NMR studies of EuB6 at low temperatures

Article abstract of DOI:10.1103/PhysRevB.60.3361

We report results of ¹⁵³Eu and ¹¹B nuclear magnetic resonance (NMR) measurements on EuB₆ at temperatures between 0.1 and 150 K and in external magnetic fields between H=0 and 7 T. The evolution of the ¹⁵³Eu NMR spectra at low temperatures and low fields provides evidence for an unexpected and complex electronic ground state of EuB₆, involving the coexistence of two magnetically very similar phases. The temperature and magnetic-field dependences of the ¹¹B spin-lattice relaxation rate T₁^-1(T,H) are very well accounted for by magnon-driven relaxation. A spin-wave theoretical interpretation of the ¹⁵³Eu-NMR spectra and the ¹¹B T₁^-1 measurements indicate that a gap of the order of 1 K exists in the magnon excitation spectrum. 1999 The American Physical Society.

Full text of DOI:10.1103/PhysRevB.60.3361

PHYSICAL REVIEW B
VOLUME 60, NUMBER 5
1 AUGUST 1999-I
NMR studies of EuB6 at low temperatures
B. Ambrosini, J. L. Gavilano, P. Vonlanthen, and H. R. Ott
¨
¨
¨
¨
Laboratorium fur Festkorperphysik, ETH-Honggerberg, CH-8093 Zurich, Switzerland
D. P. Young and Z. Fisk
National Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306
͑Received 16 February 1999͒
We report results of 153Eu and 11B nuclear magnetic resonance ͑NMR͒ measurements on EuB6 at tempera-
tures between 0.1 and 150 K and in external magnetic fields between Hϭ0 and 7 T. The evolution of the 153Eu
NMR spectra at low temperatures and low fields provides evidence for an unexpected and complex electronic
ground state of EuB6, involving the coexistence of two magnetically very similar phases. The temperature and
magnetic-field dependences of the 11B spin-lattice relaxation rate T1Ϫ1(T,H) are very well accounted for by
magnon-driven relaxation. A spin-wave theoretical interpretation of the 153Eu-NMR spectra and the 11B
TϪ1 1measurements indicate that a gap of the order of 1 K exists in the magnon excitation spectrum.
͓S0163-1829͑99͒08329-0͔
I. INTRODUCTION
to reflect the opening of a gap of Eg /kBϭ⌬ϭ45 K in the
magnon excitation spectrum.
Because of its unusual electronic and magnetic properties
the cubic compound EuB6 has now been in the focus of
research activities for more than 30 years. Since in this com-
pound the Eu ions adopt a divalent configuration, EuB6 was
initially thought to be a magnetic semiconductor.1 However,
more recent resistivity results definitely imply a semimetallic
character below 300 K with a very low charge-carrier
concentration.2,3 This interpretation is supported by self-
consistent band-structure calculations indicating a small
band overlap at the X point of the Brillouin zone.4
In order to etablish a more extended database with micro-
scopic information for a reliable interpretation of the intrigu-
ing features of EuB6, we have made a number of NMR mea-
surements on both 153Eu and 11B nuclei, at temperatures
between 0.1 and 150 K and in external magnetic fields H
between 0 and 7 T. For comparison, also NMR spectra and
spin-lattice relaxation rate measurements have been made on
the related nonmagnetic metallic hexaboride LaB6.
After giving some details concerning the samples and the
experimental procedures in Sec. II, we present and discuss
our results in Sec. III. We first display the results of the
NMR spectra and spin-spin relaxation rate measurements on
the isotope 153Eu in Secs. III A–III C. Subsequently, the
11B-NMR spectra and the corresponding spin-lattice relax-
ation rate measurements on EuB6 and LaB6 are discussed in
Secs. III D and III E. In Sec. IV we offer some conclusions
that follow from this study. Part of our NMR results on
153Eu presented here have previously been published
elsewhere.9,10
At low temperatures, EuB6 orders ferromagnetically5 via
two consecutive phase transitions at Tϭ16 and 14 K, respec-
tively, as inferred from two anomalies in the temperature
dependence of the specific heat.6,7 The ferromagnetic transi-
tion is accompanied by a significant reduction of the resis-
tivity ␳ and in the temperature range of these transitions,
EuB6 also exhibits a large negative magnetoresistance.3 The
increasingly metallic character of EuB6 below the Curie tem-
perature TC is also obvious from the results of optical reflec-
tivity measurements. A considerable blue shift of the reflec-
tivity edge is signaling a strong increase of the unscreened
plasma frequency in the ferromagnetic phase.6 The electrical
resistivity also shows a large reduction with increasing pres-
sure, concomitant with an increase of the magnetic phase-
transition temperature. These results have been claimed to
indicate a Ruderman-Kittel-Kasuya-Yosida-type coupling
between the Eu moments via the conduction electrons.8
A very detailed study of the magnetic behavior, reported
in Ref. 7, has been made in the temperature range of the two
ferromagnetic transitions by means of measurements of the
magnetization and the magnetic susceptibility. From these
results it has been argued that in the magnetically ordered
phase found between TϷ16 and 14 K the moments are fer-
romagnetically aligned along the ͓100͔ direction and that the
phase transition at TϷ14 K corresponds to a moment reori-
entation from the ͓100͔ to the ͓111͔ direction. In addition the
behavior of the low-temperature resistivity has been claimed
II. SAMPLE AND EXPERIMENT
A. Sample
The hexaboride material used in our NMR investigation
was grown in the form of small single crystals by melting the
appropriate amounts of the pure elements in aluminum flux.
After the thermal treatment the crystals were separated from
the Al in a hot NaOH solution.2,6 Hexaboride samples pre-
pared by this method have been shown to be of very high
quality in structural perfection and chemical composition.11
For our experiments the crystals have been powdered in an
argon atmosphere to grains with a typical size of less than
100 ␮m.
B. Experiment
The NMR measurements below 4 K were performed in a
dilution 3He-4He refrigerator, where the powdered sample
0163-1829/99/60͑5͒/3361͑9͒/$15.00
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©1999 The American Physical Society

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B. AMBROSINI et al.
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FIG. 2. Echo-decay curves of 153Eu in EuB6 at 1.1 K measured
in applied magnetic fields of 0.24 ͑I͒, 0.72 ͑II͒, and 1.33 T ͑III͒. The
solid lines represent the best fits to the data as described in the text.
The inset shows the 153Eu-NMR spectrum for Tϭ1.1 K with the
symbols placed where the corresponding spin-echo decays have
been measured.
FIG. 1. 153Eu-NMR spectra for EuB6 measured at various tem-
peratures and a fixed frequency of 156.36 MHz. The data below 2 K
show two broad peaks ͑A and B͒.
NMR signal. Our observations show that the shifts of the
maxima of signals A and B are both proportional to the
temperature dependence of the bulk magnetization. It seems
rather unlikely that a lowering of the symmetry that results in
two different sites, leaves the hyperfine field associated with
the high-temperature phase essentially unchanged. Still as-
suming such a scenario to be possible, one would rather ex-
pect the single peak to progressively separate into two peaks
corresponding to the two inequivalent sites. Our observation,
however, shows that peak B does not separate from peak A
but rather appears at a different field. A scenario considering
different hyperfine fields due to two crystal-field split
f-electron states, only slightly separated in energy, is not
compatible with our observation of two almost equally in-
tense lines even at the lowest temperatures. Therefore, we
conclude that our results for the NMR spectra are most likely
caused by the coexistence of two different phases at low
temperatures.
The maxima of signals A and B shown in Fig. 1 corre-
spond, below 1.1 K, to hyperfine fields Hhf at the Eu nuclei
of Ϫ34.54 and Ϫ34.18 T, respectively. The negative sign
indicates that the hyperfine fields are oriented antiparallel to
the applied magnetic field, as expected for a dominant core
polarization contribution to the hyperfine field. The differ-
ence between the average hyperfine fields of line A and B is
of the order of 1%. This small difference rules out any sig-
nificant valence change of a good part of the Eu2ϩ ions as
the origin for the unusual behavior indicated by our results.
3
was kept in contact with the liquid He-4He mixture. For the
measurements above 4 K the specimens were kept in flowing
He gas.
In all our measurements the NMR signal intensity was
obtained by integrating the spin-echo signal. The NMR spec-
tra were collected at constant frequency or field by stepwise
varying the applied magnetic field or the irradiation fre-
quency, respectively. The measurements of the spin-lattice
relaxation times T1 involved the monitoring of the nuclear
magnetization recovery m(t) of 11B as a function of the time
delay after the application of a single rf pulse or a comb of rf
pulses.
The spin-spin relaxation time T2, measured for EuB6 on
the 153Eu isotope, was obtained by monitoring the spin-echo
intensity as a function of the time delay between the pulses.
III. RESULTS AND DISCUSSION
A. 153Eu-NMR spectra
In Fig. 1 we display examples of 153Eu-NMR spectra of
EuB6 between 0.08 and 2.15 K, measured at a fixed fre-
quency of 156.36 MHz. Above 2 K the spectrum consists of
a single broad line A, but as the temperature is reduced to
below 2 K a second line B at a lower field appears and
gradually gains in relative intensity at the expense of line A
͑see Fig. 1͒. Below 1.1 K the relative intensity of signal B
saturates at approximately the same level as signal A.
The appearence of two peaks in the NMR spectrum of
153Eu is unexpected because the occurrence of inequivalent
magnetic sites in a simple cubic structure like that of EuB6
cannot a priori be anticipated. Two scenarios may, neverthe-
less, be postulated that could lead to this unexpected behav-
ior: ͑a͒ inequivalent Eu sites in a single phase or ͑b͒ the
coexistence of two different phases.
B. 153Eu-spin-spin relaxation rate
In Fig. 2 we present typical 153Eu spin-echo decay curves
of EuB6 measured at 156.36 MHz and Tϭ1.1 K in three
different fields. The labeling of the curves by I, II, and III
denotes three magnetic fields which are also displayed on the
profile of the corresponding NMR spectrum in the inset of
the same figure. The echo intensities have been normalized
to have the same value at the shortest possible time of moni-
toring. Technical limitations and very short spin-spin relax-
ation times T2 allowed us to measure only the tail end of the
echo-decay curve. The decay is nonexponential in time, sug-
gesting an inhomogeneous distribution of T2’s.
The existence of two inequivalent Eu sites in the same
magnetic phase seems very unlikely. In general even a
temperature-induced lowering of the crystal symmetry by
lattice distortion yields a single site and hence a single-peak

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In previous work Barak and co-workers12 proposed a
3363
NMR STUDIES OF EuB6 AT LOW TEMPERATURES
model for describing the spin-spin relaxation in ferromag-
netic systems where the broadening of the NMR signals is
due to microscopic and macroscopic inhomogeneities. In this
model the relaxation mechanism is provided by the indirect
Suhl-Nakamura ͑SN͒ interaction. One can visualize this ef-
fective coupling between two nuclear spins by considering
that a spin-flip of say, nuclear spin 1, creates a virtual mag-
non in the ferromagnetic background. This virtual magnon
may then be annihilated by a spin-flip of the nuclear spin 2,
hence leading to an effective spin-spin interaction.
In real ferromagnetic systems, microscopic imperfections
like strains or impurities cause the local magnetic field to
vary from site to site. In a first approximation, the resultant
microscopic distribution of Larmor frequencies is assumed to
be the same throughout the whole sample, being approxi-
FIG. 3. TϪ2 1 as a function of the applied field for 153Eu in EuB6,
measured at 156.36 MHz and temperatures ͑from top to bottom͒ of
1.82, 1.65, 1.47, 1.1, and 0.08 K. The solid lines are to guide the
eye. The inset shows the 153Eu-NMR spectrum and the correspond-
ing TϪ2 1 profile for Tϭ0.08 K. Note that the minimum of the spec-
mated by a function of the form g(␯Ϫ␯0), where ␯ repre-
0
trum does not coincide with the maximum of T2Ϫ1
.
sents the mean frequency of the microscopic domain. Con-
sider now a spin i and its neighbors, arranged in shells
labeled with index j. If the characteristic width of the Zee-
man energy distribution given by g(␯Ϫ␯0) is larger than the
effective SN coupling constant between spin i and a spin in
the jth shell, then the SN interaction will spin-flip only a
fraction of the nuclear spins in the jth shell, resulting in an
increase of the spin-spin relaxation time. The relaxation rate
TϪ2 1for a given microscopic region, characterized by g(␯
Ϫ␯0), will depend on the Larmor frequency as
First of all, the model employed to analyze the echo-
decay curves is based on reasonable assumptions and has
successfully been used in the past to analyze similar data on
related ferromagnets.12 Our estimated values of TϪ2 1 are in
qualitative agreement with the measured curves, i.e., a vis-
ibly faster relaxation corresponds to a larger TϪ2 1 ͑see Fig.
2͒. As may be seen in the inset of Fig. 3, TϪ2 1 is not constant
across the NMR spectrum and therefore its shape may be
affected by TϪ2 1effects. However, from the measured echo-
decay envelopes along the profile of the NMR signal we can,
via Eq. ͑2͒, estimate the NMR spectrum at zero delay be-
tween the pulses. An example of such a reconstructed signal
compared with the corresponding measured spectrum is
shown in Fig. 4. It is obvious that the two-peak feature is
preserved and that only the relative intensity of the two
peaks is affected. The slight shift in magnetic field of
ϳ0.05 T between the maxima of peak A of the experimental
and the reconstructed curve is insignificant if we consider the
magnitude of the hyperfine field of the order of 30 T. As a
test of confidence of this procedure we have calculated the
product of the reconstructed spectrum intensity A multiplied
TϪ2 1 ␯͒ϭCg ␯Ϫ␯ ͒,
͑1͒
͑
͑
0
where C is a constant independent of ␯. In real ferromagnets
the width of the NMR signals is expected to be much larger
than ͑i͒ the width of the microscopic distribution g(␯Ϫ␯0)
and, ͑ii͒ the characteristic irradiation width ␥H1 of the rf
pulse. In this case the integrated intensity mЌ(t) of the spin
echo, as a function of the time delay t between the pulses, is
expected to be nonexponential with
ϱ
(0)
2
mЌ t͒ϰ
g ␯ ͒eϪt([1/T ]g(␯ ))d␯ ,
͑2͒
Ј
͑
͑
Ј
Ј
͵
Ϫϱ
where ␯ ϭ␯Ϫ␯ and T(0)ϵT2 (␯ϭ␯0) as given in Eq. ͑1͒.
Ј
0
2
In what follows, we will denote the fit parameter T(20) simply
as T2. We have used Eq. ͑2͒ to fit our measured echo-decay
data with only two parameters, i.e., T2 and an overall scaling
factor, absorbing the width and the amplitude of g(␯ ). As
Ј
previously experienced by Barak and co-workers,12 the best
fits to the data are obtained using a Lorentzian distribution of
microscopic frequencies g(␯Ϫ␯0). The solid lines in Fig. 2
represent the best fits to our measurements.
In Fig. 3 we display the field dependence of TϪ2 1 at 5
different temperatures between 0.08 and 1.82 K, measured at
␯ϭ156.36 MHz. In the inset of the same figure we compare
the NMR spectrum and the corresponding TϪ2 1(H) measured
at 0.08 K. We now intend to demonstrate that, although these
measurements have only been possible at the tail ends of the
echo-decay envelope mЌ(t), our results reflect intrinsic prop-
erties of EuB6 and are not artifacts due to the experimental
limitations. Below we offer some general considerations and
several simulations that support our interpretation.
FIG. 4. 153Eu-NMR spectrum for EuB6 measured at T
ϭ1.65 K and ␯ϭ156.36 MHz ͑open circles͒. The full circles rep-
resent the reconstructed spectrum for a zero delay between the
pulses ͑see text͒. The solid line represents the best fits to the data
assuming the two Gaussian functions shown by the dotted lines.

3364
B. AMBROSINI et al.
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FIG. 6. Temperature dependence of the relative NMR integrated
intensity of peak B in the reconstructed spectrum ͑see text͒. The
solid line is to guide the eye.
FIG. 5. 153Eu-NMR spectra for EuB6 measured at 1.1 and 1.65
measurements indicate that the observed unusual features are
associated with a critical behavior of phase B.
K ͑symbols in the lower and upper figure, respectively͒. The dotted
lines represent a tentative estimate of line A and B in the overlap-
ping region. In the central figure we display the field dependence of
TϪ2 1
.
Note that the deviation of TϪ2 1 from the high-field
C. Magnetization
T-independent region approximately corresponds to the onset of
line B. The data at Tϭ1.82 K are included to demonstrate the
crossover of the measured TϪ2 1 at 1.65 K from the T-independent to
the T-dependent region.
In Fig. 8 we display the temperature dependence of the
frequency at which peak A is maximum in a constant exter-
nal field Hϭ1 T. The resonance frequency at the magnetic
site of a ferromagnet can be expressed as a sum of four
terms13
by the temperature T. For temperatures between 1 and 2 K
where signal B decreases from its saturation value and finally
vanishes, the product A•T is found, as expected, to be tem-
perature independent with an uncertainty of less than 10%.
This result represents a strong quantitative support for the
reliability of our approach.
In Fig. 3 we observe that in fields exceeding 0.9 T, where
signal A is dominant for all the chosen temperatures, TϪ2 1is
approximately T independent and increases with decreasing
field. With further decreasing field, entering the range where
peak B starts dominating, T2Ϫ1acquires a significant tempera-
ture dependence. Thus, TϪ2 1(H) reaches a maximum, whose
position and magnitude now depends on temperature, and
decreases with decreasing field at fields where peak B is in
the dominant part of the spectrum. In Fig. 5 we attempt to
demonstrate that the onset of the T dependence of TϪ2 1(H)
more or less coincides with the onset of signal B in the NMR
spectrum. This behavior, once again, indicates that the two-
peak feature is not an experimental artifact.
␥
␮
M
hf M0
␯ϭ
HϪD ␮ Mϩ 0 MϩH
,
͑3͒
ͯ
ͯ
f
0
2␲
3
where H is the applied field. The second and the third term
represent the demagnetization and the Lorentz field, respec-
tively, with Df being the demagnetization factor. Our mea-
surements were made on powdered EuB6. For simplicity we
assume an average Dfϭ1/3, implicitly implying an ‘‘average
spherical geometry’’ of our grains. Hence, the Lorentz and
the demagnetization field cancel each other. The last term of
Eq. ͑3͒ represents the hyperfine field. Except for the applied
field, all the contributions to ␯(T) are proportional to the
In Fig. 6 we display the relative intensity of peak B of
reconstructed spectra as a function of temperature. It may be
seen that signal B vanishes at approximately 2 K. In Fig. 7
we present the temperature dependence of TϪ2 1 estimated at
0.3 Tesla ͑full circles͒ and at 1.18 T ͑open circles͒, two fields
at which peak B and peak A, respectively, have been shown
to be dominant in this temperature range. At these fields,
TϪ2 1is T independent for peak A, whereas it increases rapidly
as the temperature is enhanced towards 2 K for peak B,
giving evidence that signals A and B are associated with two
coexisting phases and not with different sites in a single
phase. A comparison of Figs. 6 and 7 shows that the increase
and perhaps the divergence of TϪ2 1approximately scales in
temperature with the progressive disappearance of peak B. In
conclusion, the features of Figs. 6 and 7 of the 153Eu-NMR
FIG. 7. Temperature dependence of TϪ2 1 estimated at 0.3 T ͑full
circles͒ and at 1.18 T ͑open circles͒, two fields at which line B and
line A, respectively, are dominant in this temperature range. The
measurements were performed at 156.36 MHz. The solid lines are
to guide the eye.

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3365
NMR STUDIES OF EuB6 AT LOW TEMPERATURES
3
n
Akϭg␮BHϪ D ␮ g␮ Mϩ2S
J
1Ϫeik•r ͒ϩBk ,
͑
n
͚
f
0
B
2
n
͑9͒
and
1
Bkϭ g␮ ␮0Msin2 ␪k ,
͑10͒
B
2
where ␪k represents the angle between the magnetization and
the spin-wave propagation vector k. Jn is the exchange inte-
gral for the nth nearest neighbors. For simplicity we assume
that only J1 and J2 are different from zero and we neglect
the influence of neighbors at larger distances. A gap in the
magnon excitation spectrum associated with anisotropy will
be considered by adding a term to the external field, defining
FIG. 8. Temperature dependence of the frequency at which peak
A is maximum measured in an external field of 1 T ͑full circles͒.
The solid and broken lines represent simulations of ␯(T) according
to the spin-wave theory ͑see text͒ for J1ϭJ2ϭ0.063 K and ⌬
ϭ0 K and for J1ϭ0.157 K, J2ϭ0 and ⌬ϭ1.4 K, respectively,
͑note that for simplicity we write Jn but mean Jn /kB).
kB
HϵHextϩH⌬ϭHextϩ
⌬.
͑11͒
g␮
B
The solid and broken lines in Fig. 8 display the results of
simulations of ␯(T) assuming different values of J1 , J2, and
⌬. The best agreement between the result of our measure-
ments and our model calculation is obtained for J1ϭJ2
ϭ0.063 K and ⌬ϭ0 K or for J1ϭ0.157 K, J2ϭ0 and ⌬
ϭ1.4 K ͑note that for simplicity we write Jn but mean
Jn /kB). In fact from our simulations we cannot distinguish
between the cases ⌬ϭ1.4 K with J1ӷJ2 and a vanishingly
small gap with J1ϷJ2Ϸ0.063 K. Even an intermediate situ-
ation is possible, because J2 and ⌬ have a similar effect on
␯(T). In conclusion the results of our simulations allow us to
put an upper limit to any anisotropy gap of a few Kelvin, in
agreement with a recent neutron-diffraction result.5 From J1
and J2 we can estimate the transition temperature in a
molecular-field approximation15 ͑MFA͒ to be
magnetization M(H,T). Since for the data shown in Fig. 8,
H represents only a small fraction of about 3% of Hhf , plot-
ting ␯(T) is obviously a very useful way to establish the
temperature variation of the magnetization.
At sufficiently low temperatures the spin-wave approxi-
mation for describing the ferromagnetic state can be applied.
In our analysis of the temperature dependence of the reso-
nance frequency ␯(T) of the 153Eu nuclei we essentially rely
on the theoretical scheme developed by Holstein and Prima-
koff ͑HP͒.14 The Hamilton operator describing the system
contains in addition to the exchange term a Zeeman and a
dipole-dipole interaction and may be written as
2
␮
g␮ ͒
B
͑
0
HHPϭϪ
J S •S Ϫg␮ H
Szϩ
i
͚
͚
ij
i
j
B
4␲
i, j
i
2
kBTMFAϭ S Sϩ1͒ z J ϩz J ͒,
͑12͒
͑
͑
1
1
2 2
Si•Sj 3 S •r ͒ S •r ͒
c
͑
͑
r5ij
i
ij
j
ij
3
ϫ
Ϫ
,
͑4͒
͚
ͩ
ͪ
r3ij
i, j
where z1ϭ6 and z2ϭ12 are the number of nearest and next-
nearest Eu neighbors of the Eu sites in EuB6. In both cases
discussed above (⌬Ϸ0 or J2Ϸ0), the estimated transition
temperatures are of the order of 10 K, in fair agreement with
TC established with specific-heat and magnetization
measurements.5–7 A closer agreement cannot be expected be-
cause the shape of the specific-heat anomaly6,7 clearly sig-
nals the inadequacy of the MFA for describing the transition.
where the symbols have their usual meaning. From Eq. ͑4͒
we obtain the magnetization
M H,T͒ϭM ϪM H,T͒ϪM H͒.
͑5͒
͑
͑
͑
0
T
␤
Here, MT represents the deviation from the saturation value
M0 due to thermal excitations. It is given by
D. 11B-NMR spectra of EuB6 and LaB6
g␮
Ak
1
B
MT H,T͒ϭ
͑6͒
͑
In Fig. 9 we display examples of 11B-NMR spectra for
EuB6, measured at several temperatures and at a fixed fre-
quency of 63 MHz. From NMR measurements on the related
compound LaB6, presented in Fig. 10, we expect the quad-
rupole splitting of the 11B signal to be of the order of 0.5
MHz, much less than the characteristic width of the
11B-NMR spectrum of EuB6 at low temperatures. Therefore,
the prominent shoulder that appears in the 11B-NMR spec-
trum of EuB6 at low temperatures is attributed to differences
in the local fields. The low-temperature spectra are well ap-
proximated by the sum of two Gaussians, here denoted by I
and II, as shown in Fig. 9 for the spectrum taken at T
͚
V
⑀
exp ⑀ /k T͒Ϫ1
͑
k B
k
k
with V being the volume. In Eq. ͑5͒ M␤ is a T-independent
term associated with the dipole-dipole interaction
g␮
Ak
B
M␤ H͒ϭ
Ϫ1 .
͑7͒
͑8͒
͑
ͩ
ͪ
͚
2V
⑀
k
k
The spin-wave parameters ⑀k , Ak and Bk are defined as
2
2
ͱ
⑀kϭ AkϪBk,

3366
B. AMBROSINI et al.
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FIG. 11. Temperature dependences of the average hyperfine
fields of line I and II ͑full and open symbols, respectively͒
measured at ␯ϭ32.5 MHz ͑triangles͒ and ␯ϭ63 MHz ͑circles͒.
The solid line displays the temperature dependence of the magneti-
zation at Hϭ5 T.
FIG. 9. 11B NMR spectra of EuB6 between 2.35 and 216 K
measured at ␯ϭ63 MHz. The solid lines represent the best fits to
the data using two Gaussian functions, as indicated in the top spec-
trum with I and II.
H
͗
͘
hf
transfer hyperfine fields would be needed to explain our mea-
surements. A hypothetical orientation of the moments along
the ͓100͔ direction would also produce two inequivalent B
sites with a dipole-field difference ⌬HϭϪ0.7 T. This situ-
ation would require two transferred fields that differ in mag-
nitude and direction, hence implying a strong anisotropic
transferred hyperfine coupling for explaining our results. For
any of these orientations we cannot exclude that a slight
distortion of the crystal lattice is the cause of our observa-
tions. At any rate, the temperature dependence of the
11B-NMR spectra does not dramatically change below 3 K
and we may, therefore, exclude a moment reorientation or a
considerable alteration of electronic densities to be the cause
of the appearance of the two inequivalent phases as dis-
cussed in Sec. III A
ϭ2.35 K. The individual contributions have integrated in-
tensities of 1/3 and 2/3, respectively. At the lowest measured
temperatures the internal static fields Hhf , obtained from the
shift of the center of the Gaussian relative to the expected
position of the boron NMR line in a diamagnetic substance,
are Ϫ0.02 and Ϫ0.35 T for the minority ͑peak I͒ and major-
ity ͑peak II͒ boron sites, respectively.
The observed splitting of the maxima of peak I and II,
⌬HϷ0.33 T, is in good quantitative agreement with the dif-
ference of the dipole fields at the B sites obtained by assum-
ing the full Eu2ϩ moments being oriented along the ͓110͔
direction. Calculations assuming this orientation result in a
direct dipole field HdϭϪ0.116 T at 4 boron sites in the unit
cell and Hdϭ0.232 T at the remaining two sites, thus lead-
ing to a difference in field of ⌬Hϭ0.348 T. To reproduce
the observed maxima of peak I and II in the NMR spectrum
we have to assume an additional hyperfine transferred field
In Fig. 11 we display the temperature dependence of the
average hyperfine field H of peaks I and II with full and
͗
͘
hf
open symbols, respectively, measured at ␯ϭ32.5 MHz ͑tri-
angles͒ and ␯ϭ63 MHz ͑circles͒. The solid line displays the
temperature dependence of the magnetization of a sample
prepared from the same piece of material in a field H
ϭ5 T.16 The T dependence of H for the majority sites II
due to f-electron moments on the Eu ions of ϳϪ0.24 T.
7
¨
According to Sullow et al., the Eu moments are claimed to
be oriented along the ͓111͔ direction at low temperatures.
This moment arrangement results in a vanishing contribution
of the dipole field at all B nuclei and hence, two different
͗
͘
hf
follows the temperature dependence of the bulk magnetiza-
tion fairly well, indicating that it is a consequence of the
magnetic ordering. For the minority boron site I, H (T) is
͗
͘
hf
too small to draw any reliable conclusion.
E. 11B spin-lattice relaxation rate in EuB6 and LaB6
The 11B T1 measurements were made near the maximum
of the spin-echo intensity ͑see Fig. 9͒ and thus have the
largest contribution from peak I of the 11B-NMR spectra. In
Fig. 12 we display TϪ1 1(T) measured in applied magnetic
fields between 0.49 and 7 T ͑symbols͒. The solid lines rep-
resent the results of calculations to be discussed below and
the dashed line represents TϪ1 1(T) measured for LaB6. The
spin-lattice relaxation in the range of the spectrum, where the
11B-NMR signal is dominated by peak II, is essentially the
same as that for the boron peak I. This implies again that the
observed order-of-magnitude difference between the hyper-
fine fields at the boron sites I and II is the result of cancel-
lations of different contributions to the local fields.
FIG. 10. 11B NMR spectrum of LaB6 measured at 18 K and ␯
ϭ66.71 MHz. The solid line represents the best fit assuming a
first-order quadrupole perturbation of a Zeeman interaction domi-
nated powder pattern.

PRB 60
3367
NMR STUDIES OF EuB6 AT LOW TEMPERATURES
into a new state k by a nuclear spin-flip. In this case the
Ј
energy is easily conserved but the condition of angular mo-
mentum conservation may prohibit this process. However, if
the hyperfine Hamiltonian includes terms of the form
A[ϩz]•IϩSz , the Raman process has a nonvanishing prob-
ability. The operator Iϩ is the usual nuclear spin raising op-
erator and A[ϩz] is the hyperfine matrix component associ-
ated with the term IϩSz , where Sz is the z component of the
Eu 4f-electron spin operator. Note that for Eu2ϩ, the total
angular momentum JϭSϭ7/2. The above form of the hy-
perfine Hamiltonian is expected in the presence of the dipole
interaction or if the hyperfine transfer mechanism is aniso-
tropic. In both cases the terms IϩSz involve terms with one
creation and one annihilation boson operator14,18,19
FIG. 12. Temperature dependence of the 11B spin-lattice relax-
1
1
[ϩz]
Hhf ϭ A[ϩz]IϩSzϭ A[ϩz]Iϩ
ation rate measured in various magnetic fields from 0.49 to 7 T
͑symbols͒. The solid lines represent calculations considering two-
magnon-dominated processes at the respective field and assuming
an energy gap in the magnon excitation spectrum of 1.4 K ͑see
text͒.
2
2
1
ϫ SϪ
exp i kϪk ͒•r b†b
,
͑14͒
͓ ͑
͔
k
Ј
͚
k
Ј
ͫ
ͬ
N
k,k
Ј
and the two-magnon process is allowed. In Eq. ͑14͒, r rep-
resents the position of the spin. Using Fermi’s golden rule
the spin-lattice relaxation rate is of the form18
Except at the lowest temperatures the 11B spin-lattice re-
laxation rate for EuB6 is much larger than TϪ1 1for the refer-
ence compound LaB6. We associate this enhancement with
excitations within the f-electron system. The observed strong
temperature and field dependences of TϪ1 1 is typical for fer-
romagnets where the spin-lattice relaxation is well accounted
for by the flipping of a nuclear spin inducing a magnon ex-
citation, or vice versa.17 To estimate the efficiency of the
magnon-driven relaxation one needs detailed knowledge of
the hyperfine interaction between the magnetic moments and
the nuclear spins. As previously noted we assume that the
interaction between the f electrons and the boron nuclei con-
tains a transferred Htr and an electron-nucleus dipole Hdip
contribution. Thus the total hyperfine Hamiltonian may be
written as
2
1
␲ A[ϩz]
͒
͑
ϭ
n 1ϩn ͒D2 ⑀͒d⑀,
͑15͒
͑
͑
͵
⑀
⑀
T1
ប
where D(⑀) represents the density of spin-wave states, ⑀ is
the energy associated with a spin-wave excitation given in
Eqs. ͑8͒–͑10͒, and n⑀ is the Bose-Einstein occupation factor
n ϵ exp(⑀/k T)Ϫ1 Ϫ1. In addition to this first-order Ra-
͓
͔
⑀
B
man process, a second-order two-magnon process, which
does not require the conservation of the z component of the
angular momentum, is mediated by the dipole interaction
among ordered moments.17 In an even higher-order expan-
sion, three-magnon processes where the nuclear spin flip is
associated with the annihilation of one and the creation of
two spin waves, are encountered. In this case the energy and
momentum conservation requirements can always be ful-
filled and, therefore, this type of process is always allowed.
The solid lines in Fig. 12 represent calculations of
TϪ1 1(T,H) for a two-magnon dominated relaxation process
employing Eq. ͑15͒ and using J1ϭ0.157 K, J2ϭ0 and ⌬
ϭ1.4 K, one of the parameter sets discussed in Sec. III C.
The hyperfine coupling constant A[ϩz] provides an overall
free scaling parameter for obtaining the best agreement be-
tween our model calculations and the results of our measure-
ments. The solid lines displayed in Fig. 12 were obtained
with A[ϩz]Sz /ប␥11BϷ0.45 T, with Szϭ7/2. We note that
with these exchange couplings J1 , J2 and the energy gap ⌬
that we obtained via the analysis of the 153Eu NMR spectra
͑see Fig. 8 and corresponding discussion͒ the salient features
of the 11B spin-lattice relaxation may be reproduced rather
well. Using the same simple approximation for calculating
TϪ1 1(T,H) with the second parameter set mentioned in Sec.
III C, i.e., J1ϭ0.063 K, J2ϭ0.063 K, and ⌬ϭ0 K, the
agreement between calculation and experiment is distinctly
worse. This confirms that the magnon excitation spectrum at
low energies is dominated by a gap Eg /kB of the order of 1
K. The nontrivial agreement between the calculated and the
HϭHtrϩHdip
.
͑13͒
We mention here that if the crystal structure and the magni-
tude of the magnetic moment are known, Hdip can easily be
computed. For the transfer term the situation may be more
complicated. If Htr does not have a simple isotropic form
then a detailed account of this contribution is very difficult.
However, since the hyperfine coupling enters only as an
overall scaling factor in the relaxation, some qualitative in-
formation may still be obtained from the temperature and
field dependences of T1Ϫ1
.
The spin-wave approach, used for the analysis of the mag-
netization in Sec. III C can be applied here to estimate the
spin-lattice relaxation rate at low temperatures TӶTC . In
this approach the spin operators Sϩ,SϪ, and Sz are expanded
in terms of boson operators b† and b for the creation and
annihilation of a spin-wave excitation,14 respectively. The
first-order expansion in spin-wave operators corresponds to
the one-magnon process where a nuclear spin flip is accom-
panied by the creation of a spin wave or vice versa. Usually
this mechanism is forbidden because the energy is not con-
served. The two-magnon or Raman process represents the
situation of a spin wave with wave vector k being scattered

3368
B. AMBROSINI et al.
PRB 60
observed temperature and field dependences of TϪ1 1in the
range where the spin-wave model ought to be applicable,
gives strong evidence that the relaxation is indeed dominated
by two-magnon processes and that our approximation cap-
tures the essential ingredients. At higher temperatures, where
the deviations from the saturation magnetization grow sub-
stantially, we expect the spin-wave approximation to fail and
higher-order processes to become more significant. Here we
point out that, based on the results of previous electrical
resistivity ␳ measurements, a magnon gap of the order of 45
K has been suggested for EuB6 ͑Ref. 7͒. Our analysis of the
NMR data presented above finds no support for this claim.
We, therefore, believe that the observed features of ␳(T)
have an origin different from the electron-magnon scattering
assumed in Ref. 7.
changes in the electronic environments near the Eu sites pro-
voke the gradual development of a second ordered phase.
The small but distinct difference in the observed hyperfine
fields of the two coexisting phases is of the order of 1% with
no appreciable effect on the dynamics of magnon excitations.
Since the results for the 11B NMR give no hint for dramatic
changes around 2 K, we have to conclude that the phase B
detected in the results of 153Eu NMR does not have very
different electronic or magnetic properties than phase A,
present already at higher temperatures. This in turn leads to
the question of what causes two similar coexisting phases in
EuB6? It is natural to attribute the differences of these phases
to weak terms in the electronic Hamiltonian which either
violate an important symmetry, or which act only through a
higher-order perturbation process. Spin-orbit and crystal-
field interactions, acting weakly on the Eu2ϩ ions, are the
prime candidates for such a mechanism. Their main role
would not just be the lifting of the degeneracy of the 4f7
ground state, but to induce two slightly different ground
states. In any case our findings point to a delicately balanced
situation for the magnetic ground state of EuB6.
We now turn our attention to the hyperfine coupling
A
[ϩz]. As mentioned above the best agreement with the field
and temperature dependences of the spin-lattice relaxation
rate yields A[ϩz]Sz /ប␥11 Ϸ0.45 T. The two-magnon pro-
B
cess is allowed only if the hyperfine interaction is anisotropic
or if the electron and nuclear-spin quantization axes are not
collinear. An anisotropic interaction involved in our situation
is the direct electron-nucleus dipole interaction. The magni-
tude of A[ϩz], calculated for the dipole case for Eu moments
aligned along the three directions ͓100͔, ͓110͔, and ͓111͔, is
too small to quantitatively account for the magnitude of the
measured spin-lattice relaxation rate. Other causes that might
enhance A[ϩz] are a lattice distortion enhancing the dipole
coupling, or an anisotropic hyperfine transferred interaction
that invokes a hyperfine coupling A[ϩz] of the order of 0.45
T.
The measured temperature and field dependences of the
11B spin-lattice relaxation rate are thus consistent with a
two-magnon driven relaxation mechanism. The dipole inter-
action between the Eu moments and the B nuclei alone
seems insufficient to quantitatively account for the observed
value of TϪ1 1. This deficit may hint to either an anisotropic
transferred hyperfine interaction and/or a temperature-
induced lattice distortion. Both the lattice distortion and/or
the anisotropic transferred hyperfine interaction need to be
consistent with the two inequivalent sites observed in the
11B-NMR spectra and with the equal spin-lattice relaxation
rates observed for both these sites.
The low-temperature 11B-NMR spectra reveal two in-
equivalent B sites, experiencing different hyperfine fields. It
is difficult to unequivocally identify the cause for this differ-
ence. None of the most obvious ferromagnetic alignments of
the Eu moments can, by itself, explain both the magnitude
and the difference of the hyperfine fields. Thus very small
lattice distortions at low temperatures cannot be ruled out to
be, at least partially, the cause for two inequivalent 11B sites.
Possible lattice distortions are of interest here in connection
with electron-lattice interactions influencing the magnetic
and transport properties of EuB6. The similarity with man-
ganese oxides where large magnetoresistive effects com-
bined with ferromagnetic order and enhanced metallicity20
are, at least partially ascribed to strong electron-lattice
interactions,21,22 is intriguing.
We have attempted to analyze our results on the 153Eu
line shift, induced by the spontaneous magnetization in the
ferromagnetic phase, and of the 11B spin-lattice relaxation
well below the Curie temperature, by invoking the spin-wave
theory of Holstein and Primakoff.14 The experimental results
are fairly well accounted for by a dominant two-magnon re-
laxation process and assuming a gap Eg in the magnon exci-
tation spectrum such that Eg /kB is of the order of one
Kelvin. Finally we note that our microscopic measurements
in externally applied magnetic fields give no evidence for a
moment reorientation around 14 K, as has been suggested to
occur in zero field in Ref. 7.
IV. CONCLUSIONS
Our results for the 153Eu-NMR spectra signal that below 2
K, well within the ferromagnetic state of EuB6, slight
1 B.T. Matthias, Phys. Lett. 27A, 511 ͑1968͒.
2 Z. Fisk, D.C. Johnston, B. Cornut, S. von Molnar, S. Oseroff, and
R. Calvo, J. Appl. Phys. 50, 1911 ͑1979͒.
6 L. Degiorgi, E. Felder, H.R. Ott, J.L. Sarrao, and Z. Fisk, Phys.
Rev. Lett. 79, 5134 ͑1997͒.
7
¨
S. Sullow, I. Prasad, M.C. Aronson, J.L. Sarrao, Z. Fisk, D. Hris-
3 C.N. Guy, S. von Molnar, J. Etourneau, and Z. Fisk, Solid State
Commun. 33, 1055 ͑1980͒.
tova, A.H. Lacerda, M.F. Hundley, A. Vigliante, and D. Gibbs,
Phys. Rev. B 57, 5860 ͑1998͒.
4 S. Massida, A. Continenza, T.M. de Pascale, and R. Monnier, Z.
Phys. B 102, 83 ͑1997͒.
5 W. Henggeler, H.R. Ott, D.P. Young, and Z. Fisk, Solid State
Commun. 108, 929 ͑1998͒.
8 J.C. Cooley, M.C. Aronson, J.L. Sarrao, and Z. Fisk, Phys. Rev.
B 56, 14 541 ͑1997͒.
9 J.L. Gavilano, B. Ambrosini, P. Vonlanthen, H.R. Ott, D.P.
Young, and Z. Fisk, Phys. Rev. Lett. 81, 5648 ͑1998͒.

PRB 60
3369
NMR STUDIES OF EuB6 AT LOW TEMPERATURES
10 H.R. Ott, B. Ambrosini, J.L. Gavilano, Z. Fisk, D.P. Young, M.E.
Torelli, J.L. Sarrao, and J.D. Thompson ͑unpublished͒.
16 D. Pushin ͑private communication͒.
17 D. Beeman and P. Pincus, Phys. Rev. 166, 359 ͑1968͒.
11 H.R. Ott, M. Chernikov, E. Felder, L. Degiorgi, E.G. Moshopou-
lou, J.L. Sarrao, and Z. Fisk, Z. Phys. B 102, 337 ͑1997͒.
12 J. Barak, I. Siegelstein, A. Gabai, and N. Kaplan, Phys. Rev. B 8,
18 J. Barak, A. Gabai, and N. Kaplan, Phys. Rev. B 9, 4914 ͑1974͒.
19 A.H. Mitchell, J. Chem. Phys. 27, 17 ͑1957͒.
20 Y. Tokura, Curr. Opin. Solid State Mater. Sci. 3, 157 ͑1998͒.
21 A.J. Millis, P.B. Littlewood, and B.I. Shraiman, Phys. Rev. Lett.
74, 5144 ͑1995͒.
22 A.J. Millis, B.I. Shraiman, and R. Mueller, Phys. Rev. Lett. 77,
175 ͑1996͒.
5282 ͑1973͒.
13
¨
M.W. Pieper, J. Kotzler, and K. Nehrke, Phys. Rev. B 47, 11 962
͑1993͒.
14 T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 ͑1940͒.
15 W. Zinn, J. Magn. Magn. Mater. 3, 23 ͑1976͒.