- Email: [email protected]
A neutron scattering investigation of the magnetic phase diagram of FeI2
Journal of Magnetism and Magnetic Materials 74 (1988) 7-21 North-Holland, Amsterdam
A NEUTRON OF FeI,
SCATTERING
INVESTIGATION
A. WIEDENMANN
*, L.P. REGNAULT,
Centre d’Etudes Nucliaires
de Grenoble, DRF-SPh /MagnPtisme
0. KOUNDE
OF THE MAGNETIC
P. BURLET,
PHASE DIAGRAM
J. ROSSAT-MIGNOD
et Diffraction Neutronique,
38041 Grenoble Cedex, France
and D. BILLEREY
Universik! de Nancy, 54037 Nancy Cedex, France Received 17 February 1988
Magnetization measurements%nd neutron diffraction experiments performed on a single crystal of FeI, have revealed a H-T phase diagram which cont_aing, in addition to the paramagnetic phase, five ordered phases separated from each other by first order transition lines. At H = 0, a commensurate structure is built up below TN = 9.3 K consisting of a stacking of ferromagnetic (101) layers in** + + - - sequence with the moments (m ,, = 4.1~~) aligned along the c-axis. At the critical field Hc,, a first order transition tu a ferrimagnetic phase Fl occurs. Fl is a multi k-structure defined by three fundamental components and their second and sixth harmonics. In the magnetic unit cell with the hexagonal axes ma, \/iTa, c, one block of four nearest neighbour (n.n.) moments in the basal plane are aligned antiparallel to the c-axis surrounded by parallel aligned moments giving rise to a net magnetization of o(F1) = m,/3. N.N. moments on adjacent (001) layers are aligned parallel to each other. A second multi-k-structure F2 occurs above Hc, with a net magnetization a(F2) = 0.44ms resulting from the alignment of isolated moments and triangles of n.n. moments along the c-axis surrounded by moments of opposite direction. The moment arrangement in a magnetic unit cell 5a x 5a x c is described by three (observed) fundamental components and the fifth harmonics together with weak (and not observed) contributions of the second harmonics and of three additional vectors with their second harmonics. Another ferrimagnetic phase F4 occurs above Hc, with o(F4) = 0.6mo with the same wave vectors as observed for F2. However, F4 corresponds to a single k-structure, i.e. a stacking of ferromagnetic (110) planes in a sequence + + + + - . In a wide range between Hc, and Hc, an “amorphous”-like phase F3 was found characterized by a well defined magnetization o(F3) = 0.5m, and by the absence of any long-range periodicity. This particular H-T phase diagram results from the competition between various exchange interactions; a set of five exchange integrals is derived from mean field theory using the critical transition fields and the Neel temperature.
1. Introduction FeI, belongs to the ferrous halides which, crystallize in a hexagonal layered structure of space group Pjml. The unit cell (a = 4.05 A, c = 6.75 A) contains one Fe*+ ion located on the trigonal axis at the origin (000) and two I--ions at the positions (l/3 2/3 z) and (2/3 l/3 Z) with z = l/4 (fig. 1). The magnetic ordering of the homologous compounds FeCl, and FeBr, consists of an alternating sequence of ferromagnetic (001) layers resulting from strong ferromagnetic in-plane interactions and weak antiferromagnetic inter-plane * Present address: Hahn-Meitner-Institut Str. 100, D-1000 Berlin 39, Germany.
Berlin,
Glienicker
couplings [l-3]. These compounds exhibit a classical metamagnetic behaviour, i.e. an external magnetic field induces a transition from the antiferromagnetic state directly to the saturated paramagnetic phase [4-61. The magnetic ordering in FeI, is more complicated. At the N6el temperature TN = 9.3 K [5] FeI, exhibits, in zero field a first order phase transition [7] and developes a commensurate magnetic structure of wave vector k = [l/4 0 l/4] which corresponds to a stacking of ferromagnetic (101) planes according to a + + - - sequence and an alignment of the moments along the c-axis. Below TN, magnetization measurements in an external field applied parallel to the c-axis give evidence for several discontinuities [6] which have
0304-8853/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A. Wiedenmann et al. / Magnetic phase diagram of Fel,
resulting from the crystal field splitting [ 81 (D = 26 K [9]) makes a SF phase very unlikely, and actually a quite typical metamagnetic Ising-type behaviour is expected. The aim of the present work is to undertake a more careful analysis of the magnetic (H, T) phase diagram by means of neutron scattering and magnetization measurements in order to characterize the nature of the various intermediate magnetic phases.
2. Experimental
Fig. 1. Hexagonal
0
Fe2'
0
J-
structure of Fe1 *. Arrows indicate the relevant exchange paths.
been assigned to successive transitions from the commensurate phase (C) to a “spin flop phase” (SF) and to the paramagnetic phase, respectively. However, the high anisotropy of the Fe*+ ion
0
0
0
0
0
Magnetic Fig. 2. Magnetization of FeI,
Single crystals of FeI, were grown by a Bridgman technique. Magnetic measurements were performed at the high field facility of the S.N.C.I. at Grenoble on a single crystal by applying the magnetic field along the crystallographic c-axis. Neutron diffraction experiments were carried out on the diffractometer DN3 installed at the SILOE reactor of the CEN-Grenoble. In neutron scattering experiments, a single crystal of about 4 x 4 x 2
field
5
10
15
(Tl
as a function of a magnetic field applied along the c-axis. Several fist order phase transitions are observed defining ferrimagnetic intermediate phases.
A. Wiedenmann
et al. / Magnetic phase diagram of Fe12
9
mm3 was used and oriented with the c-axis vertical and parallel to the applied magnetic field, which was provided by a superconducting cryomagnet reaching field values up to H T 10.5 T. A short neutron wavelength of A = 0.98 A was used in order to observe more Bragg reflections out of the equatorial plane.
3. Magnetization results A series of magnetization isotherms M(H), measured below the NCel temperature as a function of the external magnetic field, is shown in fig. 2. At the lowest temperature T = 1.65 K, five magnetization steps are observed with increasing field at HC, = 4.75 T, HC, = 6.55 T, H:, = 7.9 T, Hc, = 9.85 T and Hc, = 12.35 T, respectively. With decreasing field, lower critical field values are observed, indicating some hysteresis effects. Between these magnetization steps, characteristic of first order phase transitions, the magnetization M(H) increases as a linear function of the external field, defining a constant susceptibility x0 according to M(H)
= urn,,+ x,,H.
(1)
This is still true above the highest transition at Hc, where no saturation is observed in the paramagnetic phase, and in fact the same constant susceptibility x0 is found as below the first transition at Hc,. The ferrimagnetic component u (in units of mo) associated with each of the intermediate ferrimagnetic phases Fl, F2, F3 and F4 can be derived assuming that x0 is field independent. The following values a(F1) = 0.33, a(F2) = 0.45, a(F3) = 0.5 and a(F4) = 0.6 can be deduced together with a moment value m, = 4.1~~ for a Fe*+ ion. We emphasize that this analysis is different from that in ref. [8] which deduced a ferromagnetic component mo/4 for the first ferrimagnetic phase. With increasing temperature we observe that the transition at Hc, = 7 T is smeared out and that the ferrimagnetic phase F2 (u 5: 0.45) disappears at a temperature as low as T = 2.2 K.
0
2
L 6 Temperature ( K 1
8
Fig. 3. Magnetic phase diagram of FeI, determined from magnetization (circles) and neutron diffraction (squares) experiments. In addition to the low field commensurate phase (c), four fenimagnetic phases called Fl to F4 and the paramagnetic phase (P) can be defined. (Full circles: ref. [6].)
A similar behaviour is also observed for the phases Fl and F4. From these measurements, the magnetic phase diagram (H, T), shown in fig. 3, can be deduced; the previous results [6] have also been included. Table 1 Observed intensities of the Bragg reflection measured at T= 1.47 K and H = 0 compared to the calculated values (R = 6%) hkl
Intensity (barn/tit
cell)
observed
calculated
100
0.22
0.18
110 020 200 210 120
0.22 4.03 0.18 0.18 0.16 0.15
0.18 4.04 0.18 0.18 0.18 0.18
0.93 2.95
0.90 3.49
oio
iii 021
A. Wiedenmann et al. / Magneticphase
10
This phase diagram contains six different phases, including the paramagnetic (P), the commensurate (C) and four ferrimagnetic (Fl to F4) phases. Each phase is separated from the others by a first order transition line, except between the phase F3 and the paramagnetic state P (dashed line in fig. 3) for which no experimental evidence is available. This quite complex phase diagram must result from strong frustration effects arising from the competition between intra- and inter-plane couplings.
4. Neutron scattering results 4.1. Commensurate
magnetic structure (C)
Nuclear Bragg reflections were measured in zero external field at T = 1.4 K. The full width at half maximum (fwhm) of the rocking curves was rather good, of the order of 0.3”. However, each peak was accompanied by a shoulder indicating a second peak of similar width, resulting in a large mosaic spread for the whole crystal. In order to obtain accurate integrated intensity values, it was necessary to rotate the crystal by an angle of 3” and therefore a rather poor collimation was used just in front of the detector. Apart from the most intense nuclear reflections, e.g. (llO), (0%) where extinction effects can cestainly not be neglected, the agreement between observed and calculated intensities is reasonably good as can be seen in table 1. Thus the instrumental scaling factor can be obtained with an accuracy of about 6%. In zero external field, the onset of magnetic ordering was observed below TN = (9.0 f 0.5) K in agreement with previous measurements [3,7]. Magnetic reflections corresponding to scattering vectors /I= H + k have been measured in the Brillouin zones H = [ hkO] and H = [ hkl]. Due to the three-fold symmetry, three wave-vectors kj were found: k, = [l/4 0 l/4], k, = [0 l/4 l/4] and k, = [l/4 l/4 l/4]. They lie on symmetry lines of the Brillouin zone’ and correspond to one fourth of a lattice vector kj = H/4. Such a wave vector describes a commensurate structure corresponding
diagram of FeZI
to a sequence (+ + - - ) but not a simple antiferromagnetic ordering [16]. The intensity of a magnetic reflection is given by I(h)
= (0.27)2f2(h)[
IF(h)
12
-ww~2/l~12]>
(3)
where f(h) is the magnetic form factor of Fe2+ ions and P(h) the magnetic structure factor which, in the present case of a single Bravais lattice, reduces to the Fourier component mkj associated with the wave vector kj. When only one Fourier component mk is present and it moreover transforms according to a one dimensional representation, it can be written as mk = +A,ii, (exp i$,).
(4)
Then the ordering corresponds to a pure sine-wave modulation of the moment value along the k direction with an amplitude A, and a polarization along iJk. For a given Fe’+, defined by the lattice translation R,, the magnetic moment is given by: m(R,)=A,cos(2Pk*R,++,)iik.
(5)
We emphasize that for a wave-vector k = H/4 the particular choice of the phase + = a/4, gives a + + - - sequence for which all magnetic moments have the same magnitude: m,=A,/fi
=filmkl.
(6)
For such a collinear ordering the intensity of a reflection associated with a given wave-vector kj can be written as: I(h)
= (0.27)2f2(
h) sin’clll mk, I 2,
(7)
where (Yis the angle between the scattering vector and the moment direction ii,. The intensity analysis of the reflection measured inside the Brillouin zone H = (000), (010) (110) (110) (010) clearly shows that moments are aligned along the c-axis, and then for each wave vector kj a mean value of I rni I can be deduced as reported in table 2. Exp&iments with an applied field have shown that the magnetic ordering is characterized by only one wave-vector k (single-k structure). Therefore, three domains exist, and the value of
A. Wiedenmann et al. / Magnetic phase diagram
11
of Fe12
Table 2 Commensurate magnetic structure of FeIz at T=1.4 K and H = 0. Fourier components mk corresponding to kl = [l/4 0 l/4], k, = [0 m l/4] and k, = [m l/4 0] derived from different magnetic reflections. The volume fraction u, is given for the three domains
(In the following, the magnitude of the Fourier component will be designated by mk.) Each domain will have a relative volume given by
Reflection
At T= 1.47 K a value Imk I = (2.86 f 0.11)~~ is obtained, yielding a magnetic moment value m,, = J? Imk I = (4.05 f 0.15)~~. This value is quite close to that expected for Fe2+ and to that observed by magnetization measurements. Domains are found to be almost equally distributed at low fields (see table 2). However, above H = 3 T intensities associated with each wave vector start to change in favour of the k,-domain. This behaviour certainly results from a small misalignment of the applied field direction with respect to the c-axis.
mk,
(010) + k, (llO)+ k,
1.41 1.83 1.94
(OiO)+ k,
1.83
W’O)+ k, (100) + k, (OlO)+ k,
1.55 1.64 1.77
PW+ k, (100) + k,
1.48 1.67
(iio) + k3
1.47
W')+k,
01 CQ
(PB)
1.76rtO.9
37*9
1.64kO.6’
32&5
1.54 f 0.6
2954
the Fourier component mk is obtained by summing the contributions of the three domains
(8) 1
30
.T.
I
1
10
60
Uj=WQlffl,12.
(9)
4.2. Field induced ferrimagnetic
phases
At T = 1.47 K the intensity of the nuclear reflection (100) measured as a function of the 1
I
I
1.46 K
1
I
Magnetic
field
70
8.0
1 Tesla
)
Fig. 4. Intensities of magnetic reflections of Fel, at T=1.4 K as a function of the magnetic field corresponding to the scattering vectors: [lOO]: (o), [l/4 0 l/4]: +, [l/6 l/6 01: A, [l/5 l/5 01: 0.
external field (see fig. 4) exhibits two sharp steps at the critical fields H=, and Hc,, indicating first order phase transitions from the commensurate (C) to a ferrimagnetic phase Fl and from Fl to a fe~a~etic phase F2, respectively. The magnetic contribution to the (100) reflection is in excellent agreement with magnetization measurements and gives a reduced ferromagnetic component o(F1) = 0.32 and a(F2) = 0.41, respectively. For an applied field H ==9 and 10.5 T these contributions correspond to a reduced ferromagnetic component o(F3) = 0.5 and a(F4) = 0.6, also in good agreement with magnetization data. Accurate scans performed along symmetry directions in reciprocal space indicate that the ordering in each ferrimagnetic phase is described by a ~~ensurate wave vector with k(F1) = [l/6 l/6 0], k(F2) = [l/5 l/5 0] and k(F4) = [l/5 l/5 01. A plot of the peak intensity maximum of the most intense reflections associated with each of these vectors is given in fig. 4 as a function of the applied field. At the critical fields ch~acte~stic intensity jumps are found to be in good agreement with the magnetization measurements. All these ferrimagnetic phases differ only by the in-plane ordering, the stacking along the c-axis being always of ferromagnetic type. 4.3, Magnetic structure of the ferrimagnetic pkine Fl
Apart from a change ,in the domain distribution, no variation of the wave-vector, the direction and the magnitude of mk is observed up to H = 4.6 T. However, a small additional increase of the field up to UC, = 4.7 T induces a sudden disappearence of the magnetic reflections associated with the wave-vector k1 = [l/4 0 l/4]. Siiultaneously, an increase of the intensity of the nuclear reflections (lOO), (OIO), (110) etc. have been found, which corresponds to the scattering vectors marked by closed circles in fig. 5. The most intense peaks are associated with the equivalent wave-vectors kl = [l/6 l/6 O], k, = [l/3 1/6 0] and k, = [l/6 1/3 01. In addition to these fundamental reflections, second harmonics corresponding to 2k, = [l/3 l/3 01, 2k, = [2/3 l/s 0] and 2k, a=
Fig. 5. Reciprocal space of Fe12 showing the symmetry lines (thick lines) and the points (circles) investigated at T = 1.47 K and H = 5.5 T (phase Fl). The observed intensities (full circles) correspond to k, = [l/6 l/6 01, kz = [l/3 m 0] and k3 = [l/6 v 0] and the second harmonic 2 k.
[l/3 v 0] are also observed with weaker intensity. A very careful investigation of the reciprocal space, especially for scattering vectors shown by open symbols in fig. 5, allows us to exclude the presence of any other wave-vectors, in particular no third order harmonic can be observed. Intensities of magnetic reflections, corresponding to a given wave-vector, measured in various Brillouin zones (corrected for the magnetic form factor) have quite similar values, which allows us to conclude that magnetic moments remain aligned along the c-axis. Therefore, the spin flop phase at H > 4.6 T, as proposed in ref. [4], must definitely be ruled out. In fact, the ordering must be of ferrimagnetic type since both the ferromagnetic and the co~ens~ate components are observed parallel to the c-axis. For the Fl phase, the value of the ferromagnetic component at T = 1.47 K, mkzo ?: 1.37~,, is very close to one third of the moment value m,, for Fe’+ ions. According to eq. (7), the magnetic Bragg peak intensities measured at T= 1.47 K and N = 5.5 T allow us to determine the amplitude of each of the three Fourier components. The results obtained are reported in table 3. It must be emphasized that the amplitudes
A. Wiedenmann et al. / Magnetic phase diagram of Fel, Table 3 Fenimagnetic phase Fl of FeI, at T=1.4 K and H = 5.5 T. Observed and calculated Fourier components mk associated with the wave-vector k, = [l/6 l/6 01, k2 = [l/3 1/6 0] and k, = [l/6 m 01. The calculation corresponds to a triple-k structure of the ferrimagnetic phase Fl as explained in the text h
H f k,
mk
(d
observed
(l/6 l/6 0)
VW+ k,
2.04
(5/6 1/6 0) (5/6 5/6 0)
(100) - k, (110) - k,
1.97 2.03
(l/3 l/3 (2/3mO)
(000)+2k
0.66
0)
(000)+2k
0.65
(1/3mO)
(000)+2k
0.64
(T/3 r7a 0) (213 l/6 0)
(000) + k, (100) - kz
1.56 1.37
(2/3 5/6 0)
(110) - k,
1.37
(T/6 V (5/6 l/3
0) 0)
(000) + k, (100) - k,
1.4 1.38
(5/6 273 0)
(li0) - k,
1.37
(100)
6k
calculated
2.0 f 0.10
2.05
= Crnk, exp 2dk,*R,.
domain can be determined from eqs. (8) and (9). ii) The structure is described by each of the three wave vectors kl, k, and k, and the corresponding harmonics. In that case, neutron data indicate that the associated triple-k magnetic structure does not recover the trigonal symmetry because the three Fourier components are not equal. Therefore, while the structure is of triple-k type, three domains must exist and each of these gives a contribution to a given reflection such as: (11)
0.64+ 0.03
0.66
1.37+0.05
1.21
1.37 f 0.05
1.21
1.36 f 0.05
1.37
of the Fourier components are not equal, mk, being larger than mk, and mk,. Since a neutron scattering experiment gives only the modulus of each Fourier component and not its phase, it is not a trivial problem to get the true magnetic ordering. The moment distribution is given by summing over all Fourier components such as: m(R,)
13
00)
Therefore, many possible in-plane structures can be constructed from the neutron data. In the present case, two possibilities must be considered: i) The structure is described by a single wave vector (e.g. k,). In that case, the sum contains the fundamental Fourier component mk, the harmonics and the ferromagnetic component rnhk = mk = 0. The other wave vectors k, and k, are then associated with the other magnetic domains. The existence of three K-domains arises from the loss of the three-fold symmetry axis with respect to the paramagnetic group. The relative volume of each
That means that at least six different magnetic reflections associated with each wave vector kj must be measured in order to evaluate mk and uj independently. The problem can be sim&fied if we assume that the crystal is in a single domain state favoured by the strong applied field. Then u2 = ug = 0 and the Fourier components mk,, mk, directly from the correand mk, are obtained sponding reflections. It must be noted that the intensity of the second harmonic 2k is the same in both cases i) and ii) since for the three wave vectors the second harmonic 2k = [l/3 l/3 0] overlaps. Let us examine the first possibility, i.e. a single k-structure with three domains. Then the ordering corresponds to a stacking of ferromagnetic (110) planes but only the sequences + + + + - - and + + + - + - are compatible with the observed wave-vector k = [l/6 l/6 0] and with a ferromagnetic component mkcO = m,/3. For these two square-wave modulations the Fourier components of higher harmonics can be calculated according to the relation: 6
mk=
T
c a,, exp(2Tikn), II=1
(12)
where k = l/6 and a, is the reduced magnetization per ion of a plane n, which can take the values u = f 1 according to whether the orientation of the moments is “up” or “down”. The + + + - + - sequence gives a third order harmonic with m3k = f m. in complete disagreement with experiment, indicating the absence of such a Fourier component. Thus this sequence can be ruled out immediately.
A. Wiedenmannet al. / Magneticphase diagram of Fe12
14
In the case of the + + + + - - sequence the component mJk is effectively zero. However, the values of the components mk= m,/fi= 2.37~~ using the and m 2k = m,/3 = 1.37j.~a, calculated value m, = 4.&, do not agree with the observed values of mk = (ESrn: )112 = 2.83~~ and 1mZk 1 = 0.65~~. In particular,‘the ratio m2Jmk is calculated by more than a factor of two larger than the observed one, i.e. well outside the experimental accuracy. We have further checked if this discrepancy can be explained by an incomplete squaring up of the sinewave modulation by giving the possibility for individual moments to vary slightly from the saturated value as m, = m,(l f E,), with en x 1. For this purpose we calculated m, according to the Fourier development: m, = 2m, +m,,,.
cos( +nn + &) + 2m,,
COS(
$7,
+
G2)
(13)
The phases +i and I$~were varied in order to get a modulation of moments as small as possible. The optimal value would correspond to a single kstructure with three different moment values of 2.8~s, 4.2~~ and 5.6~~ which is actually too large a deviation from the saturated value to be an acceptable solution. Therefore, we conclude that the magnetic ordering must be described in a unit cell 6a X 6a x c. The net magnetization u, per Fe2+ ion of a plane n perpendicular to a given wave-vector kj is given by
(14)
where the summation is carried out over six consecutive moments of the basal plane and wntained in the planes n which have a reduced moment a,,, = + 1. In contrast to the single kstructure, the magnitude of u,,(kj) may change from plane to plane, for instance if we consider the wave-vector k = [l/6 l/6 0] a, can take the values f 1, +2/3, + l/3 and 0. Then the Fourier components mk, and m2, associated to a given stacking of six planes with a
sequence eq. (12):
{ a,,(/~~)}, can be calculated
m0
mk, = 6
according
to
{ uI - a, + f(u2 - u3 - a5 + u6)
++ifi(u2+u3-us--u~)}, m2k,=
(15)
““{ 6 +
(II + a, -
iifi(
Taking into magnetization i
Un(kj)
;(
a2 + 63 + a, + UC)
a,-u,+u,-a,)}. account the condition of this phase is m,/3
that i.e.
the net
= l/3
(15a)
k=l
and that the third order harmonic i.e.
is not observed,
u,+u,+u,-(u2+u,+u,)=0,
(15b)
only one sequence together with the cyclic permutations is compatible with the experimental values for each wave vector kj. The result corresponds to the sequences {a,(k,)}={-l,O,l,l,l,O} {%(Q}
= {e,(Q)
and =
(0, 1, l,O,O,
O}.
However, it turned out that the remaining problem, consisting of arranging the 36 moments in the unit cell 6a X 6a with the proper sequences along k,, k, and k,, cannot easily be solved by hand. In the unit cell 6a X 6a the moments of the Fe2+ ion n = (n,, n 2) have been calculated according to the relation: m(ni,
nZ)=mk,e+2mk,
1
cos +(ni+n,)+& ]
+2mk2
cos[ :(2n,
+2mk,
cos j(n, ]
+2m,,
cos I$?( [
- rz2) + A] - 2n2) +&
n1+
n2)
+
1
1 (16)
$4
and using the observed values of the mk,. The phases +i to e4 have been varied by steps of 1”
A. Wiedenmann et al. / Magnetic phase diagram of FeIr
15
moments. This arrangement can in fact be described in a smaller hexagonal unit cell with a parameter a’ = 2&, as shown in fig. 6. This new magnetic unit cell contains three blocks of four moments, two of them being aligned along the applied field and the third one having a reversed orientation. 4.4. Magnetic F2
Fig. 6. Moment arrangement in the ferromagnetic phase Fl at H = 5.5 T. The unit cell (6a, 6a, c) (full lines) and the elementary unit cell (2fia, 2fia, c) (dashed lines) are shown. The ordering corresponds to the formations of blocks of four “up” moments (full circles) among “down” moments (open circles).
from --71< I#+< 71, until all moments m (n,,n,) have the same value within an accuracy of 10%. Within this restriction only one set of phases @i = CJd= 71, !PZ= @s = 7r/2 gives a good agreement. The corresponding moment arrangement is shown in fig. 6. First of all, we can verify easily that the sequence in the directions ki, k, and k, are really cyclic permutations of the sequences determined above. Secondly, it can be checked that the calculated Fourier components of the proposed commensurate triple-/c structure are quite close to the observed ones as can be seen in table 3. Moreover, the net magnetization of each plane a,( k,) perpendicular to any direction different from k,, k, and k, is equal to m,/3, indicating that no further Fourier components are present, in agreement with the experimental results. It must be emphasized that the proposed in-plane structure has no trigonal symmetry in which case three magnetic domains must exist. Nevertheless, the presence of a single domain state in the crystal at T = 1.47 K and H = 5.5 T is more likely since the amplitudes of the Fourier components mk calculated with this assumption, (i.e. uz = u3 = 6 in eq. (ll)), agree quite well with the observed ones as shown in table 3. So, in the ferrimagnetic phase Fl the in-plane magnetic ordering, reported in fig. 6, consists of blocks of four down moments surrounded by up
structure of the ferrimagnetic phases .a
At the critical field HC, = 6.55 T all the magnetic reflections associated with the wave-vectors k, = [l/6 l/6 01, k, = [l/3 l/6 0] and k, = [l/6 l/3 0] disappear simultaneously together with the second harmonics, whereas the intensities of nuclear reflections increase substantially. At T = 1.47 K the ferromagnetic component was found to have a value mk =0 = 1.7~~ corresponding to about 41% of m,. This value is somewhat smaller than the value a(F2) = 45% observed by magnetization measurements. By scanning the reciprocal lattice along the various symmetry lines of the Brillouin zone, new magnetic reflections were found which are associated with the wavevectors k, = [l/5 l/5 01, k, = [2/5 l/5 0] and k, = [l/5 2/5 01, defining a new commensurate phase, with moments still aligned along the c-axis. As for the Fl-phase we first examine the possibility of a single k-structure. Two sequences are compatible with the observed wave-vector k = [l/51/50]: ++++and +++--. However, these sequences give rise to a net magnetization of ur = 3/5 and l/5, respectively, in disagreement with the observed value uy = 0.4. Furthermore, these sequences have large second order harmonic mZk = 0.40 and 0.25m,, respectively, while no contribution has been found experimentally (see table 4). Thus such square-wave-modulated single-k structures must also be ruled out. Moreover, the assumption of a mixed phase composition by both sequences with a resulting average magnetization of (3/5 + l/5)/2 = 0.4, close to the observed value, must also be rejected since it cannot account for the absence of the second order harmonic. So we are forced again to consider the in-plane arrangements ,of the 25 moments within the mag-
A. Wiedenmann et al. /
16
Magneticphase diagram of Fel,
Table 4 Ferrimagnetic phase F2 of FeI, at H = 7.1 T and T = 1.4 K. Observed Fourier components mk compared with the calculated ones based on the model of hp. 7 and ma = 4.1~~ H f k,
(h)
mk,
h)
observed
calculated
(l/5 l/5 0) (4/5 4/5 0)
(OOO)+ k, (110) - kl
0.99 0.94
0.97 + 0.05
1.39
(215 l/5 0) (3/5 l/5 0) (3/5 6/5 0)
(000) + k, (lOO)- k, (110) - k,
0.76 0.68 1.07
0.84!cO.O5
1.39
(I/5 215 0) (415 215 0) (6/5 3/5 0)
(000) + k3 (NO)- k, (110) + k,
0.88 0.83 0.90
0.90+0.05
1.39
1.70 f 0.05
1.80
k=O
(100) m 2k,=m2k2=m2k3 mkt m 2kf
=
mk,, =
m2k,,
=
mkw =
m2k
111
0 0 0
0.08 0.53 0.20
Fig. 7. Moment arrangement in the ferromagnetic phase F2. The magnetic unit cell (5a x5a x c) formed by n.n. triangles of “down” moments (full circles) around one isolated “down” moment. The full and dashed lines must contain one or two down moments, respectively, in order to obtain the lowest possible values of mkt, mkt. and mk 11,.
lowest values of mkf and rnZkTcorresponds to the sequence netic unit cell 5a x 5a which account for the Fourier components mk=,, and mk, associated with the three wave vectors k,, k, and k,. This unit cell must contain 7 down and 18 up moments in order to get the required ferromagnetic component (I~= 11/25 = 0.44. The net magnetization of a plane (n = 1 to 5) perpendicular to a vector kj is again given by eq. (14) where the sum must now be extended over five consecutive moments. The possible values are u,(kj) = f 1, f 3/5 and f l/5. For all planes perpendicular to any wave-vector of the type [h/5 k/5 0] the net magnetization is always different from the ferromagnetic contribution ur = 0.44. This means that the intensity of magnetic superlattice peaks associated with wavevectors kj of this type can never be strictly zero. Consequently, in addition to the observed vectors k,, k, and k,, the vectors k’ = [l/5 0 01, k” = [0 l/5 0] and k “’ = [l/5 1/5 0] and their harmonics must also be present. Since we did not detect reflections associated with these wave-vectors and their second harmonics, the corresponding Fourier components must be very weak. We calculated mkt and rnzkt for any sequence of five planes n = (1 to 5) corresponding to the above allowed sequences { a,,( kj)}. It turned out that the
{ e(k’)}
= {3/5,3/5,3/5,
I/5, I/5}
(or their cyclic permutation) with rnp = 0_13m, and mZkJ = O.O5m,. The following graphical method was used to determine the in-plane moment distribution. In the direct lattice of basis vectors a, and a2 given in fig. 7, planes perpendicular to the wavevectors k’, k” and k “’ are represented by full or dashed lines according to whether they have a net magnetization per atom of 3/5 (one down and four up moments) or l/5 (two down and three up moments), respectively (fig. 7). Then the seven down moments are disposed in the unit cell 5a X 5a in such a manner that all full and dashed lines contain the right number of down moments. There are several solutions which satisfy the condition that rnkf ,and rnZkT must have low values when different cyclic permutations of the sequence are considered. However, only the in-plane moment arrangement reported in fig. 7 agrees with the additional conditions that the Fourier components Of the second harmonics mZk,, m2kz and m2k, must also be very weak. In the proposed in-plane magnetic structure, the down moments are arranged in six equilateral triangles around an isolated central down mo-
A. Wiedenmann et al. / Magnetic phase diagram of Fez2
ment. In this case the hexagonal symmetry is recovered and there is no K-domain. In particular, the Fourier components associated with k,, k, and k, must be equal in agreement with experimental results. From this point of view, the proposed model is the only one which is compatible with the essential experimental results. However the observed values of all Fourier components are systematically smaller than the calculated ones (table 4). This may be explained partly by experimental effects such as an increase of the extinction in the ordered phase. Another possibility is the coexistence of long-range ordering F2 with an incomplete ordering as discussed in the following section. By comparing the observed values of mk,, mk, and mk, with the calculated ones (table 4) we get a volume fraction u = 45%, i.e. only about one half of the moments are involved in the long-range ordering of the F2 type.
17
l
l
110
ii0
Fig. 8. Scans performed in reciprocal space at T=1.47 K and H = 8.3 T and H = 8.9 T which revealed the absence of long range ordering in the “amorphous” phase F3.
4.5. “Amorphous phase” F3 At T = 1.47 K all reflections associated with the wave-vector k[1/5 l/5 0] disappear at the critical field Hc, = 8.05 T and the intensities of nuclear reflections are increased by 12%. At H = 8.9 T a number of scans along the symmetry directions were performed (fig. 8) but, surprisingly, no magnetic reflections have been detected. Only the scan along the direction [llO] revealed a very large peak centered at [l/5 l/5 01. This means that, for these values of magnetic field and temperature no complete long-range ordering is established, in spite of the well defined net magnetization of the ferromagnetic phase F3 with q(F3) = 0.5. This value is much smaller than the value a,, = 0.8 expected for uncorrelated magnetic moments in an external field of H = 8.1 T, indicating the existence of some ordering. Moreover, the large peak k = [l/5 l/5 0] suggests that some kind of periodicity must be present within the plane. However, the correlation length remains rather small indicating a great variety of possible moment arrangements. For a given field value H = 7.05 T, the reflections associated with the wave-vectors k,, k, and k, of the ferrimagnetic phase F2 disappear sud-
denly when the temperature is increased, indicating that a first order phase transition occurs at T = 2.3 K (fig. 9). When the sample is cooled down again to T = 1.47 K, the intensities I(k,), 1(/c,), I(k,) have different ratios, whereas the ferromagnetic contribution keeps the same value. Similar behaviour has been observed for a lower field H = 5.5 T (see fig. 9a). At low temperature the triple-k-structure Fl defined by k, = [l/6 l/6 01, k, = [l/3 l/6 0] and k, = [l/6 l/3 0] is stable only up to T = 2.1 K where a first drastic change of the intensity ratio occurs. Then at T = 3.5 K all reflections disappear in a first order phase transition. Again, above this transition no magnetic peak associated with long-range ordering has been detected in spite of the ferromagnetic component (q = l/3) remaining unchanged, In cooling down to low temperatures, the initial intensities were never regained, implying that the “amorphous phase” F3 still coexists with the ordered ferrimagnetic phase Fl. This type of incomplete long-range ordering associated with the ferrimagnetic phase F3 is similar to that observed in amorphous systems. It may result from the competition between different in-
A. Wiedenmann et al. / Magnetic phase diagram of FeI,
18
15
H : 55T
\ CI)
x e 2 2 10 ‘0 u e B s 5
Fl-F3
A
0
1
I
Q&9+
I
--
#-I l,O,Ol
I-
q
bl x ‘5 t s S 2 10 b
F2-F3
H:%OT
qi;,:[l/S .
ciao)
11501 i
E E
0 Temperature! K Fig. 9. Thermal variation of the integrated intensities of the (100) reflection (full circles) and the superlattice reflections h = (OOO)+k, showing the transitionS in constant external fields: (a) H = 5.5 T (Fl - F3) and (b} H = 7.05 T (F2 cf F3).
plane interactions, leading to a high degree of frustration. In fact, such “amorphous” phases have been predicted for the triangular ISING model with competing interactions using a cluster-variation method [ll] and from Monte Carlo simulations [12,13]. Another possibility would be the “floating phase” predicted for frustrated anisotropic systems [14-161, i.e. a continuous change of the wave-vector as a function of temperature and applied field. However, this phase must disappear when, we& inter-plane interactions are present as in FeI,. 4.6. Ferrimagnetic phase (F4) When the external field is raised up to the experimental limit of 10.3 T, the intensity of the
(100) reflection starts to increase again (see fig. 4) and well defined magnetic superlattice peaks associated with the wave-vectors k, = [l/5 l/5 01. kA-’ = [2/5 1/5 0] and k, = [l/5 2/5 0] are agaii’obt served. The intensity does not seem, however, to be completely saturated. In a second experiment we placed the sample slightly off the central position of the superconducting coil in order to have a larger magnetic field. This increase of about 10% was sufficient to produce a single phase with a ferrimagnetic ordering called phase F4. However, it was not possible to get reliable quantitative values for the integrated intensities in this way. Qualitatively, we found that the magnetic contribution to the nuclear reflections gives roughly a net magnetization a,(F4) = 0.6, in agreement with magnetization measurements. The Fourier components associated with the wave-vector k, seem to be the dominating ones, the following values having been estimated: mk, = 0.07m0, mk, = O.lm, and mk, = 0.3m,. No other contributions associated with higher harmonics or other vectors were found. These observations allow us to make several conclusions. First of all, in a single k-structure a ferromagnetic stacking of planes with a + + + + - sequence can account for the observed ferromagnetic component af(F4) = 0.6 and for the observed wave-vector k = [l/5 l/5 01. Moreover, it can explain the estimated value ( 1FE, 1 = 0.32m,) for the fundamental Fourier component which should be equal to about 0.4m,. The only problem arises from the lack of observation of the Fourier component associated with the wave-vector 2k, which must have the same order of magnitude as the fundamental component. On the other hand, if we consider a magnetic unit cell 5a x 5a we must find a moment arrangement such as the Fourier components associated with the vectors k’ = [l/5 0 01, k” [0 l/5 0] and with k “’ = [l/5 l/5 0] and their harmonics are equal to zero. This could be achieved assuming that the planes perpendicular to these vectors have a net magnetization a,( k,) equal to the ferromagnetic component ur(F4) = 0.6, i.e. five up and one down moments. When we apply the graphical method described above, we end up with the only possible moment arrangement, shown in fig. 10, which is in
A. Wiedenmann et al. / Magnetic phase diagram of Fel,
Fig. 10. Single k-structure of the ferrimagnetic phase F4 defined by the wave-vector k, = [l/5 v 01.
fact the single k-structure already mentioned. In that case three domains can co-exist. If we accept the absence of the Fourier component mzk then we have to assume, as found in the previous case for the F2 phase, a weak but non-zero value for the Fourier components with k’, k”, k ‘I’. The lowest values were obtained for the sequences { u,(k’)} = {l/5, 3/5, l/5, 3/5, 3/5} with mkr = O.l5m, and rnzkf = 0.09. Along the vectors k,, k, and k, the sequences would be, e.g.: (1, 3/5, l/5, l/5, l} yielding the Fourier components mk = 0.25m, and mzk= O.O6m, in agreement with experimental results. A great variety of combinations was considered but in order to decide which arrangement actually occurs, one needs accurate values for mk,, mk and mk,. Since these are not available, it is impossible to proceed further with the determination of the moment arrangement in phase F4.
5. Discussion Magnetization measurements and neutron diffraction experiments performed on a single crystal revealed that the layered compound FeI, exhibits a quite original H-T phase diagram as shown in fig. 3. This phase diagram contains, in addition to the paramagnetic phase P, five ordered phases which are separated from each other by first order transition lines. In zero applied field a commensurate magnetic ordering defined by the wave-vector k = [l/4 0 l/4] is built up via a first
19
order transition at T, = 9.3 K. It consists of a stacking of ferromagnetic (101) layers in a -t + - - sequence with moments (m. = 4.1~~) aligned along the c-axis. This ordering remains stable up to a critical field HC, where a first order transition to a ferrimagnetic phase Fl takes place. This transition occurs only at temperatures smaller than the tricritical point TC, = 3.8 K. The inplane ordering of the Fl phase is not a simple single k-structure but a multi-k structure involving the three fundamental components with k, = [l/6 l/6 01, k, = [2/6 1/6 0] and k, = [l/6 l/3 0] and the second and sixth harmonics. Such an ordering (see fig. 6) consists a blocks of four magnetic neighbour moments aligned along the c-axis and antiparallel to the applied field surrounded by parallel aligned moments, yielding a net magnetization a(F1) = m,/3. Nearest neighbouring moments in adjacent (001) layers are aligned parallel to each other. A transition to a second ferrimagnetic phase F2 occurs at HC, below a tricritical temperature l& = 2.9 K. This phase has a net magnetization a(F2) = 0.44m, resulting from an in-plane moment arrangement reported in fig. 7, characterized by triangles and isolated moments aligned antiparallel to the applied field surrounded by moments having the opposite direction. This magnetic structure of a unit cell 5al5a has a trigonal symmetry and hence has no k-domains. Besides the observed commensurate wave-vectors k, = [l/5 l/5 01, k2 = [2/5 l/5 01, k, = [l/5 2/5 0] and k, = 0, for this ordering the amplitude of the Fourier components associated with the second harmonics 2k as well as that associated with the wave-vectors k’ = [l/5 0 01, k” = [0 l/5 0] and k “’ = [l/5 l/5 0] and their second harmonics are non-zero, but they are too weak to be observed experimentally. Another ferrimagnetic phase F4 exists above a critical field HC, with a net magnetization a(F4) = 0.6m, with the same wave-vectors as for phase F2. In contrast with the phase F2 the ordering of the phase F4 corresponds to a single k structure i.e. stacking of ferromagnetic (110) planes with a se: quence + + + + - (fig. 10). However, a difficulty arises because a squaring up of the sine-wave modulation yields a second order harmonic 2k which has not been observed experimentally. A
20
A. Wiedenmann et al. / Magnetic phase diagram
very interesting result is the presence of a wide range of existence of the phase F3 in the H-T diagram. In this phase magnetic moments must be ordered since the net magnetization is much lower than that expected for the paramagnetic state. However, the ordering is not long-range since no magnetic reflections have been observed, and in fact only a very broad peak along the [llO] line has been detected. An interesting result is that the magnetization remains constant at the boundary between the phases Fl and F3, or the phases F2 and F3, respectively, while the wave-vectors disappear discontinuously. On the other hand, at a given temperature the magnetization m (F3) increases continuously with increasing the magnetic field. This type of incomplete long-range ordering is similar to that found in amorphous systems. A local ordering is present but the correlation length must be small due to the great variety of possible moment arrangements. The original phase diagram obviously results from important frustration effects present in the triangular lattice due to the competition between various in-plane and inter-plane interactions. The most important in-plane couplings are the superexchange interactions via the iodide ions between first nearest neighbour (nn) Fe2+ ions at d, = 4.05 A second nn at 7.01 A, and third nn at 8.1 A. The couplings between adjacent layers at di = 6.75 A, d,’ = 7.87 A and d; = 9.73 A are expected to be much weaker. The corresponding exchange integrals are denoted as J1, J2 and J3 for the in-plane interactions and J,‘, J; and J; for the inter-plane couplings (see fig. 1). The energy of the various ferrimagnetic phases in an applied field can be written as
E,(H)
= E,, - a,m,H,
where the exchange according to
energy
E,,
is
calculated
E,, = - cJljS;S’sj’ ij by using the relevant
exchange
integrals
as defined
of Fel,
above. For S: = f 1 we obtain tions:
the following
rela-
EC= -2(J,-J,-J,-J;‘), E (Fl)
=
-{$J,-+J,-25,+2J;+2J;’ + fJ{ - $3;)
E (F2)
=
-
{
- +m,H,
$J, + &J, - %J, + 2J; + 2J;
+ pJ{
+ g-J;> - 0.44m,H,
EcF4)= -{$J,
(17)
+ ~J2+$J~+2J;+2J;’
+ YJ,’ + ?J;}
- 0.6m,H,
E (pm) = - { 6J, + 652 + 6J, + 2 J,’ + 2 Jd’ +12J,‘+
12J,‘}
- m,H.
First of all we emphasize that the magnetic energy of the zero field commensurate structure does not depend on the exchange interactions between adjacent layers (neglecting the very weak coupling J,” between ions on second neighbour layers at dh’ = 13.5 A) but is determined only by in-plane couplings! At the critical field H, defining the first order phase transition the energy of the corresponding phases must be equal, i.e. E,-E,=(u;--uj)moHc.
(18)
Furthermore, their thermal energy at the NCel temperature is equal to the ground state energy of the commensurate phase, i.e. kT,
= $S(S
+ l)(E,).
09)
From eq. (17) a set of five exchange integrals corresponding to the energy of the known magnetic structure can be derived. We use the upper and lower limits of the critical fields Hc, resulting from the hysteresis at the phase transition, the Ntel temperature TN = (9.3 f 0.2) K and the experimental value of the magnetic moment m, = (4.1 f O.l)pB, and obtain the exchange integrals in the following range: J,/k
= (1.65 f 0.03) K,
J,/k
= (-0.29
J,/k
= ( - 1.55 f 0.05) K,
Jd = (0.47 f J1’ = (- 1.37
f 0.01) K,
0.08) K, f 0.04) K.
A. Wiedenmann et al. / Magnetic phase diagram of Fel,
With these values, we calculate the paramagnetic phase and temperature according to ke, = fs(s
the free energy of the paramagnetic
+ l)Er,
leading to the range 20.6 KG 8,~ 23.7 K. This expected value agrees very well with the experimental value of 0, = 23.0 K. In summary, we conclude that in FeI, the coupling between first n.n. inside the (001) plane is ferromagnetic and is of comparable magnitude to that between third n.n. on the (001) plane and between second n.n. on adjacent layers.
References [l] M.K. Wilkinson, J.W. Cable, E.O. Wollen and W.C. Koehler, Phys. Rev. 113 (1959) 497. [2] IS. Jacobs and P.E. Lawrence, Phys. Rev. 164 (1967) 866. [3] A.R. Fert, P. Carrara, M.C. Lanusse, G. Mischler and J.P. Redoules, J. Phys. Chem. Solids 34 (1973) 223.
21
[4] A.R. Fert, Thesis (1973) Toulouse University. [5] J. Gelard, A.R. Fert, P. Meriel and Y. Allain, Solid State Commun. 14 (1974) 187. [6] A.R. Fert, J. Gelard and P. Carrara, Solid State Commun. 13 (1973) 1219. [7] D. Petitgrand, B. Hennion and C. Es&be, J. de Phys. Lett. 41 (1980) L-135. [8] J. Gelard, A.R. Fert, D. Bertrand and P. Carrara, J. de Phys. 38 (1977) 503. [9] T. Fujiata, A. Ito and K. Gno, J. Phys. Sot. Japan 21 (1966) 1734. [lo] R.E. Watson and A.J. Freeman, Acta Cryst. 14 (1961) 27. [ll] M. Kaburagi, T. Tonegawa and I. Kanamori, J. Magn. Magn. Mat. 31-34 (1983) 1037. [12] K. Wada, T. Tsukada and T. Ishikawa, J. Phys. Sot. Japan 51 (1982) 1331. [13] B. Mihura and D.P. Landau, Phys. Rev. L&t. 38 (1977) 977. [14] J. Villain and P. Bak, J. de Phys. 42 (1981) 657. [15] W. Selke and M.E. Fisher, J. Phys. B 40 (1980) 71. [16] J. Villan and M. Gordon, J. Phys. C 13 (1980) 122. [17] J. Rossat-Mignod, in: Neutron Scattering in Condensed Matter Research, eds. K. Skiild and D.L. Price (Academic Press, New York, 1986) chap. 20.
A NEUTRON OF FeI,
SCATTERING
INVESTIGATION
A. WIEDENMANN
*, L.P. REGNAULT,
Centre d’Etudes Nucliaires
de Grenoble, DRF-SPh /MagnPtisme
0. KOUNDE
OF THE MAGNETIC
P. BURLET,
PHASE DIAGRAM
J. ROSSAT-MIGNOD
et Diffraction Neutronique,
38041 Grenoble Cedex, France
and D. BILLEREY
Universik! de Nancy, 54037 Nancy Cedex, France Received 17 February 1988
Magnetization measurements%nd neutron diffraction experiments performed on a single crystal of FeI, have revealed a H-T phase diagram which cont_aing, in addition to the paramagnetic phase, five ordered phases separated from each other by first order transition lines. At H = 0, a commensurate structure is built up below TN = 9.3 K consisting of a stacking of ferromagnetic (101) layers in** + + - - sequence with the moments (m ,, = 4.1~~) aligned along the c-axis. At the critical field Hc,, a first order transition tu a ferrimagnetic phase Fl occurs. Fl is a multi k-structure defined by three fundamental components and their second and sixth harmonics. In the magnetic unit cell with the hexagonal axes ma, \/iTa, c, one block of four nearest neighbour (n.n.) moments in the basal plane are aligned antiparallel to the c-axis surrounded by parallel aligned moments giving rise to a net magnetization of o(F1) = m,/3. N.N. moments on adjacent (001) layers are aligned parallel to each other. A second multi-k-structure F2 occurs above Hc, with a net magnetization a(F2) = 0.44ms resulting from the alignment of isolated moments and triangles of n.n. moments along the c-axis surrounded by moments of opposite direction. The moment arrangement in a magnetic unit cell 5a x 5a x c is described by three (observed) fundamental components and the fifth harmonics together with weak (and not observed) contributions of the second harmonics and of three additional vectors with their second harmonics. Another ferrimagnetic phase F4 occurs above Hc, with o(F4) = 0.6mo with the same wave vectors as observed for F2. However, F4 corresponds to a single k-structure, i.e. a stacking of ferromagnetic (110) planes in a sequence + + + + - . In a wide range between Hc, and Hc, an “amorphous”-like phase F3 was found characterized by a well defined magnetization o(F3) = 0.5m, and by the absence of any long-range periodicity. This particular H-T phase diagram results from the competition between various exchange interactions; a set of five exchange integrals is derived from mean field theory using the critical transition fields and the Neel temperature.
1. Introduction FeI, belongs to the ferrous halides which, crystallize in a hexagonal layered structure of space group Pjml. The unit cell (a = 4.05 A, c = 6.75 A) contains one Fe*+ ion located on the trigonal axis at the origin (000) and two I--ions at the positions (l/3 2/3 z) and (2/3 l/3 Z) with z = l/4 (fig. 1). The magnetic ordering of the homologous compounds FeCl, and FeBr, consists of an alternating sequence of ferromagnetic (001) layers resulting from strong ferromagnetic in-plane interactions and weak antiferromagnetic inter-plane * Present address: Hahn-Meitner-Institut Str. 100, D-1000 Berlin 39, Germany.
Berlin,
Glienicker
couplings [l-3]. These compounds exhibit a classical metamagnetic behaviour, i.e. an external magnetic field induces a transition from the antiferromagnetic state directly to the saturated paramagnetic phase [4-61. The magnetic ordering in FeI, is more complicated. At the N6el temperature TN = 9.3 K [5] FeI, exhibits, in zero field a first order phase transition [7] and developes a commensurate magnetic structure of wave vector k = [l/4 0 l/4] which corresponds to a stacking of ferromagnetic (101) planes according to a + + - - sequence and an alignment of the moments along the c-axis. Below TN, magnetization measurements in an external field applied parallel to the c-axis give evidence for several discontinuities [6] which have
0304-8853/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A. Wiedenmann et al. / Magnetic phase diagram of Fel,
resulting from the crystal field splitting [ 81 (D = 26 K [9]) makes a SF phase very unlikely, and actually a quite typical metamagnetic Ising-type behaviour is expected. The aim of the present work is to undertake a more careful analysis of the magnetic (H, T) phase diagram by means of neutron scattering and magnetization measurements in order to characterize the nature of the various intermediate magnetic phases.
2. Experimental
Fig. 1. Hexagonal
0
Fe2'
0
J-
structure of Fe1 *. Arrows indicate the relevant exchange paths.
been assigned to successive transitions from the commensurate phase (C) to a “spin flop phase” (SF) and to the paramagnetic phase, respectively. However, the high anisotropy of the Fe*+ ion
0
0
0
0
0
Magnetic Fig. 2. Magnetization of FeI,
Single crystals of FeI, were grown by a Bridgman technique. Magnetic measurements were performed at the high field facility of the S.N.C.I. at Grenoble on a single crystal by applying the magnetic field along the crystallographic c-axis. Neutron diffraction experiments were carried out on the diffractometer DN3 installed at the SILOE reactor of the CEN-Grenoble. In neutron scattering experiments, a single crystal of about 4 x 4 x 2
field
5
10
15
(Tl
as a function of a magnetic field applied along the c-axis. Several fist order phase transitions are observed defining ferrimagnetic intermediate phases.
A. Wiedenmann
et al. / Magnetic phase diagram of Fe12
9
mm3 was used and oriented with the c-axis vertical and parallel to the applied magnetic field, which was provided by a superconducting cryomagnet reaching field values up to H T 10.5 T. A short neutron wavelength of A = 0.98 A was used in order to observe more Bragg reflections out of the equatorial plane.
3. Magnetization results A series of magnetization isotherms M(H), measured below the NCel temperature as a function of the external magnetic field, is shown in fig. 2. At the lowest temperature T = 1.65 K, five magnetization steps are observed with increasing field at HC, = 4.75 T, HC, = 6.55 T, H:, = 7.9 T, Hc, = 9.85 T and Hc, = 12.35 T, respectively. With decreasing field, lower critical field values are observed, indicating some hysteresis effects. Between these magnetization steps, characteristic of first order phase transitions, the magnetization M(H) increases as a linear function of the external field, defining a constant susceptibility x0 according to M(H)
= urn,,+ x,,H.
(1)
This is still true above the highest transition at Hc, where no saturation is observed in the paramagnetic phase, and in fact the same constant susceptibility x0 is found as below the first transition at Hc,. The ferrimagnetic component u (in units of mo) associated with each of the intermediate ferrimagnetic phases Fl, F2, F3 and F4 can be derived assuming that x0 is field independent. The following values a(F1) = 0.33, a(F2) = 0.45, a(F3) = 0.5 and a(F4) = 0.6 can be deduced together with a moment value m, = 4.1~~ for a Fe*+ ion. We emphasize that this analysis is different from that in ref. [8] which deduced a ferromagnetic component mo/4 for the first ferrimagnetic phase. With increasing temperature we observe that the transition at Hc, = 7 T is smeared out and that the ferrimagnetic phase F2 (u 5: 0.45) disappears at a temperature as low as T = 2.2 K.
0
2
L 6 Temperature ( K 1
8
Fig. 3. Magnetic phase diagram of FeI, determined from magnetization (circles) and neutron diffraction (squares) experiments. In addition to the low field commensurate phase (c), four fenimagnetic phases called Fl to F4 and the paramagnetic phase (P) can be defined. (Full circles: ref. [6].)
A similar behaviour is also observed for the phases Fl and F4. From these measurements, the magnetic phase diagram (H, T), shown in fig. 3, can be deduced; the previous results [6] have also been included. Table 1 Observed intensities of the Bragg reflection measured at T= 1.47 K and H = 0 compared to the calculated values (R = 6%) hkl
Intensity (barn/tit
cell)
observed
calculated
100
0.22
0.18
110 020 200 210 120
0.22 4.03 0.18 0.18 0.16 0.15
0.18 4.04 0.18 0.18 0.18 0.18
0.93 2.95
0.90 3.49
oio
iii 021
A. Wiedenmann et al. / Magneticphase
10
This phase diagram contains six different phases, including the paramagnetic (P), the commensurate (C) and four ferrimagnetic (Fl to F4) phases. Each phase is separated from the others by a first order transition line, except between the phase F3 and the paramagnetic state P (dashed line in fig. 3) for which no experimental evidence is available. This quite complex phase diagram must result from strong frustration effects arising from the competition between intra- and inter-plane couplings.
4. Neutron scattering results 4.1. Commensurate
magnetic structure (C)
Nuclear Bragg reflections were measured in zero external field at T = 1.4 K. The full width at half maximum (fwhm) of the rocking curves was rather good, of the order of 0.3”. However, each peak was accompanied by a shoulder indicating a second peak of similar width, resulting in a large mosaic spread for the whole crystal. In order to obtain accurate integrated intensity values, it was necessary to rotate the crystal by an angle of 3” and therefore a rather poor collimation was used just in front of the detector. Apart from the most intense nuclear reflections, e.g. (llO), (0%) where extinction effects can cestainly not be neglected, the agreement between observed and calculated intensities is reasonably good as can be seen in table 1. Thus the instrumental scaling factor can be obtained with an accuracy of about 6%. In zero external field, the onset of magnetic ordering was observed below TN = (9.0 f 0.5) K in agreement with previous measurements [3,7]. Magnetic reflections corresponding to scattering vectors /I= H + k have been measured in the Brillouin zones H = [ hkO] and H = [ hkl]. Due to the three-fold symmetry, three wave-vectors kj were found: k, = [l/4 0 l/4], k, = [0 l/4 l/4] and k, = [l/4 l/4 l/4]. They lie on symmetry lines of the Brillouin zone’ and correspond to one fourth of a lattice vector kj = H/4. Such a wave vector describes a commensurate structure corresponding
diagram of FeZI
to a sequence (+ + - - ) but not a simple antiferromagnetic ordering [16]. The intensity of a magnetic reflection is given by I(h)
= (0.27)2f2(h)[
IF(h)
12
-ww~2/l~12]>
(3)
where f(h) is the magnetic form factor of Fe2+ ions and P(h) the magnetic structure factor which, in the present case of a single Bravais lattice, reduces to the Fourier component mkj associated with the wave vector kj. When only one Fourier component mk is present and it moreover transforms according to a one dimensional representation, it can be written as mk = +A,ii, (exp i$,).
(4)
Then the ordering corresponds to a pure sine-wave modulation of the moment value along the k direction with an amplitude A, and a polarization along iJk. For a given Fe’+, defined by the lattice translation R,, the magnetic moment is given by: m(R,)=A,cos(2Pk*R,++,)iik.
(5)
We emphasize that for a wave-vector k = H/4 the particular choice of the phase + = a/4, gives a + + - - sequence for which all magnetic moments have the same magnitude: m,=A,/fi
=filmkl.
(6)
For such a collinear ordering the intensity of a reflection associated with a given wave-vector kj can be written as: I(h)
= (0.27)2f2(
h) sin’clll mk, I 2,
(7)
where (Yis the angle between the scattering vector and the moment direction ii,. The intensity analysis of the reflection measured inside the Brillouin zone H = (000), (010) (110) (110) (010) clearly shows that moments are aligned along the c-axis, and then for each wave vector kj a mean value of I rni I can be deduced as reported in table 2. Exp&iments with an applied field have shown that the magnetic ordering is characterized by only one wave-vector k (single-k structure). Therefore, three domains exist, and the value of
A. Wiedenmann et al. / Magnetic phase diagram
11
of Fe12
Table 2 Commensurate magnetic structure of FeIz at T=1.4 K and H = 0. Fourier components mk corresponding to kl = [l/4 0 l/4], k, = [0 m l/4] and k, = [m l/4 0] derived from different magnetic reflections. The volume fraction u, is given for the three domains
(In the following, the magnitude of the Fourier component will be designated by mk.) Each domain will have a relative volume given by
Reflection
At T= 1.47 K a value Imk I = (2.86 f 0.11)~~ is obtained, yielding a magnetic moment value m,, = J? Imk I = (4.05 f 0.15)~~. This value is quite close to that expected for Fe2+ and to that observed by magnetization measurements. Domains are found to be almost equally distributed at low fields (see table 2). However, above H = 3 T intensities associated with each wave vector start to change in favour of the k,-domain. This behaviour certainly results from a small misalignment of the applied field direction with respect to the c-axis.
mk,
(010) + k, (llO)+ k,
1.41 1.83 1.94
(OiO)+ k,
1.83
W’O)+ k, (100) + k, (OlO)+ k,
1.55 1.64 1.77
PW+ k, (100) + k,
1.48 1.67
(iio) + k3
1.47
W')+k,
01 CQ
(PB)
1.76rtO.9
37*9
1.64kO.6’
32&5
1.54 f 0.6
2954
the Fourier component mk is obtained by summing the contributions of the three domains
(8) 1
30
.T.
I
1
10
60
Uj=WQlffl,12.
(9)
4.2. Field induced ferrimagnetic
phases
At T = 1.47 K the intensity of the nuclear reflection (100) measured as a function of the 1
I
I
1.46 K
1
I
Magnetic
field
70
8.0
1 Tesla
)
Fig. 4. Intensities of magnetic reflections of Fel, at T=1.4 K as a function of the magnetic field corresponding to the scattering vectors: [lOO]: (o), [l/4 0 l/4]: +, [l/6 l/6 01: A, [l/5 l/5 01: 0.
external field (see fig. 4) exhibits two sharp steps at the critical fields H=, and Hc,, indicating first order phase transitions from the commensurate (C) to a ferrimagnetic phase Fl and from Fl to a fe~a~etic phase F2, respectively. The magnetic contribution to the (100) reflection is in excellent agreement with magnetization measurements and gives a reduced ferromagnetic component o(F1) = 0.32 and a(F2) = 0.41, respectively. For an applied field H ==9 and 10.5 T these contributions correspond to a reduced ferromagnetic component o(F3) = 0.5 and a(F4) = 0.6, also in good agreement with magnetization data. Accurate scans performed along symmetry directions in reciprocal space indicate that the ordering in each ferrimagnetic phase is described by a ~~ensurate wave vector with k(F1) = [l/6 l/6 0], k(F2) = [l/5 l/5 0] and k(F4) = [l/5 l/5 01. A plot of the peak intensity maximum of the most intense reflections associated with each of these vectors is given in fig. 4 as a function of the applied field. At the critical fields ch~acte~stic intensity jumps are found to be in good agreement with the magnetization measurements. All these ferrimagnetic phases differ only by the in-plane ordering, the stacking along the c-axis being always of ferromagnetic type. 4.3, Magnetic structure of the ferrimagnetic pkine Fl
Apart from a change ,in the domain distribution, no variation of the wave-vector, the direction and the magnitude of mk is observed up to H = 4.6 T. However, a small additional increase of the field up to UC, = 4.7 T induces a sudden disappearence of the magnetic reflections associated with the wave-vector k1 = [l/4 0 l/4]. Siiultaneously, an increase of the intensity of the nuclear reflections (lOO), (OIO), (110) etc. have been found, which corresponds to the scattering vectors marked by closed circles in fig. 5. The most intense peaks are associated with the equivalent wave-vectors kl = [l/6 l/6 O], k, = [l/3 1/6 0] and k, = [l/6 1/3 01. In addition to these fundamental reflections, second harmonics corresponding to 2k, = [l/3 l/3 01, 2k, = [2/3 l/s 0] and 2k, a=
Fig. 5. Reciprocal space of Fe12 showing the symmetry lines (thick lines) and the points (circles) investigated at T = 1.47 K and H = 5.5 T (phase Fl). The observed intensities (full circles) correspond to k, = [l/6 l/6 01, kz = [l/3 m 0] and k3 = [l/6 v 0] and the second harmonic 2 k.
[l/3 v 0] are also observed with weaker intensity. A very careful investigation of the reciprocal space, especially for scattering vectors shown by open symbols in fig. 5, allows us to exclude the presence of any other wave-vectors, in particular no third order harmonic can be observed. Intensities of magnetic reflections, corresponding to a given wave-vector, measured in various Brillouin zones (corrected for the magnetic form factor) have quite similar values, which allows us to conclude that magnetic moments remain aligned along the c-axis. Therefore, the spin flop phase at H > 4.6 T, as proposed in ref. [4], must definitely be ruled out. In fact, the ordering must be of ferrimagnetic type since both the ferromagnetic and the co~ens~ate components are observed parallel to the c-axis. For the Fl phase, the value of the ferromagnetic component at T = 1.47 K, mkzo ?: 1.37~,, is very close to one third of the moment value m,, for Fe’+ ions. According to eq. (7), the magnetic Bragg peak intensities measured at T= 1.47 K and N = 5.5 T allow us to determine the amplitude of each of the three Fourier components. The results obtained are reported in table 3. It must be emphasized that the amplitudes
A. Wiedenmann et al. / Magnetic phase diagram of Fel, Table 3 Fenimagnetic phase Fl of FeI, at T=1.4 K and H = 5.5 T. Observed and calculated Fourier components mk associated with the wave-vector k, = [l/6 l/6 01, k2 = [l/3 1/6 0] and k, = [l/6 m 01. The calculation corresponds to a triple-k structure of the ferrimagnetic phase Fl as explained in the text h
H f k,
mk
(d
observed
(l/6 l/6 0)
VW+ k,
2.04
(5/6 1/6 0) (5/6 5/6 0)
(100) - k, (110) - k,
1.97 2.03
(l/3 l/3 (2/3mO)
(000)+2k
0.66
0)
(000)+2k
0.65
(1/3mO)
(000)+2k
0.64
(T/3 r7a 0) (213 l/6 0)
(000) + k, (100) - kz
1.56 1.37
(2/3 5/6 0)
(110) - k,
1.37
(T/6 V (5/6 l/3
0) 0)
(000) + k, (100) - k,
1.4 1.38
(5/6 273 0)
(li0) - k,
1.37
(100)
6k
calculated
2.0 f 0.10
2.05
= Crnk, exp 2dk,*R,.
domain can be determined from eqs. (8) and (9). ii) The structure is described by each of the three wave vectors kl, k, and k, and the corresponding harmonics. In that case, neutron data indicate that the associated triple-k magnetic structure does not recover the trigonal symmetry because the three Fourier components are not equal. Therefore, while the structure is of triple-k type, three domains must exist and each of these gives a contribution to a given reflection such as: (11)
0.64+ 0.03
0.66
1.37+0.05
1.21
1.37 f 0.05
1.21
1.36 f 0.05
1.37
of the Fourier components are not equal, mk, being larger than mk, and mk,. Since a neutron scattering experiment gives only the modulus of each Fourier component and not its phase, it is not a trivial problem to get the true magnetic ordering. The moment distribution is given by summing over all Fourier components such as: m(R,)
13
00)
Therefore, many possible in-plane structures can be constructed from the neutron data. In the present case, two possibilities must be considered: i) The structure is described by a single wave vector (e.g. k,). In that case, the sum contains the fundamental Fourier component mk, the harmonics and the ferromagnetic component rnhk = mk = 0. The other wave vectors k, and k, are then associated with the other magnetic domains. The existence of three K-domains arises from the loss of the three-fold symmetry axis with respect to the paramagnetic group. The relative volume of each
That means that at least six different magnetic reflections associated with each wave vector kj must be measured in order to evaluate mk and uj independently. The problem can be sim&fied if we assume that the crystal is in a single domain state favoured by the strong applied field. Then u2 = ug = 0 and the Fourier components mk,, mk, directly from the correand mk, are obtained sponding reflections. It must be noted that the intensity of the second harmonic 2k is the same in both cases i) and ii) since for the three wave vectors the second harmonic 2k = [l/3 l/3 0] overlaps. Let us examine the first possibility, i.e. a single k-structure with three domains. Then the ordering corresponds to a stacking of ferromagnetic (110) planes but only the sequences + + + + - - and + + + - + - are compatible with the observed wave-vector k = [l/6 l/6 0] and with a ferromagnetic component mkcO = m,/3. For these two square-wave modulations the Fourier components of higher harmonics can be calculated according to the relation: 6
mk=
T
c a,, exp(2Tikn), II=1
(12)
where k = l/6 and a, is the reduced magnetization per ion of a plane n, which can take the values u = f 1 according to whether the orientation of the moments is “up” or “down”. The + + + - + - sequence gives a third order harmonic with m3k = f m. in complete disagreement with experiment, indicating the absence of such a Fourier component. Thus this sequence can be ruled out immediately.
A. Wiedenmannet al. / Magneticphase diagram of Fe12
14
In the case of the + + + + - - sequence the component mJk is effectively zero. However, the values of the components mk= m,/fi= 2.37~~ using the and m 2k = m,/3 = 1.37j.~a, calculated value m, = 4.&, do not agree with the observed values of mk = (ESrn: )112 = 2.83~~ and 1mZk 1 = 0.65~~. In particular,‘the ratio m2Jmk is calculated by more than a factor of two larger than the observed one, i.e. well outside the experimental accuracy. We have further checked if this discrepancy can be explained by an incomplete squaring up of the sinewave modulation by giving the possibility for individual moments to vary slightly from the saturated value as m, = m,(l f E,), with en x 1. For this purpose we calculated m, according to the Fourier development: m, = 2m, +m,,,.
cos( +nn + &) + 2m,,
COS(
$7,
+
G2)
(13)
The phases +i and I$~were varied in order to get a modulation of moments as small as possible. The optimal value would correspond to a single kstructure with three different moment values of 2.8~s, 4.2~~ and 5.6~~ which is actually too large a deviation from the saturated value to be an acceptable solution. Therefore, we conclude that the magnetic ordering must be described in a unit cell 6a X 6a x c. The net magnetization u, per Fe2+ ion of a plane n perpendicular to a given wave-vector kj is given by
(14)
where the summation is carried out over six consecutive moments of the basal plane and wntained in the planes n which have a reduced moment a,,, = + 1. In contrast to the single kstructure, the magnitude of u,,(kj) may change from plane to plane, for instance if we consider the wave-vector k = [l/6 l/6 0] a, can take the values f 1, +2/3, + l/3 and 0. Then the Fourier components mk, and m2, associated to a given stacking of six planes with a
sequence eq. (12):
{ a,,(/~~)}, can be calculated
m0
mk, = 6
according
to
{ uI - a, + f(u2 - u3 - a5 + u6)
++ifi(u2+u3-us--u~)}, m2k,=
(15)
““{ 6 +
(II + a, -
iifi(
Taking into magnetization i
Un(kj)
;(
a2 + 63 + a, + UC)
a,-u,+u,-a,)}. account the condition of this phase is m,/3
that i.e.
the net
= l/3
(15a)
k=l
and that the third order harmonic i.e.
is not observed,
u,+u,+u,-(u2+u,+u,)=0,
(15b)
only one sequence together with the cyclic permutations is compatible with the experimental values for each wave vector kj. The result corresponds to the sequences {a,(k,)}={-l,O,l,l,l,O} {%(Q}
= {e,(Q)
and =
(0, 1, l,O,O,
O}.
However, it turned out that the remaining problem, consisting of arranging the 36 moments in the unit cell 6a X 6a with the proper sequences along k,, k, and k,, cannot easily be solved by hand. In the unit cell 6a X 6a the moments of the Fe2+ ion n = (n,, n 2) have been calculated according to the relation: m(ni,
nZ)=mk,e+2mk,
1
cos +(ni+n,)+& ]
+2mk2
cos[ :(2n,
+2mk,
cos j(n, ]
+2m,,
cos I$?( [
- rz2) + A] - 2n2) +&
n1+
n2)
+
1
1 (16)
$4
and using the observed values of the mk,. The phases +i to e4 have been varied by steps of 1”
A. Wiedenmann et al. / Magnetic phase diagram of FeIr
15
moments. This arrangement can in fact be described in a smaller hexagonal unit cell with a parameter a’ = 2&, as shown in fig. 6. This new magnetic unit cell contains three blocks of four moments, two of them being aligned along the applied field and the third one having a reversed orientation. 4.4. Magnetic F2
Fig. 6. Moment arrangement in the ferromagnetic phase Fl at H = 5.5 T. The unit cell (6a, 6a, c) (full lines) and the elementary unit cell (2fia, 2fia, c) (dashed lines) are shown. The ordering corresponds to the formations of blocks of four “up” moments (full circles) among “down” moments (open circles).
from --71< I#+< 71, until all moments m (n,,n,) have the same value within an accuracy of 10%. Within this restriction only one set of phases @i = CJd= 71, !PZ= @s = 7r/2 gives a good agreement. The corresponding moment arrangement is shown in fig. 6. First of all, we can verify easily that the sequence in the directions ki, k, and k, are really cyclic permutations of the sequences determined above. Secondly, it can be checked that the calculated Fourier components of the proposed commensurate triple-/c structure are quite close to the observed ones as can be seen in table 3. Moreover, the net magnetization of each plane a,( k,) perpendicular to any direction different from k,, k, and k, is equal to m,/3, indicating that no further Fourier components are present, in agreement with the experimental results. It must be emphasized that the proposed in-plane structure has no trigonal symmetry in which case three magnetic domains must exist. Nevertheless, the presence of a single domain state in the crystal at T = 1.47 K and H = 5.5 T is more likely since the amplitudes of the Fourier components mk calculated with this assumption, (i.e. uz = u3 = 6 in eq. (ll)), agree quite well with the observed ones as shown in table 3. So, in the ferrimagnetic phase Fl the in-plane magnetic ordering, reported in fig. 6, consists of blocks of four down moments surrounded by up
structure of the ferrimagnetic phases .a
At the critical field HC, = 6.55 T all the magnetic reflections associated with the wave-vectors k, = [l/6 l/6 01, k, = [l/3 l/6 0] and k, = [l/6 l/3 0] disappear simultaneously together with the second harmonics, whereas the intensities of nuclear reflections increase substantially. At T = 1.47 K the ferromagnetic component was found to have a value mk =0 = 1.7~~ corresponding to about 41% of m,. This value is somewhat smaller than the value a(F2) = 45% observed by magnetization measurements. By scanning the reciprocal lattice along the various symmetry lines of the Brillouin zone, new magnetic reflections were found which are associated with the wavevectors k, = [l/5 l/5 01, k, = [2/5 l/5 0] and k, = [l/5 2/5 01, defining a new commensurate phase, with moments still aligned along the c-axis. As for the Fl-phase we first examine the possibility of a single k-structure. Two sequences are compatible with the observed wave-vector k = [l/51/50]: ++++and +++--. However, these sequences give rise to a net magnetization of ur = 3/5 and l/5, respectively, in disagreement with the observed value uy = 0.4. Furthermore, these sequences have large second order harmonic mZk = 0.40 and 0.25m,, respectively, while no contribution has been found experimentally (see table 4). Thus such square-wave-modulated single-k structures must also be ruled out. Moreover, the assumption of a mixed phase composition by both sequences with a resulting average magnetization of (3/5 + l/5)/2 = 0.4, close to the observed value, must also be rejected since it cannot account for the absence of the second order harmonic. So we are forced again to consider the in-plane arrangements ,of the 25 moments within the mag-
A. Wiedenmann et al. /
16
Magneticphase diagram of Fel,
Table 4 Ferrimagnetic phase F2 of FeI, at H = 7.1 T and T = 1.4 K. Observed Fourier components mk compared with the calculated ones based on the model of hp. 7 and ma = 4.1~~ H f k,
(h)
mk,
h)
observed
calculated
(l/5 l/5 0) (4/5 4/5 0)
(OOO)+ k, (110) - kl
0.99 0.94
0.97 + 0.05
1.39
(215 l/5 0) (3/5 l/5 0) (3/5 6/5 0)
(000) + k, (lOO)- k, (110) - k,
0.76 0.68 1.07
0.84!cO.O5
1.39
(I/5 215 0) (415 215 0) (6/5 3/5 0)
(000) + k3 (NO)- k, (110) + k,
0.88 0.83 0.90
0.90+0.05
1.39
1.70 f 0.05
1.80
k=O
(100) m 2k,=m2k2=m2k3 mkt m 2kf
=
mk,, =
m2k,,
=
mkw =
m2k
111
0 0 0
0.08 0.53 0.20
Fig. 7. Moment arrangement in the ferromagnetic phase F2. The magnetic unit cell (5a x5a x c) formed by n.n. triangles of “down” moments (full circles) around one isolated “down” moment. The full and dashed lines must contain one or two down moments, respectively, in order to obtain the lowest possible values of mkt, mkt. and mk 11,.
lowest values of mkf and rnZkTcorresponds to the sequence netic unit cell 5a x 5a which account for the Fourier components mk=,, and mk, associated with the three wave vectors k,, k, and k,. This unit cell must contain 7 down and 18 up moments in order to get the required ferromagnetic component (I~= 11/25 = 0.44. The net magnetization of a plane (n = 1 to 5) perpendicular to a vector kj is again given by eq. (14) where the sum must now be extended over five consecutive moments. The possible values are u,(kj) = f 1, f 3/5 and f l/5. For all planes perpendicular to any wave-vector of the type [h/5 k/5 0] the net magnetization is always different from the ferromagnetic contribution ur = 0.44. This means that the intensity of magnetic superlattice peaks associated with wavevectors kj of this type can never be strictly zero. Consequently, in addition to the observed vectors k,, k, and k,, the vectors k’ = [l/5 0 01, k” = [0 l/5 0] and k “’ = [l/5 1/5 0] and their harmonics must also be present. Since we did not detect reflections associated with these wave-vectors and their second harmonics, the corresponding Fourier components must be very weak. We calculated mkt and rnzkt for any sequence of five planes n = (1 to 5) corresponding to the above allowed sequences { a,,( kj)}. It turned out that the
{ e(k’)}
= {3/5,3/5,3/5,
I/5, I/5}
(or their cyclic permutation) with rnp = 0_13m, and mZkJ = O.O5m,. The following graphical method was used to determine the in-plane moment distribution. In the direct lattice of basis vectors a, and a2 given in fig. 7, planes perpendicular to the wavevectors k’, k” and k “’ are represented by full or dashed lines according to whether they have a net magnetization per atom of 3/5 (one down and four up moments) or l/5 (two down and three up moments), respectively (fig. 7). Then the seven down moments are disposed in the unit cell 5a X 5a in such a manner that all full and dashed lines contain the right number of down moments. There are several solutions which satisfy the condition that rnkf ,and rnZkT must have low values when different cyclic permutations of the sequence are considered. However, only the in-plane moment arrangement reported in fig. 7 agrees with the additional conditions that the Fourier components Of the second harmonics mZk,, m2kz and m2k, must also be very weak. In the proposed in-plane magnetic structure, the down moments are arranged in six equilateral triangles around an isolated central down mo-
A. Wiedenmann et al. / Magnetic phase diagram of Fez2
ment. In this case the hexagonal symmetry is recovered and there is no K-domain. In particular, the Fourier components associated with k,, k, and k, must be equal in agreement with experimental results. From this point of view, the proposed model is the only one which is compatible with the essential experimental results. However the observed values of all Fourier components are systematically smaller than the calculated ones (table 4). This may be explained partly by experimental effects such as an increase of the extinction in the ordered phase. Another possibility is the coexistence of long-range ordering F2 with an incomplete ordering as discussed in the following section. By comparing the observed values of mk,, mk, and mk, with the calculated ones (table 4) we get a volume fraction u = 45%, i.e. only about one half of the moments are involved in the long-range ordering of the F2 type.
17
l
l
110
ii0
Fig. 8. Scans performed in reciprocal space at T=1.47 K and H = 8.3 T and H = 8.9 T which revealed the absence of long range ordering in the “amorphous” phase F3.
4.5. “Amorphous phase” F3 At T = 1.47 K all reflections associated with the wave-vector k[1/5 l/5 0] disappear at the critical field Hc, = 8.05 T and the intensities of nuclear reflections are increased by 12%. At H = 8.9 T a number of scans along the symmetry directions were performed (fig. 8) but, surprisingly, no magnetic reflections have been detected. Only the scan along the direction [llO] revealed a very large peak centered at [l/5 l/5 01. This means that, for these values of magnetic field and temperature no complete long-range ordering is established, in spite of the well defined net magnetization of the ferromagnetic phase F3 with q(F3) = 0.5. This value is much smaller than the value a,, = 0.8 expected for uncorrelated magnetic moments in an external field of H = 8.1 T, indicating the existence of some ordering. Moreover, the large peak k = [l/5 l/5 0] suggests that some kind of periodicity must be present within the plane. However, the correlation length remains rather small indicating a great variety of possible moment arrangements. For a given field value H = 7.05 T, the reflections associated with the wave-vectors k,, k, and k, of the ferrimagnetic phase F2 disappear sud-
denly when the temperature is increased, indicating that a first order phase transition occurs at T = 2.3 K (fig. 9). When the sample is cooled down again to T = 1.47 K, the intensities I(k,), 1(/c,), I(k,) have different ratios, whereas the ferromagnetic contribution keeps the same value. Similar behaviour has been observed for a lower field H = 5.5 T (see fig. 9a). At low temperature the triple-k-structure Fl defined by k, = [l/6 l/6 01, k, = [l/3 l/6 0] and k, = [l/6 l/3 0] is stable only up to T = 2.1 K where a first drastic change of the intensity ratio occurs. Then at T = 3.5 K all reflections disappear in a first order phase transition. Again, above this transition no magnetic peak associated with long-range ordering has been detected in spite of the ferromagnetic component (q = l/3) remaining unchanged, In cooling down to low temperatures, the initial intensities were never regained, implying that the “amorphous phase” F3 still coexists with the ordered ferrimagnetic phase Fl. This type of incomplete long-range ordering associated with the ferrimagnetic phase F3 is similar to that observed in amorphous systems. It may result from the competition between different in-
A. Wiedenmann et al. / Magnetic phase diagram of FeI,
18
15
H : 55T
\ CI)
x e 2 2 10 ‘0 u e B s 5
Fl-F3
A
0
1
I
Q&9+
I
--
#-I l,O,Ol
I-
q
bl x ‘5 t s S 2 10 b
F2-F3
H:%OT
qi;,:[l/S .
ciao)
11501 i
E E
0 Temperature! K Fig. 9. Thermal variation of the integrated intensities of the (100) reflection (full circles) and the superlattice reflections h = (OOO)+k, showing the transitionS in constant external fields: (a) H = 5.5 T (Fl - F3) and (b} H = 7.05 T (F2 cf F3).
plane interactions, leading to a high degree of frustration. In fact, such “amorphous” phases have been predicted for the triangular ISING model with competing interactions using a cluster-variation method [ll] and from Monte Carlo simulations [12,13]. Another possibility would be the “floating phase” predicted for frustrated anisotropic systems [14-161, i.e. a continuous change of the wave-vector as a function of temperature and applied field. However, this phase must disappear when, we& inter-plane interactions are present as in FeI,. 4.6. Ferrimagnetic phase (F4) When the external field is raised up to the experimental limit of 10.3 T, the intensity of the
(100) reflection starts to increase again (see fig. 4) and well defined magnetic superlattice peaks associated with the wave-vectors k, = [l/5 l/5 01. kA-’ = [2/5 1/5 0] and k, = [l/5 2/5 0] are agaii’obt served. The intensity does not seem, however, to be completely saturated. In a second experiment we placed the sample slightly off the central position of the superconducting coil in order to have a larger magnetic field. This increase of about 10% was sufficient to produce a single phase with a ferrimagnetic ordering called phase F4. However, it was not possible to get reliable quantitative values for the integrated intensities in this way. Qualitatively, we found that the magnetic contribution to the nuclear reflections gives roughly a net magnetization a,(F4) = 0.6, in agreement with magnetization measurements. The Fourier components associated with the wave-vector k, seem to be the dominating ones, the following values having been estimated: mk, = 0.07m0, mk, = O.lm, and mk, = 0.3m,. No other contributions associated with higher harmonics or other vectors were found. These observations allow us to make several conclusions. First of all, in a single k-structure a ferromagnetic stacking of planes with a + + + + - sequence can account for the observed ferromagnetic component af(F4) = 0.6 and for the observed wave-vector k = [l/5 l/5 01. Moreover, it can explain the estimated value ( 1FE, 1 = 0.32m,) for the fundamental Fourier component which should be equal to about 0.4m,. The only problem arises from the lack of observation of the Fourier component associated with the wave-vector 2k, which must have the same order of magnitude as the fundamental component. On the other hand, if we consider a magnetic unit cell 5a x 5a we must find a moment arrangement such as the Fourier components associated with the vectors k’ = [l/5 0 01, k” [0 l/5 0] and with k “’ = [l/5 l/5 0] and their harmonics are equal to zero. This could be achieved assuming that the planes perpendicular to these vectors have a net magnetization a,( k,) equal to the ferromagnetic component ur(F4) = 0.6, i.e. five up and one down moments. When we apply the graphical method described above, we end up with the only possible moment arrangement, shown in fig. 10, which is in
A. Wiedenmann et al. / Magnetic phase diagram of Fel,
Fig. 10. Single k-structure of the ferrimagnetic phase F4 defined by the wave-vector k, = [l/5 v 01.
fact the single k-structure already mentioned. In that case three domains can co-exist. If we accept the absence of the Fourier component mzk then we have to assume, as found in the previous case for the F2 phase, a weak but non-zero value for the Fourier components with k’, k”, k ‘I’. The lowest values were obtained for the sequences { u,(k’)} = {l/5, 3/5, l/5, 3/5, 3/5} with mkr = O.l5m, and rnzkf = 0.09. Along the vectors k,, k, and k, the sequences would be, e.g.: (1, 3/5, l/5, l/5, l} yielding the Fourier components mk = 0.25m, and mzk= O.O6m, in agreement with experimental results. A great variety of combinations was considered but in order to decide which arrangement actually occurs, one needs accurate values for mk,, mk and mk,. Since these are not available, it is impossible to proceed further with the determination of the moment arrangement in phase F4.
5. Discussion Magnetization measurements and neutron diffraction experiments performed on a single crystal revealed that the layered compound FeI, exhibits a quite original H-T phase diagram as shown in fig. 3. This phase diagram contains, in addition to the paramagnetic phase P, five ordered phases which are separated from each other by first order transition lines. In zero applied field a commensurate magnetic ordering defined by the wave-vector k = [l/4 0 l/4] is built up via a first
19
order transition at T, = 9.3 K. It consists of a stacking of ferromagnetic (101) layers in a -t + - - sequence with moments (m. = 4.1~~) aligned along the c-axis. This ordering remains stable up to a critical field HC, where a first order transition to a ferrimagnetic phase Fl takes place. This transition occurs only at temperatures smaller than the tricritical point TC, = 3.8 K. The inplane ordering of the Fl phase is not a simple single k-structure but a multi-k structure involving the three fundamental components with k, = [l/6 l/6 01, k, = [2/6 1/6 0] and k, = [l/6 l/3 0] and the second and sixth harmonics. Such an ordering (see fig. 6) consists a blocks of four magnetic neighbour moments aligned along the c-axis and antiparallel to the applied field surrounded by parallel aligned moments, yielding a net magnetization a(F1) = m,/3. Nearest neighbouring moments in adjacent (001) layers are aligned parallel to each other. A transition to a second ferrimagnetic phase F2 occurs at HC, below a tricritical temperature l& = 2.9 K. This phase has a net magnetization a(F2) = 0.44m, resulting from an in-plane moment arrangement reported in fig. 7, characterized by triangles and isolated moments aligned antiparallel to the applied field surrounded by moments having the opposite direction. This magnetic structure of a unit cell 5al5a has a trigonal symmetry and hence has no k-domains. Besides the observed commensurate wave-vectors k, = [l/5 l/5 01, k2 = [2/5 l/5 01, k, = [l/5 2/5 0] and k, = 0, for this ordering the amplitude of the Fourier components associated with the second harmonics 2k as well as that associated with the wave-vectors k’ = [l/5 0 01, k” = [0 l/5 0] and k “’ = [l/5 l/5 0] and their second harmonics are non-zero, but they are too weak to be observed experimentally. Another ferrimagnetic phase F4 exists above a critical field HC, with a net magnetization a(F4) = 0.6m, with the same wave-vectors as for phase F2. In contrast with the phase F2 the ordering of the phase F4 corresponds to a single k structure i.e. stacking of ferromagnetic (110) planes with a se: quence + + + + - (fig. 10). However, a difficulty arises because a squaring up of the sine-wave modulation yields a second order harmonic 2k which has not been observed experimentally. A
20
A. Wiedenmann et al. / Magnetic phase diagram
very interesting result is the presence of a wide range of existence of the phase F3 in the H-T diagram. In this phase magnetic moments must be ordered since the net magnetization is much lower than that expected for the paramagnetic state. However, the ordering is not long-range since no magnetic reflections have been observed, and in fact only a very broad peak along the [llO] line has been detected. An interesting result is that the magnetization remains constant at the boundary between the phases Fl and F3, or the phases F2 and F3, respectively, while the wave-vectors disappear discontinuously. On the other hand, at a given temperature the magnetization m (F3) increases continuously with increasing the magnetic field. This type of incomplete long-range ordering is similar to that found in amorphous systems. A local ordering is present but the correlation length must be small due to the great variety of possible moment arrangements. The original phase diagram obviously results from important frustration effects present in the triangular lattice due to the competition between various in-plane and inter-plane interactions. The most important in-plane couplings are the superexchange interactions via the iodide ions between first nearest neighbour (nn) Fe2+ ions at d, = 4.05 A second nn at 7.01 A, and third nn at 8.1 A. The couplings between adjacent layers at di = 6.75 A, d,’ = 7.87 A and d; = 9.73 A are expected to be much weaker. The corresponding exchange integrals are denoted as J1, J2 and J3 for the in-plane interactions and J,‘, J; and J; for the inter-plane couplings (see fig. 1). The energy of the various ferrimagnetic phases in an applied field can be written as
E,(H)
= E,, - a,m,H,
where the exchange according to
energy
E,,
is
calculated
E,, = - cJljS;S’sj’ ij by using the relevant
exchange
integrals
as defined
of Fel,
above. For S: = f 1 we obtain tions:
the following
rela-
EC= -2(J,-J,-J,-J;‘), E (Fl)
=
-{$J,-+J,-25,+2J;+2J;’ + fJ{ - $3;)
E (F2)
=
-
{
- +m,H,
$J, + &J, - %J, + 2J; + 2J;
+ pJ{
+ g-J;> - 0.44m,H,
EcF4)= -{$J,
(17)
+ ~J2+$J~+2J;+2J;’
+ YJ,’ + ?J;}
- 0.6m,H,
E (pm) = - { 6J, + 652 + 6J, + 2 J,’ + 2 Jd’ +12J,‘+
12J,‘}
- m,H.
First of all we emphasize that the magnetic energy of the zero field commensurate structure does not depend on the exchange interactions between adjacent layers (neglecting the very weak coupling J,” between ions on second neighbour layers at dh’ = 13.5 A) but is determined only by in-plane couplings! At the critical field H, defining the first order phase transition the energy of the corresponding phases must be equal, i.e. E,-E,=(u;--uj)moHc.
(18)
Furthermore, their thermal energy at the NCel temperature is equal to the ground state energy of the commensurate phase, i.e. kT,
= $S(S
+ l)(E,).
09)
From eq. (17) a set of five exchange integrals corresponding to the energy of the known magnetic structure can be derived. We use the upper and lower limits of the critical fields Hc, resulting from the hysteresis at the phase transition, the Ntel temperature TN = (9.3 f 0.2) K and the experimental value of the magnetic moment m, = (4.1 f O.l)pB, and obtain the exchange integrals in the following range: J,/k
= (1.65 f 0.03) K,
J,/k
= (-0.29
J,/k
= ( - 1.55 f 0.05) K,
Jd = (0.47 f J1’ = (- 1.37
f 0.01) K,
0.08) K, f 0.04) K.
A. Wiedenmann et al. / Magnetic phase diagram of Fel,
With these values, we calculate the paramagnetic phase and temperature according to ke, = fs(s
the free energy of the paramagnetic
+ l)Er,
leading to the range 20.6 KG 8,~ 23.7 K. This expected value agrees very well with the experimental value of 0, = 23.0 K. In summary, we conclude that in FeI, the coupling between first n.n. inside the (001) plane is ferromagnetic and is of comparable magnitude to that between third n.n. on the (001) plane and between second n.n. on adjacent layers.
References [l] M.K. Wilkinson, J.W. Cable, E.O. Wollen and W.C. Koehler, Phys. Rev. 113 (1959) 497. [2] IS. Jacobs and P.E. Lawrence, Phys. Rev. 164 (1967) 866. [3] A.R. Fert, P. Carrara, M.C. Lanusse, G. Mischler and J.P. Redoules, J. Phys. Chem. Solids 34 (1973) 223.
21
[4] A.R. Fert, Thesis (1973) Toulouse University. [5] J. Gelard, A.R. Fert, P. Meriel and Y. Allain, Solid State Commun. 14 (1974) 187. [6] A.R. Fert, J. Gelard and P. Carrara, Solid State Commun. 13 (1973) 1219. [7] D. Petitgrand, B. Hennion and C. Es&be, J. de Phys. Lett. 41 (1980) L-135. [8] J. Gelard, A.R. Fert, D. Bertrand and P. Carrara, J. de Phys. 38 (1977) 503. [9] T. Fujiata, A. Ito and K. Gno, J. Phys. Sot. Japan 21 (1966) 1734. [lo] R.E. Watson and A.J. Freeman, Acta Cryst. 14 (1961) 27. [ll] M. Kaburagi, T. Tonegawa and I. Kanamori, J. Magn. Magn. Mat. 31-34 (1983) 1037. [12] K. Wada, T. Tsukada and T. Ishikawa, J. Phys. Sot. Japan 51 (1982) 1331. [13] B. Mihura and D.P. Landau, Phys. Rev. L&t. 38 (1977) 977. [14] J. Villain and P. Bak, J. de Phys. 42 (1981) 657. [15] W. Selke and M.E. Fisher, J. Phys. B 40 (1980) 71. [16] J. Villan and M. Gordon, J. Phys. C 13 (1980) 122. [17] J. Rossat-Mignod, in: Neutron Scattering in Condensed Matter Research, eds. K. Skiild and D.L. Price (Academic Press, New York, 1986) chap. 20.