Teller coupling is now
related to the orbital degrees of freedom, which are not
tightly coupled with the spins.
For example if we consider the case of Fe+ + T3 and neglect the spin orbit coupling (which is zero in
first order) the Hamiltonian equivalent to (29) for
distortions along Oz would be of the type :
where in the second term, which describes the interac-
tion between orbital multipoles
L(n - m) H possible electric multipole interactions
x(n - m) H virtual phonon coupling of the Rn
- indirect coupling of the Rn
via the macroscopic strain .
With such a Hamiltonian we may have both magnetic
transitions and crystalline transitions (induced by the
orbital order through equations (30) and (31)) but
these transitions are completely decoupled (4).
Let us now retum to the problem of coupled transi-
tions. In order to have simultaneous magnetic and crystalline transitions we must look for couplings
between S and quantities of the type
Such couplings can be provided by :
.a) Generalized exchange interactions in the pre-
sence of orbital degeneracy such as exists in F 3 [24].
b) Strain dependent Heisenberg exchange in the
presence of coupling of the orbital multipoles with
the strain. This is a more complicated form of exchange magnetostriction, perhaps more appropriate to the
case of orbital degeneracy.
c) Second order effects of spin-orbit coupling
inside r 3’ We will now examine in turn these three mechanisms.
5. 3 GENERALIZED EXCHANGE INTERACTIONS.
-As discussed by a number of authors [25, 26, 27], in the
presence of orbital degeneracy the sum of the exchange
interactions and of the orbital multipole interactions has the form :
where the first term (with k, k’ even) represents the
sum of the electric multipole interactions and of the
(4) Notice that if ç(k) and Vg (and therefore also K(n - m) and MIN) were zero, there could be an orbital ordering via the electric
multipole interactions (L(n - m)), but without a crystalline distor-
tion. The coupling of the Rn with the strain and/or phonons is a
necessary condition to have a crystalline distortion ; throughout
what follows, we will assume this condition to be satisfied. Usually
a cooperative Jahn-Teller effect is associated with the coupling
indu ce an orbital ordering by itself, we could have a situation where the orbital ordering is induced primarily by the electric multipole interactions, while ç(k) and V$ only transmit it to the lattice. We will meet a situation of this type in the next paragraph, with the diffe-
rence that L(n - m) will arise both from electric multipole interac-
tion and from a part of the generalized exchange.
coupling between orbital multipoles via virtual pho-
non exchange and via the macroscopic strain ; and the
second term (with k, k’ odd or even, but k + k’ even)
is the generalized exchange (for one electron ions the constant is 1/4). This last term indeed provides a coupling between spin and orbit. In the case of T 3, only the second order tensors [3 Li - L(L + 1)] and
are non zero inside this doublet. On the other hand
we do not know the coefficients ( ijFqqkkk) (ijG’qqkk).
Therefore in order to study the possible simulta-
neous appearance of a magnetic order and a crystallo- graphic transition we may try to replace the general
Hamiltonian (33) by the phenomenological Hamilto-
nian :
where tetragonal distortions along Oz, represented by
[3 Lz - L(L + 1)] are coupled to the spin exchange
energy Si. Si.
Inside the orbital basis states 1 (Q>> = 0 > and
This suggests the use of a fictitious spin Q = 12 with
the correspondence :
Finally we will replace Si.Sj by an Ising form
Si. S jz and we arrive at a simplified Hamiltonian of the type :
whose phase diagram we will now study in the mole- cular field approximation. Notice that Cij contains
contributions coming both from cij and also from the
exchange term (bij x const.). This means that even if
the electric multipole interactions cij were weak,
Cijj could still be comparable to Aij and Bé since
all three quantities contain exchange contributions.
Remark.
-With the Hamiltonian of equation (35)
we have magnetic ordering and tetragonal distortions
along the same direction Oz, whilst in the new phase
of FeS we have a distortion along Oz and an ordering along Ox. This is not a real limitation since S and a
have different basis states : we would obtain the same
results with an exchange Aij Six S jx and a coupling of
the form Bij Qiz (J jz Six Sjx’
Let us retum to equation (35) and assume for sim- plicity S = 12 and a coupling with first neighbours only, with A’, B’, C’ > 0, so as to favour an antifer- romagnetic magnetic ordering, together with a ferro- magnetic crystalline distortion. Let us set :
(z
=number of nearest neighbours). In the molecular field approximation, we get the self-consistent equa- tions :
(where T is the temperature), which must be solved for
Notice that the system is symmetric with respect
to A and Y (X >+-- Y, A ;:± C).
When the coupling term B is zero, we have two separate second order phase transitions at T
=A/4
for X and T
=C/4 for Y. We now switch on the coupling B and look for simultaneous discontinuous transitions of X and Y. Detailed discussion shows that such a phenomena has a chance to occur only if B/4 > I A - C I, i.e. if the coupling is strong enough.
We have computed the solutions of system (36)
for three sets of parameters (A C).
With moderate values of B, i.e. for example :
we have a second order transition of Y alone at 4 T
=C ; then at a lower temperature T M
we have a discontinuous jump of Y between two
finite values accompanied by a discontinuous jump
of X between zero and a finite value.
It is only with large values of B, i.e. for example
FIG. 8.
-Anisotropic exchange model. Weak coupling. Plots
of X, Y versus 4 T.
FIG. 9.
-Anisotropic exchange model. Strong coupling. Plots
of X, Y versus 4 T.
that we get a simultaneous first order transition of X and Y at some temperature TM such that
As has already been said A, B, C may well be compa- rable and therefore such large values of B14 are not, in principle, incompatible with reality. In FeS we
may estimate that C - one or two hundred cm-1,
from pair interactions of Fe2+ ions in ZnS, and that A is also of the order of a few hundred cm-1, from the
transition temperature TN ~ 234 K, but we have
absolutely no way of estimating B. Therefore we can
only say that anisotropic exchange is a possible
explanation of the magnetocrystalline transition in
FeS. Notice that in this model the Jahn-Teller coupl-
ing only transmits the orbital distortion to the lattice.
but it does not create it, i.e. the existence of the crys- talline distortion does not require that the Jahn-
Teller coupling be large; this feature is interesting
since we know that the Jahn-Teller coupling of Fe+ +
in ZnS is small.
where ak is a phonon destruction operator and we have omitted some constant multiplicative factors.
As previously we also assume that there exists between the orbital degrees of freedom on the one hand and the strain and phonons on the other hand, a coupling of
the form :
Finally we have to consider the elastic energy pro-
portional to e2 and the phonon energy E nrok a+ ak.
k
As previously we eliminate the terms linear in e
and ak of the Hamiltonian JCex + JCr by using the
methods of Gehring and Gehring (minimization, displaced harmonic oscillators and so on). We then
obtain indirect coupling terms of the form (schemati- cally) :
-
four spin couplings :
whose existence was already mentioned in refe-
rence [ 13a] ;
-
interaction between orbital multipoles : :
to which electric multipole interactions may add a
contribution in L(n - m) as in equation (32) ;
-
and finally cross terms
which provide the coupling between spin and orbit
which we were looking for. Notice that here this
coupling is proportional to the strength of the Jahn-
Teller coupling (Vs and Çk)’
In order to simplify the molecular field treatment
5.4 EXCHANGE MAGNETOSTRICTION IN THE PRESENCE OF ORBITAL DEGENERACY.
-Let us assume that we
.
